2009 NEHRP Recommended Seismic Provisions:
Design Examples
FEMA P751  September 2012
Prepared by the
National Institute of Building Sciences
Building Seismic Safety Council
For the
Federal Emergency Management Agency
of the Department of Homeland Security
NOTICE: Any opinions, findings, conclusions, or recommendations expressed in this publication do not
necessarily reflect the views of the Federal Emergency Management Agency. Additionally, neither
FEMA nor any of its employees make any warranty, expressed or implied, nor assume any legal liability
or responsibility for the accuracy, completeness, or usefulness of any information, product or process
included in this publication.
The opinions expressed herein regarding the requirements of the International Residential Code do not
necessarily reflect the official opinion of the International Code Council. The building official in a
jurisdiction has the authority to render interpretation of the code.
This report was prepared under Contract HSFEHQ09R0147 between the Federal Emergency
Management Agency and the National Institute of Building Sciences.
For further information on the Building Seismic Safety Council, see the Council s website
www.bssconline.org or contact the Building Seismic Safety Council, 1090 Vermont, Avenue, N.W.,
Suite 700, Washington, D.C. 20005; phone 2022897800; fax 2022891092; email bssc@nibs.org.
FOREWORD
One of the goals of the Department of Homeland Security s Federal Emergency Management Agency
(FEMA) and the National Earthquake Hazards Reduction Program (NEHRP) is to encourage design and
building practices that address the earthquake hazard and minimize the resulting risk of damage and
injury. The 2009 edition of the NEHRP Recommended Seismic Provisions for New Buildings and Other
Structures (FEMA P750) affirmed FEMA s ongoing support to improve the seismic safety of
construction in this country. The NEHRP Provisions serves as a key resource for the seismic requirements
in the ASCE/SEI 7 Standard Minimum Design Loads for Buildings and Other Structures as well as the
national model building codes, the International Building Code (IBC), International Residential Code
(IRC) and NFPA 5000 Building Construction Safety Code. FEMA welcomes the opportunity to provide
this material and to work with these codes and standards organizations.
This product provides a series of design examples that will assist the users of the 2009 NEHRP Provisions
and the ASCE/SEI 7 standard the Provisions adopted by reference.
FEMA wishes to express its gratitude to the authors listed elsewhere for their significant efforts in
preparing this material and to the BSSC Board of Direction and staff who made this possible. Their hard
work has resulted in a guidance product that will provide important assistance to a significant number of
users of the nation s seismic building codes and their reference documents.
Department of Homeland Security/
Federal Emergency Management Agency
PREFACE
This volume of design examples is intended for those experienced structural designers who are relatively
new to the field of earthquakeresistant design and to the 2009 NEHRP (National Earthquake Hazards
Reduction Program) Recommended Seismic Provisions for New Buildings and Other Structures. By
extension, it also applies to use of the current model codes and standards because the Provisions is the
key resource for updating seismic design requirements in most of those documents including ASCE 7
Standard, Minimum Design Loads for Buildings and Other Structures; and the International Building
Code (IBC). Furthermore, the 2009 NEHRP Provisions (FEMA P750) adopted ASCE705 by reference
and the 2012 International Building Code adopted ASCE710 by reference; therefore, seismic design
requirements are essentially equivalent across the Provisions, ASCE7 and the national model code.
The design examples, updated in this edition, reflect the technical changes in the 2009 NEHRP
Recommended Provisions. The original design examples were developed from an expanded version of an
earlier document (entitled Guide to Application of the NEHRP Recommended Provisions, FEMA 140)
which reflected the expansion in coverage of the Provisions and the expanding application of the
Provisions concepts in codes and standards. The widespread use of the NEHRP Recommended Provisions
in the past and the essential equivalency of ASCE7, the Provisions and the national model codes at
present attested to the success of the NEHRP at the Federal Emergency Management Agency and the
efforts of the Building Seismic Safety Council to ensure that the nation s building codes and standards
reflect the state of the art of earthquakeresistant design.
In developing this set of design examples, the BSSC initially decided on the types of structures; types of
construction and materials; and specific structural elements that needed to be included to provide the
reader with at least a beginning grasp of the new requirements and critical issues frequently encountered
when addressing seismic design problems. Many of the examples are from the previous edition of the
design examples but updated by the authors to illustrate issues or design requirements not covered or that
have changed from the past edition. Because it obviously is not possible to present, in a volume of this
type, complete building designs for all the situations and features that were selected, only portions of
designs have been used.
All users of the Design Examples are recommended to obtain and familiarize themselves with the 2003
and 2009 NEHRP Recommended Provisions (FEMA 450 and FEMA P750) or ASCE7, Copies of the
Provisions are available free of charge from FEMA by calling 18004802520 (order by FEMA
Publication Number). Currently available are the 2003 and 2009 editions as follows:
NEHRP (National Earthquake Hazards Reduction Program) Recommended Seismic Provisions for New
Buildings and Other Structures, 2009 Edition, FEMA P750, 1 volume with maps (issued as paper
document with a CD attached)
NEHRP (National Earthquake Hazards Reduction Program) Recommended Provisions for Seismic
Regulations for New Buildings and Other Structures, 2003 Edition, 2 volumes and maps, FEMA 450
(issued as a paper document).The 2003 and 2009 edition of the Provisions can also be downloaded from
the BSSC website at www.nibs.org/bssc. Also see the website for information regarding BSSC projects
and publications or write to the BSSC at bssc@nibs.org or at the National Institute of Building Sciences,
1090 Vermont Avenue, NW, Suite 700, Washington, DC 20005 (telephone 2022897800).
Updated education/training materials to supplement this set of design examples will be published as a
separate FEMA product, 2009 NEHRP Recommended Seismic Provisions: Training Material, FEMA P
752.
The BSSC is grateful to all those individuals and organizations whose assistance made the 2012 edition of
the design examples a reality:
Michael T. Valley, Magnusson Klemencic Associates, Seattle, Washington, who served as project
manager and managing technical editor for the update.
Ozgur Atlayan, Robert Bachman, Finley A. Charney, Brian Dean, Susan Dowty, John Gillengerten,
James Robert Harris, Charles A. Kircher, Suzanne Dow Nakaki, Clinton O. Rex, Frederic R. Rutz,
Rafael A. Sabelli, Peter W. Somers, Greg Soules, Adrian Tola Tola and Michael T. Valley for
editing the original chapters to prepare this update of the 2006 Edition.
Robert Pekelnicky for preparing a new Introduction; and Nicolas Luco, Michael Valley and C.B.
Crouse for preparing a new chapter on Earthquake Ground Motions for this edition.
Lawrence A. Burkett, Kelly Cobeen, Finley Charney, Ned Cleland, Dan Dolan, Jeffrey J.
Dragovich, Jay Harris, Robert D. Hanson, Neil Hawkins, Joe Maffei, Greg Soules, and Mai Tong
for their reviews of the edited, updated and expanded material.
And finally, the BSSC Board is grateful to FEMA Project Officer Mai Tong for his support and guidance
and to Deke Smith, Roger Grant and Pamela Towns of the NIBS staff for their efforts preparing the 2012
volume for publication and issuance as an edocument available for download and on CDROM.
Jim. W. Sealy, Chairman
BSSC Board of Direction
TABLE OF CONTENTS
FOREWORD iii
PREFACE iv
1 INTRODUCTION
1.1 EVOLUTION OF EARTHQUAKE ENGINEERING 13
1.2 HISTORY AND ROLE OF THE NEHRP PROVISIONS 16
1.3 THE NEHRP DESIGN EXAMPLES 18
1.4 GUIDE TO USE OF THE PROVISIONS 111
1.5 REFERENCES 138
2 FUNDAMENTALS
2.1 EARTHQUAKE PHENOMENA 23
2.2 STRUCTURAL RESPONSE TO GROUND SHAKING 25
2.2.1 Response Spectra 25
2.2.2 Inelastic Response 211
2.2.3 Building Materials 214
2.2.4 Building Systems 216
2.2.5 Supplementary Elements Added to Improve Structural Performance 217
2.3 ENGINEERING PHILOSOPHY 218
2.4 STRUCTURAL ANALYSIS 219
2.5 NONSTRUCTURAL ELEMENTS OF BUILDINGS 222
2.6 QUALITY ASSURANCE 223
3 EARTHQUAKE GROUND MOTION
3.1 BASIS OF EARTHQUAKE GROUND MOTION MAPS 32
3.1.1 ASCE 705 Seismic Maps 32
3.1.2 MCER Ground Motions in the Provisions and in ASCE 710 33
3.1.3 PGA Maps in the Provisions and in ASCE 710 37
3.1.4 Basis of Vertical Ground Motions in the Provisions and in ASCE 710 37
3.1.5 Summary 37
3.1.6 References 38
3.2 DETERMINATION OF GROUND MOTION VALUES AND SPECTRA 39
3.2.1 ASCE 705 Ground Motion Values 39
3.2.2 2009 Provisions Ground Motion Values 310
3.2.3 ASCE 710 Ground Motion Values 311
3.2.4 Horizontal Response Spectra 312
3.2.5 Vertical Response Spectra 313
3.2.6 Peak Ground Accelerations 314
3.3 SELECTION AND SCALING OF GROUND MOTION RECORDS 314
3.3.1 Approach to Ground Motion Selection and Scaling 315
3.3.2 TwoComponent Records for Three Dimensional Analysis 324
3.3.3 OneComponent Records for TwoDimensional Analysis 327
3.3.4 References 328
4 STRUCTURAL ANALYSIS
4.1 IRREGULAR 12STORY STEEL FRAME BUILDING, STOCKTON, CALIFORNIA 43
4.1.1 Introduction 43
4.1.2 Description of Building and Structure 43
4.1.3 Seismic Ground Motion Parameters 44
4.1.4 Dynamic Properties 48
4.1.5 Equivalent Lateral Force Analysis 411
4.1.6 Modal Response Spectrum Analysis 429
4.1.7 Modal Response History Analysis 439
4.1.8 Comparison of Results from Various Methods of Analysis 450
4.1.9 Consideration of Higher Modes in Analysis 453
4.1.10 Commentary on the ASCE 7 Requirements for Analysis 456
4.2 SIXSTORY STEEL FRAME BUILDING, SEATTLE, WASHINGTON 457
4.2.1 Description of Structure 457
4.2.2 Loads 460
4.2.3 Preliminaries to Main Structural Analysis 464
4.2.4 Description of Model Used for Detailed Structural Analysis 472
4.2.5 Nonlinear Static Analysis 494
4.2.6 Response History Analysis 4109
4.2.7 Summary and Conclusions 4134
5 FOUNDATION ANALYSIS AND DESIGN
5.1 SHALLOW FOUNDATIONS FOR A SEVENSTORY OFFICE BUILDING,
LOS ANGELES, CALIFORNIA 53
5.1.1 Basic Information 53
5.1.2 Design for Gravity Loads 58
5.1.3 Design for MomentResisting Frame System 511
5.1.4 Design for Concentrically Braced Frame System 516
5.1.5 Cost Comparison 524
5.2 DEEP FOUNDATIONS FOR A 12STORY BUILDING, SEISMIC DESIGN
CATEGORY D 525
5.2.1 Basic Information 525
5.2.2 Pile Analysis, Design and Detailing 533
5.2.3 Other Considerations 547
6 STRUCTURAL STEEL DESIGN
6.1 INDUSTRIAL HIGHCLEARANCE BUILDING, ASTORIA, OREGON 63
6.1.1 Building Description 63
6.1.2 Design Parameters 66
6.1.3 Structural Design Criteria 67
6.1.4 Analysis 610
6.1.5 Proportioning and Details 616
6.2 SEVENSTORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA 640
6.2.1 Building Description 640
6.2.2 Basic Requirements 642
6.2.3 Structural Design Criteria 644
6.2.4 Analysis and Design of Alternative A: SMF 646
6.2.5 Analysis and Design of Alternative B: SCBF 660
6.2.6 Cost Comparison 672
6.3 TENSTORY HOSPITAL, SEATTLE, WASHINGTON 672
6.3.1 Building Description 672
6.3.2 Basic Requirements 676
6.3.3 Structural Design Criteria 678
6.3.4 Elastic Analysis 680
6.3.5 Initial Proportioning and Details 686
6.3.6 Nonlinear Response History Analysis 693
7 REINFORCED CONCRETE
7.1 SEISMIC DESIGN REQUIREMENTS 77
7.1.1 Seismic Response Parameters 77
7.1.2 Seismic Design Category 78
7.1.3 Structural Systems 78
7.1.4 Structural Configuration 79
7.1.5 Load Combinations 79
7.1.6 Material Properties 710
7.2 DETERMINATION OF SEISMIC FORCES 711
7.2.1 Modeling Criteria 711
7.2.2 Building Mass 712
7.2.3 Analysis Procedures 713
7.2.4 Development of Equivalent Lateral Forces 713
7.2.5 Direction of Loading 719
7.2.6 Modal Analysis Procedure 720
7.3 DRIFT AND PDELTA EFFECTS 721
7.3.1 Torsion Irregularity Check for the Berkeley Building 721
7.3.2 Drift Check for the Berkeley Building 723
7.3.3 Pdelta Check for the Berkeley Building 727
7.3.4 Torsion Irregularity Check for the Honolulu Building 729
7.3.5 Drift Check for the Honolulu Building 729
7.3.6 PDelta Check for the Honolulu Building 731
7.4 STRUCTURAL DESIGN OF THE BERKELEY BUILDING 732
7.4.1 Analysis of FrameOnly Structure for 25 Percent of Lateral Load 733
7.4.2 Design of Moment Frame Members for the Berkeley Building 737
7.4.3 Design of Frame 3 Shear Wall 760
7.5 STRUCTURAL DESIGN OF THE HONOLULU BUILDING 766
7.5.1 Compare Seismic Versus Wind Loading 766
7.5.2 Design and Detailing of Members of Frame 1 769
8 PRECAST CONCRETE DESIGN
8.1 HORIZONTAL DIAPHRAGMS 84
8.1.1 Untopped Precast Concrete Units for FiveStory Masonry Buildings Located in
Birmingham, Alabama and New York, New York 84
8.1.2 Topped Precast Concrete Units for FiveStory Masonry Building Located in
Los Angeles, California (see Sec. 10.2) 818
8.2 THREESTORY OFFICE BUILDING WITH INTERMEDIATE PRECAST
CONCRETE SHEAR WALLS 826
8.2.1 Building Description 827
8.2.2 Design Requirements 828
8.2.3 Load Combinations 829
8.2.4 Seismic Force Analysis 830
8.2.5 Proportioning and Detailing 833
8.3 ONESTORY PRECAST SHEAR WALL BUILDING 845
8.3.1 Building Description 845
8.3.2 Design Requirements 848
8.3.3 Load Combinations 849
8.3.4 Seismic Force Analysis 850
8.3.5 Proportioning and Detailing 852
8.4 SPECIAL MOMENT FRAMES CONSTRUCTED USING PRECAST CONCRETE 865
8.4.1 Ductile Connections 865
8.4.2 Strong Connections 867
9 COMPOSITE STEEL AND CONCRETE
9.1 BUILDING DESCRIPTION 93
9.2 PARTIALLY RESTRAINED COMPOSITE CONNECTIONS 97
9.2.1 Connection Details 97
9.2.2 Connection MomentRotation Curves 910
9.2.3 Connection Design 913
9.3 LOADS AND LOAD COMBINATIONS 917
9.3.1 Gravity Loads and Seismic Weight 917
9.3.2 Seismic Loads 918
9.3.3 Wind Loads 919
9.3.4 Notional Loads 919
9.3.5 Load Combinations 920
9.4 DESIGN OF CPRMF SYSTEM 921
9.4.1 Preliminary Design 921
9.4.2 Application of Loading 922
9.4.3 Beam and Column Moment of Inertia 923
9.4.4 Connection Behavior Modeling 924
9.4.5 Building Drift and Pdelta Checks 924
9.4.6 Beam Design 926
9.4.7 Column Design 927
9.4.8 Connection Design 928
9.4.9 Column Splices 929
9.4.10 Column Base Design 929
10 MASONRY
10.1 WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF,
LOS ANGELES, CALIFORNIA 103
10.1.1 Building Description 103
10.1.2 Design Requirements 104
10.1.3 Load Combinations 106
10.1.4 Seismic Forces 108
10.1.5 Side Walls 109
10.1.6 End Walls 1025
10.1.7 InPlane Deflection End Walls 1044
10.1.8 Bond Beam Side Walls (and End Walls) 1045
10.2 FIVESTORY MASONRY RESIDENTIAL BUILDINGS IN
BIRMINGHAM, ALABAMA; ALBUQUERQUE, NEW MEXICO; AND
SAN RAFAEL, CALIFORNIA 1045
10.2.1 Building Description 1045
10.2.2 Design Requirements 1048
10.2.3 Load Combinations 1050
10.2.4 Seismic Design for Birmingham 1 1051
10.2.5 Seismic Design for Albuquerque 1069
10.2.6 Birmingham 2 Seismic Design 1081
10.2.7 Seismic Design for San Rafael 1089
10.2.8 Summary of Wall D Design for All Four Locations 10101
11 WOOD DESIGN
11.1 THREESTORY WOOD APARTMENT BUILDING, SEATTLE, WASHINGTON 113
11.1.1 Building Description 113
11.1.2 Basic Requirements 116
11.1.3 Seismic Force Analysis 119
11.1.4 Basic Proportioning 1111
11.2 WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF,
LOS ANGELES, CALIFORNIA 1130
11.2.1 Building Description 1130
11.2.2 Basic Requirements 1131
11.2.3 Seismic Force Analysis 1133
11.2.4 Basic Proportioning of Diaphragm Elements 1134
12 SEISMICALLY ISOLATED STRUCTURES
12.1 BACKGROUND AND BASIC CONCEPTS 124
12.1.1 Types of Isolation Systems 124
12.1.2 Definition of Elements of an Isolated Structure 125
12.1.3 Design Approach 126
12.1.4 Effective Stiffness and Effective Damping 127
12.2 CRITERIA SELECTION 127
12.3 EQUIVALENT LATERAL FORCE PROCEDURE 129
12.3.1 Isolation System Displacement 129
12.3.2 Design Forces 1211
12.4 DYNAMIC LATERAL RESPONSE PROCEDURE 1215
12.4.1 Minimum Design Criteria 1215
12.4.2 Modeling Requirements 1216
12.4.3 Response Spectrum Analysis 1218
12.4.4 Response History Analysis 1218
12.5 EMERGENCY OPERATIONS CENTER USING DOUBLECONCAVE
FRICTION PENDULUM BEARINGS, OAKLAND, CALIFORNIA 1221
12.5.1 System Description 1222
12.5.2 Basic Requirements 1225
12.5.3 Seismic Force Analysis 1234
12.5.4 Preliminary Design Based on the ELF Procedure 1236
12.5.5 Design Verification Using Nonlinear Response History Analysis 1251
12.5.6 Design and Testing Criteria for Isolator Units 1261
13 NONBUILDING STRUCTURE DESIGN
13.1 NONBUILDING STRUCTURES VERSUS NONSTRUCTURAL COMPONENTS 134
13.1.1 Nonbuilding Structure 135
13.1.2 Nonstructural Component 136
13.2 PIPE RACK, OXFORD, MISSISSIPPI 136
13.2.1 Description 137
13.2.2 Provisions Parameters 137
13.2.3 Design in the Transverse Direction 13 8
13.2.4 Design in the Longitudinal Direction 1311
13.3 STEEL STORAGE RACK, OXFORD, MISSISSIPPI 1313
13.3.1 Description 1313
13.3.2 Provisions Parameters 1314
13.3.3 Design of the System 1315
13.4 ELECTRIC GENERATING POWER PLANT, MERNA, WYOMING 1317
13.4.1 Description 1317
13.4.2 Provisions Parameters 1319
13.4.3 Design in the NorthSouth Direction 1320
13.4.4 Design in the EastWest Direction 1321
13.5 PIER/WHARF DESIGN, LONG BEACH, CALIFORNIA 1321
13.5.1 Description 1321
13.5.2 Provisions Parameters 1322
13.5.3 Design of the System 1323
13.6 TANKS AND VESSELS, EVERETT, WASHINGTON 1324
13.6.1 FlatBottom Water Storage Tank 1325
13.6.2 FlatBottom Gasoline Tank 1328
13.7 VERTICAL VESSEL, ASHPORT, TENNESSEE 1331
13.7.1 Description 1331
13.7.2 Provisions Parameters 1332
13.7.3 Design of the System 1333
14 DESIGN FOR NONSTRUCTURAL COMPONENTS
14.1 DEVELOPMENT AND BACKGROUND OF THE REQUIREMENTS
FOR NONSTRUCTURAL COMPONENTS 143
14.1.1 Approach to Nonstructural Components 143
14.1.2 Force Equations 144
14.1.3 Load Combinations and Acceptance Criteria 145
14.1.4 Component Amplification Factor 146
14.1.5 Seismic Coefficient at Grade 147
14.1.6 Relative Location Factor 147
14.1.7 Component Response Modification Factor 147
14.1.8 Component Importance Factor 147
14.1.9 Accommodation of Seismic Relative Displacements 148
14.1.10 Component Anchorage Factors and Acceptance Criteria 149
14.1.11 Construction Documents 149
14.2 ARCHITECTURAL CONCRETE WALL PANEL 1410
14.2.1 Example Description 1410
14.2.2 Design Requirements 1412
14.2.3 Spandrel Panel 1412
14.2.4 Column Cover 1419
14.2.5 Additional Design Considerations 1420
14.3 HVAC FAN UNIT SUPPORT 1421
14.3.1 Example Description 1421
14.3.2 Design Requirements 1422
14.3.3 Direct Attachment to Structure 1423
14.3.4 Support on Vibration Isolation Springs 1426
14.3.5 Additional Considerations for Support on Vibration Isolators 1431
14.4 ANALYSIS OF PIPING SYSTEMS 1433
14.4.1 ASME Code Allowable Stress Approach 1433
14.4.2 Allowable Stress Load Combinations 1434
14.4.3 Application of the Standard 1436
14.5 PIPING SYSTEM SEISMIC DESIGN 1438
14.5.1 Example Description 1438
14.5.2 Design Requirements. 1443
14.5.3 Piping System Design 1445
14.5.4 Pipe Supports and Bracing 1448
14.5.5 Design for Displacements 1453
14.6 ELEVATED VESSEL SEISMIC DESIGN 1455
14.6.1 Example Description 1455
14.6.2 Design Requirements 1458
14.6.3 Load Combinations 1460
14.6.4 Forces in Vessel Supports 1460
14.6.5 Vessel Support and Attachment 1462
14.6.6 Supporting Frame 1465
14.6.7 Design Considerations for the Vertical LoadCarrying System 1469
A THE BUILDING SEISMIC SAFETY COUNCIL
1
Introduction
Robert G. Pekelnicky, P.E., S.E. and Michael Valley, S.E.
Contents
1.1 EVOLUTION OF EARTHQUAKE ENGINEERING 3
1.2 HISTORY AND ROLE OF THE NEHRP PROVISIONS 6
1.3 THE NEHRP DESIGN EXAMPLES 8
1.4 GUIDE TO USE OF THE PROVISIONS 11
1.5 REFERENCES 38
The NEHRP Recommended Provisions: Design Examples are written to illustrate and explain the
applications of the 2009 NEHRP Recommended Seismic Provisions for Buildings and Other
Structures, ASCE 710 Minimum Design Loads for Buildings and Other Structures and the
material design standards referenced therein and to provide explanations to help understand
them. Designing structures to be resistant to major earthquake is complex and daunting to
someone unfamiliar with the philosophy and history of earthquake engineering. The target
audience for the Design Examples is broad. College students learning about earthquake
engineering, engineers studying for their licensing exam, or those who find themselves presented
with the challenge of designing in regions of moderate and high seismicity for the first time
should all find this document s explanation of earthquake engineering and the Provisions
helpful.
Fortunately, major earthquakes are a rare occurrence, significantly rarer than the other hazards,
such as damaging wind and snow storms that one must typically consider in structural design.
However, past experiences have shown that the destructive power of a major earthquake can be
so great that its effect on the built environment can be underestimated. This presents a challenge
since one cannot typically design a practical and economical structure to withstand a major
earthquake elastically in the same manner traditionally done for other hazards.
Since elastic design is not an economically feasible option for most structures where major
earthquakes can occur, there must be a way to design a structure to be damaged but still safe.
Unlike designing for strong winds, where the structural elements that resist lateral forces can be
proportioned to elastically resist the pressures generated by the wind, in an earthquake the lateral
force resisting elements must be proportioned to deform beyond their elastic range in a
controlled manner. In addition to deforming beyond their elastic range, the lateral force resisting
system must be robust enough to provide sufficient stability so the building is not at risk of
collapse.
While typical structures are designed to be robust enough to have a minimal risk of collapse in
major earthquakes, there are other structures whose function or type of occupants warrants
higher performance designs. Structures, like hospitals, fire stations and emergency operation
centers need to be designed to maintain their function immediately after or returned to function
shortly after the earthquake. Structures like schools and places where large numbers of people
assemble have been deemed important enough to require a greater margin of safety against
collapse than typical buildings. Additionally, earthquake resistant requirements are needed for
the design and anchorage of architectural elements and mechanical, electrical and plumbing
systems to prevent falling hazards and in some cases loss of system function.
Current building standards, specifically the American Society of Civil Engineers (ASCE) 7
Minimum Design Loads for Buildings and Other Structures and the various material design
standards published by the American Concrete Institute (ACI), the American Institute of Steel
Construction (AISC), the American Iron and Steel Institute (AISI), the American Forest & Paper
Association (AF&PA) and The Masonry Society (TMS) provide a means by which an engineer
can achieve these design targets. These standards represent the most recent developments in
earthquake resistant design. The majority of the information contained in ASCE 7 comes
directly from the NEHRP Recommended Seismic Provisions for New Buildings and Other
Structures. The stated intent of the NEHRP Provisions is to provide reasonable assurance of
seismic performance that will:
1. Avoid serious injury and life loss,
2. Avoid loss of function in critical facilities, and
3. Minimize structural and nonstructural repair costs where practical to do so.
The Provisions have explicit requirements to provide life safety for buildings and other
structures though the design forces and detailing requirements. The current provisions have
adopted a target risk of collapse of 1% over a 50 year lifespan for a structure designed to the
Provisions. The Provisions provide prevention of loss of function in critical facilities and
minimized repair costs in a more implicit manner though prescriptive requirements.
Having good building codes and design standards is only one action necessary to make a
community s buildings resilient to a major earthquake. A community also needs engineers who
can carry out designs in accordance with the requirements of the codes and standards and
contractors who can construct the designs in accordance with properly prepared construction
documents. The first item is what the NEHRP Recommended Provisions: Design Examples
seeks to foster. The second item is discussed briefly later in this document in Chapter 1, Section
1.6 Quality Assurance.
The purpose of this introduction is to offer general guidance for users of the design examples and
to provide an overview. Before introducing the design examples, a brief history of earthquake
engineering is presented. That is followed by a history of the NEHRP Provisions and its role in
setting standards for earthquake resistant design. This is done to give the reader a perspective of
the evolution of the Provisions and some background for understanding the design examples.
Following that is a brief summary of each chapter.
1.1 EVOLUTION OF EARTHQUAKE ENGINEERING
It is helpful to understand the evolution of the earthquake design standards and the evolution of
the field of earthquake engineering in general. Much of what is contained within the standards is
based on lessons learned from earthquake damage and the ensuing research.
Prior to 1900 there was little consideration of earthquakes in the design of buildings. Major
earthquakes were experienced in the United States, notably the 1755 Cap Ann Earthquake
around Boston, the 1811 and 1812 New Madrid Earthquakes, the 1868 Hayward California
Earthquake and the 1886 Charleston Earthquake. However, none of these earthquakes led to
substantial changes in the way buildings were constructed.
Many things changed with the Great 1906 San Francisco Earthquake. The earthquake and
ensuing fire destroyed much of San Francisco and was responsible for approximately 3,000
deaths. To date it is the most deadly earthquake the United States has ever experienced. While
there was significant destruction to the built environment, there were some important lessons
learned from those buildings that performed well and did not collapse. Most notable was the
exemplary performance of steel framed buildings which consisted of riveted wind frames and
brick infill, built in the Chicago style.
The recently formed San Francisco Section of the American Society of Civil Engineers (ASCE)
studied the effects of the earthquake in great detail. An observation was that a building
designed with a proper system of bracing wind pressure at 30 lbs. per square foot will resist
safely the stresses caused by a shock of the intensity of the recent earthquake. (ASCE, 1907)
That one statement became the first U.S. guideline on how to provide an earthquake resistant
design.
The earthquakes in Tokyo in 1923 and Santa Barbara in 1925 spurred major research efforts.
Those efforts led to the development of the first seismic recording instruments, shake tables to
investigate earthquake effects on buildings, and committees dedicated to creating code
provisions for earthquake resistant design. Shortly after these earthquakes, the 1927 Uniform
Building Code (UBC) was published (ICBO, 1927). It was the first model building code to
contain provisions for earthquake resistant design, albeit in an appendix. In addition to that, a
committee began working on what would become California s first statewide seismic code in
1939.
Another earthquake struck Southern California in Long Beach in 1933. The most significant
aspect, of that earthquake was the damage done to school buildings. Fortunately the earthquake
occurred after school hours, but it did cause concern over the vulnerabilities of these buildings.
That concern led to the Field Act, which set forth standards and regulations for earthquake
resistance of school buildings. This was the first instance of what has become a philosophy
engrained in the earthquake design standards of requiring higher levels of safety and
performance for certain buildings society deems more important that a typical building. In
addition to the Field Act, the Long Beach earthquake led to a ban on unreinforced masonry
construction in California, which in later years was extended to all areas of moderate and high
seismic risk.
Following the 1933 Long Beach Earthquake there was significant activity both in Northern and
Southern California, with the local Structural Engineers Associations of each region drafting
seismic design provisions for Los Angeles in 1943 and San Francisco in 1948. Development of
these codes was facilitated greatly by observations from the 1940 El Centro Earthquake.
Additionally, that earthquake was the first major earthquake for which the strong ground motion
shaking was recorded with an accelerograph.
A joint committee of the San Francisco Section of ASCE and the Structural Engineers
Association of Northern California began work on seismic design provisions which were
published in 1951 as ASCE ProceedingsSeparate No. 66. Separate 66, as it is commonly
referred to as, was a landmark document which set forth earthquake design provisions which
formed the basis of US building codes for almost 40 years. Many concepts and recommendations
put forth in Separate 66, such as the a period dependent design spectrum, different design forces
based on the ductility of a structure and design provisions for architectural components are still
found in today s standards.
Following Separate 66, the Structural Engineers Association of California (SEAOC) formed a
Seismology committee and in 1959 put forth the first edition of the Recommended Lateral Force
Requirements, commonly referred to as the The SEAOC Blue Book. The Blue Book became
the base document for updating and expanding the seismic design provisions of the Uniform
Building Code (UBC), the model code adopted by most western states including California.
SEAOC regularly updated the Blue Book from 1959 until 1999, with the changes made and new
recommendations in each new edition of the Blue Book being incorporated in to the subsequent
edition of the UBC.
The 1964 Anchorage Earthquake and the 1971 San Fernando Earthquake both were significant
events. Both earthquakes exposed significant issues with the way reinforced concrete structures
would behave if not detailed for ductility. There were failures of large concrete buildings which
had been designed to recent standards and those buildings had to be torn down. To most
engineers and the public this was unacceptable performance.
Following the 1971 San Fernando Earthquake, the National Science Foundation gave the
Applied Technology Council (ATC) a grant to develop more advanced earthquake design
provisions. That project engaged over 200 preeminent experts in the field of earthquake
engineering. The landmark report they produced in 1978, ATC 306, Tentative Provisions for
the Development of Seismic Regulations for Buildings (1978), has become the basis for the
current earthquake design standards. The NEHRP Provisions trace back to ATC 306, as will be
discussed in more detail in the following section.
There have been additional earthquakes since the 1971 San Fernando Earthquake which have
had significant influence on seismic design. Table 1 provides a summary of major North
American earthquakes and changes to the building codes that resulted from them through the
1997 UBC. Of specific note are the 1985 Mexico City, 1989 Loma Prieta and 1994 Northridge
Earthquakes.
Earthquake
UBC
Edition
Enhancement
1971 San Fernando
1973
Direct positive anchorage of masonry and concrete
walls to diaphragms
1976
Seismic Zone 4, with increased base shear
requirements
Occupancy Importance Factor I for certain buildings
Interconnection of individual column foundations
Special Inspection requirements
1979 Imperial Valley
1985
Diaphragm continuity ties
1985 Mexico City
1988
Requirements for column supporting discontinuous
walls
Separation of buildings to avoid pounding
Design of steel columns for maximum axial forces
Restrictions for irregular structures
Ductile detailing of perimeter frames
1987 Whittier Narrows
1991
Revisions to site coefficients
Revisions to spectral shape
Increased wall anchorage forces for flexible
diaphragm buildings
1989 Loma Prieta
1991
Increased restrictions on chevronbraced frames
Limitations on b/t ratios for braced frames
1994
Ductile detailing of piles
1994 Northridge
1997
Restrictions on use of battered piles
Requirements to consider liquefaction
Nearfault zones and corresponding base shear
requirements
Revised base shear equations using 1/T spectral shape
Redundancy requirements
Design of collectors for overstrength
Increase in wall anchorage requirements
More realistic evaluation of design drift
Steel moment connection verification by test
Table 1: Recent North American Earthquakes and Subsequent Code Changes (from SEOAC,
2009)
The 1985 Mexico City Earthquake was extremely devastating. Over 10,000 people were killed
and there was three to four billion dollars of damage. The most significant aspect of this
earthquake was that while the epicenter was located over 200 miles away from Mexico City. The
unique geologic nature, that Mexico City was sited on an old (ancient ) lake bed of silt and clay,
generated ground shaking with a much longer period and larger amplitudes than would be
expected from typical earthquakes. This long period shaking was much more damaging to mid
rise and larger structures because these buildings were in resonance with the ground motions. In
current design practice site factors based on the underlying soil are used to modify the seismic
hazard parameters.
The 1989 Loma Prieta Earthquake caused an estimated $6 billion in damage, although it was far
less deadly than other major earthquakes throughout history. Only sixtythree people lost their
lives, a testament to the over 40 years of awareness and consideration of earthquakes in the
design of structures. A majority of those deaths, 42, resulted from the collapse of the Cyprus
Street Viaduct, a nonductile concrete elevated freeway. In this earthquake the greatest damage
occurred in Oakland, parts of Santa Cruz and the Marina District in San Francisco where the soil
was soft or poorly compacted fill. As with the Mexico City experience, this indicates the
importance of subsurface conditions on the amplification of earthquake shaking. The earthquake
also highlighted the vulnerability of soft and weak story buildings because a significant number
of the collapsed buildings in the Marina District were wood framed apartment buildings with
weak first stories consisting of garages with door openings that greatly reduced the wall area at
the first story.
Five years later the 1994 Northridge earthquake struck California near Los Angeles. Fifty seven
people lost their lives and the damage was estimated at around $20 billion. The high cost of
damage repair emphasized the need for engineers to consider overall building performance, in
addition to building collapse, and spurred the movement toward PerformanceBased design. As
with the 1989 Loma Prieta earthquake, there was a disproportionate number of collapses of
soft/weak first story wood framed apartment buildings.
The 1994 Northridge Earthquake was most significant for the unanticipated damage to steel
moment frames that was discovered. Steel moment frames had generally been thought of as the
best seismic force resisting system. However, many moment frames experienced fractures of the
welds that connected the beam flange to the column flange. This led to a multiyear, FEMA
funded problemfocused study to assess and improve the seismic performance of steel moment
frames. It also led to requirements for the number of frames in a structure, and penalties for
having a lateral force resisting system that does not have sufficient redundancy.
1.2 HISTORY AND ROLE OF THE NEHRP PROVISIONS
Following the completion of the ATC 3 project in 1978, there was desire to make the ATC 306
approach the basis for new regulatory provisions and to update them periodically. FEMA, as the
lead agency of the National Earthquake Hazard Reduction Program (NEHRP) at the time,
contracted with the then newly formed Building Seismic Safety Council (BSSC) to perform trial
designs based on ATC 306 to exercise the proposed new provisions. The BSSC put together a
group of experts consisting of consulting engineers, academics, representatives from various
building industries and building officials. The result of that effort was the first (1985) edition of
the NEHRP Recommended Provisions for the Development of Seismic Regulations for New
Buildings.
Since the publication of the first edition through the 2003 edition, the NEHRP Provisions were
updated every three years. Each update incorporated recent advances in earthquake engineering
research and lessons learned from previous earthquakes. The intended purpose of the Provisions
was to serve as a code resource document. While the SEAOC Blue Book continued to serve as
the basis for the earthquake design provisions in the Uniform Building Code, the BOCA National
Building Code and the Standard Building Code both adopted the 1991 NEHRP Provisions in
their 1993 and 1994 editions respectively. The 1993 version of the ASCE 7 standard Minimum
Design Loads for Buildings and Other Structures (which had formerly been American National
Standards Institute (ANSI) Standard A58.1) also utilized the 1991 NEHRP Provisions.
In the late 1990 s the three major code organizations, ICBO (publisher of the UBC), BOCA, and
SBC decided to merge their three codes into one national model code. When doing so they
chose to incorporate the 1997 NEHRP Provisions as the seismic design requirements for the
inaugural 2000 edition of the International Building Code (IBC). Thus, the SEAOC Blue Book
was no longer the base document for the UBC/IBC. The 1997 NEHRP Provisions had a number
of major changes. Most significant was the switch from the older seismic maps of ATC 306 to
new, uniform hazard spectral value maps produced by USGS in accordance with BSSC
Provisions Update Committee (PUC) Project 97. The 1998 edition of ASCE 7 was also based on
the 1997 NEHRP Provisions.
ASCE 7 continued to incorporate the 2000 and 2003 editions of the Provisions for its 2002 and
2005 editions, respectively. However, the 2000 IBC adopted the 1997 NEHRP Provisions by
directly transferring the text from the provisions into the code. In the 2003 IBC the provisions
from the 2000 IBC were retained and there was also language, for the first time, which pointed
the user to ASCE 702 for seismic provisions instead of adopting the 2000 NEHRP Provisions
directly. The 2006 IBC explicitly referenced ASCE 7 for the earthquake design provisions, as
did the 2009 and 2012 editions.
With the shift in the IBC from directly incorporating the NEHRP Provision for their earthquake
design requirements to simply referencing the provisions in ASCE 7, the BSSC Provisions
Update Committee decided to move the NEHRP Provisions in a new direction. Instead of
providing all the seismic design provisions within the NEHRP Provisions, which would
essentially be repeating the provisions in ASCE 7, and then modifying them, the PUC chose to
adopt ASCE 705 by reference and then provide recommendations to modify it as necessary.
Therefore, Part 1 of the 2009 NEHRP Provisions contains major technical modifications to
ASCE 705 which, along with other recommendations from the ASCE 7 Seismic Subcommittee,
were the basis for proposed changes that were incorporated into ASCE 710 and included
associated commentary on those changes. The PUC also developed a detailed commentary to
the seismic provisions of ASCE 705, which became Part 2 of the 2009 NEHRP Provisions.
In addition to Part 1 and Part 2 in the 2009 NEHRP Provisions, a new section was introduced
Part 3. The intent of this new portion was to showcase new research and emerging methods,
which the PUC did not feel was ready for adoption into national design standards but was
important enough to be disseminated to the profession. This new three part format marks a
change in the Provisions from a codelanguage resource document to the key knowledgebased
resource for improving the national seismic design standards and codes.
The most significant technical change to Part 1 of the 2009 Provisions was the adoption of a
RiskTargeted approach to determine the Maximum Considered Earthquake hazard
parameters. This was a switch from the Uniform Hazard approach in the 1997, 2000, and 2003
editions. In the Risk Targeted approach, the ground motion parameters are adjusted such that
they provide a uniform 1% risk of collapse in a 50 year period for a generic building, as opposed
to a uniform return period for the seismic hazard. A detailed discussion of this can be found in
the commentary in Part 1 of the 2009 Provisions.
Today, someone needing to design a seismically resilient building in the U.S. would first go to
the local building code which has generally adopted the IBC with or without modifications by
the local jurisdiction. For seismic design requirements, the IBC then points to relevant Chapters
of ASCE 7. Those chapters of ASCE 7 set forth the seismic hazard, design forces and system
detailing requirements. The seismic forces in ASCE 7 are dependent upon the type of detailing
and specific requirements of the lateral force resisting system elements. ASCE 7 then points to
material specific requirements found in the material design standards published by ACI, AISC,
AISI, AF&PA and TMS for those detailing requirements. Within this structure, the NEHRP
Provisions serves as a consensus evaluation of the design standards and a vehicle to transfer new
knowledge to ASCE 7 and the material design standards.
1.3 THE NEHRP DESIGN EXAMPLES
Design examples were first prepared for the 1985 NEHRP Provisions in a publication entitled
Guide to Application of the NEHRP Recommended Provisions, FEMA 140. These design
examples were based on real buildings. The intent was the same as it is now, to show people
who are not familiar with seismic design of how to apply the Provisions, the standards
referenced by the Provisions and the concepts behind the Provisions.
Because of the expanded role that the Provisions were having as the basis for the seismic design
requirements for the model codes and standards, it was felt that there should be an update and
expansion of the original design examples. Following the publication of the 2003 NEHRP
Provisions, FEMA commissioned a project to update and expand the design examples. This
resulted in NEHRP Recommended Provisions: Design Examples, FEMA 451. Many of the
design problems drew heavily on the examples presented in FEMA 140, but were completely
redesigned based on first the 2000 and then the 2003 NEHRP Provisions and the materials
standards referenced therein. Additional examples were created to reflect the myriad of
structures now covered under the Provisions.
This volume is an update of the design examples in FEMA 451 to reflect the 2009 NEHRP
Provisions and the updated standards referenced therein. Many of the design examples are the
same as presented in FEMA 451, with only changes made due to changes in the provisions.
The Design Examples not only covers the application of ASCE 7, the material design standards
and the NEHRP Provisions, it also illustrates the use of analysis methods and earthquake
engineering knowledge and judgment in situations which would be encountered in real designs.
The authors of the design examples are subject matter experts in the specific area covered by the
chapter they authored. Furthermore, the companion NEHRP Recommend Provisions: Training
Materials provides greater background information and knowledge, which augment the design
examples.
It is hoped that with the Part 2 Expanded Commentary in the 2009 NEHRP Provisions, the
Design Examples and the Training Materials, an engineer will be able to understand not just how
to use the Provisions, but also the philosophical and technical basis behind the provisions.
Through this understanding of the intent of the seismic design requirements found in ASCE 7,
the material design standards and the 2009 NEHRP Provisions, it is hoped that more engineers
will find the application of those standards less daunting and thereby utilize the standards more
effectively in creating innovative and safe designs.
Chapter 1 This preceding introduction and the Guide to Use of the Provisions which
follows provides background and presents a series of flow charts to walk an engineer through the
use of the provisions.
Chapter 2 Fundamentals presents a brief but thorough introduction to the fundamentals of
earthquake engineering. While this section does not present any specific applications of the
Provisions, it provides the reader with the essential philosophical background to what is
contained within the Provisions. The concepts of idealizing a seismic dynamic load as an
equivalent static load and providing ductility instead of pure elastic strength are explained.
Chapter 3  Earthquake Ground Motion is new to this edition of the Design Examples. This
chapter explains the basis for determining seismic hazard parameters used for design in the
Provisions. It discusses the seismic hazard maps in ASCE 705 and the new Risk Targeted maps
found in the 2009 NEHRP Provisions and ASCE 710. The chapter also discusses probabilistic
seismic hazard assessment, the maximum direction response parameters and selection and
scaling of ground motion histories for use in linear and nonlinear response history analysis.
Chapter 4 Structural Analysis presents the analysis of two different buildings, a 12story
steel moment frame and a 6story steel moment frame structure. The 12story structure is
irregular and is analyzed using the three linear procedures referenced in ASCE 7 Equivalent
Lateral Force, Modal Response Spectrum, and Linear Response History. The 6story structure is
analyzed using three methods referenced in ASCE 7  Equivalent Lateral Force, Modal Response
Spectrum and Nonlinear Response History and two methods which are referenced in other
documents Plastic Strength (Virtual Work) and Nonlinear Static Pushover. The intent of this
chapter is to show the variations in predicted response based on the chosen analysis method.
Some of the examples have been updated based on advances in the state of the practice with
respect to seismic analysis.
Chapter 5 Foundation Analysis and Design presents design examples for both shallow and
deep foundations. First, a spread footing foundation for a 7story steel framed building is
presented. Second the design of a pile foundation for a 12story concrete moment frame building
is presented. Designs of the steel and concrete structures whose foundations are designed in this
chapter are presented in Chapters 6 and 7 respectively.
Chapter 6 Structural Steel Design presents the design of three different types of steel
buildings. The first building is a highbay industrial warehouse which uses an ordinary
concentric braced frame in one direction and an intermediate steel moment frame in the other
direction. The second example is a 7story office building which is designed using two alternate
framing systems, special steel moment frames and special concentric braced frames. The third
example is new to this edition of the design examples. It is a 10story hospital using buckling
restrained braced frames (BRBF). This replaces an example using eccentrically braced frames
(EBF) in the previous edition of the design examples because the profession has moved toward
favoring the BRBF system over the EBF system.
Chapter 7 Reinforced Concrete presents the designs of a 12story office building located in
moderate and high seismicity. The same building configuration is used in both cases, but in the
moderate seismicity region Intermediate member frames are used while Special moment
frames are used in the high seismicity region. Also in the high seismicity region, special
concrete walls are needed in one direction and their design is presented.
Chapter 8 Precast Concrete Design presents examples of four common cases where precast
concrete elements are a component of a seismic force resisting system. The first example
presents the design of precast concrete panels being used as horizontal diaphragms both with and
without a concrete topping slab. The second example presents the design of 3story office
building using intermediate precast concrete shear walls in a region of low or moderate
seismicity The third example presents the design of a onestory tiltup concrete industrial
building in a region of high seismicity. The last example, which is new to this edition of the
design examples, presents the design of a precast Special Moment Frame.
Chapter 9 Composite Steel and Concrete presents the design of a 4story medical office
building in a region of moderate seismicity. The building uses composite partially restrained
moment frames in both directions as the lateral force resisting system.
Chapter 10 Masonry presents the design of two common types of buildings using reinforced
masonry walls as their lateral force resisting system. The first example is a singlestory masonry
warehouse building with tall, slender walls. The second example is a fivestory masonry hotel
building with a bearing wall system designed in areas with different seismicity.
Chapter 11 Wood Design presents the design of a variety of wood elements in common
seismic force resisting applications. The first example is a threestory, woodframe apartment
building. The second example illustrates the design of the roof diaphragm and walltoroof
anchorage for the masonry building featured in the first example of Chapter 10.
Chapter 12 Seismically Isolated Structures presents both the basic concepts of seismic
isolation and then the design of an essential facility using a seismic isolation system. The
example building has a special concentrically braced frame superstructure and uses double
concave friction pendulum isolators, which have become the most common type of isolator used
in regions of high seismicity. In the previous edition of the design examples, highdamping
rubber isolators were used.
Chapter 13 Nonbuilding Structure Design presents the design of various types of structures
other than buildings that are covered by the Provisions. First there is a brief discussion about the
difference between a nonbuilding structure and a nonstructural component. The first example is
the design of a pipe rack. The second example is of an industrial storage rack. The third
example is a power generating plant with significant mass irregularities. The third example is a
pier. The fourth examples are flatbottomed storage tanks, which also illustrates how the
Provisions are used in conjunction with industry design standards. The last example is of a tall,
slender vertical storage vessel containing hazardous materials, which replaces an example of an
elevated transformer.
Chapter 14 Design for Nonstructural Components presents a discussion on the design of
nonstructural components and their anchorage plus several design examples. The examples are
of an architectural concrete wall panel, the supports for a large rooftop fan unit, the analysis and
bracing of a piping system (which is greatly expanded from FEMA 451) and an elevated vessel
(which is new).
1.4 GUIDE TO USE OF THE PROVISIONS
The flow charts that follow are provided to assist the user of the NEHRP Recommended Provisions and,
by extension, the seismic provisions of ASCE 7, Minimum Design Loads for Buildings and Other
Structures; and the International Building Code. The flow charts provide an overview of the complete
process for satisfying the Provisions, including the content of all technical chapters.
Part 1 of the 2009 NEHRP Recommended Seismic Provisions for New Buildings and Other Structures
(the Provisions) adopts by reference the national consensus design loads standard, ASCE/SEI 705,
Minimum Design Loads for Buildings and Other Structures (the Standard), including Supplements 1 and
2, and makes modifications to the seismic requirements in the Standard. Part 2 of the Provisions contains
an uptodate, user friendly commentary on the seismic design requirements of the Standard. Part 3 of
the Provisions consists of a series of resource papers that clarify aspects of the Provisions and present
new seismic design concepts and procedures.
The flow charts in this chapter are expected to be of most use to those who are unfamiliar with the scope
of the NEHRP Recommended Provisions, but they cannot substitute for a careful reading of the
Provisions. The level of detail shown varies, being greater where questions of applicability of the
Provisions are pertinent and less where a standard process of structural analysis or detailing is all that is
required. The details contained in the many standards referenced in the Provisions are not included;
therefore, the actual flow of information when proportioning structural members for the seismic load
effects specified in the Provisions will be considerably more complex.
Cited section numbers (such as Sec. 11.1.2) refer to sections of the Standard. Where reference is to a
Provisions Part 1 modification to the Standard, the citation indicates that (such as Provisions Sec. 11.1.2).
In a few rare instances, the Provisions Update Committee deferred to the ASCE 7 committee to make
needed technical changes; in those cases reference is made specifically to ASCE 710 (such as ASCE 7
10 Sec. 12.12.3).
On each chart the flow generally is from a heavyweight box at the topleft to a mediumweight box at the
bottomright. User decisions are identified by sixsided cells. Optional items and modified flow are
indicated by dashed lines.
Chart 1.1 provides an overall summary of the process which begins with consideration of the Scope of
Coverage and ends with Quality Assurance Requirements. Additions to, changes of use in and alterations
of existing structures are covered by the Provisions (see Chart 1.3), but evaluation and rehabilitation of
existing structures is not. Nearly two decades of FEMAsponsored development of technical information
to improve seismic safety in existing buildings has culminated in a comprehensive set of codes, standards
and guidelines. The International Existing Building Code references the ASCE 31 Standard, Seismic
Evaluation of Existing Buildings; and the ASCE 41 Standard, Seismic Rehabilitation of Existing
Buildings.
Chart 1.1
Overall Summary of Flow
Chart 1.2
Scope of Coverage
Chart 1.3
Application to Existing Structures
Chart 1.4
Basic Requirements
Chart 1.5
Seismic Design Category A
* The requirement to reclassify Seismic Design Category A structures to Seismic Design Category B has
been declared editorially erroneous and will be removed via errata for ASCE 713.
Chart 1.6
Structural Design
Chart 1.7
Redundancy Factor
Chart 1.8
Simplified Design Procedure
Chart 1.9
Equivalent Lateral Force (ELF) Analysis
Chart 1.10
SoilStructure Interaction (SSI)
Chart 1.11
Modal Response Spectrum Analysis
Chart 1.12
Response History Analysis
Chart 1.13
Seismically Isolated Structures
Chart 1. 14
Structures with Damping Systems
Chart 1. 15
Deformation Requirements
Chart 1.16
Design and Detailing Requirements
Chart 1.17
Steel Structures
Chart 1.18
Concrete Structures
Chart 1.19
Composite Steel and Concrete Structures
Chart 1.20
Masonry Structures
Chart 1.21
Wood Structures
Chart 1.22
Nonbuilding Structures
Chart 1.23
Foundations
Chart 1.24
Nonstructural Components
Chart 1.25
Quality Assurance
1.5 REFERENCES
American Society of Civil Engineers, 1907, The Effects of the San Francisco Earthquake of
April 18, 1906., New York, NY.
American Society of Civil Engineers, 1951, ProceedingsSeparate No. 66., New York, NY.
American Society of Civil Engineers, 2010, ASCE 710: Minimum Design Loads for Buildings
and Other Structures, Reston, VA.
Applied Technology Council, 1978, ATC 306: Tentative Provisions for the Development of
Seismic Regulations for Buildings, Redwood City, California.
Building Seismic Safety Council, 2009, 2009 NEHRP Recommended Seismic Provisions for
Buildings and Other Structures, prepared for the Federal Emergency Management Agency,
Washington, DC.
International Conference of Building Officials, 1927, Uniform Building Code. Whittier, CA.
Structural Engineers Association of California, SEAOC Blue Book: Seismic Design
Recommendations, Sacramento, CA.
2
Fundamentals
James Robert Harris, P.E., PhD
Contents
2.1 EARTHQUAKE PHENOMENA 3
2.2 STRUCTURAL RESPONSE TO GROUND SHAKING 5
2.2.1 Response Spectra 5
2.2.2 Inelastic Response 11
2.2.3 Building Materials 14
2.2.4 Building Systems 16
2.2.5 Supplementary Elements Added to Improve Structural Performance 17
2.3 ENGINEERING PHILOSOPHY 18
2.4 STRUCTURAL ANALYSIS 19
2.5 NONSTRUCTURAL ELEMENTS OF BUILDINGS 22
2.6 QUALITY ASSURANCE 23
In introducing their classic text, Fundamentals of Earthquake Engineering, Newmark and
Rosenblueth (1971) comment:
In dealing with earthquakes, we must contend with appreciable
probabilities that failure will occur in the near future. Otherwise, all the
wealth of the world would prove insufficient to fill our needs: the most
modest structures would be fortresses. We must also face uncertainty on
a large scale, for it is our task to design engineering systems about
whose pertinent properties we know little to resist future earthquakes
and tidal waves about whose characteristics we know even less. . . . In
a way, earthquake engineering is a cartoon. . . . Earthquake effects on
structures systematically bring out the mistakes made in design and
construction, even the minutest mistakes.
Several points essential to an understanding of the theories and practices of earthquake
resistant design bear restating:
1. Ordinarily, a large earthquake produces the most severe loading that a building is
expected to survive. The probability that failure will occur is very real and is
greater than for other loading phenomena. Also, in the case of earthquakes, the
definition of failure is altered to permit certain types of behavior and damage that
are considered unacceptable in relation to the effects of other phenomena.
2. The levels of uncertainty are much greater than those encountered in the design of
structures to resist other phenomena. This is in spite of the tremendous strides
made since the Federal government began strongly supporting research in
earthquake engineering and seismology following the 1964 Prince William Sound
and 1971 San Fernando earthquakes. The high uncertainty applies both to
knowledge of the loading function and to the resistance properties of the
materials, members and systems.
3. The details of construction are very important because flaws of no apparent
consequence often will cause systematic and unacceptable damage simply
because the earthquake loading is so severe and an extended range of behavior is
permitted.
The remainder of this chapter is devoted to a very abbreviated discussion of fundamentals
that reflect the concepts on which earthquakeresistant design are based. When
appropriate, important aspects of the NEHRP Recommended Seismic Provisions for New
Buildings and Other Structures are mentioned and reference is made to particularly
relevant portions of that document or the standards that are incorporated by reference.
The 2009 Provisions is composed of three parts: 1) Provisions , 2) Commentary on
ASCE/SEI 72005 and 3) Resource Papers on Special Topics in Seismic Design . Part
1 states the intent and then cites ASCE/SEI 72005 Minimum Design Loads for Buildings
and Other Structures as the primary reference. The remainder of Part 1 contains
recommended changes to update ASCE/SEI 72005; the recommended changes include
commentary on each specific recommendation. All three parts are referred to herein as
the Provisions, but where pertinent the specific part is referenced and ASCE/SEI 72005
is referred to as the Standard. ASCE/SEI 72005 itself refers to several other standards
for the seismic design of structures composed of specific materials and those standards
are essential elements to achieve the intent of the Provisions.
2.1 EARTHQUAKE PHENOMENA
According to the most widely held scientific belief, most earthquakes occur when two
segments of the earth s crust suddenly move in relation to one another. The surface along
which movement occurs is known as a fault. The sudden movement releases strain
energy and causes seismic waves to propagate through the crust surrounding the fault.
These waves cause the surface of the ground to shake violently, and it is this ground
shaking that is the principal concern of structural engineering to resist earthquakes.
Earthquakes have many effects in addition to ground shaking. For various reasons, the
other effects generally are not major considerations in the design of buildings and similar
structures. For example, seismic sea waves or tsunamis can cause very forceful flood
waves in coastal regions, and seiches (longperiod sloshing) in lakes and inland seas can
have similar effects along shorelines. These are outside the scope of the Provisions. This
is not to say, however, that they should not be considered during site exploration and
analysis. Designing structures to resist such hydrodynamic forces is a very specialized
topic, and it is common to avoid constructing buildings and similar structures where such
phenomena are likely to occur. Longperiod sloshing of the liquid contents of tanks is
addressed by the Provisions.
Abrupt ground displacements occur where a fault intersects the ground surface. (This
commonly occurs in California earthquakes but apparently did not occur in the historic
Charleston, South Carolina, earthquake or the very large New Madrid, Missouri,
earthquakes of the nineteenth century.) Mass soil failures such as landslides, liquefaction
and gross settlement are the result of ground shaking on susceptible soil formations.
Once again, design for such events is specialized, and it is common to locate structures so
that mass soil failures and fault breakage are of no major consequence to their
performance. Modification of soil properties to protect against liquefaction is one
important exception; large portions of a few metropolitan areas with the potential for
significant ground shaking are susceptible to liquefaction. Lifelines that cross faults
require special design beyond the scope of the Provisions. The structural loads specified
in the Provisions are based solely on ground shaking; they do not provide for ground
failure. Resource Paper 12 ( Evaluation of Geologic Hazards and Determination of
Seismic Lateral Earth Pressures ) in Part 3 of the Provisions includes a description of
current procedures for prediction of seismicinduced slope instability, liquefaction and
surface fault rupture.
Nearly all large earthquakes are tectonic in origin that is, they are associated with
movements of and strains in large segments of the earth s crust, called plates, and
virtually all such earthquakes occur at or near the boundaries of these plates. This is the
case with earthquakes in the far western portion of the United States where two very
large plates, the North American continent and the Pacific basin, come together. In the
central and eastern United States, however, earthquakes are not associated with such a
plate boundary, and their causes are not as completely understood. This factor, combined
with the smaller amount of data about central and eastern earthquakes (because of their
infrequency), means that the uncertainty associated with earthquake loadings is higher in
the central and eastern portions of the nation than in the West. Even in the West, the
uncertainty (when considered as a fraction of the predicted level) about the hazard level is
probably greater in areas where the mapped hazard is low than in areas where the mapped
hazard is high.
The amplitude of earthquake ground shaking diminishes with distance from the source,
and the rate of attenuation is less for lower frequencies of motion than for higher
frequencies. This effect is captured, to an extent, by the fact that the Provisions use three
parameters to define the hazard of seismic ground shaking for structures. Two are based
on statistical analysis of the database of seismological information: the SS values are
pertinent for higher frequency motion, and the S1 values are pertinent for other middle
frequencies. The third value, TL, defines an important transition point for long period
(low frequency) behavior; it is not based upon as robust an analysis as the other two
parameters.
Two basic data sources are used in establishing the likelihood of earthquake ground
shaking, or seismicity, at a given location. The first is the historical record of earthquake
effects and the second is the geological record of earthquake effects. Given the
infrequency of major earthquakes, there is no place in the United States where the
historical record is long enough to be used as a reliable basis for earthquake prediction
certainly not as reliable as with other phenomena such as wind and snow. Even on the
eastern seaboard, the historical record is too short to justify sole reliance on the historical
record. Thus, the geological record is essential. Such data require very careful
interpretation, but they are used widely to improve knowledge of seismicity. Geological
data have been developed for many locations as part of the nuclear power plant design
process. On the whole, there is more geological data available for the far western United
States than for other regions of the country. Both sets of data have been taken into
account in the Provisions seismic ground shaking maps.
The Commentary provides a more thorough discussion of the development of the maps,
their probabilistic basis, the necessarily crude lumping of parameters and other related
issues. Prior to its 1997 edition, the basis of the Provisions was to provide life safety at
the design earthquake motion, which was defined as having a 10 percent probability of
being exceeded in a 50year reference period. As of the 1997 edition, the basis became to
avoid structural collapse at the maximum considered earthquake (MCE) ground
motion, which is defined as having a 2 percent probability of being exceeded in a 50
year reference period. In the 2009 edition of the Provisions the design basis has been
refined to target a 1% probability of structural collapse for ordinary buildings in a 50 year
period. The MCE ground motion has been adjusted to deliver this level of risk combined
with a 10% probability of collapse should the MCE ground motion occur. This new
approach incorporates a fuller consideration of the nature of the seismic hazard at a
location than was possible with the earlier definitions of ground shaking hazard, which
were tied to a single level of probability of ground shaking occurrence.
2.2 STRUCTURAL RESPONSE TO GROUND SHAKING
The first important difference between structural response to an earthquake and response
to most other loadings is that the earthquake response is dynamic, not static. For most
structures, even the response to wind is essentially static. Forces within the structure are
due almost entirely to the pressure loading rather than the acceleration of the mass of the
structure. But with earthquake ground shaking, the aboveground portion of a structure is
not subjected to any applied force. The stresses and strains within the superstructure are
created entirely by its dynamic response to the movement of its base, the ground. Even
though the most used design procedure resorts to the use of a concept called the
equivalent static force for actual calculations, some knowledge of the theory of vibrations
of structures is essential.
2.2.1 Response Spectra
Figure 2.21 shows accelerograms, records of the acceleration at one point along one
axis, for several representative earthquakes. Note the erratic nature of the ground shaking
and the different characteristics of the different accelerograms. Precise analysis of the
elastic response of an ideal structure to such a pattern of ground motion is possible;
however, it is not commonly done for ordinary structures. The increasing power and
declining cost of computational aids are making such analyses more common but, at this
time, only a small minority of structures designed across the country, are analyzed for
specific response to a specific ground motion.
Figure 2.21 Earthquake Ground Acceleration in Epicentral Regions (all accelerograms are plotted to the same
scale for time and acceleration the vertical axis is % gravity). Great earthquakes extend for much longer
periods of time.
Figure 2.22 shows further detail developed from an accelerogram. Part (a) shows the
ground acceleration along with the ground velocity and ground displacement derived
from it. Part (b) shows the acceleration, velocity and displacement for the same event at
the roof of the building located where the ground motion was recorded. Note that the
peak values are larger in the diagrams of Figure 2.22(b) (the vertical scales are
essentially the same). This increase in response of the structure at the roof level over the
motion of the ground itself is known as dynamic amplification. It depends very much on
the vibrational characteristics of the structure and the characteristic frequencies of the
ground shaking at the site.
Figure 2.22 Holiday Inn Ground and Building Roof Motion During the M6.4 1971 San Fernando Earthquake:
(a) northsouth ground acceleration, velocity and displacement and (b) northsouth roof acceleration, velocity
and displacement (Housner and Jennings, 1982). The Holiday Inn, a 7story, reinforced concrete frame
building, was approximately 5 miles from the closest portion of the causative fault. The recorded building
motions enabled an analysis to be made of the stresses and strains in the structure during the earthquake.
In design, the response of a specific structure to an earthquake is ordinarily estimated
from a design response spectrum such as is specified in the Provisions. The first step in
creating a design response spectrum is to determine the maximum response of a given
structure to a specific ground motion (see Figure 2.22). The underlying theory is based
entirely on the response of a singledegreeoffreedom oscillator such as a simple one
story frame with the mass concentrated at the roof. The vibrational characteristics of
such a simple oscillator may be reduced to two: the natural period and the amount of
damping. By recalculating the record of response versus time to a specific ground
motion for a wide range of natural periods and for each of a set of common amounts of
damping, the family of response spectra for one ground motion may be determined. It is
simply the plot of the maximum value of response for each combination of period and
damping.
Figure 2.23 shows such a result for the ground motion of Figure 2.22(a) and illustrates
that the erratic nature of ground shaking leads to a response that is very erratic in that a
slight change in the natural period of vibration brings about a very large change in
response. The figure also illustrates the significance of damping. Different earthquake
ground motions lead to response spectra with peaks and valleys at different points with
respect to the natural period. Thus, computing response spectra for several different
ground motions and then averaging them, based on some normalization for different
amplitudes of shaking, will lead to a smoother set of spectra. Such smoothed spectra are
an important step in developing a design spectrum.
Figure 2.23 Response spectrum of northsouth ground acceleration (0%, 2%, 5%, 10%, 20% of critical
damping) recorded at the Holiday Inn, approximately 5 miles from the causative fault in the 1971 San Fernando
earthquake.
Figure 2.24 is an example of an averaged spectrum. Note that acceleration, velocity, or
displacement may be obtained from Figure 2.23 or 1.24 for a structure with known
period and damping.
Figure 2.24 Averaged Spectrum(In this case, the statistics are for seven ground motions representative of the
deaggregated hazard at a particular site.)
Prior to the 1997 edition of the Provisions, the maps that characterized the ground
shaking hazard were plotted in terms of peak ground acceleration (at period, T, = 0), and
design response spectra were created using expressions that amplified (or deamplified)
the ground acceleration as a function of period and damping. With the introduction of the
new maps in the 1997 edition, this procedure changed. Now the maps present spectral
response accelerations at two periods of vibration, 0.2 and 1.0 second, and the design
response spectrum is computed more directly, as implied by the smooth line in Figure
2.24. This has removed a portion of the uncertainty in predicting response accelerations.
Few structures are so simple as to actually vibrate as a singledegreeoffreedom system.
The principles of dynamic modal analysis, however, allow a reasonable approximation of
the maximum response of a multidegreeoffreedom oscillator, such as a multistory
building, if many specific conditions are met. The procedure involves dividing the total
response into a number of natural modes, modeling each mode as an equivalent single
degreeoffreedom oscillator, determining the maximum response for each mode from a
singledegreeoffreedom response spectrum and then estimating the maximum total
response by statistically summing the responses of the individual modes. The Provisions
does not require consideration of all possible modes of vibration for most buildings
because the contribution of the higher modes (lower periods) to the total response is
relatively minor.
The soil at a site has a significant effect on the characteristics of the ground motion and,
therefore, on the structure s response. Especially at low amplitudes of motion and at
longer periods of vibration, soft soils amplify the motion at the surface with respect to
bedrock motions. This amplification is diminished somewhat, especially at shorter
periods as the amplitude of basic ground motion increases, due to yielding in the soil.
The Provisions accounts for this effect by providing amplifiers that are to be applied to
the 0.2 and 1.0 second spectral accelerations for various classes of soils. (The ground
motion maps in the Provisions are drawn for sites on rock.) Thus, very different design
response spectra are specified depending on the type of soil(s) beneath the structure. The
Commentary (Part 2) contains a thorough explanation of this feature.
2.2.2 Inelastic Response
The preceding discussion assumes elastic behavior of the structure. The principal
extension beyond ordinary behavior referenced at the beginning of this chapter is that
structures are permitted to strain beyond the elastic limit in responding to earthquake
ground shaking. This is dramatically different from the case of design for other types of
loads in which stresses and therefore strains, are not permitted to approach the elastic
limit. The reason is economic. Figure 2.23 shows a peak acceleration response of about
1.0 g (the acceleration due to gravity) for a structure with moderately low damping for
only a moderately large earthquake! Even structures that resist lateral forces well will
have a static lateral strength of only 20 to 40 percent of gravity.
The dynamic nature of earthquake ground shaking means that a large portion of the
shaking energy can be dissipated by inelastic deformations if the structure is ductile and
some damage to the structure is accepted. Figure 2.25 will be used to illustrate the
significant difference between wind and seismic effects. Figure 2.25(1) would represent
a cantilever beam if the load W were small and a column if W were large. Wind
pressures create a force on the structure, which in turn produces a displacement. The
force is the independent variable and the displacement is the dependent result.
Earthquake ground motion creates displacement between the base and the mass, which in
turn produces an internal force. The displacement is the independent variable, and the
force is the dependent result. Two graphs are plotted with the independent variables on
the horizontal axis and the dependent response on the vertical axis. Thus, part (b) of the
figure is characteristic of the response to forces such as wind pressure (or gravity weight),
while part (c) is characteristic of induced displacements such as earthquake ground
shaking (or foundation settlement).
Note that the ultimate resistance (Hu) in a forcecontrolled system is marginally larger
than the yield resistance (Hy), while the ultimate displacement ( u) in a displacement
controlled system is much larger than the yield displacement ( y). The point being made
with the figures is that ductile structures have the ability to resist displacements much
larger than those that first cause yield.
The degree to which a member or structure may deform beyond the elastic limit is
referred to as ductility. Different materials and different arrangements of structural
members lead to different ductilities. Response spectra may be calculated for oscillators
with different levels of ductility. At the risk of gross oversimplification, the following
conclusions may be drawn:
1. For structures with very long natural periods, the acceleration response is reduced
by a factor equivalent to the ductility ratio (the ratio of maximum usable
displacement to effective yield displacement note that this is displacement and
not strain).
2. For structures with very short natural periods, the acceleration response of the
ductile structure is essentially the same as that of the elastic structure, but the
displacement is increased.
3. For intermediate periods (which applies to nearly all buildings), the acceleration
response is reduced, but the displacement response is generally about the same for
the ductile structure as for the elastic structure strong enough to respond without
yielding.
Figure 2.25 Force Controlled Resistance Versus Displacement Controlled Resistance
(after Housner and Jennings 1982). In part (b) the force H is the independent variable. As H is increased, the
displacement increases until the yield point stress is reached. If H is given an additional increment (about 15
percent) a plastic hinge forms, giving large displacements. For this kind of system, the force producing the yield
point stress is close to the force producing collapse. The ductility does not produce a large increase in load
capacity, although in highly redundant structures the increase is more than illustrated for this very simple
structure. In part (c) the displacement is the independent variable. As the displacement is increased, the base
moment increases until the yield point is reached. As the displacement increases still more, the resistance (H)
increases only a small amount. For a highly ductile element, the displacement can be increased 10 to 20 times
the yield point displacement before the system collapses under the weight W. (As W increases, this ductility is
decreased dramatically.) During an earthquake, the oscillator is excited into vibrations by the ground motion
and it behaves essentially as a displacementcontrolled system and can survive displacements much beyond the
yield point. This explains why ductile structures can survive ground shaking that produces displacements much
greater than yield point displacement.
Inelastic response is quite complex. Earthquake ground motions involve a significant
number of reversals and repetitions of the strains. Therefore, observation of the inelastic
properties of a material, member, or system under a monotonically increasing load until
failure can be very misleading. Cycling the deformation can cause degradation of
strength, stiffness, or both. Systems that have a proven capacity to maintain a stable
resistance to a large number of cycles of inelastic deformation are allowed to exercise a
greater portion of their ultimate ductility in designing for earthquake resistance. This
property is often referred to as toughness, but this is not the same as the classic definition
used in mechanics of materials.
Most structures are designed for seismic response using a linear elastic analysis with the
strength of the structure limited by the strength at its critical location. Most structures
possess enough complexity so that the peak strength of a ductile structure is not
accurately captured by such an analysis. Figure 2.26 shows the load versus
displacement relation for a simple frame. Yield must develop at four locations before the
peak resistance is achieved. The margin from the first yield to the peak strength is
referred to as overstrength, and it plays a significant role in resisting strong ground
motion. Note that a few key design standards (for example, American Concrete Institute
(ACI) 318 for the design of concrete structures) do allow for some redistribution of
internal forces from the critical locations based upon ductility; however, the
redistributions allowed therein are minor compared to what occurs in response to strong
ground motion.
Figure 2.26 Initial Yield Load and Failure for a Ductile Portal Frame
(The margin from initial yield to failure (mechanism in this case) is known as overstrength.)
To summarize, the characteristics important in determining a building s seismic response
are natural period, damping, ductility, stability of resistance under repeated reversals of
inelastic deformation and overstrength. The natural frequency is dependent on the mass
and stiffness of the building. Using the Provisions the designer calculates, or at least
approximates, the natural period of vibration (the inverse of natural frequency).
Damping, ductility, toughness and overstrength depend primarily on the type of building
system, but not the building s size or shape. Three coefficients R, Cd and ê0 are
provided to encompass damping, ductility, stability of resistance and overstrength. R is
intended to be a conservatively low estimate of the reduction of acceleration response in a
ductile system from that for an elastic oscillator with a certain level of damping. It is
used to compute a required strength. Computations of displacement based upon ground
motion reduced by the factor R will underestimate the actual displacements. Cd is
intended to be a reasonable mean for the amplification necessary to convert the elastic
displacement response computed for the reduced ground motion to actual displacements.
ê0 is intended to deliver a reasonably high estimate of the peak force that would develop
in the structure. Sets of R, Cd and ê0 are specified in the Provisions for the most
common structural materials and systems.
2.2.3 Building Materials
The following brief comments about building materials and systems are included as
general guidelines only, not for specific application.
2.2.3.1 Wood
Timber structures nearly always resist earthquakes very well, even though wood is a
brittle material as far as tension and flexure are concerned. It has some ductility in
compression (generally monotonic), and its strength increases significantly for brief
loadings, such as earthquake. Conventional timber structures (plywood, oriented strand
board, or board sheathing on wood framing) possess much more ductility than the basic
material primarily because the nails, and other steel connection devices yield, and the
wood compresses against the connector. These structures also possess a much higher
degree of damping than the damping that is assumed in developing the basic design
spectrum. Much of this damping is caused by slip at the connections. The increased
strength, connection ductility, and high damping combine to give timber structures a
large reduction from elastic response to design level. This large reduction should not be
used if the strength of the structure is actually controlled by bending or tension of the
gross timber cross sections. The large reduction in acceleration combined with the light
weight timber structures make them very efficient with regard to earthquake ground
shaking when they are properly connected. This is confirmed by their generally good
performance in earthquakes. Capacities and design and detailing rules for wood elements
of seismic forceresisting systems are now found in the Special Design Provisions for
Wind and Seismic supplement to the National Design Specification for Wood
Construction.
2.2.3.2 Steel
Steel is the most ductile of the common building materials. The moderatetolarge
reduction from elastic response to design response allowed for steel structures is
primarily a reflection of this ductility and the stability of the resistance of steel. Members
subject to buckling (such as bracing) and connections subject to brittle fracture (such as
partial penetration welds under tension) are much less ductile and are addressed in the
Provisions in various ways. Defects, such as stress concentrations and flaws in welds,
also affect earthquake resistance as demonstrated in the Northridge earthquake. The
basic and applied research program that grew out of that experience has greatly increased
knowledge of how to avoid low ductility details in steel construction. Capacities and
design and detailing rules for seismic design of hotrolled structural steel are found in the
Seismic Provisions for Structural Steel Buildings (American Institute of Steel
Construction (AISC) Standard 341) and similar provisions for coldformed steel are
found in the Lateral Design supplement to the North American Specification for the
Design of ColdFormed Steel Structures published by AISI (American Iron and Steel
Institute).
2.2.3.3 Reinforced Concrete
Reinforced concrete achieves ductility through careful limits on steel in tension and
concrete in compression. Reinforced concrete beams with common proportions can
possess ductility under monotonic loading even greater than common steel beams; in
which local buckling is usually a limiting factor. Providing stability of the resistance to
reversed inelastic strains, however, requires special detailing. Thus, there is a wide range
of reduction factors from elastic response to design response depending on the detailing
for stable and assured resistance. The Commentary and the commentary with the ACI
318 standard Building Code Requirements for Structural Concrete explain how to design
to control premature shear failures in members and joints, buckling of compression bars,
concrete compression failures (through confinement with transverse reinforcement), the
sequence of plastification and other factors, which can lead to large reductions from the
elastic response.
2.2.3.4 Masonry
Masonry is a more complex material than those mentioned above and less is known about
its inelastic response characteristics. For certain types of members (such as pure
cantilever shear walls), reinforced masonry behaves in a fashion similar to reinforced
concrete. The nature of masonry construction, however, makes it difficult, if not
impossible, to take some of the steps (e.g., confinement of compression members) used
with reinforced concrete to increase ductility, and stability. Further, the discrete
differences between mortar, grout and the masonry unit create additional failure
phenomena. Thus, the response reduction factors for design of reinforced masonry are
not quite as large as those for reinforced concrete. Unreinforced masonry possesses little
ductility or stability, except for rocking of masonry piers on a firm base and very little
reduction from the elastic response is permitted. Capacities and design and detailing
rules for seismic design of masonry elements are contained within The Masonry Society
(TMS) 402 standard Building Code Requirements for Masonry Structures.
2.2.3.5 Precast Concrete
Precast concrete obviously can behave quite similarly to reinforced concrete but it also
can behave quite differently. The connections between pieces of precast concrete
commonly are not as strong as the members being connected. Clever arrangements of
connections can create systems in which yielding under earthquake motions occurs away
from the connections, in which case the similarity to reinforced concrete is very real.
Some carefully detailed connections also can mimic the behavior of reinforced concrete.
Many common connection schemes, however, will not do so. Successful performance of
such systems requires that the connections perform in a ductile manner. This requires
some extra effort in design but it can deliver successful performance. As a point of
reference, the most common wood seismic resisting systems perform well yet have
connections (nails) that are significantly weaker than the connected elements (structural
wood panels). The Provisions includes guidance, some only for trial use and comment
(Part 3), for seismic design of precast structures. ACI 318 also includes provisions for
precast concrete elements resisting seismic forces, and there are also supplemental ACI
standards for specialized seismic forceresisting systems of precast concrete.
2.2.3.6 Composite Steel and Concrete
Reinforced concrete is a composite material. In the context of the Provisions, composite
is a term reserved for structures with elements consisting of structural steel and
reinforced concrete acting in a composite manner. These structures generally are an
attempt to combine the most beneficial aspects of each material. Capacities and design
and detailing rules are found in the Seismic Provisions for Structural Steel Buildings
(AISC Standard 341).
2.2.4 Building Systems
Three basic lateralloadresisting elements walls, braced frames and unbraced frames
(moment resisting frames) are used to build a classification of structural types in the
Provisions. Unbraced frames generally are allowed greater reductions from elastic
response than walls and braced frames. In part, this is because frames are more
redundant, having several different locations with approximately the same stress levels
and common beamcolumn joints frequently exhibit an ability to maintain a stable
response through many cycles of reversed inelastic deformations. Systems using
connection details that have not exhibited good ductility and toughness, such as
unconfined concrete and the welded steel joint used before the Northridge earthquake, are
penalized: the R factors permit less reduction from elastic response.
Connection details often make development of ductility difficult in braced frames, and
buckling of compression members also limits their inelastic response. The actual failure
of steel bracing often occurs because local buckling associated with overall member
buckling frequently leads to locally high strains that then lead to brittle fracture when the
member subsequently approaches yield in tension. Eccentrically braced steel frames and
new proportioning and detailing rules for concentrically braced frames have been
developed to overcome these shortcomings. But the newer and potentially more popular
bracing system is the bucklingrestrained braced frame. This new system has the
advantages of a special steel concentrically braced frame, but with performance that is
superior as brace buckling is controlled to preserve ductility. Design provisions appear in
the Seismic Provisions for Structural Steel Buildings (AISC Standard 341).
Shear walls that do not bear gravity load are allowed a greater reduction than walls that
are load bearing. Redundancy is one reason; another is that axial compression generally
reduces the flexural ductility of concrete and masonry elements (although small amounts
of axial compression usually improve the performance of materials weak in tension, such
as masonry and concrete). The 2010 earthquake in Chile is expected to lead to
improvements in understanding and design of reinforced concrete shear wall systems
because of the large number of significant concrete shear wall buildings subjected to
strong shaking in that earthquake. Systems that combine different types of elements are
generally allowed greater reductions from elastic response because of redundancy.
Redundancy is frequently cited as a desirable attribute for seismic resistance. A
quantitative measure of redundancy is included in the Provisions in an attempt to prevent
use of large reductions from elastic response in structures that actually possess very little
redundancy. Only two values of the redundancy factor, , are defined: 1.0 and 1.3. The
penalty factor of 1.3 is placed upon systems that do not possess some elementary
measures of redundancy based on explicit consideration of the consequence of failure of
a single element of the seismic forceresisting system. A simple, deemedtocomply
exception is provided for certain structures.
2.2.5 Supplementary Elements Added to Improve Structural Performance
The Standard includes provisions for the design of two systems to significantly alter the
response of the structure to ground shaking. Both have specialized rules for response
analysis and design detailing.
Seismic isolation involves placement of specialized bearings with low lateral stiffness
and large lateral displacement capacity between the foundation and the superstructure. It
is used to substantially increase the natural period of vibration and thereby decrease the
acceleration response of the structures. (Recall the shape of the response spectrum in
Figure 2.24; the acceleration response beyond a threshold period is roughly proportional
to the inverse of the period). Seismic isolation is becoming increasingly common for
structures in which superior performance is necessary, such as major hospitals and
emergency response centers. Such structures are frequently designed with a stiff
superstructure to control story drift, and isolation makes it feasible to design such
structures for lower total lateral force. The design of such systems requires a
conservative estimate of the likely deformation of the isolator. The early provisions for
that factor were a precursor of the changes in ground motion mapping implemented in the
1997 Provisions.
Added damping involves placement of specialized energy dissipation devices within
stories of the structure. The devices can be similar to a large shock absorber, but other
technologies are also available. Added damping is used to reduce the structural response,
and the effectiveness of increased damping can be seen in Figure 2.23. It is possible to
reach effective damping levels of 20 to 30 percent of critical damping, which can reduce
response by factors of 2 or 3. The damping does not have to be added in all stories; in
fact, it is common to add damping at the isolator level of seismically isolated buildings.
Isolation and damping elements require extra procedures for analysis of seismic response.
Both also require considerations beyond common building construction to assure quality
and durability.
2.3 ENGINEERING PHILOSOPHY
The Commentary, under Intent, states:
The primary intent of the NEHRP Recommended Seismic Provisions
for normal buildings and structures is to prevent serious injury and life
loss caused by damage from earthquake ground shaking. Most
earthquake injuries and deaths are caused by structural collapse. Thus,
the main thrust of the Provisions is to prevent collapse for very rare and
intense ground motion, termed the maximum considered earthquake
(MCE) motion Falling exterior walls and cladding, and falling
ceilings, light fixtures, pipes, equipment and other nonstructural
components also cause deaths and injuries.
The Provisions states:
The degree to which these goals can be achieved depends on a number
of factors including structural framing type, building configuration,
materials, asbuilt details and overall quality of design. In addition,
large uncertainties as to the intensity and duration of shaking and the
possibility of unfavorable response of a small subset of buildings or
other structures may prevent full realization of the intent.
At this point it is worth recalling the criteria mentioned earlier in describing the risk
targeted ground motions used for design. The probability of structural collapse due to
ground shaking is not zero. One percent in 50 years is actually a higher failure rate than
is currently considered acceptable for buildings subject to other natural loads, such as
wind and snow. The reason is as stated in the quote at the beginning of this chapter
all the wealth of the world would prove insufficient Damage is to be expected
when an earthquake equivalent to the design earthquake occurs. (The design
earthquake is currently taken as twothirds of the MCE ground motion). Some collapse
is to be expected when and where ground motion equivalent to the MCE ground motion
occurs.
The basic structural criteria are strength, stability and distortion. The yieldlevel strength
provided must be at least that required by the design spectrum (which is reduced from the
elastic spectrum as described previously). Structural elements that cannot be expected to
perform in a ductile manner are to have greater strength, which is achieved by applying
the ê0 amplifier to the design spectral response. The stability criterion is imposed by
amplifying the effects of lateral forces for the destabilizing effect of lateral translation of
the gravity weight (the Pdelta effect). The distortion criterion is a limit on story drift and
is calculated by amplifying the linear response to the (reduced) design spectrum by the
factor Cd to account for inelastic behavior.
Yieldlevel strengths for steel and concrete structures are easily obtained from common
design standards. The most common design standards for timber and masonry are based
on allowable stress concepts that are not consistent with the basis of the reduced design
spectrum. Although strengthbased standards for both materials have been introduced in
recent years, the engineering profession has not yet embraced these new methods. In the
past, the Provisions stipulated adjustments to common reference standards for timber and
masonry to arrive at a strength level equivalent to yield, and compatible with the basis of
the design spectrum. Most of these adjustments were simple factors to be applied to
conventional allowable stresses. With the deletion of these methods from the Provisions,
other methods have been introduced into model building codes, and the ASCE standard
Minimum Design Loads for Buildings and Other Structures to factor downward the
seismic load effects based on the Provisions for use with allowable stress design
methods.
The Provisions recognizes that the risk presented by a particular building is a
combination of the seismic hazard at the site and the consequence of failure, due to any
cause, of the building. Thus, a classification system is established based on the use and
size of the building. This classification is called the Occupancy Category (Risk Category
in the Standard). A combined classification called the Seismic Design Category (SDC)
incorporates both the seismic hazard and the Occupancy Category. The SDC is used
throughout the Provisions for decisions regarding the application of various specific
requirements. The flow charts in Chapter 2 illustrate how these classifications are used to
control application of various portions of the Provisions.
2.4 STRUCTURAL ANALYSIS
The Provisions sets forth several procedures for determining the force effect of ground
shaking. Analytical procedures are classified by two facets: linear versus nonlinear and
dynamic versus equivalent static. The two most fully constrained and frequently used are
both linear methods: an equivalent static force procedure and a dynamic modal response
spectrum analysis procedure. A third linear method, a full history of dynamic response
(previously referred to as a timehistory analysis, now referred to as a responsehistory
analysis), and a nonlinear method are also permitted, subject to certain limitations. These
methods use real or synthetic ground motions as input but require them to be scaled to the
basic response spectrum at the site for the range of periods of interest for the structure in
question. Nonlinear analyses are very sensitive to assumptions about structural behavior
made in the analysis and to the ground motions used as input, and a peer review is
required. A nonlinear static method, also known as a pushover analysis, is described in
Part 3 of the Provisions, but it is not included in the Standard. The Provisions also
reference ASCE 41, Seismic Rehabilitation of Existing Buildings, for the pushover
method. The method is instructive for understanding the development of mechanisms but
there is professional disagreement over its utility for validating a structural design.
The two most common linear methods make use of the same design spectrum. The
reduction from the elastic spectrum to design spectrum is accomplished by dividing the
elastic spectrum by the coefficient R, which ranges from 11/4 to 8. Because the design
computations are carried out with a design spectrum that is twothirds the MCE spectrum
that means the full reduction from elastic response ranges from 1.9 to 12. The specified
elastic spectrum is based on a damping level at 5 percent of critical damping, and a part
of the R factor accomplishes adjustments in the damping level. Ductility and
overstrength make up the larger part of the reduction. The Provisions define the total
effect of earthquake actions as a combination of the response to horizontal motions (or
forces for the equivalent static force method) with response to vertical ground
acceleration. The response to vertical ground motion is roughly estimated as a factor
(positive or negative) on the dead load force effect. The resulting internal forces are
combined with the effects of gravity loads and then compared to the full strength of the
members, reduced by a resistance factor, but not by a factor of safety.
With the equivalent static force procedure, the level of the design spectrum is set by
determining the appropriate values of basic seismic acceleration, the appropriate soil
profile type and the value for R. The particular acceleration for the building is
determined from this spectrum by selecting a value for the natural period of vibration.
Equations that require only the height and type of structural system are given to
approximate the natural period for various building types. (The area and length of shear
walls come into play with an optional set of equations.) Calculation of a period based on
an analytical model of the structure is encouraged, but limits are placed on the results of
such calculations. These limits prevent the use of a very flexible model in order to obtain
a large period and correspondingly low acceleration. Once the overall response
acceleration is found, the base shear is obtained by multiplying it by the total effective
mass of the building, which is generally the total permanent load.
Once the total lateral force is determined, the equivalent static force procedure specifies
how this force is to be distributed along the height of the building. This distribution is
based on the results of dynamic studies of relatively uniform buildings and is intended to
give an envelope of shear force at each level that is consistent with these studies. This set
of forces will produce, particularly in tall buildings, an envelope of gross overturning
moment that is larger than many dynamic studies indicate is necessary. Dynamic
analysis is encouraged, and the modal procedure is required for structures with large
periods (essentially this means tall structures) in the higher seismic design categories.
With one exception, the remainder of the equivalent static force analysis is basically a
standard structural analysis. That exception accounts for uncertainties in the location of
the center of mass, uncertainties in the strength and stiffness of the structural elements
and rotational components in the basic ground shaking. This concept is referred to as
horizontal torsion. The Provisions requires that the center of force be displaced from the
calculated center of mass by an arbitrary amount in either direction (this torsion is
referred to as accidental torsion). The twist produced by real and accidental torsion is
then compared to a threshold and if the threshold is exceeded, the accidental torsion must
be amplified.
In many respects, the modal analysis procedure is very similar to the equivalent static
force procedure. The primary difference is that the natural period and corresponding
deflected shape must be known for several of the natural modes of vibration. These are
calculated from a mathematical model of the structure. The procedure requires inclusion
of enough modes so that the dynamic response of the analytical model captures at least
90 percent of the mass in the structure that can vibrate. The base shear for each mode is
determined from a design spectrum that is essentially the same as that for the static
procedure. The distribution of displacements and accelerations (forces) and the resulting
story shears, overturning moments and story drifts are determined for each mode directly
from the procedure. Total values for subsequent analysis and design are determined by
taking the square root of the sum of the squares for each mode. This summation gives a
statistical estimate of maximum response when the participation of the various modes is
random. If two or more of the modes have very similar periods, more advanced
techniques for summing the values are required; these procedures must account for
coupling in the response of close modes. The sum of the absolute values for each mode
is always conservative.
A lower limit to the base shear determined from the modal analysis procedure is specified
based on the static procedure, and the approximate periods specified in the static
procedure. When this limit is violated, which is common, all results are scaled up in
direct proportion. The consideration of horizontal torsion is the same as for the static
procedure. Because the equivalent static forces applied at each floor, the story shears and
the overturning moments are separately obtained from the summing procedure, the results
are not statically compatible (that is, the moment calculated from the summed floor
forces will not match the moment from the summation of moments). Early recognition of
this will avoid considerable problems in later analysis and checking.
For structures that are very uniform in a vertical sense, the two procedures give very
similar results. The modal analysis method is better for buildings having unequal story
heights, stiffnesses, or masses. The modal procedure is required for such structures in
higher seismic design categories. Both methods are based on purely elastic behavior,
and, thus, neither will give a particularly accurate picture of behavior in an earthquake
approaching the design event. Yielding of one component leads to redistribution of the
forces within the structural system; while this may be very significant, none of the linear
methods can account for it.
Both of the common methods require consideration of the stability of the building as a
whole. The technique is based on elastic amplification of horizontal displacements
created by the action of gravity on the displaced masses. A simple factor is calculated
and the amplification is provided for in designing member strengths when the
amplification exceeds about 10 percent. The technique is referred to as the Pdelta
analysis and is only an approximation of stability at inelastic response levels.
2.5 NONSTRUCTURAL ELEMENTS OF BUILDINGS
Severe ground shaking often results in considerable damage to the nonstructural elements
of buildings. Damage to nonstructural elements can pose a hazard to life in and of itself,
as in the case of heavy partitions or facades, or it can create a hazard if the nonstructural
element ceases to function, as in the case of a fire suppression system. Some buildings,
such as hospitals and fire stations, need to be functional immediately following an
earthquake; therefore, many of their nonstructural elements must remain undamaged.
The Provisions treats damage to and from nonstructural elements in three ways. First,
indirect protection is provided by an overall limit on structural distortion; the limits
specified, however, may not offer enough protection to brittle elements that are rigidly
bound by the structure. More restrictive limits are placed upon those Occupancy
Categories (Risk Categories in the Standard) for which better performance is desired
given the occurrence of strong ground shaking. Second, many components must be
anchored for an equivalent static force. Third, the explicit design of some elements (the
elements themselves, not just their anchorage) to accommodate specific structural
deformations or seismic forces is required.
The dynamic response of the structure provides the dynamic input to the nonstructural
component. Some components are rigid with respect to the structure (light weights, and
small dimensions often lead to fundamental periods of vibration that are very short).
Application of the response spectrum concept would indicate that the response history of
motion of a building roof to which mechanical equipment is attached looks like a ground
motion to the equipment. The response of the component is often amplified above the
response of the supporting structure. Response spectra developed from the history of
motion of a point on a structure undergoing ground shaking are called floor spectra, and
are useful in understanding the demands upon nonstructural components.
The Provisions simplifies the concept greatly. The force for which components are
checked depends on:
1. The component mass;
2. An estimate of component acceleration that depends on the structural response
acceleration for short period structures, the relative height of the component
within the structure and a crude approximation of the flexibility of the component
or its anchorage;
3. The available ductility of the component or its anchorage; and
4. The function or importance of the component or the building.
Also included in the Provisions is a quantitative measure for the deformation imposed
upon nonstructural components. The inertial force demands tend to control the seismic
design for isolated or heavy components whereas the imposed deformations are important
for the seismic design for elements that are continuous through multiple levels of a
structure or across expansion joints between adjacent structures, such as cladding or
piping.
2.6 QUALITY ASSURANCE
Since strong ground shaking has tended to reveal hidden flaws or weak links in buildings,
detailed requirements for assuring quality during construction are contained in the
Provisions by reference to the Standard, where they are located in an appendix. The
actively implemented provisions for quality control are actually contained in the model
building codes, such as the International Building Code, and the material design
standards, such as Seismic Provisions for Structural Steel Buildings. Loads experienced
during construction provide a significant test of the likely performance of ordinary
buildings under gravity loads. Tragically, mistakes occasionally will pass this test only to
cause failure later, but it is fairly rare. No comparable proof test exists for horizontal
loads, and experience has shown that flaws in construction show up in a disappointingly
large number of buildings as distress and failure due to earthquakes. This is coupled with
the seismic design approach based on excursions into inelastic straining, which is not the
case for response to other loads.
The quality assurance provisions require a systematic approach with an emphasis on
documentation and communication. The designer who conceives the systems to resist the
effects of earthquake forces must identify the elements that are critical for successful
performance as well as specify the testing and inspection necessary to confirm that those
elements are actually built to perform as intended. Minimum levels of testing and
inspection are specified in the Provisions for various types of systems and components.
The Provisions also requires that the contractor and building official be aware of the
requirements specified by the designer. Furthermore, those individuals who carry out the
necessary inspection and testing must be technically qualified, and must communicate the
results of their work to all concerned parties. In the final analysis, there is no substitute
for a sound design, soundly executed.
Much of the literature on dynamic response is written in terms of frequency rather than period. The cyclic
frequency (cycles per second, or Hz) is the inverse of period. Mathematically it is often convenient to use
the angular frequency expressed as radians per second rather than Hz. The conventional symbols used in
earthquake engineering for these quantities are T for period (seconds per cycle), f for cyclic frequency (Hz)
and for angular frequency (radians per second). The word frequency is often used with no modifier; be
careful with the units.
3
Earthquake Ground Motion
Nicolas Luco , Ph.D., P.E., Michael Valley , S.E.,
C.B. Crouse , P.E., Ph.D.
Contents
3.1 BASIS OFEARTHQUAKE GROUND MOTION MAPS 2
3.1.1 ASCE 705 Seismic Maps 2
3.1.2 MCER Ground Motions in the Provisions and in ASCE 710 3
3.1.3 PGA Maps in the Provisions and in ASCE 710 7
3.1.4 Basis of Vertical Ground Motions in the Provisions and in ASCE 710 7
3.1.5 Summary 7
3.1.6 References 8
3.2 DETERMINATION OF GROUND MOTION VALUES AND SPECTRA 9
3.2.1 ASCE 705 Ground Motion Values 9
3.2.2 2009 Provisions Ground Motion Values 10
3.2.3 ASCE 710 Ground Motion Values 11
3.2.4 Horizontal Response Spectra 12
3.2.5 Vertical Response Spectra 13
3.2.6 Peak Ground Accelerations 14
3.3 SELECTION AND SCALING OF GROUND MOTION RECORDS 14
3.3.1 Approach to Ground Motion Selection and Scaling 15
3.3.2 TwoComponent Records for Three Dimensional Analysis 24
3.3.3 OneComponent Records for TwoDimensional Analysis 27
3.3.4 References 28
Most of the effort in seismic design of buildings and other structures is focused on structural design. This
chapter addresses another key aspect of the design process characterization of earthquake ground
motion. Section 3.1 describes the basis of the earthquake ground motion maps in the Provisions and in
ASCE 7. Section 3.2 has examples for the determination of ground motion parameters and spectra for use
in design. Section 3.3 discusses and provides an example for the selection and scaling of ground motion
records for use in response history analysis.
3.1 BASIS OFEARTHQUAKE GROUND MOTION MAPS
This section explains the basis of the new RiskTargeted Maximum Considered Earthquake (MCER)
ground motions specified in the 2009 Provisions and mapped in ASCE 710. In doing so, it also explains
the basis for the uniformhazard ground motion (SSUH and S1UH) maps, risk coefficient (CRS and CR1) maps
and deterministic ground motion (SSD and S1D) maps in the 2009 Provisions. These three sets of maps are
combined to form the Site Class B MCER ground motion (SS and S1) maps in ASCE 710. The use of SS
and S1 ground motions in the 2009 Provisions and ASCE 710 to derive a design response spectrum
remains the same as it is in ASCE 705.
This section also explains the basis for the new Peak Ground Acceleration (PGA) maps in the 2009
Provisions and ASCE 710 and the new equations for vertical ground motions. The basis for the long
period transition (TL) maps in the 2009 Provisions and ASCE 710, which are identical to those in ASCE
705, is also reviewed. In fact, we start with a review of these maps and the Maximum Considered
Earthquake (MCE) ground motion maps in ASCE 705.
3.1.1 ASCE 705 Seismic Maps
The bases for the seismic ground motion (MCE) and longperiod transition (TL) maps in Chapter 22 of
ASCE 705 were established by, respectively, the Building Seismic Safety Council (BSSC) Seismic
Design Procedures Group, also referred to as Project 97, and Technical Subcommittee 1 (TS1) of the
2003 Provisions Update Committee. They are reviewed briefly in the following two subsections.
3.1.1.1 Maximum Considered Earthquake (MCE) Ground Motion Maps
The MCE ground motion maps in ASCE 705 can be described as applications of its sitespecific ground
motion hazard analysis procedure in Chapter 21 (Section 21.2), using ground motion values computed by
the USGS National Seismic Hazard Mapping Project (in Golden, CO) for a grid of locations and/or
polygons that covers the US. In particular, the 1996 USGS update of the ground motion values was used
for ASCE 798 and ASCE 702; the 2002 USGS update was used for ASCE 705. The sitespecific
procedure in all three editions calculates the MCE ground motion as the lesser of a probabilistic and a
deterministic ground motion. Hence, the USGS computed both types of ground motions, whereas
otherwise it would have only computed probabilistic ground motions. Brief reviews of how the USGS
computed the probabilistic and deterministic ground motions are provided in the next few paragraphs.
For additional information, see Leyendecker et al. (2000).
The USGS computation of the probabilistic ground motions that are part of the basis of the MCE ground
motion maps in ASCE 705 is explained in detail in Frankel et al. (2002). In short, the USGS combines
research on potential sources of earthquakes (e.g., faults and locations of past earthquakes), the potential
magnitudes of earthquakes from these sources and their frequencies of occurrence, and the potential
ground motions generated by these earthquakes. Uncertainty and randomness in each of these
components is accounted for in the computation via contemporary Probabilistic Seismic Hazard Analysis
(PSHA), which was originally conceived by Cornell (1968). The primary output of PSHA computations
are socalled hazard curves, for locations on a grid covering the US in the case of the USGS computation.
Each hazard curve provides mean annual frequencies of exceeding various userspecified ground motions
amplitudes. From these hazard curves, the ground motion amplitudes for a userspecified mean annual
frequency can be interpolated and then mapped. The results are known as uniformhazard ground motion
maps, since the mean annual frequency (or corresponding probability) is uniform geographically.
For ASCE 705 (as well as ASCE 702 and ASCE 798), a mean annual exceedance frequency of 1/2475
per year, corresponding to 2% probability of exceedance in 50 years, was specified by the aforementioned
BSSC Project 97. That project also specified that the ground motion parameters be spectral response
accelerations at vibration periods of 0.2 seconds and 1 second, for 5% of critical damping. For the
average shear wave velocity at small shear strains in the upper 100 feet (30 m) of subsurface below each
location (vS,30), the USGS decided on a reference value of 760 m/s. The BSSC subsequently decided to
regard this reference value, which is at the boundary of Site Classes B and C, as corresponding to Site
Class B. Justifications for the decisions summarized in this paragraph are provided in the Commentary of
FEMA 303, FEMA 369 and FEMA 450.
The USGS computation of the deterministic ground motions for ASCE 705 is detailed in the FEMA 303
Commentary. As defined by Project 97 and subsequently specified in the sitespecific procedure of
ASCE 705 (Section 21.2.2), each deterministic ground motion is calculated as 150% of the median
spectral response acceleration for a characteristic earthquake on a known active fault within the region.
The specific characteristic earthquake is that which generates the largest median spectral response
acceleration at the given location. As for the probabilistic ground motions, the spectral response
accelerations are at vibration periods of 0.2 seconds and 1 second, for 5% of critical damping. The same
reference site class (see preceding paragraph) is used as well. Though not applied to probabilistic ground
motions, lower limits of 1.5g and 0.6g are applied to the deterministic ground motions.
As mentioned at the beginning of this section, the lesser of the probabilistic and deterministic ground
motions described above yields the MCE ground motions mapped in ASCE 705. Thus, the MCE
spectral response accelerations at 0.2 seconds and 1 second are equal to the corresponding probabilistic
ground motions wherever they are less than the lower limits of the deterministic ground motions (1.5g
and 0.6g, respectively). Where the probabilistic ground motions are greater than the lower limits, the
deterministic ground motions sometimes govern, but only if they are less than their probabilistic
counterparts. On the MCE ground motion maps in ASCE 705, the deterministic ground motions govern
mainly near major faults in California (like the San Andreas), in Reno and in parts of the New Madrid
Seismic Zone. The deterministic ground motions that govern are as small as 40% of their probabilistic
counterparts.
3.1.1.2 LongPeriod Transition Period (TL) Maps
The details of the procedure and rationale used in developing the TL maps in ASCE 705; and now in
ASCE 710 and the 2009 Provisions, are found in Crouse et al. (2006). In short, the procedure consisted
of two steps. First, a relationship between TL and earthquake magnitude was established. Second, the
modal magnitude from deaggregation of the USGS 2% in 50year ground motion hazard at a 2second
period (1 second for Hawaii) was mapped. The longperiod transition period (TL) maps that combined
these two steps delimit the transition of the design response spectrum from a constant velocity (1/T) to a
constant displacement (1/T2) shape.
3.1.2 MCER Ground Motions in the Provisions and in ASCE 710
Like the MCE ground motion maps in ASCE 705 reviewed in the preceding section, the new Risk
Targeted Maximum Considered Earthquake (MCER) ground motions in the 2009 Provisions and ASCE 7
10 can be described as applications of the sitespecific ground motion hazard analysis procedure in
Chapter 21 (Section 21.2) of both documents. For the MCER ground motions, however, the USGS values
(for a grid of site and/or polygons covering the US) that are used in the procedure are from its 2008
update. Still, the sitespecific procedure of the Provisions and ASCE 710 calculates the MCER ground
motion as the lesser of a probabilistic and a deterministic ground motion. The definitions of the
probabilistic and deterministic ground motions in ASCE 710, however, are different than in ASCE 705.
The definitions were revised for the 2009 Provisions and ASCE 710 by the BSSC Seismic Design
Procedures Reassessment Group (SDPRG), also referred to as Project 07. Three revisions were made:
1) The probabilistic ground motions are redefined as socalled risktargeted ground motions, in lieu
of the uniformhazard (2% in 50year) ground motions that underlie the ASCE 705 MCE ground
motion maps,
2) the deterministic ground motions are redefined as 84thpercentile ground motions, in lieu of
median ground motions multiplied by 1.5; and
3) the probabilistic and deterministic ground motions are redefined as maximumdirection ground
motions, in lieu of geometric mean ground motions.
In addition to these three BSSC redefinitions of probabilistic and deterministic ground motions, there is a
fourth difference in the ground motion values computed by the USGS for the 2009 Provisions and ASCE
710 versus ASCE 705:
4) The probabilistic and deterministic ground motions were recomputed using updated earthquake
source and ground motion propagation models, e.g., the Unified California Earthquake Rupture
Forecast (UCERF, Version 2; Field et al., 2008) and the Next Generation Attenuation (NGA)
ground motion models .
Each of the above four differences between the basis of the MCE ground motions (in ASCE 705) and
that of the MCER ground motions (in the 2009 Provisions and ASCE 710) is explained in more detail
below. Also explained are the differences in the presentation of MCER ground motions between the 2009
Provisions and ASCE 710; the numerical values of the MCER ground motions in the two documents are
otherwise identical.
3.1.2.1 RiskTargeted Probabilistic Ground Motions
For the MCE ground motion maps in ASCE 705, recall (from Section 3.1.1) that the underlying
probabilistic ground motions are specified to be uniformhazard ground motions that have a 2%
probability of being exceeded in 50 years. It has long been recognized, though, that it really is the
probability of structural failure with resultant casualties that is of concern; and the geographical
distribution of that probability is not necessarily the same as the distribution of the probability of
exceeding some ground motion (p. 296 of ATC 306, 1978). The primary reason that the distributions of
the two probabilities are not the same is that there are geographic differences in the shape of the hazard
curves from which uniformhazard ground motions are read. The Commentary of FEMA 303 (p. 289)
reports that because of these differences, questions were raised concerning whether definition of the
ground motion based on a constant probability for the entire United States would result in similar levels
of seismic safety for all structures .
The changeover to risktargeted probabilistic ground motions for the 2009 Provisions and ASCE 710
takes into account the differences in the shape of hazard curves across the US. Where used in design, the
risktargeted ground motions are expected to result in buildings with a geographically uniform mean
annual frequency of collapse, or uniform risk. The BSSC, via Project 07, decided on a target risk level
corresponding to 1% probability of collapse in 50 years. This target is based on the average of the mean
annual frequencies of collapse across the Western US (WUS) expected to result from design for the
probabilistic ground motions in ASCE 705. Consequently, in the WUS the risktargeted ground motions
in the 2009 Provisions and ASCE 710 are generally within 15% of the corresponding uniformhazard
(2% in 50year) ground motions. In the Central and Eastern US, where the shapes of hazard curves are
known to differ from those in the WUS, the risktargeted ground motions generally are smaller. For
instance, in the New Madrid Seismic Zone and near Charleston, South Carolina ratios of risktargeted to
uniformhazard ground motions are as small as 0.7.
The computation of risktargeted probabilistic ground motions for the MCER ground motions in the 2009
Provisions and ASCE 710 is detailed in Provisions Part 1 Sections 21.2.1.2 and C21.2.1 and in Luco et
al. (2007). While the computation of the risktargeted ground motions is different than that of the
uniformhazard ground motions specified for the MCE ground motions in ASCE 705, both begin with
USGS computations of hazard curves. As explained in Section 3.1.1, the uniformhazard ground motions
simply interpolate the hazard curves for a 2% probability of exceedance in 50 years. In contrast, the risk
targeted ground motions make use of entire hazard curves. In either case, the end results are probabilistic
spectral response accelerations at 0.2 seconds and 1 second, for 5% of critical damping and the reference
site class.
3.1.2.2 84thPercentile Deterministic Ground Motions
For the MCE ground motion maps in ASCE 705, recall (from Section 3.1.1) that the underlying
deterministic ground motions are defined as 150% of median spectral response accelerations. As
explained in the FEMA 303 Commentary (p. 296),
Increasing the median ground motion estimates by 50 percent [was] deemed to provide an
appropriate margin and is similar to some deterministic estimates for a large magnitude
characteristic earthquake using ground motion attenuation functions with one standard
deviation. Estimated standard deviations for some active fault sources have been determined to
be higher than 50 percent, but this increase in the median ground motions was considered
reasonable for defining the maximum considered earthquake ground motions for use in design.
For the MCER ground motions in the 2009 Provisions and ASCE 710, however, the BSSC decided to
define directly the underlying deterministic ground motions as those at the level of one standard
deviation. More specifically, they are defined as 84thpercentile ground motions (since it has been widely
observed that ground motions follow lognormal probability distributions). The remainder of the
definition of the deterministic ground motions remains the same as that used for the MCE ground motions
maps in ASCE 705. For example, the lower limits of 1.5g and 0.6g described in Section 3.1.1 are
retained.
The USGS applied a simplification specified by the BSSC in computing the 84thpercentile deterministic
ground motions for the 2009 Provisions and ASCE 710. The 84thpercentile spectral response
accelerations were approximated as 180% of median values. This approximation corresponds to a
logarithmic ground motion standard deviation of approximately 0.6, as demonstrated in the Provisions
Part 1 Section C21.2.2. The computation of deterministic ground motions is further described in
Provisions Part 2 Section C21.2.2.
3.1.2.3 MaximumDirection Probabilistic and Deterministic Ground Motions
Due to the ground motion attenuation models used by the USGS in computing them , overall the MCE
spectral response accelerations in ASCE 705 represent the geometric mean of two horizontal components
of ground motion. Most users of ASCE 705 are unaware of this fact, particularly since the discussion
notes on the MCE ground motion maps incorrectly state that they represent the random horizontal
component of ground motion. For the 2009 Provisions and ASCE 710, the BSSC decided that it would
be an improvement if the MCER ground motions represented the maximum direction of horizontal
spectral response acceleration. Reasons for this decision are explained in Provisions Part 1 Section
C21.2.
Since the attenuation models used in computing the 2008 update of the USGS ground motions also
represent (overall) geomean spectral response accelerations, for the 2009 Provisions and ASCE 710
the BSSC provided factors to convert approximately to maximumdirection ground motions. Based on
research by Huang et al. (2008) and others, the factors are 1.1 and 1.3 for the spectral response
accelerations at 0.2 seconds and 1.0 second, respectively. The basis for these factors is elaborated upon in
the Provisions Part 1 Section C21.2. They are applied to both the USGS probabilistic hazard curves from
which the risktargeted ground motions (described in Section 3.1.2.1) are derived and the USGS
deterministic ground motions (described in Section 3.1.2.2).
3.1.2.4 Updated Ground Motions from USGS (2008)
For the MCE ground motion maps in ASCE 705, recall (from Section 3.1.1) that the underlying
probabilistic and deterministic ground motions are from the 2002 USGS update. As mentioned above, the
MCER ground motions in the 2009 Provisions and ASCE 710 are instead based on the 2008 update of the
USGS ground motion values. This update is documented in Petersen et al. (2008) and supersedes the
1996 and 2002 USGS ground motions values. It involved interactions with hundreds of scientists and
engineers at regional and topical workshops, including advice from working groups, expert panels, state
geological surveys, other federal agencies and hazard experts from industry and academia. Based in large
part on new published studies, the 2008 update incorporated changes in both earthquake source models
(including magnitudes and occurrence frequencies) and models of ground motion propagation. The
UCERF and NGA models mentioned above are just two examples of such changes. The end results are
updated ground motions that represent the best available science as determined by the USGS from an
extensive informationgathering and review process.
It is important to note that the 2008 USGS hazard curves and uniformhazard maps (posted at
http://earthquake.usgs.gov/hazards/products/conterminous/2008/), like their 2002 counterparts, represent
the geomean ground motions discussed in the preceding subsection. Only the MCER ground motions
and their underlying probabilistic and deterministic ground motions represent the maximum direction of
horizontal spectral response acceleration.
3.1.2.5 Differing Presentation of MCER Ground Motions in the Provisions and in ASCE 710
Though their numerical values are identical, the MCER ground motions specified in the Provisions and in
ASCE 710 are presented differently. As replacements to the MCE ground motion maps in ASCE 705,
ASCE 710 presents (in Chapter 22) contour maps of the MCER ground motions for Site Class B, which
are still denoted SS and S1 for the 0.2 and 1.0second spectral response accelerations, respectively. Like
the MCE ground motions in ASCE 705, the MCER ground motions mapped in ASCE 710 are accessible
electronically via a USGS web application (see http://earthquake.usgs.gov/designmaps/).
In contrast, Provisions Section 11.4 presents equations to calculate MCER ground motions (SS and S1) for
Site Class B using maps (in Chapter 22) of uniformhazard 2% in 50year ground motions (denoted SSUH
and S1UH), socalled risk coefficients (denoted CRS and CR1); and deterministic ground motions (denoted
SSD and S1D, not to be confused with the design ground motions SDS and SD1). The risk coefficient maps
show the ratio of the risktargeted probabilistic ground motions (described in Section 3.1.2.1) to
corresponding 2% in 50year ground motions like those used to derive the MCE ground motion maps in
ASCE 705. The intent of the equations and three sets of maps presented in the Provisions is
transparency in the derivation of the MCER ground motions. The mapped values of the uniformhazard
ground motions, risk coefficients and deterministic ground motions are all accessible electronically via
http://earthquake.usgs.gov/designmaps/.
3.1.3 PGA Maps in the Provisions and in ASCE 710
The basis of the Peak Ground Acceleration (PGA) maps in the Provisions and in ASCE 710 nearly
parallels that of the MCE ground motion maps in ASCE 705 described in Section 3.1.1.1. More
specifically, the mapped PGA values for Site Class B are calculated as the lesser of uniformhazard (2%
in 50year) probabilistic and deterministic PGA values that represent the geometric mean of two
horizontal components of ground motion. Unlike in ASCE 705, though, the deterministic values are
defined as 84thpercentile ground motions rather than 150% of median ground motions. This definition of
deterministic ground motions parallels that which is described above for the MCER ground motions in the
2009 Provisions and ASCE 710. The deterministic PGA values, though, are stipulated to be no lower
than 0.5g, as opposed to 1.5g and 0.6g (respectively) for the MCER 0.2 and 1.0second spectral response
accelerations. All of these details of the basis of the PGA maps are provided in ASCE 710 Section 21.5;
the Provisions do not contain a sitespecific procedure for PGA values.
The USGScomputed PGA values for vS,30 = 760m/s that are mapped, like their MCER ground motion
counterparts in the Provisions and in ASCE 710, are from the 2008 USGS update. Also like their MCER
ground motion counterparts, the 84thpercentile PGA values have been approximated as median values
multiplied by 1.8.
While the values on and format of the PGA maps in the Provisions and in ASCE 710 are identical, the
terminology used to label the maps (and values) is different in the two documents. In the Provisions, they
are referred to as MCE Geometric Mean PGA maps. In ASCE 710, they are labeled Maximum
Considered Earthquake Geometric Mean (MCEG) PGA maps. The MCEG abbreviation is intended to
remind users of the differences between the basis of the PGA maps and the MCER maps also in ASCE 7
10, namely that the PGA values represent the geometric mean of two horizontal components of ground
motion, not the maximum direction; and that the probabilistic PGA values are not risktargeted ground
motions, but rather uniformhazard (2% in 50year) ground motions.
3.1.4 Basis of Vertical Ground Motions in the Provisions and in ASCE 710
Whereas ASCE 705 determines vertical seismic load effects via a single constant fraction of the
horizontal shortperiod spectral response acceleration SDS, the 2009 Provisions and ASCE 710 determine
a vertical design response spectrum, Sav, that is analogous to the horizontal design response spectrum, Sa.
The Sav values are determined via functions (for four different ranges of vertical period of vibration) that
each depend on SDS and a coefficient Cv representing the ratio of vertical to horizontal spectral response
acceleration. This is in contrast to determination of Sa via mapped horizontal spectral response
accelerations. The coefficient Cv, in turn, depends on the amplitude of spectral response acceleration (by
way of SS) and site class. These dependencies, as well as the period dependence of the equations for Sav,
are based on studies by Bozorgnia and Campbell (2004) and others. Those studies observed that the ratio
of vertical to horizontal spectral response acceleration is sensitive to period of vibration, site class,
earthquake magnitude (for relatively soft sites) and distance to the earthquake. The sensitivity to the
latter two characteristics is captured by the dependence of Cv on SS.
The basis of the equations for vertical response spectra in the Provisions and in ASCE 710 is explained
in more detail in the commentary to Chapter 23 of each document. Note that for vertical periods of
vibration greater than 2 seconds, Chapter 23 stipulates that the vertical spectral response accelerations be
determined via a sitespecific procedure. A sitespecific study also may be performed for periods less
than 2 seconds, in lieu of using the equations for vertical response spectra.
3.1.5 Summary
While the new RiskTargeted Maximum Considered Earthquake (MCER) ground motions in the
Provisions and in ASCE 710 are similar to the MCE ground motions in ASCE 705, in that they both
represent the lesser of probabilistic and deterministic ground motions, there are many differences in their
development. The definitions of the probabilistic and deterministic ground motions that underlie the
MCER ground motions were revised by the BSSC Seismic Design Procedures Reassessment Group
(SDPRG, or Project 07); and the hazard modeling upon which these ground motions are based was
updated by the USGS (in 2008). In particular, the underlying probabilistic ground motions were
redefined as socalled risktargeted ground motions, which led to the new MCER ground motion
terminology.
The basis of the new Peak Ground Acceleration (PGA) maps in the Provisions and in ASCE 710 nearly
parallels that of the 0.2 and 1.0second MCE spectral response accelerations in ASCE 705 (with one
important exception); new equations for vertical ground motion spectra are based on recent studies of the
ratio of vertical to horizontal ground motions. The longperiod transition (TL) maps in the new documents
are the same as those in ASCE 705.
3.1.6 References
American Society of Civil Engineers. 1998. Minimum Design Loads for Buildings and Other Structures,
ASCE/SEI 798. ASCE, Reston, Virginia.
American Society of Civil Engineers. 2002. Minimum Design Loads for Buildings and Other Structures,
ASCE/SEI 702. ASCE, Reston, Virginia.
American Society of Civil Engineers. 2006. Minimum Design Loads for Buildings and Other Structures,
ASCE/SEI 705. ASCE, Reston, Virginia.
American Society of Civil Engineers. 2010. Minimum Design Loads for Buildings and Other Structures,
ASCE/SEI 710. ASCE, Reston, Virginia.
Applied Technology Council. 1978. Tentative Provisions for the Development of Seismic Regulations for
Buildings, ATC 306. ATC, Palo Alto, California.
Bozorgnia, Y. and K.W. Campbell. 2004. The VerticaltoHorizontal Response Spectral Ratio and
Tentative Procedures for Developing Simplified V/H and Vertical Design Spectra, Journal of
Earthquake Engineering, 8:175207.
Building Seismic Safety Council. 1997. NEHRP Recommended Provisions for Seismic Regulations for
New Buildings and Other Structures, Part 2: Commentary, FEMA 303. FEMA, Washington, D.C.
Building Seismic Safety Council. 2000. NEHRP Recommended Provisions for Seismic Regulations for
New Buildings and Other Structures, Part 2: Commentary, FEMA 369. FEMA, Washington, D.C.
Building Seismic Safety Council. 2003. NEHRP Recommended Provisions and Commentary for Seismic
Regulations for New Buildings and Other Structures, FEMA 450. FEMA, Washington, D.C.
Building Seismic Safety Council. 2009. NEHRP Recommended Seismic Provisions for New Buildings
and Other Structures, FEMA P750. FEMA, Washington, D.C.
Cornell, C.A. 1968. Engineering Seismic Risk Analysis, Bulletin of the Seismological Society of
America, 58(5):15831606.
Crouse C.B., E.V. Leyendecker, P.G. Somerville, M. Power and W.J. Silva. 2006. Development of
Seismic GroundMotion Criteria for the ASCE 7 Standard, in Proceedings of the 8th US National
Conference on Earthquake Engineering. Earthquake Engineering Research Institute, Oakland,
California.
Field, E.H., T.E. Dawson, K.R. Felzer, A.D. Frankel, V. Gupta, T.H. Jordan, T. Parsons, M.D. Petersen,
R.S. Stein, R.J. Weldon and C.J. Wills. 2008. The Uniform California Earthquake Rupture Forecast,
Version 2 (UCERF 2), USGS Open File Report 20071437 (http://pubs.usgs.gov/of/2007/1437/).
USGS, Golden, Colorado.
Frankel, A.D., M.D. Petersen, C.S. Mueller, K.M. Haller, R.L. Wheeler, E.V. Leyendecker, R.L. Wesson,
S.C. Harmsen, C.H. Cramer, D.M. Perkins and K.S. Rukstales. 2002. Documentation for the 2002
Update of the United States National Seismic Hazard Maps, USGS Open File Report 02420
(http://pubs.usgs.gov/of/2002/ofr02420/). USGS, Golden, Colorado.
Huang, Y.N., A.S. Whittaker and N. Luco, 2008. Maximum Spectral Demands in the NearFault
Region, Earthquake Spectra, 24(1):319341.
Leyendecker, E.V., R.J. Hunt, A.D. Frankel and K.S. Rukstales. 2000. Development of Maximum
Considered Earthquake Ground Motion Maps, Earthquake Spectra, 16(1):2140.
Luco, N. B.R. Ellingwood, R.O. Hamburger, J.D. Hooper, J.K. Kimball and C.A. Kircher. 2007. Risk
Targeted versus Current Seismic Design Maps for the Conterminous United States, in Proceedings
of the SEAOC 76th Annual Convention. Structural Engineers Association of California, Sacramento,
California.
Petersen, M.D., A.D. Frankel, S.C. Harmsen, C.S. Mueller, K.M. Haller, R.L. Wheeler, R.L. Wesson, Y.
Zeng, O.S. Boyd, D.M. Perkins, N. Luco, E.H. Field, C.J. Wills and K.S. Rukstales. 2008.
Documentation for the 2008 Update of the United States National Seismic Hazard Maps, USGS Open
File Report 20081128 (http://pubs.usgs.gov/of/2008/1128/). USGS, Golden, Colorado.
3.2 DETERMINATION OF GROUND MOTION VALUES AND SPECTRA
This example illustrates the determination of seismic design parameters for a site in Seattle, Washington.
The site is located at 47.65§N latitude, 122.3§W longitude. Using the results of a sitespecific
geotechnical investigation and the procedure specified in Standard Chapter 20, the site is classified as Site
Class C. (This is the same site used in Design Example 6.3.)
In the sections that follow design ground motion parameters are determined using ASCE 705, the 2009
Provisions and ASCE 710. Using the 2009 Provisions, horizontal response spectra, vertical response
spectra and peak ground accelerations are computed for both design and maximum considered earthquake
ground motions.
3.2.1 ASCE 705 Ground Motion Values
ASCE 705 Section 11.4.1 requires that spectral response acceleration parameters SS and S1 be determined
using the maps in Chapter 22. Those maps are too small to permit reading values to a sufficient degree of
precision for most sites, so in practice the mapped parameters are determined using a software application
available at www.earthquake.usgs.gov/designmaps. That application requires that longitude be entered in
degrees east of the prime meridian; negative values are used for degrees west. Given the site location, the
following values may be determined using the online application (or read from Figures 221 and 222).
SS = 1.306
S1 = 0.444
Using these mapped spectral response acceleration values and the site class, site coefficients Fa and Fv are
determined in accordance with Section 11.4.3 using Tables 11.41 and 11.42. Using Table 11.41, for SS
= 1.306 > 1.25, Fa = 1.0 for Site Class C. Using Table 11.42, read Fv = 1.4 for S1 = 0.4 and Fv = 1.3 for
S1 ò 0.5 for Site Class C. Using linear interpolation for S1 = 0.444,
Using Equations 11.41 and 11.42 to determine the adjusted maximum considered earthquake spectral
response acceleration parameters,
Using Equations 11.43 and 11.44 to determine the design earthquake spectral response acceleration
parameters,
Given the site location read Figure 2215 for the longperiod transition period, TL = 6 seconds.
3.2.2 2009 Provisions Ground Motion Values
Part 1 of the 2009 Provisions modifies Chapter 11 of ASCE 705 to update the seismic design ground
motion parameters and procedures as described in Section 3.1.2 above. Given the site location, the
following values may be determined using the online application (or read from Provisions Figures 221
through 226).
SSUH = 1.305
S1UH = 0.522
CRS = 0.988
CR1 = 0.955
SSD = 1.5
S1D = 0.6
UH and D appear, respectively, in the subscripts to indicate uniform hazard and deterministic values
of the spectral response acceleration parameters at short periods and at a period of 1 second, SS and S1.
CRS and CR1 are the mapped risk coefficients at short periods and at a period of 1 second. S1D should not
be confused with SD1, which is computed below.
The spectral response acceleration parameter at short periods, SS, is taken as the lesser of the values
computed using Provisions Equations 11.41 and 11.42.
SS = CRS SSUH = 0.988(1.305) = 1.289
SS = SSD = 1.5
Therefore, SS = 1.289.
The spectral response acceleration parameter at a period of 1 second, S1, is taken as the lesser of the
values computed using Provisions Equations 11.43 and 11.44.
S1 = CR1 S1UH = 0.955(0.522) = 0.498
S1 = S1D = 0.6
Therefore, S1 = 0.498.
Using these spectral response acceleration values and the site class, site coefficients Fa and Fv are
determined in accordance with Section 11.4.3 using Tables 11.41 and 11.42 (which are identical to the
Tables in ASCE 705). Using Table 11.41, for SS = 1.289 > 1.25, Fa = 1.0 for Site Class C. Using Table
11.42, read Fv = 1.4 for S1 = 0.4 and Fv = 1.3 for S1 ò 0.5 for Site Class C. Using linear interpolation for
S1 = 0.498,
Using Provisions Equations 11.45 and 11.46 to determine the MCER spectral response acceleration
parameters,
Using Provisions Equations 11.47 and 11.48 to determine the design earthquake spectral response
acceleration parameters,
Given the site location read Provisions Figure 227 for the longperiod transition period, TL = 6 seconds.
3.2.3 ASCE 710 Ground Motion Values
The seismic design ground motion parameters and procedures in Chapter 11 of ASCE 710 are consistent
with those in the 2009 Provisions. Given the site location, the following values may be determined using
the online application (or read from ASCE 710 Figures 221 and 222).
SS = 1.289
S1 = 0.498
Using these spectral response acceleration values and the site class, site coefficients Fa and Fv are
determined in accordance with Section 11.4.3 using Tables 11.41 and 11.42 (which are identical to the
Tables in ASCE 705 and in the 2009 Provisions). Using Table 11.41, for SS = 1.289 > 1.25, Fa = 1.0 for
Site Class C. Using Table 11.42, read Fv = 1.4 for S1 = 0.4 and Fv = 1.3 for S1 ò 0.5 for Site Class C.
Using linear interpolation for S1 = 0.498,
Using Equations 11.41 and 11.42 to determine the MCER spectral response acceleration parameters,
Using Equations 11.43 and 11.44 to determine the design earthquake spectral response acceleration
parameters,
Given the site location read ASCE 710 Figure 2212 for the longperiod transition period, TL = 6
seconds.
The procedure specified in ASCE 710 produces seismic design ground motion parameters that are
identical to those produced using the 2009 Provisions but in fewer steps.
3.2.4 Horizontal Response Spectra
The design spectrum is constructed in accordance with Provisions Section 11.4.5 using Provisions Figure
11.41 and Provisions Equations 11.49, 11.410 and 11.411. The design spectral response acceleration
ordinates, Sa, may be divided into four regions based on period, T, as described below.
From T = 0 to seconds, Sa varies linearly from 0.4SDS to SDS.
From T0 to seconds, Sa is constant at SDS.
From TS to TL, Sa is inversely proportional to T, being anchored to SD1 at T = 1 second.
At periods greater than TL, Sa is inversely proportional to the square of T, being anchored to at TL.
As prescribed in Provisions Section 11.4.6, the MCER response spectrum is determined by multiplying
the design response spectrum ordinates by 1.5. Figure 31 shows the design and MCER response spectra
determined using the ground motion parameters computed in Section 3.2.3.
Figure 31 Horizontal Response Spectra for Design and MCER Ground Motions
3.2.5 Vertical Response Spectra
Part 1 of the 2009 Provisions adds a new chapter (Chapter 23) to ASCE 705 to define vertical ground
motions for seismic design. The design vertical response spectrum is constructed in accordance with
Provisions Section 23.1 using Provisions Equations 23.11, 23.12, 23.13 and 23.14. Vertical ground
motion values are related to horizontal ground motion values by a vertical coefficient, Cv, which is
determined as a function of site class and the MCER spectral response parameter at short periods, SS. The
design vertical spectral response acceleration ordinates, Sav, may be divided into four regions based on
vertical period, Tv, as described below.
Using Provisions Table 23.11, read Cv = 1.3 for SS ò 2.0 and Cv = 1.1 for SS = 1.0 for Site Class C. Using
linear interpolation for SS = 1.289,
From Tv = 0 to 0.025 seconds, Sav is constant at 0.3CvSDS = 0.3(1.158)(0.859) = 0.298. From Tv = 0.025 to
0.05 seconds, Sav varies linearly from 0.3CvSDS = 0.298 to 0.8CvSDS = 0.8(1.158)(0.859) = 0.796. From Tv
= 0.05 to 0.15 seconds, Sav is constant at 0.8CvSDS = 0.796. From Tv = 0.15 to 2.0 seconds, Sav is inversely
proportional to Tv0.75, being anchored to 0.8CvSDS = 0.796 at Tv = 0.15 seconds. For vertical periods
greater than 2.0 seconds, the vertical response spectral acceleration must be determined using sitespecific
procedures.
As prescribed in Provisions Section 23.2, the MCER vertical response spectrum is determined by
multiplying the design vertical response spectrum ordinates by 1.5. Figure 32 shows the design and
MCER vertical response spectra determined using the ground motion parameters computed in Section
3.2.3.
Figure 32 Vertical Response Spectra for Design and MCER Ground Motions
3.2.6 Peak Ground Accelerations
Part 1 of the 2009 Provisions modifies Section 11.8.3 of the ASCE 705 to update the calculation of peak
ground accelerations used for assessment of the potential for liquefaction and soil strength loss and for
determination of lateral earth pressures for design of basement and retaining walls. Given the site
location, the following value of maximum considered earthquake geometric mean peak ground
acceleration may be determined using the online application (or read from Provisions Figure 228).
PGA = 0.521 g
Using this mapped peak ground acceleration value and the site class, site coefficient FPGA is determined in
accordance with Section 11.8.3 using Table 11.81. Using Table 11.81, for PGA = 0.521 > 0.5, FPGA =
1.0 for Site Class C. Using Provisions Equation 11.81 to determine the maximum considered earthquake
geometric mean peak ground acceleration adjusted for site class effects,
PGAM = FPGA PGA = 1.0(0.521) = 0.521 g
This value is used directly to assess the potential for liquefaction or for soil strength loss. The design
peak ground acceleration used to determine dynamic seismic lateral earth pressures for design of
basement and retaining walls is computed as g.
3.3 SELECTION AND SCALING OF GROUND MOTION RECORDS
Response history analysis (whether linear or nonlinear) consists of the stepwise application of time
varying ground accelerations to a mathematical model of the subject structure. The selection and scaling
of appropriate horizontal ground motion acceleration time histories is essential to produce meaningful
results. For twodimensional or threedimensional structural analysis, singlecomponent or two
component records are used, respectively. The sections that follow discuss the approach to selection and
scaling of ground motion records as prescribed in the Provisions (and ASCE 7), illustrate the selection
and scaling of twocomponent ground motions for the structure analyzed in Design Example 6.3 located
at the site considered in Section 3.2 and discuss differences in the process for singlecomponent ground
motions.
3.3.1 Approach to Ground Motion Selection and Scaling
In the simplest terms the goal of ground motion selection and scaling is to produce acceleration histories
that are consistent with the ground shaking hazard anticipated for the subject structure at the site in
question. As difficult as it is to forecast the occurrence of an earthquake, it is even more difficult to
predict the precise waveform and phasing of the resulting accelerations at a given site. Instead it is
necessary to approximate (somewhat crudely) what ground motions can be expected based on past
observations (and, possibly, geologic modeling). Provisions Section 16.1.3 prescribes the most
commonly applied approach to this process. While some aspects of the process are quite prescriptive,
others permit considerable latitude in application.
The Pacific Earthquake Engineering Research Center makes available a database of ground motions (at
http://peer.berkeley.edu/peer_ground_motion_database/) and a web application for the selection and
scaling of ground motions (PEER, 2010). As useful as that data and application are, they do not provide a
comprehensive solution to the challenge of ground motion selection and scaling in accordance with the
Provisions for all U.S. sites. Pertinent limitations include the following.
The database is limited to shallow crustal earthquakes recorded in active tectonic regimes, like
parts of the western U.S. It does not include records from subduction zone earthquakes, deep
intraplate events, or events in less active tectonic regimes (such as the central and eastern U.S.).
The web application allows use of a code design spectrum (from the Provisions or ASCE 7) as a
target and includes powerful selection and scaling methods. However, the set of selected and
scaled records produced would still require minor adjustment (scaling up) to satisfy the
requirements in Provisions Section 16.1.3 over the period range of interest.
3.3.1.1 Number of ground motions. In recognition of the impossibility of predicting the actual ground
motion history that should be expected, Section 16.1.3 requires the use of at least three ground motions in
any response history analysis. Where at least seven ground motions are used, Sections 16.1.4 and 16.2.4
permit the use of average response quantities for design. The difference is not one of statistical
significance; in either case mean response is approximated, but an incentive is given for the use of more
records, which could identify a potential sensitivity in the response. The objective of the response history
analyses is not to evaluate the response of the building for each record (since none of the records used
will actually occur), but to determine the expected (average) response quantities for use in design
calculations. If the analysis predicts collapse for one or more ground motions, the average cannot be
computed; the structure is deemed inadequate and must be redesigned.
3.3.1.2 Recorded or synthetic ground motions. Horizontal ground motion acceleration records should
be selected as single components (for twodimensional analysis) or as orthogonal pairs (for three
dimensional analysis) from actual recorded events. Where the number of appropriate recorded ground
motions is insufficient, use of simulated records is permitted. While generation of completely artificial
records is not directly prohibited, the intent (as expressed in Provisions Section C16.1.3) is that such
simulation is limited to modification for site distance and soil conditions.
3.3.1.3 Appropriate ground motions. The measure of appropriate applied to ground motions by the
Provisions is consistency with the magnitude, fault distance and source mechanism that control the
maximum considered earthquake. (Other characteristics of ground motion, such as duration, may
influence response, but are not addressed by the Provisions.) While it is good practice to select ground
motions with these characteristics in mind, the available data are quite limited. And even where the
available records are very carefully binned and match the target characteristics quite closely, they are far
from homogeneous.
As discussed in Section 3.1 the mapped ground motion parameters reflect the likelihood that a certain
level of spectral acceleration will be exceeded in a selected period, considering numerous sources of
earthquake ground shaking. While the mapping process does not sum accelerations from various sources
it does aggregate the probabilities of occurrence from those sources. As a result, it is impossible to
determine the controlling source characteristics using only the mapped acceleration parameters. In order
to identify the magnitude, fault distance and source mechanism that control the maximum considered
earthquake at a specific spectral period, it is necessary to deaggregate the hazard, which requires
reviewing the underlying calculations to note the relative contribution of each source. The USGS
provides tools to deaggregate hazard, providing results in three formats: a text tabulation, a graphic
presentation binned by distance and magnitude and a graphic presentation projected on a map. Figure 33
shows the two graphic formats for the 2second period spectral acceleration with a 2% probability of
exceedance in 50 years (the maximum considered earthquake) at the site considered in Section 3.2.
Figure 33 Graphic results of deaggregation
At most sites deaggregation of hazard reveals that a single source controls the maximum considered
earthquake ground motions for all spectral periods. However, at some sites different sources control the
maximum considered earthquake ground motions at different spectral periods. Figure 34 shows, for one
such site, the maximum considered earthquake response spectrum generated from mapped ground motion
parameters as well as median acceleration response spectra for two of the contributing sources. Since the
shape of the uniform hazard spectrum, upon which the design spectrum is based, is artificial (arising from
the probabilistic seismic hazard analysis rather than the characteristics of recorded ground motions), there
may be conservatisms involved in providing an aggregate match for design purposes (PEER, 2010).
However, that aggregate match is exactly what the Provisions requires, so it is prudent to consider how
that conservatism may best be balanced.
Figure 34 Response spectra for a site with multiple controlling sources
In this example, source 1 can generate moderate magnitude events close to the site; Source 2 can generate
very large magnitude events far from the site. Due to differing source and attenuation characteristics,
each source can control a portion of the MCER response spectrum. The response of short period
structures or very long period structures will be governed by source 1 or source 2, respectively. However,
the controlling source is less clear for a structure with a fundamental period shown as T1 in the figure.
Source 2 appears to control at period T1, but as discussed in Section 3.3.1.5, the Provisions defines a
wider period range of interest over which the selected ground motions must be appropriate. As outlined
below, three approaches are readily apparent.
First (and arguably most technically correct), select two full sets of (seven or more) ground
motion records conditionally one set for each source, enveloping the MCER spectrum for the
portion of the period range of interest controlled by that source. Since the corresponding
portions of the actual and target spectral shapes would be similar, scale factors would be modest.
In this case, an independent series of analyses would be performed for each set of ground motion
records. Mean response parameters of interest could be computed for each set of analyses and the
more conservative of the two mean values for each response parameter could be used for design
verification. Although this approach has technical appeal, the Provisions do not outline such a
procedure that makes use of two sets of ground motions, instead requiring use of a single set that
on average envelops the entire period range of interest of the target spectrum.
Second, select a full set of ground motions consistent with Source 2 and then scale the set to
envelop the much differently shaped MCER spectrum over the specified period range of interest.
While permitted by the Provisions, this approach would require large scale factors that
unrealistically exaggerate the long period response. It may seem that this set of ground motions
has a desired degree of homogeneity, but that comes at the expense of a very poor fit for the
average.
Third, select a set with some ground motions for each controlling source type. Select individual
scale factors so that the average of their linear elastic spectra envelops the target spectrum (as
required by the Provisions) and is shaped similarly to the target. As a result of this process,
records consistent with Source 1 will control short periods and those consistent with Source 2 will
control long periods. The scale factors will be somewhat larger than those required by the first
(conditional) approach, but not excessively large like those in the second approach. Although the
record set is less homogeneous than that used in the second approach, the average is much closer
to the target. Where used for linear response history analysis, this approach will produce average
response quantities consistent with the average linear response spectrum used in the scaling
process. Where used for nonlinear response history analysis, this approach (which uses scale
factors that are larger than those for the conditional approach) will bias the average response
quantities to be slightly more conservative and may increase the prediction of response extremes
(collapse). This third method is commonly employed by seismological consultants where
multiple source types may govern.
3.3.1.4 Scale factors. The most commonly employed ground motion scaling method involves
multiplying all of the acceleration values of the timeacceleration pairs by a scalar value. This time
domain scaling modifies the amplitude of the accelerations (to approximate changes in source magnitude
and/or distance) without affecting frequency content or phasing. Although not limited by the Provisions,
the scale factors applied to recorded ground motions should be modest (usually falling between 1/3 and
3); if very small or very large scale factors are needed, some aspect of the event that produced the source
motion likely is inconsistent with the maximum considered earthquake being modeled. An identical scale
factor is applied to both components of a given ground motion to avoid unrealistically biasing one
direction of response. Since the response spectra for timedomain scaled ground motions retain their
natural jaggedness, the acceptance criterion compares their average to the target spectrum, without
imposing limits on the scaling of individual ground motions. That means that there is no single set of
scale factors that may be applied to the selected ground motions (as discussed further in Provisions Part 2
Section C16.1.3.2)
Another ground motion scaling method involves transforming the timeacceleration data into the
frequency domain (such as by means of the fast Fourier transform), making adjustments (to match exactly
the target spectrum at multiple, specific frequencies) and transforming back into the time domain. This
method affects amplitude, frequency content and phasing (and tends to increase the total input energy).
This method makes it possible to estimate mean response with fewer ground motions, but may obscure
somewhat the potential variability of response. Use of this method is permitted by the Provisions, but the
same number of records is required as for timedomain scaling. Given the jaggedness of individual
response spectra, the process of spectral matching (which produces smoother spectra) requires scale
factors that can be considerably smaller or larger than those used in timedomain scaling. Since this
method applies numerous scale factors to differing frequencies of each ground motion component in order
to match spectral ordinates, there is no requirement that the two components be scaled identically. As the
spectral ordinates of frequencydomain scaled records may fall below the target spectrum at frequencies
other than those used for matching, a second round of (minor) scaling is needed to satisfy the Provisions
requirements.
Where singlecomponent records are being selected for twodimensional analysis the design response
spectrum is used as a target; and Provisions Section 16.1.3.1 requires that the average of the response
spectra not fall below the target over the period range of interest. A different approach is needed where
twocomponent records are being selected for threedimensional analysis. The code writers selected the
square root of the sum of the squares (SRSS) of the response spectra for the two components as a measure
of the ground motion amplitude for each record. However, the SRSS of two spectra is always larger than
the average (and larger than the maximum). In practical terms for ground motion, it is reasonable to
expect that the SRSS is larger than the average by a factor of 1.4 to 1.5 and is larger than the maximum
(resultant) by a factor of about 1.2.
The code writers decided that it is sufficiently conservative to scale twocomponent records such that the
average of the SRSS spectra does not fall below the target over the period range of interest. Given the
relationship between SRSS and average, that means that scale factors for ground motions used in three
dimensional analysis are only 2/3 of those for ground motions used in twodimensional analysis. The
rationale is that a threedimensional analysis (using twocomponent ground motions) subjects the
structure to the maximum (resultant) acceleration in some direction due to the interaction of ground
motion components, while that is not possible in twodimensional analysis. Considering other
conservative criteria, such as fitting over the entire period range of interest, code writers accepted that the
resultant acceleration could be about 20 percent less than the design acceleration at some periods. Note
that Provisions Section 16.1.3.2 erroneously requires that the average of the SRSS spectra not fall below
the MCER spectrum over the period range of interest. ASCE 710 corrects this error by requiring that the
average of the SRSS spectra be compared to the response spectrum used in the design (rather than to
the MCER response spectrum).
For the special case described in Section 3.3.1.7 below, both ASCE 710 and the Provisions require
scaling so that the maximum acceleration exceeds the MCER response spectrum. Apparently, this is an
error carried forward from the Provisions to ASCE 710. Like the rest of Section 16.1.3.2, the target
spectrum used for scaling should be the response spectrum used in the design rather than the MCER
response spectrum (which is 1.5 times the design response spectrum).
3.3.1.5 Period range of interest. The smooth spectral acceleration response spectrum constructed using
mapped acceleration parameters (and site response coefficients) is a locationspecific estimate of the
ground shaking hazard. No matter how carefully recorded ground motions are selected and scaled, it is
unrealistic to expect a close match to the smooth target spectrum over all periods. On the other hand,
selecting and scaling ground motions to match the target spectrum at the natural period for the
fundamental mode of vibration of a structure is not enough to produce reasonable estimates of response;
important aspects of structural response (including collapse) are affected by both higher modes of
response and period elongation due to yielding. To balance these realities, code writers have established a
period range of interest (with respect to the fundamental period, T) that extends from 0.2T (to capture
higher mode effects) to 1.5T (to include period elongation). Although yielding and period elongation
cannot occur in linear response history analysis, for simplicity of application ground motions are selected
and scaled considering the same period range of interest as for nonlinear response history analysis.
3.3.1.6 Orientation of ground motion components. Accelerometers record earthquake ground shaking
along the vertical axis and two horizontal (orthogonal) instrument axes. Acceleration records can be used
in the asrecorded orientation, but orientation in the directions normal to and parallel to the strike of the
causative fault (termed the faultnormal and faultparallel directions, respectively) by means of a simple
trigonometric transformation permits greater seismological insights, since some ground motions recorded
very close to the causative fault contain rupture directivity effects. The differences may be meaningful
for selection and scaling, application in analysis, or both. Since the orientation of instrument axes is
arbitrary and reorientation along the faultnormal and faultparallel directions can provide additional
insight, it has become common (but not universal) to reorient all horizontal ground motion records in that
manner.
In the very common condition where a site is not within several miles of the controlling source, the
orientation of ground shaking is inconsequential, so the Provisions contain no general requirement to
consider orientation. As discussed in Section 3.3.1.7 below, there is a selection and scaling orientation
requirement (but no application orientation requirement) for sites close to active controlling faults.
Figure 35 shows the time series of two components of ground acceleration. Component 1 is fault
normal; component 2 is faultparallel. What is not apparent in such traces is the interaction of the
components. Figure 36 shows an orbit plot of ground acceleration pairs (effectively zeroperiod
response) for the same recording. The maximum resultant acceleration occurs along a diagonal direction.
Figure 35 Horizontal acceleration components for the 1989 Loma Prieta earthquake
(Saratoga Aloha Avenue recording station)
Figure 36 Horizontal acceleration orbit plot for the 1989 Loma Prieta earthquake
(Saratoga Aloha Avenue recording station)
Unfortunately, the direction of maximum ground acceleration may or may not correspond to the direction
of maximum acceleration response at any other period and the direction of maximum response generally
differs at various periods. If bilinear oscillators with various fundamental periods are subjected to the
twocomponent acceleration record, response spectra like those in Figure 37 result. The uniaxial
response spectra in that figure are identified by component. The resultant response spectrum indicates
the maximum acceleration along any direction. The SRSS response spectrum is obtained by taking the
square root of the sum of the squares of the corresponding component response spectrum ordinates. The
case illustrated reflects a possible nearsource condition: for long periods, the faultnormal component
(component 1) is much larger than the faultparallel component and is very close to the maximum
(resultant) response.
The Provisions do not require application of ground motions in multiple possible orientations. Whether
using three, seven, or more pairs, it is acceptable to consider a single, arbitrary orientation of a given two
component pair. For example, analysis can be performed with Component 1 applied in the +X
direction without considering the implications of applying that component in the X, +Y, Y, or other
directions. Since the objective of the analyses is to estimate average response quantities, it may be
advisable (but is not required) to consider whether there is an unwanted directional bias in the selected
and scaled ground motions. For instance, in the common case where the controlling source should not
produce strongly directional response, records could be oriented when applied so that the average of the
component 1 spectra is similar to the average of the component 2 spectra. The muchlesscommon case,
where strongly directional response is expected, is discussed in Section 3.3.1.7. Section 12.4.4 of these
Design Examples outlines a more involved approach that is recommended for seismically isolated
structures.
Figure 37 Horizontal acceleration response spectra for the 1989 Loma Prieta earthquake
(Saratoga Aloha Avenue recording station)
3.3.1.7 Sites close to controlling active faults. Ground motions at sites close to a causative fault can be
strongly directional. At such sites, the maximum long period ground motion often occurs in the fault
normal direction. The last paragraph of Provisions Section 16.1.3.2 addresses this case, where code
writers have judged that scaling should be more conservative than that achieved using the SRSSbased
method. Although this requirement is well intentioned, the specific language provides a degree of
additional conservatism that can vary greatly. The intent is that the maximum spectral acceleration for
the scaled motions exceeds the target response spectrum.
While it is often true that the faultnormal component is dominant at long periods, some nearfield ground
motions show no directional bias and some are dominant in the faultparallel direction. For instance, of
the 3182 records in the PEER Ground Motion Database (for shallow crustal earthquakes), only 109 have
pulselike directional effects. Of those, 60 have pulses only in the faultnormal direction, 19 have pulses
only in the faultparallel direction, and 30 have pulses in both directions. As discussed above, it is
acceptable to reorient all horizontal ground motion records to the faultnormal and faultparallel
directions. However, that does not assure that the faultnormal component will coincide with the
maximum. Figure 38 shows response spectra for a ground motion where the faultparallel direction
(component 2) dominates for long periods. Scaling such that the faultnormal component exceeds the
target response spectrum, as required in the last paragraph of Section 16.1.3.2, would force the maximum
well above the target response spectrum. To obtain the intended result, ground motions should be scaled
so that the average of the faultnormal dominant components is not less than the MCER response spectrum
used in the design for the period range from 0.2T to 1.5T. (Section 3.3.1.4 above explains why all of
Section 16.1.3.2 should refer to the response spectrum used in the design rather than to the MCER
response spectrum.)
While the Provisions set forth orientation requirements for the selection and scaling of ground motions at
sites close to controlling active faults, the orientation of ground motion components as applied in analysis
is not prescribed. After going to the effort of orienting records in the faultnormal and faultparallel
directions and applying special rules for scaling in recognition of nearsource effects, it would be prudent
(but not required) to apply the records in the analyses consistent with the faultnormal and faultparallel
directions at the actual site.
Figure 38 Horizontal acceleration response spectra for 1999 Duzce, Turkey earthquake
(Duzce recording station)
3.3.2 TwoComponent Records for Three Dimensional Analysis
Design Example 6.3 is a buckling restrained braced frame structure located at the Seattle, Washington site
considered in Section 3.2. Some aspects of the design are based on results from threedimensional
nonlinear response history analysis performed in accordance with Provisions Section 16.2. This section
illustrates application of the procedures described in Section 3.3.1 for the selection and scaling of two
component ground motion records. Pertinent information from Sections 3.2 and 6.3.6.1 is summarized as
follows.
Location: 47.65§N, 122.3§W
Site Class C
SMS = 1.289
SM1 = 0.649
TL = 6 seconds
Tx = Ty = 2.3 seconds
The period range of interest is from 0.2 2.3 = 0.46 seconds to 1.5 2.3 = 3.45 seconds. If the two
fundamental translational periods differed, the period range of interest would extend from 0.2 times the
shorter period to 1.5 times the longer period.
The next step is to deaggregate the hazard, as discussed in Section 3.3.1.3, over the period range of
interest. Figure 39 shows the MCER (target) response spectrum and the relative contributions of three
important sources to spectral acceleration at periods between 0 and 4 seconds. For periods greater than
about 1.5 seconds, ground shaking hazard is controlled by very large, but distant, subduction zone events.
At shorter periods, hazard is controlled by deep intraplate events, with substantial contributions from
shallow crustal events. It is necessary to identify not only the magnitude of the controlling event, but also
the distance and source type. Short and intermediate period response may be more important than long
period response (depending on the period of the structure).
Figure 39 MCER response spectrum and corresponding hazard contributions
Since the MCER response spectrum over the period range of interest is controlled by multiple sources
with substantially different spectral shapes, the procedure recommended in Section 3.3.1.3 is used. Table
31 provides key information for the selected ground motion records. Few large magnitude subduction
zone records are available. Records 1, 2 and 3 are for slightly smaller events than those that control the
long period hazard, but at closer distances. These differences are partially offsetting so the required scale
factors are acceptable. Record 4 is nearly a perfect match for the hazard that controls short period
response; and it is from a past occurrence of a similar event in the same region. Records 5, 6 and 7 are
from shallow crustal events with magnitude and distance appropriate for this site. Two of those records
include nearsource velocity pulses. In a manner similar to that illustrated in Figure 34, the actual
spectra for the selected ground motion records control different periods of response. Figure 310 shows
the SRSS spectra for Records 1 and 4, along with the target (MCER) spectrum. The subduction zone
event (Record No. 1) dominates long period response; the deep intraplate event (Record No. 4) dominates
short period response.
Table 31 Selected and Scaled Ground Motions for Example Site
Record
No.
Year
Earthquake name
M
Source type
Recording station
Distance
(km)
Scale
factor
1
2003
Tokachioki, Japan
8.3
Subduction zone
HKA 094
67
2.99
2
2003
Tokachioki, Japan
8.3
Subduction zone
HKD 092
46
0.96
3
1968
Tokachioki, Japan
8.2
Subduction zone
Hachinohe (S252)
71
1.28
4
1949
Western Washington
7.1
Deep intraplate
Olympia
75
1.92
5
1989
Loma Prieta
6.9
Shallow crustal
Saratoga  Aloha Ave
9
1.28
6
1999
Duzce, Turkey
7.1
Shallow crustal
Duzce
7
0.85
7
1995
Kobe, Japan
6.9
Shallow crustal
NishiAkashi
7
1.18
Figure 310 SRSS response spectra from different source types
Figure 311 compares the average of the SRSS spectra for the selected ground motions with the target
(MCER) response spectrum. It also shows the period range of interest for ground motion selection for this
structure. In an average sense the suite of ground motions provides a very good fit to the target. Since
seven records are used, average response quantities may be used in design. This suite so well matches the
target spectrum that it could be used with no modification for periods from 0.18 to 4.95 seconds, a range
much wider than the period range of interest defined in Provisions Section 16.1.3.2. Since this suite of
ground motions has been selected and scaled to match the MCER response spectrum, an additional scale
factor of 2/3 must be applied when the records are used in an analysis to represent designlevel
conditions.
Figure 311 Fit of the selected suite of ground motion records to the target spectrum
(for threedimensional analysis)
3.3.3 OneComponent Records for TwoDimensional Analysis
As discussed in Section 3.3.1.4, onecomponent records (for use in twodimensional analysis) are selected
and scaled such that their average fits the design response spectrum, which is twothirds of the MCER
response spectrum. Figure 312 compares the average of the 14 component spectra (for the records
selected and scaled in Section 3.3.2) to the design response spectrum. These records provide an excellent
fit to the target spectrum. The suite of 14 records could be used without modification. If a subset of
seven records were selected, some minor adjustment to scale factors might be required. Figure 312 also
shows the average of the SRSS spectra for those 14 scaled records. As observed in Section 3.3.1.4, the
average of the SRSS spectra is about 1.5 times the average of the component spectra. Therefore, if the
same suite of records was used for threedimensional analysis, the scale factors required would be about
2/3 of those required for twodimensional analysis, due to the difference between average and SRSS
spectra (and not due to the purely coincidental 2/3 relationship between design and MCER response
spectra).
Figure 312 Fit of the selected suite of ground motion records to the target spectrum
(for twodimensional analysis)
3.3.4 References
PEER. 2010. Technical Report for the PEER Ground Motion Database Web Application, Pacific
Earthquake Engineering Research Center, Berkeley, California.
Author of Section 3.1.
Author of Sections 3.2 and 3.3.
Reviewing author.
See the February 2008 Earthquake Spectra Special Issue on the Next Generation Attenuation
Project, Volume 24, Number 1.
See the January/February 1997 Seismological Research Letters Special Issue on Ground Motion
Attenuation Relations, Volume 68, Number 1.
4
Structural Analysis
Finley Charney, Adrian Tola Tola and Ozgur Atlayan
Contents
4.1 IRREGULAR 12STORY STEEL FRAME BUILDING, STOCKTON, CALIFORNIA 3
4.1.1 Introduction 3
4.1.2 Description of Building and Structure 3
4.1.3 Seismic Ground Motion Parameters 4
4.1.4 Dynamic Properties 8
4.1.5 Equivalent Lateral Force Analysis 11
4.1.6 Modal Response Spectrum Analysis 29
4.1.7 Modal Response History Analysis 39
4.1.8 Comparison of Results from Various Methods of Analysis 50
4.1.9 Consideration of Higher Modes in Analysis 53
4.1.10 Commentary on the ASCE 7 Requirements for Analysis 56
4.2 SIXSTORY STEEL FRAME BUILDING, SEATTLE, WASHINGTON 57
4.2.1 Description of Structure 57
4.2.2 Loads 60
4.2.3 Preliminaries to Main Structural Analysis 64
4.2.4 Description of Model Used for Detailed Structural Analysis 72
4.2.5 Nonlinear Static Analysis 94
4.2.6 Response History Analysis 109
4.2.7 Summary and Conclusions 134
This chapter presents two examples that focus on the dynamic analysis of steel frame structures:
1. A 12story steel frame building in Stockton, California. The highly irregular structure is analyzed
using three techniques: equivalent lateral force analysis, modal response spectrum analysis and
modal response history analysis. In each case, the structure is modeled in three dimensions and
only linear elastic response is considered. The results from each of the analyses are compared
and the accuracy and relative merits of the different analytical approaches are discussed.
2. A sixstory steel frame building in Seattle, Washington. This regular structure is analyzed using
both linear and nonlinear techniques. Due to the regular configuration of the structural system,
the analyses are performed for only two dimensions. For the nonlinear analysis, two approaches
are used: static pushover analysis and nonlinear response history analysis. The relative merits of
pushover analysis versus response history analysis are discussed.
Although the Seattle building, as originally designed, responds reasonably well under the design
ground motions, a second set of response history analyses is presented for the structure
augmented with added viscous fluid damping devices. As shown, the devices have the desired
effect of reducing the deformation demands in the critical regions of the structure.
In addition to the Standard, the following documents are referenced:
AISC 341 American Institute of Steel Construction. 2005. Seismic Provisions for
Structural Steel Buildings.
AISC 358 American Institute of Steel Construction. 2005. Prequalified Connections for
Special and Intermediate Steel Moment Frames for Seismic Applications.
AISC 360 American Institute of Steel Construction. 2005. Specification for Structural
Steel Buildings.
AISC Manual American Institute of Steel Construction. 2005. Manual of Steel Construction,
13th Edition.
AISC SDM American Institute of Steel Construction. 2006. Seismic Design Manual.
ASCE 41 American Society of Civil Engineers. 2006. Seismic Rehabilitation of Existing
Buildings.
ASCE 710 American Society of Civil Engineers. 2010. Minimum Design Loads for
Buildings and Other Structures
Charney & Marshall Charney, F. A. and Marshall, J. D., 2006, A comparison of the Krawinkler and
Scissors models for including beamcolumn joint deformations in the analysis of
steel frames, Engineering Journal, 43(1), 3148.
Charney (2008) Charney, F. A., 2008, Unintended consequences of modeling damping in
structures, Journal of Structural Engineering, 134(4), 581592.
Clough & Penzien Ray W. Clough and Joseph Penzien, Dynamics of Structures, 2nd Edition.
FEMA 440 Federal Emergency Management Agency. 2005. Improvement of Nonlinear
Static Seismic Analysis Procedures
FEMA P750 Federal Emergency Management Agency. 2010. 2009 NEHRP Recommended
Seismic Provisions for Buildings and Other Structures
Prakash et al. (1993) Prakash, V., Powell, G.H. and Campbell, S., 1993, Drain 2DX Base Program
Description and User s Guide, University of California, Berkeley, CA.
Uang & Bertero Uang C.M. and Bertero V.V., 1990, Evaluation of Seismic Energy in
Structures , Earthquake Engineering and Structural Dynamics, 19, 7790.
4.1 IRREGULAR 12STORY STEEL FRAME BUILDING, STOCKTON, CALIFORNIA
4.1.1 Introduction
This example presents the analysis of a 12story steel frame building under seismic effects acting alone.
Gravity forces due to dead and live load are not computed. For this reason, member stress checks,
member design and detailing are not discussed. Load combinations that include gravity effects are
considered, however. For detailed examples of the seismicresistant design of structural steel buildings,
see Chapter 6 of this volume of design examples.
The analysis of the structure, shown in Figures 4.11 through 4.13, is performed using three methods:
1. The Equivalent Lateral Force (ELF) procedure based on the requirements of Standard Section 12.8,
2. The modal response spectrum procedure based on the requirements of Standard Section 12.9 and
3. The modal response history procedure based on the requirements of Chapter 16 of ASCE 710. (The
2010 version of the Standard is used for this part of the example because it eliminates several
omissions and inconsistencies that were present in Chapter 16 of ASCE 705.)
In each case, special attention is given to applying the Standard rules for direction of loading and for
accidental torsion. All analyses were performed in three dimensions using the finite element analysis
program SAP2000 (developed by Computers and Structures, Inc., Berkeley, California).
4.1.2 Description of Building and Structure
The building has 12 stories above grade and a onestory basement below grade and is laid out on a
rectangular grid with a maximum of seven 30footwide bays in the X direction and seven 25foot bays in
the Y direction. Both the plan and elevation of the structure are irregular with setbacks occurring at
Levels 5 and 9. All stories have a height of 12.5 feet except for the first story which is 18 feet high and
the basement which extends 18 feet below grade. Reinforced 1footthick concrete walls form the
perimeter of the basement. The total height of the building above grade is 155.5 feet.
Gravity loads are resisted by composite beams and girders that support a normalweight concrete slab on
metal deck. The slab has an average thickness of 4.0 inches at all levels except Levels G, 5 and 9. The
slabs on Levels 5 and 9 have an average thickness of 6.0 inches for more effective shear transfer through
the diaphragm. The slab at Level G is 6.0 inches thick to minimize pedestrianinduced vibrations and to
support heavy floor loads. The low roofs at Levels 5 and 9 are used as outdoor patios and support heavier
live loads than do the upper roofs or typical floors.
At the perimeter of the base of the building, the columns are embedded into pilasters cast into the
basement walls, with the walls supported on reinforced concrete tie beams over drilled piers. Interior
columns are supported by concrete caps over drilled piers. A grid of reinforced concrete grade beams
connects all tie beams and pier caps.
The lateral loadresisting system consists of special steel moment frames at the perimeter of the building
and along Grids C and F. For the frames on Grids C and F, the columns extend down to the foundation,
but the lateral loadresisting girders terminate at Level 5 for Grid C and Level 9 for Grid F. Girders
below these levels are simply connected. Since the momentresisting girders terminate in Frames C and
F, much of the Y direction seismic shears below Level 9 are transferred through the diaphragms to the
frames on Grids A and H. Overturning moments developed in the upper levels of these frames are
transferred down to the foundation by axial forces in the columns. Columns in the momentresisting
frame range in size from W24x146 at the roof to W24x229 at Level G. Girders in the moment frames
vary from W30x108 at the roof to W30x132 at Level G. Members of the momentresisting frames have a
nominal yield strength of 36 ksi and floor members and interior columns that are sized strictly for gravity
forces have a nominal yield strength of 50 ksi.
4.1.3 Seismic Ground Motion Parameters
For this example the relevant seismic ground motion parameters are as follows:
SS = 1.25
S1 = 0.40
Site Class C
From Standard Tables 11.41 and 11.42:
Fa = 1.0
Fv = 1.4
Using Standard Equations 11.41 through 11.44:
SMS = FaSs = 1.0(1.25) = 1.25
SM1 = FvS1 = 1.4(0.4) = 0.56
As the primary occupancy of the building is business offices, the Occupancy Category is II (Standard
Table 11) and the Importance Factor (I) is 1.0 (Standard Table 11.51). According to Standard
Tables 11.61 and 11.62, the Seismic Design Category (SDC) for this building is D.
Figure 4.11 Various floor plans of 12story Stockton building
Figure 4.12 Sections through Stockton building
Figure 4.13 Threedimensional wireframe model of Stockton building
The lateral loadresisting system of the building is a special momentresisting frame of structural steel.
For this type of system, Standard Table 12.21 has a response modification coefficient (R) of 8 and a
deflection amplification coefficient (Cd) of 5.5. There is no height limit for special moment frames.
Section 12.2.5.5 of the standard requires that special moment frames in SDC D, where required by Table
12.11, be continuous to the foundation. While the girders of the interior moment frames are not present
at the lower levels of the interior frames, the frames are continuous to the foundation and the columns are
detailed as required for special moment frames. Additionally, there are no other structural system types
below the moment frames. Therefore, in the opinion of the author, the requirement is met.
Standard Table 12.61 is used to determine the minimum level of analysis. Because of the setbacks, the
structure clearly has a weight irregularity (Irregularity Type 2 in Standard Table 12.32). Thus, the
minimum level of analysis required for the SDC D building is modal response spectrum analysis.
However, the determination of torsional irregularities, the application of accidental torsion effects and the
assessment of Pdelta effects are based on ELF analysis procedures. For this reason and for comparison
purposes, a complete ELF analysis is carried out and described herein.
4.1.4 Dynamic Properties
Before any analysis can be carried out, it is necessary to determine the dynamic properties of the
structure. These properties include stiffness, mass and damping. The stiffness of the structure is
numerically represented by the system stiffness matrix, which is computed automatically by SAP2000.
The terms in this matrix are a function of several modeling choices that are made. These aspects of the
analysis are described later in the example. The computer can also determine the mass properties
automatically, but for this analysis they are developed by hand and are explicitly included in the computer
model. Damping is represented in different ways for the different methods of analysis, as described in
Section 4.1.4.2.
4.1.4.1 Seismic Weight. In the past it was often advantageous to model floor plates as rigid diaphragms
because this allowed for a reduction in the total number of degrees of freedom used in the analysis and a
significant reduction in analysis time. Given the speed and capacity of most personal computers, the use
of rigid diaphragms is no longer necessary and the floor plates may be modeled using 4node shell
elements. The use of such elements provides an added benefit of improved accuracy because the true
semirigid behavior of the diaphragms is modeled directly. Where it is not necessary to recover
diaphragm stresses, a very coarse element mesh may be used for modeling the diaphragm.
Where the diaphragm is modeled using finite elements, the diaphragm mass, including contributions from
structural dead weight and superimposed dead weight, is automatically represented by entering the proper
density and thickness of the diaphragm elements. The density may be adjusted to represent superimposed
dead loads (but the thickness and modulus are true values). Line mass, such as window walls and
exterior cladding, are modeled with frame element line masses. While complete building masses are
easily represented in this manner, the SAP2000 program does not automatically compute the locations of
the centers of mass, so these must be computed separately. Center of mass locations are required for the
purpose of applying lateral forces in the ELF method and for determining story drift.
Due to the various sizes and shapes of the floor plates and to the different dead weights associated with
areas within the same floor plate, the computation of mass properties is not easily carried out by hand.
For this reason, a special purpose computer program was used. The basic input for the program consists
of the shape of the floor plate, its mass density and definitions of auxiliary masses such as line,
rectangular and concentrated mass.
The uniform area and line masses (in weight units) associated with the various floor plates are given in
Tables 4.11 and 4.12. The line masses are based on a cladding weight of 15.0 psf, story heights of 12.5
or 18.0 feet and parapets 4.0 feet high bordering each roof region. Figure 4.14 shows where each mass
type occurs. The total computed floor mass, mass moment of inertia and locations of center of mass are
shown in Table 4.13. Note that the mass moments of inertia are not required for the analysis but are
provided in the table for completeness. The reference point for center of mass location is the intersection
of Grids A and 8.
Table 4.13 includes a mass computed for Level G of the building. This mass is associated with an
extremely stiff story (the basement level) and is dynamically excited by the earthquake in very high
frequency modes of response. As shown later, this mass is not included in equivalent lateral force
computations.
Table 4.11 Area Weights Contributing to Masses on Floor Diaphragms
Mass Type
Area Weight Designation
A
B
C
D
E
Slab and Deck (psf)
Structure (psf)
Ceiling and Mechanical (psf)
Partition (psf)
Roofing (psf)
Special (psf)
Total (psf)
50
20
15
10
0
0
95
75
20
15
10
0
0
120
50
20
15
0
15
0
100
75
20
15
0
15
60
185
75
50
15
10
0
25
175
See Figure 4.14 for mass location.
1.0 psf = 47.9 N/m2.
Table 4.12 Line Weights Contributing to Masses on Floor Diaphragms
Mass Type
Line Weight Designation
1
2
3
4
5
From Story Above (plf)
From Story Below (plf)
Total (plf)
60.0
93.8
153.8
93.8
93.8
187.6
93.8
0.0
93.8
93.8
135.0
228.8
135.0
1,350.0
1,485.0
See Figure 4.14 for mass location.
1.0 plf = 14.6 N/m.
Figure 4.14 Key diagram for computation of floor weights
Table 4.13 Floor Weight, Floor Mass, Mass Moment of Inertia and Center of Mass Locations
Level
Weight (kips)
Mass
(kipsec2/in.)
Mass Moment of
Inertia (in.kip
sec2//radian)
X Distance to
C.M.
(in.)
Y Distance to
C.M.
(in.)
R
12
11
10
9
8
7
6
5
4
3
2
G
1,657
1,596
1,596
1,596
3,403
2,331
2,331
2,331
4,320
3,066
3,066
3,097
6,525
36,912
4.287
4.130
4.130
4.130
8.807
6.032
6.032
6.032
11.19
7.935
7.935
8.015
16.89
2.072x106
2.017x106
2.017x106
2.017x106
5.309x106
3.703x106
3.703x106
3.703x106
9.091x106
6.356x106
6.356x106
6.437x106
1.503x107
1,260
1,260
1,260
1,260
1,638
1,553
1,553
1,553
1,160
1,261
1,261
1,262
1,265
1,050
1,050
1,050
1,050
1,175
1,145
1,145
1,145
1,206
1,184
1,184
1,181
1,149
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
4.1.4.2 Damping. Where an equivalent lateral force analysis or a modal response spectrum analysis is
performed, the structure s damping, assumed to be 5 percent of critical, is included in the development of
the spectral accelerations SS and S1. An equivalent viscous damping ratio of 0.05 is appropriate for linear
analysis of lightly damaged steel structures.
Where recombining the individual modal responses in modal response spectrum analysis, the square root
of the sum of the squares (SRSS) technique has generally been replaced in practice by the complete
quadratic combination (CQC) approach. Indeed, Standard Section 12.9.3 requires that the CQC approach
be used where the modes are closely spaced. Where using the CQC approach, the analyst must correctly
specify a damping factor. This factor, which is entered into the SAP2000 program, must match that used
in developing the response spectrum. It should be noted that if zero damping is used in CQC, the results
are the same as those for SRSS.
For modal response history analysis, SAP2000 allows an explicit damping ratio to be used in each mode.
For this structure, a damping of 5 percent of critical was specified in each mode.
4.1.5 Equivalent Lateral Force Analysis
Prior to performing modal response spectrum or response history analysis, it is necessary to perform an
ELF analysis of the structure. This analysis is used for preliminary design, for evaluating torsional
regularity, for computing torsional amplification factors (where needed), for application of accidental
torsion, for evaluation of Pdelta effects and for development of redundancy factors.
The first step in the ELF analysis is to determine the period of vibration of the building. This period can
be accurately computed from a threedimensional computer model of the structure. However, it is first
necessary to estimate the period using empirical relationships provided by the Standard.
Standard Equation 12.87 is used to estimate the building period:
where, from Standard Table 12.82, Ct = 0.028 and x = 0.8 for a steel moment frame. Using hn = the total
building height (above grade) = 155.5 ft, Ta = 0.028(155.5)0.8 = 1.59 seconds .
Even where the period is accurately computed from a properly substantiated structural analysis (such as
an eigenvalue or Rayleigh analysis), the Standard requires that the period used for ELF base shear
calculations not exceed CuTa where Cu = 1.4 (from Standard Table 12.81 using SD1 = 0.373). For the
structure under consideration, CuTa = 1.4(1.59) = 2.23 seconds.
Note that where the accurately computed period is less than CuTa, the computed period should be used. In
no case, however, is it necessary to use a period less than T = Ta = 1.59 seconds. The use of the Rayleigh
method and the eigenvalue method of determining accurate periods of vibration are illustrated in a later
part of this example.
In anticipation of the accurately computed period of the building being greater than 2.23 seconds, the ELF
analysis is based on a period of vibration equal to CuTa = 2.23 seconds . For the ELF analysis, it is
assumed that the structure is fixed at grade level. Hence, the total effective weight of the structure (see
Table 4.13) is the total weight minus the grade level weight, or 36,920 6,526 = 30,394 kips.
4.1.5.1 Base shear and vertical distribution of force. Using Standard Equation 12.81, the total
seismic base shear is:
where W is the total weight of the structure. From Standard Equation 12.82, the maximum (constant
acceleration region) spectral acceleration is:
= 0.104
Standard Equation 12.83 controls in the constant velocity region:
= 0.021
However, the acceleration must not be less than that given by Standard Equation 12.85:
= 0.037
The Cs value determined from Equation 12.85 controls the seismic base shear for this building. Using
W = 30,394 kips, V = 0.037(30,394) = 1,124 kips. The acceleration response spectrum given by the above
equations is plotted in Figure 4.15.
Figure 4.15 Computed ELF total acceleration response spectrum
Figure 4.16 Computed ELF relative displacement response spectrum
While it is reasonable to use Equation 12.85 to establish a minimum base shear, the equation should not
be used as a basis determining lateral forces used in displacement computations. The effect of using
Equation 12.85 for displacements is shown in Figure 4.16, which represents Equations 12.82, 12.83
and 12.85 in the form of a displacement spectrum. It can be seen from this figure that the dotted line,
representing Equation 12.85, will predict significantly larger displacements than Equation 12.83. The
problem with the line represented by Equation 12.85 is that it gives an exponentially increasing
displacement up to unlimited periods, whereas it is expected that the true spectral displacements will
converge towards a constant displacement (the maximum ground displacement) at large periods. In other
words, Equation 12.85 should not be considered as a branch of the response spectrum it is simply used
to represent the lower bound on design base shear. The Standard does not directly recognize this
problem. However, Section 12.8.6.2 allows the deflection analysis of the seismic forceresisting system
to be based on the accurately computed fundamental period of vibration, without the CuTa upper limit on
period. It is the authors opinion that this clause may be used to justify drift calculations with forces
based on Equation 12.83 even when Equation 12.85 controls the design base shear. ASCE 710 has
clarified this issue, by providing an exception that specifically states that Equation 12.85 need not be
considered for computing drift.
It is important to note, however, that where Equation 12.86 controls the design base shear, drifts must be
based on lateral forces consistent with Equation 12.86. This is due to the fact that Equation 12.86 is an
approximation of the long period acceleration spectrum for near field ground motions (where S1 is
likely to be greater than 0.6 g.)
In this example, all ELF analysis is performed using the forces obtained from Equation 12.85, but for the
purposes of computing drift, the story deflections are computed using the forces from Equation 12.83.
When using Equation 12.83, the upper bound period CuTa was used in lieu of the computed period. This
allows for a simple conversion of displacements where displacements computed from forces based on
Equation 12.85 are multiplied by the factor (0.021/0.037 = 0.568) to obtain displacements that would be
generated from forces based on Equation 12.83 and the CuTa limit. If it is found that the factored
computed drifts violate the drift limits (which is not the case in this example), it might be advantageous to
recompute the drifts on the basis of Equation 12.83 and the computed period T.
The seismic base shear computed according to Standard Equation 12.81 is distributed along the height of
the building using Standard Equations 12.811 and 12.812:
and
where k = 0.75 + 0.5T = 0.75 + 0.5(2.23) = 1.865. The story forces, story shears and story overturning
moments are summarized in Table 4.14.
Table 4.14 Equivalent Lateral Forces for Building Response in X and Y Directions
Level
x
wx
(kips)
hx
(ft)
wxhxk
Cvx
Fx
(kips)
Vx
(kips)
Mx
(ftkips)
R
1,657
155.5
20,272,144
0.1662
186.9
186.9
2,336
12
1,596
143.0
16,700,697
0.1370
154.0
340.9
6,597
11
1,596
130.5
14,081,412
0.1155
129.9
470.8
12,482
10
1,596
118.0
11,670,590
0.0957
107.6
578.4
19,712
9
3,403
105.5
20,194,253
0.1656
186.3
764.7
29,271
8
2,331
93.0
10,933,595
0.0897
100.8
865.5
40,090
7
2,331
80.5
8,353,175
0.0685
77.0
942.5
51,871
6
2,331
68.0
6,097,775
0.0500
56.2
998.8
64,356
5
4,324
55.5
7,744,477
0.0635
71.4
1,070.2
77,733
4
3,066
43.0
3,411,857
0.0280
31.5
1,101.7
91,505
3
3,066
30.5
1,798,007
0.0147
16.6
1,118.2
103,372
2
3,097
18.0
679,242
0.0056
6.3
1,124.5
120,694
30,394

121,937,234
1.00
1124.5
Values in column 4 based on exponent k=1.865.
1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN.
4.1.5.2 Accidental torsion and orthogonal loading effects. Where using the ELF method as the basis
for structural design, two effects must be added to the direct lateral forces shown in Table 4.14. The first
of these effects accounts for the fact that an earthquake can produce inertial forces that act in any
direction. For SDC D, E and F buildings, Standard Section 12.5 requires that the structure be
investigated for forces that act in the direction that causes the critical load effects. Since this direction
is not easily defined, the Standard allows the analyst to load the structure with 100 percent of the seismic
force in one direction (along the X axis, for example) simultaneous with the application of 30 percent of
the force acting in the orthogonal direction (along the Y axis).
The other requirement is that the structure be modeled with additional forces to account for uncertainties
in the location of center of mass and center of rigidity, uneven yielding of vertical systems and the
possibility of torsional components of ground motion. For torsionally regular buildings, this requirement,
given in Standard Section 12.8.4.2, can be satisfied by applying the equivalent lateral force at an
accidental eccentricity, where the eccentricity is equal to 5 percent of the overall dimension of the
structure in the direction perpendicular to the line of the application of force. For torsionally irregular
structures in SDC C, D, E, or F, Standard Section 12.8.4.3 requires that the accidental eccentricity be
amplified (although the amplification factor may be 1.0).
According to Standard Table 12.31, a torsional irregularity exists if:
where ëmax is the maximum story drift at the edge of the floor diaphragm and avg is the average drift at
the center of the diaphragm (see Standard Figure 12.81). If the ratio of drifts is greater than 1.4, the
torsional irregularity is referred to as extreme. In computing the drifts, the structure must be loaded
with the basic equivalent lateral forces applied at a 5 percent eccentricity.
For main loads acting in the X direction, displacements and drifts were determined on Grid Line D. For
the main loads acting in the Y direction, the story displacements on Grid Line 1 were used. Because of
the architectural setbacks, the locations for determining displacements associated with min and max are
not always vertically aligned. This situation is shown in Figure 4.17, where it is seen that three
displacement monitoring stations are required at Levels 5 and 9. The numerical values shown in
Figure 4.17 are discussed later in relation to Table 4.15b.
Figure 4.17 Drift monitoring stations for determination of torsional irregularity and torsional
amplification (deflections in inches, 1.0 in. = 25.4 mm)
The analysis of the structure for accidental torsion was performed using SAP2000. The same model was
used for ELF, modal response spectrum and modal response history analysis. The following approach
was used for the mathematical model of the structure:
1. The floor diaphragm was modeled with shell elements, providing nearly rigid behavior inplane.
2. Flexural, shear, axial and torsional deformations were included in all columns and beams.
3. Beamcolumn joints were modeled using centerline dimensions. This approximately accounts for
deformations in the panel zone.
4. Section properties for the girders were based on bare steel, ignoring composite action. This is a
reasonable assumption since most of the girders are on the perimeter of the building and are under
reverse curvature.
5. Except for those lateral loadresisting columns that terminate at Levels 5 and 9, all columns were
assumed to be fixed at their base.
6. The basement walls and grade level slab were explicitly modeled using 4node shell elements.
This was necessary to allow the interior columns to continue through the basement level. No
additional lateral restraint was applied at the grade level; thus, the basement level acts as a very
stiff first floor of the structure. This basement level was not relevant for the ELF analysis, but it
did influence the modal response spectrum and modal response history analyses as described in
later sections of this example
7. Pdelta effects were not included in the mathematical model. These effects are evaluated
separately using the procedures provided in Standard Section 12.8.7.
The results of the accidental torsion analysis are shown in Tables 4.15a and 4.15b. For loading in the
X direction, there is no torsional irregularity because all drift ratios ( max/ avg) are less than 1.2. For
loading in the Y direction, the largest ratio of maximum to average story drift is 1.24 at Level 9 of the
building. Hence, this structure has a Type 1 torsional irregularity, but only marginally so. See Figure
4.17 for the source of the dual displacement values shown for Levels 9 and 5 in Table 4.15b.
Even though the torsional irregularity is marginal, Section 12.8.4.3 of the Standard requires that torsional
amplification factors be determined for this SDC D building. The results for these calculations, which are
based on story displacement, not drift, are presented in Tables 4.16a and 4.16b for the main load applied
in the X and Y directions, respectively. As may be observed, the calculated amplification factors are
significantly less than 1.0 at all levels for both directions of loading.
Table 4.15a Computation for Torsional Irregularity with ELF Loads Acting in X Direction and
Torsional Moment Applied Counterclockwise
Level
1
(in.)
2
(in.)
1
(in.)
2
(in.)
avg
(in.)
max
(in.)
max/ avg
Irregularity
R
7.27
6.15
0.34
0.29
0.31
0.34
1.08
None
12
6.93
5.87
0.48
0.42
0.45
0.48
1.07
None
11
6.44
5.45
0.60
0.51
0.55
0.60
1.07
None
10
5.85
4.93
0.66
0.56
0.61
0.66
1.08
None
9
5.19
4.37
0.65
0.54
0.59
0.65
1.10
None
8
4.54
3.84
0.69
0.58
0.64
0.69
1.09
None
7
3.84
3.26
0.70
0.59
0.65
0.70
1.09
None
6
3.14
2.67
0.69
0.58
0.63
0.69
1.09
None
5
2.46
2.09
0.60
0.50
0.55
0.60
1.09
None
4
1.86
1.60
0.59
0.50
0.55
0.59
1.08
None
3
1.27
1.10
0.58
0.49
0.53
0.58
1.08
None
2
0.69
0.61
0.69
0.61
0.65
0.69
1.06
None
1.0 in. = 25.4 mm.
Table 4.15b Computation for Torsional Irregularity with ELF Loads Acting in Y Direction and
Torsional Moment Applied Clockwise
Level
1
(in.)
2
(in.)
1
(in.)
2
(in.)
avg
(in.)
max
(in.)
max/ avg
Irregularity
R
5.19
4.77
0.15
0.14
0.15
0.15
1.03
None
12
5.03
4.63
0.25
0.23
0.24
0.25
1.03
None
11
4.79
4.40
0.31
0.29
0.30
0.31
1.04
None
10
4.48
4.11
0.38
0.34
0.36
0.38
1.06
None
9
4.10
3.77,
3.55
0.46
0.28
0.37
0.46
1.24
Irregularity
8
3.64
3.26
0.54
0.36
0.45
0.54
1.20
None
7
3.09
2.90
0.56
0.39
0.47
0.56
1.18
None
6
2.53
2.51
0.60
0.42
0.51
0.60
1.18
None
5
1.93,
1.95
2.09
0.41
0.47
0.44
0.47
1.06
None
4
1.53
1.62
0.47
0.50
0.48
0.50
1.03
None
3
1.07
1.12
0.47
0.50
0.48
0.50
1.03
None
2
0.60
0.63
0.60
0.63
0.61
0.63
1.03
None
1.0 in. = 25.4 mm.
Table 4.16a Amplification Factor Ax for Accidental Torsional Moment Loads Acting
in the X Direction and Torsional Moment Applied Counterclockwise
Level
1
(in.)
2
(in.)
avg
(in.)
max
(in.)
Ax calculated
Ax used
R
7.27
6.15
6.71
7.27
0.81
1.00
12
6.93
5.87
6.40
6.93
0.81
1.00
11
6.44
5.45
5.95
6.44
0.82
1.00
10
5.85
4.93
5.39
5.85
0.82
1.00
9
5.19
4.37
4.78
5.19
0.82
1.00
8
4.54
3.84
4.19
4.54
0.82
1.00
7
3.84
3.26
3.55
3.84
0.81
1.00
6
3.14
2.67
2.90
3.14
0.81
1.00
5
2.46
2.09
2.27
2.46
0.81
1.00
4
1.86
1.60
1.73
1.86
0.80
1.00
3
1.27
1.10
1.18
1.27
0.80
1.00
2
0.69
0.61
0.65
0.69
0.79
1.00
1.0 in. = 25.4 mm.
Table 4.16b Amplification Factor Ax for Accidental Torsional Moment Loads
Acting in the Y Direction and Torsional Moment applied Clockwise
Level
1
(in.)
2
(in.)
avg
(in.)
max
(in.)
Ax calculated
Ax used
R
5.19
4.77
4.98
5.19
0.75
1.00
12
5.03
4.63
4.83
5.03
0.75
1.00
11
4.79
4.40
4.59
4.79
0.76
1.00
10
4.48
4.11
4.29
4.48
0.76
1.00
9
4.10
3.55
3.82
4.10
0.80
1.00
8
3.64
3.26
3.45
3.64
0.77
1.00
7
3.09
2.90
3.00
3.09
0.74
1.00
6
2.53
2.51
2.52
2.53
0.70
1.00
5
1.95
2.09
2.02
2.09
0.74
1.00
4
1.53
1.62
1.58
1.62
0.73
1.00
3
1.07
1.12
1.10
1.12
0.73
1.00
2
0.60
0.63
0.61
0.63
0.73
1.00
1.0 in. = 25.4 mm.
4.1.5.3 Drift and Pdelta effects. Using the basic structural configuration shown in Figure 4.11 and the
equivalent lateral forces shown in Table 4.14, the total story deflections were computed as described in
the previous section. In this section, story drifts are computed and compared to the allowable drifts
specified by the Standard.
For structures with significant torsional effects , Standard Section 12.12.1 requires that the maximum
drifts include torsional effects, meaning that the accidental torsion, amplified as appropriate, must be
included in the drift analysis. The same section of the Standard requires that deflections used to compute
drift should be taken at the edges of the structure if the structure is torsionally irregular. For torsionally
regular buildings, the drifts may be based on deflections at the center of mass of adjacent levels.
As the structure under consideration is only marginally irregular in torsion, the lateral loads were placed
at the center of mass and total drifts are based on center of mass deflections and not deflections at the
edges of the floor plate. Using the centers of mass of the floor plates to compute story drift is awkward
where the centers of mass of the upper and lower floor plates are not aligned vertically. For this reason,
the story drift is computed as the difference between displacements of the center of mass of the upper
level diaphragm and the displacement at a point on the lower diaphragm which is located directly below
the center of mass of the upper level diaphragm. Note that computation of drift in this manner has been
adopted in ASCE 710 Section 12.86.
The values in Column 1 of Tables 4.17 and 4.18 are the total story displacements ( at the center of
mass of the story as reported by SAP2000 and the values in Column 2 are the story drifts ( computed
from these numbers in the manner described earlier. The true elastic amplified story drift, which by
assumption is equal to Cd (= 5.5) times the SAP2000 drift, is shown in Column 3. As discussed above in
Section 4.1.5.1, the values in Column 4 are multiplied by 0.568 to scale the results to the base shear
computed using Standard Equation 12.83.
The allowable story drift of 2.0 percent of the story height per Standard Table 12.121 is shown in
Column 5. (Recall that this building is assigned to Occupancy Category II.) It is clear from Tables 4.17
and 4.18 that the allowable drift is not exceeded at any level. It is also evident that the allowable drifts
would not have been exceeded even if accidental torsion effects were included in the drift calculations,
with the drift determined at the edge of the building.
Table 4.17 ELF Drift for Building Responding in X Direction
Level
1
Total drift from
SAP2000
(in.)
2
Story drift from
SAP2000
(in.)
3
Amplified story
drift
(in.)
4
Amplified drift
times 0.568
(in.)
5
Allowable drift
(in.)
R
6.67
0.32
1.74
0.99
3.00
12
6.35
0.45
2.48
1.41
3.00
11
5.90
0.56
3.07
1.75
3.00
10
5.34
0.62
3.39
1.92
3.00
9
4.73
0.58
3.20
1.82
3.00
8
4.15
0.63
3.47
1.97
3.00
7
3.52
0.64
3.54
2.01
3.00
6
2.87
0.63
3.47
1.97
3.00
5
2.24
0.54
2.95
1.67
3.00
4
1.71
0.54
2.97
1.69
3.00
3
1.17
0.53
2.90
1.65
3.00
2
0.64
0.64
3.51
2.00
4.32
Column 4 adjusts for Standard Eq. 12.83 (for drift) vs 12.85 (for strength).
1.0 in. = 25.4 mm.
Table 4.18 ELF Drift for Building Responding in Y Direction
Level
1
Total drift from
SAP2000
(in.)
2
Story drift from
SAP2000
(in.)
3
Amplified story
drift
(in.)
4
Amplified drift
times 0.568
(in.)
5
Allowable drift
(in.)
R
4.86
0.15
0.81
0.46
3.00
12
4.71
0.24
1.30
0.74
3.00
11
4.47
0.30
1.64
0.93
3.00
10
4.17
0.36
1.96
1.11
3.00
9
3.82
0.37
2.05
1.16
3.00
8
3.44
0.46
2.54
1.44
3.00
7
2.98
0.48
2.64
1.50
3.00
6
2.50
0.48
2.62
1.49
3.00
5
2.03
0.45
2.49
1.42
3.00
4
1.57
0.48
2.66
1.51
3.00
3
1.09
0.48
2.64
1.50
3.00
2
0.61
0.61
3.35
1.90
4.32
Column 4 adjusts for Standard Eq. 12.83 (for drift) versus Eq. 12.85 (for strength).
1.0 in. = 25.4 mm.
4.1.5.3.1 Using ELF forces and drift to compute accurate period. Before continuing with the
example, it is advisable to use the computed drifts to more accurately estimate the fundamental periods of
vibration of the building. This will serve as a check on the exact periods computed by eigenvalue
extraction in SAP2000. A Rayleigh analysis will be used to estimate the periods. This procedure, which
usually is very accurate, is derived as follows:
The exact frequency of vibration (a scalar), in units of radians/second, is found from the following
eigenvalue equation:
where K is the structure stiffness matrix, M is the (diagonal) mass matrix and is a vector containing
the components of the mode shape associated with .
If an approximate mode shape ë is used instead of , where ë is the deflected shape under the
equivalent lateral forces F, the frequency can be closely approximated. Making the substitution of
ë for , premultiplying both sides of the above equation by ëT (the transpose of the displacement
vector), noting that F = Kë and M = W/g, the following is obtained:
where W is a vector containing the story weights and g is the acceleration due to gravity (a scalar).
After rearranging terms, this gives:
Using the relationship between period and frequency,
Using F from Table 4.14 and ë from Column 1 of Tables 4.17 and 4.18, the periods of vibration are
computed as shown in Tables 4.19 and 4.110 for the structure loaded in the X and Y directions,
respectively. As may be seen from the tables, the X direction period of 2.85 seconds and the Ydirection
period of 2.56 seconds are significantly greater than the approximate period of Ta = 1.59 seconds and also
exceed the upper limit on period of CuTa = 2.23 seconds.
Table 4.19 Rayleigh Analysis for X Direction Period of Vibration
Level
Drift,
in.)
Force, F
(kips)
Weight, W
(kips)
F
(in.kips)
W/g
(in.kipssec2)
R
6.67
186.9
1,657
1,247
191
12
6.35
154.0
1,596
979
167
11
5.90
129.9
1,596
767
144
10
5.34
107.6
1,596
575
118
9
4.73
186.3
3,403
881
197
8
4.15
100.8
2,331
418
104
7
3.52
77.0
2,331
271
75
6
2.87
56.2
2,331
162
50
5
2.24
71.4
4,324
160
56
4
1.71
31.5
3,066
54
23
3
1.17
16.6
3,066
19
11
2
0.64
6.3
3,097
4
3
5,536
1,138
= (5,536/1,138)0.5 = 2.21 rad/sec. T = 2 / = 2.85 sec.
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
Table 4.110 Rayleigh Analysis for Y Direction Period of Vibration
Level
Drift,
in.)
Force, F
(kips)
Weight, W
(kips)
F
W/g
R
4.86
186.9
1,657
908
101
12
4.71
154.0
1,596
725
92
11
4.47
129.9
1,596
581
83
10
4.17
107.6
1,596
449
72
9
3.82
186.3
3,403
711
128
8
3.44
100.8
2,331
347
72
7
2.98
77.0
2,331
230
54
6
2.50
56.2
2,331
141
38
5
2.03
71.4
4,324
145
46
4
1.57
31.5
3,066
49
20
3
1.09
16.6
3,066
18
9
2
0.61
6.3
3,097
4
3
4,307
716
= (4,307/716)0.5 = 2.45 rad/sec. T = 2 / = 2.56 sec.
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
4.1.5.3.2 Pdelta effects. Pdelta effects are computed for the X direction response in Table 4.111. The
last column of the table shows the story stability ratio computed according to Standard Equation 12.816 :
Standard Equation 12.817 places an upper limit on :
where is the ratio of shear demand to shear capacity for the story. Conservatively taking = 1.0 and
using Cd = 5.5, max = 0.091.
The terms in Table 4.111 are taken from the fourth column of Table 4.17 because these are consistent
with the ELF story shears of Table 4.14 and thereby represent the true lateral stiffness of the system. (If
0.568 times the story drifts were used, then 0.568 times the story shears also would need to be used.
Hence, the 0.568 factor would cancel out since it would appear in both the numerator and denominator.)
The deflections used in Pdelta stability ratio calculations must include the deflection amplifier Cd.
The live load PL in Table 4.111 is based on a 20 psf uniform live load over 100 percent of the floor and
roof area. This live load is somewhat conservative because Section 12.8.7 of the Standard states that the
gravity load should be the total vertical design load . For a 50 psf live load for office buildings, a live
load reduction factor of 0.4 would be applicable for each level (see Standard Sec. 4.8), producing a
reduced live load of 20 psf at the floor levels. This could be further reduced by a factor of 0.5 as allowed
by Section 2.3.2, bringing the effective live load to 10 psf. This value is close to the mean survey live
load of 10.9 psf for office buildings, as listed in Table C4.2 of the Standard. Several publications,
including ASCE 41 include 25 percent of the unreduced live load in Pdelta calculations. This would
result in a 12.5 psf live load for the current example.
Table 4.111 Computation of Pdelta Effects for X Direction Response
Level
hsx
(in.)
(in.)
PD
(kips)
PL
(kips)
PT
(kips)
PX
(kips)
VX
(kips)
X
R
150
1.74
1,656.5
315.0
1,971.5
1,971.5
186.9
0.022
12
150
2.48
1,595.8
315.0
1,910.8
3,882.3
340.9
0.034
11
150
3.07
1,595.8
315.0
1,910.8
5,793.1
470.8
0.046
10
150
3.39
1,595.8
315.0
1,910.8
7,703.9
578.4
0.055
9
150
3.20
3,403.0
465.0
3,868.0
11,571.9
764.7
0.059
8
150
3.47
2,330.8
465.0
2,795.8
14,367.7
865.8
0.070
7
150
3.54
2,330.8
465.0
2,795.8
17,163.5
942.5
0.078
6
150
3.47
2,330.8
465.0
2,795.8
19,959.3
998.8
0.084
5
150
2.95
4,323.8
615.0
4,938.8
24,898.1
1,070.2
0.083
4
150
2.97
3,066.1
615.0
3,681.1
28,579.2
1,101.7
0.093
3
150
2.90
3,066.1
615.0
3,681.1
32,260.3
1,118.2
0.101
2
216
3.51
3,097.0
615.0
3,712.0
35,972.3
1,124.5
0.095
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
The stability ratio just exceeds 0.091 at Levels 2 through 4. However, the live loads were somewhat
conservative and was very conservatively taken as 1.0. Because a more refined analysis would provide
somewhat lower live loads and a lower value of , we will proceed assuming that Pdelta effects are not a
problem for this structure. Calculations for Pdelta effects under Y direction loading gave no story
stability ratios greater than 0.091 and for brevity, those results are not included herein.
It is important to note that for this structure, Pdelta effects are a potential issue even though drift limits
were easily satisfied. This is often the case when drift limits are based on lateral loads that have been
computed using the computed period of vibration (without the CuTa limit, or without the use of
Equation 12.8.5). It is the authors experience that this is a typical situation in the analysis of steel special
steel moment frame systems.
Part 1 of the Provisions recommends that a significant change be made to the current Pdelta approach.
In the recommended approach, Equation 12.816 is still used to determine the magnitude of Pdelta
effects. However, if the stability ratio at any level is greater than 0.1, the designer must either redesign
the building such that the stability ratio is less than 0.1, or must perform a static pushover analysis and
demonstrate that the slope of the pushover curve of the structure is continuously positive up to the target
displacement computed in accordance with the requirements of ASCE 41.
4.1.5.4 Computation of member forces. Before member forces may be computed, the proper load
cases and combinations of load must be identified such that all critical seismic effects are captured in the
analysis.
4.1.5.4.1 Orthogonal loading effects and accidental torsion. For SDC D structures with a Type 5
horizontal structural irregularity, Section 12.5.3 of the Standard requires that orthogonal load effects be
considered. For the purposes of this example, it is assumed that such an irregularity does exist because
the layout of the frames is not symmetric. (ASCE 710 has eliminated nonsymmetry as a trigger for
invoking the nonparallel system horizontal irregularity. However, as the structural system under
consideration has several intersecting frames, it would be advisable to perform the orthogonal load
analysis as required under Section 12.5.4 of both the 2005 and 2010 versions of the Standard.)
When orthogonal load effects are included in the analysis, four directions of seismic force (+X, X,
+Y, Y) must be considered and for each direction of force, there are two possible directions in which the
accidental eccentricity can apply (causing positive or negative torsion). This requires a total of eight
possible combinations of direct force plus accidental torsion. Where the 30 percent orthogonal loading
rule is applied (see Standard Sec. 12.5.3 Item a ), the number of load combinations increases to 16
because, for each direct application of load, a positive or negative orthogonal loading can exist.
Orthogonal loads are applied without accidental eccentricity.
Figure 4.18 illustrates the basic possibilities of application of load. Although this figure shows 16
different load conditions, it may be observed that eight of these conditions 7, 8, 5, 6, 15, 16, 13 and
14 are negatives (opposite signs throughout) of conditions 1, 2, 3, 4, 9, 10, 11 and 12, respectively.
Figure 4.18 Basic load conditions used in ELF analysis
4.1.5.4.2 Load combinations. The basic load combinations for this structure are designated in Chapter 2
of the Standard. Two sets of combinations are provided: one for strength design and the other for
allowable stress design. The strengthbased combinations that are related to seismic effects are the
following:
1.2D + 1.0E + 1.0L + 0.2S
0.9D + 1.0E+1.6H
The factor on live load, L, may be reduced to 0.5 if the nominal live load is less than 100 psf. The load
due to lateral earth pressure, H, may need to be considered when designing the basement walls.
Section 12.4 of the Standard divides the earthquake load, E, into two components, Eh and Ev, where the
subscripts h and v represent horizontal and vertical seismic effects, respectively. These components are
defined as follows:
Eh = QE
Ev = 0.2SDSD
where QE is the earthquake load effect and is a redundancy factor, described later.
When the above components are substituted into the basic load combinations, the load combinations for
strength design with a factor of 0.5 used for live load and with the H load removed are as follows:
(1.2 + 0.2SDS)D+ QE + 0.5L + 0.2S
(0.90.2SDS)D + QE
Using SDS = 0.833 and assuming the snow load is negligible in Stockton, California, the basic load
combinations for strength design become:
1.37D + 0.5L + QE
0.73D + QE
The redundancy factor, is determined in accordance with Standard Section 12.3.4. This factor will
take a value of 1.0 or 1.3, with the value depending on a variety of conditional tests. None of the
conditions specified in Section 12.3.4.1 are applicable, so may not be automatically taken as 1.0, and
the more detailed evaluation specified in Section 12.3.4.2 is required. Subparagraph b of
Section 12.3.4.2 applies to this building (because of the plan irregularities) and therefore, the evaluation
described in the second row of Table 12.33 must be performed. It can be seen from inspection that the
removal of a single beam from the perimeter moment frames will not cause a reduction in strength of
33 percent, nor will an extreme torsional irregularly result from the removal of the beam. Hence, the
redundancy factor may be taken as 1.0 for this structure.
Hence, the final load conditions to be used for design are as follows:
1.37D + 0.5L + 1.0QE
0.73D + 1.0QE
The first load condition will produce the maximum negative moments (tension on the top) at the face of
the supports in the girders and maximum compressive forces in columns. The second load condition will
produce the maximum positive moments (or minimum negative moment) at the face of the supports of the
girders and maximum tension (or minimum compression) in the columns. In addition to the above load
condition, the gravityonly load combinations as specified in the Standard also must be checked. Due to
the relatively short spans in the moment frames, however, it is not expected that the nonseismic load
combinations will control.
4.1.5.4.3 Setting up the load combinations in SAP2000. The load combinations required for the
analysis are shown in Table 4.112.
It should be noted that 32 different load combinations are required only if one wants to maintain the signs
in the member force output, thereby providing complete design envelopes for all members. As mentioned
later, these signs are lost in response spectrum analysis and as a result, it is possible to capture the effects
of dead load plus live load plusorminus earthquake load in a single SAP2000 run containing only four
load combinations.
Table 4.112 Seismic and Gravity Load Combinations as Run on SAP2000
Run
Combination
Lateral*
Gravity
A
B
1 (Dead)
2 (Live)
One
1
[1]
1.37
0.5
2
[1]
0.73
0.0
3
[7]
1.37
0.5
4
[7]
0.73
0.0
5
[2]
1.37
0.5
6
[2]
0.73
0.0
7
[8]
1.37
0.5
8
[8]
0.73
0.0
Two
1
[3]
1. 37
0.5
2
[3]
0.73
0.0
3
[4]
1. 37
0.5
4
[4]
0.73
0.0
5
[5]
1. 37
0.5
6
[5]
0.73
0.0
7
[6]
1. 37
0.5
8
[6]
0.73
0.0
Three
1
[9]
1. 37
0.5
2
[9]
0.73
0.0
3
[10]
1. 37
0.5
4
[10]
0.73
0.0
5
[15]
1. 37
0.5
6
[15]
0.73
0.0
7
[16]
1. 37
0.5
8
[16]
0.73
0.0
Four
1
[11]
1. 37
0.5
2
[11]
0.73
0.0
3
[12]
1. 37
0.5
4
[12]
0.73
0.0
5
[13]
1. 37
0.5
6
[13]
0.73
0.0
7
[14]
1. 37
0.5
8
[14]
0.73
0.0
*Numbers in brackets [ ] in represent load cases shown in Figure 4.18.
4.1.5.4.4 Member forces. For this portion of the analysis, the earthquake shears in the girders along
Gridline 1 are computed. This analysis considers 100 percent of the X direction forces applied in
combination with 30 percent of the (positive or negative) Y direction forces. The accidental torsion is not
included and will be considered separately. The results of the member force analysis are shown in
Figure 4.19a. In a later part of this example, the girder shears are compared to those obtained from
modal response spectrum and modal response history analyses.
Beam shears in the same frame, due to accidental torsion only, are shown in Figure 4.19b. The
eccentricity was set to produce clockwise torsions (when viewed from above) on the floor plates. These
shears would be added to the shears shown in Figure 4.19a to produce the total seismic beam shears in
the frame. The same torsional shears (from Table 4.19b) will be used in the modal response spectrum
and modal response history analyses.
8.99
10.3
10.3
R12
17.3
18.9
19.0
1211
27.7
28.1
29.5
1110
33.4
33.1
35.7
109
34.8
34.7
32.2
30.3
13.2
98
36.4
35.9
33.9
37.8
23.7
87
41.2
40.1
38.4
41.3
25.8
76
43.0
40.6
39.3
41.7
26.4
65
14.1
33.1
33.8
36.5
35.5
37.2
24.9
54
24.1
37.9
32.0
34.6
33.9
34.9
23.9
43
24.1
37.0
33.3
35.1
34.6
35.4
24.6
32
22.9
36.9
34.1
35.3
34.9
35.9
23.3
2  G
Figure 4.19a Seismic shears (kips) in girders on Frame Line 1 as computed using ELF
analysis (analysis includes orthogonal loading but excludes accidental torsion)
0.56
0.56
0.58
R12
1.13
1.13
1.16
1211
1.87
1.77
1.89
1110
2.26
2.12
2.34
109
2.07
1.97
1.89
1.54
0.76
98
1.89
1.81
1.72
1.84
1.36
87
2.17
2.05
1.99
2.06
1.49
76
2.29
2.09
2.04
2.09
1.51
65
0.59
1.33
1.65
1.72
1.68
1.72
1.27
54
1.04
1.45
1.34
1.41
1.39
1.42
1.07
43
1.07
1.51
1.45
1.48
1.45
1.47
1.10
32
1.04
1.58
1.52
1.54
1.53
1.56
1.06
2  G
Figure 4.19b Seismic shears in girders (kips) from clockwise torsion only
4.1.6 Modal Response Spectrum Analysis
The first step in the modal response spectrum analysis is the computation of the natural mode shapes and
associated periods of vibration. Using the structural masses from Table 4.14 and the same mathematical
model as used for the ELF and the Rayleigh analyses, the mode shapes and frequencies are automatically
computed by SAP2000. This mathematical model included the basement as a separate level. (See
Section 4.1.5.2 of this example for a description of the mathematical model used in the analysis). The
basement walls were fixed at the base but were unrestrained at grade level. Thus, the basement level is
treated as a separate story in the analysis. However, the lateral stiffness of the basement level is
significantly greater than that of the upper levels and this causes complications when interpreting the
requirements of Standard Section 12.9.1. As shown later, the explicit modeling of the basement can also
lead to some unexpected results in the modal response history analysis of the structure.
The periods of vibration for the first 12 modes, computed from an eigenvalue analysis, are summarized in
the second column of Table 4.113. The first eight mode shapes are shown in Figure 4.110. The first
mode period, 2.87 seconds, corresponds to vibration primarily in the X direction and the second period,
2.60 seconds, corresponds to vibration in the Y direction. The third mode, with a period of 1.57 seconds,
is almost purely torsion. The directionality of the modes may be inferred from the effective mass values
shown in Columns 3 through 5 of the tables, as well as from the mode shapes. There is very little lateral
torsional coupling in any of the first 12 modes, which is somewhat surprising because of the shifted
centers of mass associated with the plan offsets.
The X and Ytranslation periods of 2.87 and 2.60 seconds, respectively, are somewhat longer than the
upper limit on the approximate period, CuTa, of 2.23 seconds.
The first and second mode periods are virtually identical to the periods compute by Rayleigh analysis
(2.85 and 2.56 seconds in the X and Y directions, respectively). The closeness of the Rayleigh and
eigenvalue periods for this building arises from the fact that the first and second modes of vibration act
primarily along the orthogonal axes. Had the first and second modes not acted along the orthogonal axes,
the Rayleigh periods (based on loads and displacements in the X and Y directions) would have been
somewhat less accurate.
Standard Section 12.9.1 specifies that the analysis shall include a sufficient number of modes to obtain a
combined modal mass participation of at least 90 percent of the total mass in each of the orthogonal
horizontal directions of response considered by the model . Usually, this is a straightforward requirement
and the first twelve modes would be sufficient for a 12story building. For this building, however, twelve
modes capture only about 82 percent of the X and Y direction mass. (The effective mass as a fraction of
total mass is shown in brackets [ ] in Columns 3 through 5 of Tables 4.113 and 4.114.) Most of the
remaining effective mass is in the gradelevel slab and in the basement walls. This mass does not show
up until Mode 112 in the Y direction and Mode 118 in the X direction. This is shown in Table 12.114,
which provides the periods and effective modal masses in Modes 108 through 119. The intermediate
modes (13 through 107) represent primarily vertical vibration of various portions of the floor diaphragms.
Analyzing the system with 120 or more modes might provide useful information on the response of the
basement level, including shears through the basement and total system base shears at the base of the
basement. However, there would be some difficulty in interpreting the results because the model did not
include subgrade soil that would be in contact with the basement walls and which would absorb part of
the base shear. Additionally, the computed response of the upper 12 levels of the building, which is the
main focus of this analysis, is virtually identical for the 12 and the 120 mode analyses. For this reason,
the modal response spectrum analysis discussed in this example was run with only the first 12 modes
listed in Table 4.113.
Mode 1: T = 2.87 sec
Mode 2: T = 2.60 sec
Mode 3: T = 1.57 sec
Mode 4: T = 1.15 sec
Mode 5: T = 0.98 sec
Mode 6: T = 0.71 sec
Mode 7: T = 0.68 sec
Mode 8: T = 0.57 sec
Figure 4.110 First eight mode shapes
Table 4.113 Computed Periods and Effective Mass Factors (Lower Modes)
Mode
Period
(sec.)
Effective Mass Factor [Accum Mass Factor]
X Translation
Y Translation
Z Rotation
1
2.87
0.6446 [0.64]
0.0003 [0.00]
0.0028 [0.00]
2
2.60
0.0003 [0.65]
0.6804 [0.68]
0.0162 [0.02]
3
1.57
0.0035 [0.65]
0.0005 [0.68]
0.5806 [0.60]
4
1.15
0.1085 [0.76]
0.0000 [0.68]
0.0000 [0.60]
5
0.975
0.0000 [0.76]
0.0939 [0.78]
0.0180 [0.62]
6
0.705
0.0263 [0.78]
0.0000 [0.78]
0.0271 [0.64]
7
0.682
0.0056 [0.79]
0.0006 [0.79]
0.0687 [0.71]
8
0.573
0.0000 [0.79]
0.0188 [0.79]
0.0123 [0.73]
9
0.434
0.0129 [0.80]
0.0000 [0.79]
0.0084 [0.73]
10
0.387
0.0048 [0.81]
0.0000 [0.79]
0.0191 [0.75]
11
0.339
0.0000 [0.81]
0.0193 [0.81]
0.0010 [0.75]
12
0.300
0.0089 [0.82]
0.0000 [0.81]
0.0003 [0.75]
Table 4.114 Computed Periods and Effective Mass Factors (Higher Modes)
Mode
Period
(sec.)
Effective Mass Factor [Accum Effective Mass]
X Translation
Y Translation
Z Rotation
108
0.0693
0.0000 [0.83]
0.0000 [0.83]
0.0000 [0.79]
109
0.0673
0.0000 [0.83]
0.0000 [0.83]
0.0000 [0.79]
110
0.0671
0.0000 [0.83]
0.0354 [0.86]
0.0000 [0.79]
111
0.0671
0.0000 [0.83]
0.0044 [0.87]
0.0000 [0.79]
112
0.0669
0.0000 [0.83]
0.1045 [0.97]
0.0000 [0.79]
113
0.0663
0.0000 [0.83]
0.0000 [0.97]
0.0000 [0.79]
114
0.0646
0.0000 [0.83]
0.0000 [0.97]
0.0000 [0.79]
115
0.0629
0.0000 [0.83]
0.0000 [0.97]
0.0000 [0.79]
116
0.0621
0.0008 [0.83]
0.0010 [0.97]
0.0000 [0.79]
117
0.0609
0.0014 [0.83]
0.0009 [0.97]
0.0000 [0.79]
118
0.0575
0.1474 [0.98]
0.0000 [0.97]
0.0035 [0.80]
119
0.0566
0.0000 [0.98]
0.0000 [0.97]
0.0000 [0.80]
4.1.6.1 Response spectrum coordinates and computation of modal forces. The coordinates of the
response spectrum are based on Standard Section 11.4.5. This spectrum consists of three parts (for
periods less than TL = 8.0 seconds) as follows:
For periods less than T0:
For periods between T0 and TS:
For periods greater than TS:
where T0 = 0.2SD1/SDS and TS = SD1/SDS.
Using SDS = 0.833 and SD1 = 0.373, TS = 0.448 seconds and T0 = 0.089 seconds. The computed response
spectrum coordinates for several period values are shown in Table 4.115 and the response spectrum,
shown with and without the I/R = 1/8 modification, is plotted in Figure 4.111. The spectrum does not
include the high period limit on Cs (0.044ISDS), which controlled the ELF base shear for this structure and
which ultimately will control the scaling of the results from the response spectrum analysis. (Recall that
if the computed base shear falls below 85 percent of the ELF base shear, the computed response must be
scaled up such that the computed base shear equals 85 percent of the ELF base shear.)
Table 4.115 Response Spectrum Coordinates
Tm
(sec.)
Sa
Sa(I/R)
0.000
0.333
0.0416
0.089 (T0)
0.833
0.104
0.448 (TS)
0.833
0.104
1.000
0.373
0.0446
1.500
0.249
0.0311
2.000
0.186
0.0235
2.500
0.149
0.0186
3.000
0.124
0.0155
I = 1, R = 8.0.
Figure 4.111 Total acceleration response spectrum used in analysis
Using the response spectrum coordinates listed in Column 3 of Table 4.115, the response spectrum
analysis was carried out using SAP2000. As mentioned above, the first 12 modes of response were
computed and superimposed using the CQC approach. A modal damping ratio of 5 percent of critical was
used in the CQC calculations.
Two analyses were carried out. The first directed the seismic motion along the X axis of the structure and
the second directed the motion along the Y axis. Combinations of these two loadings plus accidental
torsion are discussed later.
4.1.6.1.1 Dynamic base shear. After specifying member groups, SAP2000 automatically computes the
CQC story shears. Groups were defined such that total shears would be obtained for each story of the
structure. The shears at the base of the first story above grade are reported as follows:
X direction base shear = 438.1 kips
Y direction base shear = 492.8 kips
These values are much lower that the ELF base shear of 1,124 kips. Recall that the ELF base shear was
controlled by Standard Equation 12.85. The modal response spectrum shears are less than the ELF
shears because the fundamental periods of the structure used in the response spectrum analysis
(2.87 seconds and 2.6 seconds in the X and Y directions, respectively) are greater than the upper limit
empirical period, CuTa, of 2.23 seconds and because the response spectrum of Figure 4.111 does not
include the minimum base shear limit imposed by Standard Equation 12.85.
According to Standard Section 12.9.4, the base shears from the modal response spectrum analysis must
not be less than 85 percent of that computed from the ELF analysis. If the response spectrum shears are
lower than the ELF shear, then the computed shears must be scaled up such that the response spectrum
base shear is 85 percent of that computed from the ELF analysis.
Hence, the required scale factors are as follows:
X direction scale factor = 0.85(1124)/438.1 = 2.18
Y direction scale factor = 0.85(1124)/492.8 = 1.94
The computed and scaled story shears are as shown in Table 4.116. Since the base shears for the ELF
and the modal analysis are different (due to the 0.85 factor), direct comparisons cannot be made between
Table 4.116 and Table 4.14. However, it is clear that the vertical distribution of forces is somewhat
similar where computed by ELF and modal response spectrum.
Table 4.116 Story Shears from Modal Response Spectrum Analysis
Story
X Direction (SF = 2.18)
Y Direction (SF = 1.94)
Unscaled Shear
(kips)
Scaled Shear
(kips)
Unscaled Shear
(kips)
Scaled Shear
(kips)
R12
82.7
180
77.2
150
1211
130.9
286
132.0
256
1110
163.8
357
170.4
330
109
191.4
418
201.9
392
98
240.1
524
265.1
514
87
268.9
587
301.4
585
76
292.9
639
328.9
638
65
316.1
690
353.9
686
54
359.5
784
405.1
786
43
384.8
840
435.5
845
32
401.4
895
462.8
898
2G
438.1
956
492.8
956
1.0 kip = 4.45 kN.
4.1.6.2 Drift and Pdelta effects. According to Standard Section 12.9.4, the computed displacements
and drift (as based on the response spectrum of Figure 4.111) need not be scaled by the base shear factors
(SF) of 2.18 and 1.94 for the structure loaded in the X and Y directions, respectively. This provides
consistency with Section 12.8.6.2, which allows drift from an ELF analysis to be based on the computed
period without the upper limit CuTa.
Section 12.9.4.2 of ASCE 710 requires that drifts from a response spectrum analysis be scaled only if
Equation 12.86 controls the value of CS. In this example, Equation 12.85 controlled the base shear, so
drifts need not be scaled in ASCE 710.
In Tables 4.117 and 4.118, the story displacement from the response spectrum analysis, the story drift,
the amplified story drift (as multiplied by Cd = 5.5) and the allowable story drift are listed. As before the
story drifts represent the differences in the displacement at the center of mass of one level, and the
displacement at vertical projection of that point at the level below. These values were determined in each
mode and then combined using CQC. As may be observed from the tables, the allowable drift is not
exceeded at any level.
Table 4.117 Response Spectrum Drift for Building Responding in X Direction
Level
Total Drift from
R.S. Analysis
(in.)
Story Drift
(in.)
Story Drift
Cd
(in.)
Allowable
Story Drift
(in.)
R
2.23
0.12
0.66
3.00
12
2.10
0.16
0.89
3.00
11
1.94
0.19
1.03
3.00
10
1.76
0.20
1.08
3.00
9
1.56
0.18
0.98
3.00
8
1.38
0.19
1.06
3.00
7
1.19
0.20
1.08
3.00
6
0.99
0.20
1.08
3.00
5
0.80
0.18
0.97
3.00
4
0.62
0.19
1.02
3.00
3
0.43
0.19
1.05
3.00
2
0.24
0.24
1.34
4.32
1.0 in. = 25.4 mm.
Table 4.118 Response Spectrum Drift for Building Responding in Y Direction
Level
Total Drift from
R.S. Analysis
(in.)
Story Drift
(in.)
Story Drift
Cd
(in.)
Allowable
Story Drift
(in.)
R
1.81
0.06
0.32
3.00
12
1.76
0.09
0.49
3.00
11
1.67
0.11
0.58
3.00
10
1.56
0.12
0.67
3.00
9
1.44
0.13
0.70
3.00
8
1.31
0.16
0.87
3.00
7
1.15
0.17
0.91
3.00
6
0.99
0.17
0.92
3.00
5
0.92
0.17
0.93
3.00
4
0.65
0.19
1.04
3.00
3
0.46
0.20
1.08
3.00
2
0.26
0.26
1.44
4.32
1.0 in. = 25.4 mm.
According to Standard Section 12.9.6, Pdelta effects should be checked using the ELF method. This
implies that such effects should not be determined using the results from the modal response spectrum
analysis. Thus, the results already shown and discussed in Table 4.111 of this example are applicable.
Nevertheless, Pdelta effects can be assessed using the results of the modal response spectrum analysis if
the displacements, drifts and story shears are used as computed from the response spectrum analysis,
without the base shear scale factors. However, when computing the stability ratio, the drifts must include
the amplifier Cd (because of the presence of Cd in the denominator of Standard Equation 12.816). Using
this approach, Pdelta effects were computed for the X direction response as shown in Table 4.119. Note
that the stability factors are very similar to those given in Table 4.111. As with Table 4.111, the
stability factors from Table 4.119 exceed the limit ( max = 0.091) only at the bottom three levels of the
structure and are only marginally above the limit. Since the factor was conservatively set at 1.0 inch for
computing the limit, it is likely that a refined analysis for would indicate that Pdelta effects are not of
particular concern for this structure.
Table 4.119 Computation of Pdelta Effects for X Direction Response
Level
hsx
(in.)
(in.)
PD
(kips)
PL
(kips)
PT
(kips)
PX
(kips)
VX
(kips)
X
R
150
0.66
1,656.5
315.0
1,971.5
1,971.5
82.7
0.019
12
150
0.89
1,595.8
315.0
1,910.8
3,882.3
130.9
0.032
11
150
1.03
1,595.8
315.0
1,910.8
5,793.1
163.8
0.044
10
150
1.08
1,595.8
315.0
1,910.8
7,703.9
191.4
0.053
9
150
0.98
3,403.0
465.0
3,868.0
11,571.9
240.1
0.057
8
150
1.06
2,330.8
465.0
2,795.8
14,367.7
268.9
0.069
7
150
1.08
2,330.8
465.0
2,795.8
17,163.5
292.9
0.077
6
150
1.08
2,330.8
465.0
2,795.8
19,959.3
316.1
0.083
5
150
0.97
4,323.8
615.0
4,938.8
24,898.1
359.5
0.081
4
150
1.02
3,066.1
615.0
3,681.1
28,579.2
384.8
0.092
3
150
1.05
3,066.1
615.0
3,681.1
32,260.3
401.9
0.102
2
216
1.34
3,097.0
615.0
3,712.0
35,972.3
438.1
0.093
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
4.1.6.3 Torsion, orthogonal loading and load combinations. To determine member design forces, it is
necessary to add the effects of accidental torsion and orthogonal loading into the analysis. When
including accidental torsion in modal response spectrum analysis, there are generally two approaches that
can be taken:
Displace the center of mass of the floor plate plus or minus 5 percent of the plate dimension
perpendicular to the direction of the applied response spectrum. As there are four possible mass
locations, this will require four separate modal analyses for torsion with each analysis using a
different set of mode shapes and frequencies.
Compute the effects of accidental torsion by creating a load condition with the accidental story
torques applied as static forces. Member forces created by the accidental torsion are then added
directly to the results of the response spectrum analysis. As with the displaced mass method,
there are four possible ways to apply the accidental torsion: plus and minus torsion for primary
loads in the X and Y directions. Where scaling of the modal response spectrum design forces is
required, the torsional loading used for accidental torsion analysis should be multiplied by 0.85.
Each of the above approaches has advantages and disadvantages. The primary disadvantage of the first
approach is a practical one: most computer programs do not allow for the extraction of member force
maxima from more than one run where the different runs incorporate a different set of mode shapes and
frequencies. An advantage of the approach stipulated in Standard Section 12.9.5 is that accidental torsion
need not be amplified (when otherwise required by Standard Section 12.8.4.3) because the accidental
torsion effect is amplified within the dynamic analysis.
For structures that are torsionally regular and which will not require amplification of torsion, the second
approach may be preferred. A disadvantage of the approach is the difficulty of combining member forces
from a CQC analysis (all results positive), and a separate static torsion analysis (member forces have
positive and negative signs as appropriate).
In the analysis that follows, the second approach has been used because the structure has excellent
torsional rigidity, and amplification of accidental torsion is not required (all amplification factors = 1.0).
There are two possible methods for applying the orthogonal loading rule:
Run two separate response spectrum analyses, one in the X direction and one in the Y direction,
with CQC being used for modal combinations in each analysis. Using a direct sum, combine
100 percent of the scaled X direction results with 30 percent of the scaled Y direction results.
Perform a similar analysis using 100 percent of the scaled Y direction forces and 30 percent of
the scaled X direction forces. All seismic effects can be considered in only two dynamic load
cases (one response spectrum analysis in each direction) and two torsion cases (resulting from
loads applied at a 5 percent eccentricity in each direction). These are shown in Figure 4.112.
Run two separate response spectrum analyses, one in the X direction and one in the Y direction,
with CQC being used for modal combinations in each analysis. Using SRSS, combine
100 percent of the scaled X direction results with 100 percent of the scaled Y direction results
(Wilson, 2004).
Figure 4.112 Load combinations for response spectrum analysis
4.1.6.4 Member design forces. Earthquake shear forces in the beams of Frame 1 are given in
Figure 4.113. These member forces are based on 2.18 times the spectrum applied in the X direction and
1.94 times of the spectrum applied independently in the Y direction. Individual member forces from the
X and Y directions are obtained by CQC for that analysis and these forces are combined by SRSS. To
account of accidental torsion, the forces in Figure 4.113 should be added to 0.85 times the forces shown
in Figure 4.19b.
8.41
8.72
8.91
R12
14.9
15.6
15.6
1211
21.5
21.6
22.5
1110
24.2
24.0
25.8
109
23.3
23.3
21.8
20.0
8.9
98
23.7
23.5
22.4
24.5
15.8
87
26.9
26.1
25.4
26.7
17.2
76
28.4
26.8
26.2
27.3
17.8
65
10.1
22.4
23.6
25.3
24.8
25.5
17.0
54
17.4
26.6
23.7
24.9
24.6
25.1
17.0
43
18.5
27.5
25. 9
26.6
26.4
26.8
18.5
32
18.5
29.1
27.8
28.2
28.1
28.7
18.5
2  G
Figure 4.113 Seismic shears in girders (kips) as computed using response spectrum analysis
(analysis includes orthogonal loading but excludes accidental torsion)
4.1.7 Modal Response History Analysis
Before beginning this section, it is important to note that the analysis performed here is based on
the requirements of Chapter 16 of ASCE 710. This version contains several important updates
that removed inconstancies and omissions that were present in Chapter 16 of ASCE 705.
In modal response history analysis, the original set of coupled equations of motion is transformed into a
set of uncoupled modal equations, an explicit displacement history is computed for each mode, the
modal histories are transformed back into the original coordinate system, and these responses are added
together to produce the response history of the displacements at each of the original degrees of freedom.
These displacement histories may then be used to determine histories of story drift, member forces, or
story shears.
Requirements for response history analysis are provided in Chapter 16 of ASCE 710. The same
mathematical model of the structure used for the ELF and response spectrum analysis is used for the
response history analysis. Five percent damping was used in each mode and as with the response
spectrum method, 12 modes were used in the analysis. These 12 modes captured more than 90 percent of
the mass of the structure above grade. Several issues related to a reanalysis of the structure with 120
modes are described later.
As allowed by ASCE 710 Section 16.1, the structure is analyzed using three different pairs of ground
acceleration histories. The development of a proper suite of ground motions is one of the most critical
and difficult aspects of response history approaches. The motions should be characteristic of the site and
should be from real (or simulated) ground motions that have a magnitude, distance and source mechanism
consistent with those that control the maximum considered earthquake (MCE).
For the purposes of this example, however, the emphasis is on the implementation of the response history
approach rather than on selection of realistic ground motions. For this reason, the motion suite developed
for Example 4.2 is also used for the present example.5 The structure for Example 4.2 is situated in
Seattle, Washington and uses three pairs of motions developed specifically for the site. The use of the
Seattle motions for a Stockton building analysis is, of course, not strictly consistent with the requirements
of the Standard. However, a realistic comparison may still be made between the ELF, response spectrum
and response history approaches.
4.1.7.1 The Seattle ground motion suite. It is beneficial to provide some basic information on the
Seattle motion suites in Table 4.120a below. Refer to Figures 4.240 through 4.242 for additional
information, including plots of the ground acceleration histories and 5percent damped response spectra
for each component of each motion.
The acceleration histories for each source motion were downloaded from the PEER NGA Strong Ground
Motion Database:
http://peer.berkeley.edu/products/strong_ground_motion_db.html
The PEER NGA record number is provided in the first column of the table. Note that the magnitude,
epicenter distance and site class were obtained from the NGA Flatfile (a large Excel file that contains
information about each NGA record).
Table 4.120a Suite of Ground Motions Used for Response History Analysis
NGA
Record
Number
Magnitude
[Epicenter
Distance,
km]
Site
Class
Number of
Points and
Digitization
Increment
Component Source
Motion
PGA
(g)
Record
Name
(This
Example)
0879
7.28
C
9625 @
0.005 sec
Landers/LCN260*
0.727
A00
[44]
Landers/LCN345*
0.789
A90
0725
6.54
D
2230 @
0.01 sec
SUPERST/BPOE270
0.446
B00
[11.2]
SUPERST/BPOE360
0.300
B90
0139
7.35
C
1192 @
0.02 sec
TABAS/DAYLN
0.328
C00
[21]
TABAS/DAYTR
0.406
C90
*Note that the two components of motion for the Landers earthquake are apparently separated by an
85 degree angle, not 90 degrees as is traditional. It is not known whether these are true orientations or
whether there is an error in the descriptions provided in the NGA database.
Before the ground motions may be used in the response history analysis, they must be scaled for
compatibility with the design spectrum. The scaling procedures for threedimensional dynamic analysis
are provided in Section 16.1.3.2 of ASCE 710. These requirements are provided verbatim as follows:
For each pair of horizontal ground motion components a square root of the sum of the squares
(SRSS) spectrum shall be constructed by taking the SRSS of the 5percent damped response
spectra for the scaled components (where an identical scale factor is applied to both
components of a pair). Each pair of motions shall be scaled such that for each period in the
range from 0.2T to 1.5T, the average of the SRSS spectra from all horizontal component pairs
does not fall below the corresponding ordinate of the design response spectrum, determined in
accordance with Section 11.4.5 or 11.4.7.
ASCE 710 does not provide clear guidance as to which fundamental period, T, should be used for
determining 0.2T and 1.5T when the periods of vibration are different in the two orthogonal directions of
analysis. This issue is resolved herein by taking T as the average of the computed periods in the two
principal directions. For this example, the average period, referred to a TAvg, is 0.5(2.87 +
2.60) = 2.74 seconds. (Another possibility would be to use the shorter of the two fundamental periods for
computing 0.2T and the longer of the two fundamental periods for computing 1.5T.)
It is also noted that the scaling procedure provided by ASCE 710 does not provide a unique set of scale
factors for each set of ground motions. This degree of freedom in the scaling process may be
eliminated by providing a sixstep procedure, as described below:
1. Compute the 5 percent damped pseudoacceleration spectrum for each unscaled component of
each pair of ground motions in the set and produce the SRSS spectrum for each pair of motions
within the set.
2. Using the same period values used to compute the ground motion spectra, compute the design
spectrum following the procedures in Standard Section 11.4.5. This spectrum is designated as
the target spectrum .
3. Scale each SRSS spectrum such that the spectral ordinate of the scaled spectrum at TAvg is equal
to the spectral ordinate of the design spectrum at the same period. Each SRSS spectrum will have
a unique scale factor, S1i, where i is the number of the pair (i ranges from 1 to 3 for the current
example).
4. Create a new spectrum that is the average of the S1 scaled SRSS spectra. This spectrum is
designated as the average S1 scaled SRSS spectrum and should have the same spectral ordinate
as the target spectrum at the period TAvg.
5. For each spectral ordinate in the period range 0.2TAvg to 1.5TAvg, divide the ordinate of the target
spectrum by the corresponding ordinate of the average S1 scaled SRSS spectrum, producing a set
of spectral ratios over the range 0.2TAvg to 1.5TAvg. The largest value among these ratios is
designated as S2.
6. Multiply the factor S1i determined in Step 3 for each pair in the set by the factor S2 determined in
Step 5. This product, SSi = S1i S2 is the scale factor that should be applied to each component
of ground motion in pair i of the set.
The results of the scaling process are summarized in Table 4.120b and in Figures 4.114 through 4.118.
Table 4.120b Result of 3D Scaling Process
Set
Number
Designation
SRSS Ordinate
at T = TAvg
(g)
Target
Ordinate at
T = TAvg
(g)
S1
S2
SS
1
A00 & A90
0.335
0.136
0.407
1.184
0.482
2
B00 & B90
0.191
0.136
0.712
1.184
0.843
3
C00 & C90
0.104
0.136
1.310
1.184
1.551
Figure 4.114 shows the unscaled SRSS spectra for each component pair, together with the target
spectrum. Figure 4.115 shows the average S1 scaled SRSS spectrum and the target spectrum, where it
may be seen that both spectra have a common ordinate at the average period of 2.74 seconds. Figure 4.1
16 is a plot of the spectral ratios computed in Step 5. Figure 4.117 is a plot of the SS scaled average
SRSS spectrum, together with the target spectrum. From this plot it may be seen that all ordinates of the
SS scaled average SRSS spectrum are greater than or equal to the ordinate of the target spectrum over the
period range 0.2TAvg to 1.5TAvg. The controlling period at which the two spectra in Figure 4.117 have
exactly the same ordinate is approximately 1.6 seconds.
Figure 4.118a shows the SS scaled spectra for the 00 components of each earthquake, together with the
target spectrum. Figure 4.18b is similar, but shows the 90 components of the ground motions. Also
shown in these plots are vertical lines that represent the first 12 periods of vibration for the structure under
consideration. Two additional vertical lines are shown that represent the periods for Modes 112 and 118,
at which the basement walls and gradelevel slab become dynamically effective. Three important points
are noted from Figures 4.118:
The match for the lower few modes (T > 1.0 sec) is good for the 00 components, but not as
good for the 90 components. In particular, the ground motion coordinates for motions A90 and
B90 are considerably less than those for the target spectrum.
Higher mode responses (T < 1.0 sec) will be significantly greater in Earthquake C than in
Earthquake A or B. In Modes 10 through 12, the response for Earthquake A is several times
greater than for Earthquake B.
In Modes 112 and 118, the response for Earthquake A is approximately three times that for the
code spectrum.
The impact of these points on the computed response of the structure will be discussed in some detail later
in this example.
Figure 4.114 Unscaled SRSS spectra and target spectrum
Figure 4.115 Average S1 scaled SRSS spectrum and target spectrum
Figure 4.116 Ratio of target spectrum to average S1 scaled SRSS spectrum
Figure 4.117 SS scaled average SRSS spectrum and target spectrum
(a) 00 Components
(b) 90 Components
Figure 4.118 SS scaled individual spectra and target spectrum
Another detail not directly specified by Chapter 16 of ASCE 710 is how ground motions should be
oriented when applied. In the analysis presented herein, 12 dynamic analyses were performed with scaled
ground motions applied only in one direction, as follows:
A00X: SS scaled component A00 applied in X direction
A00Y: SS scaled component A00 applied in Y direction
A90X: SS scaled component A90 applied in X direction
A90Y: SS scaled component A90 applied in Y direction
B00X: SS scaled component B00 applied in X direction
B00Y: SS scaled component B00 applied in Y direction
B90X: SS scaled component B90 applied in X direction
B90Y: SS scaled component B90 applied in Y direction
C00X: SS scaled component C00 applied in X direction
C00Y: SS scaled component C00 applied in Y direction
C90X: SS scaled component C90 applied in X direction
C90Y: SS scaled component C90 applied in Y direction
The scaled motions, without the (I/R) factor, were applied at the base of the basement walls. Accidental
torsion effects are included in a separate static analysis, as described later. All 12 individual response
history analyses were carried out using SAP2000. As with the response spectrum analysis, 12 modes
were used in the analysis. Five percent of critical damping was used in each mode. The integration time
step used in all analyses was equal to the digitization interval of the ground motion used (see
Table 4.120a). The results from the analyses are summarized Tables 4.121.
A summary of base shear and roof displacement results from the analyses using the SS scaled ground
motions is provided in Table 4.121. As may be observed, the base shears range from a low of 1,392 kips
for analysis A90Y to a high of 5,075 kips for analysis C90Y. Roof displacements range from a low of
5.16 inches for analysis A90Y to a high of 20.28 inches for analysis A00X. This is a remarkable range
of behavior when one considers that the ground motions were scaled for consistency with the design
spectrum.
Table 4.121 Result Maxima from Response History Analysis Using SS
Scaled Ground Motions
Analysis
Maximum
base shear
(kips)
Time of
maximum
shear
(sec.)
Maximum
roof
displacement
(in.)
Time of
maximum
displacement
(sec.)
A00X
3507
11.29
20.28
11.38
A00Y
3573
11.27
14.25
11.28
A90X
1588
12.22
7.32
12.70
A90Y
1392
13.56
5.16
10.80
B00X
3009
8.28
12.85
9.39
B00Y
3130
9.37
11.20
10.49
B90X
2919
8.85
11.99
7.11
B90Y
3460
7.06
11.12
8.20
C00X
3130
13.5
9.77
13.54
C00Y
2407
4.64
6.76
8.58
C90X
3229
6.92
15.61
6.98
C90Y
5075
6.88
14.31
7.80
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
The analysis was performed without the (I/R) factor, so in conformance with Section 16.1.4 of
ASCE 710, all force quantities produced from the analysis were multiplied by this factor. All
displacements from the analysis were multiplied by the factor Cd/R.
Additionally, the 2010 version of the Standard requires that forces be scaled by the factor 0.85V/Vi where
the base shears from the response history analysis, Vi, are less than 0.85 times the base shears, V,
produced by the ELF method when either Equation 12.85 or 12.86 controls the seismic base shear. The
displacements must be scaled by the same factor only if Equation 12.86 controls when computing the
seismic base shear. (It is noted that these requirements are similar to the scaling requirements provided
for modal response spectrum analysis [Sections 12.9.4.1 and 12.9.4.2] except that forces from modal
response spectrum analysis would be scaled if the shear from the response spectrum analysis is less than
0.85V, regardless of the Cs equation which controls V.)
The base shears from the SS scaled motions with the I/R = 1/8 scaling are provided in the first column of
Table 4.122. These forces are all significantly less than 0.85 times the ELF base shear, which is
0.85(112.5) = 956 kips. The required scale factors to bring the base shears up to the 85 percent
requirement are shown in Column 2 of Table 4.222.
Before proceeding, it is important to remind the reader that three separate sets of scale factors apply to the
response history analysis of this structure when member design forces are being obtained:
1. The ground motion SS scale factors
2. The I/R scale factor
3. The 0.85V/Vi factor because the base shear from modal response history analysis (including scale
factors 1 and 2 above) is less than 85 percent of that determined from ELF when ELF is governed
by Equation 12.85 or 12.86.
Table 4.122 I/R Scaled Shears and Required 85% Rule Scale Factors
Analysis
(I/R) times maximum base
shear from analysis
(kips)
Required additional scale factor for
V = 0.85VELF = 956 kips
A00X
438.4
2.18
A00Y
446.7
2.14
A90X
198.5
4.81
A90Y
173.9
5.49
B00X
376.1
2.54
B00Y
391.2
2.44
B90X
364.8
2.62
B90Y
432.5
2.21
C00X
391.2
2.44
C00Y
300.9
3.18
C90X
403.6
2.37
C90Y
634.4
1.51
1.0 kip = 4.45 kN
4.1.7.2 Drift and Pdelta effects. Only two scale factors are required for displacement and drift because
Equation 12.86 did not control the base shear for this structure:
The ground motion scale factors SS
The Cd/R scale factor
Drift is checked for each individual component of motion acting in the X direction and the envelope
values of drift are taken as the design drift values. The procedure is repeated for motions applied in the
Y direction. As with the ELF and Modal Response Spectrum analyses, drifts are taken as the difference
between the displacement at the center of mass of one level and the displacement at the projection of this
point on the level below. The results of the analysis, shown in Table 4.123 for the X direction only,
indicate that the allowable drift is not exceeded at any level of the structure. Similar results were obtained
for Y direction loading.
Table 4.123 Response History Drift for Building Responding in X Direction for All of the
Ground Motions in the X Directions
Level
Envelope of drift (in.) for each ground motion
Envelope
of drift for
all the
ground
motions
Envelope
of drift
Cd/R
Allowable
drift
(in.)
A00X
A90X
B00X
B90X
C00X
C90X
R
1.17
0.49
0.95
0.81
0.91
1.23
1.23
0.85
3.00
12
1.64
0.66
1.22
0.95
1.16
1.27
1.64
1.13
3.00
11
1.97
0.78
1.32
0.99
1.25
1.52
1.97
1.35
3.00
10
2.05
0.86
1.42
1.04
1.20
1.68
2.05
1.41
3.00
9
1.79
0.82
1.26
1.25
0.99
1.41
1.79
1.23
3.00
8
1.83
0.87
1.22
1.42
1.23
1.50
1.83
1.26
3.00
7
1.82
0.83
1.27
1.36
1.21
1.67
1.82
1.25
3.00
6
1.77
0.74
1.36
1.35
1.06
1.94
1.94
1.33
3.00
5
1.50
0.59
1.19
1.21
1.09
1.81
1.81
1.24
3.00
4
1.55
0.62
1.22
1.32
1.23
1.76
1.76
1.21
3.00
3
1.56
0.64
1.24
1.30
1.33
1.60
1.60
1.10
3.00
2
1.97
0.86
1.64
1.58
1.73
1.85
1.97
1.35
4.32
1.0 in. = 25.4 mm.
ASCE 710 does not provide information on how Pdelta effects should be addressed in response history
analysis. It would appear reasonable to use the same procedure as specified for ASCE 705 and
ASCE 710 for Modal Response Spectrum Analysis (Sec. 12.9.6), where it is stated that the Equivalent
Lateral Force method of analysis be used. Such an analysis was performed in Section 4.1.5.3.2 of this
example, with results provided in Table 4.111. These results indicate that allowable stability ratios are
marginally exceeded at Levels 2, 3 and 4, but that rigorous analysis with less than 1.0 would show that
the allowable stability ratios are not exceeded.
4.1.7.3 Torsion, orthogonal loading and member design forces. As with ELF or response spectrum
analysis, it is necessary to add the effects of accidental torsion and orthogonal loading into the analysis.
Accidental torsion is applied separately with a static analysis in exactly the same manner as done for the
response spectrum approach. Member shears for this torsiononly analysis are shown separately in
Figure 4.19b. These shears must be multiplied by 0.85 before adding to the scaled shears produced by
the dynamic response history analysis.
Orthogonal loading is automatically accounted for by applying the 100 percent of the ground motions in
the X and Y direction simultaneously. For each ground motion pair, these forces are applied in the
orientations shown in Figure 4.119. The figure also shows the scale factor that was used in each
analysis.
Load Combination for Response History Analysis
Earthquake
Load
Combination
Loading X Direction
Loading Y Direction
Record
Scale
Factor
Record
Scale
Factor
A
1
A00X
2.18
A00Y
5.49
2
A90X
4.81
A90Y
2.14
3
A00X
2.18
A00Y
5.49
4
A90X
4.81
A90Y
2.14
B
5
B00X
2.54
B00Y
2.21
6
B90X
2.62
B90Y
2.44
7
B00X
2.54
B00Y
2.21
8
B90X
2.62
B90Y
2.44
C
9
C00X
2.44
C00y
1.50
10
C90X
2.36
C90Y
3.18
11
C00X
2.44
C00Y
1.50
12
C90X
2.36
C90Y
3.18
Figure 4.119 Orthogonal Loading in Response History Analysis
Using the load combinations described above, the individual beam shear maxima developed in Frame 1
were computed for each load combination. Envelope values from all combinations are shown in
Figure 4.120.
14.15
12.82
14.17
R12
21.5
20.6
21.5
1211
29.5
29.4
30.6
1110
33.7
33.2
35.5
109
32.9
32.0
29.5
28.2
12.1
98
33.6
32.3
30.7
34.0
21.0
87
36.3
34.5
33.2
35. 7
22.0
76
39.0
35.3
34.5
36.2
22.8
65
15.1
32.9
33.9
35.8
35.6
36.0
24.6
54
25.0
38.5
33.6
35.6
35.5
35.7
24.7
43
23.7
35.7
33.1
34.3
34.2
34.3
24.0
32
21.6
34.3
32.3
33.1
33.0
33.5
21.9
2  G
Figure 4.120 Envelope of seismic shears in girders (kips) as computed using response history
analysis (analysis includes orthogonal loading but excludes accidental torsion)
4.1.8 Comparison of Results from Various Methods of Analysis
A summary of the results from all of the analyses is provided in Tables 4.124 through 4.128.
4.1.8.1 Comparison of base shear and story shear. The maximum story shears are shown in Table 4.1
24. For the response history analysis, the shears are the envelope values of story shears for all twelve
individual analyses. Note that the modal response spectrum and modal response history shears for the
lowest level are both equal to 956 kips, which is 0.85 times the ELF base shear.
The story shear is basically of the same character lower values in upper stories, larger values in lower
stories. It appears, however, that the maximum shears from the modal response history analysis occur at
stories 2, 3 and 4. This must be due to the amplified energy in the higher modes in the actual ground
motions (when compared to the design spectrum).
4.1.8.2 Comparison of drift. Table 4.125 summarizes the drifts computed from each of the analyses.
The modal response history drifts are the envelopes among all analyses. The ELF drifts are significantly
greater than those determined using modal response spectrum analysis. The drifts from the modal
response history analysis are slightly greater than those from the response spectrum analysis.
4.1.8.3 Comparison member forces. The shears developed in Bay DE of Frame 1 are compared in
Table 4.126. The shears from the response history analysis are envelope values among all analyses,
including torsion and orthogonal load effects. The response history approach produced beam shears
similar to those from ELF analysis and somewhat greater than those produced by response spectrum
analysis.
Table 4.124 Summary of Results of Various Methods of Analysis:
Story Shear
Level
ELF
Modal
response
spectrum
Modal response history
R
187
180
295
12
341
286
349
11
471
357
462
10
578
418
537
9
765
524
672
8
866
587
741
7
943
639
753
6
999
690
943
5
1,070
784
1,135
4
1,102
840
1,099
3
1,118
895
1,008
2
1,124
956
956
Table 4.125 Summary of Results from Various Methods of Analysis:
Story Drifts
Level
X Direction Drift
(in.)
ELF
Modal
response
spectrum
Modal
response
history
R
0.99
0.66
0.85
12
1.41
0.89
1.13
11
1.75
1.03
1.35
10
1.92
1.08
1.41
9
1.82
0.98
1.23
8
1.97
1.06
1.26
7
2.01
1.08
1.25
6
1.97
1.08
1.33
5
1.67
0.97
1.24
4
1.69
1.02
1.21
3
1.65
1.05
1.10
2
2.00
1.34
1.35
1.0 in. = 25.4 mm.
Table 4.126 Summary of Results from Various Methods of Analysis:
Beam Shear
Level
Beam Shear Force in Bay DE of Frame 1
(kips)
ELF
Modal response
spectrum
Modal response
history
R
10.27
8.72
12.82
12
18.91
15.61
20.61
11
28.12
21.61
29.45
10
33.15
24.02
33.22
9
34.69
23.32
32.02
8
35.92
23.47
32.30
7
40.10
26.15
34.53
6
40.58
26.76
35.29
5
36.52
25.29
35.82
4
34.58
24.93
35.65
3
35.08
26.60
34.27
2
35.28
28.25
33.07
1.0 kip = 4.45 kN.
4.1.8.4 Which analysis method is best In this example, an analysis of an irregular steel moment frame
was performed using three different techniques: equivalent lateral force analysis, modal response
spectrum analysis and modal response history analysis. Each analysis was performed using a linear
elastic model of the structure even though it is recognized that the structure will repeatedly yield during
the earthquake. Hence, each analysis has significant shortcomings with respect to providing a reliable
prediction of the actual response of the structure during an earthquake.
The purpose of analysis, however, is not to predict response but rather to provide information that an
engineer can use to proportion members and to estimate whether or not the structure has sufficient
stiffness to limit deformations and avoid overall instability. In short, the analysis only has to be good
enough for design. If, on the basis of any of the above analyses, the elements are properly designed for
strength, the stiffness requirements are met and the elements and connections of the structure are detailed
for inelastic response according to the requirements of ASCE 7 and AISC 341, the structure will likely
survive an earthquake consistent with the MCE ground motion. The exception would be if a highly
irregular structure were analyzed using the ELF procedure. Fortunately, ASCE 7 safeguards against this
by requiring threedimensional dynamic analysis for highly irregular structures.
For the structure analyzed in this example, the irregularities were probably not so extreme such that the
ELF procedure would produce a bad design. However, where computer programs that can perform
modal response spectrum analysis with only marginally increased effort over that required for ELF are
available (e.g., SAP2000 and ETABS), the modal analysis should always be used for final design in lieu
of ELF (even if ELF is allowed by the Provisions). As mentioned in the example, this does not negate the
need for or importance of ELF analysis because such an analysis is useful for preliminary design and
several components of the ELF analysis are necessary for application of accidental torsion.
Modal response history analysis is of limited practical use where applied to a linear elastic model of the
structure. The amount of additional effort required to select and scale the ground motions, perform the
modal response history analysis, scale the results and determine envelope values for use in design simply
is not warranted where compared to the effort required for modal response spectrum analysis. This might
change in the future where standard suites of ground motions are developed and are made available to
the earthquake engineering community. Also, significant improvement is needed in the software
available for the preprocessing and, particularly, for the postprocessing of the huge amounts of
information that produced by the analysis.
Scaling the ground motions used for modal response history analysis is also an issue. The Standard
requires that the selected motions be consistent with the magnitude, distance and source mechanism of the
MCE expected at the site. If the ground motions satisfy this criterion, then why scale at all Distant
earthquakes may have a lower peak acceleration but contain a frequency content that is more significant.
Nearsource earthquakes may display single damaging pulses. Scaling these two earthquakes to the
Standard design spectrum seems to eliminate some of the most important characteristics of the ground
motions. The fact that there is a degree of freedom in the ASCE 7 scaling requirements compensates for
this effect, but only for very knowledgeable users.
The main benefit of modal response history analysis is in the nonlinear dynamic analysis of structures or
in the analysis of nonproportionally damped linear systems. This type of analysis is the subject of
Example 4.2.
4.1.9 Consideration of Higher Modes in Analysis
All of the computed results for the modal response spectrum and modal response history methods of
analysis were based on the first 12 modes of the model with the basement level explicitly modeled.
Recall that the basement walls were modeled with 1.0foot shell elements, that the gradelevel diaphragm
was modeled using 6.0inchthick shell elements and that the grade level was not laterally restrained. The
weight associated with the basementlevel walls and gradelevel slab is 6,526 kips, which is
approximately 15 percent of the total weight of the structure (see Table 4.13).
The accumulated effective modal mass for the first 12 modes (see Table 4.114a) is in the neighborhood
of 82 percent of the total mass of the structure, which is less than the 90 percent required by
Section 12.91 of the Standard. However, the first 12 modes capture more than 90 percent of the mass
above grade, so it was deemed sufficient to run the analysis with only 12 modes. If the requirement of
Section 12.91 were satisfied for the structure as modeled, it would have taken 119 modes to capture more
than 90 percent of the effective mass of the entire system (see Table 4.114b).
In the analysis presented so far, all of the seismic base shears were computed at the base of the first story
above grade, not the base of the entire structure (the base of the basement walls). It is of some interest to
examine how the results of the analysis would change if 120 modes were to be used in the analysis. This
would definitely satisfy the requirements of Section 12.91 for the full structure.
4.1.9.1 Modal response spectrum analysis with higher modes. Table 4.127a provides the seismic
shears through the basement level and through the first floor above grade for the analysis run with 12, 18,
120 and 200 modes. In this part of the table, the modes are the natural mode shapes from an eigenvalue
analysis. As may be seen, the shear through the first story above grade is unchanged as the number of
modes increases above 12 modes. However, the shears though the basement level are substantially
increased when 120 or more modes are used. In the X direction, for example, the ratio of the basement
level shear for 120 modes to that for 12 modes is 630/439 = 1.44. Thus, in terms of the shear at the base
of the structure, the activation of the higher modes increases the shears 44 percent, while the added
weight associated with the basement level is only 15 percent.
This increase in shear was rather unexpected, so the analysis was rerun using Ritz vectors in lieu of the
natural mode shapes. Ritz vectors automatically include the static corrections that are sometimes
needed for very high frequency modes. As may be seen from Table 4.127b, the results using Ritz
Vectors are virtually identical to those obtained using the natural mode shapes.
4.1.9.2 Modal response history analysis with higher modes. The comparison of shears using modal
response history analysis with 12, 18, 120 and 200 modes are presented in Table 4.128. The results are
based on the use of natural mode shapes. For brevity, results are given only for motions A00, B00 and
C00 applied in the X and Y directions. The analyses include SS ground motion scaling, I/R scaling, but
not the 85 percent scaling.
As may be observed from Table 4.127, the use of the higher modes produces virtually no change in the
shears through the first level above grade. However, very significant increases in shear are developed
through the basement. The most extreme increase in shears is for ground motion A00, wherein the shears
in the basement increase from 439 kips to 744 kips for loading in the X direction and increase from
440 kips to 862 kips for loading in the Y direction. These increases in shear are not unexpected because
of the spectral amplitudes of the ground motions at periods associated with modes 112 and 118 (see
Figure 4.118).
4.1.9.3 Discussion on use of higher modes. Many structures have stiff lower stories or have one or
more levels of basement. If the basement is modeled explicitly and if full lateral restraint is not provided
at the top of the basement or at subgrade slab levels, the phenomena described herein will likely result.
Based on the results presented above, there is some question as to which results should be used for the
85 percent scaling requirements of Standard Section 12.9.4. If the basement level were included in the
ELF analysis, the computed period would not significantly change, and the base shear would increase
15 percent to accommodate the added weight associated with the basement walls and gradelevel slab.
However, the scale factors required to bring the modal response spectrum or modal response history
shears up to 85 percent of the ELF shears (with 15 percent increase) could be significantly less than those
obtained when the basement level is not included in the model. The net result would be significantly
reduced for design shears in the upper levels of the structure.
Given these results, it is recommended that scaling always be based on the shears determined at the first
level above grade. The question of how many modes to use in the analysis is not as easy to answer.
Certainly, a sufficient number of modes must be used to capture at least 90 percent of the abovegrade
mass. In cases where it is desired to explicitly determine the shears at the base of unrestrained basements,
enough modes should be used to capture 90 percent of the mass of the entire structure.
Table 4.127 Comparison of Modal Response Spectrum Shears Using 12, 18, 120 and
200 Modes
(a) Using Natural Mode Shapes
(values from SAP2000 without 85% scaling)
Shear Location
Load Case
Shear (kips) for
number of modes =
12
18
120
200
Base of 1st story
Code spectrum X direction
438
438
439
439
Base of structure
Code spectrum X direction
439
439
630
631
Base of 1st story
Code spectrum Y direction
492
493
493
493
Base of structure
Code spectrum Y direction
493
493
686
686
(b) Using Ritz Vectors
(values from SAP2000 without 85% scaling)
Shear Location
Load Case
Shear (kips) for
number of modes =
12
18
120
200
Base of 1st story
Code spectrum X direction
435
439
439
439
Base of structure
Code spectrum X direction
435
439
467
630
Base of 1st story
Code spectrum Y direction
485
494
494
494
Base of structure
Code spectrum Y direction
485
494
497
686
Table 4.128 Comparison of Modal Response History Shears Using 12, 18, 120 and 200
Modes
Using Natural Mode Shapes
(values from SAP2000 without 85% scaling)
Shear Location
Load Case
Shear (kips) for
number of modes=
12
18
120
200
Base of 1st story
A00 in X direction
438
438
445
445
Base of structure
A00 in X direction
439
439
744
744
Base of 1st story
B00 in X direction
376
376
377
490
Base of structure
B00 in X direction
377
377
529
530
Base of 1st story
C00 in X direction
391
391
392
391
Base of structure
C00 in X direction
392
392
438
440
Base of 1st Story
A00 in Y direction
447
447
452
452
Base of Structure
A00 in Y direction
440
447
861
862
Base of 1st Story
B00 in Y direction
391
391
395
395
Base of Structure
B00 in Y direction
397
392
508
508
Base of 1st Story
C00 in Y direction
301
301
301
301
Base of Structure
C00 in Y direction
307
302
561
562
4.1.10 Commentary on the ASCE 7 Requirements for Analysis
As mentioned in this example, ASCE 705 contained several inconsistencies in scaling requirements for
modal response spectrum analysis and for modal response history analysis. The main source of problems
was in Chapter 16 of ASCE 705 and fortunately, most of these problems have been eliminated in
ASCE 710.
There are still a few issues that need to be clarified. Some of these are listed below:
Accidental torsion: The Standard needs to be more specific on how accidental torsion should be
applied where used with modal response spectrum and modal response history analyses. The
method suggested herein, to apply such torsions as part of a static loading, is easy to implement.
However, automatic methods based on shifting center of mass need to be explored and, if
effective, standardized.
Amplification of accidental torsion: Currently, accidental torsion need be amplified only for
torsionally irregular structures in SDC C and higher (Sec. 12.8.4.3). However, the torsion need
not be amplified if a dynamic analysis is performed (Sec. 12.9.5). This implies that the
amplification of torsion is a dynamic phenomenon, but the author has found no published
technical basis for such amplification. Indeed, most references to amplification are based on
problems associated with uneven yielding of lateral loadresisting components. This issue needs
to be clarified and resolved.
PDelta effects: It appears that the most efficient method for handling Pdelta effects is to
perform the analysis without such effects, use a separate ELF analysis to determine if such effects
are significant and if so, magnify forces and displacements to include such effects. It would be
much more reasonable to include such effects in the analysis directly and establish procedures to
determine if such effects are excessive. Comparison of analysis results with and without Pdelta
effects included is an effective means to assess the significance of the effects.
Computing drift: When threedimensional analysis is performed, drift should be checked at the
corners of the building, not the center of mass. Consideration should be given to eliminating the
use of story drift in favor of computing shear strain in damageable components. Such
calculations can be easily automated.
Scaling ground motions for linear response history analysis: The need to scale ground motions
over a period range of 0.2T to 1.5T is not appropriate for elastic analysis because the effective
period in any mode will never exceed 1.0T. Additionally, placing equal weight on scaling
spectral ordinates at higher modes does not seem rational. In some cases a high mode that is only
barely contributing to response can dominate the scaling process.
Development of standard ground motion histories: The requirement that analysts scan through
thousands of ground motion records to find appropriate suites for analysis is unnecessarily
burdensome. The Standard should provide tables of ground motion suites that are appropriate to
simple parameters such as magnitude, site class and distance.
Finally, it is suggested that requirements for linear response history analysis be removed from Chapter 16
and placed in Chapter 12 (as Section 12.10, for example). Requirements for performing such analysis
should be as consistent as possible with those of modal response spectrum analysis.
4.2 SIXSTORY STEEL FRAME BUILDING, SEATTLE, WASHINGTON
In this example, the behavior of a simple, sixstory structural steel momentresisting frame is investigated
using a variety of analytical techniques. The structure was initially proportioned using a preliminary
analysis and it is this preliminary design that is investigated. The analysis will show that the structure
falls short of several performance expectations. In an attempt to improve performance, viscous fluid
dampers are considered for use in the structural system. Analysis associated with the added dampers is
performed in a very preliminary manner.
The following analytical techniques are employed:
Linear static analysis
Plastic strength analysis (using virtual work)
Nonlinear static (pushover) analysis
Linear dynamic (response history) analysis
Nonlinear dynamic (response history) analysis
The primary purpose of this example is to highlight some of the more advanced analytical techniques;
hence, more detail is provided on these methods. It is also noted that the linear dynamic analysis was
performed only as a precursor and check on the analytical model used for nonlinear dynamic analysis and
is not discussed in the example.
The 2005 and 2010 versions of the Standard do not provide any guidance on pushover analysis because it
is not a permitted method of analysis in Table 12.61. Some guidance for pushover analysis is provided
in Resource Paper 2 in Part 3 of the Provisions. More detailed information on pushover analysis is
provided in FEMA 440 and in ASCE 41. The procedures outlined in ASCE 41 are used in this example.
Chapter 16 of the Standard provides some guidance and requirements for linear and nonlinear response
history analysis. Certain aspects of these requirements are clarified in ASCE 710, but the basic
methodology is unchanged. More detailed requirements for response history analysis are provided in
Resource Paper 3 of the Provisions. This example follows the recommendations in Resource Paper 3,
with certain exceptions, which are noted as the example proceeds.
4.2.1 Description of Structure
The structure analyzed for this example is a sixstory office building in Seattle, Washington. According
to the descriptions in Standard Table 11, the building is assigned to Occupancy Category II. From
Standard Table 11.51, the importance factor (I) is 1.0. A plan and elevation of the building are shown in
Figures 4.21 and 4.22, respectively. The lateral loadresisting system consists of steel momentresisting
frames on the perimeter of the building. There are five bays at 28 feet on center in the northsouth (NS)
direction and six bays at 30 feet on center in the eastwest (EW) direction. The typical story height is
12 feet6 inches with the exception of the first story, which has a height of 15 feet. There is a 5foottall
perimeter parapet at the roof and one basement level that extends 15 feet below grade. For this example,
it is assumed that the columns of the momentresisting frames are embedded into pilasters formed into the
reinforced concrete basement wall.
For the momentresisting frames in the NS direction (Frames A and G), all of the columns bend about
their strong axes and the girders are attached with fully welded momentresisting connections. The
expected plastic hinge regions of the girders have reduced flange sections, detailed in accordance with
AISC 341.
For the frames in the EW direction (Frames 1 and 6), momentresisting connections are used only at the
interior columns. At the exterior bays, the EW girders are connected to the weak axis of the exterior
(corner) columns using nonmomentresisting connections. All interior columns are gravity columns and
are not intended to resist lateral loads. A few of these columns, however, would be engaged as part of the
added damping system described in the last part of this example. With minor exceptions, all of the
analyses in this example are for lateral loads acting in the NS direction. Analysis for lateral loads acting
in the EW direction would be performed in a similar manner.
Figure 4.21 Plan of structural system
Figure 4.22 Elevation of structural system
Prior to analyzing the structure, a preliminary design was performed in accordance with AISC 341. All
members, including miscellaneous plates, were designed using steel with a nominal yield stress of 50 ksi
and expected yield strength of 55 ksi. Detailed calculations for the design are beyond the scope of this
example. Table 4.21 summarizes the members selected for the preliminary design.1
Table 4.21 Member Sizes Used in NS Moment Frames
Member supporting
level
Column
Girder
Doubler plate
thickness
(in.)
R
W21x122
W24x84
1.00
6
W21x122
W24x84
1.00
5
W21x147
W27x94
1.00
4
W21x147
W27x94
1.00
3
W21x201
W27x94
0.875
2
W21x201
W27x94
0.875
The sections shown in Table 4.21 meet the widthtothickness requirements for special moment frames
and the size of the column relative to the girders should ensure that plastic hinges initially will form in the
girders. Due to strain hardening, plastic hinges will eventually form in the columns. However, these
form under lateral displacements that are in excess of those allowed under the Design Basis Earthquake
(DBE). Doubler plates of 0.875 inch thick are used at each of the interior columns at Levels 2 and 3 and
1.00 inch thick plates are used at the interior columns at Levels 4, 5, 6 and R. Doubler plates were not
used in the exterior columns.
4.2.2 Loads
4.2.2.1 Gravity loads. It is assumed that the floor system of the building consists of a normalweight
composite concrete slab formed on metal deck. The slab is supported by floor beams that span in the NS
direction. These floor beams have a span of 28 feet and are spaced 10 feet on center.
The dead weight of the structural floor system is estimated at 70 psf. Adding 15 psf for ceiling and
mechanical units, 10 psf for partitions at Levels 2 through 6 and 10 psf for roofing at Level R, the total
dead load at each level is 95 psf. The cladding system is assumed to weigh 15 psf.
A basic live load of 50 psf is used at Levels 2 through 6. The roof live load is 20 psf and (based on
calculations not shown here) the roof snow load is 25 psf. The reduced floor loads are taken as 0.4(50),
or 20 psf. Only half of this load is required in seismic load combinations (see Standard Section 2.3), so
the design live loads for the floor is 0.5(20) = 10 psf. The roof live load is not reducible, but never
appears in seismic load combinations. The snow load for seismic load combinations is 0.2(25) = 5 psf,
which is half of the floor live load.
Based on these loads, the total dead load, live or snow load and dead plus live or snow load applied to
each level of the entire building are given in Table 4.22. The slight difference in dead loads at Levels R
and 2 is due to the parapet and the tall first story, respectively.
Tributary areas for columns and girders as well as individual element gravity loads used in the analysis
are illustrated in Figure 4.23. These loads are based on a total dead load of 95 psf, a cladding weight of
15 psf and a live load of 10 psf.
Table 4.22 Gravity Loads on Seattle Building*
Dead load
(kips)
Reduced live or snow load
(kips)
Total load
(kips)
Level
Story
Accumulated
Story
Accumulated
Story
Accumulated
R
2,596
2,596
131
131
2,727
2,727
6
2,608
5,204
262
393
2,739
5,597
5
2,608
7,813
262
655
2,739
8,468
4
2,608
10,421
262
917
2,739
11,338
3
2,608
13,029
262
1,179
2,739
14,208
2
2,621
15,650
262
1,441
2,752
17,091
*Loads are for the entire building.
4.2.2.2 Equivalent static earthquake loads. Although the main analysis in this example is nonlinear,
equivalent static forces are computed in accordance with Standard Section 12.8. These forces are used in
a preliminary static analysis to determine whether the structure, as designed, conforms to the drift
requirements limitations imposed by Standard Section 12.2.
The structure is situated in Seattle, Washington. The short period and the 1second mapped spectral
acceleration parameters for the site are as follows:
SS = 1.63
S1 = 0.57
The structure is situated on Site Class C materials. From Standard Tables 11.41 and 11.42:
Fa = 1.00
Fv = 1.30
From Standard Equations 11.41 and 11.42, the maximum considered spectral acceleration parameters
are as follows:
SMS = FaSS = 1.00(1.63) = 1.63
SM1 = FvS1 = 1.30(0.57) = 0.741
And from Standard Equations 11.43 and 11.44, the design acceleration parameters are as follows:
SDS = (2/3)SM1 = (2/3)1.63 = 1.09
SD1 = (2/3)SM1 = (2/3)0.741 = 0.494
Figure 4.23 Element loads used in analysis
Based on the above coefficients and on Standard Tables 11.61 and 11.62, the structure is assigned to
Seismic Design Category D. For the purpose of analysis, it is assumed that the structure complies with
the requirements for a special moment frame, which, according to Standard Table 12.21, has the
following design values:
R = 8
Cd = 5.5
ê0 = 3.0
Note that the overstrength factor, ê0, is not needed for the analysis presented herein.
4.2.2.2.1 Approximate period of vibration. Standard Equation 12.87 is used to estimate the building
period:
where, from Standard Table 12.82, Ct = 0.028 and x = 0.8 for a steel moment frame. Using hn (the total
building height above grade) = 77.5 feet, Ta = 0.028(77.5)0.8 = 0.91 sec/cycle .
Where the period is determined from a properly substantiated analysis, the Standard requires that the
period used for computing base shear not exceed CuTa where, from Standard Table 12.81 (using
SD1 = 0.494), Cu = 1.4. For the structure under consideration, CuTa = 1.4(0.91) = 1.27 seconds. This
period is used for base shear calculation as it is expected that the period computed for the actual structure
will be greater than 1.27 seconds.
4.2.2.2.2 Computation of base shear. Using Standard Equation 12.81, the total seismic base shear is:
where W is the total seismic weight of the structure. From Standard Equation 12.82, the maximum
(constant acceleration region) seismic response coefficient is:
Equation 12.83 controls in the constant velocity region:
The seismic response coefficient, however, must not be less than that given by Equation 12.85:
Thus, the value from Equation 12.83 controls for this building. Using W = 15,650 kips,
V = 0.0485(15,650) = 759 kips.
4.2.2.2.2 Vertical distribution of forces. The seismic base shear is distributed along the height of the
building using Standard Equations 12.811 and 12.812:
and
where k = 0.75 + 0.5T = 0.75 + 0.5(1.27) = 1.385. The lateral forces acting at each level and the story
shears acting at the bottom of the story below the indicated level are summarized in Table 4.23. These
are the forces acting on the whole building. For analysis of a single frame, onehalf of the tabulated
values are used.
Table 4.23 Equivalent Lateral Forces for Building Responding in NS Direction
Level x
wx
(kips)
hx
(ft)
wxhxk
Cvx
Fx
(kips)
Vx
(kips)
R
2,596
77.5
1,080,327
0.321
243.6
243.6
6
2,608
65.0
850,539
0.253
191.8
435.4
5
2,608
52.5
632,564
0.188
142.6
578.0
4
2,608
40.0
433,888
0.129
97.8
675.9
3
2,608
27.5
258,095
0.077
58.2
734.1
2
2,621
15.0
111,909
0.033
25.2
759.3
15,650
3,367,323
1.000
759.3
4.2.3 Preliminaries to Main Structural Analysis
Performing a nonlinear analysis of a structure is an incremental process. The analyst should first perform
a linear analysis to obtain some basic information on expected behavior and to serve later as a form of
verification for the more advanced analysis. Once the linear behavior is understood (and extrapolated to
expected nonlinear behavior), the anticipated nonlinearities are introduced. If more than one type of
nonlinear behavior is expected to be of significance, it is advisable to perform a preliminary analysis with
each nonlinearity considered separately and then to perform the final analysis with all nonlinearities
considered. This is the approach employed in this example.
4.2.3.1 The computer programs NONLINPro and DRAIN 2Dx
The computer program NONLINPro was used for all of the analyses described in this example. This
program is basically a pre and postprocessor to DRAIN 2Dx (Prakash et al., 1993). While DRAIN is not
the most robust program currently available for performing nonlinear response history analysis, it was
used because many of the details of the analysis (e.g., panel zone modeling) must be done explicitly. This
detail provides insight into the modeling process which is not available when using the automated
features of the more robust software. Note that a full version of NONLINPro, as well as input files used
for this example, is provided on the CD.
DRAIN has several shortcomings that are related specifically to the example at hand. These
shortcomings are listed below. Also provided is a brief explanation of the influence the shortcoming may
have on the analysis.
It is not possible to model strength loss when using the ASCE 41 model for girder plastic hinges.
However, as discussed later in the example, this loss of strength generally occurs at plastic hinge
rotations well beyond the rotational demands produced under the DBE ground motions.
Maximum plastic rotation angles of plastic hinges were checked with the values in Table 56 of
ASCE 4106.
The DRAIN model for axialflexural interaction in columns is not particularly accurate. This is
of some concern in this example because hinges form at the base of the columns in all of the
analyses and in some of the upper columns during analysis with MCE level ground motions.
Only twodimensional analysis may be performed. Such an analysis is reasonable for the
structure considered in this example because of its regular shape and because full moment
connections are provided only in the NS direction for the corner columns (see Fig. 4.21).
As with any finite element analysis program, DRAIN models the structure as an assembly of nodes and
elements. While a variety of element types is available, only three element types were used in the
analysis:
Type 1 inelastic bar (truss) element
Type 2 beamcolumn element
Type 4 connection element
Two models of the structure were prepared for DRAIN. The first model, used for preliminary analysis
and for verification of the second (more advanced) model, consisted only of Type 2 elements for the main
structure and Type 1 elements for modeling Pdelta effects. All analyses carried out using this model
were linear.
For the second, more detailed model, Type 1 elements were used for modeling Pdelta effects and the
dampers in the damped system. It was assumed that these elements would remain linear elastic
throughout the response. Type 2 elements were used to model the beams, the columns and the braces in
the damped system, as well as the rigid links associated with the panel zones. Plastic hinges were allowed
to form in all columns. The column hinges form through the mechanism provided in DRAIN s Type 2
element. Plastic behavior in girders and in the panel zone region of the structure was modeled explicitly
through the use of Type 4 connection elements. Girder yielding was forced to occur in the Type 4
elements (in lieu of the main span represented by the Type 2 elements) to provide more control in hinge
location and modeling. A complete description of the implementation of these elements is provided later.
4.2.3.2 Description of preliminary model and summary of preliminary results
The preliminary DRAIN model is shown in Figure 4.24. Important characteristics of the model are as
follows:
Only a single frame (Frame A or G) is modeled. Hence onehalf of the loads shown in
Tables 4.22 and 4.23 are applied.
Columns are fixed at their base (at grade level; the basement is not modeled).
Each beam or column element is modeled using a Type 2 element. For the columns, axial,
flexural and shear deformations are included. For the girders, flexural and shear deformations are
included but, because of diaphragm slaving, axial deformation is not included. Composite action
in the floor slab is ignored for all analysis.
All members are modeled using centerline dimensions without rigid end offsets. This approach
allows for the effects of panel zone deformation to be included in an approximate but reasonably
accurate manner. Note that this model does not provide any increase in beamcolumn joint
stiffness due to the presence of doubler plates. The stiffness of the girders was decreased by
7 percent (in preliminary analyses) to account for the reduced flange sections. Moment rotation
properties of the reduced flange sections are used in the detailed analyses.
Pdelta effects are modeled using the leaner ghost column shown in Figure 4.24 at the right of the main
frame. This column is modeled with an axially rigid truss element. Pdelta effects are activated for this
column only (Pdelta effects are turned off for the columns of the main frame). The lateral degree of
freedom at each level of the Pdelta column is slaved to the floor diaphragm at the matching elevation.
Where Pdelta effects are included in the analysis, a special initial load case was created and executed.
This special load case consists of a vertical force equal to onehalf of the total story weight (dead load
plus 50 percent of the fully reduced live load) applied to the appropriate node of the Pdelta column.
When Pdelta effects are included, modal analysis should be performed after the Pdelta load case is
applied so that stiffness modification of Pdelta effects will increase the period of the structure. Pdelta
effects are modeled in this manner to provide true column axial forces for assessing strength.
Figure 4.24 Simple wire frame model used for preliminary analysis
4.2.3.2.1 Results of preliminary analysis: period of vibration and drift. The computed periods for
the first three natural modes of vibration are shown in Table 4.24. As expected, the period including
Pdelta effects is slightly larger than that produced by the analysis without such effects. More significant
is the fact that the first mode period is considerably longer than that predicted from Standard
Equation 12.87. Recall from previous calculations that this period (Ta) is 0.91 seconds and the upper
limit on the computed period CuTa is 1.4(0.91) = 1.27 seconds. Where doubler plate effects are included
in the detailed analysis, the period will decrease slightly, but it remains obvious that the structure is quite
flexible.
Table 4.24 Periods of Vibration From Preliminary Analysis (sec/cycle)
Mode
Pdelta excluded
Pdelta included
1
2.054
2.130
2
0.682
0.698
3
0.373
0.379
The results of the preliminary analysis for drift are shown in Tables 4.25 and 4.26 for the computations
excluding and including Pdelta effects, respectively. In each table, the deflection amplification factor
(Cd) equals 5.5 and the acceptable story drift (story drift limit) is taken as 2 percent of the story height,
which is the limit provided by Standard Table 12.121. In the Standard it is permitted to determine the
elastic drifts using seismic design forces based on the computed fundamental period of the structure
without the upper limit CuTa. Thus a new set of lateral loads based on the computed period of the actual
structure is applied to the structure to calculate the elastic drifts.
Where Pdelta effects are not included, the computed story drift is less than the allowable story drift at
each level of the structure. The largest magnified story drift, including Cd = 5.5, is 2.26 inches in
Stories 2 and 3. As a preliminary estimate of the importance of Pdelta effects, story stability
coefficients, , were computed in accordance with Standard Section 12.87. These are shown in the last
column of Table 4.25. At Story 2, the stability coefficient is 0.0862. According to the Standard, Pdelta
effects may be ignored where the stability coefficient is less than 0.10. For this example, however,
analyses are performed with and without Pdelta effects.
When Pdelta effects are included (Table 4.26), the drifts can also be estimated as the drifts without
Pdelta times the quantity 1/(1 ), where is the stability coefficient for the story. As can be seen in
Table 4.26, drifts calculated in this manner are consistent with the results obtained by running the
analyses with Pdelta effects. The difference is always less than 2 percent.
Table 4.25 Results of Preliminary Analysis Excluding Pdelta Effects
Story
Total drift
(in.)
Story drift
(in.)
Magnified
story drift
(in.)
Drift limit
(in.)
Story stability
ratio,
6
2.08
0.22
1.21
3.00
0.0278
5
1.86
0.32
1.76
3.00
0.0453
4
1.54
0.38
2.09
3.00
0.0608
3
1.16
0.41
2.26
3.00
0.0749
2
0.75
0.41
2.26
3.00
0.0862
1
0.34
0.34
1.87
3.60
0.0691
Table 4.26 Results of Preliminary Analysis Including Pdelta Effects
Story
Total drift
(in.)
Story drift
(in.)
Magnified
story drift
(in.)
Drift from
(in.)
Drift limit
(in.)
6
2.23
0.23
1.27
1.24
3.00
5
2.00
0.34
1.87
1.84
3.00
4
1.66
0.40
2.20
2.23
3.00
3
1.26
0.45
2.48
2.44
3.00
2
0.81
0.45
2.48
2.47
3.00
1
0.36
0.36
1.98
2.01
3.60
4.2.3.2.2 Results of preliminary analysis: demandtocapacity ratios. To determine the likelihood
and possible order of yielding, demandtocapacity ratios (DCR) are computed for each element. The
results are shown in Figure 4.25. For this analysis, the structure is subjected to full dead load plus 0.5
times the fully reduced live load, followed by equivalent lateral forces computed without the R factor. P
delta effects are included. Figure 4.25(a) displays the DCR of columns and girders and Figure 4.25(b)
displays the DCR of panel zones with and without doubler plates. In Figure 4.25b, the values in
parentheses represent the DCRs without doubler plates. Since the DCRs in Figure 4.25 are found from
preliminary analyses, in which the centerline model is used, doubler plates aren t added into the model.
Thus, the demand values of Figure 4.25(b) are the same with and without doubler plates. However, since
the capacity of the panel zone increases with added doubler plates, the DCRs decrease at the interior beam
column joints as the doubler plates are used only at the interior joints. As may be seen in Figure 4.25(b),
the DCR at the exterior joints are the same with and without doubler plates added.
For girders, the DCR is simply the maximum moment in the member divided by the member s plastic
moment capacity where the plastic capacity is ZeFye. Ze is the plastic section modulus at center of reduced
beam section and Fye is the expected yield strength. For columns, the ratio is similar except that the
plastic flexural capacity is estimated to be Zcol(FyePu/Acol) where Pu is the total axial force in the column.
The ratios are computed at the center of the reduced section for beams and at the face of the girder for
columns.
To find the shear demand at the panel zones, the total moment in the girders (at the left and right sides of
the joint) is divided by the effective beam depth to produce the panel shear due to beam flange forces.
Then the column shear at above or below the panel zone joint was subtracted from the beam flange shears
and the panel zone shear force is obtained. This force is divided by the shear strength capacity,
(which is discussed in Section 4.2.4.2) to determine the DCR of the panel zones.
Several observations are made regarding the likely inelastic behavior of the frame:
The structure has considerable overstrength, particularly at the upper levels.
The sequence of yielding will progress from the lowerlevel girders to the upperlevel girders.
Because of relatively low live load, the DCRs in the girders are almost uniform at each level.
Hence, all the hinges in the girders in a level will form almost simultaneously.
With the possible exception of the first level, the girders should yield before the columns. While
not shown in the table, the DCRs for the lowerstory columns are controlled by the moment at the
base of the column. It is usually very difficult to prevent yielding of the base of the firststory
columns in moment frames and this frame is no exception. The column on the leeward (right)
side of the building will yield first because of the additional axial compressive force arising from
the seismic effects.
The maximum DCR of the columns and girders is 3.475, while the maximum DCR for the panel
zones without doubler plates is 4.339. Thus, if doubler plates aren t used, the first yield in the
structure is in the panel zones. However, with doubler plates added, the first yield is at the
girders as the maximum DCR of the panel zones reduces to 2.405.
(a) DCRs of columns and girders
(b) DCRs of panel zones with and without doubler plates
(DCR values in parentheses are without doubler plates)
Figure 4.25 DCRs for elements from preliminary analysis with Pdelta effects included
4.2.3.2.3 Results of preliminary analysis: overall system strength. The last step in the preliminary
analysis is to estimate the total lateral strength (collapse load) of the frame using virtual work. In the
analysis, it is assumed that plastic hinges are perfectly plastic. Girders hinge at a value ZeFye and the
hinges form at the center of the reduced section (approximately 15 inches from the face of the column).
Columns hinge only at the base and the plastic moment capacity is assumed to be Zcol(FyePu/Acol). The
fully plastic mechanism for the system is illustrated in Figure 4.26. The inset to the figure shows how
the angle modification term, å, was computed. The strength, V, for the total structure is computed from
the following relationships (see Figure 4.26 for nomenclature):
Internal Work = External Work
Internal Work = 2[20å MPA + 40å MPB + (MPC + 4MPD + MPE)]
Three lateral force patterns are used: uniform, upper triangular and Standard (where the Standard pattern
is consistent with the vertical force distribution of Table 4.23 in this volume of design examples). The
results of the analysis are shown in Table 4.27. As expected, the strength under uniform load is
significantly greater than under triangular or Standard load. The closeness of the Standard and triangular
load strengths results from the verticalloaddistributing parameter (k = 1.385) being close to 1.0.
The ELF base shear, 759 kips (see Table 4.23), when divided by the Standard pattern capacity,
2,616 kips, is 0.29. This is reasonably consistent with the DCRs shown in Figure 4.25.
Table 4.27 Lateral Strength on Basis of RigidPlastic Mechanism
Lateral Load Pattern
Lateral strength for
entire structure (kips)
Lateral strength
single frame (kips)
Uniform
3,332
1,666
Upper Triangular
2,747
1,373
Standard
2,616
1,308
Figure 4.26 Plastic mechanism for computing lateral strength
Three important points concerning the virtual work analysis are as follows:
The rigidplastic analysis does not include strain hardening, which is an additional source of
overstrength.
The rigidplastic analysis does not consider the true behavior of the panel zone region of the
beamcolumn joint. Yielding in this area can have a significant effect on system strength.
Slightly more than 15 percent of the system strength comes from plastic hinges that form in the
columns. If the strength of the column is taken simply as Mp (without the influence of axial
force), the difference in total strength is less than 2 percent.
4.2.4 Description of Model Used for Detailed Structural Analysis
Nonlinear static and nonlinear dynamic analyses require a much more detailed model than was used in the
linear analysis. The primary reason for the difference is the need to explicitly represent yielding in the
girders, columns and panel zone region of the beamcolumn joints.
The DRAIN model used for the nonlinear analysis is shown in Figure 4.27. A detail of a girder and its
connection to two interior columns is shown in Figure 4.28. The detail illustrates the two main features
of the model: an explicit representation of the panel zone region and the use of concentrated plastic
hinges in the girders.
In Figure 4.27, the column shown to the right of the structure is used to represent Pdelta effects. See
Section 4.2.3.2 for details.
Figure 4.27 Detailed analytical model of sixstory frame
Figure 4.28 Model of girder and panel zone region
The development of the numerical properties used for panel zone and girder hinge modeling is not
straightforward. For this reason, the following theoretical development is provided before proceeding
with the example.
4.2.4.1 Plastic hinge modeling and compound nodes. In the analysis described below, much use is
made of compound nodes. These nodes are used to model plastic hinges in girders and deformations in
the panel zone region of beamcolumn joints.
A compound node typically consists of a pair of single nodes with each node sharing the same point in
space. The X and Y degrees of freedom of the first node of the pair (the slave node) are constrained to be
equal to the X and Y degrees of freedom of the second node of the pair (the master node), respectively.
Hence, the compound node has four degrees of freedom: an X displacement, a Y displacement and two
independent rotations.
In most cases, one or more rotational spring connection elements (DRAIN element Type 4) are placed
between the two single nodes of the compound node and these springs develop bending moment in
resistance to the relative rotation between the two single nodes. If no spring elements are placed between
the two single nodes, the compound node acts as a momentfree hinge. A typical compound node with a
single rotational spring is shown in Figure 4.29. The figure also shows the assumed bilinear, inelastic
momentrotation behavior for the spring.
Figure 4.29 A compound node and attached spring
4.2.4.2 Modeling of beamcolumn joint regions. A very significant portion of the total story drift of a
momentresisting frame is due to deformations that occur in the panel zone region of the beamcolumn
joint. In this example, panel zones are modeled explicitly using an approach developed by Krawinkler
(1978) and described in more detail in Charney and Marshall (2006). Only a brief overview is presented
here.
This model, illustrated in Figure 4.210, represents the panel zone stiffness and strength by an assemblage
of four rigid links and two rotational springs. The links form the boundary of the panel and the springs
are used to provide the desired inelastic behavior. The model has the advantage of being conceptually
simple, yet robust. The disadvantage of the model is that the number of degrees of freedom required to
model a structure is significantly increased.4
Figure 4.210 Krawinkler beamcolumn joint model
The Krawinkler model assumes that the panel zone has two resistance mechanisms acting in parallel:
Shear resistance of the web of the column, including doubler plates
Flexural resistance of the flanges of the column
These two resistance mechanisms, apparent in AISC 360 Section J1011, are used for determining panel
zone shear strength:
The equation can be rewritten as:
In ASCE 41, the first term of the above equation is taken as and the second term is neglected
conservatively. In this study, the following equation in which the first term is taken as the same as in
ASCE 41 and the second term is taken from AISC 360, with the exception of replacing nominal yield
stress with expected yield strength (for consistency) is used to calculate the panel zone shear strength:
where the first term is the panel shear resistance and the second term is the plastic flexural resistance of
the column flange. The terms in the equations are defined as follows:
Fye = expected yield strength of the column and the doubler plate
dc = total depth of column
tp = thickness of panel zone region = column web thickness plus doubler plate thickness
bcf = width of column flange
tcf = thickness of column flange
db = total depth of girder
Additional terms used in the subsequent discussion are:
tbf = girder flange thickness
G = shear modulus of steel
Figure 4.211 Column flange component of panel zone resistance
The panel zone shear resistance, VPanel, is simply the effective shear area of the panel, dctp, multiplied by
the yield stress in shear, assumed as 0.55Fye. (The 0.55 factor is a simplification of the Von Mises yield
criterion that gives the yield stress in shear as times the strength in tension.) The additional
plastic flexural resistance provided by yielding in the column flange is neglected in ASCE 4106 but is
included herein.
The second term, 1.8VFlanges, is based on experimental observation. Testing of simple beamcolumn
subassemblies show that a kink forms in the column flanges as shown in Figure 4.211(a). If it can be
assumed that the kink is represented by a plastic hinge with a plastic moment capacity of Mp = FyeZ =
Fyebcftcf2/4, it follows from virtual work (see Figure 4.211b) that the equivalent shear strength of the
column flanges is:
and by simple substitution for Mp:
This value does not include the 1.8 multiplier that appears in the AISC equation. This multiplier is based
on calibration of experimental results. It should be noted that the flange component of strength is small
compared to the panel component unless the column has very thick flanges.
The shear stiffness of the panel is derived as shown in Figure 4.212:
noting that the displacement can be written as follows:
Figure 4.212 Column web component of panel zone resistance
Krawinkler assumes that the column flange component yields at four times the yield deformation of the
panel component, where the panel yield deformation is:
At this deformation, the panel zone strength is VPanel + 0.25 Vflanges; at four times this deformation, the
strength is VPanel + VFlanges. The inelastic forcedeformation behavior of the panel is illustrated in
Figure 4.213. This figure applies also to exterior joints (girder on one side only), roof joints (girders on
both sides, column below only) and corner joints (girder on one side only, column below only).
Figure 4.213 Forcedeformation behavior of panel zone region
The actual Krawinkler model is shown in Figure 4.210. This model consists of four rigid links,
connected at the corners by compound nodes. The columns and girders frame into the links at right
angles at Points I through L. These are momentresisting connections. Rotational springs are used at the
upper left (Point A) and lower right (Point D) compound nodes. These springs are used to represent the
panel resistance mechanisms described earlier. The upper right and lower left corners (Points B and C),
without rotational springs, act as real hinges.
The finite element model of the joint requires 12 individual nodes: one node each at Points I through L
and two nodes (compound node pairs) at Points A through D. It is left to the reader to verify that the total
number of degrees of freedom in the model is 28 (if the only constraints are associated with the corner
compound nodes).
The rotational spring properties are related to the panel shear resistance mechanisms by a simple
transformation, as shown in Figure 4.214. From the figure it may be seen that the moment in the
rotational spring is equal to the applied shear times the beam depth. Using this transformation, the
properties of the rotational spring representing the panel shear component of resistance are as follows:
Figure 4.214 Transforming shear deformation to rotational deformation
in the Krawinkler model
It is interesting to note that the shear strength in terms of the rotation spring is simply 0.55Fye times the
volume of the panel and the shear stiffness in terms of the rotational spring is equal to G times the panel
volume.
The flange component of strength in terms of the rotational spring is determined in a similar manner:
Because of the equivalence of rotation and shear deformation, the yield rotation of the panel is the same
as the yield strain in shear:
To determine the initial stiffness of the flange spring, it is assumed that this spring yields at four times the
yield deformation of the panel spring. Hence:
The complete resistance mechanism, in terms of rotational spring properties, is shown in Figure 4.213.
This trilinear behavior is represented by two elasticperfectly plastic springs at the opposing corners of the
joint assemblage.
If desired, strainhardening may be added to the system. ASCE 41 suggests use of a strain hardening
stiffness equal to 6 percent of the initial stiffness of the joint. In this analysis, the strainhardening
component was simply added to both the panel and the flange components:
Before continuing, one minor adjustment is made to the above derivations. Instead of using the nominal
total beam and girder depths in the calculations, the distance between the center of the flanges was used
as the effective depth. Hence:
where the nom part of the subscript indicates the property listed as the total depth in the AISC Manual.
The Krawinkler properties are now computed for a typical interior subassembly of the sixstory frame. A
summary of the properties used for all connections is shown in Table 4.28.
Table 4.28 Properties for the Krawinkler BeamColumn Joint Model
Connection
Girder
Column
Doubler plate
(in.)
Mpanel,
(in.k)
Kpanel,
(in.k/rad)
Mflanges,
(in.k)
Kflanges,
(in.k/rad)
A
W24x84
W21x122
8,782
3,251,567
1,131
104,721
B
W24x84
W21x122
1.00
23,419
8,670,846
1,131
104,721
C
W27x94
W21x147
11,934
4,418,647
1,637
151,486
D
W27x94
W21x147
1.00
28,510
10,555,656
1,637
151,486
E
W27x94
W21x201
15,386
5,696,639
3,314
306,771
F
W27x94
W21x201
0.875
30,180
11,174,176
3,314
306,771
Example calculations shown for row in bold type.
The sample calculations below are for Connection D in Table 4.28.
Material Properties:
Fye = 55.0 ksi (girder, column and doubler plate)
G = 11,200 ksi
Girder:
W27x94
db,nom = 26.90 in.
tbf = 0.745 in.
db = 26.16 in.
Column:
W21x147
dc,nom = 22.10 in.
tw = 0.72 in.
tcf = 1.150 in.
dc = 20.95 in.
bcf = 12.50 in.
Doubler plate: 1.00 in.
Total panel zone thickness = tp = 0.72 + 1.00 = 1.72 in.
= 1,090 kips
= 62.6 kips
= 403,581 kips/unit shear strain
= 0.0027
= 28,510 in.kips
= 10,555,656 in.kips/radian
= 1,637 in.kips
= 151,486 in.kips/radian
4.2.4.3 Modeling girders. Because this structure is designed in accordance with the strong
column/weakbeam principle, it is anticipated that the girders will yield in flexure. Although DRAIN
provides special yielding beam elements (Type 2 elements), more control over behavior is obtained
through the use of the Type 4 connection element.
The modeling of a typical girder is shown in Figure 4.28. This figure shows an interior girder, together
with the panel zones at the ends. The portion of the girder between the panel zones is modeled as four
segments with one simple node at midspan and one compound node near each end. The midspan node
is used to enhance the deflected shape of the structure.5 The compound nodes are used to represent
inelastic behavior in the hinging region.
The following information is required to model each plastic hinge:
The initial stiffness (moment per unit rotation)
The effective yield moment
The secondary stiffness
The location of the hinge with respect to the face of the column
AISC SDM recommends design practices to force the plastic hinge forming in the beam away from the
face of the column. There are two methods used to move the plastic hinges of the beam away from the
column face. The first one aims to reduce the crosssectional properties of the beam at a specific location
away from the column, and the second one focuses on special detailing of the beamcolumn connection to
provide adequate strength and toughness in the connection so that inelasticity will be forced into the beam
adjacent to the column face. In this study the reduced beam section (RBS) was used.
A side view of the reduced beam sections is shown in Figure 4.215. The distance between the column
face and the edge of the reduced beam section was chosen as and the reduced section length
was assumed as . Both of these values are just at the middle of the limits stated in AISC 358.
Plastic hinges of the beams are modeled at the center of the reduced section length.
Figure 4.215 Side view of beam element and beam modeling
To determine the plastic hinge capacities of the girder cross section, a momentcurvature analysis, which
is dependent on the stressstrain curve of the steel, was implemented. The idealized stressstrain curve is
shown in Figure 4.216. This curve does not display a yield plateau, which is consistent with the
assumption that the section has yielded in previous cycles, with the Bauschinger effect eliminating any
trace of the yield plateau. The strain hardening ratio is taken as 3 percent of the initial stiffness and the
curvature ductility limit used is 20.
To compute the momentcurvature relationship, the girder is divided into 50 slices through it s depth,
with 10 slices in each flange and 30 slices at the web. By gradually increasing the rotation, fiber strains,
fiber stresses, fiber forces and then the resisting moment are found consecutively. Figure 4.217 shows
the top view of the assumed reduced beam section in this study. The reduced beam length is divided into
seven equal sections and flange widths of each section are calculated using the radius of the cut. The
radius of the cut, R, is calculated using the formulas in AISC 358.
where:
= depth of cut at center of the RBS, in.
= width of beam flange, in.
= length of RBS cut, in.
= beam depth, in.
= distance from face of the column to start of RBS cut, in.
= radius of cut, in.
Figure 4.216 Assumed stressstrain curve for modeling girders
Figure 4.217 Top view of RBS
Figure 4.218 shows the momentcurvature diagram for the W27x94 girder. As may be seen in the figure,
the momentcurvature relationship is different at each segment of the reduced length. The locations of the
different reduced beam sections used in Figure 4.218, named as bf1 , bf2 and bf3 , can be seen in
Figure 4.217. Because of the closely adjacent locations chosen for 0.65bf and bf3 (see Figure 4.2
17), their momentcurvature plots are nearly indistinguishable from each other in Figure 4.218.
Figure 4.218 Momentcurvature diagram for W27x94 girder
A tip loaded cantilever beam analysis using half of the clear span length is used to generate the moment
rotation relationship for the inelastic hinges. For regular beams, where a cantilever beam is tip loaded, the
moment diagram is linear and the curvature diagram is also linear as long as the moment along the beam
remains in the elastic region (see Figure 4.219). If the moment along the beam exceeds the yield
moment, the curvature along the beam will be as shown in Figure 4.220.
Figure 4.219 Tip loaded cantilever beam and moment diagram for cantilever beam
Figure 4.220 Curvature diagram for cantilever beam
Because a RBS is used in this study, the curvature diagrams are different from those for regular beams.
As may be seen in Figure 4.221, the curvatures in the reduced flange region of the beam have a
distinctive bump . Because the moment diagram of the tip loaded cantilever beam is always linear, the
moment values can be found easily at the different sections of the reduced flange, and then the
corresponding curvature values can be assigned from the moment curvature diagram (Figure 4.218) to
the curvature diagram along half of the clear span length (Figure 4.221).
Figure 4.221 Curvature diagram for cantilever beam with reduced beam section
Figure 4.221 shows the curvature diagram when the curvature ductility reaches 20. The curvature
difference (the bump at the center of RBS in Figure 4.221) section is less prominent when the ductility is
smaller. Given the curvature distribution along the cantilever beam length, the deflections at the point of
load (tip deflections) can be found using the moment area method. Figure 4.222 illustrates the force
displacement relationship at the end of the half span cantilever for the W27x94 with the reduced flange
section.
Figure 4.222 Force displacement diagram for W27x94 with RBS
To convert the forcetip displacement diagram into momentrotation of the plastic hinge, the following
procedure is followed:
1. Using the trilinear force displacement relationship shown in Figure 4.222, find the moment at the
plastic hinge for P1, P2 and P3 load levels and name them M1, M2 and M3. To find the moments,
the tip forces (P1, P2 and P3) are multiplied by the distance from the center of the reduced section
to the tip of the cantilever.
2. Calculate the change in moment for each added load (for example: dM1 = M2  M1).
3. Find the flexural rigidity (EI) of the beam given a tip displacement of 1 inch under the first load
(P1 in Figure 4.222).
4. Calculate the required rotational stiffnesse of the hinge between M1 and M2 and then M2 and M3.
5. Calculate the change in rotation from M1 to M2 and from M2 to M3, by dividing the change in
moment found at Step 2 by the required rotational stiffness values calculated at Step 4.
6. Find the specific rotations at M1, M2 and M3 using the change in rotation values found in Step 5.
Note that the rotation is zero at M1.
7. Plot a momentrotation diagram of the plastic hinge using the values calculated at Step 1 and
Step 6.
Figure 4.223 shows the momentrotation diagrams for the plastic hinges of both of the girders used in the
models. Note that two bilinear springs (Components 1 and 2) are needed to represent the trilinear
behavior shown in the figure.
Figure 4.223 Momentrotation diagram for girder hinges with RBS
The properties for the W24x84 and W27x94 girder are shown in Table 4.29. Note that the first yield of
the model is the yield moment from Component 1.
s
Table 4.29 Girder Properties as Modeled in DRAIN
Property
Section
W24x84
W27x94
Elastic Properties
Moment of Inertia (in.4)
2,370
3,270
Shear Area (in.2)
11.3
13.2
Inelastic Component 1
Yield Moment (in.kip)
8,422
10,458
Initial Stiffness (in.kip/radian)
1 1010
1 1010
S.H. Ratio
0.0
0.0
Inelastic Component 2
Yield Moment (in.kip)
2,075
2,615
Initial Stiffness (in.kip/radian)
287,550
337,020
S.H. Ratio
0.217
0.232
Comparative Property
Plastic Moment = ZeFye
9,200
11,539
4.2.4.4 Modeling columns. All columns in the analysis are modeled in DRAIN with Type2 elements.
Preliminary analysis indicated that columns should not yield, except at the base of the first story.
Subsequent analysis shows that the columns will yield in the upper portion of the structure as well. For
this reason, column yielding must be activated in all of the Type2 column elements. The columns are
modeled using the builtin yielding functionality of the DRAIN program, wherein the yield moment is a
function of the axial force in the column. The yield surfaces used by DRAIN for all the columns in the
model are shown in Figure 4.224.
Figure 4.224 Yield surface used for modeling columns
The rules employed by DRAIN to model column yielding are adequate for eventtoevent nonlinear static
pushover analysis, but leave much to be desired where dynamic analysis is performed. The greatest
difficulty in the dynamic analysis is adequate treatment of the column when unloading and reloading. An
assessment of the effect of these potential problems is beyond the scope of this example.
4.2.4.5 Results of detailed analysis.
4.2.4.5.1 Period of vibration. Table 4.210 tabulates the first three natural modes of vibration for
models with and without doubler plates. While the Pdelta effects increase the period, the doubler plates
decrease the period because the model becomes stiffer with doubler plates. As may be seen, different
period values are obtained from preliminary and detailed analyses (see Table 4.24). The detailed model
results in a slightly stiffer structure than the preliminary model especially when doubler plates are added.
Table 4.210 Periods of Vibration From Detailed Analysis (sec/cycle)
Model
Mode
Pdelta excluded
Pdelta included
Strong Panel with
Doubler Plates
1
1.912
1.973
2
0.627
0.639
3
0.334
0.339
Weak Panel without
Doubler Plates
1
2.000
2.069
2
0.654
0.668
3
0.344
0.349
4.2.4.5.2 Demandtocapacity ratios. DCRs are found for the detailed analyses with the same load
combination used for the preliminary analyses. The main reason for repeating the DCR for the detailed
model is to make a comparison with the DCR of the preliminary model. The detailed analyses include the
advanced panel zone, girder and column modeling discussed in Section 4.2.4. Figures 4.225(a) and 4.2
25(b) illustrate the DCR of the beams with columns and panel zones of the detailed model respectively.
In both figures, the values in the parentheses represent the DCR with no doubler plates added to the
structure.
The DCR values of the detailed analyses are similar to those of the preliminary analysis displayed in
Figure 4.25. The girders of the first and fifth bays at the third level have the maximum DCR in both the
preliminary and detailed analyses. Similar trends are also observed for the DCR of the columns in both
analyses. Note that the flexural stiffness of the girders is decreased by 7 percent in the preliminary
analyses to compensate for the effect of reduced beam sections (with 35 percent flange reduction) which
are included in the detailed analyses.
Similar to the preliminary DCR, the panel zone DCR increases significantly when doubler plates aren t
used. Since the doubler plates are used only at the interior columns, that is where the difference of the
DCRs changes significantly with and without doubler plates. See Figure 4.25(b) and Figure 4.225 (b).
(a) DCRs of columns and girders with and without doubler plates
(DCR values in parentheses are for without doubler plates)
(b) DCRs of panel zones with and without doubler plates
(DCR values in parentheses are for without doubler plates)
Figure 4.225 DCRs for elements from detailed analysis with Pdelta effects included
4.2.5 Nonlinear Static Analysis
As mentioned in the introduction to this example, nonlinear static (pushover) analysis is not an allowed
analysis procedure in the Standard, nor does it appear in ASCE 710. The method is allowed in analysis
related to rehabilitation of existing buildings and guidance for that use is provided in ASCE 41.
The Provisions makes at least two references to pushover analysis. In Section 12.8.7 of Part 1 pushover
analysis is used to determine if structures with stability ratios (see Equation 12.816) greater than 0.1 are
allowed. Such systems have a potential for dynamic instability and the pushover curve is used to
determine if the slope of the pushover curve is continuously positive up to the target displacement. If the
slope is positive, the system is deemed acceptable. If not, it must be redesigned such that either the
stability ratio is less than 0.1, or the slope stays positive. The analysis carried out for this purpose must be
performed according to the requirements of ASCE 41.
Pushover analysis is also mentioned in Provisions Part 3 Resource Paper 2. The intent of the procedure
outlined there is to determine whether lateral strength is nominally less than that required by the ELF
procedure. The use of nonlinear static analysis for this purpose is limited to structures with a height of
less than 40 feet. The building under consideration has a height of 77.5 feet and violates this limit.
In this example, pushover analysis is used simply to establish an estimate of the inelastic behavior of the
structure under gravity and lateral loads. Of particular interest is the sequence of yielding in the beams,
columns and panel zones; the lateral strength of the structure; the expected inelastic displacement; and the
basic shape of the pushover curve. In the authors opinion, such analysis should always be used as a
precursor to nonlinear response history analysis. Without pushover analysis as a precursor, it is difficult
to determine if the response history analysis is producing reasonable results.
The nonlinear static analysis illustrated in this example follows the recommendations of ASCE 41. The
reader is also referred to FEMA 440.
The pushover curve obtained from a nonlinear static analysis is a function of both modeling and load
application. For this example, the structure is subjected to the full dead load plus 50 percent of the fully
reduced live load, followed by the lateral loads.
The Provisions states that the lateral load pattern should follow the shape of the first mode. In this
example, three different load patterns are initially considered:
UL = uniform load (equal force at each level)
ML = modal load (lateral loads proportional to first mode shape)
BL = Provisions load distribution (using the forces indicated in Table 4.23)
Relative values of these load patterns are summarized in Table 4.211. The loads have been normalized
to a value of 15 kips at Level 2.
DRAIN analyses are run with Pdelta effects included and, for comparison purposes, with such effects
excluded. This effect is represented through linearized geometric stiffness, which is the basis of the
outrigger column shown in Figure 4.24. Consistent geometric stiffness, which may be used to represent
the influence of axial forces on the flexural flexibility of individual columns, may not be used directly in
DRAIN. Such effects may be approximated in DRAIN by subdividing columns into several segments
and activating the linearized geometric stiffness on a columnbycolumn basis.
Table 4.211 Lateral Load Patterns Used in Nonlinear Static Analysis
Level
Uniform load, UL
(kips)
Modal load, ML
(kips)
Provisions load,
BL
(kips)
R
15.0
85.1
144.8
6
15.0
77.3
114.0
5
15.0
64.8
84.8
4
15.0
49.5
58.2
3
15.0
32.2
34.6
2
15.0
15.0
15.0
As described later, the pushover analysis indicates most of the yielding in the structure occurs in the clear
span of the girders and columns. Panel zone hinging occurs only at the exterior columns where doubler
plates weren t used. To see the effect of doubler plates, the ML analysis is repeated for a structure
without doubler plates. These structures are referred to as the strong panel (SP) and weak panel (WP)
structures, respectively.
The analyses are carried out using the DRAIN2Dx computer program. Using DRAIN, an analysis may
be performed under load control or under displacement control. Under load control, the structure is
subjected to gradually increasing lateral loads. If, at any load step, the tangent stiffness matrix of the
structure has a negative on the diagonal, the analysis is terminated. Consequently, loss of strength due to
Pdelta effects cannot be tracked. Using displacement control, one particular point of the structure (the
control point) is forced to undergo a monotonically increasing lateral displacement, and the lateral forces
are constrained to follow the desired pattern. In this type of analysis, the structure can display loss of
strength because the displacement control algorithm adds artificial stiffness along the diagonal to
overcome the stability problem. This approach is meaningful because structures subjected to dynamic
loading can display strength loss and remain stable incrementally. It is for this reason that the post
strengthloss realm of the pushover response is of interest.
Where performing a displacement controlled pushover analysis in DRAIN with Pdelta effects included,
one must be careful to recover the baseshear forces properly.7 At any displacement step in the analysis,
the true base shear in the system consists of two parts:
where the first term represents the sum of all the column shears in the first story, and the second term
represents the destabilizing Pdelta shear in the first story. The Pdelta effects for this structure are
included through the use of the outrigger column shown at the right of Figure 4.24. Figure 4.226 plots
two base shear components of the pushover response for the SP structure subjected to the ML loading.
Also shown is the total response. The kink in the line representing Pdelta forces occurs because these
forces are based on firststory displacement, which, for an inelastic system, generally will not be
proportional to the roof displacement.
For all of the pushover analyses reported in this example, the structure is pushed to a displacement of
37.5 inches at the roof level. This value is approximately 4 percent of the total height.
Figure 4.226 Two base shear components of pushover response
4.2.5.1 Pushover response of strong panel structure. Figure 4.227 shows the pushover response of
the SP structure to all three lateral load patterns where Pdelta effects are excluded. In each case, gravity
loads are applied first, and then the lateral loads are applied using the displacement control algorithm.
Figure 4.228 shows the response curves if Pdelta effects are included. In Figure 4.229, the response of
the structure under ML loading with and without Pdelta effects is illustrated. Clearly, Pdelta effects are
an extremely important aspect of the response of this structure and the influence grows in significance
after yielding. This is particularly interesting in the light of the Standard, which ignores Pdelta effects in
elastic analysis if the maximum stability ratio is less than 0.10 (see Sec. 12.87). For this structure, the
maximum computed stability ratio is 0.0862 (see Table 4.25), which is less than 0.10 and is also less
than the upper limit of 0.0909. The upper limit is computed according to Standard Equation 12.817 and
is based on the very conservative assumption that = 1.0. While the Standard allows the analyst to
exclude Pdelta effects in an elastic analysis, this clearly should not be done in the pushover analysis (or
in response history analysis). (In the Provisions the upper limit for the stability ratio is eliminated.
Where the calculated is greater than 0.10, a pushover analysis must be performed in accordance with
ASCE 41, and it must be shown that that the slope of the pushover curve is positive up to the target
displacement. The pushover analysis must be based on the MCE spectral acceleration and must include
Pdelta effects [and loss of strength, as appropriate]. If the slope of the pushover curve is negative at
displacements less than the target displacement, the structure must be redesigned such that is less than
0.10 or the pushover slope is positive up to the target displacement.)
Figure 4.227 Response of strong panel model to three load patterns,
excluding Pdelta effects
Figure 4.228 Response of strong panel model to three load patterns,
including Pdelta effects
Figure 4.229 Response of strong panel model to ML loads,
with and without Pdelta effects
In Figure 4.230, a plot of the tangent stiffness versus roof displacement is shown for the SP structure
with ML loading and with Pdelta effects excluded or included. This plot, which represents the slope of
the pushover curve at each displacement value, is more effective than the pushover plot in determining
when yielding occurs. As Figure 4.230 illustrates, the first significant yield occurs at a roof displacement
of approximately 6.5 inches and that most of the structure s original stiffness is exhausted by the time the
roof displacement reaches 13 inches.
Figure 4.230 Tangent stiffness history for structure under ML loads,
with and without Pdelta effects
For the case with Pdelta effects excluded, the final tangent stiffness shown in Figure 4.230 is
approximately 10.2 kips/in., compared to an original value of 139 kips/in. Hence, the strainhardening
stiffness of the structure is 0.073 times the initial stiffness. This is somewhat greater than the 0.03 (3.0
percent) strain hardening ratio used in the development of the model because the entire structure does not
yield simultaneously.
Where Pdelta effects are included, the final tangent stiffness is 1.6 kips per inch. The structure attains
this negative tangent stiffness at a displacement of approximately 23 inches.
4.2.5.1.1 Sequence and pattern of plastic hinging. The sequence of yielding in the structure with ML
loading and with Pdelta effects included is shown in Figure 4.231. Part (a) of the figure shows an
elevation of the structure with numbers that indicate the sequence of plastic hinge formation. For
example, the numeral 1 indicates that this was the first hinge to form. Part (b) of the figure shows a
pushover curve with several hinge formation events indicated. These events correspond to numbers
shown in Part (a) of the figure. The pushover curve only shows selected events because an illustration
showing all events would be difficult to read. Comparing Figure 4.231(a) with Figures 4.25 and 4.225,
it can be seen how the DCRs indicate the plastic hinge formation sequence. The highest ratios in
Figure 4.25 are observed at the girders of the third and the second levels beginning from the bays at the
leeward (right) side. As may be seen from Figure 4.231(a), first plastic hinges form at the same locations
of the building. Similarly, the first panel zone hinge forms at the beamcolumn joint of the sixth column
at the fourth level, and this is where the highest DCR values are obtained for the panel zones in both
preliminary and detailed DCR analyses.
(a)
(b)
Figure 4.231 Patterns of plastic hinge formation: SP model under ML load,
including Pdelta effects
Several important observations are made from Figure 4.231:
There is no hinging in Levels 6 and R.
There is panel zone hinging only at the exterior columns at Levels 4 and 5. Panel zone hinges do
not form at the interior joints where doubler plates are used.
Hinges form at the base of all the Level 1 columns.
Plastic hinges form in all columns on Level 3 and all the interior columns on Level 4.
Both ends of all the girders at Levels 2 through 5 yield.
It appears the structure is somewhat weak in the middle two stories and is relatively strong at the upper
stories. The doubler plates added to the interior columns prevent panel zone yielding.
The presence of column hinging at Levels 3 and 4 is a bit troublesome because the structure is designed
as a strongcolumn/weakbeam system. This design philosophy, however, is intended to prevent the
formation of complete story mechanisms, not to prevent individual column hinging. While hinges do
form at the top of each column in the third story, hinges do not form at the bottom of these columns and a
complete story mechanism is avoided.
Even though the pattern of hinging is interesting and useful as an evaluation tool, the performance of the
structure in the context of various acceptance criteria cannot be assessed until the expected inelastic
displacement can be determined. This is done below in Section 4.2.5.3.
4.2.5.1.2 Comparison with strength from plastic analysis. It is interesting to compare the strength of
the structure from pushover analysis with that obtained from the rigidcollapse analysis performed using
virtual work. These values are summarized in Table 4.212. The strength from the case with Pdelta
excluded was estimated from the curves shown in Figure 4.227 and is taken as the strength at the
principal bend in the curve (the estimated yield from a bilinear representation of the pushover curve).
Consistent with the upper bound theorem of plastic analysis, the strength from virtual work is greater than
that from pushover analysis. The reason for the difference in predicted strengths is related to the pattern
of yielding that actually formed in the structure, compared to that assumed in the rigidplastic analysis.
Table 4.212 Strength Comparisons: Pushover versus Rigid Plastic
Pattern
Lateral Strength (kips)
Pdelta Excluded
Pdelta Included
RigidPlastic
Uniform
1,340
1,270
1,666
Modal (Triangular)
1,200
1,130
1,373
BSSC
1,170
1,105
1,308
4.2.5.2 Pushover response of weak panel structure. Before continuing, the structure should be re
analyzed without panel zone reinforcing, and the behavior compared with that determined from the
analysis described above. For this exercise, only the modal load pattern is considered, but the analysis is
performed with and without Pdelta effects.
The pushover curves for the structure under modal loading and with weak panels are shown in
Figure 4.232. Curves for the analyses run with and without Pdelta effects are included. Figures 4.233
and 4.234 are more informative because they compare the response of the structures with and without
panel zone reinforcement. Figure 4.235 shows the tangent stiffness history comparison for the structures
with and without doubler plates. In both cases Pdelta effects have been included.
Figures 4.232 through 4.235 show that the doubler plates, which represent approximately 2.0 percent of
the volume of the structure, increase the strength and initial stiffness by approximately 10 percent.
Figure 4.232 Weak panel zone model under ML load
Figure 4.233 Comparison of weak panel zone model with strong panel zone model,
excluding Pdelta effects
Figure 4.234 Comparison of weak panel zone model with strong panel zone model,
including Pdelta effects
Figure 4.235 Tangent stiffness history for structure under ML loads,
strong versus weak panels, including Pdelta effects
The difference between the behavior of the structures with and without doubler plates is attributed to the
yielding of the panel zones in the structure without panel zone reinforcement. The sequence of hinging is
illustrated in Figure 4.236. Part (a) of this figure indicates that panel zone yielding occurs early. (Panel
zone yielding is indicated by a numeric sequence label in the corner of the panel zone.) In fact, the first
yielding in the structure is due to yielding of a panel zone at the fourth level of the structure, which is
consistent with panel zone DCR calculated before where no doubler plates were added to the structure.
Under very large displacements the flange component of the panel zone yields. Girder and column
hinging also occurs, but the column hinging appears relatively late in the response. It is also significant
that the upper two levels of the structure display yielding in several of the panel zones.
Aside from the relatively marginal loss in stiffness and strength due to removal of the doubler plates, it
appears that the structure without panel zone reinforcement behaves adequately. Of course, actual
performance cannot be evaluated without predicting the maximum inelastic panel shear strain and
assessing the stability of the panel zones under these strains.
(a)
(b)
Figure 4.236 Patterns of plastic hinge formation: weak panel zone model under ML load,
including Pdelta effects
4.2.5.3 Target displacement. In this section, the only loading pattern considered is the modal load
pattern discussed earlier. This is consistent with the requirements of ASCE 41 and FEMA 440. The
structures with strong and weak panel zones are analyzed including Pdelta effects.
ASCE 41 uses the coefficient method for calculating target displacement. The target displacement is
computed as follows:
where:
modification factor to relate roof displacement of a multiple degree of freedom building
system to the spectral displacement of an equivalent single degree of freedom system
the ordinate of mode shape 1 at the roof (control node)
the first mode participation factor
modification factor to relate expected maximum inelastic displacements to displacements
calculated for linear elastic response
modification factor to represent the effect of pinched hysteresis shape, cyclic stiffness
degradation and strength deterioration on maximum displacement response
response spectrum acceleration, at the effective fundamental period and damping ratio of the
building in the direction under consideration
effective fundamental period of the building in the direction under consideration
elastic fundamental period in the direction under consideration calculated by elastic dynamic
analysis
elastic lateral stiffness of the building in the direction under consideration
effective lateral stiffness of the building in the direction under consideration
acceleration due to gravity
To find the coefficient , the general horizontal response spectrum defined in ASCE 41 Section 1.6.1.5
is used. The damping of the spectrum is chosen as 2 percent for this study. The same damping ratio is
used in the dynamic analysis. The parameters and are chosen as the same values as SMS and SM1
which are defined in Section 4.2.2.2 of this study. Note that these are the MCE spectral acceleration
parameters. Figure 4.237 shows the horizontal response spectrum obtained from ASCE 41. The
parameters of this spectrum are discussed further with the dynamic analyses. This spectrum is for the
Basic Safety Earthquake 2 (BSE2) hazard level which has a 2 percent probability of exceedance in
50 years.
Coefficient is found using the first mode shape of the model with the mass matrix. For both strong
and weak panel models, the coefficient is found a bit higher than 1.3, which is the value provided in
Table 3.2 of ASCE41 for the shear buildings with triangular load pattern.
and are equal to 1.0 for periods greater than 1.0 second and 0.7 second respectively. Since the
first mode periods of the strong and weak panel models are both approximately 2 seconds, these
coefficients are taken as 1.0.
Figure 4.237 Two percent damped horizontal response spectrum from ASCE 41
To find the target displacement, the procedure described in ASCE 41 is followed. The nonlinear force
displacement relationship between the base shear and displacement of the control node are replaced with
an idealized forcedisplacement curve. The effective lateral stiffness and the effective period depend on
the idealized forcedisplacement curve. The idealized forcedisplacement curve is developed using an
iterative graphical procedure where the areas below the actual and idealized curves are balanced
approximately up to a displacement value of . is the displacement at the end of second line
segment of the idealized curve and is the base shear at the same displacement. ( , ) should be a
point on the actual force displacement curve either at the calculated target displacement or at the
displacement corresponding to the maximum base shear, whichever is the least. The first line segment of
the idealized forcedisplacement curve should begin at the origin and finish at ( , ), where is the
effective yield strength and is the yield displacement of idealized curve. The slope of the first line
segment is equal to the effective lateral stiffness, , which should be taken as the secant stiffness
calculated at a base shear force equal to 60 percent of the effective yield strength of the structure. See
Figures 4.238 and 4.239 for the actual and idealized forcedisplacement curves of strong and weak panel
models which are under ML loading and both include Pdelta effects.
Figure 4.238 Actual and idealized force displacement curves for strong panel model,
under ML load, including Pdelta effects
Figure 4.239 Actual and idealized force displacement curves for weak panel model,
under ML load, including Pdelta effects
Table 4.213 shows the target displacement values of SP and WP models. Story drifts are also shown at
the load level of target displacement for both models.
Table 4.213 Target Displacement for Strong and Weak Panel Models
Strong Panel
Weak Panel
1.303
1.310
1.000
1.000
1.000
1.000
(g)
0.461
0.439
(sec)
1.973
2.069
(in.) at Roof Level
22.9
24.1
Drift R6 (in.)
0.96
1.46
Drift 65 (in.)
1.76
2.59
Drift 54 (in.)
2.87
3.73
Drift 43 (in.)
4.84
4.84
Drift 32 (in.)
5.74
5.35
Drift 21 (in.)
6.73
6.12
Negative tangent stiffness starts at 22.9 inches and 29.3 inches for strong and weak panel models,
respectively. Thus negative tangent stiffness starts after target displacements for both models. Again
note that these displacements are computed from the 2 percentdamped MCE horizontal response
spectrum of ASCE 41.
4.2.6 Response History Analysis
The response history analysis method, with three ground motions, is used to estimate the inelastic
deformation demands for the structure. While an analysis with seven or more ground motions generally
is preferable, that was not done here due to time and space limitations.
The analysis did consider a number of parameters, as follows:
Scaling of ground motions to the DBE and MCE level
Analysis with and without Pdelta effects
Two percent and five percent inherent damping
Added linear viscous damping
All of the models analyzed have Strong Panels (with doubler plates included in the interior beam
column joints).
4.2.6.1 Modeling and analysis procedure.
The DRAIN2Dx program is used for each of the response history analyses. With the exception of
requirements for including inherent damping, the structural model is identical to that used in the nonlinear
static analysis. Secondorder effects are included through the use of the leaning column element shown to
the right of the actual frame in Figure 4.24. Only onehalf of the building (a single frame in the NS
direction) is modeled.
Inelastic hysteretic behavior is represented through the use of a bilinear model. This model exhibits
neither a loss of stiffness nor a loss of strength and for this reason, it will generally have the effect of
overestimating the hysteretic energy dissipation in the yielding elements. Fortunately, the error produced
by such a model will not be of great concern for this structure because the hysteretic behavior of panel
zones and flexural plastic hinges should be very robust where inelastic rotations are less than about 0.03
radians.
Rayleigh proportional damping was used to represent viscous energy dissipation in the structure. The
mass and stiffness proportional damping factors are set initially to produce 2.0 percent damping in the
first and third modes. It is generally recognized that this level of damping (in lieu of the 5 percent
damping that is traditionally used in elastic analysis) is appropriate for nonlinear response history
analysis. Two percent damping is also consistent with that used in the pushover analysis (see
Section 4.2.5 of this example).
In Rayleigh proportional damping, the damping matrix, C, is a linear combination of the mass matrix, M
and the initial stiffness matrix, K:
where à and are mass and stiffness proportionality factors, respectively. If the first and third mode
frequencies, 1 and 3, are known, the proportionality factors may be computed from the following
expression (Clough & Penzien):
Note that à and are directly proportional to . To increase the target damping from 2 percent to
5 percent of critical, all that is required is a multiplying factor of 2.5 on à and .
The targeted structural frequencies and the resulting damping proportionality factors are shown in
Table 4.214. The frequencies shown in the table are computed from the detailed model shown in
Figure 4.27.
Table 4.214 Structural frequencies and damping factors used in response history analysis
(damping factors that produce 2 percent damping in modes 1 and 3)
Model/Damping Parameters
1
(rad/sec)
3
(rad/sec)
à
Strong Panel with Pdelta
3.184
18.55
0.109
0.00184
Strong Panel without Pdelta
3.285
18.81
0.112
0.00181
The stiffness proportional damping factor must not be included in the Type 4 elements used to represent
rotational plastic hinges in the structure. These hinges, particularly those in the girders, have a very high
initial stiffness. Before the hinge yields, there is virtually no rotational velocity in the hinge. After
yielding, the rotational velocity is significant. If a stiffness proportional damping factor is used for the
hinge, a viscous moment will develop in the hinge. This artificial viscous moment the product of the
rotational velocity, the initial rotational stiffness of the hinge and the stiffness proportional damping
factor can be quite large. In fact, the viscous moment may even exceed the intended plastic capacity of
the hinge. These viscous moments occur in phase with the plastic rotation; hence, the plastic moment and
the viscous moments are additive. These large moments transfer to the rest of the structure, affecting the
sequence of hinging in the rest of the structure and produce artificially high base shears. The use of
stiffness proportional damping in discrete plastic hinges can produce a totally inaccurate analysis result.
See Charney (2008) for details.
The structure is subjected to dead load and half of the fully reduced live load, followed by ground
acceleration. The incremental differential equations of motion are solved in a stepbystep manner using
the Newmark constant average acceleration approach. Time steps and other integration parameters are
carefully controlled to minimize errors. The minimum time step used for analysis is as small as 0.0005
second for the first earthquake and 0.001 second for the second and third earthquakes. A smaller
integration time step is required for the first earthquake because of its impulsive nature.
4.2.6.2 Development of ground motion records. The ground motion acceleration histories used in the
analysis are developed specifically for the site. Basic information for the records is shown in Table 4.1
20a.
Ground acceleration histories and 2 and 5percentdamped pseudoacceleration spectra for each of the
motions are shown in Figures 4.240 through 4.242. For these twodimensional analyses performed using
DRAIN, single ground motion components areapplied one at a time. For this example, the component
that produces the larger spectral acceleration at the structure s fundamental period (A00, B90 and C90) is
used. A complete analysis would require consideration of both components of ground motions and
possibly of a rotated set of components.
When analyzing structures in two dimensions, Section 16.1.3.1 of the Standard (as well as ASCE 710)
gives the following instructions for scaling:
The ground motions shall be scaled such that the average value of the 5 percent damped
response spectra for the suite of motions is not less than the design response spectrum
for the site for periods ranging from 0.2T to 1.5T where T is the natural period of the
structure in the fundamental mode for the direction of response being analyzed.
The scaling requirements in Provisions Part 3 Resource Paper 3are similar, except that the target spectrum
for scaling is the MCER spectrum. In this example, the only adjustment is made for scaling when the
inherent damping is taken as 2 percent of critical. In this case, the ground motion spectra are based on
2 percent damping and the DBE or MCE spectrum is adjusted from 5 percent damping to 2 percent
damping using the modification factors given in ASCE 41.
The scaling procedure described above has a degree of freedom in that there are an infinite number of
scaling factors that can fit the criterion. To avoid this, a twostep scaling process is used wherein each
spectrum is initially scaled to match the target spectrum at the structure s fundamental period and then the
average of the scaled spectra are rescaled such that no ordinate of the scaled average spectrum falls
below the target spectrum in the range of periods between 0.2T and 1.5T. The final scale factor for each
motion consists of the product of the initial scale factor and the second scale factor.
The initial scale factors, referred to as S1i (for each ground motion, i), are different for the three ground
motions. The second scale factor, S2, is the same for each ground motion. The scale factors used in the
response history analyses are shown in Table 4.215. Factors are determined for 2 percent and 5 percent
damping and for the DBE and MCE motions. The 2 percent damped target MCE spectrum corresponds
to ASCE 4106 spectrum used in the pushover analysis. If a scale factor of 1.367 is used for the structure
with 2 percent damping, Figure 4.243 indicates that the scaling criteria specified by the Standard are met
for all periods in the range 0.2(1.973) = 0.4 second to 1.5(1.973) = 3.0 seconds. 1.973 seconds is the
period of the SP model with Pdelta effects included (See Table 4.210).
Figure 4.240 Ground acceleration histories and response spectra for Record A
Figure 4.241 Ground acceleration histories and response spectra for Record B
Figure 4.242 Ground acceleration histories and response spectra for Record C
Table 4.215 Ground Motion Scale Factors Used in the Analyses
Scale Factor
2% Damped
DBE
2% Damped
MCE
5%Damped
DBE
5% Damped
MCE
Motion A00
S1
0.919
1.380
0.765
1.147
S2
1.367
1.367
1.428
1.428
SS
1.257
1.886
1.092
1.638
Motion B90
S1
1.495
2.245
1.439
2.159
S2
1.367
1.367
1.428
1.428
SS
2.045
3.068
2.056
3.084
Motion C90
S1
1.332
2.000
1.359
2.039
S2
1.367
1.367
1.428
1.428
SS
1.822
2.734
1.941
2.911
(a) Comparison of average of scaled spectra and target spectrum (SF = 1.367)
(b) Ratio of average of scaled spectra to target spectrum (SF = 1.367)
Figure 4.243 Ground motion scaling parameters
4.2.6.3 Results of response history analysis. The following parameters are varied to determine the
sensitivity of the response to that parameter:
Analyses are run with and without Pdelta effects for all three ground motions.
Analyses are run with 2 percent and 5 percent inherent damping. The ground motion scale
factors are correlated with the corresponding inherent damping of the structure.
Added dampers are used for the structure with 2 percent inherent damping. Various added
damper configurations are used. These analyses are performed to assess the potential benefit of
added viscous fluid damping devices. The SP model with Pdelta effects included is used for this
analysis and only Ground Motions A00 and B90 are used.
The results from the first series of analyses, all run with 2 or 5 percent of critical damping with and
without Pdelta effects, are summarized in Tables 4.216 through 4.223. Selected time history traces are
shown in Figures 4.244 through 4.248.
The tabulated shears in the tables are for the single frame analyzed and should be doubled to obtain the
total shear in the structure. The tables of story shear also provide two values for each ground motion.
The first value is the maximum total elastic column story shear, including Pdelta effects if applicable.
The second value represents the maximum total inertial force for the structure. The inertial base shear,
which is not necessarily concurrent with the column shears, was obtained as a sum of the products of the
total horizontal accelerations and nodal mass of each joint. For a system with no damping, the story
shears obtained from the two methods should be identical. For a system with damping, the base shear
obtained from column forces generally will be less than the shear from inertial forces because the viscous
component of column shear is not included. Additionally, the force absorbed by the mass proportional
component of damping will be lost (as this is not directly recoverable in DRAIN).
The total roof displacement and the story drifts listed in the tables are peak (envelope) values and are not
necessarily concurrent.
4.2.6.3.1 Response of structure with 2 and 5 percent of critical damping. Tables 4.216 and 4.217
summarize the results of the DBE analyses with 2 percent inherent damping, including and excluding
Pdelta effects. Part (a) of each table provides the maximum base shears, computed either as the sum of
column forces (including Pdelta effects as applicable), or as the sum of the products of the total
acceleration and mass at each level. In each case, the shears computed using the two methods are similar,
which serves as a check on the accuracy of the analysis. Had the analysis been run without damping, the
shears computed by the two methods should be identical. As expected base shears decrease when Pdelta
effects are included.
The maximum story drifts are shown in the (b) parts of each table. The drift limits in the table, equal to
2 percent of the story height, are the same as provided in Standard Table 12.121. Standard
Section 16.2.4.3 provides for the allowable drift to be increased by25 percent where nonlinear response
history analysis is used; these limits are shown in the tables in parentheses. Provisions Part 2 states that
the increase in drift limit is attributed to the more accurate analysis and the fact that drifts are computed
explicitly. Drifts that exceed the increased limits are shown in bold text in the tables.
When a SP frame with 2 percent inherent damping is analyzed under MCE spectrum scaled motions
excluding Pdelta effects, earthquake A00 results in 62.40inch displacement at the roof level and
approximately between 15 to 20inch drifts at the first three stories of the structure. These story drifts
are well above the limits. When Pdelta effects are included with the same level of motion, roof
displacement increases to 101.69 inches with approximately 20 to 40inch displacement at the first three
stories.
It is clear from Part (b) of Tables 4.216 and 4.217 that Ground Motion A00 is much more demanding
with respect to drift than are the other two motions. The drifts produced by Ground Motion A00 are
particularly large at the lower levels, with the more liberal drift limits being exceeded in the lower four
stories of the building. When Pdelta effects are included, the drifts produced by Ground Motion A00
increase significantly;drifts produced by Ground Motions B90 and C90 change only slightly.
Tables 4.218 and 4.219 provide result summaries for the structure analyzed with the MCE ground
motions. Damping is still set at 2 percent of critical and analysis is run with and without Pdelta effects.
The drift limits listed in the (b) parts of Tables 4.218 and 4.219 are based on Provisions Part 3 Resource
Paper 3 Section 16.4.5. These limits are 1.5 times those allowed by Standard Section 12.2.1. The
50 percent increase in drift limits is consistent with the increase in ground motion intensity when moving
from DBE to MCE ground motions. If all of the increase in drift limit is attributed to the DBEMCE
scaling, there is no apparent adjustment related to the more accurate analysis and explicit computation of
drift .
When Pdelta effects are included maximum story shears decrease and the drifts in lower stories increase
for all motions. The drifts predicted for Ground Motion A00 (as much as 40 inches) indicate probable
collapse of the structure. Loss of strength associated with such large drifts is not included in the
analytical model (since DRAIN does not provide a mechanism for decreasing moment capacity under
large plastic rotations). It is highly likely, however, that collapse would be predicted by more accurate
modeling.
Similar trends in response are produced when the inherent damping is increased from 2 percent critical to
5 percent. The results for the 5 percent damped analysis are provided in Tables 4.220 through 4.223.
The first two of these tables, Tables 4.120 and 4.121, are for the analysis using the DBE ground
motions. When compared to the results using 2 percent damping, it is seen that both the base shears and
the story drifts decrease significantly. DBElevel drifts at lower stories due to Ground Motion A00
exceed the drift limit but may not indicate collapse. MCElevel drifts produced by the A00 ground
motion indicate likely collapse.
Table 4.216 DBE Results for 2% Damped Strong Panel Model with P Delta Excluded
(a) Maximum Base Shear (kips)
Level
Motion A00
Motion B90
Motion C90
Column forces
1,780
1,649
1,543
Inertial forces
1,848
1,650
1,540
(b) Maximum Displacment and Story Drift (in.)
Level
Motion A00
Motion B90
Motion C90
Limit*
Roof displacement
26.80
14.57
13.55
NA
Drift R6
1.85
1.92
1.71
3.00 (3.75)
Drift 65
2.51
2.60
2.33
3.00 (3.75)
Drift 54
3.75
3.08
3.03
3.00 (3.75)
Drift 43
5.62
2.98
3.03
3.00 (3.75)
Drift 32
6.61
3.58
2.82
3.00 (3.75)
Drift 2G
8.09
4.68
3.29
3.60 (4.50)
*Values in ( ) reflect increased drift limits provided by Standard Sec. 16.2.4.3.
Table 4.217 DBE Results for 2% Damped Strong Panel Model with PDelta Included
(a) Maximum Base Shear (kips)
Level
Motion A00
Motion B90
Motion C90
Column forces
1,467
1,458
1,417
Inertial forces
1,558
1,481
1,419
(b) Maximum Displacement and Story Drift (in.)
Level
Motion A00
Motion B90
Motion C90
Limit*
Roof
displacement
32.65
14.50
14.75
NA
Drift R6
1.86
1.82
1.70
3.00 (3.75)
Drift 65
2.64
2.50
2.41
3.00 (3.75)
Drift 54
4.08
2.81
3.19
3.00 (3.75)
Drift 43
6.87
3.21
3.33
3.00 (3.75)
Drift 32
8.19
3.40
2.90
3.00 (3.75)
Drift 2G
10.40
4.69
3.44
3.60 (4.50)
*Values in ( ) reflect increased drift limits provided by Standard Sec. 16.2.4.3.
Table 4.218 MCE Results for 2% Damped Strong Panel Model with PDelta Excluded
(a) Maximum Base Shear (kips)
Level
Motion A00
Motion B90
Motion C90
Column forces
2,181
1,851
1,723
Inertial forces
2,261
1,893
1,725
(b) Maximum Displacement and Story Drift (in.)
Level
Motion A00
Motion B90
Motion C90
Limit
Roof
displacement
62.40
22.45
20.41
NA
Drift R6
1.98
2.30
3.05
4.5
Drift 65
3.57
2.77
3.69
4.5
Drift 54
7.36
3.33
4.43
4.5
Drift 43
14.61
4.61
4.45
4.5
Drift 32
16.29
5.21
3.97
4.5
Drift 2G
19.76
6.60
5.11
5.4
Table 4.219 MCE Results for 2% Damped Strong Panel Model with PDelta Included
(a) Maximum Base Shear (kips)
Level
Motion A00
Motion B90
Motion C90
Column Forces
1,675
1,584
1,507
Inertial Forces
1,854
1,633
1,515
(b) Maximum Story Drifts (in.)
Level
Motion A00
Motion B90
Motion C90
Limit
Total Roof
101.69
26.10
20.50
NA
R6
1.95
2.32
2.93
4.5
65
2.97
2.60
3.49
4.5
54
6.41
3.62
4.32
4.5
43
20.69
5.61
4.63
4.5
32
31.65
6.32
4.18
4.5
2G
40.13
7.03
5.11
5.4
Table 4.220 DBE Results for 5% Damped Strong Panel Model with PDelta Excluded
(a) Maximum Base Shear (kips)
Level
Motion A00
Motion B90
Motion C90
Column Forces
1,622
1,568
1,483
Inertial Forces
1,773
1,576
1,482
(b) Maximum Story Drifts (in.)
Level
Motion A00
Motion B90
Motion C90
*Limit
Total Roof
19.17
14.09
13.14
NA
R6
1.33
1.73
1.77
3.00 (3.75)
65
2.18
2.52
2.32
3.00 (3.75)
54
3.06
2.98
2.89
3.00 (3.75)
43
3.97
2.86
2.78
3.00 (3.75)
32
5.02
3.19
2.72
3.00 (3.75)
2G
6.13
4.05
3.01
3.60 (4.50)
*Values in ( ) reflect increased drift limits provided by Standard Sec. 16.2.4.3.
Table 4.221 DBE Results for 5% Damped Strong Panel Model with PDelta Included
(a) Maximum Base Shear (kips
Level
Motion A00
Motion B90
Motion C90
Column forces
1,374
1,419
1,355
Inertial forces
1,524
1,448
1,361
(b) Maximum Displacement and Story Drift (in.)
Level
Motion A00
Motion B90
Motion C90
*Limit
Roof
displacement
21.76
14.07
14.16
NA
Drift R6
1.40
1.56
1.73
3.00 (3.75)
Drift 65
2.25
2.42
2.33
3.00 (3.75)
Drift 54
3.23
2.80
3.00
3.00 (3.75)
Drift 43
4.38
3.04
3.09
3.00 (3.75)
Drift 32
5.60
3.28
2.77
3.00 (3.75)
Drift 2G
7.12
4.33
3.15
3.60 (4.50)
*Values in ( ) reflect increased drift limits provided by Standard Sec. 16.2.4.3.
Table 4.222 MCE Results for 5% Damped Strong Panel Model with PDelta Excluded
(a) Maximum Base Shear (kips)
Level
Motion A00
Motion B90
Motion C90
Column forces
1,918
1,760
1,630
Inertial forces
2,139
1,861
1,633
(b) Maximum Displacement and Story Drift (in.)
Level
Motion A00
Motion B90
Motion C90
Limit
oof displacement
40.84
20.17
21.10
NA
Drift R6
1.68
1.94
2.97
4.5
Drift 65
2.91
2.61
3.75
4.5
Drift 54
4.86
3.12
4.50
4.5
Drift 43
9.04
4.18
4.43
4.5
Drift 32
10.48
4.77
3.98
4.5
Drift 2G
13.04
6.09
4.93
5.4
Table 4.223 MCE Results for 5% Damped Strong Panel Model with PDelta Included
(a) Maximum Base Shear (kips)
Level
Motion A00
Motion B90
Motion C90
Column forces
1,451
1,486
1,413
Inertial forces
1,798
1,607
1,419
(b) Maximum Displacement and Story Drift (in.)
Level
Motion A00
Motion B90
Motion C90
Limit
Roof
displacement
54.33
23.12
21.83
NA
Drift R6
1.66
2.01
2.88
4.5
Drift 65
2.65
2.38
3.64
4.5
Drift 54
4.88
3.31
4.49
4.5
Drift 43
12.63
5.09
4.72
4.5
Drift 32
15.27
5.66
4.28
4.5
Drift 2G
19.31
6.14
5.07
5.4
4.2.6.3.2 Discussion of response history analyses. The computed structural response to Ground Motion
A00 is clearly quite different from that for Ground Motions B90 and C90. This difference in behavior
occurs even though the records are all scaled to produce exactly the same spectral acceleration at the
structure s fundamental period. A casual inspection of the ground acceleration histories and response
spectra (Figures 4.240 through 4.242) does not reveal the underlying reason for this difference in
behavior.
Figure 4.244 shows response histories of roof displacement and first story drift for the 2 percent damped
SP model subjected to the DBEscaled A00 ground motion. Two trends are readily apparent. First, the
vast majority of the roof displacement results in residual deformation in the first story. Second, the P
delta effect increases residual deformations by about 50 percent. Such extreme differences in behavior do
not appear in plots of base shear, as provided in Figure 4.245.
The residual deformations shown in Figure 4.244 may be real (due to actual system behavior) or may
reflect accumulated numerical errors in the analysis. Numerical errors are unlikely because the shears
computed from member forces and from inertial forces are similar. The energy response history can
provide further validation.. Figure 4.246 shows the energy response history for the 2 percent damped
DBE analysis with Pdelta effects included. If the analysis is accurate, the input energy will coincide with
the total energy (sum of kinetic, damping and structural energy). DRAIN 2D produces individual energy
values as well as the input energy. See the article by Uang and Bertero for background on computing
energy curves.
As evident from Figure 4.246, the total and input energy curves coincide, so the analysis is numerically
accurate. Where this accuracy is in doubt, the analysis should be rerun using a smaller integration time
step. A time step of 0.0005 second is required to produce the energy balance shown in Figure 4.246. A
time step of 0.001 second is sufficient for analyses with Ground Motions B90 and C90.
The trends observed for the DBE analysis are even more extreme when the MCE ground motion is used.
Figure 4.247 shows the displacement histories for the 2 percent damped structure under the MCE scaled
A00 ground motion. As may be seen, residual deformations again dominate and in this case the total
residual roof displacement with Pdelta effects included is five times that without Pdelta effects. This
behavior indicates dynamic instability and eventual collapse.
It is interesting to compare the response computed for Ground Motion B90 with that obtained for ground
motion A00. Displacements occurring for the 2 percent damped model under the MCEscaled B90
ground motions are shown in Figure 4.248. While there is some small residual deformation in this
system, it is not extreme, and it appears that the structure is not in danger of collapse. (The corresponding
plastic rotations are less than those that would be associated with significant strength loss.)
The characteristic of the ground motion that produces the residual deformations shown in Figures 4.244
and 4.248 (the DBE and MCE scaled A00 ground motions, respectively) is not evident from the ground
acceleration history or from the acceleration response spectrum. The source of the behavior is quite
obvious from plots of the ground velocity and ground displacement histories, shown in Figure 4.249(a)
and (b), respectively. The ground velocity history shows that a very large velocity pulse occurs
approximately 10 seconds into the earthquake. This leads to a surge in ground displacement, also
occurring approximately 10 seconds into the response. The surge in ground displacement is more than
8 feet, which is somewhat unusual. Recall from Table 4.120(a) that the distance between the epicenter
and the recording site for this ground motion is 44 kilometers; so, the motion would not be considered as
nearfield. The unusual characteristics of Ground Motion A00 may be seen in Figure 4.249 (c), which is
a tripartite spectrum.
Figure 4.244 Response history of roof and firststory displacement,
Ground Motion A00 (DBE)
Figure 4.245 Response history of total base shear,
Ground Motion A00 (DBE)
Figure 4.246 Energy response history, Ground Motion A00 (DBE),
including Pdelta effects
Figure 4.247 Response history of roof and firststory displacement,
Ground Motion A00 (MCE)
Figure 4.248 Response history of roof and firststory displacement,
Ground Motion B90 (MCE)
(a) Ground velocity history
(b) Ground displacement history
(c) Tripartite spectrum
Figure 4.249 Ground velocity and displacement histories and tripartite spectrum of
Ground Motion A00 (unscaled)
Figure 4.250 shows the pattern of yielding in the structure subjected to a 2 percent damped MCEscaled
Ground Motion B90 including Pdelta effects. Recall that the model incorporates panel zone
reinforcement at the interior beamcolumn joints. The circles on the figure represent yielding at any time
during the response; consequently, yielding does not necessarily occur at all locations simultaneously.
The circles shown at the upper left corner of the beamcolumn joint region indicate yielding in the
rotational spring, which represents the web component of panel zone behavior. There is no yielding in
the flange component of the panel zones, as seen in Figure 4.250.
Yielding patterns for the other ground motions and for analyses run with and without Pdelta effects are
similar but are not shown here. As expected, there is more yielding in the columns when the structure is
subjected to the A00 ground motion.
Figure 4.250 shows that yielding occurs at both ends of each of the girders at Levels 2, 3, 4 and 5.
Yielding occurs at the bottom of all the firststory columns as well as at the top of the interior columns at
the third and fourth stories and at bottom of the fifthstory interior columns. The panel zones at the
exterior joints of Levels 4 and 5 also yield. The maximum plastic hinge rotations are shown where they
occur for the columns, girders and panel zones; values are shown in Table 4.224. The maximum plastic
shear strain in the web of the panel zone is identical to the computed hinge rotation in the panel zone
spring. For the DBEscaled B90 ground motion, the maximum rotations occurring at the plastic hinges
are less than 0.02 radians.
Figure 4.250 Yielding locations for structure with strong panels subjected to
MCEscaled B90 motion, including Pdelta effects
4.2.6.3.3 Comparison with results from other analyses. Table 4.224 compares the results from the
response history analysis with those from the ELF and the nonlinear static analyses. Base shears in the
table are half of the total shear. The nonlinear static analysis results are for the 2 percent damped MCE
target displacement so, for consistency, the tabulated dynamic analysis results are for the 2 percent
damped MCEscaled B90 ground motion. In addition, the lateral forces used to find the ELF drifts in
Table 4.26 are multiplied by 1.5 forconsistency with MCElevel shaking; the ELF analysis drift values
include the deflection amplification factor of 5.5. The results show some similarities and some striking
differences, as follows:
The base shear from nonlinear dynamic analysis is approximately three times the value from ELF
analysis. The predicted displacements and story drifts are similar at the top three stories but are
significantly different at the bottom three stories. Due to the highly empirical nature of the ELF
approach, it is difficult to explain these differences. The ELF method also has no mechanism to
include the overstrength that will occur in the structure, although it is represented explicitly in the
static and dynamic nonlinear analyses.
The nonlinear static analysis predicts base shears and story displacements that are less than those
obtained from response history analysis. Excessive drift occurred at the bottom three stories as a
result of both pushover and response history analyses.
Table 4.224 Summary of All Analyses for Strong Panel Structure, Including Pdelta
Effects
Response Quantity
Analysis Method
Equivalent
Lateral Forces
Nonlinear Static
Pushover
Nonlinear
Dynamic
Base shear (kips)
569
1,208
1,633
Roof disp. (in.)
18.4
22.9
26.1
Drift R6 (in.)
1.86
0.96
2.32
Drift 65 (in.)
2.78
1.76
2.60
Drift 54 (in.)
3.34
2.87
3.62
Drift 43 (in.)
3.73
4.84
5.61
Drift 32 (in.)
3.67
5.74
6.32
Drift 21 (in.)
2.98
6.73
7.03
Girder hinge rot. (rad)
NA
0.03304
0.03609
Column hinge rot. (rad)
NA
0.02875
0.02993
Panel hinge rot. (rad)
NA
0.00335
0.00411
Panel plastic shear strain
NA
0.00335
0.00411
Note: Shears are for half of total structure.
Some of the difference between pushover and nonlinear response history results is due to the scale factor
(1.367) used to satisfy ground motion scaling requirements for the nonlinear response history analysis,
but most of the difference is due to higher mode effects. Figure 4.251 shows the inertial forces from the
nonlinear response history analyses at the time of peak base shear and the loads applied to the nonlinear
static analysis model at the target displacement. The higher mode effects apparent in Figure 4.251 likely
are the cause of the different hinging patterns and certainly are the reason for the very high base shear
developed in the response history analysis. (If the inertial forces were constrained to follow the first
mode response, the maximum base shear that could be developed in the system would be in the range of
1200 kips. See, for example, Figure 4.228.)
Figure 4.251 Comparison of inertial force patterns
4.2.6.3.4 Effect of increased damping on response. The nonlinear response history analysis of the
structure with panel zone reinforcement indicates first story drifts in excess of the allowable limits. The
most costeffective measure to enhance the performance of the structure probably would be to provide
additional strength and/or stiffness at this story. However, added damping is also a viable approach.
To investigate the viability of added damping, additional analysis that treats individual dampers explicitly
is required. Linear viscous damping can be modeled in DRAIN using the stiffness proportional
component of Rayleigh damping. Base shear increases with added damping, so in practice added
damping systems usually employ viscous fluid devices with a softening nonlinear relationship between
the deformational velocity in the device and the force in the device, to limit base shears when
deformational velocities become large.
A linear viscous fluid damping device (Figure 4.252) in a selected story can be modeled using a Type 1
(truss bar) element. A damping constant for the device, , is obtained as follows:
The elastic stiffness of the damper element is simply as follows:
where:
= the cross sectional area
= the modulus of elasticity
= the length of the Type 1 damper element
As stiffness proportional damping is used, the damping constant for the element is:
The damper elastic stiffness should be negligible, so consider = 0.001 kips/in. Thus:
Where modeling added dampers in this manner, it is convenient to consider Edevice = 0.001 and Adevice = the
damper length Ldevice.
This value of device is for the added damper element only. Different dampers may require different
values. Also, a different (global) value of is required to model the stiffness proportional component of
damping in the remaining nondamper elements.
Modeling the dynamic response using Type 1 elements is exact within the typical limitations of finite
element analysis. Using the modal strain energy approach, DRAIN reports a damping value in each
mode. These modal damping values are approximate and may be poor estimates of actual modal
damping, particularly where there is excessive flexibility in the mechanism that connects the damper to
the structure.
To determine the effect of added damping on the behavior of the structure, dampers are added to the SP
frame with 2 percent inherent damping, and the structure is subjected to the DBEscaled A00 and B90
ground motions. Pdelta effects are included in the analyses. Table 4.225 shows the base shear and story
drifts of the SP frame with 2 percent inherent damping when it is subjected to DBEscaled A00 and B90
ground motions. The results summarized in this table can also be found in the tables of Section 4.2.6.3.1.
Figure 4.252 Modeling a simple damper
Table 4.225 Maximum Story Drifts (in.) and Base Shear (kips) when SP Model
with 2% Inherent Damping is Subjected to DBE Scaled A00 and B90
Ground Motions, including Pdelta Effects
Level
Motion A00
Motion B90
Limit
Roof displacement
32.65
14.50
NA
Drift R6
1.86
1.82
3.75
Drift 65
2.64
2.50
3.75
Drift 54
4.08
2.81
3.75
Drift 43
6.87
3.21
3.75
Drift 32
8.19
3.40
3.75
Drift 2G
10.40
4.69
4.50
Column forces
1467
1458
NA
Inertial forces
1558
1481
NA
As can be seen in Table 4.225, drift limits are exceeded at the bottom four stories for the A00 ground
motion and only for the bottom story for the B90 ground motion.
Four different added damper configurations are used to assess their effect on story drifts and base shear,
as summarized in Tables 4.226 and 4.227.
Table 4.226 Effect of Different Added Damper Configurations when SP Model is Subjected to
DBEScaled A00 Ground Motion, including Pdelta Effects
First Config
Second Config
Third Config
Fourth Config
Level
Damper
Coeff.
(kip
sec/in.)
Drift
(in.)
Damper
Coeff.
(kip
sec/in.)
Drift
(in.)
Damper
Coeff.
(kip
sec/in.)
Drift
(in.)
Damper
Coeff.
(kip
sec/in.)
Drift
(in.)
Drift
Limit
(in.)
R6
10.5
1.10
60
1.03

1.82

1.47
3.75
65
33.7
1.90
60
1.84

3.56

2.41
3.75
54
38.4
2.99
70
2.88

4.86
56.25
3.46
3.75
43
32.1
5.46
70
4.42

5.24
56.25
4.47
3.75
32
36.5
6.69
80
5.15
160
4.64
112.5
4.76
3.75
2G
25.6
8.39
80
5.87
160
4.40
112.5
4.96
4.50
Column base
shear (kips)
1,629
2,170
2,134
2,267
Inertial base
shear (kips)
1,728
2,268
2,215
2,350
Total
damping (%)
10.1
20.4
20.2
20.4
Table 4.227 Effect of Different Added Damper Configurations when SP Model is Subjected to
DBEScaled B90 Ground Motion, including Pdelta Effects
First Config
Second Config
Third Config
Fourth Config
Level
Damper
Coeff.
(kip
sec/in.)
Drift
(in.)
Damper
Coeff.
(kip
sec/in.)
Drift
(in.)
Damper
Coeff.
(kip
sec/in.)
Drift
(in.)
Damper
Coeff.
(kip
sec/in.)
Drift
(in.)
Drift
Limit
(in.)
R6
10.5
1.11
60
0.86

1.53

1.31
3.75
65
33.7
1.76
60
1.35

2.11

1.83
3.75
54
38.4
2.33
70
1.75

2.51
56.25
2.07
3.75
43
32.1
2.67
70
2.11

2.37
56.25
2.16
3.75
32
36.5
2.99
80
2.25
160
2.09
112.5
2.13
3.75
2G
25.6
3.49
80
1.96
160
1.87
112.5
1.82
4.50
Column base
shear (kips)
1,481
1,485
1,697
1,637
Inertial base
shear (kips)
1,531
1,527
1,739
1,680
Total
damping (%)
10.1
20.4
20.2
20.4
These configurations increase total damping of the structure from 2 percent (inherent) to 10 and
20 percent. In the first configuration added dampers are distributed proportionally to approximate story
stiffnesses. In the second configuration, dampers are added at all six stories, with larger dampers in lower
stories. Since the structure seems to be weak at the bottom stories (where it exceeds drift limits), dampers
are concentrated at the bottom stories in the last two configurations. Added dampers are used only at the
first and second stories in the third configuration and at the bottom four stories in the fourth
configuration.
Based on this supplemental damper study, it appears to be impossible to decrease the story drifts for the
A00 ground motion below the limits. This is because of the incremental velocity of Ground Motion A00
causes such significant structural damage. The drift limits could be satisfied if the total damping ratio is
increased to 33.5 percent, but since that is impractical the results are not reported here. The third
configuration of added dampers reduces the firststory drift from 10.40 inches to 4.40 inches
All of the configurations easily satisfy drift limits for the B90 ground motion.While the system with
10 percent total damping is sufficient for drift limits, systems with 20 percent damping further improve
performance. Although configurations 3 and 4 have the same amount of total damping as configuration
2, story drifts are higher at the top stories since dampers are added only at lower stories.
Figures 4.253 through 4.255 show the effect of added damping of roof displacement, inertial base shear
and energy history for the A00 ground motion. As Figure 4.253 shows added dampers reduce roof
displacement significantly but do not prevent residual displacement. Figure 4.254 shows how added
damping increases peak base shear. Figure 4.255 is an energy response history for the structure with
damping configuration 4. It should be compared to Figure 4.246, which is the energy history for the
structure with 2 percent inherent damping but with no added damping. As should be expected, adding
discrete damping reduces the hysteretic energy demand in the structure (designated as structural energy in
Figure 4.255). A reduction in hysteretic energy demand for the system with added damping corresponds
to a reduction in structural damage.
Figures 4.256 through 4.258 display the same response plots for Ground Motion B90. As for Ground
Motion A00 roof displacement decreases with added damping, peak base shear increases and hysteretic
energy demand (which is related to structural damage) decreases.
Figure 4.253 Roof displacement response histories with added damping (20% total)
and inherent damping (2%) for Ground Motion A00
Figure 4.254 Inertial base shear response histories with added damping (20% total)
and inherent damping (2%) for Ground Motion A00
Figure 4.255 Energy response history with added damping of fourth configuration
(20% total damping) for Ground Motion A00
Figure 4.256 Roof displacement response histories with added damping (20% total)
and inherent damping (2%) for Ground Motion B90
Figure 4.257 Inertial base shear response histories with added damping (20% total)
and inherent damping (2%) for Ground Motion B90
Figure 4.258 Energy response history with added damping of fourth configuration
(20% total damping) for Ground Motion B90
4.2.7 Summary and Conclusions
In this example, five different analytical approaches are used to estimate the deformation demands in a
simple structural steel momentresisting frame structure:
1. Linear static analysis (the equivalent lateral force method)
2. Plastic strength analysis (using virtual work)
3. Nonlinear static (pushover) analysis
4. Linear dynamic (modal response history) analysis
5. Nonlinear dynamic (response history) analysis
The nonlinear structural model includes careful representation of possible inelastic behavior in the panel
zone regions of the beamcolumn joints.
The results obtained from the three different analytical approaches 1, 3 and 5 are quite dissimilar. Except
for preliminary design, the ELF approach should not be used in explicit performance evaluation since it
cannot reflect the location and extent of yielding in the structure. Due to higher mode effects, pushover
analysis, where used alone, is inadequate.
This leaves nonlinear response history analysis as the most viable approach. Given the speed and
memory capacity of personal computers, nonlinear response history analysis is increasingly common in
the seismic analysis of buildings. However, significant shortcomings, limitations and uncertainties in
response history analysis still exist.
Among the most pressing problems is the need for a suitable suite of ground motions. All ground
motions must adequately reflect site conditions and where applicable, the suite must include nearfield
effects. Through future research and the efforts of code writing bodies, it may be possible to develop
standard suites of ground motions that could be published together with selection tools and scaling
methodologies. The scaling techniques currently recommended in the Standard are a start but need
improvement.
Systematic methods need to be developed for identifying uncertainties in the modeling of the structure
and for quantifying the effect of such uncertainties on the response. While probabilistic methods for
dealing with such uncertainties seem like a natural extension of the analytical approach, the authors
believe that deterministic methods should not be abandoned entirely.
In the context of performancebased design, improved methods for assessing the effect of inelastic
response and acceptance criteria based on such measures need to be developed. Methods based on
explicit quantification of damage should be considered seriously.
The ideas presented above certainly are not original. They have been presented by many academics and
practicing engineers. What is still lacking is a comprehensive approach to seismicresistant design based
on these principles.
The proper computational units for period of vibration are theoretically seconds/cycle . However, it is traditional
to use units of seconds, and this is done in the remainder of this example.
As shown later in this example, the computed period is indeed greater than CuTa.
Note that the I in the numerator of Equation 12.816 was inadvertently omitted in early printings of the Standard.
5See Sec. 3.2.6.2 of this volume of design examples for a detailed discussion of the selection and scaling of ground motions.
Elimination of the degree of freedom results in consistent scale factors for all persons using the process. This
consistency is not required by ASCE 7 and experienced analysts may wish to use the degree of freedom to reduce
or increase the influence of a given ground motion.
1The term Level is used in this example to designate a horizontal plane at the same elevation as the centerline of a girder. The top
level, Level R, is at the roof elevation; Level 2 is the first level above grade; and Level 1 is at grade. The term Story represents
the distance between adjacent levels. The story designation is the same as the designation of the level at the bottom of the story.
Hence, Story 1 is the lowest story (between Levels 2 and 1) and Story 6 is the uppermost story (between Levels R and 6).
The correct computational units for period of vibration is seconds per cycle . However, the traditional units of
seconds are used in the remainder of this example.
4 The numbers of degrees of freedom in the Krawinkler model may be reduced to only four if the rigid links around the perimeter
of the model are represented by mathematical constraints instead of stiff elements. Most commercial programs employ this
approach for the Krawinkler model.
5A graphic postprocessor was used to display the deflected shape of the structure. The program represents each element as a
straight line. Although the computational results are unaffected, a better graphical representation is obtained by subdividing the
member.
DRAIN uses the axial forces at the end of the gravity load analysis to set geometric stiffness for the structure. This
is reasonably accurate where consistent geometric stiffness is used, but is questionable where linearized geometric
stiffness is used.
7If Pdelta effects have been included, this procedure needs to be used where recovering base shear from column shear forces.
This is true for displacement controlled static analysis, force controlled static analysis and dynamic response history analysis.
5
Foundation Analysis and Design
Michael Valley, S.E.
Contents
5.1 SHALLOW FOUNDATIONS FOR A SEVENSTORY OFFICE BUILDING, LOS
ANGELES, CALIFORNIA 3
5.1.1 Basic Information 3
5.1.2 Design for Gravity Loads 8
5.1.3 Design for MomentResisting Frame System 11
5.1.4 Design for Concentrically Braced Frame System 16
5.1.5 Cost Comparison 24
5.2 DEEP FOUNDATIONS FOR A 12STORY BUILDING, SEISMIC DESIGN CATEGORY D
25
5.2.1 Basic Information 25
5.2.2 Pile Analysis, Design and Detailing 33
5.2.3 Other Considerations 47
This chapter illustrates application of the 2009 Edition of the NEHRP Recommended Provisions to the
design of foundation elements. Example 5.1 completes the analysis and design of shallow foundations for
two of the alternative framing arrangements considered for the building featured in Example 6.2.
Example 5.2 illustrates the analysis and design of deep foundations for a building similar to the one
highlighted in Chapter 7 of this volume of design examples. In both cases, only those portions of the
designs necessary to illustrate specific points are included.
The forcedisplacement response of soil to loading is highly nonlinear and strongly time dependent.
Control of settlement is generally the most important aspect of soil response to gravity loads. However,
the strength of the soil may control foundation design where large amplitude transient loads, such as those
occurring during an earthquake, are anticipated.
Foundation elements are most commonly constructed of reinforced concrete. As compared to design of
concrete elements that form the superstructure of a building, additional consideration must be given to
concrete foundation elements due to permanent exposure to potentially deleterious materials, less precise
construction tolerances and even the possibility of unintentional mixing with soil.
Although the application of advanced analysis techniques to foundation design is becoming increasingly
common (and is illustrated in this chapter), analysis should not be the primary focus of foundation design.
Good foundation design for seismic resistance requires familiarity with basic soil behavior and common
geotechnical parameters, the ability to proportion concrete elements correctly, an understanding of how
such elements should be detailed to produce ductile response and careful attention to practical
considerations of construction.
In addition to the Standard and the Provisions and Commentary, the following documents are either
referenced directly or provide useful information for the analysis and design of foundations for seismic
resistance:
ACI 318 American Concrete Institute. 2008. Building Code Requirements and
Commentary for Structural Concrete.
Bowles Bowles, J. E. 1988. Foundation Analysis and Design. McGrawHill.
CRSI Concrete Reinforcing Steel Institute. 2008. CRSI Design Handbook. Concrete
Reinforcing Steel Institute.
ASCE 41 ASCE. 2006. Seismic Rehabilitation of Existing Buildings.
Kramer Kramer, S. L. 1996. Geotechnical Earthquake Engineering. Prentice Hall.
LPILE Reese, L. C. and S. T. Wang. 2009. Technical Manual for LPILE Plus 5.0 for
Windows. Ensoft.
Rollins et al. (a) Rollins, K. M., Olsen, R. J., Egbert, J. J., Jensen, D. H., Olsen, K. G.and Garrett,
B. H. (2006). Pile Spacing Effects on Lateral Pile Group Behavior: Load Tests.
Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 132,
No. 10, p. 12621271.
Rollins et al. (b) Rollins, K. M., Olsen, K. G., Jensen, D. H, Garrett, B. H., Olsen, R. J.and Egbert,
J. J. (2006). Pile Spacing Effects on Lateral Pile Group Behavior: Analysis.
Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 132,
No. 10, p. 12721283.
Wang & Salmon Wang, C.K. and C. G. Salmon. 1992. Reinforced Concrete Design .
HarperCollins.
Several commercially available programs were used to perform the calculations described in this chapter.
SAP2000 is used to determine the shears and moments in a concrete mat foundation; LPILE, in the
analysis of laterally loaded single piles; and spColumn, to determine concrete pile section capacities.
5.1 SHALLOW FOUNDATIONS FOR A SEVENSTORY OFFICE BUILDING, LOS ANGELES,
CALIFORNIA
This example features the analysis and design of shallow foundations for two of the three framing
arrangements for the sevenstory steel office building described in Section 6.2 of this volume of design
examples. Refer to that example for more detailed building information and for the design of the
superstructure.
5.1.1 Basic Information
5.1.1.1 Description. The framing plan in Figure 5.11 shows the gravity loadresisting system for a
representative level of the building. The site soils, consisting of medium dense sands, are suitable for
shallow foundations. Table 5.11 shows the design parameters provided by a geotechnical consultant.
Note the distinction made between bearing pressure and bearing capacity. If the longterm, servicelevel
loads applied to foundations do not exceed the noted bearing pressure, differential and total settlements
are expected to be within acceptable limits. Settlements are more pronounced where large areas are
loaded, so the bearing pressure limits are a function of the size of the loaded area. The values identified
as bearing capacity are related to gross failure of the soil mass in the vicinity of loading. Where loads are
applied over smaller areas, punching into the soil is more likely.
Figure 5.11 Typical framing plan
Because bearing capacities are generally expressed as a function of the minimum dimension of the loaded
area and are applied as limits on the maximum pressure, foundations with significantly nonsquare loaded
areas (tending toward strip footings) and those with significant differences between average pressure and
maximum pressure (as for eccentrically loaded footings) have higher calculated bearing capacities. The
recommended values are consistent with these expectations.
Table 5.11 Geotechnical Parameters
Parameter
Value
Basic soil properties
Medium dense sand
(SPT) N = 20
= 125 pcf
Angle of internal friction = 33 degrees
Net bearing pressure (to control
settlement due to sustained loads)
ó 4,000 psf for B ó 20 feet
ó 2,000 psf for B ò 40 feet
(may interpolate for intermediate dimensions)
Bearing capacity (for plastic
equilibrium strength checks with
factored loads)
2,000B psf for concentrically loaded square footings
3,000B' psf for eccentrically loaded footings
where B and B' are in feet, B is the footing width and B' is
an average width for the compressed area.
Resistance factor, = 0.7
[This factor for cohesionless soil is specified in
Provisions Part 3 Resource Paper 4; the value is set at 0.7
for vertical, lateral and rocking resistance.]
Lateral properties
Earth pressure coefficients:
Active, KA = 0.3
Atrest, K0 = 0.46
Passive, KP = 3.3
Ultimate friction coefficient at base of footing = 0.65
Resistance factor, = 0.7
The structural material properties assumed for this example are as follows:
f'c = 4,000 psi
fy = 60,000 psi
5.1.1.2 Seismic Parameters. The complete set of parameters used in applying the Provisions to design of
the superstructure is described in Section 6.2.2.1 of this volume of design examples. The following
parameters, which are used during foundation design, are duplicated here.
Site Class = D
SDS = 1.0
Seismic Design Category = D
5.1.1.3 Design Approach.
5.1.1.3.1 Selecting Footing Size and Reinforcement. Most foundation failures are related to excessive
movement rather than loss of loadcarrying capacity. In recognition of this fact, settlement control should
be the first issue addressed. Once service loads have been calculated, foundation plan dimensions should
be selected to limit bearing pressures to those that are expected to provide adequate settlement
performance. Maintaining a reasonably consistent level of service loadbearing pressures for all of the
individual footings is encouraged since it will tend to reduce differential settlements, which are usually of
more concern than are total settlements.
Once a preliminary footing size that satisfies serviceability criteria has been selected, bearing capacity can
be checked. It would be rare for bearing capacity to govern the size of footings subjected to sustained
loads. However, where large transient loads are anticipated, consideration of bearing capacity may
become important.
The thickness of footings is selected for ease of construction and to provide adequate shear capacity for
the concrete section. The common design approach is to increase footing thickness as necessary to avoid
the need for shear reinforcement, which is uncommon in shallow foundations.
Design requirements for concrete footings are found in Chapters 15 and 21 of ACI 318. Chapter 15
provides direction for the calculation of demands and includes detailing requirements. Section capacities
are calculated in accordance with Chapters 10 (for flexure) and 11 (for shear). Figure 5.12 illustrates the
critical sections (dashed lines) and areas (hatched) over which loads are tributary to the critical sections.
For elements that are very thick with respect to the plan dimensions (as at pile caps), these critical section
definitions become less meaningful and other approaches (such as strutandtie modeling) should be
employed. Chapter 21 provides the minimum requirements for concrete foundations in Seismic Design
Categories D, E and F, which are similar to those provided in prior editions of the Provisions.
For shallow foundations, reinforcement is designed to satisfy flexural demands. ACI 318 Section 15.4
defines how flexural reinforcement is to be distributed for footings of various shapes.
Section 10.5 of ACI 318 prescribes the minimum reinforcement for flexural members where tensile
reinforcement is required by analysis. Provision of the minimum reinforcement assures that the strength
of the cracked section is not less than that of the corresponding unreinforced concrete section, thus
preventing sudden, brittle failures. Less reinforcement may be used as long as the area of tensile
reinforcement provided is at least onethird greater than that required by analysis. Section 10.5.4 relaxes
the minimum reinforcement requirement for footings of uniform thickness. Such elements need only
satisfy the shrinkage reinforcement requirements of Section 7.12. Section 10.5.4 also imposes limits on
the maximum spacing of bars.
5.1.1.3.2 Additional Considerations for Eccentric Loads. The design of eccentrically loaded footings
follows the approach outlined above with one significant addition: consideration of overturning stability.
Stability calculations are sensitive to the characterization of soil behavior. For sustained eccentric loads,
a linear distribution of elastic soil stresses is generally assumed and uplift is usually avoided. If the
structure is expected to remain elastic when subjected to shortterm eccentric loads (as for wind loading),
uplift over a portion of the footing is acceptable to most designers. Where foundations will be subjected
to shortterm loads and inelastic response is acceptable (as for earthquake loading), plastic soil stresses
may be considered. It is most common to consider stability effects on the basis of statically applied loads
even where the loading is actually dynamic; that approach simplifies the calculations at the expense of
increased conservatism. Figure 5.13 illustrates the distribution of soil stresses for the various
assumptions. Most textbooks on foundation design provide simple equations to describe the conditions
shown in Parts b, c and d of the figure; finite element models of those conditions are easy to develop.
Simple hand calculations can be performed for the case shown in Part f. Practical consideration of the
case shown in Part e would require modeling with inelastic elements, but that offers no advantage over
direct consideration of the plastic limit. (All of the discussion in this section focuses on the common case
in which foundation elements may be assumed to be rigid with respect to the supporting soil. For the
interested reader, Chapter 4 of ASCE 41 provides a useful discussion of foundation compliance, rocking
and other advanced considerations.)
Figure 5.12 Critical sections for isolated footings Figure 5.13 Soil pressure distributions
5.1.2 Design for Gravity Loads
Although most of the examples in this volume do not provide detailed design for gravity loads, it is
provided in this section for two reasons. First, most of the calculation procedures used in designing
shallow foundations for seismic loads are identical to those used for gravity design. Second, a complete
gravity design is needed to make the cost comparisons shown in Section 5.1.5 below meaningful.
Detailed calculations are shown for a typical interior footing. The results for all three footing types are
summarized in Section 5.1.2.5.
5.1.2.1 Demands. Dead and live load reactions are determined as part of the threedimensional analysis
described in Section 6.2 of this volume of design examples. Although there are slight variations in the
calculated reactions, the foundations are lumped into three groups (interior, perimeter and corner) for
gravity load design and the maximum computed reactions are applied to all members of the group, as
follows:
Interior: D = 387 kips
L = 98 kips
Perimeter: D = 206 kips
L = 45 kips
Corner: D = 104 kips
L = 23 kips
The service load combination for consideration of settlement is D + L. Considering the load
combinations for strength design defined in Section 2.3.2 of the Standard, the controlling gravity load
combination is 1.2D + 1.6L.
5.1.2.2 Footing Size. The preliminary size of the footing is determined considering settlement. The
service load on a typical interior footing is calculated as:
P = D + L = 387 kips + 98 kips = 485 kips
Since the footing dimensions will be less than 20 feet, the allowable bearing pressure (see Table 5.11) is
4,000 psf. Therefore, the required footing area is 487,000 lb/4,000 psf = 121.25 ft2.
Check a footing that is 11'0" by 11'0":
Pallow = 11 ft(11 ft)(4,000 psf) = 484,000 lb = 484 kips ÷ 485 kips (demand) OK
The strength demand is:
Pu = 1.2(387 kips) + 1.6(98 kips) = 621 kips
As indicated in Table 5.11, the bearing capacity (qc) is 2,000B = 2,000 11 = 22,000 psf = 22 ksf.
The design capacity for the foundation is:
Pn = qcB2 = 0.7(22 ksf)(11 ft)2 = 1,863 kips > 621 kips OK
For use in subsequent calculations, the factored bearing pressure qu = 621 kips/(11 ft)2 = 5.13 ksf.
5.1.2.3 Footing Thickness. Once the plan dimensions of the footing are selected, the thickness is
determined such that the section satisfies the oneway and twoway shear demands without the addition of
shear reinforcement. Demands are calculated at critical sections, shown in Figure 5.12, which depend on
the footing thickness.
Check a footing that is 26 inches thick:
For the W14 columns used in this building, the side dimensions of the loaded area (taken halfway
between the face of the column and the edge of the base plate) are approximately 16 inches.
Accounting for cover and expected bar sizes, d = 26  (3 + 1.5(1)) = 21.5 in.
Oneway shear:
= 172 kips
= 269 kips > 172 kips OK
Twoway shear:
= 571 kips
= 612 kips > 571 kips OK
5.1.2.4 Footing Reinforcement. Footing reinforcement is selected considering both flexural demands
and minimum reinforcement requirements. The following calculations treat flexure first because it
usually controls:
Try nine #8 bars each way. The distance from the extreme compression fiber to the center of the top layer
of reinforcement, d = t  cover  1.5db = 26  3  1.5(1) = 21.5 in.
T = As fy = 9(0.79)(60) = 427 kips
Noting that C = T and solving the expression C = 0.85 f'c b a for a produces a = 0.951 in.
= 673 ftkips > 659 ftkips OK
The ratio of reinforcement provided is = 9(0.79)/[(11)(12)(26)] = 0.00207. The distance between bars
spaced uniformly across the width of the footing is s = [(11)(12)2(3+0.5)]/(91) = 15.6 in.
According to ACI 318 Section 7.12, the minimum reinforcement ratio = 0.0018 < 0.00207 OK
and the maximum spacing is the lesser of 5 26 in. and 18 = 18 in. > 15.6 in. OK
5.1.2.5 Design Results. The calculations performed in Sections 5.1.2.2 through 5.1.2.4 are repeated for
typical perimeter and corner footings. The footing design for gravity loads is summarized in Table 5.12;
Figure 5.14 depicts the resulting foundation plan.
Table 5.12 Footing Design for Gravity Loads
Location
Loads
Footing Size and Reinforcement;
Soil Capacity
Critical Section Demands and
Design Strengths
Interior
D = 387 kip
L = 98 kip
P = 485 kip
Pu = 621 kip
11'0" 11'0" 2'2" deep
9#8 bars each way
Pallow = 484 kip
Pn = 1863 kip
Oneway shear: Vu = 172 kip
Vn = 269 kip
Twoway shear: Vu = 571 kip
Vn = 612 kip
Flexure: Mu = 659 ftkip
Mn = 673 ftkip
Perimeter
D = 206 kip
L = 45 kip
P = 251 kip
Pu = 319 kip
8'0" 8'0" 1'6" deep
9#6 bars each way
Pallow = 256 kip
Pn = 716 kip
Oneway shear: Vu = 88.1 kip
Vn = 123 kip
Twoway shear: Vu = 289 kip
Vn = 302 kip
Flexure: Mu = 222 ftkip
Mn = 234 ftkip
Corner
D = 104 kip
L = 23 kip
P = 127 kip
Pu = 162 kip
6'0" 6'0" 1'2" deep
6#5 bars each way
Pallow = 144 kip
Pn = 302 kip
Oneway shear: Vu = 41.5 kip
Vn = 64.9 kip
Twoway shear: Vu = 141 kip
Vn = 184 kip
Flexure: Mu = 73.3 ftkip
Mn = 75.2 ftkip
Figure 5.14 Foundation plan
5.1.3 Design for MomentResisting Frame System
Framing Alternate A in Section 6.2 of this volume of design examples includes a perimeter moment
resisting frame as the seismic forceresisting system. A framing plan for the system is shown in
Figure 5.15. Detailed calculations are provided in this section for a combined footing at the corner and
focus on overturning and sliding checks for the eccentrically loaded footing; settlement checks and design
of concrete sections would be similar to the calculations shown in Section 5.1.2. The results for all
footing types are summarized in Section 5.1.3.4.
Figure 5.15 Framing plan for momentresisting frame system
5.1.3.1 Demands. A threedimensional analysis of the superstructure, in accordance with the
requirements for the equivalent lateral force (ELF) procedure, is performed using the ETABS program.
Foundation reactions at selected grids are reported in Table 5.13.
Table 5.13 Demands from MomentResisting Frame System
Location
Load
Fx
Fy
Fz
Mxx
Myy
A5
D
203.8
L
43.8
Ex
13.8
4.6
3.8
53.6
243.1
Ey
0.5
85.1
21.3
1011.5
8.1
A6
D
103.5
L
22.3
Ex
14.1
3.7
51.8
47.7
246.9
Ey
0.8
68.2
281.0
891.0
13.4
Note: Units are kips and feet. Load Ex is for loads applied toward the east, including appropriately
amplified counterclockwise accidental torsion. Load Ey is for loads applied toward the north,
including appropriately amplified clockwise accidental torsion.
Section 6.2.3.5 of this volume of design examples outlines the design load combinations, which include
the redundancy factor as appropriate. A large number of load cases result from considering two senses of
accidental torsion for loading in each direction and including orthogonal effects . The detailed
calculations presented here are limited to two primary conditions, both for a combined foundation for
columns at Grids A5 and A6: the downward case (1.4D + 0.5L + 0.3Ex + 1.0Ey) and the upward case
(0.7D + 0.3Ex + 1.0Ey).
Before loads can be computed, attention must be given to Standard Section 12.13.4. That Section states
that overturning effects at the soilfoundation interface are permitted to be reduced by 25 percent where
the ELF procedure is used and by 10 percent where modal response spectrum analysis is used. Because
the overturning effect in question relates to the global overturning moment for the system, judgment must
be used in determining which design actions may be reduced. If the seismic forceresisting system
consists of isolated shear walls, the shear wall overturning moment at the base best fits that description.
For a perimeter momentresisting frame, most of the global overturning resistance is related to axial loads
in columns. Therefore, in this example column axial loads (Fz) from load cases Ex and Ey are multiplied
by 0.75 and all other load effects remain unreduced.
5.1.3.2 Downward Case (1.4D + 0.5L + 0.3Ex + 1.0Ey). In order to perform the overturning checks, a
footing size must be assumed. Preliminary checks (not shown here) confirmed that isolated footings
under single columns were untenable. Check overturning for a footing that is 9 feet wide by 40 feet long
by 5 feet thick. Furthermore, assume that the top of the footing is 2 feet below grade (the overlying soil
contributes to the resisting moment). (In these calculations the 0.2SDSD modifier for vertical accelerations
is used for the dead loads applied to the foundation but not for the weight of the foundation and soil. This
is the author s interpretation of the Standard. The footing and soil overburden are not subject to the same
potential for dynamic amplification as the dead load of the superstructure and it is not common practice to
include the vertical acceleration on the weight of the footing and the overburden. Furthermore, for
footings that resist significant overturning, this issue makes a significant difference in design.)
Combining the loads from columns at Grids A5 and A6 and including the weight of the foundation and
overlying soil produces the following loads at the foundationsoil interface:
P = applied loads + weight of foundation and soil
= 1.4(203.8  103.5) + 0.5(43.8  22.3) +0.75[0.3(3.8 + 51.8) + 1.0(21.3 + 281)]
 1.2[9(40)(5)(0.15) + 9(40)(2)(0.125)]
= 688 kips.
Mxx = direct moments + moment due to eccentricity of applied axial loads
= 0.3(53.6 + 47.7) + 1.0(1011.5  891.0)
+ [1.4(203.8) + 0.5(43.8) + 0.75(0.3)(3.8) + 0.75(1.0)(21.3)](12.5)
+ [1.4(103.5) + 0.5(22.3) + 0.75(0.3)(51.8) + 0.75(1.0)(281)](12.5)
= 6,717 ftkips.
Myy = 0.3(243.1  246.9) + 1.0(8.1 + 13.4)
= 126 ftkips. (The resulting eccentricity is small enough to neglect here, which simplifies the
problem considerably.)
Vx = 0.3(13.8  14.1) + 1.0(0.5 + 0.8)
= 7.11 kips.
Vy = 0.3(4.6 + 3.7) + 1.0(85.1 68.2)
= 149.2 kips.
Note that the above load combination does not yield the maximum downward load. Reversing the
direction of the seismic load results in P = 1,103 kips and Mxx = 2,964 ftkips. This larger axial load does
not control the design because the moment is so much less that the resultant is within the kern and no
uplift occurs.
The following soil calculations use a different sign convention than that in the analysis results noted
above; compression is positive for the soil calculations. The eccentricity is as follows:
e = M/P = 6,717/688 = 9.76 ft
Figure 5.13 shows the elastic and plastic design conditions and their corresponding equations. Where e
is less than L/2, a solution to the overturning problem exists; however, as e approaches L/2, the bearing
pressures increase without bound. Since e is greater than L/6 = 40/6 = 6.67 feet, uplift occurs and the
maximum bearing pressure is:
and the length of the footing in contact with the soil is:
The bearing capacity qc = 3,000B' = 3,000 min(B, L'/2) = 3,000 min(9, 30.7/2) = 27,000 psf = 27 ksf.
(L'/2 is used as an adjustment to account for the gradient in the bearing pressure in that dimension.)
The design bearing capacity qc = 0.7(27 ksf) = 18.9 ksf > 4.98 ksf OK
The foundation satisfies overturning and bearing capacity checks. The upward case, which follows, will
control the sliding check.
5.1.3.3 Upward Case (0.7D + 0.3Ex + 1.0Ey). For the upward case the loads are:
P = 332 kips
Mxx = 5,712 ftkips
Myy = 126 ftkips (negligible)
Vx = 7.1 kips
Vy = 149 kips
The eccentricity is:
e = M/P = 5,712/332 = 17.2 feet
Again, e is greater than L/6, so uplift occurs and the maximum bearing pressure is:
and the length of the footing in contact with the soil is:
The bearing capacity qc = 3,000 min(9, 8.4/2) = 12,500 psf = 12.5 ksf.
The design bearing capacity qc = 0.7(12.5 ksf) = 8.78 ksf < 8.82 ksf. NG
Using an elastic distribution of soil pressures, the foundation fails the bearing capacity check (although
stability is satisfied). Try the plastic distribution. Using this approach, the bearing pressure over the
entire contact area is assumed to be equal to the design bearing capacity. In order to satisfy vertical
equilibrium, the contact area times the design bearing capacity must equal the applied vertical load P.
Because the bearing capacity used in this example is a function of the contact area and the value of P
changes with the size, the most convenient calculation is iterative.
By iteration, the length of contact area is L' = 4.19 feet.
The bearing capacity qc = 3,000 min(10, 4.19) = 12,570 psf = 12.57 ksf. (No adjustment to L' is needed
as the pressure is uniform.)
The design bearing capacity qc = 0.7(12.6 ksf) = 8.80 ksf.
(8.80)(4.19)(9) = 332 kips = 332 kips, so equilibrium is satisfied.
The resisting moment, MR = P (L/2L'/2) = 33 (40/2  4.19/2) = 5,944 ftkip > 5,712 ftkip. OK
Therefore, using a plastic distribution of soil pressures, the foundation satisfies overturning and bearing
capacity checks.
The calculation of demands on concrete sections for strength checks should use the same soil stress
distribution as the overturning check. Using a plastic distribution of soil stresses defines the upper limit
of static loads for which the foundation remains stable, but the extreme concentration of soil bearing tends
to drive up shear and flexural demands on the concrete section. It should be noted that the foundation
may remain stable for larger loads if they are applied dynamically; even in that case, the strength demands
on the concrete section will not exceed those computed on the basis of the plastic distribution.
For the sliding check, initially consider base traction only. The sliding demand is:
As calculated previously, the total compression force at the bottom of the foundation is 332 kips. The
design sliding resistance is:
Vc = friction coefficient P = 0.7(0.65)(332 kips) = 151 kips > 149.4 kips OK
If base traction alone had been insufficient, resistance due to passive pressure on the leading face could be
included. Section 5.2.2.2 below illustrates passive pressure calculations for a pile cap.
5.1.3.4 Design Results. The calculations performed in Sections 5.1.3.2 and 5.1.3.3 are repeated for
combined footings at middle and side locations. Figure 5.16 shows the results.
Figure 5.16 Foundation plan for momentresisting frame system
One last check of interest is to compare the flexural stiffness of the footing with that of the steel column,
which is needed because the steel frame design was based upon flexural restraint at the base of the
columns. Using an effective moment of inertia of 50 percent of the gross moment of inertia and also
using the distance between columns as the effective span, the ratio of EI/L for the smallest of the
combined footings is more than five times the EI/h for the steel column. This is satisfactory for the
design assumption.
5.1.4 Design for Concentrically Braced Frame System
Framing Alternate B in Section 6.2 of this volume of design examples employs a concentrically braced
frame system at a central core to provide resistance to seismic loads. A framing plan for the system is
shown in Figure 5.17.
Figure 5.17 Framing plan for concentrically braced frame system
5.1.4.1 Check Mat Size for Overturning. Uplift demands at individual columns are so large that the
only practical shallow foundation is one that ties together the entire core. The controlling load
combination for overturning has minimum vertical loads (which help to resist overturning), primary
overturning effects (Mxx) due to loads applied parallel to the short side of the core and smaller moments
about a perpendicular axis (Myy) due to orthogonal effects. Assume mat dimensions of 45 feet by 95 feet
by 7 feet thick, with the top of the mat 3'6" below grade. Combining the factored loads applied to the
mat by all eight columns and including the weight of the foundation and overlying soil produces the
following loads at the foundationsoil interface:
P = 7,849 kips
Mxx = 148,439 ftkips
Myy = 42,544 ftkips
Vx = 765 kips
Vy = 2,670 kips
Figure 5.18 shows the soil pressures that result from application in this controlling case, depending on
the soil distribution assumed. In both cases the computed uplift is significant. In Part a of the figure, the
contact area is shaded. The elastic solution shown in Part b was computed by modeling the mat in
SAP2000 with compression only soil springs (with the stiffness of edge springs doubled as recommended
by Bowles). For the elastic solution, the average width of the contact area is 11.1 feet and the maximum
soil pressure is 16.9 ksf.
The bearing capacity qc = 3,000 min(95, 11.1/2) = 16,650 psf = 16.7 ksf.
The design bearing capacity qc = 0.7(16.7 ksf) = 11.7 ksf < 16.9 ksf. NG
Figure 5.18 Soil pressures for controlling bidirectional case
As was done in Section 5.1.3.3 above, try the plastic distribution. The present solution has an additional
complication as the offaxis moment is not negligible. The bearing pressure over the entire contact area is
assumed to be equal to the design bearing capacity. In order to satisfy vertical equilibrium, the contact
area times the design bearing capacity must equal the applied vertical load P. The shape of the contact
area is determined by satisfying equilibrium for the offaxis moment. Again the calculations are iterative.
Given the above constraints, the contact area shown in Figure 5.18 is determined. The length of the
contact area is 4.13 feet at the left side and 8.43 feet at the right side. The average contact length, for use
in determining the bearing capacity, is (4.13 + 8.43)/2 = 6.27 feet. The distances from the center of the
mat to the centroid of the contact area are as follows:
The bearing capacity is qc = 3,000 min(95, 6.27) = 18,810 psf = 18.81 ksf.
The design bearing capacity is qc = 0.7(18.8 ksf) = 13.2 ksf.
(13.2)(6.27)(95) = 7,863 kips ÷ 7,849 kips, confirming equilibrium for vertical loads.
(7,849)(5.42) = 42,542 ftkips ÷ 42,544 ftkips, confirming equilibrium for offaxis moment.
The resisting moment, >148,439 ftkips. OK
So, the checks of stability and bearing capacity are satisfied. The mat dimensions are shown in
Figure 5.19.
Figure 5.19 Foundation plan for concentrically braced frame system
5.1.4.2 Design Mat for Strength Demands. As was previously discussed, the computation of strength
demands for the concrete section should use the same soil pressure distribution as was used to satisfy
stability and bearing capacity. Because dozens of load combinations were considered and hand
calculations were used for the plastic distribution checks, the effort required would be considerable. The
same analysis used to determine elastic bearing pressures yields the corresponding section demands
directly. One approach to this dilemma would be to compute an additional factor that must be applied to
selected elastic cases to produce section demands that are consistent with the plastic solution. Rather than
provide such calculations here, design of the concrete section will proceed using the results of the elastic
analysis. This is conservative for the demand on the concrete for the same reason that it was
unsatisfactory for the soil: the edge soil pressures are high (that is, we are designing the concrete for a
peak soil pressure of 16.9 ksf, even though the plastic solution gives 13.2 ksf).
Standard Section 12.13.3 requires consideration of parametric variation for soil properties where
foundations are modeled explicitly. This example does not illustrate such calculations.
Concrete mats often have multiple layers of reinforcement in each direction at the top and bottom of their
thickness. Use of a uniform spacing for the reinforcement provided in a given direction greatly increases
the ease of construction. The minimum reinforcement requirements defined in Section 10.5 of ACI 318
were discussed in Section 5.1.1.3 above. Although all of the reinforcement provided to satisfy
Section 7.12 of ACI 318 may be provided near one face, for thick mats it is best to compute and provide
the amount of required reinforcement separately for the top and bottom halves of the section. Using a bar
spacing of 10 inches for this 7footthick mat and assuming one or two layers of bars, the section
capacities indicated in Table 5.14 (presented in order of decreasing strength) may be precomputed for
use in design. The amount of reinforcement provided for Marks B, C and D are less than the basic
minimum for flexural members, so the demands should not exceed threequarters of the design strength
where those reinforcement patterns are used. The amount of steel provided for Mark D is the minimum
that satisfies ACI 318 Section 7.12.
Table 5.14 Mat Foundation Section Capacities
Mark
Reinforcement
As (in.2 per ft)
Mn (ftkip/ft)
3/4 Mn (ftkip/ft)
A
2 layers of #10 bars at
10 in. o.c.
3.05
1,012
Not used
B
2 layers of #9 bars at
10 in. o.c.
2.40
Not used
601
C
2 layers of #8 bars at
10 in. o.c.
1.90
Not used
477
D
#8 bars at 10 in. o.c.
0.95
Not used
254
Note: Where the area of steel provided is less than the minimum reinforcement for flexural
members as indicated in ACI 318 Sec. 10.5.1, demands are compared to 3/4 of Mn as
permitted in Sec. 10.5.3.
To facilitate rapid design, the analysis results are processed in two additional ways. First, the flexural and
shear demands computed for the various load combinations are enveloped. Then the enveloped results
are presented (see Figure 5.110) using contours that correspond to the capacities shown for the
reinforcement patterns noted in Table 5.14.
Figure 5.110 Envelope of mat foundation flexural demands
Using the noted contours permits direct selection of reinforcement. The reinforcement provided within a
contour for a given mark must be that indicated for the next higher mark. For instance, all areas within
Contour B must have two layers of #10 bars. Note that the reinforcement provided will be symmetric
about the centerline of the mat in both directions. Where the results of finite element analysis are used in
the design of reinforced concrete elements, averaging of demands over short areas is appropriate. In
Figure 5.111, the selected reinforcement is superimposed on the demand contours. Figure 5.112 shows
a section of the mat along Gridline C.
Figure 5.111 Mat foundation flexural reinforcement
Figure 5.112 Section of mat foundation
Figure 5.113 presents the envelope of shear demands. The contours used correspond to the design
strengths computed assuming Vs = 0 for oneway and twoway shear. In the hatched areas the shear stress
exceeds and in the shaded areas it exceeds . The critical sections for twoway shear (as
discussed in Section 5.1.1.3) also are shown. The only areas that need more careful attention (to
determine whether they require shear reinforcement) are those where the hatched or shaded areas are
outside the critical sections. At the columns on Gridline D, the hatched area falls outside the critical
section, so closer inspection is needed. Because the perimeter of the hatched area is substantially smaller
than the perimeter of the critical section for punching shear, the design requirements of ACI 318 are
satisfied.
Oneway shears at the edges of the mat exceed the criterion. Note that the high shear stresses are
not produced by loads that create high bearing pressures at the edge. Rather, they are produced by loads
that create large bending stresses parallel to the edge. The distribution of bending moments and shears is
not uniform across the width (or breadth) of the mat, primarily due to the torsion in the seismic loads and
the orthogonal combination. It is also influenced by the doubled spring stiffnesses used to model the soil
condition. However, when the shears are averaged over a width equal to the effective depth (d), the
demands are less than the design strength.
In this design, reinforcement for punching or beam shear is not required. If shear reinforcement cannot be
avoided, standee bars may be used both to chair the upper decks of reinforcement and to provide
resistance to shear in which case they may be bent thus: .
Figure 5.113 Critical sections for shear and envelope of mat foundation shear demands
5.1.5 Cost Comparison
Table 5.15 provides a summary of the material quantities used for all of the foundations required for the
various conditions considered. Corresponding preliminary costs are assigned. The gravityonly condition
does not represent a realistic case because design for wind loads would require changes to the
foundations; it is provided here for discussion. It is obvious that design for lateral loads adds cost as
compared to a design that neglects such loads. However, it is also worth noting that braced frame systems
usually have substantially more expensive foundation systems than do moment frame systems. This
condition occurs for two reasons. First, braced frame systems are stiffer, which produces shorter periods
and higher design forces. Second, braced frame systems tend to concentrate spatially the demands on the
foundations. In this case the added cost amounts to approximately $0.80/ft2, which is an increase of
perhaps 4 or 5 percent to the cost of the structural system.
Table 5.15 Summary of Material Quantities and Cost Comparison
Design Condition
Concrete at Gravity
Foundations
Concrete at Lateral
Foundations
Total Excavation
Total Cost
Gravity only
(see Figure 5.14)
310 cy at $350/cy
= $108,600
310 cy at $30/cy
= $9,300
$117,900
Moment frame
(see Figure 5.16)
233 cy at $350/cy
= $81,600
507 cy at $400/cy
= $202,900
770 cy at $30/cy
= $23,100
$307,600
Braced frame
(see Figure 5.19)
233 cy at $350/cy
= $81,600
1,108 cy at $400/cy
= $443,300
1895 cy at $30/cy
= $56,800
$581,700
5.2 DEEP FOUNDATIONS FOR A 12STORY BUILDING, SEISMIC DESIGN CATEGORY D
This example features the analysis and design of deep foundations for a 12story reinforced concrete
momentresisting frame building similar to that described in Chapter 7 of this volume of design examples.
5.2.1 Basic Information
5.2.1.1 Description. Figure 5.21 shows the basic design condition considered in this example. A 2 2
pile group is designed for four conditions: for loads delivered by a corner and a side column of a
momentresisting frame system for Site Classes C and E. Geotechnical parameters for the two sites are
given in Table 5.21.
Figure 5.21 Design condition: Column of concrete momentresisting frame
supported by pile cap and castinplace piles
Table 5.21 Geotechnical Parameters
Depth
Class E Site
Class C Site
0 to 3 feet
Loose sand/fill
= 110 pcf
Angle of internal friction = 28 degrees
Soil modulus parameter, k = 25 pci
Neglect skin friction
Neglect end bearing
Loose sand/fill
= 110 pcf
Angle of internal friction = 30 degrees
Soil modulus parameter, k = 50 pci
Neglect skin friction
Neglect end bearing
3 to 30 feet
Soft clay
= 110 pcf
Undrained shear strength = 430 psf
Soil modulus parameter, k = 25 pci
Strain at 50 percent of maximum stress,
50 = 0.01
Skin friction (ksf) = 0.3
Neglect end bearing
Dense sand (one layer: 3 to 100foot depth)
= 130 pcf
Angle of internal friction = 42 degrees
Soil modulus parameter, k = 125 pci
Skin friction (ksf)* = 0.3 + 0.03/ft ó 2
End bearing (ksf)* = 65 + 0.6/ft ó 150
30 to 100 feet
Medium dense sand
= 120 pcf
Angle of internal friction = 36 degrees
Soil modulus parameter, k = 50 pci
Skin friction (ksf)* = 0.9 + 0.025/ft ó 2
End bearing (ksf)* = 40 + 0.5/ft ó 100
Pile cap
resistance
300 pcf, ultimate passive pressure
575 pcf, ultimate passive pressure
Resistance
factor,
0.8 for vertical, lateral and rocking
resistance of cohesive soil
0.7 for vertical, lateral and rocking
resistance of cohesionless soil
Safety factor
for settlement
2.5
2.5
*Skin friction and end bearing values increase (up to the maximum value noted) for each additional foot
of depth below the top of the layer. (The values noted assume a minimum pile length of 20 ft.)
The structural material properties assumed for this example are as follows:
f'c = 3,000 psi
fy = 60,000 psi
5.2.1.2 Seismic Parameters.
Site Class = C and E (both conditions considered in this example)
SDS = 1.1
Seismic Design Category = D (for both conditions)
5.2.1.3 Demands. The unfactored demands from the moment frame system are shown in Table 5.22.
Table 5.22 Gravity and Seismic Demands
Location
Load
Vx
Vy
P
Mxx
Myy
Corner
D
460.0
L
77.0
Vx
55.5
0.6
193.2
4.3
624.8
Vy
0.4
16.5
307.5
189.8
3.5
ATx
1.4
3.1
26.7
34.1
15.7
ATy
4.2
9.4
77.0
103.5
47.8
Side
D
702.0
L
72.0
Vx
72.2
0.0
0.0
0.0
723.8
Vy
0.0
13.9
181.6
161.2
1.2
ATx
0.4
1.8
2.9
18.1
4.2
ATy
1.2
5.3
8.3
54.9
12.6
Note: Units are kips and feet. Load Vy is for loads applied toward the east. ATx is the
corresponding accidental torsion case. Load Vx is for loads applied toward the north. ATy is the
corresponding accidental torsion case.
Using Load Combinations 5 and 7 from Section 12.4.2.3 of the Standard (with 0.2SDSD = 0.22D and
taking = 1.0), considering orthogonal effects as required for Seismic Design Category D and including
accidental torsion, the following 32 load conditions must be considered.
1.42D + 0.5L ñ 1.0Vx ñ 0.3Vy ñ max(1.0ATx, 0.3ATy)
1.42D + 0.5L ñ 0.3Vx ñ 1.0Vy ñ max(0.3ATx, 1.0ATy)
0.68D ñ 1.0Vx ñ 0.3Vy ñ max(1.0ATx, 0.3ATy)
0.68D ñ 0.3Vx ñ 1.0Vy ñ max(0.3ATx, 1.0ATy)
5.2.1.4 Design Approach. For typical deep foundation systems, resistance to lateral loads is provided by
both the piles and the pile cap. Figure 5.22 shows a simple idealization of this condition. The relative
contributions of these piles and pile cap depend on the particular design conditions, but often both effects
are significant. Resistance to vertical loads is assumed to be provided by the piles alone regardless of
whether their axial capacity is primarily due to end bearing, skin friction, or both. Although the behavior
of foundation and superstructure are closely related, they typically are modeled independently.
Earthquake loads are applied to a model of the superstructure, which is assumed to have fixed supports.
Then the support reactions are seen as demands on the foundation system. A similar substructure
technique is usually applied to the foundation system itself, whereby the behavior of pile cap and piles are
considered separately. This section describes that typical approach.
Figure 5.22 Schematic model of deep foundation system
5.2.1.4.1 Pile Group Mechanics. With reference to the free body diagram (of a 2 2 pile group) shown in
Figure 5.23, demands on individual piles as a result of loads applied to the group may be determined as
follows:
and M = V , where is a characteristic length determined from analysis of a
laterally loaded single pile.
, where s is the pile spacing, h is the height of the pile cap
and hp is the height of Vpassive above Point O.
and P = Pot + Pp
Figure 5.23 Pile cap free body diagram
5.2.1.4.2 Contribution of Piles. The response of individual piles to lateral loads is highly nonlinear. In
recent years it has become increasingly common to consider that nonlinearity directly. Based on
extensive testing of fullscale specimens and smallscale models for a wide variety of soil conditions,
researchers have developed empirical relationships for the nonlinear py response of piles that are suitable
for use in design. Representative py curves (computed for a 22inchdiameter pile) are shown in
Figure 5.24. The stiffness of the soil changes by an order of magnitude for the expected range of
displacements (the vertical axis uses a logarithmic scale). The py response is sensitive to pile size (an
effect not apparent in the figure, which is based on a single pile size); soil type and properties; and, in the
case of sands, vertical stress, which increases with depth. Pile response to lateral loads, like the py
curves on which the calculations are based, is usually computed using computer programs like LPILE.
Figure 5.24 Representative py curves
(note that a logarithmic scale is used on the vertical axis)
5.2.1.4.3 Contribution of Pile Cap. Pile caps contribute to the lateral resistance of a pile group in two
important ways: directly as a result of passive pressure on the face of the cap that is being pushed into the
soil mass and indirectly by producing a fixed head condition for the piles, which can significantly reduce
displacements for a given applied lateral load. Like the py response of piles, the passive pressure
resistance of the cap is nonlinear. Figure 5.25 shows how the passive pressure resistance (expressed as a
fraction of the ultimate passive pressure) is related to the imposed displacement (expressed as a fraction
of the minimum dimension of the face being pushed into the soil mass).
Figure 5.25 Passive pressure mobilization curve (after ASCE 41)
5.2.1.4.4 Group Effect Factors. The response of a group of piles to lateral loading will differ from that
of a single pile due to pilesoilpile interaction. (Group effect factors for axial loading of very closely
spaced piles may also be developed but are beyond the scope of the present discussion.)
Fullsize and model tests show that the lateral capacity of a pile in a pile group versus that of a
single pile (termed efficiency ) is reduced as the pile spacing is reduced. The observed group
effects are associated with shadowing effects. Various researchers have found that leading piles
are loaded more heavily than trailing piles when all piles are loaded to the same deflection. The
lateral resistance is primarily a function of row location within the group, rather than pile location
within a row. Researchers recommend that these effects may be approximated by adjusting the
resistance value on the single pile py curves (that is, by applying a pmultiplier).
Based on fullscale testing and subsequent analysis, Rollins et al. recommend the following p
multipliers (fm), where D is the pile diameter or width and s is the centertocenter spacing
between rows of piles in the direction of loading.
First (leading) row piles:
Second row piles:
Third or higher row piles:
Because the direction of loading varies during an earthquake and the overall efficiency of the group is the
primary point of interest, the average efficiency factor is commonly used for all members of a group in
the analysis of any given member. In that case, the average preduction factor is as follows:
For a 2 2 pile group thus with s = 3D, the group effect factor is calculated as follows:
For piles 1 and 2, in the leading row, .
For piles 3 and 4, in the second row, .
So, the group effect factor (average pmultiplier) is .
Figure 5.26 shows the group effect factors that are calculated for pile groups of various sizes with piles at
several different spacings.
Figure 5.26 Calculated group effect factors
5.2.2 Pile Analysis, Design and Detailing
5.2.2.1 Pile Analysis. For this design example, it is assumed that all piles will be fixedhead, 22inch
diameter, castinplace piles arranged in 2 2 pile groups with piles spaced at 66 inches centertocenter.
The computer program LPILE Plus 5.0 is used to analyze single piles for both soil conditions shown in
Table 5.21 assuming a length of 50 feet. Pile flexural stiffness is modeled using onehalf of the gross
moment of inertia because of expected flexural cracking. The response to lateral loads is affected to some
degree by the coincident axial load. The full range of expected axial loads was considered in developing
this example, but in this case the lateral displacements, moments and shears were not strongly affected;
the plots in this section are for zero axial load. A pmultiplier of 0.68 for group effects (as computed at
the end of Section 5.2.1.4) is used in all cases. Figures 5.27, 5.28 and 5.29 show the variation of shear,
moment and displacement with depth (within the top 30 feet) for an applied lateral load of 15 kips on a
single pile with the group reduction factor. It is apparent that the extension of piles to depths beyond
30 feet for the Class E site (or approximately 25 feet for the Class C site) does not provide additional
resistance to lateral loading; piles shorter than those lengths would have reduced lateral resistance. The
trends in the figures are those that should be expected. The shear and displacement are maxima at the pile
head. Because a fixedhead condition is assumed, moments are also largest at the top of the pile.
Moments and displacements are larger for the soft soil condition than for the firm soil condition.
Figure 5.27 Results of pile analysisshear versus depth
(applied lateral load is 15 kips)
Figure 5.28 Results of pile analysismoment versus depth
(applied lateral load is 15 kips)
Figure 5.29 Results of pile analysisdisplacement versus depth
(applied lateral load is 15 kips)
The analyses performed to develop Figures 5.27 through 5.29 are repeated for different levels of applied
lateral load. Figures 5.210 and 5.211 show how the moment and displacement at the head of the pile are
related to the applied lateral load. It may be seen from Figure 5.210 that the head moment is related to
the applied lateral load in a nearly linear manner; this is a key observation. Based on the results shown,
the slope of the line may be taken as a characteristic length that relates head moment to applied load.
Doing so produces the following:
= 46 in. for the Class C site
= 70 in. for the Class E site
Figure 5.210 Results of pile analysis applied lateral load versus head moment
Figure 5.211 Results of pile analysis head displacement versus applied lateral load
A similar examination of Figure 5.211 leads to another meaningful insight. The loaddisplacement
response of the pile in Site Class C soil is essentially linear. The response of the pile in Site Class E soil
is somewhat nonlinear, but for most of the range of response a linear approximation is reasonable (and
useful). Thus, the effective stiffness of each individual pile is:
k = 175 kip/in. for the Class C site
k = 40 kip/in. for the Class E site
5.2.2.2 Pile Group Analysis. The combined response of the piles and pile cap and the resulting strength
demands for piles are computed using the procedure outlined in Section 5.2.1.4 for each of the 32 load
combinations discussed in Section 5.2.1.3. Assume that each 2 2 pile group has a 9'2" 9'2" 4'0"
thick pile cap that is placed 1'6" below grade.
Check the Maximum Compression Case under a Side Column in Site Class C
Using the sign convention shown in Figure 5.23, the demands on the group are as follows:
P = 1,224 kip
Myy = 222 ftkips
Vx = 20 kips
Myy = 732 ftkips
Vy = 73 kips
From preliminary checks, assume that the displacements in the x and y directions are sufficient to
mobilize 30 percent and 35 percent, respectively, of the ultimate passive pressure:
and
and conservatively take hp = h/3 = 16 inches.
Since Vpassive,x > Vx, passive resistance alone is sufficient for this case in the x direction. However, in order
to illustrate the full complexity of the calculations, reduce Vpassive,x to 4 kips and assign a shear of 4.0 kips
to each pile in the x direction. In the y direction, the shear in each pile is as follows:
The corresponding pile moments are:
M = 4.0(46) = 186 in.kips for xdirection loading
and
M = 11.8(46) = 543 in.kips for ydirection loading
The maximum axial load due to overturning for xdirection loading is:
and for ydirection loading (determined similarly), Pot = 106.4 kips.
The axial load due to direct loading is Pp = 1224/4 = 306 kips.
Therefore, the maximum load effects on the most heavily loaded pile are the following:
Pu = 32.5 + 106.4 + 306 = 445 kips
The expected displacement in the y direction is computed as follows:
= V/k = 11.8/175 = 0.067 in., which is 0.14 percent of the pile cap height (h)
Reading Figure 5.25 with /H = 0.0014, P/Pult ÷ 0.34, so the assumption that 35 percent of Pult would be
mobilized was reasonable.
5.2.2.3 Design of Pile Section. The calculations shown in Section 5.2.2.2 are repeated for each of the 32
load combinations under each of the four design conditions. The results are shown in Figures 5.212 and
5.213. In these figures, circles indicate demands on piles under side columns and squares indicate
demands on piles under corner columns. Also plotted are the P M design strengths for the 22inch
diameter pile sections with various amounts of reinforcement (as noted in the legends). The appropriate
reinforcement pattern for each design condition may be selected by noting the innermost capacity curve
that envelops the corresponding demand points. The required reinforcement is summarized in Table 5.2
4, following calculation of the required pile length.
Figure 5.212 PM interaction diagram for Site Class C
Figure 5.213 PM interaction diagram for Site Class E
5.2.2.4 Pile Length for Axial Loads. For the calculations that follow, recall that skin friction and end
bearing are neglected for the top 3 feet in this example. The design is based on having 1 6 of soil over a
4 0 deep pile cap.
5.2.2.4.1 Length for Settlement. Service loads per pile are calculated as P = (PD + PL)/4.
Check the pile group under the side column in Site Class C, assuming L = 52.5 feet 5.5 feet = 47 feet:
P = (752 + 114)/4 = 217 kips.
Pskin = average friction capacity pile perimeter pile length for friction
= 0.5[0.3 + 2.5(0.03) + 0.3 + 49.5(0.03)] (22/12)(44) = 292 kips
Pend = end bearing capacity at depth end bearing area
= [65 + 49.5(0.6)]( /4)(22/12)2 = 250 kips
Pallow = (Pskin + Pend)/S.F. = (292 + 250)/2.5 = 217 kips = 217 kips (demand) OK
Check the pile group under the corner column in Site Class E, assuming L = 49 feet:
P = (460 + 77)/4 = 134 kips
Pskin = [friction capacity in first layer + average friction capacity in second layer] pile perimeter
= [24.5(0.3) + (24.5/2)(0.9 + 0.9 + 24.5[0.025])] (22/12) = 212 kips
Pend = [40 + 24.5(0.5)]( /4)(22/12)2 = 138 kips
Pallow = (212 + 138)/2.5 = 140 kips > 134 kips OK
5.2.2.4.2 Length for Compression Capacity. All of the strengthlevel load combinations (discussed in
Section 5.2.1.3) must be considered.
Check the pile group under the side column in Site Class C, assuming L = 49 feet:
As seen in Figure 5.112, the maximum compression demand for this condition is Pu = 394 kips.
Pskin = 0.5[0.3 + 0.3 + 47(0.03)] (22/12)(47) = 272 kips
Pend = [65 + 47(0.6)]( /4)(22/12)2 = 246 kips
Pn = (Pskin + Pend) = 0.75(272 + 246) = 389 kips ÷ 390 kips OK
Check the pile group under the corner column in Site Class E, assuming L = 64 feet:
As seen in Figure 5.213, the maximum compression demand for this condition is Pu = 340 kips.
Pskin = [27(0.3) + (34/2)(0.9 + 0.9 + 34[0.025])] (22/12) = 306 kips
Pend = [40 + 34(0.5)]( /4)(22/12)2 = 150 kips
Pn = (Pskin + Pend) = 0.75(306 + 150) = 342 kips > 340 kips OK
5.2.2.4.3 Length for Uplift Capacity. Again, all of the strengthlevel load combinations (discussed in
Section 5.2.1.3) must be considered.
Check the pile group under side column in Site Class C, assuming L = 5 feet:
As seen in Figure 5.212, the maximum tension demand for this condition is Pu = 1.9 kips.
Pskin = 0.5[0.3 + 0.3 + 2(0.03)] (22/12)(2) = 3.8 kips
Pn = (Pskin) = 0.75(3.8) = 2.9 kips > 1.9 kips OK
Check the pile group under the corner column in Site Class E, assuming L = 52 feet:
As seen in Figure 5.213, the maximum tension demand for this condition is Pu = 144 kips.
Pskin = [27(0.3) + (22/2)(0.9 + 0.9 + 22[0.025])] (22/12) = 196 kips
Pn = (Pskin) = 0.75(196) = 147 kips > 144 kips OK
5.2.2.4.4 Graphical Method of Selecting Pile Length. In the calculations shown above, the adequacy of
the soilpile interface to resist applied loads is checked once a pile length is assumed. It would be
possible to generate mathematical expressions of pile capacity as a function of pile length and then solve
such expressions for the demand conditions. However, a more practical design approach is to pre
calculate the capacity for piles for the full range of practical lengths and then select the length needed to
satisfy the demands. This method lends itself to graphical expression as shown in Figures 5.214 and 5.2
15.
Figure 5.214 Pile axial capacity as a function of length for Site Class C
Figure 5.215 Pile axial capacity as a function of length for Site Class E
5.2.2.4.5 Results of Pile Length Calculations. Detailed calculations for the required pile lengths are
provided above for two of the design conditions. Table 5.23 summarizes the lengths required to satisfy
strength and serviceability requirements for all four design conditions.
Table 5.23 Pile Lengths Required for Axial Loads
Piles Under Corner Column
Piles Under Side Column
Site Class
Condition
Load
Min Length
Condition
Load
Min Length
Site Class C
Compression
369 kip
46 ft
Compression
394 kip
49 ft
Uplift
108 kip
32 ft
Uplift
13.9 kip
8 ft
Settlement
134 kip
27 ft
Settlement
217 kip
47 ft
Site Class E
Compression
378 kip
61 ft
Compression
406 kip
64 ft
Uplift
119 kip
42 ft
Uplift
23.6 kip
17 ft
Settlement
134 kip
48 ft
Settlement
217 kip
67 ft
5.2.2.5 Design Results. The design results for all four pile conditions are shown in Table 5.24. The
amount of longitudinal reinforcement indicated in the table is that required at the pilepile cap interface
and may be reduced at depth as discussed in the following section.
Table 5.24 Summary of Pile Size, Length and Longitudinal Reinforcement
Site Class
Piles Under Corner Column
Piles Under Side Column
Site Class C
22 in. diameter by 46 ft long
22 in. diameter by 49 ft long
8#6 bars
6#5 bars
Site Class E
22 in. diameter by 61 ft long
22 in. diameter by 67 ft long
8#7 bars
6#6 bars
5.2.2.6 Pile Detailing. Standard Sections 12.13.5, 12.13.6, 14.2.3.1 and 14.2.3.2 contain special pile
requirements for structures assigned to Seismic Design Category C or higher and D or higher. In this
section, those general requirements and the specific requirements for uncased concrete piles that apply to
this example are discussed. Although the specifics are affected by the soil properties and assigned site
class, the detailing of the piles designed in this example focuses on consideration of the following
fundamental items:
All pile reinforcement must be developed in the pile cap (Standard Sec. 12.13.6.5).
In areas of the pile where yielding might be expected or demands are large, longitudinal and
transverse reinforcement must satisfy specific requirements related to minimum amount and
maximum spacing.
Continuous longitudinal reinforcement must be provided over the entire length resisting design
tension forces (ACI 318 Sec. 21.12.4.2).
The discussion that follows refers to the detailing shown in Figures 5.216 and 5.217.
5.2.2.6.1 Development at the Pile Cap. Where neither uplift nor flexural restraint are required, the
development length is the full development length for compression. Where the design relies on head
fixity or where resistance to uplift forces is required (both of which are true in this example), pile
reinforcement must be fully developed in tension unless the section satisfies the overstrength load
condition or demands are limited by the uplift capacity of the soilpile interface (Standard Sec. 12.13.6.5).
For both site classes considered in this example, the pile longitudinal reinforcement is extended straight
into the pile cap a distance that is sufficient to fully develop the tensile capacity of the bars. In addition to
satisfying the requirements of the Standard, this approach offers two advantages. By avoiding lap splices
to fieldplaced dowels where yielding is expected near the pile head (although such would be permitted
by the Standard), more desirable inelastic performance would be expected. Straight development, while
it may require a thicker pile cap, permits easier placement of the pile cap s bottom reinforcement
followed by the addition of the spiral reinforcement within the pile cap. Note that embedment of the
entire pile in the pile cap facilitates direct transfer of shear from pile cap to pile but is not a requirement of
the Standard. (Section 1810.3.11 of the 2009 International Building Code requires that piles be
embedded at least 3 inches into pile caps.)
Figure 5.216 Pile detailing for Site Class C (under side column)
Figure 5.217 Pile detailing for Site Class E (under corner column)
5.2.2.6.2 Longitudinal and Transverse Reinforcement Where Demands Are Large. Requirements
for longitudinal and transverse reinforcement apply over the entire length of pile where demands are
large. For uncased concrete piles in Seismic Design Category D, at least four longitudinal bars (with a
minimum reinforcement ratio of 0.005) must be provided over the largest region defined as follows: the
top onehalf of the pile length, the top 10 feet below the ground, or the flexural length of the pile. The
flexural length is taken as the length of pile from the cap to the lowest point where 0.4 times the concrete
section cracking moment (see ACI 318 Section 9.5.2.3) exceeds the calculated flexural demand at that
point. For the piles used in this example, onehalf of the pile length governs. (Note that providing a
given reinforcement ratio means that the reinforcement in question must be developed at that point. Bar
development and cutoff are discussed in more detail in Chapter 7 of this volume of design examples.)
Transverse reinforcement must be provided over the same length for which minimum longitudinal
reinforcement requirements apply. Because the piles designed in this example are larger than 20 inches in
diameter, the transverse reinforcement may not be smaller than 0.5 inch diameter. For the piles shown in
Figures 5.216 and 5.217, the spacing of the transverse reinforcement in the top half of the pile length
may not exceed the least of the following: 12db (7.5 in. for #5 longitudinal bars and 10.5 in. for #7
longitudinal bars), 22/2 = 11 in., or 12 in.
Where yielding may be expected, even more stringent detailing is required. For the Class C site, yielding
can be expected within three diameters of the bottom of the pile cap (3D = 3 22 = 66 in.). Spiral
reinforcement in that region must not be less than onehalf of that required in Section 21.4.4.1(a) of
ACI 318 (since the site is not Class E, Class F, or liquefiable) and the requirements of Sections 21.4.4.2
and 21.4.4.3 must be satisfied. Note that Section 21.4.4.1(a) refers to Equation 105, which often will
govern. In this case, the minimum volumetric ratio of spiral reinforcement is onehalf that determined
using ACI 318 Equation 105. In order to provide a reinforcement ratio of 0.01 for this pile section, a #4
spiral must have a pitch of no more than 4.8 inches, but the maximum spacing permitted by
Section 21.4.4.2 is 22/4 = 5.5 inches or 6db = 3.75 inches, so a #4 spiral at 3.75inch pitch is used.
(Section 1810.3.2.1.2 of the 2009 International Building Code clarifies that ACI 318 Equation 105 need
not be applied to piles.)
For the Class E site, the more stringent detailing must be provided within seven diameters of the pile cap
and of the interfaces between strata that are hard or stiff and strata that are liquefiable or are composed of
soft to mediumstiff clay (Standard Sec. 14.2.3.2.1). The author interprets within seven diameters of ...
the interface as applying in the direction into the softer material, which is consistent with the expected
location of yielding. Using that interpretation, the Standard does not indicate the extent of such detailing
into the firmer material. Taking into account the soil layering shown in Table 5.21 and the pile cap depth
and thickness, the tightly spaced transverse reinforcement shown in Figure 5.217 is provided within 7D
of the bottom of pile cap and top of firm soil and is extended a little more than 3D into the firm soil.
Because the site is Class E, the full amount of reinforcement indicated in ACI 318 Section 21.6.4 must be
provided. In order to provide a reinforcement ratio of 0.02 for this pile section, a #5 spiral must have a
pitch of no more than 3.7 inches. The maximum spacing permitted by Section 21.6.4.3 is 22/4 =
5.5 inches or 6db = 5.25 inches, so a #5 spiral at 3.5inch pitch is used.
5.2.2.6.3 Continuous Longitudinal Reinforcement for Tension. Table 5.23 shows the pile lengths
required for resistance to uplift demands. For the Site Class E condition under a corner column
(Figure 5.217), longitudinal reinforcement must resist tension for at least the top 42 feet (being
developed at that point). Extending four longitudinal bars for the full length and providing widely spaced
spirals at such bars is practical for placement, but it is not a specific requirement of the Standard. For the
Site Class C condition under a side column (Figure5.216), design tension due to uplift extends only
approximately 5 feet below the bottom of the pile cap. Therefore, a design with Section C of
Figure 5.216 being unreinforced would satisfy the Provisions requirements, but the author has decided to
extend very light longitudinal and nominal transverse reinforcement for the full length of the pile.
5.2.3 Other Considerations
5.2.3.1 Foundation Tie Design and Detailing. Standard Section 12.13.5.2 requires that individual pile
caps be connected by ties. Such ties are often grade beams, but the Standard would permit use of a slab
(thickened or not) or calculations that demonstrate that the site soils (assigned to Site Class A, B, or C)
provide equivalent restraint. For this example, a tie beam between the pile caps under a corner column
and a side column is designed. The resulting section is shown in Figure 5.218.
For pile caps with an assumed centertocenter spacing of 32 feet in each direction and given Pgroup =
1,224 kips under a side column and Pgroup = 1,142 kips under a corner column, the tie is designed as
follows.
As indicated in Standard Section 12.13.5.2, the minimum tie force in tension or compression equals the
product of the larger column load times SDS divided by 10 = 1224(1.1)/10 = 135 kips.
The design strength for six #6 bars is as follows
As fy = 0.9(6)(0.44)(60) = 143 kips > 135 kips OK
According to ACI 318 Section 21.12.3.2, the smallest crosssectional dimension of the tie beam must not
be less than the clear spacing between pile caps divided by 20 = (32'0"  9'2")/20 = 13.7 inches. Use a
tie beam that is 14 inches wide and 16 inches deep. ACI 318 Section 21.12.3.2 further indicates that
closed ties must be provided at a spacing of not more than onehalf the minimum dimension, which is
14/2 = 7 inches.
Assuming that the surrounding soil provides restraint against buckling, the design strength of the tie beam
concentrically loaded in compression is as follows:
Pn = 0.8 [0.85f'c(Ag  Ast) + fyAst]
= 0.8(0.65)[0.85(3){(16)(14) 6(0.44)}+ 60(6)(0.44)] = 376 kips > 135 kips OK
Figure 5.218 Foundation tie section
5.2.3.2 Liquefaction. For Seismic Design Categories C, D, E and F, Standard Section 11.8.2 requires
that the geotechnical report address potential hazards due to liquefaction. For Seismic Design
Categories D, E and F, Standard Section 11.8.3 further requires that the geotechnical report describe the
likelihood and potential consequences of liquefaction and soil strength loss (including estimates of
differential settlement, lateral movement, lateral loads on foundations, reduction in foundation soil
bearing capacity, increases in lateral pressures on retaining walls and flotation of buried structures) and
discuss mitigation measures. During the design of the structure, such measures (which can include
ground stabilization, selection of appropriate foundation type and depths and selection of appropriate
structural systems to accommodate anticipated displacements and forces) must be considered. Provisions
Part 3, Resource Paper 12 contains a calculation procedure that can be used to evaluate the liquefaction
hazard.
5.2.3.3 Kinematic Interaction. Piles are subjected to curvature demands as a result of two different
types of behavior: inertial interaction and kinematic interaction. The term inertial interaction is used to
describe the coupled response of the soilfoundationstructure system that arises as a consequence of the
mass properties of those components of the overall system. The structural engineer s consideration of
inertial interaction is usually focused on how the structure loads the foundation and how such loads are
transmitted to the soil (as shown in the pile design calculations that are the subject of most of this
example) but also includes assessment of the resulting foundation movement. The term kinematic
interaction is used to describe the manner in which the stiffness of the foundation system impedes
development of freefield ground motion. Consideration of kinematic interaction by the structural
engineer is usually focused on assessing the strength and ductility demands imposed directly on piles by
movement of the soil. Although it is rarely done in practice, Standard Section 12.13.6.3 requires
consideration of kinematic interaction for foundations of structures assigned to Seismic Design
Category D, E, or F. Kramer discusses kinematic and inertial interaction and the methods of analysis
employed in consideration of those effects and demonstrates that the solution to the entire soilstructure
interaction problem is equal to the sum of the solutions of the kinematic and inertial interaction analyses.
One approach that would satisfy the requirements of the Standard would be as follows:
The geotechnical consultant performs appropriate kinematic interaction analyses considering
freefield ground motions and the stiffness of the piles to be used in design.
The resulting pile demands, which generally are greatest at the interface between stiff and soft
strata, are reported to the structural engineer.
The structural engineer designs piles for the sum of the demands imposed by the vibrating
superstructure and the demands imposed by soil movement.
A more practical, but less rigorous, approach is to provide appropriate detailing in regions of the pile
where curvature demands imposed directly by earthquake ground motions are expected to be significant.
Where such a judgmentbased approach is used, one must decide whether to provide only additional
transverse reinforcement in areas of concern to improve ductility or whether additional longitudinal
reinforcement should also be provided to increase strength. Section 18.10.2.4.1 of the 2009 International
Building Code permits application of such deemedtocomply detailing in lieu of explicit calculations and
prescribes a minimum longitudinal reinforcement ratio of 0.005.
5.2.3.4 Design of Pile Cap. Design of pile caps for large pile loads is a very specialized topic for which
detailed treatment is beyond the scope of this volume of design examples. CRSI notes that most pile
caps are designed in practice by various shortcut ruleofthumb procedures using what are hoped to be
conservative allowable stresses. Wang & Salmon indicates that pile caps frequently must be designed
for shear considering the member as a deep beam. In other words, when piles are located inside the
critical sections d (for oneway action) or d/2 (for twoway action) from the face of column, the shear
cannot be neglected. They go on to note that there is no agreement about the proper procedure to use.
Direct application of the special provisions for deep flexural members as found in ACI 318 is not possible
since the design conditions are somewhat different. CRSI provides a detailed outline of a design
procedure and tabulated solutions, but the procedure is developed for pile caps subjected to concentric
vertical loads only (without applied overturning moments or pile head moments). Strutandtie models
(as described in Appendix A of ACI 318) may be employed, but their application to elements with
important threedimensional characteristics (such as pile caps for groups larger than 2 1) is so involved
as to preclude hand calculations.
5.2.3.5 Foundation Flexibility and Its Impact on Performance
5.2.3.5.1 Discussion. Most engineers routinely use fixedbase models. Nothing in the Provisions or
Standard prohibits that common practice; the consideration of foundation flexibility and of soilstructure
interaction effects (Standard Section 12.13.3 and Chapter 19) is permitted but not required. Such
fixedbase models can lead to erroneous results, but engineers have long assumed that the errors are
usually conservative. There are two obvious exceptions to that assumption: soft soil siteresonance
conditions (e.g., as in the 1985 Mexico City earthquake) and excessive damage or even instability due to
increased displacement response.
Site resonance can result in significant amplification of ground motion in the period range of interest. For
sites with a fairly long predominant period, the result is spectral accelerations that increase as the
structural period approaches the site period. However, the shape of the general design spectrum used in
the Standard does not capture that effect; for periods larger than T0, accelerations remain the same or
decrease with increasing period. Therefore, increased system period (as a result of foundation flexibility)
always leads to lower design forces where the general design spectrum is used. Sitespecific spectra may
reflect longperiod siteresonance effects, but the use of such spectra is required only for Class F sites.
Clearly, an increase in displacements, caused by foundation flexibility, does change the performance of a
structure and its contents raising concerns regarding both stability and damage. Earthquakeinduced
instability of buildings has been exceedingly rare. The analysis and acceptance criteria in the Standard
are not adequate to the task of predicting real stability problems; calculations based on linear, static
behavior cannot be used to predict instability of an inelastic system subjected to dynamic loading. While
Provisions Part 2 Section 12.12 indicates that structural stability was considered in arriving at the
consensus judgment reflected in the drift limits, such considerations were qualitative. In point of fact,
the values selected for the drift limits were selected considering damage to nonstructural systems (and,
perhaps in some cases, control of structural ductility demands). For most buildings, application of the
Standard is intended to satisfy performance objectives related to life safety and collapse prevention, not
damage control or postearthquake occupancy. Larger design forces and more stringent drift limits are
applied to structures assigned to Occupancy Category III or IV in the hope that those measures will
improve performance without requiring explicit consideration of such performance. Although foundation
flexibility can affect structural performance significantly, since all consideration of performance in the
context of the Standard is approximate and judgmentbased, it is difficult to define how such changes in
performance should be characterized. Explicit consideration of performance measures also tends to
increase engineering effort substantially, so mandatory performance checks often are resisted by the user
community.
The engineering framework established in ASCE 41 is more conducive to explicit use of performance
measures. In that document (Sections 4.4.3.2.1 and 4.4.3.3.1), the use of fixedbased structural models is
prohibited for buildings being rehabilitated for the Immediate Occupancy Performance Level that are
sensitive to base rotations or other types of foundation movement. In this case the focus is on damage
control rather than structural stability.
5.2.3.5.2 Example Calculations. To assess the significance of foundation flexibility, one may compare
the dynamic characteristics of a fixedbase model to those of a model in which foundation effects are
included. The effects of foundation flexibility become more pronounced as foundation period and
structural period approach the same value. For this portion of the example, use the Site Class E pile
design results from Section 5.2.2.1 and consider the northsouth response of the concrete moment frame
building located in Berkeley (Section 7.2) as representative for this building.
5.2.3.5.2.1 Stiffness of the Structure. Calculations of the effect of foundation flexibility on the dynamic
response of a structure should reflect the overall stiffness of the structure (e.g., that associated with the
fundamental mode of vibration) rather than the stiffness of any particular story. Table 72 shows that the
total weight of the structure is 43,919 kips. Table 73 shows that the calculated period of the fixedbase
structure is 2.02 seconds and Table 77 indicates that 83.6 percent of the mass participates in that mode.
Using the equation for the undamped period of vibration of a singledegreeoffreedom oscillator, the
effective stiffness of the structure is as follows:
5.2.3.5.2.2 Foundation Stiffness. As seen in Figure 71, there are 36 moment frame columns. Assume
that a 2 2 pile group supports each column. As shown in Section 5.2.2.1, the stiffness of each pile is
40 kip/in. Neglecting both the stiffness contribution from passive pressure resistance and the flexibility
of the beamslab system that ties the pile caps, the stiffness of each pile group is 4 40 = 160 kip/in. and
the stiffness of the entire foundation system is 36 160 = 5,760 kip/in.
5.2.3.5.2.3 Effect of Foundation Flexibility. Because the foundation stiffness is much greater than the
structural stiffness, period elongation is expected to be minimal. To confirm this expectation, the period
of the combined system is computed. The total stiffness for the system (springs in series) is as follows:
Assume that the weight of the foundation system is 4,000 kips and that 100 percent of the corresponding
mass participates in the new fundamental mode of vibration. The period of the combined system is as
follows:
which is an increase of 13 percent over that predicted by the fixedbase model. For systems responding in
the constantvelocity portion of the spectrum, accelerations (and thus forces) are a function of 1/T and
relative displacements are a function of T. Therefore, with respect to the fixedbased model, the
combined system would have forces that are 12 percent smaller and displacements that are 13 percent
larger. In the context of earthquake engineering, those differences are not significant.
6
Structural Steel Design
Rafael Sabelli, S.E. and Brian Dean, P.E.
Originally developed by
James R. Harris, P.E., PhD, Frederick R. Rutz, P.E., PhD and Teymour Manouri, P.E., PhD
Contents
6.1 INDUSTRIAL HIGHCLEARANCE BUILDING, ASTORIA, OREGON 3
6.1.1 Building Description 3
6.1.2 Design Parameters 6
6.1.3 Structural Design Criteria 7
6.1.4 Analysis 10
6.1.5 Proportioning and Details 16
6.2 SEVENSTORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA 40
6.2.1 Building Description 40
6.2.2 Basic Requirements 42
6.2.3 Structural Design Criteria 44
6.2.4 Analysis and Design of Alternative A: SMF 46
6.2.5 Analysis and Design of Alternative B: SCBF 60
6.2.6 Cost Comparison 72
6.3 TENSTORY HOSPITAL, SEATTLE, WASHINGTON 72
6.3.1 Building Description 72
6.3.2 Basic Requirements 76
6.3.3 Structural Design Criteria 78
6.3.4 Elastic Analysis 80
6.3.5 Initial Proportioning and Details 86
6.3.6 Nonlinear Response History Analysis 93
This chapter illustrates how the 2009 NEHRP Recommended Provisions (the Provisions) is applied to the
design of steel framed buildings. The following three examples are presented:
1. An industrial warehouse structure in Astoria, Oregon
2. A multistory office building in Los Angeles, California
3. A midrise hospital in Seattle, Washington
The discussion examines the following types of structural framing for resisting horizontal forces:
Ordinary concentrically braced frames (OCBF)
Special concentrically braced frames
Intermediate moment frames
Special moment frames
Bucklingrestrained braced frames, with momentresisting beamcolumn connections
The examples cover design for seismic forces in combination with gravity they are presented to illustrate
only specific aspects of seismic analysis and design such as lateral force analysis, design of concentric
and eccentric bracing, design of moment resisting frames, drift calculations, member proportioning
detailing.
All structures are analyzed using threedimensional static or dynamic methods. ETABS (Computers &
Structures, Inc., Berkeley, California, v.9.5.0, 2008) is used in Examples 6.1 and 6.2.
In addition to the 2009 NEHRP Recommended Provisions, the following documents are referenced:
AISC 341 American Institute of Steel Construction. 2005. Seismic Provisions for
Structural Steel Buildings, including Supplement No. 1.
AISC 358 American Institute of Steel Construction. 2005. Prequalified Connections
for Special and Intermediate Steel Moment Frames for Seismic Applications.
AISC 360 American Institute of Steel Construction. 2005. Specification for Structural
Steel Buildings.
AISC Manual American Institute of Steel Construction. 2005. Manual of Steel
Construction, 13th Edition.
AISC SDM American Institute of Steel Construction. 2006. Seismic Design Manual.
IBC International Code Council, Inc. 2006. 2006 International Building Code.
AISC SDGS4 AISC Steel Design Guide Series 4. Second Edition. 2003. Extended End
Plate Moment Connections, 2003.
SDI Luttrell, Larry D. 1981. Steel Deck Institute Diaphragm Design Manual.
Steel Deck Institute.
The symbols used in this chapter are from Chapter 11 of the Standard, the above referenced documents,
or are as defined in the text. U.S. Customary units are used.
6.1 INDUSTRIAL HIGHCLEARANCE BUILDING, ASTORIA, OREGON
This example utilizes a transverse intermediate steel moment frame and a longitudinal ordinary concentric
steel braced frame. The following features of seismic design of steel buildings are illustrated:
Seismic design parameters
Equivalent lateral force analysis
Threedimensional analysis
Drift check
Check of compactness and spacing for moment frame bracing
Moment frame connection design
Proportioning of concentric diagonal bracing
6.1.1 Building Description
This building has plan dimensions of 180 feet by 90 feet and a clear height of approximately 30 feet. It
includes a 12foothigh, 40footwide mezzanine area at the east end of the building. The structure
consists of 10 gable frames spanning 90 feet in the transverse (northsouth) direction. Spaced at 20 feet
on center, these frames are braced in the longitudinal (eastwest) direction in two bays at the east end.
The building is enclosed by nonstructural insulated concrete wall panels and is roofed with steel decking
covered with insulation and roofing. Columns are supported on spread footings.
The elevation and transverse sections of the structure are shown in Figure 6.11. Longitudinal struts at
the eaves and at the mezzanine level run the full length of the building and therefore act as collectors for
the distribution of forces resisted by the diagonally braced bays and as weakaxis stability bracing for the
moment frame columns.
The roof and mezzanine framing plans are shown in Figure 6.12. The framing consists of a steel roof
deck supported by joists between transverse gable frames. The mezzanine represents both an additional
load and additional strength and stiffness. Because all the frames resist lateral loading, the steel deck
functions as a diaphragm for distribution of the effects of eccentric loading caused by the mezzanine floor
when the building is subjected to loads acting in the transverse direction.
The mezzanine floor at the east end of the building is designed to accommodate a live load of 125 psf. Its
structural system is composed of a concrete slab over steel decking supported by floor beams spaced at
10 feet on center. The floor beams are supported on girders continuous over two intermediate columns
spaced approximately 30 feet apart and are attached to the gable frames at each end.
The member sizes in the main frame are controlled by serviceability considerations. Vertical deflections
due to snow were limited to 3.5 inches and lateral sway due to wind was limited to 2 inches.
Figure 6.11 Framing elevation and sections
(1.0 ft = 0.3048 m; 1.0 in. = 25.4 mm)
Earthquake rather than wind governs the lateral design due to the mass of the insulated concrete panels.
The panels are attached with long pins perpendicular to the concrete surface. These slender, flexible pins
isolate the panels from acting as shear walls.
The building is supported on spread footings based on moderately deep alluvial deposits (i.e., medium
dense sands). The foundation plan is shown in Figure 6.13. Transverse ties are placed between the
footings of the two columns of each moment frame to provide restraint against horizontal thrust from the
moment frames. Grade beams carrying the enclosing panels serve as ties in the longitudinal direction as
well as across the end walls. The design of footings and columns in the braced bays requires
consideration of combined seismic loadings. The design of foundations is not included here.
Figure 6.12 Roof framing and mezzanine framing plan
(1.0 ft = 0.3048 m; 1.0 in. = 25.4 mm)
Figure 6.13 Foundation plan
(1.0 ft = 0.3048 m; 1.0 in. = 25.4 mm)
6.1.2 Design Parameters
6.1.2.1 Ground motion and system parameters. See Section 3.2 for an example illustrating the
determination of design ground motion parameters. For this example the parameters are as follows.
SDS = 1.0
SD1 = 0.6
Occupancy Category II
Seismic Design Category D
Note that Standard Section 12.2.5.6 permits an ordinary steel moment frame for buildings that do not
exceed one story and 65 feet tall with a roof dead load not exceeding 20 psf. Intermediate steel moment
frames with stiffened bolted end plates and ordinary steel concentrically braced frames are used in this
example.
Northsouth (NS) direction:
Momentresisting frame system = intermediate steel moment frame (Standard Table 12.21)
R = 4.5
0 = 3
Cd = 4
Eastwest (EW) direction:
Braced frame system = ordinary steel concentrically braced frame (Standard Table 12.21)
R = 3.25
0 = 2
Cd = 3.25
6.1.2.2 Loads
Roof live load (L), snow = 25 psf
Roof dead load (D) = 15 psf
Mezzanine live load, storage = 125 psf
Mezzanine slab and deck dead load = 69 psf
Weight of wall panels = 75 psf
Roof dead load includes roofing, insulation, metal roof deck, purlins, mechanical and electrical equipment
that portion of the main frames that is tributary to the roof under lateral load. For determination of the
seismic weights, the weight of the mezzanine will include the dead load plus 25 percent of the storage
load (125 psf) in accordance with Standard Section 12.7.2. Therefore, the mezzanine seismic weight is
69 + 0.25(125) = 100 psf.
6.1.2.3 Materials
Concrete for footings: fc' = 2.5 ksi
Slabsongrade: fc' = 4.5 ksi
Mezzanine concrete on metal deck: fc' = 3.0 ksi
Reinforcing bars: ASTM A615, Grade 60
Structural steel (wide flange sections): ASTM A992, Grade 50
Plates (except continuity plates): ASTM A36
Bolts: ASTM A325
Continuity Plates: ASTM A572, Grade 50
6.1.3 Structural Design Criteria
6.1.3.1 Building configuration. Because there is a mezzanine at one end, vertical weight irregularities
might be considered to apply (Standard Sec. 12.3.2.2). However, the upper level is a roof and the
Standard exempts roofs from weight irregularities. There also are no plan irregularities in this building
(Standard Sec. 12.3.2.1).
6.1.3.2 Redundancy. In the NS direction, the moment frames do not meet the requirements of Standard
Section 12.3.4.2b since the frames are only one bay long. Thus, Standard Section 12.3.4.2a must be
checked. A copy of the threedimensional model is made, with the moment frame beam at Gridline A
pinned. The structure is checked to make sure that an extreme torsional irregularity (Standard Table
12.31) does not occur:
where:
A = maximum displacement at knee along Gridline A, in.
K = maximum displacement at knee along gridline K, in.
Thus, the structure does not have an extreme torsional irregularity when a frame loses moment resistance.
Additionally, the structure must be checked in the NS direction to ensure that the loss of moment
resistance at Beam A has not resulted in more than a 33 percent reduction in story strength. This can be
checked using elastic methods (based on first yield) as shown below, or using strength methods. The
original model is run with the NS load combinations to determine the member with the highest demand
capacity ratio. This demandcapacity ratio, along with the applied base shear, is used to calculate the base
shear at first yield:
where:
Vbase = base shear from Equivalent Lateral Force (ELF) analysis
A similar analysis can be made using the model with no moment resistance at Frame A:
Thus, the loss of resistance at both ends of a single beam only results in a 6 percent reduction in story
strength. The moment frames can be assigned a value of = 1.0.
In the EW direction, the OCBF system meets the prescriptive requirements of Standard
Section 12.3.4.2a. As a result, no further calculations are needed and this system can be assigned a value
of = 1.0.
6.1.3.3 Orthogonal load effects. A combination of 100 percent seismic forces in one direction plus 30
percent seismic forces in the orthogonal direction must be applied to the columns of this structure in
Seismic Design Category D (Standard Sec. 12.5.4).
6.1.3.4 Structural component load effects. The effect of seismic load (Standard Sec. 12.4.2) is:
SDS = 1.0 for this example. The seismic load is combined with the gravity loads as shown in Standard
Sec. 12.4.2.3, resulting in the following:
Note that 1.0L is for the storage load on the mezzanine; the coefficient on L is 0.5 for many common live
loads.
6.1.3.5 Drift limits. For a building assigned to Occupancy Category II, the allowable story drift
(Standard Table 12.121) is:
a = 0.025hsx in the EW direction
a/ = 0.025hsx/1.0 in the NS direction
At the roof ridge, hsx = 34 ft3 in. and = 10.28 in.
At the knee (columnroof intersection), hsx = 30 ft6 in. and a = 9.15 in.
At the mezzanine floor, hsx = 12 ft and a = 3.60 in.
Footnote c in Standard Table 12.121 permits unlimited drift for singlestory buildings with interior
walls, partitions, etc., that have been designed to accommodate the story drifts. See Section 6.1.4.3 for
further discussion. The main frame of the building can be considered to be a onestory building for this
purpose, given that there are no interior partitions except below the mezzanine. (The definition of a story
in building codes generally does not require that a mezzanine be considered a story unless its area exceeds
onethird the area of the room or space in which it is placed; this mezzanine is less than onethird of the
footprint of the building.)
6.1.3.6 Seismic weight. The weights that contribute to seismic forces are:
EW direction NS direction
Roof D = (0.015)(90)(180) = 243 kips 243 kips
Panels at sides = (2)(0.075)(32)(180)/2 = 0 kips 437 kips
Panels at ends = (2)(0.075)(35)(90)/2 = 224 kips 0 kips
Mezzanine slab and 25% LL = 360 kips 360 kips
Mezzanine framing = 35 kips 35 kips
Main frames = 27 kips 27 kips
Seismic weight = 889 kips 1,102 kips
The weight associated with the main frames accounts for only the main columns, because the weight
associated with the remainder of the main frames is included in the roof dead load above. The computed
seismic weight is based on the assumption that the wall panels offer no shear resistance for the structure
but are selfsupporting when the load is parallel to the wall of which the panels are a part. Additionally,
snow load does not need to be included in the seismic weight per Standard Section 12.7.2 because it does
not exceed 30 psf.
6.1.4 Analysis
Base shear will be determined using an ELF analysis.
6.1.4.1 Equivalent Lateral Force procedure. In the longitudinal direction where stiffness is provided
only by the diagonal bracing, the approximate period is computed using Standard Equation 12.87:
where hn is the height of the building, taken as 34.25 feet at the midheight of the roof. In accordance
with Standard Section 12.8.2, the computed period of the structure must not exceed the following:
The subsequent threedimensional modal analysis finds the computed period to be 0.54 seconds. For
purposes of determining the required base shear strength, Tmax will be used in accordance with the
Standard; drift will be calculated using the period from the model.
In the transverse direction where stiffness is provided by momentresisting frames (Standard Eq. 12.87):
and
Also note that the dynamic analysis finds a computed period of 1.03 seconds. As in the longitudinal
direction, Tmax will be used for determining the required base shear strength.
The seismic response coefficient (Cs) is computed in accordance with Standard Section 12.8.1.1. In the
longitudinal direction:
but need not exceed:
Therefore, use Cs = 0.308 for the longitudinal direction.
In the transverse direction:
but need not exceed:
Therefore, use Cs = 0.202 for the transverse direction.
In both directions the value of Cs exceeds the minimum value (Standard Eq. 12.85) computed as:
The seismic base shear in the longitudinal direction (Standard Eq. 12.81) is:
The seismic base shear in the transverse direction is:
Standard Section 12.8.3 prescribes the vertical distribution of lateral force in a multilevel structure. Even
though the building is considered to be one story for some purposes, it is clearly a twolevel structure.
Using the data in Section 6.1.3.6 of this example and interpolating the exponent k as 1.08 for the period of
0.66 second, the distribution of forces for the NS analysis is shown in Table 6.11.
Table 6.11 ELF Vertical Distribution for NS Analysis
Level
Weight (wx)
(kips)
Height (hx)
(ft)
wxhxk
(kips)
Roof
707
32.375
30,231
0.84
187
Mezzanine
395
12
5,782
0.16
36
Total
1,102
36,013
223
It is not immediately clear whether the roof (a 22gauge steel deck with conventional roofing over it) will
behave as a flexible, semirigid, or rigid diaphragm. For this example, a threedimensional model was
created in ETABS including frame and diaphragm stiffness.
6.1.4.2 Threedimensional ELF analysis. The threedimensional analysis is performed for this example
to account for the following:
The differing stiffness of the gable frames with and without the mezzanine level
The different centers of mass for the roof and the mezzanine
The flexibility of the roof deck
The significance of braced frames in controlling torsion due to NS ground motions
The gabled moment frames, the tension bracing, the moment frames supporting the mezzanine and the
diaphragm chord members are explicitly modeled using threedimensional beamcolumn elements. The
tapered members are approximated as short, discretized prismatic segments. Thus, combined axial
bending checks are performed on a prismatic element, as required by AISC 360 Chapter H. The collector
at the knee level is included, as are those at the mezzanine level in the two east bays. The mezzanine
diaphragm is modeled using planar shell elements with their inplane rigidity being based on actual
properties and dimensions of the slab. The roof diaphragm also is modeled using planar shell elements,
but their inplane rigidity is based on a reduced thickness that accounts for compression buckling
phenomena and for the fact that the edges of the roof diaphragm panels are not connected to the wall
panels. SDI s Diaphragm Design Manual is used for guidance in assessing the stiffness of the roof deck.
The analytical model includes elements with onetenth the stiffness of a plane plate of 22 gauge steel.
The ELF analysis of the threedimensional model in the transverse direction yields an important result:
the roof diaphragm behaves as a rigid diaphragm. Accidental torsion is applied at the center of the roof as
a moment whose magnitude is the roof lateral force multiplied by 5 percent of 180 feet (9 feet). A
moment is also applied to the mezzanine level in a similar fashion. The resulting displacements are
shown in Table 6.12.
Table 6.12 ELF Analysis Displacements in
NS Direction
Grid
Roof Displacement (in.)
A
4.98
B
4.92
C
4.82
D
4.68
E
4.56
F
4.46
G
4.34
H
4.19
J
4.05
K
3.92
The average of the extreme displacements is 4.45 inches. The displacement at the centroid of the roof is
4.51 inches. Thus, the deviation of the diaphragm from a straight line is 0.06 inch, whereas the average
frame displacement is approximately 75 times that. Clearly, then, the diaphragm flexibility is negligible
and the deck behaves as a rigid diaphragm. The ratio of maximum to average displacement is 1.1, which
does not exceed the 1.2 limit given in Standard Table 12.31 and torsional irregularity is not triggered.
The same process needs to be repeated for the EW direction.
Table 6.13 ELF Analysis Displacements
in NS Direction
Grid
Roof Displacement (in.)
1
0.88
2/3
0.82
4
0.75
The ratio of the maximum to average displacement is 1.07, well under the torsional irregularity threshold
ratio of 1.2.
The demands from the threedimensional ELF analysis are combined to meet the orthogonal combination
requirement of Standard Section 12.5.3 for the columns:
EW: (1.0)(EW direction spectrum) + (0.3)(NS direction spectrum)
NS: (0.3)(EW direction spectrum) + (1.0)(NS direction spectrum)
6.1.4.3 Drift. The lateral deflection cited previously must be multiplied by Cd = 4 to find the transverse
drift:
This exceeds the limit of 10.28 inches computed previously. However, there is no story drift limit for
singlestory structures with interior wall, partitions, ceilings and exterior wall systems that have been
designed to accommodate the story drifts. Detailing for this type of design may be problematic.
In the longitudinal direction, the lateral deflection is much smaller and is within the limits of Standard
Section 12.12.1. The deflection computations do not include the redundancy factor.
6.1.4.4 Pdelta effects. The Pdelta effects on the structure may be neglected in analysis if the provisions
of Standard Section 12.8.7 are followed. First, the stability coefficient maximum should be determined
using Standard Equation 12.817. may be assumed to be 1.0.
Next, the stability coefficient is calculated using Standard Equation 12.816. The stability coefficient is
calculated at both the roof and mezzanine levels in both orthogonal directions. For purposes of
illustration, the roof level check in the NS direction will be shown as:
The three other stability coefficients were all determined to be less than max, thus allowing Pdelta effects
to be excluded from the analysis.
6.1.4.5 Force summary. The maximum moments and axial forces caused by dead, live and earthquake
loads on the gable frames are listed in Tables 6.13 and 6.14. The frames are symmetrical about their
ridge and the loads are either symmetrical or can be applied on either side on the frame because the forces
are given for only half of the frame extending from the ridge to the ground. The moments are given in
Table 6.14 and the axial forces are given in Table 6.15. The moment diagram for the combined load
condition is shown in Figure 6.14. The load combination is 1.4D + L + 0.2S + QE, which is used
throughout the remainder of calculations in this section, unless specifically noted otherwise.
The size of the members is controlled by gravity loads, not seismic loads. The design of connections will
be controlled by the seismic loads.
Forces in the design of the braces are discussed in Section 6.1.5.5.
Table 6.14 Moments in Gable Frame Members
Location
D
(ftkips)
L
(ftkips)
S
(ftkips)
QE
(ftkips)
Combined*
(ftkips)
1  Ridge
61
0
128
0
112 (279)
2  Knee
161
0
333
162
447 (726)
3  Mezzanine
95
83
92
137
79
4  Base
0
0
0
0
0
* Combined Load = 1.4D + L + 0.2S + QE (or 1.2D + 1.6S). Individual maxima are not necessarily on the
same frame; combined load values are maximum for any frame.
1.0 ft = 0.3048 m, 1.0 kip = 1.36 kNm.
Table 6.15 Axial Forces in Gable Frame Members
Location
D
(ftkips)
L
(ftkips)
S
(ftkips)
QE
(ftkips)
Combined*
(ftkips)
1  Ridge
14
3.5
25
0.8
39
2  Knee
16
4.5
27
7.0
37
3  Mezzanine
39
39
23
26
127
4  Base
39
39
23
26
127
* Combined Load = 1.4D + L + 0.2S + QE. Individual maxima are not necessarily on the same frame;
combined load values are maximum for any frame.
1.0 ft = 0.3048 m, 1.0 kip = 1.36 kNm.
Figure 6.14 Moment diagram for seismic load combinations
(1.0 ftkip = 1.36 kNm)
6.1.5 Proportioning and Details
The gable frame is shown schematically in Figure 6.15. Using the load combinations presented in
Section 6.1.3.4 and the loads from Tables 6.14 and 6.15, the proportions of the frame are checked at the
roof beams and the variabledepth columns (at the knee). The mezzanine framing, also shown in
Figure 6.11, was proportioned similarly. The diagonal bracing, shown in Figure 6.11 at the east end of
the building, is proportioned using tension forces determined from the threedimensional ELF analysis.
Figure 6.15 Gable frame schematic: Column tapers from 12 in. at base to 36 in. at knee;
roof beam tapers from 36 in. at knee to 18 in. at ridge; plate sizes are given in Figure 6.17
(1.0 in. = 25.4 mm)
Additionally, the bolted, stiffened, extended endplate connections must be sized correctly to conform to
the prequalification standards. AISC 358 Table 6.1 provides parametric limits on the beam and
connection sizes. Table 6.16 shows these limits as well as the values used for design.
Table 6.16 Parametric Limits for Moment Frame Connection
Parameter
Minimum (in.)
As Designed (in.)
Maximum (in.)
tp
3/4
1 1/4
2 1/2
bp
9
9
15
g
5
5
6
pfi, pfo
1 3/4
1 3/4
2
pb
3 1/2
3 1/2
3 3/4
d
18 1/2
36
36
tbf
19/32
5/8
1
bbf
7 3/4
8
12 1/4
6.1.5.1 Frame compactness and brace spacing. According to Standard Section 14.1.3, steel structures
assigned to Seismic Design Category D, E, or F must be designed and detailed per AISC 341. For an
intermediate moment frame (IMF), AISC 341, Part I, Section 1, Scope, stipulates that those
requirements are to be applied in conjunction with AISC 360. Part I, Section 10 of AISC 341 itemizes a
few additional items from AISC 360 for intermediate moment frames, but otherwise the intermediate
moment frames are to be designed per AISC 360.
AISC 341 requires IMFs to have compact widththickness ratios per AISC 360, Table B4.1.
All widththickness ratios are less than the limiting p from AISC 360, Table B4.1. All PM ratios
(combined compression and flexure) are less than 1.00. This is based on proper spacing of lateral
bracing.
Lateral bracing is provided by the roof joists. The maximum spacing of lateral bracing is determined
using beam properties at the ends and AISC 341, Section 10.8:
Lb is 48 inches; therefore, the spacing is OK.
Also, the required brace strength and stiffness are calculated per AISC 360, Equations A67 and A68:
where:
ho = distance between flange centroids, in.
Lb = distance between braces or Lp (from AISC 360 Eq. F25), whichever is greater, in.
Adjacent to the plastic hinge regions, lateral bracing must have additional strength as defined in
AISC 341 10.8
The Cjoists used in this structure likely are not adequate to brace the moment frames. Instead, tube brace
members will be used, but they are not analyzed in this example.
At the negative moment regions near the knee, lateral bracing is necessary on the bottom flange of the
beams and inside the flanges of the columns (Figure 6.16).
Figure 6.16 Arrangement at knee
(1.0 in. = 25.4 mm)
6.1.5.2 Knee of the frame. The knee detail is shown in Figures 6.16 and 6.17. The vertical plate
shown near the upper left corner in Figure 6.16 is a gusset providing connection for Xbracing in the
longitudinal direction. The beamtocolumn connection requires special consideration. The method of
AISC 358 for bolted, stiffened end plate connections is used. Refer to Figure 6.18 for the configuration.
Highlights from this method are shown for this portion of the example. Refer to AISC 358 for a
discussion of the entire procedure.
Figure 6.17 Bolted stiffened connection at knee
(1.0 in. = 25.4 mm)
The AISC 358 method for bolted stiffened end plate connection requires the determination of the
maximum moment that can be developed by the beam. The steps in AISC 358 for bolted stiffened end
plates follow:
Step 1. Determine the maximum moment at the plastic hinge location. The end plate stiffeners at the
top and bottom flanges increase the local moment of inertia of the beam, forcing the plastic
hinge to occur away from the welds at the end of beam/face of column. The stiffeners should
be long enough to force the plastic hinge to at least d/2 away from the end of the beam. With
the taper of the section, the depth will be slightly less than 36 inches at the location of the
hinge, but that reduction will be ignored here. The probable maximum moment, Mpe, at the
plastic hinge is computed using AISC 358 Equation 6.92 as follows:
Where, per AISC 358 Equation 2.4.32:
where:
Ry = 1.1 from AISC 341 Table I61
Ze = 309 in.3
Fy = 50 ksi
Therefore:
The moment at the column flange, Mf , which drives the connection design, is determined from
AISC 358 Equation 6.92 as follows:
where:
Vu = shear at location of plastic hinge
L = distance between plastic hinges
Sh = distance from the face of the column to the plastic hinge, ft.
where:
Lst = length of endplate stiffener, as shown in AISC 358 Figure 6.2.
tp = thickness of end plate, in.
where:
hst = height of the endplate from the outside face of the beam flange to the end of the endplate
Use Lst = 13 in.
Step 2. Find bolt size for end plates. For a connection with two rows of two bolts inside and outside
the flange, AISC 358 Equation 6.97 indicates the following:
where:
Fnt = nominal tensile stress of bolt, ksi
hi = distance from the centerline of the beam compression flange to the centerline of the ith
tension bolt row, in.
Try A490 bolts. See Figure 6.17 for bolt geometry.
Use 1 in. diameter A490N bolts.
Step 3. Determine the minimum endplate thickness from AISC 358 Equation 6.98.
where:
Fyp = specified minimum yield stress of the end plate material, ksi
Yp = the endplate yield line mechanism parameter from AISC 358 Table 6.4
d = resistance factor for ductile limit states, taken as 1.0
From AISC 358 Table 6.4:
where:
bp = width of the end plate, in.
g = horizontal distance between bolts on the end plate, in.
de = 7 in. (see Figure 6.17)
Use Case 1 from AISC 358 Table 6.4, since de > s
where:
pfo = vertical distance between beam flange and the nearest outer row of bolts, in.
pfi = vertical distance between beam flange and the nearest inner row of volts, in.
pb = distance between the inner and our row of bolts, in.
Yp = 499 in.
Use 1.25inch thick endplates.
Step 4. Calculate the factored beam flange force from AISC 358 Equation 6.99.
where:
d = depth of the beam, in.
tbf = thickness of beam flange, in.
Step 5. Determine the endplate stiffener thickness from AISC 358 Equation 6.913.
where:
tbw = thickness of the beam web, in.
Fyb = specified minimum yield stress of beam material, ksi
Fys = specified minimum yield stress of stiffener material, ksi
Use 5/8inch plates.
The stiffener widththickness ratio must also comply with AISC 358 Equation 6.914.
hst = 7 in. OK
Step 6. Check bolt shear rupture strength at the compression flange by AISC 358 Equation 6.915.
where:
n = resistance factor for nonductile limit states, taken as 0.9
nb = number of bolts at compression flange
Fv = nominal shear stress of bolts from AISC 360 Table J3.2, ksi
Ab = nominal bolt area, in.
OK
Step 7. Check bolt bearing/tearout of the endplate and column flange by AISC 358 Equation 6.917.
where:
ni = number of inner bolts
no = number of outer bolts
for each inner bolt
for each outer bolt
Lc = clear distance, in the direction of force, between the edge of the hole and the edge of the
adjacent hole or edge of the material, in.
t = endplate or column flange thickness, in.
Fu = specified minimum tensile strength of endplate or column flange material, ksi
db = diameter of bolt, in.
where:
de = effective area of bolt hole, in.
Le = edge spacing of the bolts, in.
OK
Step 8. Check the column flange for flexural yielding by AISC 358 Equation 6.920.
where:
Fyc = specified minimum yield stress of column flange material, ksi
Yc = stiffened column flange yield line from AISC 358 Table 6.6
tcf = column flange thickness, in.
where:
bcf = column flange width, in.
psi = distance from column stiffener to inner bolts, in.
pso = distance from column stiffener to outer bolts, in.
Column flange of 2 inches is OK.
Step 9. Determine the required stiffener force by AISC 358 Equation 6.921.
The equivalent column flange design force used for stiffener design by AISC 358
Equation 6.922.
2,808 kips > 576 kips OK
Step 10. Check local column web yielding strength of the unstiffened column web at the beam flanges
by AISC 358 Equations 6.923 and 6.924.
where:
Ct = 0.5 if the distance from the column top to the top of the beam flange is less than the depth
of the column: otherwise 1.0
kc = distance from outer face of the column flange to web toe of fillet weld, in.
tp = endplate thickness, in.
Fyc = specified yield stress of the column web material, ksi
tcw = column web thickness, in.
tbf = beam flange thickness, in.
The design is not acceptable. Column stiffeners need to be provided.
Step 11. Check the unstiffened column web buckling strength at the beam compression flange by
AISC 358 Equations 6.925 and 6.927.
where:
h = clear distance between flanges when welds are used for builtup shapes, in.
42 kips < 576 kips NOT OK
Step 12. Check the unstiffened column web crippling strength at the beam compression flange by
AISC 358 Equation 630.
where:
N = thickness of beam flange plus 2 times the groove weld reinforcement leg size, in.
dc = overall depth of the column, in.
184 kips < 576 kips NOT OK
Step 13. Check the required strength of the stiffener plates by AISC 358 Equation 632.
where:
= the minimum design strength value from column flange bending check, column web
yielding, column web buckling and column web crippling check
Although AISC 358 says to use this value of 534 kips to design the continuity plate, a different
approach will be used in this example. In compression, the continuity plate will be designed to
take the full force delivered by the beam flange, Fsu. In tension, however, the compressive limit
states (web buckling and web yielding) are not applicable and column web yielding will control
the design instead. The tension design force can be taken as follows:
Step 14. Design the continuity plate for required strength by AISC 360 Section J10.
Find the crosssectional area required by the continuity plate acting in tension:
Use a 13/8inch continuity plate. As it will be shown later, net section rupture (not gross
yielding) will control the design of this plate.
From AISC Section J10.8, calculate member properties using an effective length of 0.75h and a
column web length of 12tw = 6 in.:
Check the continuity plate in compression from AISC 360 Equation J4.4:
Strength in the other direction does not need to be checked because the cruciform section will
not buckle in the plane of the column web.
Since KL/r is less than 25, use AISC 360 Equation J46 to determine compression strength:
However, torsional buckling may control. Therefore, check flexuraltorsional buckling using
AISC 360 Equation E44:
Check the continuity plate in tension. The continuity plate had been previously sized for
adequacy to tensile yielding of the gross section. Now tensile rupture of the net section must be
checked using AISC 360 Section D22. The critical section will be analyzed where the
continuity plates are clipped adjacent to the kregion of the column.
OK
Step 15. Check the panel zone for required strength per AISC 341 Equation J109.
where:
A = column cross sectional area, in2.
Therefore, use AISC 360 Equation J10111. Note that panel zone flexibility was accounted for
in the ETABS model.
NOT OK
The column web is not sufficient to resist the panel zone shear. Although doubler plates can be
added to the panel zone to increase strength, this may be an expensive solution. A more
economical solution would be to simply upsize the column web to a sufficient thickness, such
as 5/8 inch.
Note that changing the column member properties might affect the analysis results. In this
example, this is not the case, although the slight difference in web thickness would result in
marginally different values for some of the endplate connection calculations. For simplicity,
these changes are not undertaken in this example.
6.1.5.3 Frame at the ridge. The ridge joint detail is shown in Figure 6.18. An unstiffened bolted
connection plate is selected.
Figure 6.18 End plate connection at ridge
This is an AISC 360 designed connection, not an AISC 358 designed connection because there should not
be a plastic hinge forming in this vicinity. Lateral seismic forces produce no moment at the ridge until
yielding takes place at one of the knees. Vertical accelerations on the dead load do produce a moment at
this point; however, the value is small compared to all other moments and does not appear to be a
concern. Once seismic loads produce a plastic hinge at one knee, further lateral displacement produces
positive moment at the ridge. Under the condition on which the AISC 358 design is based (a full plastic
moment is produced at each knee), the moment at the ridge will simply be the static moment from the
gravity loads less the horizontal thrust times the rise from knee to ridge. Analyzing this frame under the
gravity load case 1.2D + 0.2S, the static moment is 406 ftkip and the reduction for the thrust is
128 ftkip, leaving a net positive moment of 278 ftkip, coincidentally close to the design moment for the
factored gravity loads.
6.1.5.4 Design of mezzanine framing. The design of the framing for the mezzanine floor at the east end
of the building is controlled by gravity loads. The concretefilled 3inch, 20gauge steel deck of the
mezzanine floor is supported on steel beams spaced at 10 feet and spanning 20 feet (Figure 6.12). The
steel beams rest on threespan girders connected at each end to the portal frames and supported on two
intermediate columns (Figure 6.11). The girder spans are approximately 30 feet each. Those lateral
forces that are received by the mezzanine are distributed to the frames and diagonal bracing via the floor
diaphragm. A typical beamcolumn connection at the mezzanine level is provided in Figure 6.19. The
design of the end plate connection is similar to that at the knee, but simpler because the beam is horizontal
and not tapered. Also note that demands on the endplate connection will be less because this connection
is not at the end of the column.
Figure 6.19 Mezzanine framing (1.0 in. = 25.4 mm)
6.1.5.5 Braced frame diagonal bracing
Although the force in the diagonal Xbraces can be either tension or compression, only the tensile value is
considered because it is assumed that the diagonal braces are capable of resisting only tensile forces.
See AISC 341 Section 14.2 for requirements on braces for OCBFs. The strength of the members and
connections, including the columns in this area but excluding the brace connections, must be based on
Standard Section 12.4.2.3:
1.4D + 1.0L + 0.2S + QE
0.7D + QE
Recall that a 1.0 factor is applied to L when the live load is greater than 100 psf (Standard Sec. 2.3.2).
For the case discussed here, the tension only brace does not carry any live or dead load, so the load
factor does not matter.
For simplicity, we can assume that the lateral force is equally divided among the roof level braces and is
slightly amplified to account for torsional effects. Thus the brace force can be approximated using the
following equation:
All braces at this level will have the same design. Choose a brace member based on tensile yielding of
the gross section by AISC 360 Equation D21:
This also needs to be checked for tensile rupture of the net section. Demand will be taken as either the
expected yield strength of the brace or the amplified seismic load. Try a 2L3«x3x 7/16, which is the
smallest seismically compact angle shape available.
Ag = 5.34 in.2
The Kl/r requirement of AISC 341 Section 14.2 does not apply because this is not a V or an inverted V
configuration.
Check net rupture by AISC 360 Equation D22 and D31:
Determine the shear lag factor, U, from AISC 360, Table D3.1, Case 2. In ordered to calculate U, the
weld length along the double angles needs to be determined.
Brace connection demand is given as the expected yield strength of the brace in tension per AISC 341
Section 14.4.
Expected yield strength of the brace is 288 kips. However, AISC 341 Section 14.4b limits the brace
connection design force to the amplified seismic load.
Use four fillet welds, two on each angle. Try 1/4inch welds using AISC 360 Equation J23:
Use four 1/4inch fillet welds 8 inches long.
Check the base metal:
Shear yielding from AISC 360 Equation J43:
OK
Shear rupture from AISC 360 Equation J44:
OK
Calculate the shear lag factor and the effective net area:
Calculate the tensile rupture strength:
OK
Additionally, the capacity of the eave strut at the roof must be checked. The eave strut, part of the braced
frame, also acts as a collector element and must be designed using the overstrength factor per Standard
Section 12.10.2.1.
6.1.5.6 Roof deck diaphragm. Figure 6.110 shows the inplane shear force in the roof deck diaphragm
for both seismic loading conditions. There are deviations from simple approximations in both directions.
In the EW direction, the base shear is 274 kips (Section 6.1.4.2) with 77 percent or 211 kips at the roof.
Torsion is not significant, so a simple approximation is to take half the force to each side and divide by
the length of the building, which yields (211,000/2)/180 feet = 586 plf. The plot shows that the shear in
the edge of the diaphragm is significantly higher in the two braced bays. This is a shear lag effect; the
eave strut in the threedimensional model is a HSS 6x6x1/4. In the NS direction, the shear is generally
highest in the bay between the mezzanine frame and the first frame without the mezzanine. This is
expected given the significant change in stiffness. There is no simple approximation to estimate the shear
here without a threedimensional model. The shear is also high at the longitudinal braced bays because
they tend to resist the horizontal torsion. However, the shear at the braced bays is lower than observed for
the EW motion.
Figure 6.110 Shear force in roof deck diaphragm;
upper diagram is for EW motion and lower is for NS motion
6.2 SEVENSTORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA
Two alternative framing arrangements for a sevenstory office building are illustrated.
6.2.1 Building Description
6.2.1.1 General description. This sevenstory office building of rectangular plan configuration is 177
feet, 4 inches long in the EW direction and 127 feet, 4 inches wide in the NS direction (Figure 6.21).
The building has a penthouse. It is framed in structural steel with 25foot bays in each direction. The
typical story height is 13 feet, 4 inches; the first story is 22 feet, 4 inches high. The penthouse extends
16 feet above the roof level of the building and covers the area bounded by Gridlines C, F, 2 5 in
Figure 6.21. Floors consist of 31/4inch lightweight concrete over composite metal deck. The elevators
and stairs are located in the central three bays.
6.2.1.2 Alternatives. This example includes two alternatives a steel momentresisting frame and a
concentrically braced frame:
Alternative A: Seismic force resistance is provided by special moment frames (SMF) with
prequalified Reduced Beam Section (RBS) connections located on the perimeter of the building
(on Gridlines A, H, 1 6 in Figure 6.21, also illustrated in Figure 6.22). There are five bays of
moment frames on each line.
Alternative B: Seismic force resistance is provided by four special concentrically braced frames
(SCBF) in each direction. They are located in the elevator core walls between Columns 3C and
3D, 3E and 3F, 4C and 4D 4E and 4F in the EW direction and between Columns 3C and 4C, 3D
and 4D, 3E and 4E 3F and 4F in the NS direction (Figure 6.21). The braced frames are in a
twostory X configuration. The frames are identical in brace size and configuration, but there are
some minor differences in beam and column sizes. Braced frame elevations are shown in
Figures 6.210 through 6.212.
6.2.1.3 Scope. The example covers:
Seismic design parameters (Sec. 6.2.2.1)
Analysis of perimeter moment frames (Sec. 6.2.4.1)
Beam and column proportioning (Sec. 6.2.4.2.3)
Moment frame connection design (Sec. 6.2.4.2.5)
Analysis of concentrically braced frames (Sec. 6.2.5.1)
Proportioning of braces (Sec. 6.2.5.2.1)
Braced frame connection design (Sec. 6.2.5.2.5)
Figure 6.21 Typical floor framing plan and building section
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
Figure 6.22 Framing plan for special moment frame
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
6.2.2 Basic Requirements
6.2.2.1 Provisions parameters. See Section 3.2 for an example illustrating the determination of design
ground motion parameters. For this example, the parameters are as follows
SDS = 1.0
SD1 = 0.6
Occupancy Category II
Seismic Design Category D
For Alternative A, Special Steel Moment Frame (Standard Table 12.21)
R = 8
0 = 3
Cd = 5.5
For Alternative B, Special Steel Concentrically Braced Frame (Standard Table 12.21):
R = 6
0 = 2
Cd = 5
6.2.2.2 Loads.
Roof live load (L): 25 psf
Penthouse roof dead load (D): 25 psf
Exterior walls of penthouse: 25 psf of wall
Roof DL (roofing, insulation, deck beams, girders, fireproofing, ceiling, mechanical, electrical
plumbing): 55 psf
Exterior wall cladding: 25 psf of wall
Penthouse floor D: 65 psf
Penthouse Equipment: 39 psf
Floor L: 50 psf
Floor D (deck, beams, girders, fireproofing, ceiling, mechanical electrical, plumbing, partitions):
68 psf
Floor L reductions: per the IBC
6.2.2.3 Basic gravity loads.
Penthouse roof:
Roof slab = (0.025 ksf)(25 ft)(75 ft) = 47 kips
Walls = (0.025 ksf)(8 ft)(200 ft) = 40 kips
Columns = (0.110 ksf)(8 ft)(8 ft) = 7 kips
Total = 94 kips
Lower roof:
Roof slab = (0.055 ksf)[(127.33 ft)(177.33 ft)  (25 ft)(75 ft) = 1139 kips
Penthouse floor = (0.065 ksf)(25 ft)(75 ft) = 122 kips
Walls = 40 kips + (0.025 ksf)(609 ft)(6.67 ft) = 142 kips
Columns = 7 kips + (0.170 ksf)(6.67 ft)(48 ft) = 61 kips
Equipment = (0.039 ksf)(25 ft)(75 ft) = 73 kips
Total = 1,537 kips
Typical floor:
Floor = (0.068 ksf)(127.33 ft)(177.33 ft) = 1,535 kips
Walls = (0.025 ksf)(609 ft)(13.33 ft) = 203 kips
Columns = (0.285 ksf)(13.33 ft)(48 ft) = 182 kips
Total = 1,920 kips
Total weight of building = 94 kips + 1,537 kips + 6 (1,920 kips) = 13,156 kips
6.2.2.4 Materials
Concrete for floors: fc' = 3 ksi, lightweight (LW)
All other concrete: fc' = 4 ksi, normal weight (NW)
Structural steel:
Wide flange sections: ASTM A992, Grade 50
HSS: ASTM A500, Grade B
Plates: ASTM A36
6.2.3 Structural Design Criteria
6.2.3.1 Building configuration. The building has no vertical irregularities despite the relatively tall
height of the first story. The exception of Standard Section 12.3.2.2 is taken, in which the drift ratio of
adjacent stories are compared rather than the stiffness of the stories. In the threedimensional analysis,
the first story drift ratio is less than 130 percent of that for the story above. Because the building is
symmetrical in plan, plan irregularities would not be expected. Analysis reveals that Alternative B is
torsionally irregular, which is not uncommon for corebraced buildings.
6.2.3.2 Orthogonal load effects. A combination of 100 percent of the seismic forces in one direction
with 30 percent of the seismic forces in the orthogonal direction is required for structures in Seismic
Design Category D for certain elements namely, the shared columns in the SCBF (Standard Sec.
12.5.4). In using modal response spectrum analysis (MRSA), the bidirectional case is handled by using
the square root of the sum of the squares (SRSS) of the orthogonal spectra.
6.2.3.3 Structural component load effects. The effect of seismic load is defined by Standard
Section 12.4.2 as:
Using Standard Section 12.3.4.2, is 1.0 for Alternative A and 1.3 for Alternative B. (For simplicity, is
taken as 1.3; the design does not comply with the prescriptive requirements of Standard Sec. 12.3.4.2. It
is assumed that the design would fail the calculationbased requirements of Standard Sec. 12.3.4.2.)
Substitute for (and for SDS = 1.0).
For Alternative A:
E = QE ñ 0.2D
Alternative B:
E = 1.3QE ñ 0.2D
6.2.3.4 Load combinations
Load combinations from ASCE 705 are as follows:
1.4D
1.2D + 1.6L + 0.5Lr
1.2D + L + 1.6Lr
(1.2 + 0.2SDS)D + 0.5L + QE
(0.9 0.2 SDS)D + QE
For each of these load combinations, substitute E as determined above, showing the maximum additive
and minimum subtractive. QE acts both east and west (or north and south):
Alternative A:
1.4D
1.2D + 1.6L + 0.5Lr
1.2D + L + 1.6Lr
1.4D + 0.5L + QE
0.7D + QE
Alternative B"
1.4D
1.2D + 1.6L + 0.5Lr
1.2D + L + 1.6Lr
1.4D + 0.5L + 1.3QE
0.7D + 1.3QE
For both cases, six scaled response spectrum cases are used:
1) Spectrum in X direction
2) Spectrum in X direction with 5 percent eccentricity
3) Spectrum in Y direction
4) Spectrum in Y direction with 5 percent eccentricity
5) SRSS combined spectra in X and Y directions
6) SRSS combined spectra in X and Y directions with 5 percent eccentricity.
6.2.3.5 Drift limits. The allowable story drift per Standard Section 12.12.1 is a = 0.02hsx.
The allowable story drift for the first floor is a (0.02)(22.33 ft)(12 in./ft) = 5.36 in.
The allowable story drift for a typical story is a (0.02)(13.33 ft)(12 in./ft) = 3.20 in.
Adjust calculated story drifts by the appropriate Cd factor from Standard Table 12.21.
6.2.4 Analysis and Design of Alternative A: SMF
6.2.4.1 Modal Response Spectrum Analysis. Determine the building period (T) per Standard
Equation 12.87:
where hn, the height to the main roof, is conservatively taken as 102.3 feet. The height of the penthouse
will be neglected since its seismic mass is negligible. CuTa, the upper limit on the building period, is
determined per Standard Table 12.81:
It is assumed that the calculated period will exceed CuTa; this is verified after member selection. The
seismic response coefficient (Cs) is determined from Standard Equation 12.82 as follows:
However, Standard Equation 12.83 indicates that the value for Cs need not exceed:
and the minimum value for Cs per Standard Equation 12.85 is:
Therefore, use Cs = 0.047.
Seismic base shear is computed per Standard Equation 12.81 as:
where W is the seismic mass of the building as determined above.
In evaluating the building in ETABS, twelve modes are analyzed, resulting in a total modal mass
participation of 97 percent. The code requires at least 90 percent participation for strength. A scaling
factor is used to take the response spectrum to 85 percent of the base shear, with a minimum scale factor
for strength calculations of I/R. Typical software utilizes a spectrum presented as a coefficient of g, thus
requiring scaling by g, thus the scaling factor used here is g/(R/I) = 386/(8/1) = 48.3. For drift, results are
scaled by Cd/(R/I); for a spectrum using a coefficient times g, this factor is gCd/(R/I).
6.2.4.2 Size members. The method used is as follows:
1. Select preliminary member sizes
2. Check deflection and drift (Standard Sec. 12.12)
3. Check the columnbeam moment ratio rule (AISC 341 Sec. 9.6)
4. Check beam strength
5. Check connection design (AISC 341 Sec. 9.7)
6. Check shear requirement at panelzone (AISC 341 Sec. 9.3; AISC 358 Sec. 5.4)
After the weight and stiffness have been modified by changing member sizes, the response spectrum must
be rescaled for strength. The most significant criteria for the design are drift limits, relative strengths of
columns and beams the panelzone shear. Member strength must be checked but rarely governs for this
system.
1. Select Preliminary Member Sizes: The preliminary member sizes are shown for the moment frame in
the Xdirection in Figure 6.23 and in the Y direction in Figure 6.24. These sections are selected
from AISC SDM Table 12, ensuring that they are seismically compact. Members are sized to meet
the prequalification limits of AISC 358 Section 5.3 for spandepth ratios, weight flange thickness.
Members are also sized for drift limitations and to satisfy strong column weak beam requirements by
using a target ratio of:
This proportioning does not guarantee compliance with AISC 341 Section 9.6, but is a useful target
that makes conformance likely. Using a ratio of 2.0 may save on detailing costs, such as continuity
plates, doublers bracing.
The software used accounts explicitly for the increase in beam flexibility due to the RBS cuts. For
every beam, RBS parameters were chosen as follows:
In accordance with AISC 341 Table I81, beam flange slenderness ratios are limited to
(7.22 for Fy = 50 ksi) beam web heighttothickness ratios are limited to (59.0 for Fy =
50 ksi). Since all members selected are seismically compact per AISC SDM Table 12, they conform
to these limits.
For columns in special steel moment frames such as this example, AISC 341 Table I81 Footnote b
requires that where the ratio of column moment strength to beam moment strength is less than or
equal to 2.0, the more stringent p requirements apply for b/t (given above) when Pu/ bPy is greater
than or equal to 0.125, the more stringent h/t requirements apply.
Per AISC 341 Table I81, consider the W14x132 column at Gridline B:
Therefore, the column is seismically compact.
Strength checks are performed using ETABS; all members are satisfactory for strength
Figure 6.23 SMRF frame in EW direction (penthouse not shown)
2. Check Drift: Check drift is in accordance with Standard Section 12.12.1. The building is modeled in
three dimensions using ETABS. Displacements at the building corners under the 5 percent accidental
torsion load cases are used here. Calculated story drifts, response spectrum scaling factors Cd
amplification factors are summarized in Table 6.21 below. Pdelta effects are included.
All story drifts are within the allowable story drift limit of 0.020hsx per Standard Section 12.12 and
Section 6.2.3.6 of this chapter.
Figure 6.24 SMRF frame in NS direction (penthouse not shown)
Table 6.21 Alternative A (Moment Frame) Story Drifts under Seismic Loads
Elastic Displacement
at Building Corner,
From Analysis
Expected
Displacement (=ëeCd)
Design Story Drift Ratio
Allowable
Story Drift
Ratio
Level
e EW
(in.)
e NS
(in.)
EW
(in.)
NS
(in.)
EW/h
(%)
NS/h
(%)
/h
(%)
Level 7
2.92
3.18
16.0
17.5
1.2
1.2
2.0
Level 6
2.66
2.89
14.7
15.9
1.4
1.7
2.0
Level 5
2.33
2.47
12.8
13.6
1.6
2.0
2.0
Level 4
1.91
1.95
10.5
10.7
1.9
2.0
2.0
Level 3
1.41
1.40
7.76
7.70
1.8
1.8
2.0
Level 2
0.90
0.88
4.96
4.85
1.2
1.2
2.0
Level 1
0.55
0.52
3.04
2.89
1.1
1.1
2.0
1.0 in. = 25.4 mm.
3. Check the ColumnBeam Moment Ratio: Check the columnbeam moment ratio per AISC 341
Section 9.6. The expected moment strength of the beams is projected from the plastic hinge location
to the column centerline per the requirements of AISC 341 Section 9.6. This is illustrated in
Figure 6.25. For the columns, the moments at the location of the beam flanges are projected to the
columnbeam intersection as shown in Figure 6.26.
Figure 6.25 Projection of expected moment strength of beam
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
The columnbeam strength ratio calculation is illustrated for the lower level in the EW direction,
Level 2, at Gridline D (W24x146 column and W21x73 beam).
For the beams:
where:
Mpr = CprRyFy Ze = (1.15)(1.1)(50) (122) = 7,361 in.kips
Ry = 1.1 for Grade 50 steel
Ze = Zx 2ctbf (d  tbf) = 172 2(1.659)(0.74)(21.240.74) = 122 in.3
Sh = Distance from column face to centerline of plastic hinge (see Figure 6.29) = a + b/2 = 13.2 in.
for the RBS
L = Distance between plastic hinges = 248.8 in.
wu = Factored uniform gravity load along beam
= 1.4D + 0.5L = 1.4[(0.068 ksf)(12.5 ft)+(0.025)(13.3 ft)] + 0.5(0.050 ksf)(12.5 ft)
= 2.42 klf
Figure 6.26 Moment in the column
The shear at the plastic hinge (Figure 6.27) is computed as:
where:
Vp = Shear at plastic hinge location
Figure 6.27 Free body diagram bounded by plastic hinges
Figure 6.28 Forces at beamcolumn connection
Figure 6.29 Reduced beam section dimensions
(1.0 in. = 25.4 mm)
Therefore:
For the beam on the right, with gravity moments adding to seismic:
= 9,517 in.kips
For the beam on the left, with gravity moments subtracting from seismic:
= 8,233 in.kips
Note that in most cases, the gravity moments cancel out and can be ignored for this check.
For the columns, the sum of the moments at the top and bottom flanges of the beam is:
where:
MBF = column moment at beam flange elevation
Referring to Figure 6.26, the moment at the beam centerline is:
where:
= /hc, based on the expected yielding of the spliced column assuming an
inflection point at column midheight (e.g., a portal frame) and not the expected shear when the
mechanism forms, which is:
, where h is the story height
hc = clear column height between beams = (13.33 ft)(12 in./ft) 21.24 in. = 139 in.
=102 kips
Thus:
The ratio of column moment strengths to beam moment strengths is computed as:
OK
Since the ratio is greater than 2, bracing is only required at the top flange per AISC 341 Section 9.7a.
4. Check the Beam Strength: Per AISC 358 Equation 5.84, the beam strength at the reduced section is:
From analysis, Mu = 4072 inkips. Therefore, Mpr ò Mu; the beam has adequate strength.
The moment at the column face is:
OK
To check the shear in the beam, first the appropriate equation must be selected:
Therefore:
where Cv = 1.0.
Comparing this to Vp:
OK
Check the beam lateral bracing. Per AISC 341 Section 9.8, the maximum spacing of the lateral
bracing is:
The braces near the plastic hinges are required to have a minimum strength of:
where:
Mu = RyFyZ
ho = the distance between flange centroids
The required brace stiffness is:
Lb is taken as Lp. These values are for the typical lateral braces. No supplemental braces are required
at the reduced section per AISC 358 Section 5.3.1.
5. Check Connection Design:
Check the need for continuity plates. Continuity plates are required per AISC 358 Section 2.4.4
unless:
And:
Since tcf = 1.09 inches, continuity plates are required. See below for the design of the plates.
Checking web crippling per AISC 360 Section J10.3:
OK
Checking web local yielding per Specification Section J10.2:
Therefore, since Rn ó Ru, as well as due to the check above, continuity plates are required. The force
that the continuity plates must take is 413  293 = 120 kips. Therefore, each plate takes 60 kips. The
minimum thickness of the plates is the thickness of the beam flanges, 0.74 inch. The minimum width
of the plates per AISC 341 Section 7.4 is:
Checking the strength of the plate with minimum dimensions:
Therefore, since Rn = 206 kips > 60 kips, the minimum continuity plates have adequate strength.
Alternatively, a W24x192 section will work in lieu of adding continuity plates.
6. Check Panel Zone: The Standard defers to AISC 341 for the panel zone shear calculation.
The panel zone shear calculation for Story 2 of the frame in the EW direction at Grid C (column:
W24x176; beam: W21x73) is from AISC 360 Section J10.6. Check the shear requirement at the
panel zone in accordance with AISC 341 Section 9.3. The factored shear Ru is determined from the
flexural strength of the beams connected to the column. This depends on the style of connection. In
its simplest form, the shear in the panel zone (Ru) is as follows for W21x73 beams framing into each
side of a W24x146 column (such as Level 2 at Grid C):
Mf is the moment at the column face determined by projecting the expected moment at the plastic
hinge points to the column faces (see Figure 6.25):
Note that in most cases, the gravity moments cancel out and can be ignored for this check. The total
moment at the column face is:
The shear transmitted to the joint from the story above, Vc, opposes the direction of Ru and may be
used to reduce the demand. Previously calculated, this is 102 kips at this location. Thus the frame
Ru = 794  102 = 692 kips.
The column axial force (Load Combination: 1.2D + 0.5L + êoE) is Pr = 228 kips.
Since Pr ó 0.75 Pc, using AISC 360 Equation J1011:
= 547 kips
Since v is 1, vRn = 547 kips.
Therefore, doubler plates are required. The required additional strength from the doubler plates is
692  547 = 145 kips. The strength of the doubler plates is:
Therefore, to satisfy the demand the doubler plate must be at least 1/4 inch thick. Plug welds are
required as:
Use four plug welds spaced 12 inches apart. Alternatively, the use of a W24x192 column will not
require doubler plates ( vRn = 737 kips).
6.2.5 Analysis and Design of Alternative B: SCBF
6.2.5.1 Modal Response Spectrum Analysis. As with the SMF, find the approximate building period
(Ta) using Standard Equation 12.87:
CuTa, the upper limit on the building period, is determined per Standard Table 12.81:
It is assumed that the calculated period will exceed CuTa; this is verified after member selection. The
seismic response coefficient (Cs) is determined from Standard Equation 12.82 as follows:
However, Standard Equation 12.83 indicates that the value for Cs need not exceed:
and the minimum value for Cs per Standard Equation 12.85 is:
Use Cs = 0.112.
Seismic base shear is computed using Standard Equation 12.81 as:
where W is the seismic mass of the building as determined above.
In evaluating the building in ETABS, twelve modes are analyzed, resulting in a total modal mass
participation of 99 percent. The Standard Sec. 12.9.1 requires at least 90 percent participation. As before
with Alternative A, strength is scaled to 85 percent of the equivalent lateral force base shear and drift is
scaled by gCd/(R/I).
6.2.5.2 Size members. The method used to size members is as follows:
1. Select brace sizes based on strength
2. Select column sizes based on special seismic load combinations (Standard Sec. 12.4.3.2)
3. Select beam sizes based on the load imparted by the expected strength of the braces
4. Check drift (Standard Sec. 12.12)
5. Design the connection
Reproportion member sizes as necessary after each check. After the weight and stiffness have been
modified by changing member sizes, the response spectrum must be rescaled. Torsional amplification is
a significant consideration in this alternate.
1. Select Preliminary Member Sizes and Check Strength: The preliminary member sizes are shown for
the braced frame in the EW direction (seven bays) in Figure 6.210 and in the NS direction (five
bays) in Figures 6.211 and 6.212.
Figure 6.210 Braced frame in EW direction
Figure 6.211 Braced frame in NS direction on Gridlines C and F
Figure 6.212 Braced frame in NS direction on Gridlines D and E
Check slenderness and widthtothickness ratios the geometrical requirements for local stability. In
accordance with AISC 341 Section C13.2a, bracing members must satisfy the following:
All members are seismically compact for SCBF per AISC SDM Table 12, thus satisfying
slenderness requirements.
Columns: Wide flange members must comply with the widthtothickness ratios contained in
AISC 341 Table I81. Flanges must satisfy the following:
Webs in combined flexural and axial compression (where Pu/ bPy = 0.385 > 0.125) must satisfy the
following:
Braces: Rectangular HSS members must satisfy the following:
Using a redundancy factor of 1.3 on the earthquake loads, the braces are checked for strength using
ETABS and found to be satisfactory.
2. Select Column Sizes: Columns are checked using special seismic load combinations; does not
apply in these combinations (see Standard Sec. 12.3.4.1 Item 6). The columns are then checked for
strength using ETABS and found to be satisfactory.
3. Select Beam Sizes: The beams are sized to be able to resist the expected plastic and postbuckling
capacity of the braces. In the computer model, the braces are removed and replaced with forces
representing their capacities. These loads are applied for four cases reflecting earthquake loads
applied both left and right in the two orthogonal directions (T1x, T2x, T1y, T2y). For instance, in T1x, the
earthquake load is imagined to act left to right; the diagonal braces expected to be in tension under
this loading are replaced with the force RyFyAg and the braces expected to be in compression are
replaced with the force 0.3Pn. For T2x, the tension braces are now in compression and vice versa. T1y
and T2y apply in the other orthogonal direction.
The load cases applied are as follows:
(four combinations; use all four T s)
(four combinations; use all four T s)
Beam strength is checked for each of these eight load combinations using ETABS and found to be
satisfactory.
4. Check Story Drift: After designing the members for strength, the ETABS model is used to determine
the design story drift. The results are summarized in Table 6.22.
Table 6.22 Alternative B Story Drifts under Seismic Load
Elastic Displacement
at Building Corner,
From Analysis
Expected
Displacement (=ëeCd)
Design Story Drift Ratio
Allowable
Story Drift
Ratio
Level
e EW
(in.)
e NS
(in.)
EW
(in.)
NS
(in.)
EW/h
(%)
NS/h
(%)
/h
(%)
Level 7
1.63
1.75
8.14
8.76
0.72
0.93
2.0
Level 6
1.41
1.48
7.07
7.38
0.74
0.94
2.0
Level 5
1.19
1.20
5.97
5.99
0.76
0.84
2.0
Level 4
0.96
0.94
4.80
4.72
0.81
0.85
2.0
Level 3
0.71
0.69
3.56
3.43
0.72
0.71
2.0
Level 2
0.49
0.47
2.44
2.33
0.60
0.59
2.0
Level 1
0.30
0.28
1.49
1.40
0.56
0.52
2.0
1.0 in. = 25.4 mm.
All story drifts are within the allowable story drift limit of 0.020hsx in accordance with Standard
Section 12.12 and the allowable deflections for this building from Section 6.2.3.6 above. As shown
in the table above, the drift is far from being the governing design consideration.
5. Design the Connection: Figure 6.213 illustrates a typical connection design at a column per
AISC 341 Section 13.
Figure 6.213 Bracing connection detail (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
The connection designed in this example is at the fourth floor on Gridline C. The required strength of
the connection is to be the nominal axial tensile strength of the bracing member. For an HSS6x6x5/8,
the expected axial tensile strength is computed using AISC 341 Section 13.3a:
Ru = RyFyAg = (1.4)(46 ksi)(11.7 in.2) = 753 kips
The area of the gusset is determined using the plate thickness and section width (based on geometry).
See Figure 6.213 for the determination of this dimension. The thickness of the gusset is chosen to be
1 inch.
For tension yielding of the gusset plate:
Tn = FyAg = (0.90)(50 ksi)(1 in. 17 in.) = 765 kips > 753 kips OK
For fracture in the net section:
Tn = FuAn = (0.75)(65 ksi )(1 in. 17 in.) = 829 kips > 753 kips OK
For a tube slotted to fit over a connection plate, there will be four welds. The demand in each weld
will be 753 kips/4 = 188 kips. The design strength for a fillet weld per AISC 360 Table J2.5 is:
Fw = (0.6Fexx) = (0.75)(0.6)(70 ksi) = 31.5 ksi
For a 1/2inch fillet weld, the required length of weld is determined to be:
Therefore, use 17 inches of weld.
In accordance with the exception of AISC 341 Section 13.3b, the design of brace connections need
not consider flexure if the gusset can accommodate the inelastic rotation associated with brace post
buckling deformations. This is typically done by providing a hinge zone"; the gusset plate is
detailed such that it can form a plastic hinge over a distance of 2t (where t = thickness of the gusset
plate) from the end of the brace. The gusset plate must be permitted to flex about this hinge,
unrestrained by any other structural member. See also AISC 341 Section C13.3b. With such a
pinnedend condition, the compression brace tends to buckle outofplane. During an earthquake,
there will be alternating cycles of compression and tension in a single bracing member and its
connections. Proper detailing is imperative so that tears or fractures in the steel do not initiate during
the cyclic loading.
While the gusset is permitted to hinge, it must not buckle. To prevent buckling, the gusset
compression strength must exceed the expected brace strength in compression per AISC 341
Section 13.3c. Determine the nominal compressive strength of the brace member. The effective
brace length (kL) is the distance between the hinge zones on the gusset plates at each end of the brace
member. This length is somewhat dependent on the gusset design. For the brace being considered,
kL = 161 inches the expected compressive strength is determined using expected (not specified
minimum) material properties per AISC 360 Section E3:
where:
Ag = gross area of the brace
Fcr = flexural buckling stress, determined as follows
When:
Otherwise,
where:
The equations have been recalibrated to use the expected stress rather than the specified minimum
yield stress. Note that the 0.877 factor, which represents outofstraightness, is not used here in order
to calculate an upper bound brace strength and thereby ensure adequate gusset compression strength.
Here, kL/r = (1)(161)/(2.17) = 74.2, thus:
Now, using the expected compressive load from the brace of 449 kips, check the buckling capacity of
the gusset plate using the section above. By this method, illustrated by Figure 6.213, the
compressive force per unit length of gusset plate is (478 kips/23.5 in.) = 20.3 kips/in.
Try a plate thickness of 1 inch:
fa = P/A = 20.3 kips/(1 in. 1 in.) = 20.3 ksi
The gusset plate is modeled as a 1inchwide by 1inchdeep rectangular section, fixed at both ends.
The length, from geometry, is 17.2 inches. The effective length factor, k, for this column is 1.2 per
AISC 360 Table CC2.2. The radius of gyration, r, for a plate is .
Per AISC 360 Section E3:
OK
Next, check the reduced section of the tube, which has a 11/8inchwide slot for the gusset plate (the
thickness of the gusset plus an extra 1/8 inch for ease of construction). The reduction in HSS6x6x5/8
section due to the slot is (0.581 in. 1.125 in. 2) = 1.31 in.2 the net section, Anet = (11.7  1.31) =
10.4 in.2
To ensure gross section yielding governs, reinforcement is added over the area of the slot. The shear
lag factor is computed per AISC 360 Table D3.1:
where:
and l is the length of the weld as determined above.
Thus, the effective area of the section is:
Try a reinforcing plate 1/2 inch thick and 31/2 inches wide on each side of the brace. (The necessary
width can be computed from the effective area, but that calculation is not performed here.) Grade 50
material is used in order to match or exceed the brace material strength, thus allowing for treatment of
the material as homogenous. The area of the section is (2 0.5 in. 3.5 in.) = 3.5 in.2. The distance
of its center of gravity from the center of gravity of the slotted brace is:
Thus, the area of the reinforced section is:
The weighted average of the x s is 2.53 inches. Thus, the shear lag factor for the reinforced section
is:
Thus, the effective area of the section is:
Now, check the effective area of the reinforced section against the original section of the brace per
AISC 341 Section 13.2b:
OK
The reinforcement is attached to the brace such that its expected yield strength is developed.
The plate will be developed with two 5/16inch fillet welds, 14 inches long:
The force must be developed into the plate, carried past the reduced section developed out of the
plate. To accomplish this, the reinforcement plate will be 33 inches: 14 inches on each side of the
reduced section, 2 inches of anticipated over slot, plus 1 inch to provide erection tolerance.
The complete connection design includes the following checks (which are not demonstrated here):
Attachment of reinforcement to brace
Brace shear rupture
Brace shear yield
Gusset block shear
Gusset yield, tension rupture, shear rupture weld at both the column and the beam
Web crippling and yielding for both the column and the beam
Gusset edge buckling
Beamtocolumn connection
6.2.6 Cost Comparison
For each case, the total structural steel was estimated. The takeoffs are based on all members, but do not
include an allowance for plates and bolts at connections. The result of the material takeoffs are as
follows:
Alternative A, Special Steel Moment Resisting Frame: 640 tons
Alternative B, Special Steel Concentrically Braced Frame: 646 tons
The higher weight of the systems with bracing is primarily due to the placement of the bracing in the core,
where resistance to torsion is poor. Torsional amplification and drift limitations both increased the
weight of steel in the bracing. The weight of the momentresisting frame is controlled by drift and the
strong column rule.
6.3 TENSTORY HOSPITAL, SEATTLE, WASHINGTON
This example features a bucklingrestrained braced frame (BRBF) building. The example covers:
Analysis issues specific to bucklingrestrained braced frames
Proportioning of bucklingrestrained braces
Capacity design principles
Nonlinear response history analysis
Bucklingrestrained brace connections
6.3.1 Building Description
This tenstory hospital includes a twostory podium structure beneath an eightstory tower, as shown in
Figures 6.31 and 6.32. The podium is 211.3 feet by 121.3 feet in plan, while the tower s floorplate is
square with 91.3foot sides. Story heights are 18 feet in the podium and reduce to 15 feet throughout the
tower, bringing the total building height to 156 feet. As the tower is centered horizontally on the podium
below, the entire building is symmetric about a single axis. Both the podium and the tower have large
roof superimposed dead loads due to heavy HVAC equipment located there.
Figure 6.31 Typical tower plan
Figure 6.32 Level 3 podium plan
The structure exemplifies a common situation for hospital facilities. The combination of a stiff podium
structure beneath a more flexible tower results in significant force transfer at the floor level between
them.
The verticalloadcarrying system consists of lightweight concrete fill on steel deck floors supported by
steel beams and girders that span to steel columns. The bay spacing is 30 feet each way. There are three
floor beams per bay. All beams and girders are composite.
BRBFs have been selected for this building because they provide high stiffness paired with a high degree
of ductility and stable hysteretic properties. The building has a thick mat foundation. The foundation soil
is representative of Site Class C conditions identical to those discussed in Section 3.2. The design of
foundations is not included here.
6.3.1.1 Design method. Seismic forces, rather than wind forces, govern the building s lateral design (in
part due to the mass of the thick concretefilled decks). The lateral forceresisting system throughout the
tower consists of BRBFs in the middle bay along each side of the perimeter Gridlines 3, 6, A D, as can
be seen in the representative elevation of Figure 6.33. These BRBFs deliver lateral loads to the
collectors and diaphragm at the third floor where both inplane and outofplane discontinuities exist.
This transfer occurs inplane along Gridlines A and D to two braced bays nearer the ends of the podium
and outofplane from a single braced bay in the tower along Gridlines 3 and 6 to braces in the two
adjacent bays along Gridlines 2 and 7 in the podium below. The podium bracing configuration is
illustrated in Figure 6.32. A typical bracing elevation in the transverse direction of the podium
(illustrating the outofplane offset) is shown in Figure 6.34.
Figure 6.33 Longitudinal elevation at Gridline D
Figure 6.34 Transverse elevations
An ELF analysis is first performed to scale the base shear for the subsequent MRSA used for strength
design of the bucklingrestrained braces (BRB). Each BRB is designed for its share of 100 percent of the
horizontal component of the earthquake lateral load without considering additional tributary vertical
loads. This is done to encourage distributed yielding of braces up the height of the structure and is
justified because the braces will shed any gravity load upon first yield and transfer it to the connecting
beams and columns, which are designed to accommodate gravity loads without support provided by the
braces. Beams, columns collectors are preliminarily sized using capacity design principles considering
plastic mechanisms that develop based on the brace sizes determined using elastic MRSA. Finally, a
nonlinear response history analysis (NRHA) is executed to verify BRB strains remain at acceptable
levels, check that story drifts do not exceed allowable limits possibly reduce column sizes from what the
plastic mechanism analyses require. The details of this design procedure are summarized in Table 6.31.
Table 6.31 Design Philosophy
Element
Action
Analysis Method
Software
Acceptance Criteria
BRBs
Strength
MRSA
ETABS
AISC 341
Deformation
NRHA
PERFORM
ASCE 41
Columns
Strength
NRHA
PERFORM
AISC 360
Rotation
NRHA
PERFORM
ASCE 41
Diaphragms
Story Drift
NRHA
PERFORM
ASCE 7
Collector
beams, braced
bay beams, etc.
Strength
MRSA
ETABS
AISC 360
6.3.2 Basic Requirements
6.3.2.1 Provisions parameters. Section 3.2 illustrates the determination of design ground motion
parameters for this example. They are as follows:
SDS = 0.859
SD1 = 0.433
Occupancy Category IV
Seismic Design Category D
For BRBFs (Standard Table 12.21):
R = 8
o = 2.5
Cd = 5
These values are representative of a seismic forceresisting system that is a unification of two previously
different BRBF system classifications: those with nonmomentresisting beamcolumn connections and
those with momentresisting beamcolumn connections. Modifications to Section 15.7 of AISC 341 now
require the beamtocolumn connections for a building frame system either to meet the requirements for
fully restrained (FR) moment connections as specified in AISC 341 Section 11.2a or to possess sufficient
rotation capacity to accommodate the rotation required to achieve a story drift of 2.5 percent. The later
compliance path is selected for this design example. This requirement effectively amounts to a braced
frame with simple beamtocolumn connections per AISC 360 Section B3.6a with rotation specified at
2.5 percent. A standard detail illustrating this connection is presented in Figure 6.35.
Figure 6.35 Pinned beamtocolumn connection
6.3.2.2 Loads.
Roof live load, Lr: 25 psf
Roof dead load, D: 135 psf
Exterior wall cladding: 300 plf of spandrel beams
Floor live load, L: 60 psf
Partitions: 10 psf
Floor dead load, D: 104 psf
Floor live load reductions: per the IBC
Roof dead load includes roofing, insulation, lightweight concretefilled metal deck, concrete ponding
allowance, framing, mechanical and electrical equipment, ceiling fireproofing. Floor dead load includes
lightweight concretefilled metal deck, ponding allowance, framing, mechanical and electrical equipment,
ceiling and fireproofing. Due to potential for rearrangement, partition loads are considered live loads per
Standard Section 4.2.2 but are also included in the effective seismic weight in accordance with Standard
Section 12.7.2. Therefore, the seismic weight of a typical tower floor, whose footprint is a square 8,342
ft2 in area, is 104 + 10 + 300(30)(3)(4)/8,342 = 127 psf.
6.3.2.3 Materials
Concrete for drilled piers: fc' = 5 ksi, normal weight (NW)
Concrete for floors: fc' = 3 ksi, lightweight (LW)
All other concrete: fc' = 4 ksi, NW
Structural steel:
Wide flange sections: ASTM A992, Grade 50
Plates: ASTM A36
6.3.3 Structural Design Criteria
6.3.3.1 Building configuration. The hospital building does not possess any stiffness, strength, or weight
irregularities despite the relatively tall height of the podium stories. At the podium levels, the two braced
bays corresponding to each line of singlebay chevron bracing in the tower above provide more than
enough additional strength to compensate for the slight increase in floortofloor height. The story drift
ratio increases up the full height of the structure, meeting the exception of Standard Section 12.3.2.2 for
assessing vertical stiffness and weight irregularities. However, the structure does possess both a vertical
geometric irregularity (Type 3) and an inplane discontinuity in vertical lateral forceresisting element
irregularity (Type 4) since the lateral forceresisting system transitions from a single chevron braced bay
in the tower to two chevron braced bays at the podium levels. The two chevron braced bays in the
podium occur two bays away from the tower braced bay in the longitudinal direction. The Type 4 vertical
irregularity triggers an increase in certain design forces per Standard Sections 12.3.3.3 and 12.3.3.4.
Together, the Type 3 and Type 4 vertical irregularities preclude the use of an equivalent lateral force
analysis as defined in Standard Section 12.8 based on the permissions in Standard Table 12.61. Note
that this analysis prohibition is also triggered by the flexibility of the structure, as its fundamental period
(see Sec. 6.3.4.1) exceeds 3.5Ts = 3.5(SD1/SDS) seconds = 3.5 (0.433/ 0.859) seconds = 1.76 seconds.
Nevertheless, the design base shear still must be determined using the equivalent lateral force analysis
procedures to ensure that the design base shear for a modal response spectrum analysis meets the
requirements of Standard Section 12.9.4.
Due to the building s symmetry and the strong torsional resistance provided by the layout of the vertical
lateral forceresisting elements, numerous plan irregularities are not expected. Analysis reveals that the
structure is torsionally regular the only horizontal structural irregularity present is an outofplane offset
irregularity (Type 4) triggered by the shift in the vertical lateral forceresisting system from Gridlines 3
and 6 in the tower to Gridlines 2 and 7 in the podium structure below. The only additional provisions
triggered by the Type 4 horizontal structural irregularity relate to threedimensional modeling
requirements.
6.3.3.2 Redundancy. The limited number of braced bays in each direction of the tower require the
redundancy factor ( ) to be taken as 1.3 per Standard Section 12.3.4.2 Item a and Table 12.33. Because
there are only two BRBF chevrons in each direction throughout the tower, removal of a single brace
would dramatically increase flexural demands in the beam at that location and would certainly result in at
least a 33 percent reduction in story strength even if the resulting system does not have an extreme
torsional irregularity. The 1.3 redundancy factor ( ) is incorporated as a load factor on the seismic loads
used in the design of the braces.
6.3.3.3 Orthogonal load effects. Standard Section 12.5.4 stipulates a combination of 100 percent of the
seismic forces in one direction plus 30 percent of the seismic forces in the orthogonal direction, at a
minimum, for structures in Seismic Design Category D. However, it has been shown (Wilson, 2004) that
use of the 100/30 percentage combination rule can result in member designs that are not equally resistant
to earthquake ground motions originating from different directions. Instead, a SRSS combination of
seismic forces from two fullmagnitude response spectra analyses conducted along each principal axis of
the building is performed to ensure the design forces remain independent of the selected reference
coordinate system (in this case, the building s main orthogonal axes).
In the context of NRHA, orthogonal pairs of ground motion acceleration histories are applied
simultaneously in accordance with the requirements of Standard Section 12.5.4 for structures in Seismic
Design Category D.
6.3.3.4 Structural component load effects. The effect of seismic load as defined by Standard
Section 12.4.2 is as follows:
E = QE ñ 0.2SDSD
In this example, SDS = 0.859. The seismic load is combined with the gravity loads in elastic analyses as
shown in Standard Section 12.4.3.2, resulting in the following load combinations:
1.37D + 0.5L + 0.2S + QE
0.73D + 1.6H + QE
The 0.5 coefficient on L is permitted for all occupancies in which Lo in Standard Table 4.1 is less than or
equal to 100 psf per Exception 1 to Standard Section 2.3.2. The braces are designed without considering
additional tributary vertical loads to encourage distributed yielding up the height of the structure.
However, the surrounding beams and columns that are part of the lateral forceresisting system are
designed for the above gravity loads in conjunction with the earthquake effect as specified in AISC 341
Section 16.5b. Again, the redundancy factor, , is taken as 1.3 for design of the braces themselves.
In a NRHA, the structure is analyzed for the effects of the scaled pairs of ground motions simultaneously
with the effects of dead load and 25 percent of the required live loads per Standard Section 16.2.3.
6.3.3.5 Drift limits. For a building assigned to Occupancy Category IV, the allowable story drift
(Standard Sec. 12.12.1 and Table 12.121) is a = 0.010hsx.
The allowable story drift for a typical podium floor is a = (0.01)(18 ft)(12 in./ft) = 2.16 in.
The allowable story drift for a typical tower floor is a = (0.01)(15 ft)(12 in./ft) = 1.80 in.
The calculated design story drifts are amplified by the appropriate Cd factor from Standard Table 12.21
in elastic analysis procedures that employ seismic response coefficients reduced by the appropriate
response modification factor, R.
Standard Section 16.2.4.3 permits the allowable story drift obtained from a nonlinear response history
analysis to be increased by 25 percent relative to the drift limit specified in Section 12.12.1.
The maximum allowable value of story drifts summed to the roof of the tenstory hospital building
(156 feet) obtained from an elastic analysis is 18.72 inches. This same figure extracted from a nonlinear
response history analysis cannot exceed 1.25(18.72 in.) = 23.40 in.
6.3.3.6 Seismic weight. The area of the tower floorplate is approximately equal to [(3)(30 ft) +
(2)(8 in.)(1 ft/12 in.)]2 = 8,342 ft2, while the area of the podium floorplate is approximately [(7)(30 ft) +
(2)(8 in.)(1 ft/12 in.)] [(4)(30 ft) + (2)(8 in.)(1 ft/12 in.)] = 25,642 ft2. Thus, the weights that contribute
to seismic forces are as follows:
Tower roof:
Roof D = (0.135)(8,342) = 1,126 kips
Cladding = (4)(3)(30)(0.300) = 108 kips
Total = 1,234 kips
Tower floor:
Floor D = (0.104)(8,342) = 868 kips
Partitions = (0.010)(8,342) = 83 kips
Cladding = (4)(3)(30)(0.300) = 108 kips
Total = 1,059 kips
Podium roof:
Roof D = (0.135)(25,642  8,342) = 2,336 kips
Floor D = (0.104)(8,342) = 868 kips
Partitions = (0.010)(8,342) = 83 kips
Cladding = (2)(11)(30)(0.300) = 198 kips
Total = 3,485 kips
Podium floor:
Floor D = (0.104)(25,642) = 2,667 kips
Partitions = (0.010)(25,642) = 256 kips
Cladding = (2)(11)(30)(0.300) = 198 kips
Total = 3,121 kips
Total effective seismic weight of building = 1,234 + 7(1,059) + 3,485 + 3,121 = 15,253 kips
6.3.4 Elastic Analysis
The base shear is determined using an ELF analysis; the base shear so computed is needed later when
evaluating the scaling of the base shears obtained from the modal response spectrum analysis.
In a subsequent section (Section 6.3.6.3.3), columns are designed using forces obtained from nonlinear
response history analyses that are intended to represent the maximum force that can develop in these
elements per the exception to Standard Section 12.4.3.1. Compliance with story drift limits is also
evaluated using the results of the nonlinear response history analyses.
6.3.4.1 Equivalent Lateral Force procedure. First, the ELF base shear will be determined, followed by
its vertical distribution up the height of the building.
6.3.4.1.1 ELF base shear. Compute the approximate building period, Ta, using Standard
Equation 12.87:
In accordance with Standard Section 12.8.2, the building period used to determine the design base shear
must not exceed the following:
The subsequent threedimensional modal analysis finds the computed period to be approximately
2.30 seconds in each principal direction. Thus the upper limit on the fundamental period Tmax applies.
The seismic response coefficient, Cs, is computed in accordance with Standard Section 12.8.1.1.
Equation 12.82 provides the value of Cs that generally governs at short periods:
However, Standard Equation 12.83 indicates that the value for Cs need not exceed the following:
and the minimum value for Cs per Standard Equation 12.85 is:
Therefore, use Cs = 0.057.
The seismic base shear is computed per Standard Equation 12.81 as follows:
The redundancy factor ( ) is accounted for by setting the coefficient on the horizontal seismic load effect
to 1.3 in all earthquake load combinations used for strength design of the BRBs. The redundancy factor
is not applicable to the determination of deflections.
6.3.4.1.2 Vertical distribution of ELF seismic forces. Standard Section 12.8.3 prescribes the vertical
distribution of lateral force in a multilevel structure. The floor force, Fx, is calculated using Standard
Equation 12.811 as:
where (per Standard Eq. 12.812):
Using the data in Section 6.3.3.5 of this example and interpolating the exponent k as 1.68 for the period of
1.85 seconds, the vertical distribution of forces for the ELF analysis is shown in Table 6.32. The seismic
design shear in any story is computed as follows (per Standard Eq. 12.813):
Table 6.32 ELF Vertical Seismic Load Distribution
Level
Weight (wx)
Height (hx)
wxhxk
Cvx
Fx
Vx
Roof
1,234 kips
156 ft
5,818,337
0.24
210 kips
210 kips
Story 10
1,059 kips
141 ft
4,215,392
0.18
152 kips
362 kips
Story 9
1,059 kips
126 ft
3,491,537
0.15
126 kips
488 kips
Story 8
1,059 kips
111 ft
2,823,657
0.12
102 kips
590 kips
Story 7
1,059 kips
96 ft
2,214,116
0.09
80 kips
670 kips
Story 6
1,059 kips
81 ft
1,665,744
0.07
60 kips
730 kips
Story 5
1,059 kips
66 ft
1,182,040
0.05
43 kips
772 kips
Story 4
1,059 kips
51 ft
767,496
0.03
28 kips
800 kips
Story 3
3,485 kips
36 ft
1,409,320
0.06
51 kips
851 kips
Story 2
3,121 kips
18 ft
395,253
0.02
14 kips
865 kips
Total
15,253 kips
23,982,891
1.00
865 kips
1.0 kip = 4.45 kN
1.0 ft = 30.5 cm
All floor decks, including the roofs, are constructed of 3 in. lightweight concrete over 3 in. metal deck.
Standard Sec. 12.3.1.2 allows for such diaphragms to be modeled as rigid as long as their spantodepth
ratios do not exceed three. Since the floor spantodepth ratio is a maximum of 1.75 at the podium levels,
the hospital diaphragms meet these conditions. However, Standard Sec. 12.3.1.2 also requires the
structure to have no horizontal irregularities for its diaphragms to be modeled as rigid. Due to the outof
plane offsets irregularity (Type 4) in the transverse lateral frames at the podiumtotower interface, the
hospital does not meet this restriction. As such, the effect on the vertical lateral force distribution of
explicitly considering the stiffness of the diaphragm at the podium roof level was examined in a three
dimensional computer model of the structure and found to be insignificant (i.e., results closely matching a
model with rigid diaphragms) due to the stiff nature of the thick concrete floor. Thus, it was deemed
acceptable to model all diaphragms as rigid for subsequent analyses. The rigid diaphragm assumption is
especially helpful in the context of nonlinear response history analysis, where the additional degrees of
freedom needed to model diaphragm stiffness explicitly can render analysis times prohibitive.
Assessment of load transfer in the level 3 diaphragm would require a separate, explicit analysis. Another
reasonable approach to the primary model would be to include the level 3 diaphragm explicitly and model
all other diaphragms as rigid.
6.3.4.2 Threedimensional static and Modal Response Spectrum Analysis. The threedimensional
analysis is performed for this example to accurately account for the following:
The different centers of mass for the podium and tower levels
The varying stiffness of the braced frames as they transition from their wide configuration at the
podium levels to a singlebay arrangement throughout the tower
The effects of bidirectional frame interaction on the columns engaged by orthogonal braced
frames in the podium
The ability of the braced frames to control torsion
The braced frames and diaphragm chords and collectors, together with all gravity system beams and
columns, are explicitly modeled using threedimensional beamcolumn elements. The floor diaphragms
are modeled as rigid.
As mentioned previously, the ELF analysis procedure of Standard Section 12.8 is not admissible for this
structure due to the restrictions on fundamental period in Standard Table 12.61. However, the ELF
analysis of the threedimensional model is still useful in assessing whether torsional irregularities are
present. The ELF seismic forces derived in Table 6.32 above are applied to each diaphragm at 5 percent
eccentricity orthogonal to the direction of loading. The maximum and average story drifts along an edge
transverse to the direction of loading for the critical direction of eccentricity at each level are then
compared. This ratio of max/ avg never exceeds 1.11, which is below the 1.2 limit that defines torsional
irregularity. For this torsionally regular structure, the accidental torsion amplification factor, Ax, is equal
to 1.0.
A threedimensional modal response spectrum analysis is performed per Standard Section 12.9 using the
threedimensional computer model. The design response spectrum is based on Standard Section 11.4.5
and is shown in Figure 6.36.
Figure 6.36 Design response spectrum, 5 percent damped
Within this model, the first twelve modes of vibration and the corresponding mode shapes of the structure
were determined. Twelve modes provide more than enough participation to capture 90 percent of the
actual mass in each direction of response as required by Standard Section 12.9.1.
The design value for modal base shear, Vt, is determined by combining the individual modal values for
base shear after dividing the design response spectrum by the quantity R/I = 8/1.5 = 5.33 as prescribed by
Standard Section 12.9.2. The complete quadratic combination (CQC) modal combination rule was
selected for this task to account for coupling of closelyspaced modes that are likely present in
symmetrical structures. Five percent modal damping in all modes is specified for the response spectrum
analysis to match the assumption used in deriving the design response spectrum of Figure 6.36. Base
shears thus obtained from the model having an effective seismic weight of 15,253 kips are as follows:
Longitudinal: Vt = 627 kips
Transverse: Vt = 637 kips
In accordance with Standard Section 12.9.4, the design values of modal base shear are compared to the
base shear determined by the ELF method. If the design value for modal base shear is less than
85 percent of the ELF base shear calculated using a period of CuTa (see Sec. 6.3.4.1.1 above), a factor
greater than unity must be applied to the design forces to raise the modal base shear up to this minimum
ELF comparison value. Accordingly:
Multiplier: 0.85 (V/Vt)
Longitudinal multiplier: 0.85 (865 kips / 627 kips) = 1.17
Transverse multiplier: 0.85 (865 kips / 637 kips) = 1.15
In a typical elastic analysis, it is recommended to examine lateral displacements early in the design
process, as seismic (or wind) drift often controls the design of taller structures. For this building,
compliance with drift limits will ultimately be checked using a nonlinear response history analysis.
However, lateral displacements are still examined in the elastic analysis to ensure they remain reasonably
close to the limits derived in Section 6.3.3.5 (that is, in order to prevent wasting analysis effort on a
design that is unlikely to meet the drift limits). This check is illustrated in Section 6.3.4.3 below.
To obtain elastic design forces for the BRBs, the results from the two orthogonal MRSAs in the three
dimensional model are combined via SRSS, exceeding the requirements of Standard Section 12.5.4.
6.3.4.3 Preliminary drift assessment. Seismic drift is examined in accordance with Standard
Section 12.12.1. The design story drift in each translational direction was extracted from the three
dimensional ETABS model corresponding to the response spectrum case, including 5 percent accidental
torsion, exciting that same direction. Although only strictly required for structures possessing torsional
irregularities as defined by Standard Table 12.31, story drift was nonetheless examined at the building
corners rather than the centers of mass for this structure because the location of cladding attachment is the
most critical location for this check.
The lateral deflections obtained from the response spectrum analysis must be multiplied by Cd/I(I/R) =
Cd//R = 5/8 = 0.625 to find the design story drift. However, the response spectrum used in the analysis
has already been scaled twice. The spectrum was first scaled by R/I = 8 / 1.5 = 5.33 to obtain designlevel
forces; thus the resulting displacements can be amplified by Cd/I = 5 / 1.5 = 3.33 to obtain expected drifts.
The second scaling was by a factor (in each direction) to ensure that the design base shear forces in each
direction to meet the minimum 85 percent of ELF base shear. This latter scaling does not apply to drifts,
per Standard Section 12.9.4. Thus, the 1.17 and 1.15 scale factors applied in the longitudinal and
transverse directions, respectively, must be divided back out of the drifts extracted from the model used to
obtain design forces for the braces. The resulting scale factors applied to the results of the scaled spectra
are 3.33/1.17 = 2.85 and 3.33/1.15 = 2.90 applied in the longitudinal and transverse directions,
respectively. (It would also be possible to simply perform an additional response spectrum analysis with
the design spectrum multiplied by 0.625 and use the resulting story drifts directly.)
Story drifts in all ten stories of the hospital building are within the allowable story drift limit of 0.010hsx
per Standard Section 12.12.1 and Section 6.3.5.5 of this chapter. Although story drifts calculated using
MRSA reach a maximum value at the roof level that is just 89 percent of the 0.010hsx limit (and hence
acceptable), a nonlinear response history analysis is nevertheless used to confirm that all story drifts
indeed remain within prescribed limits.
A comparison of story drift ratios also confirms that no story drift ratio is more than 130 percent of that
for the story above, as required to prove certain vertical irregularities are not present in the structure via
the exception to Standard Section 12.3.2.2.
6.3.4.4 Secondorder (Pdelta) effects. AISC 360 requires consideration of second order effects. Such
effects were investigated by conducting a threedimensional Pdelta analysis, which determined that
secondary Pdelta effects on the frame accounted for less than 10 percent of the primary demand.
Furthermore, Standard Section 12.8.7 gives a different means of determining the significance of Pdelta
effects through the stability coefficient, , defined in Standard Equation 12.86. In either case, Pdelta
effects were found to be insignificant for this particular braced frame structure.
6.3.4.5 Brace design force summary. The maximum axial forces in each level s individual BRBs
caused by horizontal earthquake loads are listed in Table 6.33. Again, each brace is designed for its
share of 100 percent of the horizontal earthquake load effect times the redundancy factor ( ) of 1.3
without considering additional vertical loads to encourage distributed yielding of braces up the height of
the structure. Nonetheless, the braces are also checked to ensure they do not yield under maximum live
load (i.e., the load combination of 1.2D + 1.6L + 0.5Lr).
Because the length of the yielding segment of a BRB is significantly less than its workpointtoworkpoint
length (see Sec. 6.5.3.1.1 below), the axial stiffness of the brace elements in the threedimensional elastic
analysis model must be adjusted to account for the nonprismatic nature of these elements. The modulus
of elasticity of the steel in the brace elements was increased by a factor of 1.51 for singlediagonal and
chevron bracing throughout the tower and 1.45 for chevron or V bracing configurations in the podium to
match the true elastic stiffness of these elements as they are defined in the nonlinear response history
analysis.
The sizes of the BRBF members are controlled by seismic loads, always bidirectional and with
eccentricity, rather than wind loads. Standard Section 12.8.4.2 only requires that the 5 percent
displacement of the center of mass associated with accidental torsion be applied in the direction that
generates the greater effect when earthquake forces are applied simultaneously in two orthogonal
directions. However, due to the intricacies of SRSS directional combination of response spectra in
ETABS, the 5 percent offset is applied in both orthogonal directions at the same time, which is slightly
conservative for torsional response (and is not a significant penalty for this particular building due to its
regular nature in plan). The design of connections will be governed by the seismic requirements of
AISC 341.
Table 6.33 Design Axial Forcesa in BucklingRestrained Bracing Membersb
Location
Gridline A (kips)
Gridline D (kips)
Gridline 2, 3, 6, or 7 (kips)
Roofc
139
144
142
Story 10
117
127
126
Story 9
135
136
133
Story 8
142
141
141
Story 7
152
151
151
Story 6
167
175
170
Story 5
193
195
197
Story 4
212
214
210
Story 3
152
197d
187
Story 2
162
233
195
aIndividual maxima are not necessarily on the same frame; values are maximum for any frame.
bAll braces are oriented in the chevron configuration except for single diagonalc or Vd.
1.0 kip = 4.45 kN
6.3.5 Initial Proportioning and Details
The BRBFs occur on Gridlines 3, 6, A D in the tower and transfer their loads at the third floor to two
BRBFs per line on Gridlines 2, 7, A D in the podium. These frames are shown schematically in plan in
Figures 6.31 and 6.32 and in elevation in Figures 6.33 and 6.34. Using the horizontal component of
the seismic load (amplified by the redundancy factor) as determined by response spectrum analysis and
the loads from Table 6.33, the proportions of the braces are checked for adequacy. Then, initial sizes for
the lateral columns, beams collectors are determined from the threedimensional elastic analysis model
using capacity design principles and relevant plastic mechanism analyses. In the preliminary elastic
design stage, generic BRB properties are used to derive expected brace strengths. This allows for a
specific BRB supplier to be selected further downstream in the project schedule, as is usually done in
traditional project delivery methods. Design forces for columns are obtained from a summation of the
vertical component of the adjusted brace strengths above the level of interest, while those for horizontal
elements are derived from two different plastic mechanisms, always using adjusted brace strengths in
tension and compression as required by AISC 341 Section 16.5b. All lateral columns are then subject to
resizing according to the force and displacement demands determined using nonlinear response history
analysis.
6.3.5.1 BucklingRestrained Brace sizes
6.3.5.1.1 BucklingRestrained Brace mechanics. The mechanics of BRBs are such that compression
buckling need not be considered in their selection. Their required strength is controlled by yielding of the
steel core material only. A BRB consists of a steel core that resists imposed axial stresses together with a
mortarfilled sleeve that resists buckling. The steel core has both a yielding portion and two nonyielding
portions at its ends where the crosssection enlarges to facilitate connection to a gusset plate. A de
bonding agent, often proprietary, decouples the axial behavior of the core from the buckling behavior of
the sleeve. In compression, a BRB acts as a sleeved column the steel core is able to achieve the full
magnitude of its squash load while, at the same time, the sleeve can provide its full Euler buckling
resistance without taking on any axial load. From a performance standpoint, such a component produces
very desirable balanced hysteretic behavior that exhibits both isotropic and kinematic (cyclic) strain
hardening. Unlike conventional bracing, BRB behavior is much more symmetric with respect to tension
and compression and is not subject to strength and stiffness degradation.
6.3.5.1.2 Steel core area. According to AISC 341 Section 16.2a, the steel core must resist the entire
axial force in the brace. This force is tabulated in Table 6.33. The brace design axial strength, Pysc, in
either tension or compression, as controlled by the limit state of yielding, is equal to the following:
where:
= 0.90
Fysc = specified minimum yield stress (or actual from coupon test) of the steel core
Asc = net area of steel core
Setting Pysc equal to the Pu values in Table 6.33 and rearranging terms, the required net area of steel
core can be expressed as follows:
This required area, together with the actual steel core area provided, is shown for each brace in
Table 6.34. Rarely do designers know the actual yield stress of the steel core during the design phase;
hence, a minimum yield stress (Fysc) of the steel core equal to 38 ksi is assumed for this example. This
minimum yield stress value would be specified on the design drawings.
Table 6.34 Steel Core Areas for BucklingRestrained Bracing Membersa
Location
Gridline A (kips)
Gridline D (kips)
Gridline 2, 3, 6, or 7 (kips)
Asc req d
(in.2)
Asc
(in.2)
Asc req d
(in.2)
Asc
(in.2)
Asc req d
(in.2)
Asc
(in.2)
Roofb
4.06
4.5
4.21
4.5
4.15
4.5
Story 10
3.42
3.5
3.71
4.0
3.68
4.0
Story 9
3.95
4.0
3.98
4.0
3.89
4.0
Story 8
4.15
4.5
4.12
4.5
4.12
4.5
Story 7
4.44
4.5
4.42
4.5
4.42
4.5
Story 6
4.88
5.0
5.12
5.5
4.97
5.0
Story 5
5.64
6.0
5.70
6.0
5.76
6.0
Story 4
6.20
6.5
6.26
6.5
6.14
6.5
Story 3
4.44
4.5
5.76c
6.0c
5.47
5.5
Story 2
4.74
5.0
6.81
7.0
5.70
6.0
aAll braces are oriented in the chevron configuration except for single diagonalb or Vc
1.0 in = 25.4 mm
6.3.5.2 Lateral forceresisting columns. To design the frame containing the BRBs, unless using a
nonlinear analysis, the designer should assume a plastic mechanism in which all BRBs are yielding in
tension or compression and have reached their strainhardened adjusted strengths, including all sources of
overstrength. These adjusted brace strengths per AISC 341 Section 16.2d are as follows:
Compression: RyFyscAsc
Tension: RyFyscAsc
The adjusted brace strength values represent the yield strength of the steel core adjusted for material
overstrength (Ry), strainhardening ( ) compression overstrength ( ). Whereas conventional bracing
usually buckles in compression well before reaching its yield strength, BRBs are often slightly stronger in
compression than in tension. The strainhardening ( ) and compression overstrength ( ) factors
traditionally are provided by BRB manufacturers and are calculated from cyclic subassemblage testing to
a brace deformation equivalent to twice the design story drift per AISC 341 Appendix T.
For the initial proportioning of bracedframe columns and beams using the threedimensional elastic
analysis model, generic values of = 1.05, = 1.36 Ry = 1.21 are assumed. A material overstrength
factor, Ry, of 1.21 is selected to bring the design yield strength of the steel core, Fysc = 38 ksi, up to the
typical maximum specified steel core yield strength of 46 ksi. Note that all of these values are subject to
revision for use in the nonlinear response history analysis once the BRB calibration has been performed in
Section 6.3.6.2.
This capacity design methodology can easily be implemented to design lateral columns in a BRBF once
the threedimensional analysis model is constructed. The designer simply needs to generate an axial force
in each brace corresponding to its adjusted brace strength as defined above, either by deleting the braces
from the model and replacing them with their associated forces directly or by some other method that
achieves the same result (for example, hand calculations or spreadsheets). Two different load cases must
be examined for each principal direction: one with all braces in either tension or compression based on
lateral load originating from one side of the frame a second with the brace forces determined by lateral
load originating from the other side of the frame.
Appropriate consideration should be given to bidirectional combination of the resulting brace loads on
columns engaged by two orthogonal frames. While AISC 341 is unwavering in its requirement to design
columns for the full adjusted brace strengths, the displacement corresponding to the adjusted brace
strength should remain constant in any direction. Thus, such a displacement imposed at 45 degrees to the
principal building axes will cause yielding of all braced frames, but not full strain hardening. This
reduction factor for bidirectional loading is dependent on the brace s postyield behavior and will not be
much less than one for a ductile system such as a BRBF. The 100%/30% orthogonal combination
procedure defined in Standard Section 12.5.3 is not applicable in the context of capacity design as
mandated for columns by AISC 341 Section 16.5b.
To complete the preliminary column design, the column axial loads resulting from the maximum
expected brace forces defined above are substituted for the earthquake load effect and combined with
vertical loads as specified in Section 5.3.3.4. The translation and twist of all diaphragms should be locked
when performing the column design for stability of the model. This will ensure each column is designed
for the axial force equal to the summation of the vertical components of the adjusted brace strengths of all
braces above it. Column flexural forces are not considered in their design, consistent with AISC 341
Section 8.3 (1). Such an approximation is valid because localized flexural yielding of a column at
locations where it receives a brace is deemed acceptable from a performance standpoint. The preliminary
column designs can be seen in Figures 6.33 and 6.34.
To illustrate this process, a detailed calculation of the column design forces for the column at D4 can be
seen in Table 6.35. Because column strengths are governed by compression buckling rather than
yielding, brace actions that induce compression on the columns are considered critical. For the uppermost
column below the tower roof, the 228kip design force is equal to the sum of the design load due to
gravity/vertical earthquake effect of 102 kips and the vertical component of the adjusted brace strength in
tension at that level, equal to 126 kips. The adjusted brace strength in tension is used here because
tension in the single diagonal brace at the roof will impose compressive forces on the column below, as
can be seen in the elevation of Figure 6.33. At lower levels with chevron bracing, the design axial load
in the column is calculated as follows:
1. Start with the vertical component of the roof brace in tension (126 kips).
2. Add the associated design gravity and vertical earthquake effect loading at that level.
3. Add the sum of the vertical components of the adjusted brace strengths in compression of all
chevron bracing at levels above.
4. Subtract half of the sum of the unbalanced vertical loads (difference in vertical components of
adjusted brace strengths in compression and tension) on the beams intersecting all chevron
bracing at that level and above.
This procedure recognizes that for levels with chevron bracing, the adjusted brace strength in
compression will always control over that in tension and will enter the column below at the base of that
level. Additionally, the unbalanced vertical load from the chevron transmits shear to the beam above and
ultimately the column that works to alleviate the downward gravity and brace compression forces.
Table 6.35 Determination of Column Design Forces for Column at Gridline D4
Level
Bracea
Area
(1.2+0.2SDS)D +
0.5L + 0.2S
Brace
Angle b
Vertical Component of
Adjusted Brace Strengths
Column
Required
Strength Pu
Tensionc
Compressiond
Roof
4.5 in2
102 kips
26.6ø
126 kips
132 kips
228 kips
Story 10
4.0 in2
196 kips
45ø
177 kips
186 kips
318 kips
Story 9
4.0 in2
293 kips
45ø
177 kips
186 kips
596 kips
Story 8
4.5 in2
390 kips
45ø
199 kips
209 kips
873 kips
Story 7
4.5 in2
486 kips
45ø
199 kips
209 kips
1174 kips
Story 6
5.5 in2
583 kips
45ø
243 kips
255 kips
1474 kips
Story 5
6.0 in2
681 kips
45ø
265 kips
279 kips
1821 kips
Story 4
6.5 in2
780 kips
45ø
288 kips
302 kips
2191 kips
Story 3
none
947 kips



2660 kips
Story 2
none
1106 kips



2819 kips
aAll braces are oriented in the chevron configuration except for single diagonal at the roof level.
bMeasured from the horizontal.
c RyFyscAscsinà
d RyFyscAscsinà
1.0 kip = 4.45 kN
1.0 in = 25.4 mm
Columns are spliced at every other level to simplify erection. Column sizes are subject to revision due to
results from nonlinear response history analysis. While tension forces in the columns may not control
their design, tension demands can certainly affect the design of base plates, anchor rods drilled piers. The
design tension force at the base of this same column calculated in a similar manner (using the appropriate
load combination) is equal to 1,222 kips.
6.3.5.3 Lateral forceresisting beams. The braced frame beams were designed for gravity loads
corresponding to 1.37D + 0.5L + 0.2S, without accounting for any midspan support provided by chevron
bracing, together with earthquake loads extracted from two different plastic mechanism analyses. The
first of these mechanisms assumes both the brace(s) above and below the beam of interest have reached
their full adjusted brace strengths. The beam is then designed for its share (depending on its location
along the line of framing) of the horizontal component of the resulting story force together with the
largest drag force from the brace(s) above. In the second mechanism, the brace(s) below the beam of
interest is assumed to have reached their full adjusted brace strengths at the same time the diaphragm
reaches its design force (this mechanism requires a rough diaphragm analysis to derive collector forces).
The beam must resist the same share of the horizontal component of the diaphragm force at that line of
framing plus the largest drag force from the brace(s) above, determined by the difference between the
force in the brace(s) below minus the diaphragm force at that line of framing. In either mechanism, the
braced frame beam is also designed for any unbalanced upward component of the BRBs that would arise
in a chevron or Vbracing configuration. These mechanisms and their associated forces are illustrated in
Figure 6.37.
Figure 6.37 BRB plastic mechanism
As an example, consider the beam engaged in a chevron bracing configuration along Gridline D above the
ninth story. The shear and moment in the beam corresponding to the load combination of 1.37D + 0.5L +
0.2S, calculated by neglecting the midspan support provided by the chevron bracing below, are equal to
18 kips and 128 kipft, respectively. The brace above is a 4.5 in2 single diagonal while two 4 in.2 braces
frame into the midpoint of the beam from below in a chevron configuration. Hence the upward force at
midspan resulting from the unbalanced upward component of the chevron braces below reaching their
adjusted strengths is given by the following:
RyFyscAsc(  1)sinà = (1.36)(1.21)(38)(4)(1.05  1)(sin45ø) = 9 kips
This unbalanced upward component reduces the shear demand in the beam by P/2 = 9/2 = 5 kips and the
moment by PL/4 = (9)(30)/4 = 66 kipft. Hence the moment demand that will eventually be combined
with the critical axial demand determined by plastic mechanism analysis is:
128 kipft  66 kipft = 62 kipft
The first plastic mechanism shown in Figure 6.37 considers both the brace(s) above and below the beam
of interest to have reached their adjusted strengths. Hence, the plastic story shear below the tenth floor is
equal to:
RyFyscAsc(1 + )cosà = (1.36)(1.21)(38)(4)(1 + 1.05)(cos45ø) = 363 kips
The plastic story shear above the tenth floor is:
RyFyscAsccosà = (1.05) (1.36)(1.21)(38)(4.5)(cos26.6ø) = 264 kips
(The brace at the tenth story is a singlediagonal.)
The story force corresponding to this yielding mechanism is equal to the difference between these two
values, or 99 kips. Since the chevron braces below the tenth floor beam receive the lateral force at the
midpoint of the associated line of framing, half of this story force (attributed to inertial mass) is presumed
to come from each half of the braced frame beam, or approximately 50 kips. This 50kip force is added to
the 264kip horizontal component from the yielding singlediagonal brace above that must be dragged
through the braced frame beam to the chevron braces below, resulting in a design axial force of 314 kips
for the first plastic mechanism.
The second plastic mechanism illustrated in Figure 6.37 requires estimation of the force entering the
braced frame beam from the diaphragm at that floor in accordance with Standard Section 12.10. As
shown in Figure 6.37 this is a collector force thus the value of Fpx calculated using Standard
Equation 12.101 requires amplification by the overstrength value, 0, of 2.5; this gives 2.5 142 kips =
355 kips. This value cannot be taken as less than the minimum of 0.2SDSIwpx, which gives a value of
270 kips. (Note that the overstrength factor does not apply to this minimum, even for collectors.) Thus,
the 355 kips governs the corresponding frame force can be taken as 55 percent of this force (that is taking
1/2 adding 10 percent to account for accidental eccentricity): 195 kips. The chevron braces below the
tenthfloor braced frame beam are still assumed to have reached their yield strength, resulting in the same
363kip frame shear at that level. Hence the statically consistent force in the (now elastic) singlediagonal
brace above the tenth floor is equal to the difference between these two values, or 168 kips. Just as is
done in the calculation of the design axial force resulting from the first plastic mechanism, half of the
195kip story force (the maximum from the diaphragm) is added to the 168kip horizontal component
from the singlediagonal brace above that must be dragged through the braced frame beam to the chevron
braces below, resulting in a design axial force of 265 kips for the second plastic mechanism.
The 314 kips obtained from the first plastic mechanism (both braces above and below the beam at their
adjusted strengths) controls. Thus, the braced frame beam along Gridline D above the ninth story must be
designed for a 62 kipft moment in combination with a 314 kip axial force. The axial strength is typically
determined without accounting for the benefits of composite action with the concretefilled deck above.
Although some minor benefit can be obtained from considering the composite contribution to flexural
strength, this is often neglected for simplicity; flexural forces due to brace unbalanced loading are
typically small in the invertedV configuration they oppose gravity forces. The axial capacity of braced
frame beams is often controlled by flexuraltorsional buckling (as opposed to buckling about the weak
axis).
6.3.5.4 Third floor/low roof collector forces. Collector elements that transfer forces between the single
bay of chevron bracing in the tower to the multiple, offset bays of chevron bracing in the podium are
sized in a manner identical to the lateral forceresisting beams. The same two plastic mechanisms one
involving braces above and below the level of interest reaching their full adjusted strengths the second
involving the braces below the level of interest reaching their full adjusted strengths in conjunction with
the diaphragm delivering its maximum force to the framing in line with the braces are assumed the
forces are traced from the single bays of chevron bracing above through the collector lines to the chevron
bracing below. Story forces accumulate in the collectors and braced frame beams based on the fraction of
the full length of the line of framing represented by the particular beam section of interest. In the case of
the outofplane offset that occurs between Gridlines 2 and 3 (and 6 and 7), the horizontal component of
the adjusted chevron brace strengths above the third floor must be distributed into the diaphragm via the
adjacent collector elements, then collected by collector elements along the outer line of framing (at
Gridlines 2 or 7) to be channeled to the chevron braces in the podium levels below. As with the braced
frame beams, the axial capacity of these collector elements is based on that of the bare steel section and
usually is governed by flexuraltorsional buckling. It is acceptable to calculate the flexural capacity of the
collector elements considering composite action with the concretefilled deck above.
6.3.5.5 Connection design. According to AISC 341 Section 16.3, gussets and beamcolumn connections
must be designed for 1.1 Cmax, where Cmax is the adjusted brace strength in compression as defined in
AISC 341 Section 16.2d. Connection design is not illustrated here since this topic is more thoroughly
treated elsewhere and is not unique to BRBF systems. The connection of the 4.5 in.2 single diagonal
brace to the frame beam below it at the tenth floor would need to be designed for 1.1 RyFyscAsc =
1.1(1.05)(1.36)(1.21)(38)(4.5) = 325 kips in tension and compression based the full expected and strain
hardened brace capacity. However, it should be mentioned that a designer might consider using NRHA to
design for potentially reduced connection force demands; such a reduction would likely be limited to
establishing a lower value for the strainhardening factor for each brace.
6.3.6 Nonlinear Response History Analysis
After completing a preliminary design using threedimensional modal response spectrum analysis,
nonlinear response history analysis is performed to:
Establish brace deformation demands (and verify the adequacy of specified braces for the
application)
Determine expected drifts
Reexamine the required strength of column members in the BRBF
The braced frames and diaphragm chords and collectors, together with all primary gravity system beams
and columns, are explicitly modeled using threedimensional beamcolumn elements. Secondary gravity
framing is omitted from the nonlinear model for simplicity. The floor diaphragms are still modeled as
rigid.
The specific goals of the nonlinear response history analysis are threefold. First, even though the original
elastic design is found to comply with the drift limitations of Standard Section 12.12.1, the nonlinear
response history analysis can more accurately predict results such as story drift. Hence, the acceptability
of the design in satisfying story drift requirements is evaluated using the procedures of Standard
Section 16.2.4. Second, the ability of the BRBs to perform at the Immediate Occupancy (IO)
performance level in a design basis earthquake (DBE) event will be verified explicitly. Third, the
required strength of column elements in the BRBF system is reevaluated according to the maximum
force that can be developed by the system as permitted by AISC 34105 Section 16.5b. The use of a
nonlinear response history analysis to determine this maximum force for individual elements is justified
in the provisions of Standard Section 12.4.3.1. Specifically, the exception to this section permits the
determination of the maximum force that can develop in the element by a rational, plastic mechanism
analysis or nonlinear response analysis utilizing realistic expected values of material strengths. Standard
Section 16.2 then defines the requirements for nonlinear response history analysis. In this design
example, only the columns in the BRBF system are examined. This is due to the limited savings potential
of economizing the relatively small number of BRBF beam and collector elements in the structure that
have already been reasonably sized (attributable to the absence of large BRB elements) using a rational
plastic mechanism analysis. The designer of a taller building with bulky BRB elements should probably
consider using the same procedure to potentially reduce design force demands on the BRBF beams and
collectors as well as the columns.
A second, separate threedimensional building model is assembled in the PERFORM program its
similarity to the ETABS model used for the elastic design is confirmed by comparing fundamental
periods and loads.
6.3.6.1 Design ground motions. In Chapter 3, risktargeted maximum considered earthquake (MCE)
response spectra are determined in accordance with Standard Section 11.4 using NGA attenuation
relations. Seven pairs of time histories are selected and scaled to be consistent with the event magnitudes,
fault distances source mechanisms controlling the MCE spectrum for the Seattle, Washington, hospital
site. The base ground motions in the suite are scaled to the MCE response spectrum so as to satisfy the
requirements of Standard Section 16.1.3.2 for periods between 0.18 and 4.95 seconds. The translational
structural periods for the hospital facility are found to be approximately 2.3 seconds, so the period range
of interest is from 0.2 2.3 = 0.46 seconds to 1.5 2.3 = 3.45 seconds, which is narrower than the
preliminary range used in the ground motion selection and scaling. Designlevel ground motions are
obtained by multiplying MCE motions by 2/3 per Standard Section 16.2.3.
6.3.6.2 Basis of nonlinear design. In keeping with the intent of the building code to protect essential
facilities in a seismic event, the hospital should perform at the IO performance level under ground
shaking corresponding to the DBE. As is traditionally the case in elastic designs, an explicit performance
check at the MCE is not done here. The building is assumed to meet Life Safety (LS) performance
objectives at the MCE if it meets IO performance criteria at the DBE.
At the present time, there are three international providers of BRBs: CoreBrace and StarSeismic in the
United States Nippon Steel in Japan. In the preliminary design stage, it is often the case that any one of
these three brace manufacturers may ultimately be chosen as the supplier. Thus, generic BRB properties
may be assumed until the later stages of a project. However, a specific BRB supplier must be selected to
accurately model actual brace behavior in the PERFORM NRHA model.
BRBs in the NRHA are modeled using expected properties based on test results provided by CoreBrace;
acceptance criteria are based on Section 2.8.3 of ASCE 41 for deformationcontrolled elements. There
are three different types of bracetogusset connections used for BRBs: bolted, welded pinned. However,
generic ACME braces are used throughout the structure hence the nature of the bracetogusset
connection is not considered. All other structural elements are modeled using expected properties.
Because the lateral beams and columns are connected by simple pin connections in this example (see
Section 6.3.2.1), the beams will not require consideration of inelastic behavior. Unless some column
element sizes are reduced based on NRHA results, inelastic column actions are not likely, although it is
possible that uneven (or highermode) story drifts will result in inelastic flexural demands.
The initial gravity load condition is 1.0D + 0.25L per Standard Section 16.2.3.
6.3.6.3 BucklingRestrained Brace calibration. In order to capture the nonlinear BRB behavior as
accurately as possible in the NRHA computer model, brace properties are calibrated to match test results
provided by CoreBrace. Typically, a number of calibrations are performed on a range of different brace
sizes. Critical modeling parameters are then interpolated between (or extrapolated from) those
established for a brace or braces of similar size. For illustration purposes, one calibration is shown and
modeling parameters are matched to corresponding values for the flat core plate test specimen. (Note that
the specific series of BRBs tested possess a postyield modulus of elasticity that is unusually stiff.)
Data to which critical brace parameters are calibrated is obtained from a standard cyclic testing protocol
that is in line with the requirements of AISC 341 Appendix T and ASCE 41 Section 2.8.3. Elastic
displacement components in both the connection region and the nonyielding brace segments (the larger
elastic bar segment in PERFORM) of the BRB specimens are subtracted out of measured displacements
in the test data. As a result, the inelastic core plate s forcedisplacement behavior is isolated to facilitate
calibration. The result of the calibration is a backbone curve that reasonably envelopes the observed
cyclic test hysteretic behavior.
BRB elements typically are modeled in nonlinear computer analysis software as three components in
series: a stiff connection region, an elastic bar segment an inelastic yielding segment. The workpointto
workpoint length of the BRB element in the threedimensional computer model must be divided
reasonably into these three components. Thus, the stiff connection zone size and relative stiffness are
determined by a rough gusset plate design based on a representative brace size and geometry in the
building. The elastic bar segment s length and crosssectional area are set such that their proportions
relative to the inelastic yielding segment remain identical to those of the test specimen. The remainder of
this section is devoting to developing modeling parameters for the inelastic yielding portion of the BRB
element.
One key modeling parameter for the inelastic portion of the BRB elements is its initial elastic stiffness,
Ko. This value is simply set to equal AscE / Ly, where Ly is the length of the yielding segment set to equal
the same proportion of the brace length outside the connection region as the test specimen. Next, the
designer must select a postyield stiffness, Kf, for the inelastic portion of the backbone curve. This value
is chosen such that the postyield slope of the hysteresis loops obtained from the computer model is
similar to that in the test data is typically a percentage of the initial stiffness, Ko. For the hospital building
in this example, Kf = 1.5 percent of Ko to match the observed hysteretic behavior of the test specimen.
Other key modeling inputs are the strainhardening and compression overstrength factors that define the
ultimate strength of the BRB elements. Each brace in the hospital computer analysis model is assigned
strainhardening ( ) and compression overstrength ( ) factors equal to 1.63 and 1.05, respectively. These
values are selected to match the full hardened strength of the test specimen in tension and compression
and are different from the generic values assumed earlier for the plastic mechanism analysis used to
design beams in columns in the BRBF. Because the compression overstrength factor does not equal 1.0,
the BRB behavior is not symmetric in tension and compression. The material strength of the braces in the
model is set to equal the minimum specified strength for this project (i.e., RyFy = 38 ksi). Alternatively,
one could use the actual material strength as determined by coupon test of BRB specimens tested
specifically for that particular project.
All nonlinear analysis software programs require an ultimate deformation or strain value corresponding to
the BRB having reached its fullyhardened strength as an input in the inelastic BRB component s
properties. This figure was set such that the strain matched that observed in the test at full compression
hardening. Additionally, the program requires information about the rate of isotropic hardening between
cycles. In the nonlinear analysis software used for this design example, the rate of isotropic hardening is
captured by inputting the maximum brace deformation corresponding to the average of the initial BRB
yield strength and its strength after full hardening (this cyclic hardening parameter can also be defined in
terms of accumulated deformations but is not done so here). To match the vertical progression of
hysteretic behavior (i.e., hardening between progressive loading cycles) observed in the test specimen,
this hardening parameter was set to 2/3 of the deformation from initial yield to fullyhardened strength.
Finally, the software must know at what point the deformation in the nonlinear BRB component has
exceeded its capacity. Since the BRBs in this example are expected to perform well below their capacity
and this value was never reached during the standard testing protocol, an artificially high number was
selected for this input individual brace performance was subject to review as outlined in Section 6.3.6.3.1.
The above parameters are sufficient to define a bilinear elasticplastic backbone curve for the BRB
elements in the computer analysis model. The hysteretic behavior of the BRB component matching the
test specimen extracted from the computer model is compared with the experimental test data in
Figure 6.38. One can see that the backbone curve modeling parameters accurately capture the observed
experimental hysteresis properties for this size component. With some additional effort, a trilinear
backbone curve can also be calibrated to the test data to better capture the rounding of the actual BRB
hysteretic loops. The trilinear curve considers a higher component stiffness value just after yield before
the ultimate postyield stiffness value, Kf, prevails. This third stiffness value, together with the force
deformation point at which the ultimate postyield stiffness (Kf) takes over, must be calibrated to the
test data as well. Figure 6.39 shows the inelastic BRB component s hysteretic behavior as modeled
using the trilinear backbone curve together with the experimental test data to which it is calibrated.
Subsequent analysis results are based on BRB inelastic components modeled using the trilinear backbone
curve because the additional accuracy of the trilinear backbone curve can be realized without causing
excessive analysis run times for this example building.
Figure 6.38 Bilinear BRB calibration (Asc = 12 in2)
(1.0 in. = 25.4 mm; 1.0 kip = 4.45 kN)
Figure 6.39 Trilinear BRB calibration (Asc = 12 in2)
(1.0 in. = 25.4 mm; 1.0 kip = 4.45 kN)
6.3.6.4 Results.
6.3.6.4.1 BRB strains. One goal of the NRHA is to confirm the ability of the BRBs to perform at the IO
performance level in a DBE event. The acceptance criteria for BRB strain, as dictated by ASCE 41
Section 2.8.3, can be seen in Figure 6.310 (overlaid on the corresponding test results). No permanent,
visible damage was observed during the standard experimental testing protocol used to calibrate the BRB
elements the test was terminated at or near Point 2 as defined in the Type 1 and Type 2 component force
versus deformation curves for deformationcontrolled actions in ASCE 41 Section 2.4.4.3. Consequently,
the IO, LS Collapse Prevention (CP) acceptance criteria are equal to 0.67 0.75, 0.75 1.0 times the
deformation at Point 2 on the component force versus deformation curves, respectively. Observe that the
limiting BRB strains for IO performance in tension and compression are 6.64 y / Ly = 6.64 0.00131 =
0.008704 and 6.47 y / Ly = 6.47 0.00131 = 0.008482, in turn.
Figure 6.310 BRB strain acceptance criteria
The ratio of maximum inelastic deformation demand = / y observed along a subset of the 92 total
BRB elements during each of the seven ground motion time histories to the relevant IO performance point
in tension or compression is shown in Table 6.36. Braces for which results are presented are chosen to
represent BRB elements in all ten levels of the structure. As permitted by Standard Section 16.2.4 for
analyses including at least seven ground motion pairs, the average BRB max / IO value across the seven
earthquake records is calculated for each brace. In assessing the performance of the entire structure, the
maximum of these 92 average BRB inelastic deformation demand values is extracted and compared with
the relevant IO performance point in tension or compression. Table 6.36 shows that this critical max/ IO
value for the hospital structure is equal to 0.566, indicating acceptable performance of the BRB elements
in the DBE event according to the methodology set forth in Standard Section 16.2.4. The procedure set
forth in the Standard may lead to designs that fail a criterion in some element or measure for every
ground motion but still pass the criteria on average.
Table 6.36 Ratio of Maximum BRB Inelastic Deformation Demand to Immediate Occupancy (IO)
Performance Limit ( max / IO)
BRB ID
Record ID
Average
1
2
3
4
5
6
7
1
0.245
0.350
0.173
0.159
0.125
0.250
0.486
0.255
2
0.331
0.472
0.179
0.267
0.201
0.282
0.732
0.351
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
16
0.187a
0.189a
0.154
0.185
0.266
0.294
0.215
0.212
17
0.574
0.258
0.319
0.251
0.350
0.450
0.314
0.359
18
0.770
0.314
0.321
0.217
0.313
0.369
0.316
0.374
19
0.956
0.537
0.542
0.245
0.343
0.615
0.373
0.515
20
1.082
0.532
0.588
0.268
0.329
0.753
0.310
0.551
21
1.001
0.392
0.574
0.351
0.231
0.609
0.465
0.518
22
0.875
0.402
0.520
0.264
0.357
0.718
0.534
0.524
23
0.545
0.414
0.371
0.258
0.399
0.644
0.455
0.440
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
89
0.642
1.064a
0.572a
0.254a
0.695
0.478
0.263
0.566b
90
0.605
0.959a
0.328a
0.180
0.425
0.262
0.215
0.424
91
0.518
0.592a
0.382
0.371
0.347
0.212
0.181
0.372
92
0.241
0.366
0.364
0.631
0.368
0.248
0.410
0.375
Max
1.086
1.286
0.834
0.631
0.857
0.759
0.792
0.566b
a All max / IO values for the individual braces are controlled by the compression strain limit except those denoted
by the superscript a (for which the tension strain limit controls).
b Note that the 0.566 structure usage ratio is equal to the maximum of the average max / IO values across the
seven ground motion pairs for each BRB element, not the average of the maximum max / IO values for each of
the seven ground motion pairs across all 92 BRB elements.
6.3.6.4.2 Drift assessment. Although seismic drift was preliminarily examined in Section 6.3.4.3 to
ensure the design possessed reasonable stiffness before modeling its nonlinear behavior, conformance
with story drift limits is assessed using the procedures of Standard Section 16.2. The maximum story
drift ratio in each translational direction was extracted from the threedimensional PERFORM model for
each of the seven ground motion pairs. As before, story drift was examined at the building corners rather
than the center of mass.
Once the maximum story drift ratio at every story and corner of the floorplate is identified in each of the
seven time histories, the resulting seven values for each story are averaged (as allowed by Standard
Section 16.2.4) the maximum average story drift ratio is identified to assess compliance with the
allowable story drift, which Standard Section 16.2.4.3 permits to be increased by 25 percent relative to
the drift limit specified in Section 12.12.1 in the context of nonlinear response history analysis. Thus, the
relevant story drift limit for nonlinear response history analysis is (1 + 0.25)(0.010hsx) = 0.0125hsx.
Figure 6.311 shows, for each story, the story drift ratios in the longitudinal direction for each time
history, the average of all seven the allowable story drift ratio. (Also shown are the analysis results and
drift limit for MRSA, which are discussed below.) The controlling story drift for the NRHA occurs in the
seventh story, where it is about 93 percent of the allowable story drift. Table 6.37 illustrates how this
value is calculated using drift ratios at the critical corner of the floorplate. Drifts in the transverse
direction are somewhat smaller.
Figure 6.311 Longitudinal story drift ratios (and drift limits)
Table 6.37 Maximum Longitudinal Story Drift Ratio at Critical Corner of Building Floorplate
Level
Record ID
Average
1
2
3
4
5
6
7
Roof
0.0084
0.0088
0.0076
0.0108
0.0062
0.0060
0.0083
0.0080
Story 10
0.0064
0.0079
0.0079
0.0108
0.0067
0.0052
0.0062
0.0073
Story 9
0.0095
0.0164
0.0082
0.0073
0.0077
0.0056
0.0062
0.0087
Story 8
0.0109
0.0212
0.0108
0.0069
0.0084
0.0063
0.0055
0.0100
Story 7
0.0110
0.0225
0.0145
0.0090
0.0116
0.0085
0.0045
0.0117a
Story 6
0.0096
0.0166
0.0110
0.0070
0.0109
0.0072
0.0041
0.0095
Story 5
0.0084
0.0098
0.0068
0.0049
0.0108
0.0066
0.0042
0.0074
Story 4
0.0063
0.0062
0.0047
0.0050
0.0131
0.0047
0.0039
0.0063
Story 3
0.0057
0.0045
0.0054
0.0058
0.0132
0.0038
0.0050
0.0062
Story 2
0.0048
0.0032
0.0051
0.0047
0.0084
0.0035
0.0054
0.0050
a Note that the 0.0117 structure story drift ratio is equal to the maximum of the average story drift ratio values
across the seven ground motion pairs for each story, not the average of the maximum story drift ratio values for
each of the seven ground motion pairs across all ten stories.
The maximum story drift calculated earlier using MRSA reached a maximum value at the roof level equal
to 89 percent of the 0.010hsx limit prescribed in Standard Section 12.12.1. The magnitude of this
maximum was similar in both principal building axes. The design story drifts obtained from the MRSA
were also found to increase from story to story up the height of the structure. Contrast the preliminary
elastic MRSA drift results with those just presented from the NRHA. First, the maximum story drift no
longer always occurs at the uppermost level of the building; the specific location of the maximum varies
depending on the ground motion. Second, the maximum story drift is no longer consistent between the
two primary structural axes. This is an important point and is a direct result of the way the ground motion
pairs were applied to the structure. While not required by the Standard, some engineers elect to reanalyze
the structure with the ground motions rotated, in order to investigate sensitivity to groundmotion
orientation; this results in a total of 14 analyses, with a corresponding increase in effort. However, a
detailed discussion of this and other ground motion issues is beyond the scope of this design example.
Finally perhaps most importantly, the maximum story drift from the NRHA, 0.0117hsx, is 31 percent
higher than that from the MRSA, 0.89 0.010hsx = 0.0089hsx.
The results in this section highlight an important misconception of NRHA in general. There is no
guarantee of economizing a design with respect to the required strength or stiffness of a frame simply by
performing a NRHA. Rather, when executed correctly, a NRHA simply assures a more accurate
representation of actual structural performance in a particular seismic event. This increase in the accuracy
of seismic response parameters can actually increase the required frame strength or stiffness in some
instances.
6.3.6.4.3 Column design forces. All BRBF columns were initially designed in Section 6.3.5.2 to resist
the vertical component of the adjusted strengths of any braces above, using capacity design principles and
generic values of the brace material overstrength (Ry), strainhardening ( ) compression overstrength ( )
parameters. Thus, the lateral columns are expected to remain nominally elastic in the DBE event. More
realistic expected BRB behavior specific to a particular BRB product line and supplier is modeled in
Section 6.3.6.2 based on experimental test data. The required strengths of the columns in the BRBF as
specified in AISC 341 Section 16.5b are subject to revision based on results from the NRHA, as is
permitted in the exception to Standard Section 12.4.3.1. To this end, Table 6.38 shows the maximum
compressive axial force attained at every story in the BRBF column at Gridline D/4 during each of the
seven time history analyses. This force represents the summation of the gravity load prescribed in
Standard Section 16.2.3 (1.0D + 0.25L) plus the additional force imposed by the relevant earthquake time
history pair. As above, compression forces are the most critical for column design because column
strengths are governed by compression buckling rather than yielding.
The model included column potential hinges with axialmoment interaction relationships determined from
ASCE 41. Inelastic rotations were limited to IO values per that standard. However, virtually no inelastic
rotation was recorded in the analyses.
Table 6.38 also identifies the overall maximum compression force in the column at every story across all
seven ground motion pairs, together with the column design force determined by plastic mechanism
analysis in Section 6.3.5.2. While Standard Section 16.2.4 permits the use of average member forces in
determining design values with at least seven ground motions, maximum member force values are
selected for design of the columns due to their critical role in sustaining the vertical loadcarrying
capacity of the structure. Even when using maximum values of member forces extracted from the time
histories, substantial savings in column design forces and hence steel tonnage are facilitated by NRHA in
this example.
Table 6.38 Comparison of Column Design Forces from NRHA and Plastic Analysis for Column at
Gridline D/4
Level
Maximum Compression Force (kips)
Design Axial Force QE
(kips)
Percent
Reduction
in Design
Forcec
Record ID
1
2
3
4
5
6
7
NRHAa
Plasticb
Roof
152
154
152
156
148
155
168
168
228
26%
Story 10
214
216
216
219
211
217
231
231
318
27%
Story 9
404
412
393
393
382
397
416
416
596
30%
Story 8
603
616
589
567
571
578
590
616
873
29%
Story 7
812
842
800
743
774
768
781
842
1174
28%
Story 6
986
1068
1018
948
996
948
972
1068
1474
28%
Story 5
1198
1320
1264
1183
1214
1113
1145
1320
1821
28%
Story 4
1414
1566
1504
1430
1421
1276
1264
1566
2191
29%
Story 3
1750
1851
1780
1707
1651
1486
1421
1851
2660
30%
Story 2
1861
1965
1894
1818
1764
1598
1535
1965
2819
30%
aMaximum of individual maxima from the seven time histories.
bAs determined in Table 6.35.
cRelative to that determined using plastic analysis.
1.0 kip = 4.45 kN1.0 in = 25.4 mm
In addition to these significant reductions in column design forces, the same procedure reveals a design
tension uplift force at this same column of 336 kips, which is just 28 percent of the comparable 1,222kip
force obtained from plastic analysis. This force reduction enables tremendous savings in the design of
foundation elements such as base plates, anchor rods drilled piers.
With the possible exception of the foundation elements just mentioned, despite the fact that the column
design forces have been sharply reduced relative to those obtained from the plastic mechanism analysis in
Section 6.3.5.2, none of the column sizes are actually reduced in this example. This is because the
structure is currently very close to the allowable story drift limit. In other words, the NRHA reveals that
stiffness, not strength, governs the design of the BRBF examined here. A prudent designer might
consider enlarging select braces in conjunction with reducing column sizes as allowed by the NRHA
design forces in Table 6.38 to increase cost savings on the project, depending on the relative prices of
BRBs and structural steel. However, the tradeoff between BRB and column stiffness would have to
preserve the overall stiffness of the frame in order to ensure the allowable drift limit is still satisfied. In
particular, it may be possible to increase the brace size in Story 7 (and possibly Stories 6 and 8) while
reducing the column size considerably.
6.3.6.4.4 Bracetogusset connection design. As mentioned earlier, bracetogusset connections for
BRBs take on three different forms: bolted, welded pinned. The specific nature of this connection is not
considered in this design example. However, the possible additional benefit of economizing on material
by using NRHA to determine connection design forces will be demonstrated. As shown in
Section 6.3.5.5, the connection of the single diagonal brace below the roof to its gusset would need to be
designed for 325 kips in tension and compression. Using the maximum BRB force results from NRHA
(obtained in the same manner that column design forces were determined), this design value can be
reduced to 210 kips in tension and 213 kips in compression, representing a reduction of approximately
35 percent in both cases. This corresponds to utilizing a lower strainhardening factor, corresponding
to a more refined method of establishing deformation demands. It should be noted that this reduction is
not explicitly allowed by AISC 341; therefore, it would constitute alternate means and methods and be
subject to approval by the building official.
6.3.6.4.5 Summary of NRHA goals. Before concluding, the exact extent to which NRHA was used in
this design example merits emphasis one last time. Based on the fundamental period of the structure, the
minimum level of sophistication required for its seismic lateral analysis is an elastic MRSA. Thus, in
keeping with code requirements, a threedimensional model of the structure was created the BRBs were
designed to accommodate force demands determined by MRSA. BRBF beams, columns collectors were
then designed using a rational plastic mechanism analysis with the assumption that any earthquake load
effect is determined from the full adjusted brace strengths in tension and compression. This example then
goes beyond elastic analysis and relies on a NRHA for three additional aspects of the design. First,
NRHA is used to verify that the strains in the BRBs designed to MRSA forces indeed satisfy the IO
performance requirements in DBElevel shaking using the methodology in ASCE 41 Section 2.8.3.
Second, compliance with the allowable story drift limit was evaluated in the context of NRHA and the
provisions of Standard Section 16.2.4. Finally, as a demonstration, NRHA was used to establish reduced
design axial forces for the BRBF columns using the exception to Standard Section 12.4.3.1 to justify the
use of NRHA in determining their required strength as stipulated in AISC 341 Section 16.5b. However,
because the structure was found to be at the allowable drift limit as designed using plastic mechanism
analysis, the potential savings offered by these reduced column design forces went unrealized.
Designers could elect to place an even greater emphasis on NRHA during the design of such a structure.
For example, design forces for BRBF beams and collectors might also be set using NRHA, as was
illustrated for the columns, rather than by plastic mechanism analysis. Depending on the relative prices of
BRBs and structural steel, designers could economize the design by enlarging select BRB elements
(relative to what MRSA finds is necessary) in order to reduce some column sizes. Justifying such a
design would require iterative NRHA runs to ensure the structure s overall stiffness is such that allowable
drift limits are not exceeded. Finally, the entire structure might be designed using NRHA exclusively.
BRB sizes would be established not by MRSA but instead by NRHA and hence a different force
distribution that takes into account the building s inelastic characteristics.
7
Reinforced Concrete
By Peter W. Somers, S.E.
Originally developed by Finley A. Charney, PhD, P.E.
Contents
7.1 SEISMIC DESIGN REQUIREMENTS 7
7.1.1 Seismic Response Parameters 7
7.1.2 Seismic Design Category 8
7.1.3 Structural Systems 8
7.1.4 Structural Configuration 9
7.1.5 Load Combinations 9
7.1.6 Material Properties 10
7.2 DETERMINATION OF SEISMIC FORCES 11
7.2.1 Modeling Criteria 11
7.2.2 Building Mass 12
7.2.3 Analysis Procedures 13
7.2.4 Development of Equivalent Lateral Forces 13
7.2.5 Direction of Loading 19
7.2.6 Modal Analysis Procedure 20
7.3 DRIFT AND PDELTA EFFECTS 21
7.3.1 Torsion Irregularity Check for the Berkeley Building 21
7.3.2 Drift Check for the Berkeley Building 23
7.3.3 Pdelta Check for the Berkeley Building 27
7.3.4 Torsion Irregularity Check for the Honolulu Building 29
7.3.5 Drift Check for the Honolulu Building 29
7.3.6 PDelta Check for the Honolulu Building 31
7.4 STRUCTURAL DESIGN OF THE BERKELEY BUILDING 32
7.4.1 Analysis of FrameOnly Structure for 25 Percent of Lateral Load 33
7.4.2 Design of Moment Frame Members for the Berkeley Building 37
7.4.3 Design of Frame 3 Shear Wall 60
7.5 STRUCTURAL DESIGN OF THE HONOLULU BUILDING 66
7.5.1 Compare Seismic Versus Wind Loading 66
7.5.2 Design and Detailing of Members of Frame 1 69
In this chapter, a 12story reinforced concrete office building with some retail shops on the first floor is
designed for both high and moderate seismic loading. For the more extreme loading, it is assumed that
the structure will be located in Berkeley, California and for the moderate loading, in Honolulu, Hawaii.
The basic structural configuration for both locations is shown in Figures 71 and 72, which show a
typical floor plan and building section, respectively. The building has 12 stories above grade and one
basement level. The typical bays are 30 feet long in the northsouth (NS) direction and either 40 or 20
feet long in the eastwest (EW) direction.
The main gravity framing system consists of seven continuous 30foot spans of pan joists. These joists
are spaced at 36 inches on center and have an average web thickness of 6 inches and a depth below slab
of 16 inches. Due to fire code requirements, a 4inchthick floor slab is used, giving the joists a total
depth of 20 inches. The joists are supported by concrete beams running in the EW direction. The
building is constructed of normalweight concrete.
Concrete walls are located around the entire perimeter of the basement level.
For both locations, the seismic forceresisting system in the NS direction consists of four 7bay moment
resisting frames. At the Berkeley location, these frames are detailed as special momentresisting frames.
Due to the lower seismicity and lower demand for system ductility, the frames of the Honolulu building
are detailed as intermediate momentresisting frames as permitted by ASCE 7.
In the EW direction, the seismic forceresisting system for the Berkeley building is a dual system
composed of a combination of moment frames and framewalls (walls integrated into a momentresisting
frame). Along Grids 1 and 8, the frames have five 20foot bays. Along Grids 2 and 7, the frames consist
of two exterior 40foot bays and one 20foot interior bay. At Grids 3, 4, 5 and 6, the interior bay consists
of shear walls infilled between the interior columns. The exterior bays of these frames are similar to
Grids 2 and 7. For the Honolulu building, the structural walls are not necessary, so EW seismic
resistance is supplied by the moment frames along Grids 1 through 8. The frames on Grids 1 and 8 are
fivebay frames and those on Grids 2 through 7 are threebay frames with the exterior bays having a 40
foot span and the interior bay having a 20foot span. Hereafter, frames are referred to by their gridline
designation (e.g., Frame 1 is located on Grid 1).
The foundation system is not considered in this example, but it is assumed that the structure for both the
Berkeley and Honolulu locations is founded on very dense soil (shear wave velocity of approximately
2,000 feet per second).
Figure 71 Typical floor plan of the Berkeley building; the Honolulu building is
similar but without structural walls (1.0 ft = 0.3048 m)
Figure 72 Typical elevations of the Berkeley building; the Honolulu building is
similar but without structural walls (1.0 ft = 0.3048 m)
The calculations herein are intended to provide a reference for the direct application of the design
requirements presented in the 2009 NEHRP Recommended Provisions (hereafter, the Provisions) and its
primary reference document ASCE 705 Minimum Design Loads for Buildings and Other Structures
(hereafter, the Standard) and to assist the reader in developing a better understanding of the principles
behind the Provisions and ASCE 7.
Because a single building configuration is designed for both high and moderate levels of seismicity, two
different sets of calculations are required. Instead of providing one full set of calculations for the
Berkeley building and then another for the Honolulu building, portions of the calculations are presented in
parallel. For example, the development of seismic forces for the Berkeley and Honolulu buildings are
presented before structural design is considered for either building. The design or representative elements
then is given for the Berkeley building followed by the design of the Honolulu building. Each major
section (development of forces, structural design, etc.) is followed by discussion. In this context, the
following portions of the design process are presented in varying amounts of detail for each structure:
1. Seismic design criteria
2. Development and computation of seismic forces
3. Structural analysis and drift checks
4. Design of structural members including typical beams, columns and beamcolumn joints in Frame 1;
and for the Berkeley building only, the design of the shear wall on Grid 3
In addition to the Provisions and the Standard, ACI 31808 is the other main reference in this example.
Except for very minor exceptions, the seismic forceresisting system design requirements of ACI 318
have been adopted in their entirety by the Provisions. Cases where requirements of the Provisions, the
Standard and ACI 318 differ are pointed out as they occur. In addition to serving as a reference standard
for seismic design, the Standard is also cited where discussions involve gravity loads, live load reduction,
wind loads and load combinations.
The following are referenced in this chapter:
ACI 318 American Concrete Institute. 2008. Building Code Requirements and Commentary for
Structural Concrete.
ASCE 7 American Society of Civil Engineers. 2005. Minimum Design Loads for Buildings and
Other Structures.
ASCE 41 American Society of Civil Engineers. 2006. Seismic Rehabilitation of Existing
Buildings, including Supplement #1.
Moehle Moehle, Jack P., Hooper, John D and Lubke, Chris D. 2008. Seismic design of
reinforced concrete special moment frames: a guide for practicing engineers, NEHRP
Seismic Design Technical Brief No. 1, produced by the NEHRP Consultants Joint
Venture, a partnership of the Applied Technology Council and the Consortium of
Universities for Research in Earthquake Engineering, for the National Institute of
Standards and Technology, Gaithersburg, MD., NIST GCR 89171
The structural analysis for this chapter was carried out using the ETABS Building Analysis Program,
version 9.5, developed by Computers and Structures, Inc., Berkeley, California. Axialflexural
interaction for column and shear wall design was performed using the PCA Column program, version 3.5,
created and developed by the Portland Cement Association.
7.1 SEISMIC DESIGN REQUIREMENTS
7.1.1 Seismic Response Parameters
For Berkeley, California, the short period and onesecond period spectral response acceleration
parameters SS and S1 are 1.65 and 0.68, respectively. For the very dense soil conditions, Site Class C is
appropriate as described in Standard Section 20.3. Using SS = 1.65 and Site Class C, Standard
Table 11.41 lists a short period site coefficient, Fa, of 1.0. For S1 > 0.5 and Site Class C, Standard Table
11.42 gives a velocity based site coefficient, Fv, of 1.3. Using Standard Equation 11.41 and 11.42, the
adjusted maximum considered spectral response acceleration parameters for the Berkeley building are:
SMS = FaSS = 1.0(1.65) = 1.65
SM1 = FvS1 = 1.3(0.68) = 0.884
The design spectral response acceleration parameters are given by Standard Equation 11.43 and 11.44:
SDS = 2/3 SMS = 2/3 (1.65) = 1.10
SD1 = 2/3 SM1 = 2/3 (0.884) = 0.589
The transition period, Ts, for the Berkeley response spectrum is:
Ts is the period where the horizontal (constant acceleration) portion of the design response spectrum
intersects the descending (constant velocity or acceleration inversely proportional to T) portion of the
spectrum. It is used later in this example as a parameter in determining the type of analysis that is
required for final design.
For Honolulu, the shortperiod and onesecond period spectral response acceleration parameters are 0.61
and 0.18, respectively. For Site Class C soils and interpolating from Standard Table 11.41, the Fa is 1.16
and from Standard Table 11.41, the interpolated value for Fv is 1.62. The adjusted maximum considered
spectral response acceleration parameters for the Honolulu building are:
SMS = FaSS = 1.16(0.61) = 0.708
SM1 = FvS1 = 1.62(0.178) = 0.288
and the design spectral response acceleration parameters are:
SDS = 2/3 SMS = 2/3 (0.708) = 0.472
SD1 = 2/3 SM1 = 2/3 (0.288) = 0.192
The transition period, Ts, for the Honolulu response spectrum is:
7.1.2 Seismic Design Category
According to Standard Section 1.5, both the Berkeley and the Honolulu buildings are classified as
Occupancy Category II. Standard Table 11.51 assigns an occupancy importance factor, I, of 1.0 to all
Occupancy Category II buildings.
According to Standard Tables 11.61 and 11.62, the Berkeley building is assigned to Seismic Design
Category D and the Honolulu building is assigned to Seismic Design Category C.
7.1.3 Structural Systems
The seismic forceresisting systems for both the Berkeley and the Honolulu buildings consist of moment
resisting frames in the NS direction. EW loading is resisted by a dual framewall system in the
Berkeley building and by a set of momentresisting frames in the Honolulu building. For the Berkeley
building, assigned to Seismic Design Category D, Standard Table 12.21 requires that all concrete
momentresisting frames be designed and detailed as special moment frames. Similarly, Standard Table
12.21 requires shear walls in dual systems to be detailed as special reinforced concrete shear walls. For
the Honolulu building assigned to Seismic Design Category C, Standard Table 12.21 permits the use of
intermediate moment frames for all building heights.
Standard Table 12.21 provides values for the response modification coefficient, R, the system
overstrength factor, ê0 and the deflection amplification factor, Cd, for each structural system type. The
values determined for the Berkeley and Honolulu buildings are summarized in Table 71.
Table 71 Response Modification, Overstrength and Deflection Amplification Coefficients for
Structural Systems Used
Location
Response
Direction
Building Frame Type
R
ê0
Cd
Berkeley
NS
Special moment frame
8
3
5.5
EW
Dual system incorporating special moment
frame and special shear wall
7
2.5
5.5
Honolulu
NS
Intermediate moment frame
5
3
4.5
EW
Intermediate moment frame
5
3
4.5
For the Berkeley building dual system, Standard Section 12.2.5.1 requires that the moment frame portion
of the system be designed to resist at least 25 percent of the total seismic force. As discussed below, this
requires that a separate analysis of a frameonly system be carried out for loading in the EW direction.
7.1.4 Structural Configuration
Based on the plan view of the building shown in Figure 71, the only potential horizontal irregularity is a
Type 1a or 1b torsional irregularity (Standard Table 12.31). While the actual presence of such an
irregularity cannot be determined without analysis, it appears unlikely for both the Berkeley and the
Honolulu buildings because the lateral forceresisting elements of both buildings are distributed evenly
over the floor. However, this will be determined later.
As for the vertical irregularities listed in Standard Table 12.32, the presence of a soft or weak story
cannot be determined without analysis. In this case, however, the first story is suspect, because its height
of 18 feet is well in excess of the 13foot height of the story above. However, it is assumed (but verified
later) that a vertical irregularity does not exist.
7.1.5 Load Combinations
The combinations of loads including earthquake effects are provided in Standard Section 12.4. Load
combinations for other loading conditions are in Standard Chapter 2.
For the Berkeley structure, the basic strength design load combinations that must be considered are:
1.2D + 1.6L (or 1.6Lr)
1.2D + 0.5L ñ 1.0E
0.9D ñ 1.0E
In addition to the combinations listed above, for the Honolulu building wind loads govern the design of a
portion of the building (as determined later), so the following strength design load combinations should
also be considered:
1.2D + 0.5L ñ 1.6W
0.9D ñ 1.6W
The load combination including only 1.4 times dead load will not control for any condition in these
buildings.
In accordance with Standard Section 12.4.2 the earthquake load effect, E, be defined as:
where gravity and seismic load effects are additive and
where the effects of seismic load counteract gravity.
The earthquake load effect requires the determination of the redundancy factor, , in accordance with
Standard Section 12.3.4. For the Honolulu building (Seismic Design Category C), = 1.0 per Standard
Section 12.3.4.1.
For the Berkeley building, must be determined in accordance with Standard Section 12.3.4.2. For the
purpose of the example, the method in Standard Section 12.3.4.2, Method b, will be utilized. Based on
the preliminary design, it is assumed that = 1.0 because the structure has a perimeter moment frame and
is assumed to be regular based on the plan layout. As discussed in the previous section, this will be
verified later.
For the Berkeley building, substituting E and with taken as 1.0, the following load combinations must
be used for seismic design:
(1.2 + 0.2SDS)D + 0.5L ñ QE
(0.9  0.2 SDS)D ñ QE
Finally, substituting 1.10 for SDS, the following load combinations must be used:
1.42D + 0.5L ñ QE
0.68D ñ QE
For the Honolulu building, substituting E and with taken as 1.0, the following load combinations must
be used for seismic design:
(1.2 + 0.2SDS)D + 0.5L ñ QE
(0.9  0.2SDS)D ñ QE
Finally, substituting 0.472 for SDS, the following load combinations must be used:
1.30D + 0.5L ñ QE
0.80D ñ QE
The seismic load combinations with overstrength given in Standard Section 12.4.3.2 are not utilized for
this example because there are no discontinuous elements supporting stiffer elements above them and
collector elements are not addressed.
7.1.6 Material Properties
For the Berkeley building, normalweight concrete of 5,000 psi strength is used everywhere (except as
revised for the lower floor shear walls as determined later). All reinforcement has a specified yield
strength of 60 ksi. As required by ACI 318 Section 21.1.5.2, the longitudinal reinforcement in the
moment frames and shear walls either must conform to ASTM A706 or be ASTM A615 reinforcement, if
the actual yield strength of the steel does not exceed the specified strength by more than 18 ksi and the
ratio of actual ultimate tensile stress to actual tensile yield stress is greater than 1.25.
The Honolulu building also uses 5,000 psi concrete and ASTM A615 Grade 60 reinforcing steel. ASTM
706 reinforcing is not required for an intermediate moment frame.
7.2 DETERMINATION OF SEISMIC FORCES
The determination of seismic forces requires an understanding of the magnitude and distribution of
structural mass and the stiffness properties of the structural system. Both of these aspects of design are
addressed in the mathematical modeling of the structure.
7.2.1 Modeling Criteria
Both the Berkeley and Honolulu buildings will be analyzed with a threedimensional mathematical model
using the ETABS software. Modeling criteria for the seismic analysis is covered in Standard
Section 12.7. This section covers how to determine the effective seismic weight (addressed in the next
section) and provides guidelines for the modeling of the building. Of most significance in a concrete
building is modeling realistic stiffness properties of the structural elements considering cracked sections
in accordance with Standard Section 12.7.3, Item a. Neither the Standard nor ACI 318 provides
requirements for modeling cracked sections for seismic analysis, but the typical practice is to use a
reduced moment of inertia for the beams, columns and walls based on the expected level of cracking.
This example utilizes the following effective moment of inertia, Ieff, for both buildings:
Beams: Ieff = 0.3Igross
Columns: Ieff = 0.5Igross
Walls: Ieff = 0.5Igross
The effective stiffness of the moment frame elements is based on the recommendations in Moehle and
ASCE 41 and account for the expected axial loads and reinforcement levels in the members. The value
for the shear walls is based on the recommendations in ASCE 41 for cracked concrete shear walls.
The following are other significant aspects of the mathematical model that should be noted:
1. The structure is modeled with 12 levels above grade and one level below grade. The perimeter
basement walls are modeled as shear panels as are the main structural walls at the Berkeley building.
The walls are assumed to be fixed at their base, which is at the basement level.
2. All floor diaphragms are modeled as infinitely rigid in plane and infinitely flexible outofplane,
consistent with common practice for a regularshaped concrete diaphragm (see Standard
Section 12.3.1.2).
3. Beams, columns and structural wall boundary members are represented by twodimensional frame
elements. The beams are modeled as Tbeams using the effective slab width per ACI 318
Section 8.10, as recommended by Moehle.
4. The structural walls of the Berkeley building are modeled as a combination of boundary columns and
shear panels with composite stiffness.
5. Beamcolumn joints are modeled in accordance with Moehle, which references the procedure in
ASCE 41. Both the beams and columns are modeled with end offsets based on the geometry, but the
beam offset is modeled as 0 percent rigid, while the column offset is modeled as 100 percent rigid.
This provides effective stiffness for beamcolumn joints consistent with the expected behavior of the
joint: strong columnweak beam condition. (While the recommendations in Moehle are intended for
special moment frames, the same joint rigidities are used for Honolulu for consistency.)
6. Pdelta effects are neglected in the analysis for simplicity. This assumption is verified later in this
example.
7. While the base of the model is located at the basement level, the seismic base for determination of
forces is assumed to be at the first floor, which is at the exterior grade.
7.2.2 Building Mass
Before the building mass can be determined, the approximate size of the different members of the seismic
forceresisting system must be established. For special moment frames, limitations on beamcolumn joint
shear and reinforcement development length usually control. An additional consideration is the amount
of vertical reinforcement in the columns. ACI 318 Section 21.4.3.1 limits the vertical steel reinforcing
ratio to 6 percent for special moment frame columns; however, 3 to 4 percent vertical steel is a more
practical upperbound limit.
Based on a series of preliminary calculations (not shown here), it is assumed that for the Berkeley
building all columns and structural wall boundary elements are 30 inches by 30 inches, beams are 24
inches wide by 32 inches deep and the panel of the structural wall is 16 inches thick. It has already been
established that pan joists are spaced at 36 inches on center, have an average web thickness of 6 inches
and, including a 4inchthick slab, are 20 inches deep. For the Berkeley building, these member sizes
probably are close to the final sizes. For the Honolulu building (which does not have the weight of
concrete walls and ends up with slightly smaller frame elements: 28 by 28inch columns and 20 by 30
inch beams), the masses computed from the Berkeley member sizes are slightly high but are used for
consistency.
In addition to the building structural weight, the following superimposed dead loads are assumed:
Roofing = 10 psf
Partition = 10 psf (see Standard Section 12.7.2, Item 2)
Ceiling and M/E/P = 10 psf
Curtain wall cladding = 10 psf (on vertical surface area)
Based on the above member sizes and superimposed dead load, the individual story weights and masses
are listed in Table 72. These masses are used for the analysis of both the Berkeley and the Honolulu
buildings. Note from Table 72 that the roof and lowest floor have masses slightly different from the
typical floors. It is also interesting to note that the average density of this building is 12.4 pcf, which is in
the range of typical concrete buildings with relatively high floortofloor heights.
Table 72 Story Weights and Masses
Level
Weight (kips)
Mass (kipssec2/in.)
Roof
3,352
8.675
12
3,675
9.551
11
3,675
9.551
10
3,675
9.551
9
3,675
9.551
8
3,675
9.551
7
3,675
9.551
6
3,675
9.551
5
3,675
9.551
4
3,675
9.551
3
3,675
9.551
2
3,817
9.879
Total
43,919
113.736
(1.0 kip = 4.45 kN, 1.0 in. = 25.4 mm)
In the ETABS model, these masses are applied as uniform distributed masses across the extent of the
floor diaphragms in order to provide a realistic distribution of mass in the dynamic model as described
below. The structural framing is modeled utilizing massless elements since their mass is included with
the floor mass. Note that for relatively heavy cladding systems, it would be more appropriate to model
the cladding mass linearly along the perimeter in order to more correctly model the mass moment of
inertia. This has little impact in relatively light cladding systems as is the case here, so the cladding
masses are distributed across the floor diaphragms for convenience.
7.2.3 Analysis Procedures
The selection of analysis procedures is in accordance with Standard Table 12.61. Based on the initial
review, it appears that the Equivalent Lateral Force (ELF) procedure is permitted for both the Berkeley
and Honolulu buildings. However, as we shall see, the analysis demonstrates that the Berkeley building
is torsionally irregular, meaning that the Model Response Spectrum Analysis (MRSA) procedure is
required. Regardless of irregularities, it is common practice to use the MRSA for buildings in regions of
high seismic hazard since the more rigorous analysis method tends to provide lower seismic forces and
therefore more economical designs. For the Honolulu building, located in a region of lower seismic
hazard and with wind governing in some cases, the ELF will be used. However, a dynamic model of the
Honolulu building is used for determining the structural periods.
It should be noted that even though the Berkeley building utilizes the MRSA, the ELF must be used for at
least determining base shear for scaling of results as discussed below.
7.2.4 Development of Equivalent Lateral Forces
This section covers the ELF procedure for both the Berkeley and Honolulu buildings. Since the final
analysis of the Berkeley building utilizes the MRSA procedure, the ELF is illustrated for determining
base shear only. The complete ELF procedure is illustrated for the Honolulu building.
7.2.4.1 Period Determination. Requirements for the computation of building period are given in
Standard Section 12.8.2. For the preliminary design using the ELF procedure, the approximate period,
Ta, computed in accordance with Standard Equation 12.8.7 can be used:
The method for determining approximate period will generally result in periods that are lower (hence,
more conservative for use in predicting base shear) than those computed from a more rigorous
mathematical model. If a more rigorous analysis is carried out, the resulting period may be too high due
to a variety of possible modeling simplifications and assumptions. Consequently, the Standard places an
upper limit on the period that can be used for design. The upper limit is T = CuTa where Cu is provided in
Standard Table 12.81.
For the NS direction of the Berkeley building, the structure is a reinforced concrete momentresisting
frame and the approximate period is calculated according to Standard Equation 12.87 using Ct = 0.016
and x = 0.9 per Standard Table 12.82. For hn = 161 feet, Ta = 1.55 seconds and SD1 > 0.40 for the
Berkeley building, Cu = 1.4 and the upper limit on the analytical period is T = 1.4(1.55) = 2.17 seconds.
For EW seismic activity in Berkeley, the structure is a dual system, so Ct = 0.020 and x =0.75 for other
structures. The approximate period, Ta = 0.90 second and the upper limit on the analytical period is
1.4(0.90) = 1.27 seconds.
For the Honolulu building, the Ta = 1.55 second period computed above for concrete moment frames is
applicable in both the NS and EW directions. For Honolulu, SD1 is 0.192 and, from Standard Table
12.81, Cu can be taken as 1.52. The upper limit on the analytical period is T = 1.52(1.55) = 2.35 seconds.
For the detailed period determination at both the Berkeley and Honolulu buildings, computer models were
developed based on the criteria in Section 7.2.1.
A summary of the Berkeley analysis is presented in Section 7.2.6, but the fundamental periods are
presented here. The computed NS period of vibration is 2.02 seconds. This is between the approximate
period, Ta = 1.55 seconds and CuTa = 2.17 seconds. In the EW direction, the computed period is 1.42
seconds, which is greater than both Ta = 0.90 second and CuTa = 1.27 seconds. Therefore, the periods
used for the ELF procedure are 2.02 seconds in the NS direction and 1.27 seconds in the EW direction.
For the Honolulu building, the computed periods in the NS and EW directions are 2.40 seconds and
2.33 seconds, respectively. The NS period is similar to the Berkeley building because there are no walls
in the NS direction of either building, but the Honolulu period is higher due to the smaller framing
member sizes. In the EW direction, the increase in period from 1.42 seconds at the Berkeley building to
2.33 seconds indicates a significant reduction in stiffness due to the lack of the walls in the Honolulu
building. For both the EW and the NS directions, Ta for the Honolulu building is 1.55 seconds and CuTa
is 2.35 seconds. Therefore, for the purpose of computing ELF forces, the periods are 2.35 seconds and
2.33 seconds in the NS and EW directions, respectively.
A summary of the approximate and computed periods is given in Table 73.
Table 73 Comparison of Approximate and Computed Periods (in seconds)
Method of Period
Computation
Berkeley
Honolulu
NS
EW
NS
EW
Approximate Ta
1.55
0.90
1.55
1.55
Approximate Cu
2.17
1.27
2.35
2.35
ETABS
2.02
1.42
2.40
2.33
*Bold values should be used in the ELF analysis.
7.2.4.2 Seismic Base Shear. For the ELF procedure, seismic base shear is determined using the short
period and 1second period response acceleration parameters, the computed structural period and the
system response modification factor (R).
Using Standard Equation 12.81, the design base shear for the structure is:
V = CsW
where W is the total effective seismic weight of the building and Cs is the seismic response coefficient
computed in accordance with Standard Section 12.8.1.1.
The seismic design base shear for the Berkeley is computed as follows:
For the moment frame system in the NS direction with W = 43,919 kips (see Table 72), SDS = 1.10, SD1 =
0.589, R = 8, I = 1 and T = 2.02 seconds:
Cs,min = 0.01
Cs,min = 0.0484 controls and the design base shear in the NS direction is V = 0.0484 (43,919) = 2,126
kips.
In the EW direction with the dual system, Cs,max and Cs,min are as before, T = 1.27 seconds and
In this case, Cs = 0.0670 controls and V = 0.0670 (43,919) = 2,922 kips.
For the Honolulu building, base shears are computed in a similar manner and are nearly the same for the
NS and the EW directions. With W = 43,919 kips, SDS = 0.474, SD1 = 0.192, R = 5, I = 1 and T = 2.35
seconds in the NS direction:
Cs,min = 0.01
Cs = 0.0207 controls and V = 0.0207 (43,919) = 908 kips.
Due to rounding, the EW base shear is also 908 kips. A summary of the Berkeley and Honolulu seismic
design parameters are provided in Table 74.
Table 74 Comparison of Periods, Seismic Shears Coefficients and Base Shears for the Berkeley and
Honolulu Buildings
Location
Response
Direction
Building Frame Type
T (sec)
Cs
V (kips)
Berkeley
NS
Special moment frame
2.02
0.0485
2,126
EW
Dual system incorporating special moment
frame and structural wall
1.27
0.0670
2,922
Honolulu
NS
Intermediate moment frame
2.35
0.0207
908
EW
Intermediate moment frame
2.33
0.0207
908
(1.0 kip = 4.45 kN)
7.2.4.3 Vertical Distribution of Seismic Forces. The vertical distribution of seismic forces for the ELF
is computed from Standard Equations 12.811 and 12.812.:
Fx = CvxV
where:
k = 1.0 for T < 0.5 second
k = 2.0 for T > 2.5 seconds
k = 0.75 + 0.5T for 1.0 < T < 2.5 seconds
Based on the equations above, the seismic story forces, shears and overturning moments are easily
computed using a spreadsheet. Since the analysis of the Berkeley building utilizes the MRSA procedure,
the vertical force distribution for the ELF procedure will not be used for the design and are not shown
here. The vertical force distribution computations for the Honolulu building are shown in Table 75. The
table is presented with as many significant digits to the left of the decimal as the spreadsheet generates but
that should not be interpreted as real accuracy; it is just the simplest approach.
Table 75 Vertical Distribution of NS and EW Seismic Forces for the Honolulu Building*
Level
Height h
(ft)
Weight W
(kips)
Whk
Whk/ä
Force Fx
(kips)
Story
Shear Vx
(kips)
Overturning
Moment Mx
(ftk)
R
161.00
3,352
59,048,176
0.196
177.8
177.8
12
148.00
3,675
55,053,755
0.183
165.8
343.6
2,312
11
135.00
3,675
46,128,207
0.153
138.9
482.5
6,779
10
122.00
3,675
37,963,112
0.126
114.3
596.9
13,052
9
109.00
3,675
30,564,359
0.101
92.0
688.9
20,811
8
96.00
3,675
23,938,555
0.079
72.1
761.0
29,767
7
83.00
3,675
18,093,222
0.060
54.5
815.5
39,660
6
70.00
3,675
13,037,074
0.043
39.3
854.8
50,262
5
57.00
3,675
8,780,453
0.029
26.4
881.2
61,374
4
44.00
3,675
5,336,045
0.018
16.1
897.3
72,830
3
31.00
3,675
2,720,196
0.009
8.2
905.5
84,494
2
18.00
3,817
992,774
0.003
3.0
908.5
96,265
Total
43,919
301,655,927
1.000
908
112,617
*Table based on T = 2.35 sec and k = 1.92.
(1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN, 1.0 ftkip = 1.36 kNm)
The computed seismic story shears for the Honolulu buildings are shown graphically in Figure 73. Also
shown in this figure are the wind load story shears determined in accordance with the Standard based on a
3second gust of 105 mph and Exposure Category B. The wind shears have been multiplied by the 1.6
load factor to make them comparable to the strength design seismic loads (with a 1.0 load factor).
As can be seen, the NS seismic shears are significantly greater than the corresponding wind shears, but
the EW seismic and wind shears are closer. In the lower stories of the building, wind controls the
strength demands and, in the upper levels, seismic forces control the strength demands. (A somewhat
more detailed comparison is given later when the Honolulu building is designed.) With regards to
detailing the Honolulu building, all of the elements must be detailed for inelastic deformation capacity as
required by ACI 318 for intermediate moment frames.
Figure 73 Comparison of wind and seismic story shears for the Honolulu building
(1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN)
As expected, wind loads do not control the design of the Berkeley building based on calculations not
presented here. (Note that the comparison between wind and seismic forces should be based on more
than just the base shear values. For buildings where the wind and seismic loads are somewhat similar, it
is possible that overturning moment for wind could govern even where the seismic base shear is greater,
in which case a more detailed analysis of specific member forces would need to be performed to
determine the controlling load case.)
7.2.4.4 Horizontal Force Distribution and Torsion. The story forces are distributed to the various
vertical elements of the seismic forceresisting system based on relative rigidity using the ETABS model.
As described previously, the buildings are modeled using rigid diaphragms at each floor. Since the
structures are symmetric in both directions and the distribution of mass is assumed to be uniform, there is
no inherent torsion (Standard Section 12.8.4.1) at either building. However, accidental torsion needs to
be considered in accordance with Standard Section 12.8.4.2.
For this example, accidental torsion is applied to each level as a moment equal to the story shear
multiplied by 5 percent of the story width perpendicular to the direction of loading. The applied moment
is based on the ELF forces for both the Berkeley building (analyzed using the MRSA) and Honolulu
building (ELF). The computation of the accidental torsion moments for the Honolulu building is shown
in Table 76.
Table 76 Accidental Torsion for the Honolulu Building
Level
Force Fx
(kips)
NS
Building
Width (ft)
NS Torsion
(ftkips)
EW
Building
Width (ft)
EW
Torsion
(ftkips)
R
177.8
103
911
216
1,915
12
165.8
103
850
216
1,787
11
138.9
103
712
216
1,499
10
114.3
103
586
216
1,234
9
92.0
103
472
216
995
8
72.1
103
369
216
780
7
54.5
103
279
216
590
6
39.3
103
201
216
426
5
26.4
103
136
216
287
4
16.1
103
82
216
175
3
8.2
103
42
216
90
2
3.0
103
15
216
33
(1.0 kip = 4.45 kN, 1.0 ftkip = 1.36 kNm)
Amplification of accidental torsion, which needs to be considered for buildings with torsional
irregularities in accordance with Standard Section 12.8.4.3, will be addressed if required after the
irregularities are determined.
7.2.5 Direction of Loading
For the initial analysis, the seismic loading is applied in two directions independently as permitted by
Standard Section 12.5. This assumption at the Berkeley building will need to be verified later since
Standard Section 12.5.4 requires consideration of multidirectional loading (the 100 percent30 percent
procedure) for columns that form part of two intersection systems and have a high seismic axial load.
Note that rather than checking whether or not multidirectional loading needs to be considered, some
designers apply the seismic forces using the 100 percent30 percent rule (or an SRSS combination of the
two directions) as common practice when intersecting systems are utilized since today s computer
analysis programs can make the application of multidirectional loading easier than checking each
specific element. Since multidirectional loading is not a requirement of the Standard, the Berkeley
building will not be analyzed in this manner unless required for specific columns.
The Honolulu building, in Seismic Design Category C, does not require consideration of multidirectional
loading since it does not contain the nonparallel system irregularity (Standard Section 12.5.3).
7.2.6 Modal Analysis Procedure
The Berkeley building will be analyzed using the MRSA procedure of Standard Section 12.9 and the
ETABS software. The building is modeled based on the criteria discussed in Section 7.2.1 and analyzed
using a response spectrum generated by ETABS based on the seismic response parameters presented in
Section 7.1.1. The modal parameters were combined using the complete quadratic combination (CQC)
method per Standard Section 12.9.3.
The computed periods and the modal response characteristics of the Berkeley building are presented in
Table 77. In order to capture higher mode effects, 12 modes were selected for the analysis and with 12
modes, the accumulated modal mass in each direction is more than 90 percent of the total mass as
required by Standard Section 12.9.1.
Table 77 Periods and Modal Response Characteristics for the Berkeley Building
Mode
Period
(sec)
% of Effective Mass Represented by Mode*
Description
NS
EW
1
2.02
83.62 (83.62)
0.00 (0.00)
First Mode NS
2
1.46
0.00 (83.62)
0.00 (0.00)
First Mode Torsion
3
1.42
0.00 (83.62)
74.05 (74.05)
First Mode EW
4
0.66
9.12 (92.74)
0.00 (74.05)
Second Mode NS
5
0.38
2.98 (95.72)
0.00 (74.05)
Third Mode NS
6
0.35
0.00 (95.72)
16.02 (90.07)
Second Mode EW
7
0.25
1.36 (97.08)
0.00 (90.07)
Fourth Mode NS
8
0.18
0.86 (97.94)
0.00 (90.07)
Fifth Mode NS
9
0.17
0.00 (97.94)
0.09 (90.16)
Second Mode Torsion
10
0.15
0.00 (97.94)
5.28 (95.44)
Third Mode EW
11
0.10
0.59 (98.53)
0.00 (95.44)
Sixth Mode NS
12
0.08
0.00 (98.53)
3.14 (98.58)
Fourth Mode EW
*Accumulated modal mass in parentheses.
One of the most important aspects of the MRSA procedure is the scaling requirement. In accordance with
Standard Section 12.9.4, the results of the MRSA cannot be less than 85 percent of the results of the ELF.
This is commonly accomplished by running the MRSA to determine the modal base shear. If the modal
base shear is more than 85 percent of the ELF base shear in each direction, then no scaling is required.
However, if the model base shear is less than the ELF base shear, then the response spectrum is scaled
upward so that the modal base shear is equal to 85 percent of the ELF base shear. This is illustrated in
Table 78.
Table 78 Scaling of MRSA results for the Berkeley Building
Direction
VELF (kips)
0.85VELF
(kips)
VMRSA (kips)
Scale Factor
NS
2,126
1,807
1,462
1.24
EW
2,922
2,483
2,296
1.08
(1.0 kip = 4.45 kN)
Therefore, the response spectrum functions for the Berkeley analysis will be scaled by 1.24 and 1.08 in
the NS and EW directions, respectively, which will result in the modal base shears being equal to
85 percent of the static base shears.
As discussed previously, the accidental torsion requirement for the model analysis will be satisfied by
applying the torsional moments computed for the ELF procedure as a static load case that will be
combined with the dynamic load case for the MRSA forces.
7.3 DRIFT AND PDELTA EFFECTS
The checks of story drift and Pdelta effect are contained in this section, but first, deflectionrelated
configuration checks are performed for each building. As discussed previously, these structures could
contain torsional or softstory irregularities. The output from the drift analysis will be used to determine
if either of these irregularities is present in the buildings. However, the presence of a soft story
irregularity impacts only the analysis procedure limitations for the Berkeley building and has no impact
on the design procedures for the Honolulu building.
7.3.1 Torsion Irregularity Check for the Berkeley Building
In Section 7.1.4 it was mentioned that torsional irregularities are unlikely for the Berkeley building
because the elements of the seismic forceresisting system were well distributed over the floor area. This
will now be verified by comparing the story drifts at each end of the building in accordance with Standard
Table 12.31. For this check, drifts are computed using the ETABS program using the ELF procedure (to
avoid having to obtain modal combinations of drifts at multiple points) and including accidental torsion
with Ax = 1.0. Note that since this check is only for relative drifts, the Cd factor is not included.
The drift computations and torsion check for the EW direction are shown in Table 79. The drift values
are shown only for one direction of accidental torsion (positive torsion moment) since the other direction
is the opposite due to symmetry.
Table 79 Torsion Check for Berkeley Building Loaded in the EW Direction
Story
Story Drift
North End (in)
Story Drift
South End(in)
Average Story
Drift (in)
Max Drift /
Average Drift
Roof
0.180
0.230
0.205
1.12
12
0.184
0.246
0.215
1.14
11
0.188
0.261
0.225
1.16
10
0.192
0.276
0.234
1.18
9
0.193
0.288
0.241
1.20
8
0.192
0.296
0.244
1.21
7
0.188
0.298
0.243
1.23
6
0.178
0.292
0.235
1.24
5
0.164
0.278
0.221
1.26
4
0.144
0.254
0.199
1.28
3
0.117
0.218
0.167
1.30
2
0.119
0.229
0.174
1.32
(1.0 in = 25.4 mm)
As can be seen from the table, a torsional irregularity (Type 1a) does exist at Story 8 and below because
the ratio of maximum to average drift exceeds 1.2. This is counterintuitive for a symmetric building but
can happen for a building in which the lateral elements are located towards the center of a relatively long
floor plate, as occurs here. This configuration results in a relatively large accidental torsion load but
relatively low torsional resistance.
For loading in the NS direction, similar computations (not shown here) demonstrate that the structure is
torsionally regular.
The presence of the torsional irregularity in the EW direction has several implications for the design:
The qualitative determination for using the redundancy factor, , equal to 1.0 is not applicable per
Standard Section 12.3.4.2, Item b, as previously assumed in Section 7.1.5. For the purposes of
this example, we will assume = 1.0 based on Standard Section 12.3.4.2, Item a. Due to the
number of shear walls and moment frames in the EW direction, the loss of individual wall or
frame elements would still satisfy the criteria of Standard Table 12.33. This would have to be
verified independently and if those criteria were not met, then analysis would have to be revised
with = 1.3.
The ELF procedure is not permitted per Standard Table 12.61. This does not change the
analysis since we are utilizing the MRSA procedure.
The amplification of accidental torsion needs to be considered per Standard Section 12.8.4.3.
The Ax factor is computed for each floor in this direction and the analysis is revised. See below.
Story drifts need to be checked at both ends of the building rather than at the floor centroid, per
Standard Section 12.12.1. This is covered in Section 7.3.2 below.
The initial determination of accidental torsion was based on Ax = 1.0. Due to the torsional irregularity,
accidental torsion for the EW direction of loading needs to be computed again with the amplification
factor. This is shown in Table 710. Note that while the determination of the torsional irregularity is
based on story drifts, the computation of the torsional amplification factor is based on story
displacements.
Table 710 Accidental Torsion for the Berkeley Building
Level
Force Fx
(kips)
EW
Building
Width (ft)
EW
Torsion
(ftk)
Max Displ
(in)
Ave Displ
(in)
Ax
EW
Torsion,
AxMta (ftk)
Roof
474.8
213
5,045
3.17
2.60
1.03
5,186
12
463.3
213
4,923
2.94
2.40
1.04
5,128
11
408.0
213
4,335
2.69
2.18
1.06
4,575
10
354.7
213
3,769
2.43
1.96
1.07
4,029
9
303.5
213
3,225
2.15
1.72
1.08
3,493
8
254.6
213
2,706
1.87
1.48
1.10
2,970
7
208.2
213
2,213
1.57
1.24
1.11
2,463
6
164.5
213
1,748
1.27
1.00
1.13
1,976
5
123.8
213
1,316
0.98
0.76
1.15
1,511
4
86.6
213
920
0.70
0.54
1.17
1,075
3
53.3
213
567
0.45
0.34
1.19
674
2
26.1
213
278
0.23
0.17
1.20
333
(1.0 kip = 4.45 kN, .0 ft = 0.3048 m, 1.0 ftkip = 1.36 kNm)
With the revised accidental torsion values for the EW direction of loading, the ETABS model is rerun for
the drift checks and member design in subsequent sections.
7.3.2 Drift Check for the Berkeley Building
Story drifts are computed in accordance with Standard Section 12.9.2 and then checked for acceptance
based on Standard Section 12.12.1. According to Standard Table 12.121, the story drift limit for this
Occupancy Category II building is 0.020hsx, where hsx is the height of story x. This limit may be thought
of as 2 percent of the story height. Quantitative results of the drift analysis for the NS and EW
directions are shown in Tables 711a and 711b, respectively. The story drifts are taken directly from the
modal combinations in ETABS. Due to the torsional irregularity in the EW direction, drifts are checked
at both ends of the structure, while NS drifts are checked at the building centroid.
In neither case does the computed drift ratio (amplified story drift divided by hsx) exceed 2 percent of the
story height. Therefore, the story drift requirement is satisfied. A plot of the total deflection in both the
NS and EW directions is shown in Figure 74 and a plot of story drifts is in Figure 75.
An example calculation for drift in Story 5 loaded in the NS direction is given below. Note that the
relevant row is highlighted in Table 711a.
Story drift = 5e = 0.208 inch
Deflection amplification factor, Cd = 5.5
Importance factor, I = 1.0
Amplified story drift = 5 = Cd 5e/I = 5.5(0.208)/1.0 = 1.14 inches
Amplified drift ratio = 5/h5 = (1.14/156) = 0.00733 = 0.733% < 2.0% OK
Figure 74 Deflected shape for Berkeley building
(1.0 ft = 0.3048 m, 1.0 in = 25.4 mm)
Figure 75 Drift profile for Berkeley building
(1.0 ft = 0.3048 m)
Table 711a Drift Computations for the Berkeley Building Loaded in the NS Direction
Story
Story Drift (in)
Story Drift Cd * (in)
Drift Ratio** (%)
Roof
0.052
0.29
0.184
12
0.083
0.46
0.293
11
0.112
0.61
0.393
10
0.134
0.74
0.474
9
0.153
0.84
0.540
8
0.168
0.93
0.593
7
0.182
1.00
0.641
6
0.195
1.07
0.688
5
0.208
1.14
0.733
4
0.221
1.21
0.778
3
0.232
1.28
0.819
2
0.287
1.58
0.732
*Cd = 5.5 for loading in this direction.
**Story height = 156 inches for Stories 3 through roof and 216 inches for Story 2.
(1.0 in = 25.4 mm)
Table 711b Drift Computations for the Berkeley Building Loaded in the EW Direction
Story
Story Drift
North End (in)
Story Drift
South End (in)
Max Story Drift
Cd * (in)
Max Drift Ratio**
(%)
Roof
0.163
0.163
0.90
0.576
12
0.177
0.177
0.97
0.623
11
0.188
0.188
1.03
0.663
10
0.199
0.199
1.10
0.702
9
0.208
0.208
1.14
0.734
8
0.214
0.214
1.18
0.755
7
0.217
0.217
1.19
0.763
6
0.215
0.215
1.18
0.756
5
0.207
0.207
1.14
0.729
4
0.192
0.192
1.05
0.675
3
0.167
0.168
0.92
0.591
2
0.167
0.167
0.92
0.425
*Cd = 5.5 for loading in this direction.
**Story height = 156 inches for Stories 3 through roof and 216 inches for Story 2.
(1.0 in = 25.4 mm)
The story deflection information will be used to determine whether or not a soft story irregularity exists.
As indicated previously, a soft story irregularity (Vertical Irregularity Type 1a) would not impact the
design since we are utilizing the MRSA. However, an extreme soft story irregularity (vertical irregularity
Type 1b) is prohibited in Seismic Design Category D building per Standard Section 12.3.3.1.
However, Standard Section 12.3.2.2 lists an exception:
Structural irregularities of Types 1a, 1b, or 2 in Table 12.32 do not apply where no story drift
ratio under design lateral load is less than or equal to 130 percent of the story drift ratio of the
next story above . The story drift ratios of the top two stories of the structure are not required to
be evaluated.
To determine whether the exception applies to the Berkeley building, the ratio of the drift ratios are
reported in Table 711c.
Table 711c Drift Ratio Comparisons for Stiffness Irregularity Check
Story
NorthSouth
Drift Ratio
Ratio to
Story Above
EastWest
Drift Ratio
Ratio to
Story Above
Roof
0.184

0.184

12
0.293
1.59
0.293
1.09
11
0.393
1.34
0.393
1.06
10
0.474
1.21
0.474
1.06
9
0.540
1.14
0.540
1.05
8
0.593
1.10
0.593
1.03
7
0.641
1.08
0.641
1.01
6
0.688
1.07
0.688
0.99
5
0.733
1.07
0.733
0.96
4
0.778
1.06
0.778
0.93
3
0.819
1.05
0.819
0.88
2
0.732
0.89
0.732
0.72
As can be seen the vertical irregularity does not apply in the EW direction since the ratio is less than 1.3
at all stories. In the NS direction, however, the ratio exceeds 1.3 at the two upper stories. While the top
stories are excluded from this check, the ratio of 1.34 at Story 11 means that the story stiffness s need to
be evaluated to determine whether there is a stiffness irregularity based on Standard Table 12.32.
Since this controlling ratio of drift ratios is at an upper floor and just exceeds the 1.3 limit, it could be
reasonable to conclude that a stiffness irregularity does not exist. For the purposes of this example, as
long as an extreme stiffness irregularity is not present (which seems highly unlikely given the relative
drift ratios), the presence of a nonextreme stiffness irregularity does not have a substantive impact on the
design since this example utilizes the MRSA procedure anyway. In accordance with Standard Table
12.61, the ELF procedure would not be permitted if there were to be a stiffness irregularity. Therefore,
the required stiffness checks for the NS direction are not shown in this example.
7.3.3 Pdelta Check for the Berkeley Building
In accordance with Standard Section 12.8.7 (as referenced by Standard Section 12.9.6 for the MRSA), P
delta effects need not be considered in the analysis if the stability coefficient, , is less than 0.10 for each
story. However, the Standard also limits to a maximum value determined by Standard Equation 12.8
17 as:
Taking as 1.0 (see Standard Section 12.8.7), the limit on stability coefficient for both directions is
0.5/(1.0)5.5 = 0.091.
The Pdelta analysis for each direction of loading is shown in Tables 712a and 712b. For this Pdelta
analysis a story live load of 20 psf (50 psf for office occupancy reduced to 40 percent per Standard
Section 4.8.1) was included in the total story load calculations. Deflections and story shears are based on
the MRSA with no upper limit on period in accordance with Standard Sections 12.9.6 and 12.8.6.2. As
can be seen in the last column of each table, does not exceed the maximum permitted value computed
above and Pdelta effects can be neglected for both drift and strength analyses.
An example Pdelta calculation for the Story 5 under NS loading is shown below. Note that the relevant
row is highlighted in Table 712a.
Amplified story drift = 5 = 1.144 inches
Story shear = V5 = 1,240 kips
Accumulated story weight P5 = 36,532 kips
Story height = hs5= 156 inches
I = 1.0
Cd = 5.5
= (P5I 5/(V5hs5Cd) = (36,532)(1.0)(1.144)/(6.5)(1,240)(156) = 0.0393 < 0.091 OK
Table 712a PDelta Computations for the Berkeley Building Loaded in the NS Direction
Story
Story Drift
(in)
Story Shear
(kips)
Story Dead
Load (kips)
Story Live
Load (kips)
Total Story
Load (kips)
Accum. Story
Load (kips)
Stability
Coeff,
Roof
0.287
261
3,352
420
3,772
3,772
0.0048
12
0.457
495
3,675
420
4,095
7,867
0.0085
11
0.613
672
3,675
420
4,095
11,962
0.0127
10
0.740
807
3,675
420
4,095
16,057
0.0172
9
0.842
914
3,675
420
4,095
20,152
0.0216
8
0.926
1,003
3,675
420
4,095
24,247
0.0261
7
1.000
1,083
3,675
420
4,095
28,342
0.0305
6
1.073
1,161
3,675
420
4,095
32,437
0.0349
5
1.144
1,240
3,675
420
4,095
36,532
0.0393
4
1.214
1,322
3,675
420
4,095
40,627
0.0435
3
1.278
1,400
3,675
420
4,095
44,722
0.0476
2
1.581
1,462
3,817
420
4,237
48,959
0.0446
(1.0 in = 25.4 mm, 1.0 kip = 4.45 kN)
Table 712b PDelta Computations for the Berkeley Building Loaded in the EW Direction
Story
Story Drift
(in)
Story Shear
(kips)
Story Dead
Load (kips)
Story Live
Load (kips)
Total Story
Load (kips)
Accum. Story
Load (kips)
Stability
Coeff,
Roof
0.899
463
3,352
420
3,772
3,772
0.0085
12
0.972
843
3,675
420
4,095
7,867
0.0106
11
1.035
1,104
3,675
420
4,095
11,962
0.0131
10
1.096
1,275
3,675
420
4,095
16,057
0.0161
9
1.145
1,396
3,675
420
4,095
20,152
0.0193
8
1.177
1,512
3,675
420
4,095
24,247
0.0220
7
1.191
1,645
3,675
420
4,095
28,342
0.0239
6
1.180
1,787
3,675
420
4,095
32,437
0.0250
5
1.137
1,927
3,675
420
4,095
36,532
0.0251
4
1.054
2,073
3,675
420
4,095
40,627
0.0241
3
0.921
2,215
3,675
420
4,095
44,722
0.0217
2
0.918
2,296
3,817
420
4,237
48,959
0.0165
(1.0 in = 25.4 mm, 1.0 kip = 4.45 kN)
7.3.4 Torsion Irregularity Check for the Honolulu Building
A test for torsional irregularity for the Honolulu building can be performed in a manner similar to that for
the Berkeley building. Based on computations not shown here, the Honolulu building is not torsionally
irregular. This is the case because the walls, which draw the torsional resistance towards the center of the
Berkeley building, do not exist in the Honolulu building. Therefore, the torsional amplification factor,
Ax = 1.0 for all levels and the accidental torsion moments used for the analysis do not need to be revised.
7.3.5 Drift Check for the Honolulu Building
The story drift computations for the Honolulu building deforming under the NS and EW seismic loading
are shown in Tables 713a and 713b.
These tables show that the story drift at all stories is less than the allowable story drift of 0.020hsx
(Standard Table 12.121). Even though it is not pertinent for Seismic Design Category C buildings, a soft
first story does not exist for the Honolulu building because the ratio of first story drift to second story drift
does not exceed 1.3.
Table 713a Drift Computations for the Honolulu Building Loaded in the NS Direction
Story
Total Drift (in)
Story Drift (in)
Story Drift Cd *
(in)
Drift Ratio (%)
Roof
1.938
0.057
0.259
0.166
12
1.880
0.087
0.391
0.251
11
1.793
0.115
0.517
0.331
10
1.678
0.138
0.623
0.399
9
1.540
0.157
0.708
0.454
8
1.382
0.172
0.773
0.496
7
1.210
0.182
0.821
0.526
6
1.028
0.190
0.854
0.547
5
0.838
0.194
0.873
0.559
4
0.644
0.195
0.881
0.565
3
0.449
0.198
0.889
0.570
2
0.251
0.247
1.113
0.515
*Cd = 4.5 for loading in this direction; total drift is at top of story, story height = 156 inches
for Stories 3 through roof and 216 inches for Story 2.
(1.0 in. = 25.4 mm)
Table 713b Drift Computations for the Honolulu Building Loaded in the EW Direction
Story
Total Drift (in)
Story Drift (in)
Story Drift Cd *
(in)
Drift Ratio (%)
Roof
2.034
0.051
0.230
0.147
12
1.983
0.083
0.376
0.241
11
1.899
0.115
0.518
0.332
10
1.784
0.142
0.639
0.410
9
1.642
0.164
0.736
0.473
8
1.478
0.181
0.814
0.522
7
1.297
0.194
0.874
0.559
6
1.103
0.203
0.915
0.586
5
0.900
0.209
0.942
0.604
4
0.691
0.213
0.958
0.614
3
0.478
0.216
0.970
0.622
2
0.262
0.261
1.173
0.543
Cd = 4.5 for loading in this direction; total drift is at top of story, story height = 156 inches
for Levels 2 through roof and 216 inches for Level 1.
(1.0 in = 25.4 mm)
A sample calculation for Story 5 of Table 713b (highlighted in the table) is as follows:
Deflection at top of story = ë5e =0.900 inches
Deflection at bottom of story = ë4e = 0.691 inch
Story drift = 5e = ë5e  ë4e = 0.900  0.0691 = 0.209 inch
Deflection amplification factor, Cd = 4.5
Importance factor, I = 1.0
Amplified story drift = 5 = Cd 5e/I = 4.5(0.209)/1.0 = 0.942 inch
Amplified drift ratio = 5 / h5 = (0.942/156) = 0.00604 = 0.604% < 2.0% OK
Therefore, story drift satisfies the drift requirements.
7.3.6 PDelta Check for the Honolulu Building
Calculations for Pdelta effects are shown in Tables 714a and 714b for NS and EW loading,
respectively.
Table 714a PDelta Computations for the Honolulu Building Loaded in the NS Direction
Story
Story Drift
(in)
Story Shear
(kips)
Story Dead
Load (kips)
Story Live
Load (kips)
Total Story
Load (kips)
Accum. Story
Load (kips)
Stability
Coeff,
Roof
0.259
177.8
3,352
420
3,772
3,772
0.0069
12
0.391
343.6
3,675
420
4,095
7,867
0.0123
11
0.517
482.5
3,675
420
4,095
11,962
0.0183
10
0.623
596.9
3,675
420
4,095
16,057
0.0245
9
0.708
688.9
3,675
420
4,095
20,152
0.0307
8
0.773
761.0
3,675
420
4,095
24,247
0.0370
7
0.821
815.5
3,675
420
4,095
28,342
0.0432
6
0.854
854.8
3,675
420
4,095
32,437
0.0494
5
0.873
881.2
3,675
420
4,095
36,532
0.0556
4
0.881
897.3
3,675
420
4,095
40,627
0.0618
3
0.889
905.5
3,675
420
4,095
44,722
0.0682
2
1.113
908.5
3,817
420
4,237
48,959
0.0650
(1.0 in = 25.4 mm, 1.0 kip = 4.45 kN)
Table 714b PDelta Computations for the Honolulu Building Loaded in the EW Direction
Story
Story Drift
(in)
Story Shear
(kips)
Story Dead
Load (kips)
Story Live
Load (kips)
Total Story
Load (kips)
Accum. Story
Load (kips)
Stability
Coeff,
Roof
0.230
177.8
3,352
420
3,772
3,772
0.0079
12
0.376
343.6
3,675
420
4,095
7,867
0.0128
11
0.518
482.5
3,675
420
4,095
11,962
0.0183
10
0.639
596.9
3,675
420
4,095
16,057
0.0239
9
0.736
688.9
3,675
420
4,095
20,152
0.0296
8
0.814
761.0
3,675
420
4,095
24,247
0.0351
7
0.874
815.5
3,675
420
4,095
28,342
0.0407
6
0.915
854.8
3,675
420
4,095
32,437
0.0462
5
0.942
881.2
3,675
420
4,095
36,532
0.0515
4
0.958
897.3
3,675
420
4,095
40,627
0.0568
3
0.970
905.5
3,675
420
4,095
44,722
0.0626
2
1.173
908.5
3,817
420
4,237
48,959
0.0617
(1.0 in = 25.4 mm, 1.0 kip = 4.45 kN)
The stability ratio at Story 5 from Table 714b is computed as follows:
Amplified story drift = 5 = 0.942 inch
Story shear = V5 = 881.2 = kips
Accumulated story weight P5 = 36,532 kips
Story height = hs5 = 156 inches
Cd = 4.5
= [P5 ( 5/Cd)]/(V5hs5) = 36,532(0.942/4.5)/(881.2)(156) = 0.0515
The requirements for maximum stability ratio (0.5/Cd = 0.5/4.5 = 0.111) are satisfied. Because the
stability ratio is less than 0.10 at all floors, Pdelta effects need not be considered (Standard
Section 12.8.7).
7.4 STRUCTURAL DESIGN OF THE BERKELEY BUILDING
Framewall interaction plays an important role in the behavior of the structure loaded in the EW
direction. This behavior has the following attributes:
1. For frames without walls (Frames 1, 2, 7 and 8), the shears developed in the beams (except for the
first story) do not differ greatly from story to story. This allows for uniformity in the design of the
beams.
2. For frames containing structural walls (Frames 3 through 6), the overturning moments in the
structural walls are reduced as a result of interaction with the remaining frames (Frames 1, 2, 7 and
8).
3. For the frames containing structural walls, the 40footlong girders act as outriggers further reducing
the overturning moment resisted by the structural walls.
4. A significant load reversal occurs at the top of frames with structural walls. This happens because the
structural wall pulls back on (supports) the top of Frame 1. The deflected shape of the structure
loaded in the EW direction also shows the effect of framewall interaction because the shape is
neither a cantilever mode (wall alone) nor a shear mode (frame alone). It is the straightening out of
the deflected shape of the structure that causes the story shears in the frames without walls to be
relatively equal.
Some of these attributes are shown graphically in Figure 76, which illustrates the total story force
resisted by Frames 1, 2 and 3.
Figure 76 Story shears in the EW direction
(1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN)
7.4.1 Analysis of FrameOnly Structure for 25 Percent of Lateral Load
Where a dual system is utilized, Standard Section 12.2.5.1 requires that the moment frames themselves
are designed to resist at least 25 percent of the total base shear. This provision ensures that the dual
system has sufficient redundancy to justify the increase from R = 6 for a special reinforced concrete
structural wall to R = 7 for a dual system (see Standard Table 122). This 25 percent analysis was carried
out using the ETABS program with the mathematical model of the building being identical to the
previous version except that the panels of the structural walls were removed. The boundary elements of
the walls were retained in the model so that behavior of the interior frames (Frames 3, 4, 5 and 6) would
be analyzed in a rational way. (It could be argued that keeping the boundary columns in the 25 percent
model violates the intent of the provision since they are an integral part of the shear walls. However, in
this condition, the columns are needed for the moment frames adjacent to the walls and those in
longitudinal direction (which resist a small amount of torsion). Since these eight boundary columns resist
only a small portion (just over 15 percent) the total base shear for the 25 percent model, the intent of the
dual system requirements is judged to be satisfied. It should be noted that it is not the intent of the
Standard to allow dual systems of coplanar and integral moment frames and shear walls.)
The seismic demands for this frameonly analysis were scaled such that the spectra base shear is equal to
25 percent of the design base shear for the dual system. In this case, the response spectrum was scaled
such that the frameonly base shear is equal to (0.25)(0.85)VELF. While this may not result in story forces
exactly equal to 25 percent of the story forces from the MRSA of the dual system, the method used is
assumed to meet the intent of this provision of the Standard.
The results of the analysis are shown in Figures 77, 78 and 79 for the frames on Grids 1, 2 and 3,
respectively. The frames on Grids 6, 7 and 8 are similar by symmetry and Grids 4 and 5 are similar to
Grid 3. In these figures, the original analysis (structural wall included) is shown by a heavy line and the
25 percent (frameonly) analysis is shown by a light, dashed line. As can be seen, the 25 percent rule
controls only at the lower level of the building. Therefore, for the design of the beams and columns at the
lower two levels (not part of this example), the greater of the dual system and frameonly analysis should
be used. For the purposes of this example, which includes representative designs for the framing at a
middle level, design forces from the dual system analysis will satisfy the 25 percent requirement.
Figure 77 25 percent story shears, Frame 1 EW direction
(1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN)
Figure 78 25 percent story shears, Frame 2 EW direction
(1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN)
Figure 79 25 percent story shear, Frame 3 EW direction
(1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN)
7.4.2 Design of Moment Frame Members for the Berkeley Building
For this part of the example, the design and detailing of five beams and one interior column along Grid 1
on Level 5 are presented in varying amounts of detail. The beams are designed first because the flexural
capacity of the asdesigned beams is a factor in the design and detailing of the column and the beam
column joint.
Before continuing with the example, it should be mentioned that the design of ductile reinforced concrete
moment frame members is controlled by the flexural reinforcement in the beams. The percentage and
placement of beam flexural reinforcement governs the flexural rebar cutoff locations, the size and spacing
of beam shear reinforcement, the crosssectional characteristics of the column, the column flexural
reinforcement and the column shear reinforcement. The beam reinforcement is critical because the basic
concept of ductile frame design is to force most of the energydissipating deformation to occur through
inelastic rotation in plastic hinges at the ends of the beams.
In carrying out the design calculations, three different flexural strengths are used for the beams. These
capacities are based on the following:
Design strength: = 0.9, tensile stress in reinforcement at 1.00 fy
Nominal strength: = 1.0, tensile stress in reinforcement at 1.00 fy
Probable strength: = 1.0, tensile stress in reinforcement at 1.25 fy
Various aspects of the design of the beams and other members depend on the above capacities are as
follows:
Beam rebar cutoffs: Design strength
Beam shear reinforcement: Probable strength of beam
Beamcolumn joint strength: Probable strength of beam
Column flexural strength: 6/5 nominal strength of beam
Column shear strength: Probable strength of column or beam
In addition, beams in ductile frames will always have top and bottom longitudinal reinforcement
throughout their length. In computing flexural capacities, only the tension steel will be considered. This
is a valid design assumption because reinforcement ratios are quite low, yielding a depth to the neutral
axis similar to the depth of the compression reinforcement.
Finally, a sign convention for bending moments is required in flexural design. In this example, where the
steel at the top of a beam section is in tension, the moment is designated as a negative moment. Where
the steel at the bottom is in tension, the moment is designated as a positive moment. All moment
diagrams are drawn using the reinforced concrete or tensionside convention. For beams, this means
negative moments are plotted on the top and positive moments are plotted on the bottom. For columns,
moments are drawn on the tension side of the member.
7.4.2.1 Preliminary Calculations. Before the quantity and placement of reinforcement is determined, it
is useful to establish, in an overall sense, how the reinforcement will be distributed. The preliminary
design established that the moment frame beams would be 24 inches wide by 32 inches deep and the
columns would be 30 inches by 30 inches. Note that the beam widths were selected to consider the beam
column joints confined per ACI 318 Section 21.7.4.1, which requires beam widths of at least 75 percent
of the column width.
In order to determine the effective depth used for the design of the beams, it is necessary to estimate the
size and placement of the reinforcement that will be used. In establishing this depth, it is assumed that #8
bars will be used for longitudinal reinforcement and that hoops and stirrups will be constructed from #4
bars. In all cases, clear cover of 1.5 inches is assumed. Since this structure has beams spanning in two
orthogonal directions, it is necessary to layer the flexural reinforcement as shown in Figure 710. The
reinforcement for the EW spanning beams was placed in the upper and lower layers because the strength
demand for these members is somewhat greater than that for the NS beams.
Figure 710 Layout for beam reinforcement
(1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm)
Given Figure 710, compute the effective depth for both positive and negative moment as follows:
Beams spanning in the EW direction, d = 32  1.5  0.5  1.00/2 = 29.5 inches
Beams spanning in the NS direction, d = 32  1.5  0.5  1.0  1.00/2 = 28.5 inches
For negative moment bending, the effective width is 24 inches for all beams. For positive moment, the
slab is in compression and the effective Tbeam width varies according to ACI 318 Section 8.12. The
effective widths for positive moment are as follows (with the parameter controlling effective width shown
in parentheses):
20foot beams in Frames 1 and 8: b = 24 + 20(12)/12 = 44 inches (span length)
20foot beams in Frames 2 and 7: b = 20(12)/4 = 60 inches (span length)
40foot beams in Frames 2 through 7: b = 24 + 2[8(4)] = 88 inches (slab thickness)
30foot beams in Frames A and D: b = 24 + [6(4)] = 48 inches (slab thickness)
30foot beams in Frames B and C: b = 24 + 2[8(4)] = 88 inches (slab thickness)
ACI 318 Section 21.5.2 controls the longitudinal reinforcement requirements for beams. The minimum
reinforcement to be provided at the top and bottom of any section is as follows:
This amount of reinforcement can be supplied by three #8 bars with As = 2.37 in2. Since the three #8 bars
will be provided continuously top and bottom, reinforcement required for strength will include these #8
bars.
Before getting too far into member design, it is useful to check the required tension development length
for hooked bars since the required length may control the dimensions of the exterior columns and the
boundary elements of the structural walls.
From Equation 216 of ACI 318 Section 21.7.5.1, the required development length is as follows:
For normalweight (NW) concrete, the computed length cannot be less than 6 inches or 8db.
For straight typical bars, ld = 2.5ldh and for straight top bars, ld = 3.25ldh (ACI 318 Sec. 21.7.5.2). These
values are applicable only where the bars are anchored in wellconfined concrete (e.g., column cores and
plastic hinge regions with confining reinforcement). The development length for the portion of the bar
extending into unconfined concrete must be increased by a factor of 1.6 per ACI 318 Section 12.7.5.3.
Development length requirements for hooked and straight bars are summarized in Table 715.
Where hooked bars are used, the hook must be 90 degrees and be located within the confined core of the
column or boundary element. For bars hooked into 30inchsquare columns with 1.5 inches of cover and
#4 ties, the available development length is 30  1.50  0.5 = 28.0 inches. With this amount of available
length, there will be no problem developing hooked bars in the columns.
Table 715 is applicable to bars anchored in joint regions only. For development of bars outside of joint
regions, ACI 318 Chapter 12 should be used.
Table 715 Tension Development Length Requirements for Hooked Bars and Straight Bars
in 5,000 psi NW Concrete
Bar Size
db (in)
ldh hook (in)
ld typ (in)
ld top (in)
#4
0.500
6.5
16.3
21.2
#5
0.625
8.2
20.4
26.5
#6
0.750
9.8
24.5
31.8
#7
0.875
11.4
28.6
37.1
#8
1.000
13.1
32.6
42.4
#9
1.128
14.7
36.8
47.9
#10
1.270
16.6
41.4
53.9
#11
1.410
18.4
46.0
59.8
(1.0 in = 25.4 mm)
Another requirement to consider prior to establishing column sizes is ACI 381 Section 21.7.2.3 which
sets a minimum ratio of 20 for the column width to the diameter of the largest longitudinal beam bar
passing through the joint. This requirement is easily satisfied for the 30inch columns in this example.
7.4.2.2 Design of Representative Frame 1 Beams. The preliminary design of the beams of Frame 1 was
based on members with a depth of 32 inches and a width of 24 inches. The effective depth for positive
and negative bending is 29.5 inches and the effective widths for positive and negative bending are 44 and
24 inches, respectively. This assumes the stress block in compression is less than the 4.0inch flange
thickness.
The layout of the geometry and gravity loading on the three easternmost spans of Level 7 of Frame 1 as
well as the unfactored gravity and seismic moments are illustrated in Figure 711. The seismic and
gravity moments are taken directly from the ETABS program output. The seismic forces are from the
EW spectral load case plus the controlling accidental torsion case (the torsional moment where
translation and torsion are additive). Note that all negative moments are given at the face of the column
and that seismic moments are considerably greater than those due to gravity.
Factored bending moment envelopes for all five spans are shown in Figure 711. Negative moment at the
supports is controlled by the 1.42D + 0.5L + 1.0E load combination and positive moment at the support is
controlled by 0.68D  1.0E. Midspan positive moments are based on the load combination 1.2D + 1.6L.
Figure 711 Bending moments for Frame 1
(1.0 ft = 0.3048 m, 1.0 inkip = 0.113 kNm)
7.4.2.2.1 Longitudinal Reinforcement. The design process for determining longitudinal reinforcement
is illustrated as follows for Span AA .
1. Design for Negative Moment at the Face of the Exterior Support (Grid A):
Mu = 1.42(550) + 0.5(251) + 1.0(3,383) = 4,290 inchkips
Try one #8 bar in addition to the three #8 bars required for minimum steel:
As = 4(0.79) = 3.16 in2
fc' = 5,000 psi
fy = 60 ksi
Width b for negative moment = 24 inches
d = 29.5 in.
Depth of compression block, a = Asfy/0.85fc'b
a = 3.16 (60)/[0.85 (5) 24] = 1.86 inches
Design strength, Mn = Asfy(d  a/2)
Mn = 0.9(3.16)60(29.5 1.86/2) = 4,875 inchkips > 4,290 inchkips OK
2. Design for Positive Moment at Face of Exterior Support (Grid A):
Mu = [0.68(550)] + [1.0(3,383)] = 3,008 inchkips
Try the three #8 bars required for minimum steel:
As = [3(0.79)] = 2.37 in 2
Width b for positive moment = 44 inches
d = 29.5 inches
a = [2.37(60)]/[0.85(5)44] = 0.76 inch
Mn = 0.9(2.37) 60(29.5 0.76/2) = 3,727 inchkips > 3,008 inchkips OK
3. Positive Moment at Midspan:
Mu = [1.2(474)] + [1.6(218)] = 918.1 inchkips
Minimum reinforcement (three #8 bars) controls by inspection.
4. Design for Negative Moment at the Face of the Interior Support (Grid A ):
Mu = 1.42(602) + 0.5(278) + 1.0(3,177) = 4,172 inchkips
Try one #8 bars in addition to the three #8 bars required for minimum steel:
Mn = 4,875 inchkips > 4,172 inchkips OK
5. Design for Positive Moment at Face of Interior Support (Grid A ):
Mu = [0.68(602)] + [1.0(3,177)] = 2,767 inchkips
Three #8 bars similar to the exterior support location are adequate by inspection.
Similar calculations can be made for the Spans A'B and BC and then the remaining two spans are
acceptable via symmetry. A summary of the preliminary flexural reinforcing is shown in Table 716.
In addition to the computed strength requirements and minimum reinforcement ratios cited above, the
final layout of reinforcing steel also must satisfy the following from ACI 318 Section 21.5.2:
Minimum of two bars continuous top and
bottom
OK (three #8 bars continuous top OK (three #8 bars continuous top and
bottom)
Positive moment strength greater than 50
percent negative moment strength at a joint
OK (at all joints)
Minimum strength along member greater
than 0.25 maximum strength
OK (As provided = three #8 bars is
more than 25 percent of
reinforcement provided at joints)
The preliminary layout of reinforcement is shown in Figure 712. The arrangement of bars is based on
the above computations and Table 716 summary of the other spans. Note that a slightly smaller amount
of reinforcing could be used at the top of the exterior spans, but #8 bars are selected for consistency. In
addition, the designer could opt to use four #8 bars continuous throughout the span for uniformity and
ease of placement.
Figure 712 Preliminary rebar layout for Frame 1
(1.0 ft = 03.048 m)
As mentioned above, later phases of the frame design will require computation of the design strength and
the maximum probable strength at each support. The results of these calculations are shown in
Table 716.
Table 716 Design and Maximum Probable Flexural Strength For Beams in Frame 1
Item
Location*
A
A'
B
C
C'
D
Negative
Moment
Moment Demand
(inchkips)
4,290
4,672
4,664
4,664
4,672
4,290
Reinforcement
four #8
four #8
four #8
four #8
four #8
four #8
Design Strength
(inchkips)
4,875
4,875
4,875
4,875
4,875
4,875
Probable Strength
(inchkips)
7,042
7,042
7,042
7,042
7,042
7,042
Positive
Moment
Moment Demand
(inchkips)
3,009
3,288
3,255
3,255
3,289
3,009
Reinforcement
three #8
three #8
three #8
three #8
three #8
three #8
Design Strength
(inchkips)
3,727
3,727
3,727
3,727
3,727
3,727
Probable Strength
(inchkips)
5,159
5,159
5,159
5,159
5,159
5,159
*Moment demand is taken as the larger of the beam moments on each side of the column.
(1.0 inkip = 0.113 kNm)
As an example of computation of probable strength, consider the case of four #8 top bars plus the portion
of slab reinforcing within the effective beam flange width computed above, which is assumed to be
0.002(4 inches)(4424)=0.16 square inches. (The slab reinforcing, which is not part of this example, is
assumed to be 0.002 for minimum steel.)
As = 4(0.79) + 0.16 = 3.32 in2
Width b for negative moment = 24 inches
d = 29.5 inches
Depth of compression block, a = As(1.25fy)/0.85fc'b
a = 3.32(1.25)60/[0.85(4)24] = 2.44 inches
Mpr = 1.0As(1.25fy)(d  a/2)
Mpr = 1.0(3.32)1.25(60)(29.6 2.44/2) = 7,042 inchkips
For the case of three #8 bottom bars:
As = 3(0.79) = 2.37 in2
Width b for positive moment = 44 inches
d = 29.5 inches
a = 2.37(1.25)60/[0.85(5)44] = 0.95 inch
Mpr = 1.0(2.37)1.25(60)(29.5 0.95/2) = 5,159 inchkips
At this point in the design process, the layout of reinforcement has been considered preliminary because
the quantity of reinforcement placed in the beams has a direct impact on the magnitude of the stresses
developed in the beamcolumn joint. If the computed joint stresses are too high, the only remedies are
increasing the concrete strength, increasing the column area, changing the reinforcement layout, or
increasing the beam depth. The complete check of the beamcolumn joint is illustrated in Section 7.4.2.3
below, but preliminary calculations indicate that the joint is adequate, so the design can progress based on
the reinforcing provided.
Because the arrangement of steel is acceptable from a joint strength perspective, the cutoff locations of
the various bars may be determined (see Figure 712 for a schematic of the arrangement of
reinforcement). The three #8 bars (top and bottom) required for minimum reinforcement are supplied in
one length that runs continuously across the two end spans and are spliced in the center span. An
additional #8 top bar is placed at each column.
To determine where added top bars should be cut off in each span, it is assumed that theoretical cutoff
locations correspond to the point where the continuous top bars develop their design flexural strength.
Cutoff locations are based on the members developing their design flexural capacities (fy = 60 ksi and =
0.9). Using calculations similar to those above, it has been determined that the design flexural strength
supplied by a section with only three #8 bars is 3,686 inchkips for negative moment.
Sample cutoff calculations are given for Span BC. To determine the cutoff location for negative
moment, both the additive and counteractive load combinations must be checked to determine the
maximum required cutoff length. In this case, the 1.42D + 0.5L ñ QE load combination governs. Loading
diagrams for determining cutoff locations are shown in Figure 713.
For negative moment cutoff locations, refer to Figure 714, which is a free body diagram of the left end of
the member in Figure 713. Since the goal is to develop a negative moment capacity of 3,686 inchkips in
the continuous #8 bars, summing moments about Point A in Figure 714 can be used to determine the
location of this moment demand. The moment summation is as follows:
In the above equation, 4,653 (inchkips) is the negative moment demand at the face of column, 0.318
(kips/inch) is the factored gravity load, 68.2 kips is the end shear and 3,686 inchkips is the design
strength of the section with three #8 bars. Solving the quadratic equation results in x = 14.7 inches. ACI
318 Section 12.10.3 requires an additional length equal to the effective depth of the member or 12 bar
diameters (whichever is larger). Consequently, the total length of the bar beyond the face of the support
is 14.7 + 29.5 = 44.2 inches and a 3 9 extension beyond the face of the column could be used at this
location. Similar calculations should be made for the other beam ends.
Figure 713 Loading for determination of rebar cutoffs
(1.0 ft = 0.3048 m, 1.0 klf = 14.6 kN/m, 1.0 inkip = 0.113 kNm)
Figure 714 Free body diagram
(1.0 kip = 4.45kN, 1.0 klf = 14.6 kN/m, 1.0 inkip = 0.113 kNm)
As shown in Figure 715, another requirement in setting cutoff length is that the bar being cut off must
have sufficient length to develop the strength required in the adjacent span. From Table 715, the
required development length of the #8 top bars in tension is 42.4 inches if the bar is anchored in a
confined joint region. The confined length in which the bar is developed is shown in Figure 715 and
consists of the column depth plus twice the depth of the beam (the length for beam hoops per ACI 318
Section 21.5.3.1). This length is 30 + 32 + 32 = 94 inches, which is greater than the 42.4 inches required.
The column and beam are considered confined because of the presence of closed hoop reinforcement as
required by ACI 318 Sections 21.5.3 and 21.6.4.
Figure 715 Development length for top bars
(1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm)
The continuous top bars are spliced at the center of Span BC and the bottom bars at Spans A B and
CC as shown in Figure 716. The splice length is taken as Class B splice length for #8 bars. These
splice locations satisfy the requirements of ACI 318 Section 21.5.2.3 for permitted splice locations.
The splice length is determined in accordance with ACI 318 Section 12.15, which indicates that the splice
length is 1.3 times the development length. From ACI 318 Section 12.2.2, the development length, ld, is
computed as:
using t = 1.3 (top bar), e =1.0 (uncoated), s = 1.0 (#8 bar), = 1.0 (NW concrete) and taking (c + Ktr) /
db as 2.5 (based on clear cover and confinement), the development length for one #8 top bar is:
The splice length = 1.3(33.1) = 43 inches. Therefore, use a 43inch contact splice for the top bars.
Computed in a similar manner, the required splice length for the #8 bottom bars is 34 inches. According
to ACI 318 Section 21.5.2.3, the entire region of the splice must be confined by closed hoops spaced at
the smaller of d/4 or 4 inches.
The final bar placement and cutoff locations for all five spans are shown in Figure 716.
Figure 716 Final bar arrangement
(1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm)
7.4.2.2.2 Transverse Reinforcement. The requirements for transverse reinforcement in special moment
frame beams, include shear strength requirements (ACI 318 Sec. 21.5.4) covered here first and then
detailing requirements (ACI 318 Sec. 21.5.3).
To avoid nonductile shear failures, the shear strength demand is computed as the sum of the factored
gravity shear plus the maximum earthquake shear. The maximum earthquake shear is computed based on
the maximum probable beam moments described and computed previously. The probable moment
strength at each support is shown in Table 716.
Figure 717 illustrates the development of the design shear strength envelopes for Spans AA', A'B and
BC. In Figure 717a, the maximum probable earthquake moments are shown for seismic forces acting to
the east (solid lines) and to the west (dashed lines). The moments shown occur at the face of the supports.
The earthquake shears produced by the maximum probable moments are shown in Figure 717b. For
Span BC, the values shown in the figure are:
where lclear = 17 feet6 inches = 210 inches
Note that the magnitude of the earthquake shear can vary with direction (if the beam moment capacities
are different at each end). However, in this case the shears are the same in both directions and are
computed as:
VE = (7,042 + 5,159) / 210 = 58.1 kips
Figure 717 Shear forces for transverse reinforcement
(1.0 in. = 25.4 mm, 1.0 kip = 4.45kN, 1.0 inkip = 0.113 kNm)
The gravity shears shown in Figure 717c are taken from the ETABS model:
Factored gravity shear = VG = 1.42Vdead + 0.5Vlive
Vdead = 20.2 kips
Vlive = 9.3 kips
VG = 1.42(20.2) + 0.5(9.3) = 33.3 kips
Total design shears for each span are shown in Figure 717d. The strength envelope for Span BC is
shown in detail in Figure 718, which indicates that the maximum design shears is 58.1 + 33.3 = 91.4
kips. While this shear acts at one end, a shear of 58.1 33.3 = 24.8 kips acts at the opposite end of the
member. In the figure the sloping lines indicate the shear demands along the beam and the horizontal
lines indicate the shear capacities at the end and center locations.
Figure 718 Detailed shear force envelope in Span BC
(1.0 in. = 25.4 mm, 1.0 kip = 4.45kN)
In designing shear reinforcement, the shear strength can consist of contributions from concrete and from
steel hoops or stirrups. However, according to ACI 318 Section 21.5.4.2, the design shear strength of the
concrete must be taken as zero where the axial force is small (Pu/Agf c < 0.05) and the ratio VE/Vu is
greater than 0.5. From Figure 717, this ratio is VE/Vu = 58.1/91.4 = 0.64, so concrete shear strength must
be taken as zero.
Compute the required shear strength provided by reinforcing steel at the face of the support:
Vu = Vs = 91.4 kips
Vs = Avfyd/s
For reasons discussed below, assume four #4 vertical legs (Av = 0.8 in2), fy = 60 ksi and d = 29.5 inches
and compute the required spacing as follows:
s = Avfyd/Vu = 0.75[4(0.2)](60)(29.5/91.4) = 11.6 inches
At midspan, the design shear Vu = (91.4 + 24.8)/2 = 58.1 kips, which is the same as the earthquake shear
since gravity shear is nominally zero. Compute the required spacing assuming two #4 vertical legs:
s = 0.75[2(0.2)](60)(29.5/58.1) = 9.13 inches
In terms of detailing requirements, ACI 318 Section 21.5.3.1 states that closed hoops at a tighter spacing
are required over a distance of twice the member depth from the face of the support and ACI 318
Section 21.5.3.4 indicates that stirrups are permitted away from the ends.
Therefore, the shear strength requirements at this transition point should be computed. At a point equal to
twice the beam depth, or 64 inches from the support, the shear is computed as:
Vu = 91.4  (64/210)(91.4 24.8) = 71.1 kips
Compute the required spacing assuming two #4 vertical legs:
s = 0.75[2(0.2)](60)(29.5/71.1) = 7.4 inches
Before the final layout can be determined, the detailing requirements need to be considered. The first
hoop must be placed 2 inches from the face of the support and the maximum hoop spacing at the beam
ends is per ACI 318 Section 21.5.3.2 as follows:
d/4 = 29.5/4 = 7.4 inches
8db = 8(1.0) = 8.0 inches
24dh = 24(0.5) = 12.0 inches
Outside of the region at the beam ends, ACI 318 Section 21.5.3.4 permits stirrups with seismic hooks to
be spaced at a maximum of d/2.
Therefore, at the beam ends, overlapped close hoops with four legs will be spaced at 7 inches and in the
middle, closed hoops with two legs will be spaced at 7 inches. This satisfies both the strength and
detailing requirements and results in a fairly simple pattern. Note that hoops are being used along the
entire member length. This is being done because the earthquake shear is a large portion of the total
shear, the beam is relatively short and the economic premium is negligible.
This arrangement of hoops will be used for Spans AA', BC and C'D. In Spans A'B and CC', the
bottom flexural reinforcement is spliced and hoops must be placed over the splice region at d/4 or a
maximum of 4 inches on center per ACI 318 Section 21.5.2.3.
One additional requirement at the beam ends is that where hoops are required (the first 64 inches from the
face of support), longitudinal reinforcing bars must be supported as specified in ACI 318 Section 7.10.5.3
as required by ACI 318 Section 21.5.3.3. Hoops should be arranged such that every corner and alternate
longitudinal bar is supported by a corner of the hoop assembly and no bar should be more than 6 inches
clear from such a supported bar. This will require overlapping hoops with four vertical legs as assumed
previously. Details of the transverse reinforcement layout for all spans of Level 5 of Frame 1 are shown
in Figure 716.
7.4.2.3 Check BeamColumn Joint at Frame 1. Prior to this point in the design process, preliminary
calculations were used to check the beamcolumn joint, since the shear force developed in the beam
column joint is a direct function of the beam longitudinal reinforcement. These calculations are often
done early in the design process because if the computed joint shear is too high, the only remedies are
increasing the concrete strength, increasing the column area, changing the reinforcement layout, or
increasing the beam depth. At this point in the design, the joint shear is checked for the final layout of
beam reinforcing.
The design of the beamcolumn joint is based on the requirements of ACI 318 Section 21.7. While ACI
318 provides requirements for joint shear strength, it does not specify how to determine the joint shear
demand, other than to indicate that the joint forces are computed using the probable moment strength of
the beam (ACI 318 Sec. 21.7.2.1). This example utilizes the procedure for determining joint shear
demand contained in Moehle. The shear in the joint is a function of the shear in the column and the
tension/compression couple contributed by the beam moments. The method for determining column
shear is illustrated in Figure 719. In this freebody diagram, the column shear, Vcol, is determined from
equilibrium as follows:
Figure 719 Column shear free body diagram
The determination of the forces in the joint of the column on Grid C of Frame 1 is based on Figure 716a,
which shows how plastic moments are developed in the various spans for equivalent lateral forces acting
to the east. An isolated subassemblage from the frame showing moments is shown in Figures 720b.
The beam shears shown in Figure 720c are based on the probable moment strengths shown in
Table 716.
For forces acting from west to east, compute the earthquake shear in Span BC as follows:
VE = (Mpr + Mpr+ )/lclear = (7,042 + 5,159)/(240  30) = 58.1 kips
For Span CC', the earthquake shear is the same since the probable moments are equal and opposite.
Figure 720 Diagram for computing column shears
(1.0 ft = 0.3048 m, 1.0 kip = 4.45kN, 1.0 inkip = 0.113 kNm)
With h = 30 inches and lc = 156 inches, the column shear is computed as follows:
With equal spans, gravity loads do not produce significant column shears, except at the end column,
where the seismic shear is much less. Therefore, gravity loads are not included in this computation.
The forces in the beam reinforcement for negative moment are based on four #8 bars at 1.25 fy:
T = C = 1.25(60)[(4(0.79)] = 237.0 kips
For positive moment, three #8 bars also are used, assuming C = T, C = 177.8 kips.
As illustrated in Figure 721, the joint shear force Vj is computed as follows:
Vj = T + C Vcol
= 237.0 + 177.8 89.4
= 325.4 kips
Figure 721 Computing joint shear stress (1.0 kip = 4.45kN)
For joints confined on three faces or on two opposite faces, the nominal shear strength is based on ACI
318 Section 21.7.4 as follows:
kips
For joints of special moment frames, ACI 318 Section 9.3.4 permits = 0.85, so Vn = 0.85(954.6
kips) = 811.4 kips, which exceeds the computed joint shear, so the joint is acceptable. Joint stresses
would be checked for the other columns in a similar manner.
ACI 318 Section 21.7.3.1 specifies the amount of transverse reinforcement required in the joint. Since
the joint is not confined on all four sides by a beam, the total amount of transverse reinforcement required
by ACI 318 Section 21.6.4.4 will be placed within the depth of the joint. As shown later, this
reinforcement consists of fourleg #4 hoops at 4 inches on center.
7.4.2.4 Design of a Typical Interior Column of Frame 1. This section illustrates the design of a typical
interior column on Gridline A'. The column, which supports Level 7 of Frame 1, is 30 inches square and
is constructed from 5,000 psi concrete and 60 ksi reinforcing steel. An isolated view of the column is
shown in Figure 722. The flexural reinforcement in the beams framing into the column is shown in
Figure 716. Using simple tributary area calculations (not shown), the column supports an unfactored
axial dead load of 367 kips and an unfactored axial reduced live load of 78 kips. The ETABS analysis
indicates that the maximum axial earthquake force is 33.7 kips, tension or compression. The load
combination used to compute this force consists of the full earthquake force in the EW direction plus
amplified accidental torsion. Since this column is not part of a NS moment frame, orthogonal effects
need not be considered per Standard Section 12.5.4. Hence, the column is designed for axial force plus
uniaxial bending.
Figure 722 Layout and loads on column of Frame A
(1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm, 1.0 kip = 4.45kN)
7.4.5.3.1 Longitudinal Reinforcement. To determine the axial design loads, use the controlling basic
load combinations:
1.42D + 0.5L + 1.0E
0.68D  1.0E
The combination that results in maximum compression is:
Pu = 1.42(367.2) + 0.5(78.0) + 1.0(33.7) = 595 kips (compression)
The combination for minimum compression (or tension) is:
Pu = 0.68(367.2)  1.0(33.7) = 216 kips (compression)
The maximum axial compression force of 595 kips is greater than 0.1fc'Ag = 0.1(5)(302) = 450 kips, so the
design is based on ACI 318 Section 21.6 for columns (see ACI 318 Sec. 21.6.1). According to ACI 318
Section 21.6.2, the sum of nominal column flexural strengths at the joint must be at least 6/5 of the sum of
nominal flexural strength of the beams framing into the column. Beam moments at the face of the
support are used for this computation. These capacities are provided in Table 716.
Nominal (negative) moment strength at end A' of Span AA' = 4,875/0.9 = 5,417 inchkips
Nominal (positive) moment strength at end A' of Span A' B = 3,727/0.9 = 4,141 inchkips
Sum of beam moment at the joint = 5,417 + 4,141 = 9,558 inchkips
Required sum of column design moments = 6/5 9,558 = 11,469 inchkips.
Individual column design moment = 11,469/2 = 5,735 inchkips
Knowing the factored axial load and the required design flexural strength, a column with adequate
capacity must be selected. Figure 723 shows a PM interaction curve for a 30 by 30inch column with
longitudinal reinforcing consisting of twelve #8 bars (1.05 percent steel). Computed using PCA Column,
the curve is based on a factor of 1.0 as required for nominal strength. At axial forces of 595 kips and
216 kips, solid horizontal lines are drawn. The dots on the lines represent the required average nominal
flexural strength (5,735 inchkips) at each axial load level. These dots must lie to the left of the curve
representing the nominal column strengths. Since the dots are within the capacity curve for both design
and nominal moments strengths at both the minimum and maximum axial forces, this column design is
clearly adequate.
Figure 723 Design interaction diagram for column on Gridline A'
(1.0 kip = 4.45kN, 1.0 ftkip = 1.36 kNm)
7.4.2.4.2 Transverse Reinforcement. The design of transverse reinforcement for columns of special
moment frames must consider confinement requirements (ACI 318 Sec. 21.6.4) and shear strength
requirements (ACI 318 Sec. 21.6.5). The confinement requirements are typically determined first.
Based on ACI 318 Section 21.6.4.1, tighter spacing of confinement is generally required at the ends of the
columns, over a distance, lo, equal to the larger of the following:
Column depth = 30 inches
Onesixth of the clear span = (15632)/6 = 20.7 inches
18 inches
There are both spacing and quantity requirements for the reinforcement. ACI 318 Section 21.6.4.3
specifies the spacing as the minimum of the following:
Onefourth the minimum column dimension = 30/4=7.5 inches
Six longitudinal bar diameters = 6(1.0) = 6.0 inches
Dimension so = 4 + (14  hx) / 3, where so is between 4 inches and 6 inches and hx is the maximum
horizontal spacing of hoops or cross ties.
For the column with twelve #8 bars and #4 hoops and cross ties, hx = 8.833 inches and so = 5.72 inches,
which controls the spacing requirement.
ACI 318 Section 21.6.4.4 gives the requirements for minimum transverse reinforcement in terms of cross
sectional area. For rectangular sections with hoops, ACI 318 Equations 214 and 215 are applicable:
The first of these equations controls when Ag/Ach > 1.3. For the 30 by 30inch columns:
Ach = (30  1.5  1.5)2 = 729 in2
Ag = 30 (30) = 900 in2
Ag/Ach = 900/729 = 1.24
Therefore, ACI 318 Equation 215 controls. Try hoops with four #4 legs:
bc = 30  1.5  1.5 = 27.0 inches
s = [4 (0.2)(60,000)]/[0.09 (27.0)(5,000)] = 3.95 inches
This spacing controls the design, so hoops consisting of four #4 bars spaced at 4 inches will be considered
acceptable.
ACI 318 Section 21.6.4.5 specifies the maximum spacing of transverse reinforcement in the region
beyond the lo zones. The maximum spacing is the smaller of 6.0 inches or 6db, which for #8 bars is also 6
inches. Hoops and crossties with the same details as those placed in the critical regions of the column
will be used.
7.4.2.4.3 Check Column Shear Strength. The amount of transverse reinforcement computed in the
previous section is the minimum required for confinement. The column also must be checked for shear
strength in based on ACI 318 Sec. 21.6.5.1. According to that section, the column shear is based on the
probable moment strength of the columns, but need not be more than what can be developed into the
column by the beams framing into the joint. However, the design shear cannot be less than the factored
shear determined from the analysis.
The shears computed based on the probable moment strength of the column can be conservative since the
actual column moments are limited by the moments that can be delivered by the beams. For this example,
however, the shear from the column probable moments will be checked first and then a determination will
be made if a more detailed limit state analysis should be used.
As determined from PCA Column, the maximum probable moment of the column in the range of factored
axial load is 14,940 in.kips. With a clear height of 124 inches, the column shear can be determined as
2(14,940)/124 = 241 kips. This shear will be compared to the capacity provided by the 4leg #4 hoops
spaced at 6 inches on center. If this capacity is in excess of the demand, the columns will be acceptable
for shear.
For the design of column shear capacity, the concrete contribution to shear strength may be considered
because the minimum Pu > Agf c/20. The design shear strength contributed by concrete and reinforcing
steel are as follows:
kips
kips
Vn = (Vc + Vs) = 0.75(116.7 + 220.0) = 252.5 kips > 241 kips OK
The column with the minimum transverse steel is therefore adequate for shear. The final column detail
with both longitudinal and transverse reinforcement is given in Figure 724. The spacing of
reinforcement through the joint has been reduced to 4 inches on center. This is done for practical reasons
only. Column bar splices, where required, should be located in the center half of the column and must be
proportioned as Class B tension splices.
Figure 724 Details of reinforcement for column
(1.0 in. = 25.4 mm)
7.4.3 Design of Frame 3 Shear Wall
This section addresses the design of a representative shear wall. The shear wall includes the 16inch wall
panel in between two 30 by 30inch columns. The design includes shear, flexureaxial interaction and
boundary elements.
The factored forces acting on the structural wall of Frame 3 are summarized in Table 717. The axial
compressive forces are based on the selfweight of the wall, a tributary area of 1,800 square feet of floor
area for the entire wall (includes column selfweight), an unfactored floor dead load of 139 psf and an
unfactored (reduced) floor live load of 20 psf. Based on the assumed 16inch wall thickness, the wall
between columns weighs (1.33 feet)(17.5 feet)(13 feet)(150 pcf) = 45.4 kips per floor. The total axial
force for a typical floor is:
Pu = 1.42D + 0.5L = 1.42[1,800(0.139) + 45,400] + 0.5[1,800(0.02)] = 456 kips for maximum
compression
Pu = 0.68D = 0.68[1,800(0.139) + 45,400] = 201 kips for minimum compression
The bending moments come from the ETABS analysis, using a section cut to combine forces in the wall
panel and end columns.
Note that the gravity moments and the earthquake axial loads on the shear wall are assumed to be
negligible given the symmetry of the system, so neither of these load effects are considered in the shear
wall design.
Table 717 Design Forces for Grid 3 Shear Wall
Supporting
Level
Axial Compressive Force Pu (kips)
Shear Vu
(kips)
Moment Mu (inch
kips)
1.42D + 0.5L
0.68D
R
420
201
173.2
35,375
12
876
402
133.9
50,312
11
1,332
603
156.7
63,337
10
1,788
804
195.7
73,993
9
2,243
1,005
221.8
81,646
8
2,699
1,206
252.4
86,298
7
3,155
1,408
294.6
90,678
6
3,611
1,609
344.9
102,405
5
4,067
1,810
400.7
132,941
4
4,523
2,011
467.5
178,321
3
4,979
2,212
546.0
241,021
2
5,435
2,413
663.3
366,136
1
5,891
2,614
580.3 (use 663.3)
258,851
(1.0 kip = 4.45 kN, 1.0 inchkip = 0.113 kNm)
7.4.3.1 Design for Shear Loads. First determine the required shear reinforcement in the wall panel and
then design the wall for combined bending and axial force. The nominal shear strength of the wall is
given by ACI 318 Equation 217:
where àc = 2.0 because hw/lw = 161/22.5 = 7.15 > 2.0, where the 161 feet is the wall height and 22.5 feet is
the overall wall length from the edges of the 30inch boundary columns.
Using fc' = 5,000 psi, fy = 60 ksi, = 1.0, Acv = (22.5)(12)(16) = 4,320 in2, the required amount of shear
reinforcement, t , can be determined by setting Vn = Vu. In accordance with ACI 318 Section 9.3.4,
the factor for shear is 0.60 for special structural walls unless the wall is specifically designed to be
governed by flexure yielding. If the walls were designed to be flexurecritical, then the factor for
shear would be 0.75, consistent with typical shear design. Unlike special moment frames, shearcritical
special shear walls are permitted (with the reduced ), although it should be noted that in areas of high
seismic hazard many practitioners recommend avoiding shearcritical shear walls where practical. In this
case, = 0.60 will be used for design.
The required reinforcement ratio for strength is determined as:
Since this is less than the minimum ratio of 0.0025 required by ACI 318 Section 21.9.2.1, that minimum
will apply to all levels of the wall. (This is a good indication that the actual wall thickness can be
reduced, but this example will proceed with the 16inch wall thickness.) Assuming two curtains of #5
bars spaced at 15 inches on center, t = 0.0026 and Vn = 768 kips, which exceeds the required shear
capacity at all levels.
Vertical reinforcing will be the same as the horizontal reinforcing based on the minimum reinforcing ratio
requirements of ACI 318 Section 21.9.2.1
7.4.3.2 Design for Flexural and Axial Loads. The flexural and axial design of special shear walls
includes two parts: design of the wall for flexural and axial loads and the design of boundary elements
where required. This section covers the design loads and the following section covers the boundary
elements.
The wall analysis was performed using PCA Column and considers the wall panel plus the boundary
columns. For axial and flexural loads, = 0.65 and 0.90, respectively. Figure 725 shows the
interaction diagram for the wall section below Level 2, considering the range of possible factored axial
loads. The wall panel is 16 inches thick and has two curtains of #5 bars at 15 inches on center. The
boundary columns are 30 by 30 inches with twelve #9 bars at this location. The section is clearly
adequate because the interaction curve fully envelopes the design values.
Figure 725 Interaction diagram for structural wall
(1.0 kip = 4.45kN, 1.0 inkip = 0.113 kNm)
7.4.3.3 Design of Boundary Elements. An important consideration in the ductility of special reinforced
concrete shear walls is the determination of where boundary elements are required and the design of them
where they are required. ACI 318 provides two methods for this. The first approach, specified in
ACI 318 Section 21.9.6.2, uses a displacement based procedure. The second approach, described in ACI
318 Section 21.9.6.3 uses a stressbased procedure and will be illustrated for this example.
In accordance with ACI 318 Section 21.9.6.3, special boundary elements are required where the
maximum extreme fiber compressive stress exceeds 0.2f c and they can be terminated where the stress is
less than 0.15f c. The stresses are determined based on the factored axial and flexure loads as shown in
Table 718. The stresses are determined using a wall area of 5,160 in2, a section modulus of 284,444 in3
and f c = 5,000 psi.
Table 718 Grid 3 Shear Wall Boundary Element Check
Supporting
Level
Axial Force
Pu (kips)
Moment Mu
(inchkips)
Maximum stress
Boundary
Element
Required
(ksi)
( fc )
R
420
35,375
0.206
0.04
No
12
876
50,312
0.347
0.07
No
11
1,332
63,337
0.481
0.10
No
10
1,788
73,993
0.607
0.12
No
9
2,243
81,646
0.722
0.14
No
8
2,699
86,298
0.827
0.17
Yes
7
3,155
90,678
0.930
0.19
Yes
6
3,611
102,405
1.060
0.21
Yes
5
4,067
132,941
1.256
0.25
Yes
4
4,523
178,321
1.503
0.30
Yes
3
4,979
241,021
1.812
0.36
Yes
2
5,435
366,136
2.340
0.47
Yes
1
5,891
258,851
2.052
0.41
Yes
(1.0 kip = 4.45 kN, 1.0 inkip = 0.113 kNm)
As can be seen, special boundary elements are required at the base of the wall and can be terminated
above Level 8.
Where they are required, the detailing of the special boundary element is based on ACI 318
Section 21.9.6.4.
According to ACI 318 Section 21.9.6.4 Item (a), the special boundary elements must have a minimum
plan length equal to the greater of c  0.1lw, or c/2, where c is the neutral axis depth and lw is the wall
length. The neutral axis depth is a function of the factored axial load and the nominal ( = 1.0) flexural
capacity of the wall section. This value is obtained from the PCA Column analysis for the wall section
and range of axial loads. For the Level 2 wall with twelve #9 vertical bars at each boundary column and
two curtains of #5 bars at 15 inches at vertical bars, the computed neutral axis depths are 32.4 inches and
75.3 inches for axial loads of 5,434 and 2,513 kips, respectively. For the governing case of 75.3 inches
and a wall length of 270 inches, the boundary element length is the greater of 75.3  0.1(270) = 48.3
inches and the second is 75.3/2 = 37.7 inches.
It is clear, therefore, that the special boundary element needs to extend beyond the 30inch edge columns
at least at the lower levels. For the wall below Level 5 where the maximum factored axial load is
4,067 kips, c = 53.6 inches and the required length is 27 inches, which fits within the boundary column.
For the walls from the basement to below Level 4, the boundary element can be detailed to extend into the
wall panel, or the concrete strength could be increased. Based on the desire to simply the reinforcing, the
wall concrete could be increased to fc' = 7,000 psi and the required boundary element length below
Level 2 is 26.9 inches. Figure 726 illustrates the variation in neutral axis depth based on factored axial
load and concrete strength. Although there is a cost premium for the higher strength concrete, this is still
in the range of commonly supplied concrete and will save costs by allowing the column rebar cage to
serve as the boundary element and have only distributed reinforcing in the wall panel itself. The use of
7,000 psi concrete at the lower levels will impact the calculations for maximum extreme fiber stress per
Table 718, but since the 7,000 psi concrete extends up to Level 4, not Level 8, the vertical extent of the
boundary elements is unchanged.
It is expected that the increase in concrete strength (and thus the modulus of elasticity) at the lower floors
will have a slight impact on the overall building stiffness, but this will not impact the overall design.
However, this should be verified.
Figure 726 Variations of neutral axis depth
(1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN)
Where special boundary elements are required, transverse reinforcement must conform to ACI 318
Section 21.9.6.4(c), which refers to ACI 318 Sections 21.6.4.2 through 21.6.4.4. In addition, this section
indicates that ACI 318 Equation 214 need not apply and the transverse reinforcing spacing limit of
ACI 318 Section 21.6.4.3(a) can be onethird of the least dimension of the element. Similar to columns
of special moment frames, there are requirements for spacing and total area of transverse reinforcing.
The spacing is determined as follows:
Onethird of least dimension = 30/3 = 10 inches
Six longitudinal bar diameters = 6(1.125) = 6.75 inches
Dimension so = 4 + (14  hx) / 3, where so is between 4 inches and 6 inches and hx is the maximum
horizontal spacing of hoops or cross ties.
Where hoops are used, the transverse reinforcement must satisfy ACI 318 Equation 215:
If #4 hoops with two crossties in each direction are used similar to the moment frame columns, Ash = 0.80
in2 and bc = 27 inches. For fc' = 7,000 psi and fyt = 60 ksi,
s = [(0.8)(60,000)]/[0.09(27.0)(7,000)] = 2.82 inches
which is impractical. Therefore, use #5 hoops and cross ties for the 7,000 psi concrete below Level 4, so
Ash = 4(0.31) = 1.24 in2 and s = 4.4 inches.
Where the concrete strength is 5,000 psi above Level 4, use #4 hoops and cross ties and the spacing, s =
3.95 inches.
Therefore, for the special boundary elements, use hoops with two cross ties spaced at 4 inches. The
hoops and cross ties are #5 below Level 4 and #4 above Level 4. ACI 318 Section 21.9.6.4(d) also
requires that the boundary element transverse reinforcement be extended beyond the base of the wall a
distance equal to the tension development length of the longitudinal reinforcement in the boundary
elements unless there is a mat or footing, in which case the transverse reinforcement extends down at least
12 inches.
Details of the boundary element and wall panel reinforcement are shown in Figures 727 and 728,
respectively. The vertical reinforcement in the boundary elements will be spliced as required using either
Class B lap splices or Type 2 mechanical splices at all locations. According to Table 715 (prepared for
5,000 psi concrete), there should be no difficulty in developing the horizontal wall panel steel into the 30
by 30inch boundary elements.
Figure 727 Details of structural wall boundary element
(1.0 in. = 25.4 mm)
Figure 728 Overall details of structural wall
(1.0 in. = 25.4 mm)
7.5 STRUCTURAL DESIGN OF THE HONOLULU BUILDING
The structure illustrated in Figures 71 and 72 is now designed and detailed for the Honolulu building.
Because of the relatively moderate level of seismicity, the lateral loadresisting system will consist of a
series of intermediate momentresisting frames in both the EW and NS directions. This is permitted for
Seismic Design Category C buildings in accordance with Standard Table 12.21. Design guidelines for
the reinforced concrete framing members are provided in ACI 318 Section 21.3.
As noted previously, the beams are assumed to be 30 inches deep by 20 inches wide and the columns are
28 inches by 28 inches. These are slightly smaller than the Berkeley building, reflecting the lower
seismicity.
7.5.1 Compare Seismic Versus Wind Loading
As has been discussed and as illustrated in Figure 73, wind forces appear to govern the strength
requirements of the structure at the lower floors and seismic forces control at the upper floors. The
seismic and wind shears, however, are so close at the middle levels of the structure that a careful
evaluation must be made to determine which load governs for strength. This determination requires
consideration of several load cases for both wind and seismic loads.
Because the Honolulu building is in Seismic Design Category C and does not have a Type 5 horizontal
irregularity (Standard Table 12.31); orthogonal loading effects need not be considered per Standard
Section 12.5.3. However, as required by Standard Section 12.8.4.2, accidental torsion must be
considered. Torsional amplification is not required per Provisions Section 12.8.4.3 because the building
does not have a torsional irregularity as determined previously.
For wind, the Standard requires that buildings over 60 feet in height be checked for four loading cases
under the Method 2 Analytical Procedure of Standard Section 6.5. The required load cases are shown in
Figure 729, which is reproduced directly from Standard Figure 69. In Cases 1 and 2, load is applied
separately in the two orthogonal directions, but Case 2 adds a torsional component. Cases 3 and 4
involve wind loads in two directions simultaneously and Case 4 adds a torsional component.
Figure 729 Wind loading requirements from ASCE 7
In this example, only loading in the EW direction is considered. Hence, the following lateral load
conditions are applied to the ETABS model:
EW seismic with accidental torsion
Wind Case 1 applied in EW direction only
Wind Case 2 applied in EW direction only
Wind Case 3
Wind Case 4
All cases with torsion are applied in such a manner as to maximize the shears in the elements of Frame 1,
for whose members the design is illustrated in the following section.
A simple method for determining which load case is likely to govern is to compare the beam shears for
each story. For the five load cases indicated above, the beam shears produced from seismic effects
control at the sixth level, with the next largest forces coming from direct EW wind Case 1. This is
shown graphically in Figure 730, where the beam shears at the center bay of Frame 1 are plotted versus
story height. Note that this comparison is based on 1.0 times seismic loads and 1.6 times wind loads
consistent with the strength design load combinations. Wind controls load at the lower four stories and
seismic controls for all other stories. This is somewhat different from that shown in Figure 73, wherein
the total story shears are plotted and where wind controlled for the lower five stories. A basic difference
between Figures 73 and 730 is that Figure 730 includes torsion effects.
Figure 730 Wind versus seismic shears in center bay of Frame 1
(1.0 ft = 0.3048 m, 1.0 kip = 4.45kN)
7.5.2 Design and Detailing of Members of Frame 1
In this section, the beams and a typical interior column of Level 6 of Frame 1 are designed and detailed.
7.5.2.1 Initial Calculations. The girders of Frame 1 are 30 inches deep and 20 inches wide. For positive
moment bending, the effective width of the compression flange is taken as 20 + 20(12)/12 = 40.0 inches.
Assuming 1.5inch cover, #4 stirrups and #9 longitudinal reinforcement, the effective depth for
computing flexural and shear strength is 30  1.5  0.5  1.125 / 2 = 27.4 inches.
7.5.2.2 Design of Representative Beams. ACI 318 Section 21.3.4 provides the minimum requirements
for longitudinal and transverse reinforcement in the beams of intermediate moment frames. The
requirements for longitudinal steel are as follows:
1. The positive moment strength at the face of a joint shall be at least onethird of the negative moment
strength at the same joint.
2. Neither the positive nor the negative moment strength at any section along the length of the member
shall be less than onefifth of the maximum moment strength supplied at the face of either joint.
The second requirement has the effect of requiring top and bottom reinforcement along the full length of
the member. The minimum reinforcement ratio at any section is taken from ACI 318 Section 10.5.1 as
200/fy or 0.0033 for fy = 60 ksi. However, according to ACI 318 Section 10.5.3, the minimum
reinforcement provided need not exceed 1.33 times the amount of reinforcement required for strength.
The gravity loads and design moments for the first three spans of Frame 1 are shown in Figure 731. The
seismic and gravity moments are determined from ETABS analysis, similar to the Berkeley building. All
moments are given at the face of the support. The gravity moments shown in Figures 731c and 731d are
slightly different from those shown for the Berkeley building (Figure 711) because the beam selfweight
is less and the clear span is longer due to the reduction in column size.
Figure 731 Bending moment envelopes at Level 6 of Frame 1
(1.0 ft = 0.3048 m, 1.0 kip/ft = 14.6 kN/m, 1.0 inkip = 0.113 kNm)
7.5.2.2.1 Longitudinal Reinforcement. Based on a minimum amount of longitudinal reinforcing of
0.0033bwd = 0.0033(20)(27.4)=1.83 in2, provide two #9 bars continuous top and bottom as a starting point
and provide additional reinforcing as required.
1. Design for Negative Moment at the Face of the Exterior Support (Grid A)
Mu = 1.3 (567)  0.5 (259)  1.0 (2,557) = 3,423 inchkips
Try two #9 bars plus one #7 bar.
As = 2 (1.00) + 0.60 = 2.60 in2
Depth of compression block, a = [2.6 (60)]/[0.85 (5) 20] = 1.83 inches
Nominal strength, Mn = [2.60 (60)] [27.4  1.83/2] = 4,131 inchkips
Design strength, Mn = 0.9 (4,131) = 3,718 inchkips > 3,423 inchkips OK
This reinforcement also will work for negative moment at all other supports.
2. Design for Positive Moment at the Face of the Exterior Support (Grid A)
Mu = 0.8 (567) + 1.0 (2,557) = 2,114 inchkips
Try the minimum of two #9 bars.
As = 2 (1.00) = 2.00 in2
a = 2.00 (60)/[0.85 (5) 40] = 0.71 inch
Mn = [2.00 (60)] [27.4 0.71/2] = 3,246 inchkips
Mn = 0.9 (3,246) = 2,921 inchkips > 2,114 inchkips OK
This reinforcement also will work for positive moment at all other supports.
The layout of flexural reinforcement layout is shown in Figure 732. The top short bars are cut off
5 feet0 inch from the face of the support. The bottom bars are spliced in Spans A'B and CC' with a
Class B lap length of 37 inches. Unlike special moment frames, there are no requirements that the spliced
region of the bars in intermediate moment frames be confined by hoops over the length of the splice.
Note that the steel clearly satisfies the detailing requirements of ACI 318 Section 21.3.4.1.
Figure 732 Longitudinal reinforcement layout for Level 6 of Frame 1
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
7.5.2.2.2 Transverse Reinforcement. The requirements for transverse reinforcement in intermediate
moment frames are somewhat different from those in special moment frames, both in terms of detailing
and shear design. The shear strength requirements will be covered first, followed by the detailing
requirements.
In accordance with ACI 318 Section 21.3.3, the design earthquake shear for the design of intermediate
moment frame beams must be larger than the smaller of the following:
a. The sum of the shears associated with the nominal moment strength at the ends of the
members. Nominal moment strengths are computed with a flexural reinforcement tensile
strength of 1.0fy and a flexural factor of 1.0.
b. Two times the factored earthquake shear force determined from the structural analysis.
In either case, the earthquake shears are combined with the factored gravity shears to determine the total
design shear.
Consider the end span between Grids A and A . For determining earthquake shears per Item a above, the
nominal strengths at the ends of the beam were computed earlier as 3,246 inchkips for positive moment
at Support A and 4,131 inchkips for negative moment at Support A. Compute the design earthquake
shear VE:
where 212 inches is the clear span of the member. The shear is the same for earthquake forces acting in
the other direction.
For determining earthquake shears per Item b above, the shear is taken from the ETABS analysis as 23.4
kips. The design earthquake shear for this method is 2(23.4 kips) = 46.8 kips.
Since the design shear using Item a is the smaller value, it is used for computing the design shear.
The gravity load shears are taken from the ETABS model. Since the gravity shears at Grid A are similar
but slightly larger than those at Grid A, Grid A will be used for the design. From the ETABS analysis,
VD = 20.7 kips and VL = 9.5 kips.
The factored design shear Vu = 1.3(20.7) + 0.5(9.5) + 1.0(46.8) = 66.5 kips. This shear force applies for
earthquake forces coming from either direction as shown in the shear strength design envelope in
Figure 733.
The design shear force is resisted by a concrete component, Vc and a steel component, Vs. Note that the
concrete component may be used regardless of the ratio of earthquake shear to total shear. The required
design strength is:
Vu ó Vc + Vs
where = 0.75 for shear.
The shear to be resisted by reinforcing steel, assuming two #4 vertical legs (Av = 0.4) and fy = 60 ksi is:
Using Vs = Av fyd/s:
Minimum transverse steel requirements are given in ACI 318 Section 21.3.4.2. At the ends of the beam,
hoops are required. The first hoop must be placed 2 inches from the face of the support and within a
distance 2h from the face of the support, the spacing should be not greater than d/4, eight times the
smallest longitudinal bar diameter, 24 times the hip bar diameter, or 12 inches. For the beam under
consideration d/4 controls minimum transverse steel, with the maximum spacing being 27.4/4 =
6.8 inches, which is less than what is required for shear strength.
In the remainder of the span, stirrups are permitted and must be placed at a maximum of d/2 (ACI 318
Sec. 21.3.4.3).
Because the earthquake shear (at midspan where the gravity shear is essentially zero) is greater than
50 percent of the shear strength provided by concrete alone, the minimum requirements of ACI 318
Section 11.4.6.1 must be checked:
This spacing does not control over the d/2 requirement. The layout of transverse reinforcement for the
beam is shown in Figure 732. This spacing is used for all other spans as well.
Figure 733 Shear strength envelopes for Span AA' of Frame 1
(1.0 in. = 25.4 mm, 1.0 kip = 4.45kN, 1.0 inkip = 0.113 kNm)
7.5.2.3 Design of Representative Column of Frame 1. This section illustrates the design of a typical
interior column on Gridline A'. The column, which supports Level 6 of Frame 1, is 28 inches square and
is constructed from 5,000 psi concrete and 60 ksi reinforcement. An isolated view of the column is
shown in Figure 734.
The column supports an unfactored axial dead load of 506 kips and an unfactored axial live load
(reduced) of 117 kips. The ETABS analysis indicates that the axial earthquake force is ñ19.9 kips, the
earthquake shear force is ñ37.7 kips and the earthquake moments at the top and the bottom of the column
are ñ2,408 and ñ2,340 inchkips, respectively. Moments and shears due to gravity loads are assumed to
be negligible.
Figure 734 Isolated view of Column A'
(1.0 ft = 0.3048 m, 1.0 kip = 4.45kN)
7.5.2.3.1 Longitudinal Reinforcement. The factored gravity force for maximum compression (without
earthquake) is:
Pu = 1.2(506) + 1.6(117) = 794 kips
This force acts with no significant gravity moment.
The factored gravity force for maximum compression (including earthquake) is:
Pu = 1.3(506) + 0.5(117) + 19.9 = 736 kips
The factored gravity force for minimum compression (including earthquake) is:
Pu = 0.8(506) 19.9 = 385 kips
Before proceeding with the flexural strength calculations, first determine whether or not slenderness
effects need to be considered. For a frame that is unbraced against sideway, ACI 318 Section 10.10.1
allows slenderness effects to be neglected where klu/r < 22. For a 28 by 28inch column with a clear
unbraced length, lu = 126 inches, r = 0.3(28) = 8.4 inches (ACI 318 Sec. 10.10.1.2) and lu/r = 126/8.4 =
15.0. Therefore, as long as the effective length factor k for this column is less than 22/15.0 = 1.47, then
slenderness effects can be ignored. It is reasonable to assume that k is less than 1.47 and this can be
confirmed using the commentary to ACI 318 Section 10.10.1.
Continuing with the design, an axialflexural interaction diagram for a 28 by 28inch column with 12 #8
bars ( = 0.0121) is shown in Figure 735. The column clearly has the strength to support the applied
loads (represented as solid dots in the figure).
Figure 735 Interaction diagram for column
(1.0 kip = 4.45kN, 1.0 ftkip = 1.36 kNm)
7.5.2.3.2 Transverse Reinforcement. The design earthquake shear for columns in determined in the
same manner as for beams in accordance with ACI 318 Section 21.3.3 as described in Section 7.5.2.2.2.
Assuming two times the shear from analysis will produce the smaller design shear, the ETABS analysis
indicates that the shear force is 37.7 kips and the design shear is 2.0(37.7) = 75.4 kips.
The concrete supplies a capacity of:
> 75.4 kips OK
Therefore, steel reinforcement is not required for strength, but shear reinforcement is required per
ACI 318 Section 11.4.6.1 since the design shear exceeds onehalf of the design capacity of the concrete
alone. First, however, determine the detailing requirements for transverse reinforcement in intermediate
moment frame columns in accordance ACI 318 Section 21.3.5.
Within a region lo from the face of the support, the tie spacing must not exceed:
8db = 8(1.0) = 8.0 inches (using #8 longitudinal bars)
24dtie = 24 (0.5) = 12.0 inches (using #4 ties)
1/2 the smallest dimension of the frame member = 28/2 = 14 inches
12 inches
The 8inch maximum spacing controls. Ties at this spacing are required over a length lo of:
1/6 clearspan of column = 126/6 = 21 inches
Maximum cross section dimension = 28 inches
18 inches
Given the above, a fourlegged #4 tie spaced at 8 inches over a depth of 28 inches will be used. The top
and bottom ties will be provided at 4 inches from the beam soffit and floor slab.
Beyond the end regions, ACI 318 Section 21.3.5.4 requires that tie spacing satisfy ACI 318 Sections 7.10
and 11.4.5.1, but the minimum shear reinforcing requirement of ACI 318 Section 11.4.6.1 also applies.
Of the above requirements, ACI 318 Section 11.4.5.1, which requires ties at d/2 maximum spacing,
governs. Therefore, use a 12inch tie spacing in the middle region of the column.
The layout of the transverse reinforcing for the subject column is shown in Figure 736
Figure 736 Column reinforcement
(1.0 in. = 25.4 mm)
7.5.2.4 Design of BeamColumn Joint. Joint reinforcement for intermediate moment frames is
addressed in ACI 318 Section 21.3.5.5, which refers to ACI 318 Section 11.10. ACI 318 Section 11.10
requires that all beamcolumn connections have a minimum amount of transverse reinforcement through
the beamcolumn joints. The only exception is in nonseismic frames where the column is confined on all
four sides by beams framing into the column. The amount of reinforcement required is given by ACI 318
Equation 1113:
This is the same equation used to proportion minimum transverse reinforcement in beams. Assuming Av
is supplied by four #4 ties and fy = 60 ksi:
32.4 inches
This essentially permits no ties to be located in the joint. Since it is good practice to provide transverse
reinforcing in moment frame joints, ties will be provided at the same 8inch spacing as at the ends of the
columns. The arrangement of ties within the beamcolumn joint is shown in Figure 736.
8
Precast Concrete Design
Suzanne Dow Nakaki, S.E.
Originally developed by
Gene R. Stevens, P.E. and James Robert Harris, P.E., PhD
Contents
8.1 HORIZONTAL DIAPHRAGMS 4
8.1.1 Untopped Precast Concrete Units for FiveStory Masonry Buildings Located in
Birmingham, Alabama and New York, New York 4
8.1.2 Topped Precast Concrete Units for FiveStory Masonry Building Located in Los Angeles,
California (see Sec. 10.2) 18
8.2 THREESTORY OFFICE BUILDING WITH INTERMEDIATE PRECAST CONCRETE
SHEAR WALLS 26
8.2.1 Building Description 27
8.2.2 Design Requirements 28
8.2.3 Load Combinations 29
8.2.4 Seismic Force Analysis 30
8.2.5 Proportioning and Detailing 33
8.3 ONESTORY PRECAST SHEAR WALL BUILDING 45
8.3.1 Building Description 45
8.3.2 Design Requirements 48
8.3.3 Load Combinations 49
8.3.4 Seismic Force Analysis 50
8.3.5 Proportioning and Detailing 52
8.4 SPECIAL MOMENT FRAMES CONSTRUCTED USING PRECAST CONCRETE 65
8.4.1 Ductile Connections 65
8.4.2 Strong Connections 67
This chapter illustrates the seismic design of precast concrete members using the NEHRP Recommended
Provisions (referred to herein as the Provisions) for buildings in several different seismic design
categories. Over the past several years there has been a concerted effort to coordinate the requirements in
the Provisions with those in ACI 318, so that now there are very few differences between the two. Very
briefly, the Provisions set forth the following requirements for precast concrete structural systems.
Precast seismic systems used in structures assigned to Seismic Design Category C must be
intermediate or special moment frames, or intermediate precast or special structural walls.
Precast seismic systems used in structures assigned to Seismic Design Category D must be
special moment frames, or intermediate precast (up to 40 feet) or special structural walls.
Precast seismic systems used in structures assigned to Seismic Design Category E or F must be
special moment frames or special structural walls.
Prestress provided by prestressing steel resisting earthquakeinduced flexural and axial loads in
frame members must be limited to 700 psi or f c/6 in plastic hinge regions. These values are
different from the ACI 318 limitations, which are 500 psi or f c/10.
An ordinary precast structural wall is defined as one that satisfies ACI 318 Chapters 118.
An intermediate precast structural wall must meet additional requirements for its connections
beyond those defined in ACI 318 Section 21.4. These include requirements for the design of wall
piers that amplify the design shear forces and prescribe wall pier detailing and requirements for
explicit consideration of the ductility capacity of yielding connections.
A special structural wall constructed using precast concrete must satisfy the acceptance criteria
defined in Provisions Section 9.6 if it doesn t meet the requirements for special structural walls
constructed using precast concrete contained in ACI 318 Section 21.10.2.
Examples are provided for the following concepts:
The example in Section 8.1 illustrates the design of untopped and topped precast concrete floor
and roof diaphragms of the fivestory masonry buildings described in Section 10.2 of this volume
of design examples. The two untopped precast concrete diaphragms of Section 8.1.1 show the
requirements for Seismic Design Categories B and C using 8inchthick hollow core precast,
prestressed concrete planks. Section 8.1.2 shows the same precast plank with a 21/2inchthick
composite lightweight concrete topping for the fivestory masonry building in Seismic Design
Category D described in Section 10.2. Although untopped diaphragms are commonly used in
regions of low seismic hazard, their design is not specifically addressed in the Provisions, the
Standard, or ACI 318.
The example in Section 8.2 illustrates the design of an intermediate precast concrete shear wall
building in a region of low or moderate seismicity, which is where many precast concrete seismic
forceresisting systems are constructed. The precast concrete walls in this example resist the
seismic forces for a threestory office building located in southern New England (Seismic Design
Category B). The Provisions have a few requirements beyond those in ACI 318 and these
requirements are identified in this example. Specifically, ACI 318 requires that in connections
that are expected to yield, the yielding be restricted to steel elements or reinforcement. The
Provisions also require that the deformation capacity of the connection be compared to the
deformation demand on the connection unless Type 2 mechanical splices are used. There are
additional requirements for intermediate precast structural walls relating to wall piers; however,
due to the geometry of the walls used in this design example, this concept is not described in the
example.
The example in Section 8.3 illustrates the design of a special precast concrete shear wall for a
singlestory industrial warehouse building in Los Angeles. For buildings assigned to Seismic
Design Category D, the Provisions require that the precast seismic forceresisting system be
designed and detailed to meet the requirements for either an intermediate or special precast
concrete structural wall. The detailed requirements in the Provisions regarding explicit
calculation of the deformation capacity of the yielding element are shown here.
The example in Section 8.4 shows a partial example for the design of a special moment frame
constructed using precast concrete per ACI 318 Section 21.8. Concepts for ductile and strong
connections are presented and a detailed description of the calculations for a strong connection
located at the beamcolumn interface is presented.
Tiltup concrete wall buildings in all seismic zones have long been designed using the precast wall panels
as concrete shear walls for the seismic forceresisting system. Such designs usually have been performed
using design force coefficients and strength limits as if the precast walls emulated the performance of
castinplace reinforced concrete shear walls, which they usually do not. Tiltup buildings assigned to
Seismic Design Category C or higher should be designed and detailed as intermediate or special precast
structural wall systems as defined in ACI 318.
In addition to the Provisions, the following documents are either referred to directly or are useful design
aids for precast concrete construction:
ACI 318 American Concrete Institute. 2008. Building Code Requirements for
Structural Concrete.
AISC 360 American Institute of Steel Construction. 2005. Specification for Structural
Steel Buildings.
AISC Manual American Institute of Steel Construction. 2005. Manual of Steel
Construction, Thirteen Edition.
Moustafa Moustafa, Saad E. 1981 and 1982. Effectiveness of ShearFriction
Reinforcement in Shear Diaphragm Capacity of HollowCore Slabs.
PCI Journal, Vol. 26, No. 1 (Jan.Feb. 1981) and the discussion contained in
PCI Journal, Vol. 27, No. 3 (MayJune 1982).
PCI Handbook Precast/Prestressed Concrete Institute. 2004. PCI Design Handbook, Sixth
Edition.
PCI Details Precast/Prestressed Concrete Institute. 1988. Design and Typical Details of
Connections for Precast and Prestressed Concrete, Second Edition.
SEAA Hollow Core Structural Engineers Association of Arizona, Central Chapter. Design and
Detailing of Untopped HollowCore Slab Systems for Diaphragm Shear.
The following style is used when referring to a section of ACI 318 for which a change or insertion is
proposed by the Provisions: Provisions Section xxx (ACI 318 Sec. yyy) where xxx is the section in the
Provisions and yyy is the section proposed for insertion into ACI 318.
8.1 HORIZONTAL DIAPHRAGMS
Structural diaphragms are horizontal or nearly horizontal elements, such as floors and roofs, that transfer
seismic inertial forces to the vertical seismic forceresisting members. Precast concrete diaphragms may
be constructed using topped or untopped precast elements depending on the Seismic Design Category.
Reinforced concrete diaphragms constructed using untopped precast concrete elements are not addressed
specifically in the Standard, in the Provisions, or in ACI 318. Topped precast concrete elements, which
act compositely or noncompositely for gravity loads, are designed using the requirements of ACI 318
Section 21.11.
8.1.1 Untopped Precast Concrete Units for FiveStory Masonry Buildings Located in
Birmingham, Alabama and New York, New York
This example illustrates floor and roof diaphragm design for fivestory masonry buildings located in
Birmingham, Alabama, on soft rock (Seismic Design Category B) and in New York, New York (Seismic
Design Category C). The example in Section 10.2 provides design parameters used in this example. The
floors and roofs of these buildings are to be untopped 8inchthick hollow core precast, prestressed
concrete plank. Figure 10.21 shows the typical floor plan of the diaphragms.
8.1.1.1 General Design Requirements. In accordance with ACI 318, untopped precast diaphragms are
permitted only in Seismic Design Categories A through C. Static rational models are used to determine
shears and moments on joints as well as shear and tension/compression forces on connections. Dynamic
modeling of seismic response is not required. Per ACI 318 Section 21.1.1.6, diaphragms in Seismic
Design Categories D through F are required to meet ACI 318 Section 21.11, which does not allow
untopped diaphragms. In previous versions of the Provisions, an appendix was presented that provided a
framework for the design of untopped diaphragms in higher Seismic Design Categories in which
diaphragms with untopped precast elements were designed to remain elastic and connections designed for
limited ductility. However, in the 2009 Provisions, that appendix has been removed. Instead, a white
paper describing emerging procedures for the design of such diaphragms has been included in Part 3 of
the Provisions.
The design method used here is that proposed by Moustafa. This method makes use of the shear friction
provisions of ACI 318 with the friction coefficient, æ, being equal to 1.0. To use æ = 1.0, ACI 318
requires grout or concrete placed against hardened concrete to have clean, laitance free and intentionally
roughened surfaces with a total amplitude of approximately 1/4 inch (peak to valley). Roughness for
formed edges is provided either by sawtooth keys along the length of the plank or by hand roughening
with chipping hammers. Details from the SEAA Hollow Core reference are used to develop the
connection details. Note that grouted joints with edges not intentionally roughened can be used with
= 0.6.
The terminology used is defined in ACI 318 Section 2.2.
8.1.1.2 General InPlane Seismic Design Forces for Untopped Diaphragms. For Seismic Design
Categories B through F, Standard Section 12.10.1.1 defines a minimum diaphragm seismic design force.
For Seismic Design Categories C through F, Standard Section 12.10.2.1 requires that collector elements,
collector splices and collector connections to the vertical seismic forceresisting members be designed in
accordance with Standard Section 14.4.3.2, which amplifies design forces by means of the overstrength
factor, o.
For Seismic Design Categories D, E and F, Standard Section 12.10.1.1 requires that the redundancy
factor, , be used on transfer forces only where the vertical seismic forceresisting system is offset and the
diaphragm is required to transfer forces between the elements above and below, but need not be applied to
inertial forces defined in Standard Equation 12.101.
Parameters from the example in Section 10.2 used to calculate inplane seismic design forces for the
diaphragms are provided in Table 8.11.
Table 8.11 Design Parameters from Example 10.2
Design Parameter
Birmingham 1
New York City
1.0
1.0
êo
2.5
2.5
Cs
0.12
0.156
wi (roof)
861 kips
869 kips
wi (floor)
963 kips
978 kips
SDS
0.24
0.39
I
1.0
1.0
1.0 kip = 4.45 kN.
8.1.1.3 Diaphragm Forces for Birmingham Building 1. The weight tributary to the roof and floor
diaphragms (wpx) is the total story weight (wi) at Level i minus the weight of the walls parallel to the
direction of loading.
Compute diaphragm weight (wpx) for the roof and floor as follows:
Roof:
Total weight = 861 kips
Walls parallel to force = (45 psf)(277 ft)(8.67 ft / 2) = 54 kips
wpx = 807 kips
Floors:
Total weight = 963 kips
Walls parallel to force = (45 psf)(277 ft)(8.67 ft) = 108 kips
wpx = 855 kips
Compute diaphragm demands in accordance with Standard Equation 12.101:
Calculations for Fpx are provided in Table 8.12.
Table 8.12 Birmingham 1 Fpx Calculations
Level
wi
(kips)
(kips)
Fi
(kips)
(kips)
wpx
(kips)
Fpx
(kips)
Roof
861
861
175
175
807
164
4
963
1,824
156
331
855
155
3
963
2,787
117
448
855
137
2
963
3,750
78
526
855
120
1
963
4,713
39
565
855
103
1.0 kip = 4.45 kN.
The values for Fi and Vi used in Table 8.12 are listed in Table 10.22.
The minimum value of Fpx = 0.2SDSIwpx = 0.2(0.24)1.0(807 kips) = 38.7 kips (roof)
= 0.2(0.24)1.0(855 kips) = 41.0 kips (floors)
The maximum value of Fpx = 0.4SDSIwpx = 2(38.7 kips) = 77.5 kips (roof)
= 2(41.0 kips) = 82.1 kips (floors)
Note that the calculated Fpx in Table 8.12 is substantially larger than the specified maximum limit value
of Fpx. This is generally true at upper levels if the R factor is less than 5.
To simplify the design, the diaphragm design force used for all levels will be the maximum force at any
level, 82 kips.
8.1.1.4 Diaphragm Forces for New York Building. The weight tributary to the roof and floor
diaphragms (wpx) is the total story weight (wi) at Level i minus the weight of the walls parallel to the
force.
Compute diaphragm weight (wpx) for the roof and floor as follows:
Roof:
Total weight = 870 kips
Walls parallel to force = (48 psf)(277 ft)(8.67 ft / 2) = 58 kips
wpx = 812 kips
Floors:
Total weight = 978 kips
Walls parallel to force = (48 psf)(277 ft)(8.67 ft) = 115 kips
wpx = 863 kips
Calculations for Fpx are provided in Table 8.13.
Table 8.13 New York Fpx Calculations
Level
wi
(kips)
(kips)
Fi
(kips)
(kips)
wpx
(kips)
Fpx
(kips)
Roof
870
870
229
229
812
214
4
978
1,848
207
436
863
204
3
978
2,826
155
591
863
180
2
978
3,804
103
694
863
157
1
978
4,782
52
746
863
135
1.0 kip = 4.45 kN.
The values for Fi and Vi used in Table 8.13 are listed in Table 10.27.
The minimum value of Fpx = 0.2SDSIwpx = 0.2(0.39)1.0(870 kips) = 67.9 kips (roof)
= 0.2(0.39)1.0(978 kips) = 76.3 kips (floors)
The maximum value of Fpx = 0.4SDSIwpx = 2(67.9 kips) = 135.8 kips (roof)
= 2(76.3 kips) = 152.6 kips (floors)
As for the Birmingham example, note that the calculated Fpx given in Table 8.13 is substantially larger
than the specified maximum limit value of Fpx.
To simplify the design, the diaphragm design force used for all levels will be the maximum force at any
level, 153 kips.
8.1.1.5 Static Analysis of Diaphragms. The balance of this example will use the controlling diaphragm
seismic design force of 153 kips for the New York building. In the transverse direction, the loads will be
distributed as shown in Figure 8.11.
Figure 8.11 Diaphragm force distribution and analytical model
(1.0 ft = 0.3048 m)
The Standard requires that structural analysis consider the relative stiffness of the diaphragms and the
vertical elements of the seismic forceresisting system. Since a pretopped precast diaphragm doesn t
satisfy the conditions of either the flexible or rigid diaphragm conditions identified in the Standard,
maximum inplane deflections of the diaphragm must be evaluated. However, that analysis is beyond the
scope of this document. Therefore, with a rigid diaphragm assumption, assuming the four shear walls
have the same stiffness and ignoring torsion, the diaphragm reactions at the transverse shear walls (F as
shown in Figure 8.11) are computed as follows:
F = 153 kips/4 = 38.3 kips
The uniform diaphragm demands are proportional to the distributed weights of the diaphragm in different
areas (see Figure 8.11).
W1 = [67 psf (72 ft) + 48 psf (8.67 ft)4](153 kips / 863 kips) = 1,150 lb/ft
W2 = [67 psf (72 ft)](153 kips / 863 kips) = 855 lb/ft
Figure 8.12 identifies critical regions of the diaphragm to be considered in this design. These regions
are:
Joint 1: Maximum transverse shear parallel to the panels at paneltopanel joints
Joint 2: Maximum transverse shear parallel to the panels at the paneltowall joint
Joint 3: Maximum transverse moment and chord force
Joint 4: Maximum longitudinal shear perpendicular to the panels at the paneltowall connection
(exterior longitudinal walls) and anchorage of exterior masonry wall to the diaphragm for outof
plane forces
Joint 5: Collector element and shear for the interior longitudinal walls
Figure 8.12 Diaphragm plan and critical design regions
(1.0 ft = 0.3048 m)
Joint forces are as follows:
Joint 1 Transverse forces:
Shear, Vu1 = 1.15 kips/ft (36 ft) = 41.4 kips
Moment, Mu1 = 41.4 kips (36 ft / 2) = 745 ftkips
Chord tension force, Tu1 = M/d = 745 ftkips / 71 ft = 10.5 kips
Joint 2 Transverse forces:
Shear, Vu2 = 1.15 kips/ft (40 ft) = 46 kips
Moment, Mu2 = 46 kips (40 ft / 2) = 920 ftkips
Chord tension force, Tu2 = M/d = 920 ftkips / 71 ft = 13.0 kips
Joint 3 Transverse forces:
Shear, Vu3 = 46 kips + 0.86 kips/ft (24 ft) 38.3 kips = 28.3 kips
Moment, Mu3 = 46 kips (44 ft) + 20.6 kips (12 ft)  38.3 kips (24 ft) = 1,352 ftkips
Chord tension force, Tu3 = M/d = 1,352 ftkips / 71 ft = 19.0 kips
Joint 4 Longitudinal forces:
Wall force, F = 153 kips / 8 = 19.1 kips
Wall shear along wall length, Vu4 = 19.1 kips (36 ft)/(152 ft / 2) = 9.0 kips
Collector force at wall end, Tu4 = Cu4 = 19.1 kips  9.0 kips = 10.1 kips
Joint 4 Outofplane forces:
The Standard has several requirements for outofplane forces. None are unique to precast
diaphragms and all are less than the requirements in ACI 318 for precast construction regardless of
seismic considerations. Assuming the planks are similar to beams and comply with the minimum
requirements of Standard Section 12.14 (Seismic Design Category B and greater), the required out
ofplane horizontal force is:
0.05(D+L)plank = 0.05(67 psf + 40 psf)(24 ft / 2) = 64.2 plf
According to Standard Section 12.11.2 (Seismic Design Category B and greater), the minimum
anchorage for masonry walls is:
400(SDS)(I) = 400(0.39)1.0 = 156 plf
According to Standard Section 12.11.1 (Seismic Design Category B and greater), bearing wall
anchorage must be designed for a force computed as:
0.4(SDS)(Wwall) = 0.4(0.39)(48 psf)(8.67 ft) = 64.9 plf
Standard Section 12.11.2.1 (Seismic Design Category C and greater) requires masonry wall
anchorage to flexible diaphragms to be designed for a larger force. Due to its geometry, this
diaphragm is likely to be classified as rigid. However, the relative deformations of the wall and
diaphragm must be checked in accordance with Standard Section 12.3.1.3 to validate this assumption.
Fp = 0.85(SDS)(I)(Wwall) = 0.85(0.39)1.0[(48 psf)(8.67 ft)] = 138 plf
(Note that since this diaphragm is not flexible, this load is not used in the following calculations.)
The force requirements in ACI 318 Section 16.5 will be described later.
Joint 5 Longitudinal forces:
Wall force, F = 153 kips / 8 = 19.1 kips
Wall shear along each side of wall, Vu5 = 19.1 kips [2(36 ft) / 152 ft]/2 = 4.5 kips
Collector force at wall end, Tu5 = Cu5 = 19.1 kips  2(4.5 kips) = 10.1 kips
Joint 5 Shear flow due to transverse forces:
Shear at Joint 2, Vu2 = 46 kips
Q = A d
A = (0.67 ft) (24 ft) = 16 ft2
d = 24 ft
Q = (16 ft2) (24 ft) = 384 ft3
I = (0.67 ft) (72 ft)3 / 12 = 20,840 ft4
Vu2Q/I = (46 kip) (384 ft3) / 20,840 ft4 = 0.847 kip/ft maximum shear flow
Joint 5 length = 40 ft
Total transverse shear in joint 5, Vu5 = 0.847 kip/ft) (40 ft)/2 = 17 kips
ACI 318 Section 16.5 also has minimum connection force requirements for structural integrity of precast
concrete bearing wall building construction. For buildings over two stories tall, there are force
requirements for horizontal and vertical members. This building has no vertical precast members.
However, ACI 318 Section 16.5.1 specifies that the strengths ... for structural integrity shall apply to all
precast concrete structures. This is interpreted to apply to the precast elements of this masonry bearing
wall structure. The horizontal tie force requirements for a precast bearing wall structure three or more
stories in height are:
1,500 pounds per foot parallel and perpendicular to the span of the floor members. The
maximum spacing of ties parallel to the span is 10 feet. The maximum spacing of ties
perpendicular to the span is the distance between supporting walls or beams.
16,000 pounds parallel to the perimeter of a floor or roof located within 4 feet of the edge at all
edges.
ACI s tie forces are far greater than the minimum tie forces given in the Standard for beam supports and
anchorage of masonry walls. They do control some of the reinforcement provided, but most of the
reinforcement is controlled by the computed connections for diaphragm action.
8.1.1.6 Diaphragm Design and Details. The phi factors used for this example are as follows:
Tension control (bending and ties): = 0.90
Shear: = 0.75
Compression control in tied members: = 0.65
The minimum tie force requirements given in ACI 318 Section 16.5 are specified as nominal values,
meaning that = 1.00 for those forces.
Note that although buildings assigned to Seismic Design Category C are not required to meet ACI 318
Section 21.11, some of the requirements contained therein are applied below as good practice but shown
as optional.
8.1.1.6.1 Joint 1 Design and Detailing. The design must provide sufficient reinforcement for chord
forces as well as shear friction connection forces, as follows:
Chord reinforcement, As1 = Tu1/ fy = (10.5 kips)/[0.9(60 ksi)] = 0.19 in2 (The collector force from
the Joint 4 calculations at 10.1 kips is not directly additive.)
Shear friction reinforcement, Avf1 = Vu1/ æfy = (41.4 kips)/[(0.75)(1.0)(60 ksi)] = 0.92 in2
Total reinforcement required = 2(0.19 in2) + 0.92 in2 = 1.30 in2
ACI tie force = (1.5 kips/ft)(72 ft) = 108 kips; reinforcement = (108 kips)/(60 ksi) = 1.80 in2
Provide four #5 bars (two at each of the outside edges) plus four #4 bars (two each at the interior joint at
the ends of the plank) for a total area of reinforcement of 4(0.31 in2) + 4(0.2 in2) = 2.04 in2.
Because the interior joint reinforcement acts as the collector reinforcement in the longitudinal direction
for the interior longitudinal walls, the cover and spacing of the two #4 bars in the interior joints will be
provided to meet the requirements of ACI 318 Section 21.11.7.6 (optional):
Minimum cover = 2.5(4/8) = 1.25 in., but not less than 2.00 in.
Minimum spacing = 3(4/8) = 1.50 in., but not less than 1.50 in.
Figure 8.13 shows the reinforcement in the interior joints at the ends of the plank, which is also the
collector reinforcement for the interior longitudinal walls (Joint 5). The two #4 bars extend along the
length of the interior longitudinal walls as shown in Figure 8.13.
Figure 8.13 Interior joint reinforcement at the ends of plank and collector reinforcement
at the end of the interior longitudinal walls  Joints 1 and 5
(1.0 in. = 25.4 mm)
Figure 8.14 shows the extension of the two #4 bars of Figure 8.13 into the region where the plank is
parallel to the bars (see section cut on Figure 8.12). The bars will need to be extended the full length of
the diaphragm unless supplemental plank reinforcement is provided. This detail makes use of this
supplement plank reinforcement (two #4 bars or an equal area of strand) and shows the bars anchored at
each end of the plank. The anchorage length of the #4 bars is calculated using ACI 318 Chapter 12:
Using #4 bars, the required ld = 37.9(0.5 in.) = 18.9 in. Therefore, use ld = 4 ft, which is the width of the
plank.
Figure 8.14 Anchorage region of shear reinforcement for Joint 1 and
collector reinforcement for Joint 5
(1.0 in. = 25.4 mm)
8.1.1.6.2 Joint 2 Design and Detailing. The chord design is similar to the previous calculations:
Chord reinforcement, As2 = Tu2/ fy = (13.0 kips)/[0.9(60 ksi)] = 0.24 in2
The shear force may be reduced along Joint 2 by the shear friction resistance provided by the
supplemental chord reinforcement (2Achord  As2) and by the four #4 bars projecting from the interior
longitudinal walls across this joint. The supplemental chord bars, which are located at the end of the
walls, are conservatively excluded here. The shear force along the outer joint of the wall where the plank
is parallel to the wall is modified as follows:
= 36.0 kips
This force must be transferred from the planks to the wall. Using the arrangement shown in Figure 8.15,
the required shear friction reinforcement (Avf2) is computed as:
Use two #3 bars placed at 26.6 degrees (2to1 slope) across the joint at 6 feet from the ends of the plank
(two sets per plank). The angle (àf) used above provides development of the #3 bars while limiting the
grouting to the outside core of the plank. The total shear reinforcement provided is 6(0.11 in2) = 0.66 in2.
Note that the spacing of these connectors will have to be adjusted at the stair location.
The shear force between the other face of this wall and the diaphragm is:
Vu2F = 4638.3 = 7.7 kips
The shear friction resistance provided by #3 bars in the grout key between each plank (provided for the
1.5 klf requirement of ACI 318) is computed as:
Avffyæ = (0.75)(10 bars)(0.11 in2)(60 ksi)(1.0) = 49.5 kips
The development length of the #3 bars will now be checked. For the 180 degree standard hook, use
ACI 318 Section 12.5, ldh times the factors of ACI 318 Section 12.5.3, but not less than 8db or 6 inches.
Side cover exceeds 21/2 inches and cover on the bar extension beyond the hook is provided by the grout
and the planks, which is close enough to 2 inches to apply the 0.7 factor of ACI 318 Section 12.5.3. For
the #3 hook:
= 4.98 in. (ó 6 in. minimum)
The available distance for the perpendicular hook is approximately 51/2 inches. The bar will not be fully
developed at the end of the plank because of the 6inch minimum requirement. The full strength is not
required for shear transfer. By inspection, the diagonal #3 hook will be developed in the wall as required
for the computed diaphragmtoshearwall transfer. The straight end of the #3 bar will now be checked.
The standard development length of ACI 318 Section 12.2 is used for ld.
= 14.2 in.
Figure 8.15 shows the reinforcement along each side of the wall on Joint 2.
Figure 8.15 Joint 2 transverse wall joint reinforcement
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
8.1.1.6.3 Design and Detailing at Joint 3. Compute the required amount of chord reinforcement at
Joint 3 as:
As3 = Tu3/ fy = (19.0 kips)/[0.9(60 ksi)] = 0.35 in2
Use two #4 bars, As = 2(0.20) = 0.40 in2 along the exterior edges (top and bottom of the plan in
Figure 8.12). Require cover for chord bars and spacing between bars at splices and anchorage zones per
ACI 318 Section 21.11.7.6 (optional).
Minimum cover = 2.5(4/8) = 1.25 in., but not less than 2.00 in.
Minimum spacing = 3(4/8) = 1.50 in., but not less than 1.50 in.
Figure 8.16 shows the chord element at the exterior edges of the diaphragm. The chord bars extend
along the length of the exterior longitudinal walls and act as collectors for these walls in the longitudinal
direction (see the Joint 4 collector reinforcement calculations and Figure 8.17).
Figure 8.16 Joint 3 chord reinforcement at the exterior edge
(1.0 in. = 25.4 mm)
Joint 3 must also be checked for the minimum ACI tie forces. The chord reinforcement obviously
exceeds the 16 kip perimeter force requirement. To satisfy the 1.5 kips per foot requirement, a 6 kip tie is
needed at each joint between the planks, which is satisfied with a #3 bar in each joint (0.11 in2 at 60 ksi =
6.6 kips). This bar is required at all bearing walls and is shown in subsequent details.
8.1.1.6.4 Joint 4 Design and Detailing. The required shear friction reinforcement along the wall length
is computed as:
Avf4 = Vu4/ æfy = (9.0 kips)/[(0.75)(1.0)(60 ksi)] = 0.20 in2
Based upon the ACI tie requirement, provide #3 bars at each planktoplank joint. For eight bars total, the
area of reinforcement is 8(0.11) = 0.88 in2, which is more than sufficient even considering the marginal
development length, which is less favorable at Joint 2. The bars are extended 2 feet into the grout key,
which is more than the development length and equal to half the width of the plank.
The required collector reinforcement is computed as:
As4 = Tu4/ fy = (10.1 kips)/[0.9(60 ksi)] = 0.19 in2
The two #4 bars, which are an extension of the transverse chord reinforcement, provide an area of
reinforcement of 0.40 in2.
The reinforcement required by the Standard for outofplane force (156 plf) is far less than the ACI 318
requirement.
Figure 8.17 shows this joint along the wall.
Figure 8.17 Joint 4 exterior longitudinal walls to diaphragm reinforcement
and outofplane anchorage
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
8.1.1.6.5 Joint 5 Design and Detailing. The required shear friction reinforcement along the wall length
is computed as:
Avf5 = Vu5/ æfy = (16.9 kips)/[(0.75)(1.0)(0.85)(60 ksi)] = 0.44 in2
Provide #3 bars at each planktoplank joint for a total of 8 bars.
The required collector reinforcement is computed as:
As5 = Tu5/ fy = (10.1 kips)/[0.9(60 ksi)] = 0.19 in2
Two #4 bars specified for the design of Joint 1 above provide an area of reinforcement of 0.40 in2.
Figure 8.18 shows this joint along the wall.
Figure 8.18 Walltodiaphragm reinforcement along interior longitudinal walls  Joint 5
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
8.1.2 Topped Precast Concrete Units for FiveStory Masonry Building Located in Los
Angeles, California (see Sec. 10.2)
This design shows the floor and roof diaphragms using topped precast units in the fivestory masonry
building in Los Angeles, California. The topping thickness exceeds the minimum thickness of 2 inches as
required for composite topping slabs by ACI 318 Section 21.11.6. The topping is lightweight concrete
(weight = 115 pcf) with a 28day compressive strength (f'c) of 4,000 psi and is to act compositely with the
8inchthick hollowcore precast, prestressed concrete plank. Design parameters are provided in
Section 10.2. Figure 10.21 shows the typical floor and roof plan.
8.1.2.1 General Design Requirements. Topped diaphragms may be used in any Seismic Design
Category. ACI 318 Section 21.11 provides design provisions for topped precast concrete diaphragms.
Standard Section 12.10 specifies the forces to be used in designing the diaphragms.
8.1.2.2 General InPlane Seismic Design Forces for Topped Diaphragms. The inplane diaphragm
seismic design force (Fpx) is calculated using Standard Equation 12.101 but must not be less than
0.2SDSIwpx and need not be more than 0.4SDSIwpx. Vx must be added to Fpx calculated using
Equation 12.101 where:
SDS = the spectral response acceleration parameter at short periods
I = occupancy importance factor
wpx = the weight tributary to the diaphragm at Level x
Vx = the portion of the seismic shear force required to be transferred to the components of the
vertical seismic forceresisting system due to offsets or changes in stiffness of the vertical
resisting member at the diaphragm being designed
For Seismic Design Category C and higher, Standard Section 12.10.2.1 requires that collector elements,
collector splices and collector connections to the vertical seismic forceresisting members be designed in
accordance with Standard Section 12.4.3.2, which combines the diaphragm forces times the overstrength
factor ( 0) and the effects of gravity forces. The parameters from the example in Section 10.2 used to
calculate inplane seismic design forces for the diaphragms are provided in Table 8.14.
Table 8.14 Design Parameters from Section 10.2
Design Parameter
Value
o
2.5
wi (roof)
1,166 kips
wi (floor)
1,302 kips
SDS
1.0
I
1.0
Seismic Design Category
D
1.0 kip = 4.45 kN.
8.1.2.3 Diaphragm Forces. As indicated previously, the weight tributary to the roof and floor
diaphragms (wpx) is the total story weight (wi) at Level i minus the weight of the walls parallel to the
force.
Compute diaphragm weight (wpx) for the roof and floor as follows:
Roof:
Total weight = 1,166 kips
Walls parallel to force = (60 psf)(277 ft)(8.67 ft / 2) = 72 kips
wpx = 1,094 kips
Floors:
Total weight = 1,302 kips
Walls parallel to force = (60 psf)(277 ft)(8.67 ft) = 144 kips
wpx = 1,158 kips
Compute diaphragm demands in accordance with Standard Equation 12.101:
Calculations for Fpx are provided in Table 8.15. The values for Fi and Vi are listed in Table 10.217.
Table 8.15 Fpx Calculations from Section 10.2
Level
wi
(kips)
(kips)
Fi
(kips)
(kips)
wpx
(kips)
Fpx
(kips)
Roof
1,166
1,166
564
564
1,094
529
4
1,302
2,468
504
1,068
1,158
501
3
1,302
3,770
378
1,446
1,158
444
2
1,302
5,072
252
1,698
1,158
387
1
1,302
6,374
126
1,824
1,158
331
1.0 kip = 4.45 kN.
The minimum value of Fpx = 0.2SDSIwpx = 0.2(1.0)1.0(1,094 kips) = 219 kips (roof)
= 0.2(1.0)1.0(1,158 kips) = 232 kips (floors)
The maximum value of Fpx = 0.4SDSIwpx = 2(219 kips) = 438 kips (roof)
= 2(232 kips) = 463 kips (floors)
The value of Fpx used for design of the diaphragms is 463 kips, except for collector elements where forces
will be computed below.
8.1.2.4 Static Analysis of Diaphragms. The seismic design force of 463 kips is distributed as in
Section 8.1.1.6 (Figure 8.11 shows the distribution). The force is three times higher than that used to
design the untopped diaphragm for the New York design due to the higher seismic demand. Figure 8.12
shows critical regions of the diaphragm to be considered in this design. Collector elements will be
designed for 2.5 times the diaphragm force based on the overstrength factor (ê0).
Joint forces taken from Section 8.1.1.5 times 3.0 are as follows:
Joint 1 Transverse forces:
Shear, Vu1 = 41.4 kips 3.0 = 124 kips
Moment, Mu1 = 745 ftkips 3.0 = 2,235 ftkips
Chord tension force, Tu1 = M/d = 2,235 ftkips / 71 ft = 31.5 kips
Joint 2 Transverse forces:
Shear, Vu2 = 46 kips 3.0 = 138 kips
Moment, Mu2 = 920 ftkips 3.0 = 2,760 ftkips
Chord tension force, Tu2 = M/d = 2,760 ftkips / 71 ft = 38.9 kips
Joint 3 Transverse forces:
Shear, Vu3 = 28.3 kips 3.0 = 84.9 kips
Moment, Mu2 = 1,352 ftkips 3.0 = 4,056 ftkips
Chord tension force, Tu3 = M/d = 4,056 ftkips / 71 ft = 57.1 kips
Joint 4 Longitudinal forces:
Wall force, F = 19.1 kips 3.0 = 57.3 kips
Wall shear along wall length, Vu4 = 9 kips 3.0 = 27.0 kips
Collector force at wall end, ê0Tu4 = 2.5(10.1 kips)(3.0) = 75.8 kips
Joint 4 Outofplane forces:
Just as with the untopped diaphragm, the outofplane forces are controlled by ACI 318
Section 16.5, which requires horizontal ties of 1.5 kips per foot from floor to walls.
Joint 5 Longitudinal forces:
Wall force, F = 463 kips / 8 walls = 57.9 kips
Wall shear along each side of wall, Vu4 = 4.5 kips 3.0 = 13.5 kips
Collector force at wall end, ê0Tu4 = 2.5(10.1 kips)(3.0) = 75.8 kips
Joint 5 Shear flow due to transverse forces:
Shear at Joint 2, Vu2 = 138 kips
Q = A d
A = (0.67 ft) (24 ft) = 16 ft2
d = 24 ft
Q = (16 ft2) (24 ft) = 384 ft3
I = (0.67 ft) (72 ft)3 / 12 = 20,840 ft4
Vu2Q/I = (138 kip) (384 ft3) / 20,840 ft4 = 2.54 kips/ft maximum shear flow
Joint 5 length = 40 ft
Total transverse shear in joint 5, Vu5 = 2.54 kips/ft) (40 ft)/2 = 50.8 kips
8.1.2.5 Diaphragm Design and Details
8.1.2.5.1 Minimum Reinforcement for 2.5inch Topping. ACI 318 Section 21.11.7.1 references
ACI 318 Section 7.12, which requires a minimum As = 0.0018bd for grade 60 welded wire reinforcement.
For a 2.5inch topping, the required As = 0.054 in2/ft. WWR 10 10  W4.5 W4.5 provides 0.054 in2/ft.
The minimum spacing of wires is 10 inches and the maximum spacing is 18 inches. Note that the
ACI 318 Section 7.12 limit on spacing of five times thickness is interpreted such that the topping
thickness is not the pertinent thickness.
8.1.2.5.2 Boundary Members. Joint 3 has the maximum bending moment and is used to determine the
boundary member reinforcement of the chord along the exterior edge. The need for transverse boundary
member reinforcement is reviewed using ACI 318 Section 21.11.7.5. Calculate the compressive stress in
the chord with the ultimate moment using a linear elastic model and gross section properties of the
topping. It is conservative to ignore the precast units, but this is not necessary since the joints between
precast units are grouted. As developed previously, the chord compressive stress is:
6Mu3/td2 = 6(4,056 12)/(2.5)(72 12)2 = 157 psi
The chord compressive stress is less than 0.2f'c = 0.2(4,000) = 800 psi. Transverse reinforcement in the
boundary member is not required.
The required chord reinforcement is:
As3 = Tu3/ fy = (57.1 kips)/[0.9(60 ksi)] = 1.06 in2
8.1.2.5.3 Collectors. The design for Joint 4 collector reinforcement at the end of the exterior longitudinal
walls and for Joint 5 at the interior longitudinal walls is the same.
As4 = As5 = ê0Tu4/ fy = (75.8 kips)/[0.9(60 ksi)] = 1.40 in2
Use two #8 bars (As = 2 0.79 = 1.58 in2) along the exterior edges, along the length of the exterior
longitudinal walls and along the length of the interior longitudinal walls. Provide cover for chord and
collector bars and spacing between bars per ACI 318 Section 21.11.7.6.
Minimum cover = 2.5(8/8) = 2.5 in., but not less than 2.0 in.
Minimum spacing = 3(8/8) = 3.0 in., but not less than 1.5 in.
Figure 8.19 shows the diaphragm plan and section cuts of the details and Figure 8.110 shows the
boundary member and chord/collector reinforcement along the edge. Given the close margin on cover,
the transverse reinforcement at lap splices also is shown.
Figure 8.19 Diaphragm plan and section cuts
Figure 8.110 Boundary member and chord and collector reinforcement
(1.0 in. = 25.4 mm)
Figure 8.111 shows the collector reinforcement for the interior longitudinal walls. The side cover of
21/2 inches is provided by casting the topping into the cores and by the stems of the plank. A minimum
space of 1 inch is provided between the plank stems and the sides of the bars.
Figure 8.111 Collector reinforcement at the end of the interior longitudinal walls  Joint 5
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
8.1.2.5.4 Shear Resistance. In thin composite and noncomposite topping slabs on precast floor and roof
members, joints typically are tooled during construction, resulting in cracks forming at the joint between
precast members. Therefore, the shear resistance of the topping slab is limited to the shear friction
strength of the reinforcing crossing the joint.
ACI 318 Section 21.11.9.1 provides an equation for the shear strength of the diaphragm, which includes
both concrete and reinforcing components. However, for noncomposite topping slabs on precast floors
and roofs where the only reinforcing crossing the joints is the field reinforcing in the topping slab, the
shear friction capacity at the joint will always control the design. ACI 318 Section 21.11.9.3 defines the
shear strength at the joint as follows:
Vn = Avffy = 0.75(0.054 in2/ft)(60 ksi)(1.0)(0.85) = 2.07 kips/ft
Note that = 1.0 is used since the joint is assumed to be precracked.
The shear resistance in the transverse direction is:
2.07 kips/ft (72 ft) = 149 kips
which is greater than the Joint 1 shear (maximum transverse shear) of 124 kips.
At the plank adjacent to Joint 2, the shear strength of the diaphragm in accordance with ACI 318
Section 21.11.9.1 is:
= 348 kips
Number 3 dowels are used to provide continuity of the topping slab welded wire reinforcement across the
masonry walls. The topping is to be cast into the masonry walls as shown in Figure 8.112 and the
spacing of the #3 bars is set to be modular with the CMU.
Figure 8.112 Walltodiaphragm reinforcement along interior longitudinal walls  Joint 5
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
The required shear reinforcement along the exterior longitudinal wall (Joint 4) is:
Avf4 = Vu4/ æfy = (27.0 kips)/[(0.75)(1.0)(0.85)(60 ksi)] = 0.71 in2
The required shear reinforcement along the interior longitudinal wall (Joint 5) is:
Avf5 = Vu5/ æfy = (50.8 kips)/[(0.75)(1.0)(0.85)(60 ksi)] = 1.32 in2
Number 3 dowels spaced at 16 o.c. provide
Av = (0.11 in2) (40 ft x 12 in/ft) / 16 in = 3.30 in2
8.1.2.5.5 Check of OutofPlane Forces. At Joint 4, the outofplane forces are checked as follows:
Fp = 0.85 SDS I Wwall = 0.85(1.0)(1.0)(60 psf)(8.67 ft) = 442 plf
With bars at 4 feet on center, Fp = 4 ft (442 plf) = 1.77 kips.
The required reinforcement, As = 1.77 kips/(0.9)(60ksi) = 0.032 in2. Provide #3 bars at 4 feet on center,
which provides a nominal strength of 0.11 60/4 = 1.7 klf. This detail satisfies the ACI 318 Section 16.5
required tie force of 1.5 klf. The development length was checked in the prior example. Using #3 bars at
4 feet on center will be adequate and the detail is shown in Figure 8.113. The detail at Joint 2 is similar.
Figure 8.113 Exterior longitudinal walltodiaphragm reinforcement and
outofplane anchorage  Joint 4
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
8.2 THREESTORY OFFICE BUILDING WITH INTERMEDIATE PRECAST CONCRETE
SHEAR WALLS
This example illustrates the seismic design of intermediate precast concrete shear walls. These walls can
be used up to any height in Seismic Design Categories B and C but are limited to 40 feet for Seismic
Design Categories D, E and F.
ACI 318 Section 21.4.2 requires that yielding between wall panels or between wall panels and the
foundation be restricted to steel elements. However, the Provisions are more specific in their means to
accomplish the objective of providing reliable postelastic performance. Provisions Section 21.4.3
(ACI 318 Sec. 21.4.4) requires that connections that are designed to yield be capable of maintaining
80 percent of their design strength at the deformation induced by the design displacement. Alternatively,
they can use Type 2 mechanical splices.
Additional requirements are contained in the Provisions for intermediate precast walls with wall piers
(Provisions Sec. 14.2.2.4 [ACI 318 Sec. 21.4.5]); however, these requirements do not apply to the solid
wall panels used for this example.
8.2.1 Building Description
This precast concrete building is a threestory office building (Occupancy Category II) in southern New
England on Site Class D soils. The structure utilizes 10footwide by 18inchdeep prestressed double
tees (DTs) spanning 40 feet to prestressed inverted tee beams for the floors and the roof. The DTs are to
be constructed using lightweight concrete. Each of the abovegrade floors and the roof are covered with a
2inchthick (minimum), normalweight castinplace concrete topping. The vertical seismic force
resisting system is to be constructed entirely of precast concrete walls located around the stairs and
elevator/mechanical shafts. The only features illustrated in this example are the rational selection of the
seismic design parameters and the design of the reinforcement and connections of the precast concrete
shear walls. The diaphragm design is not illustrated.
As shown in Figure 8.21, the building has a regular plan. The precast shear walls are continuous from
the ground level to 12 feet above the roof. The walls of the elevator/mechanical pits are castinplace
below grade. The building has no vertical irregularities. The story height is 12 feet.
Figure 8.21 Threestory building plan
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
The precast walls are estimated to be 8 inches thick for building mass calculations. These walls are
normalweight concrete with a 28day compressive strength, f'c, of 5,000 psi. Reinforcing bars used at the
ends of the walls and in welded connectors are ASTM A706 (60 ksi yield strength). The concrete for the
foundations and belowgrade walls has a 28day compressive strength, f'c, of 4,000 psi.
8.2.2 Design Requirements
8.2.2.1 Seismic Parameters. The basic parameters affecting the design and detailing of the building are
shown in Table 8.21.
Table 8.21 Design Parameters
Design Parameter
Value
Occupancy Category II
I = 1.0
SS
0.266
S1
0.08
Site Class
D
Fa
1.59
Fv
2.4
SMS = FaSS
0.425
SM1 = FvS1
0.192
SDS = 2/3 SMS
0.283
SD1 = 2/3 SM1
0.128
Seismic Design Category
B
Basic Seismic ForceResisting System
Bearing Wall System
Wall Type
Intermediate Precast Shear Walls
R
4
ê0
2.5
Cd
4
A Bearing Wall System is defined in the Standard as a structural system with bearing walls providing
support for all or major portions of the vertical loads. In the 2006 International Building Code, this
requirement is clarified by defining a concrete Load Bearing Wall as one which supports more than 200
pounds per linear foot of vertical load in addition to its own weight. While the IBC definition is much
more stringent, this interpretation is used in this example. Note that if a Building Frame Intermediate
Precast Shear Wall system were used, the design would be based on R=5, o=2 « and Cd=4«.
Note that in Seismic Design Category B an ordinary precast shear wall could be used to resist seismic
forces. However, the design forces would be 33 percent higher since they would be based on R = 3,
o = 2.5 and Cd = 3. Ordinary precast structural walls need not satisfy any provisions in ACI 318
Chapter 21.
8.2.2.2 Structural Design Considerations
8.2.2.2.1 Precast Shear Wall System. This system is designed to yield in bending at the base of the
precast shear walls without shear slip at any of the joints. The remaining connections (shear connectors
and flexural connectors away from the base) are then made strong enough to ensure that the inelastic
action is forced to the intended location.
Although it would be desirable to force yielding to occur in a significant portion of the connections, it
frequently is not possible to do so with common configurations of precast elements and connections. The
connections are often unavoidable weak links. Careful attention to detail is required to assure adequate
ductility in the location of first yield and to preclude premature yielding of other connections. For this
particular example, the vertical bars at the ends of the shear walls (see Figure 8.26) act as flexural
reinforcement for the walls and are selected as the location of first yield. The yielding will not propagate
far into the wall vertically due to the unavoidable increase in flexural strength provided by unspliced
reinforcement within the panel. The issue of most significant concern is the performance of the shear
connections (see Figure 8.27) at the same joint. The connections are designed to provide the necessary
shear resistance and avoid slip without providing increased flexural strength at the connection since such
an increase would also increase the maximum shear force on the joint. At the base of the panel, welded
steel angles are designed to be flexible for uplift but stiff for inplane shear.
8.2.2.2.2 Building System. No height limits are imposed (Standard Table 12.21).
For structural design, the floors are assumed to act as rigid horizontal diaphragms to distribute seismic
inertial forces to the walls parallel to the motion. The building is regular both in plan and elevation, for
which, according to Standard Table 12.61, use of the Equivalent Lateral Force (ELF) procedure
(Standard Sec. 12.8) is permitted.
Orthogonal load combinations are not required for this building (Standard Sec. 12.5.2).
Ties, continuity and anchorage must be considered explicitly when detailing connections between the
floors and roof and the walls and columns.
This example does not include consideration of nonstructural elements.
Collector elements are required due to the short length of shear walls as compared to the diaphragm
dimensions, but they are not designed in this example.
Diaphragms need to be designed for the required forces (Standard Sec. 12.10), but that design is not
illustrated here.
The bearing walls must be designed for a force perpendicular to their plane (Standard Sec. 12.11), but
design for that requirement is not shown for this building.
The drift limit is 0.025hsx (Standard Table 12.121), but drift is not computed here.
ACI 318 Section 16.5 requires minimum strengths for connections between elements of precast building
structures. The horizontal forces were described in Section 8.1; the vertical forces will be described in
this example.
8.2.3 Load Combinations
The basic load combinations require that seismic forces and gravity loads be combined in accordance
with the factored load combinations presented in Standard Section 12.4.2.3. Vertical seismic load effects
are described in Standard Section 12.4.2.2.
According to Standard Section 12.3.4.1, = 1.0 for structures in Seismic Design Categories A, B and C,
even though this seismic forceresisting system is not particularly redundant.
The relevant load combinations from ASCE 7 are as follows:
(1.2 + 0.2SDS)D ñ QE + 0.5L
(0.9  0.2SDS)D ñ QE
Into each of these load combinations, substitute SDS as determined above:
1.26D + QE + 0.5L
0.843D  QE
These load combinations are for loading in the plane of the shear walls.
8.2.4 Seismic Force Analysis
8.2.4.1 Weight Calculations. For the roof and two floors:
18inch double tees (32 psf) + 2inch topping (24 psf) = 56.0 psf
Precast beams at 40 feet = 12.5 psf
16inch square columns = 4.5 psf
Ceiling, mechanical, miscellaneous = 4.0 psf
Exterior cladding (per floor area) = 5.0 psf
Partitions = 10.0 psf
Total = 92.0 psf
Note that since the design snow load is 30 psf, it can be ignored in calculating the seismic weight
(Standard Sec. 12.7.2). The weight of each floor including the precast shear walls is:
(120 ft)(150 ft)(92 psf / 1,000) + [(15 ft)4 + (25 ft)2](12 ft)(0.10 ksf) = 1,788 kips
Considering the roof to be the same weight as a floor, the total building weight is W = 3(1,788 kips) =
5,364 kips.
8.2.4.2 Base Shear. The seismic response coefficient, Cs, is computed using Standard Equation 12.82:
= 0.0708
except that it need not exceed the value from Standard Equation 12.83 computed as:
= 0.110
where T is the fundamental period of the building computed using the approximate method of Standard
Equation 12.87:
= 0.29 sec
Therefore, use Cs = 0.0708, which is larger than the minimum specified in Standard Equation 12.85:
Cs = 0.044(SDS)(I) ò 0.01 = 0.044(0.283)(1.0) = 0.012
The total seismic base shear is then calculated using Standard Equation 12.81 as:
V = CsW = (0.0708)(5,364) = 380 kips
Note that this force is substantially larger than a design wind would be. If a nominal 20 psf were applied
to the long face and then amplified by a load factor of 1.6, the result would be less than half this seismic
force already reduced by an R factor of 4.
8.2.4.3 Vertical Distribution of Seismic Forces. The seismic lateral force ,Fx, at any level is determined
in accordance with Standard Section 12.8.3:
where:
Since the period, T, is less than 0.5 seconds, k = l in both building directions. With equal weights at each
floor level, the resulting values of Cvx and Fx are as follows:
Roof: Cvr = 0.50; Fr = 190 kips
Third Floor: Cv3 = 0.333; F3 = 127 kips
Second Floor: Cv2 = 0.167; F2 = 63 kips
8.2.4.4 Horizontal Shear Distribution and Torsion
8.2.4.4.1 Longitudinal Direction. Design each of the 25footlong walls at the elevator/mechanical
shafts for half the total shear. Since the longitudinal walls are very close to the center of rigidity, assume
that torsion will be resisted by the 15footlong stairwell walls in the transverse direction. The forces for
each of the longitudinal walls are shown in Figure 8.22.
Figure 8.22 Forces on the longitudinal walls
(1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m)
8.2.4.4.2 Transverse Direction. Design the four 15footlong stairwell walls for the total shear including
5 percent accidental torsion (Standard Sec. 12.8.4.2). A rough approximation is used in place of a more
rigorous analysis considering all of the walls. The maximum force on the walls is computed as follows:
V = 380/4 + 380(0.05)(150)/[(100 ft moment arm) (2 walls in each set)] = 109 kips
Thus:
Fr = 109(0.50) = 54.5 kips
F3 = 109(0.333) = 36.3 kips
F2 = 109(0.167) = 18.2 kips
Seismic forces on the transverse walls of the stairwells are shown in Figure 8.23.
Figure 8.23 Forces on the transverse walls
(1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m)
8.2.5 Proportioning and Detailing
The strength of members and components is determined using the strengths permitted and required in
ACI 318 Chapters 1 through 19, plus Sections 21.1.2 and 21.4.
8.2.5.1 Overturning Moment and End Reinforcement. Design shear panels to resist overturning by
means of reinforcing bars at each end with a direct tension coupler at the joints. A commonly used
alternative is a threaded posttensioning (PT) bar inserted through the stack of panels, but the behavior is
different than assumed by ACI 318 Section 21.4 since the PT bars don t yield. If PT bars are used, the
system should be designed as an Ordinary Precast Shear Wall (allowed in SDC B.) For a building in a
higher seismic design category, a post tensioned wall would need to be qualified as a Special Precast
Structural Wall Based on Validation Testing per 14.2.4.
8.2.5.1.1 Longitudinal Direction. The freebody diagram for the longitudinal walls is shown in
Figure 8.24. The tension connection at the base of the precast panel to the belowgrade wall is governed
by the seismic overturning moment and the dead loads of the panel and supported floors and roof. In this
example, the weights for an elevator penthouse, with a floor and equipment load at 180 psf between the
shafts and a roof load at 20 psf, are included. The weight for the floors includes double tees, ceiling and
partitions (total load of 70 psf) but not beams and columns. Floor live load is 50 psf, except 100 psf is
used in the elevator lobby. Roof snow load is 30 psf. (The elevator penthouse is so small that it was
ignored in computing the gross seismic forces on the building, but it is not ignored in the following
calculations.)
Figure 8.24 Freebody diagram for longitudinal walls
(1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m)
At the base:
ME = (95 kips)(36 ft) + (63.5 kips)(24 ft) + (31.5 kips)(12 ft) = 5,320 ftkips
D = wall + exterior floors/roof + lobby floors + penthouse floor + penthouse roof
= (25 ft)(48 ft)(0.1 ksf) + (25 ft)(48 ft / 2)(0.070 ksf)(3) + (25 ft)(8 ft / 2)(0.070 ksf)(2)
+ (25 ft)(8 ft / 2)(0.18 ksf) + (25ft )(24 ft / 2)(0.02 ksf)
= 120 + 126 + 14 + 18 + 6 = 284 kips
L = (25 ft)(48 ft / 2)(0.05 ksf)(2) + (25 ft)(8 ft / 2)(0.1 ksf) = 60 + 10 = 70 kips
S = (25ft)(48 ft + 24 ft)(0.03 ksf)/2 = 27 kips
Using the load combinations described above, the vertical loads for combining with the overturning
moment are computed as:
Pmax = 1.26D + 0.5L + 0.2S = 397 kips
Pmin = 0.843D = 239 kips
The axial load is quite small for the wall panel. The average compression Pmax/Ag = 0.165 ksi (3.3 percent
of f'c). Therefore, the tension reinforcement can easily be found from the simple couple shown in
Figure 8.24.
The effective moment arm is:
jd = 25  1.5 = 23.5 ft
and the net tension on the uplift side is:
= 107 kips
The required reinforcement is:
As = Tu/ fy = (107 kips)/[0.9(60 ksi)] = 1.98 in2
Use two #9 bars (As = 2.0 in2) at each end with Type 2 couplers for each bar at each panel joint. Since the
flexural reinforcement must extend a minimum distance, d, (the flexural depth) beyond where it is no
longer required, use both #9 bars at each end of the panel at all three levels for simplicity. Note that if it
is desired to reduce the bar size up the wall, the design check of ACI 318 Section 21.4.3 must be applied
to the flexural strength calculation at the upper wall panel joints.
At this point a check per ACI 318 Section 16.5 will be made. Bearing walls must have vertical ties with a
nominal strength exceeding 3 kips per foot and there must be at least two ties per panel. With one tie at
each end of a 25foot panel, the demand on the tie is:
Tu = (3 kip/ft)(25 ft)/2 = 37.5 kips
The two #9 bars are more than adequate for the ACI requirement.
Although no check for confinement of the compression boundary is required for intermediate precast
shear walls, it is shown here for interest. Using the check from ACI 318 Section 21.9.6, the depth to the
neutral axis is:
Total compression force, As fy + Pmax = (2.0)(60) + 397 = 517 kips
Compression block, a = (517 kips)/[(0.85)(5 ksi)(8 in. width)] = 15.2 in.
Neutral axis depth, c = a/(0.80) = 19.0 in.
The maximum depth (c) with no boundary member per ACI 318 Equation 218 is:
where the term (ëu/hw) shall not be taken as less than 0.007.
Once the base joint yields, it is unlikely that there will be any flexural cracking in the wall more than a
few feet above the base. An analysis of the wall for the design lateral forces using 50 percent of the gross
moment of inertia, ignoring the effect of axial loads and applying the Cd factor of 4 to the results gives a
ratio (ëu/hw) far less than 0.007. Therefore, applying the 0.007 in the equation results in a distance, c, of
71 inches, far in excess of the 19 inches required. Thus, ACI 318 would not require transverse
reinforcement of the boundary even if this wall were designed as a special reinforced concrete shear wall.
For those used to checking the compression stress as an index:
= 694 psi
The limiting stress is 0.2f'c, which is 1,000 psi, so no transverse reinforcement is required at the ends of
the longitudinal walls.
8.2.5.1.2 Transverse Direction. The freebody diagram of the transverse walls is shown in Figure 8.25.
The weight of the precast concrete stairs is 100 psf and of the roof over the stairs is 70 psf.
Figure 8.25 Freebody diagram of the transverse walls
(1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m)
The transverse wall is similar to the longitudinal wall.
At the base:
ME = (54.5 kips)(36 ft) + (36.3 kips)(24 ft) + (18.2 kips)(12 ft) = 3,052 ftkips
D = (15 ft)(48 ft)(0.1 ksf) + 2(12.5 ft / 2)(10 ft / 2)(0.07 ksf)(3) + (15 ft)(8 ft / 2)[(0.1 ksf)(3) +
(0.07 ksf)]
= 72 + 13 + 18 + 4 = 107 kips
L = 2(12.5 ft / 2)(10 ft / 2)(0.05 ksf)(2) + (15 ft)(8 ft / 2)(0.1 ksf)(3)
= 6 + 18 = 24 kips
S = [2(12.5 ft / 2)(10 ft / 2) + (15 ft)(8 ft / 2)](0.03 ksf) = 3.7 kips
Pmax = 1.26(107) + 0.5(24) + 0.2(4) = 148 kips
Pmin = 0.843(107) = 90.5 kips
jd = 15  1.5 = 13.5 ft
Tu = (Mnet/jd)  Pmin/2 = (3,052/13.5)  90.5/2 = 181 kips
As = Tu/ fy = (181 kips)/[0.9(60 ksi)] = 3.35 in2
Use two #10 and one #9 bars (As = 3.54 in2) at each end of each wall with a Type 2 coupler at each bar for
each panel joint. All three bars at each end of the panel will also extend up through all three levels for
simplicity. Following the same method for boundary member check as on the longitudinal walls:
Total compression force, As fy + Pmax = (3.54)(60) + 148 = 360 kips
Compression block, a = (360 kips)/[(0.85)(5 ksi)(8 in. width)] = 10.6 in.
Neutral axis depth, c = a/(0.80) = 13.3 in.
Even though this wall is more flexible and the lateral loads will induce more flexural cracking, the
computed deflections are still small and the minimum value of 0.007 is used for the ratio (ëu/hw). This
yields a maximum value of c = 42.9 inches, thus confinement of the boundary would not be required.
The check of compression stress as an index gives:
= 951 psi
Since å < 1,000 psi, no transverse reinforcement is required at the ends of the transverse walls. Note how
much closer to the criterion this transverse wall is by the compression stress check.
The overturning reinforcement and connection are shown in Figure 8.26.
Figure 8.26 Overturning connection detail at the base of the walls
(1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)
ACI 318 Section 21.4.3 requires that elements of the connection that are not designed to yield develop at
least 1.5Sy. This requirement applies to the anchorage of the coupled bars.
The bar in the panel is made continuous to the roof; therefore, no calculation of development length is
necessary in the panel. The dowel from the foundation will be hooked; otherwise the depth of the
foundation would be more than required for structural reasons. The size of the foundation will provide
adequate cover to allow the 0.7 factor on ACI s standard development length for hooked bars. For the #9
bar:
= 22.5 in.
Similarly, for the #10 bar, the length is 25.3 inches.
Like many shear wall designs, this design does concentrate a demand for overturning resistance on the
foundation. In this instance the resistance may be provided by a large footing (on the order of 20 feet by
28 feet by 3 feet thick) under the entire stairwell or by deep piers or piles with an appropriate cap for load
transfer. Refer to Chapter 4 for examples of design of each type of foundation, although not for this
particular example. Note that the Standard permits the overturning effects at the soilfoundation interface
to be reduced under certain conditions.
8.2.5.2 Shear Connections and Reinforcement. Panel joints often are designed to resist the shear force
by means of shear friction, but that technique is not used for this example because the joint at the
foundation will open due to flexural yielding. This opening would concentrate the shear stress on the
small area of the drypacked joint that remains in compression. This distribution can be affected by the
shims used in construction. With care taken to detail the grouted joint, shear friction can provide a
reliable mechanism to resist this shear. Alternatively, the joint can be designed with direct shear
connectors that will prevent slip along the joint. That concept is developed here.
8.2.5.2.1 Longitudinal Direction. The design shear force is based on the yield strength of the flexural
connection. The flexural strength of the connection can be approximated as follows:
= 1.41
Therefore, the design shear, Vu, at the base is 1.5(1.41)(190 kips) = 402 kips.
The base shear connection is shown in Figure 8.27 and is to be flexible vertically but stiff horizontally in
the plane of the panel. The vertical flexibility is intended to minimize the contribution of these
connections to overturning resistance, which would simply increase the shear demand.
Figure 8.27 Shear connection at base
(1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)
In the panel, provide an assembly with two face plates measuring 3/8" 4" 12" connected by a
C8x18.75 and with diagonal #5 bars as shown in the figure. In the foundation, provide an embedded plate
1/2" 12" 1'6" with six 3/4inchdiameter headed anchor studs as shown. In the field, weld an
L4 3 5/16 0'8", long leg horizontal, on each face. The shear capacity of this connection is checked as
follows:
Shear in the two loose angles:
Vn = (0.6Fu)tl(2) = (0.75)(0.6)(58 ksi)(0.3125 in.)(8 in.)(2) = 130.5 kip
Weld at toe of loose angles:
Vn = (0.6Fu)tel(2) = (0.75)(0.6)(70 ksi)(0.25 in. / û2)(8 in.)(2) = 89.1 kip
Weld at face plates, using Table 88 in AISC Manual (13th edition):
Vn = CC1Dl (2 sides)
=0.75
C1 = 1.0 for E70 electrodes
L = 8 in.
D = 4 (sixteenths of an inch)
K = 2 in. / 8 in. = 0.25
a = eccentricity, summed vectorally: horizontal component is 4 in.; vertical component is
2.67 in.; thus, al = 4.80 in. and a = 4.8 in. / 8 in. = 0.6 from the table. By interpolation,
C = 1.73
Vn = 0.75(1.73)(1.0)(4)(8)(2) = 83.0 kip
Weld from channel to plate has at least as much capacity, but less demand.
Bearing of concrete at steel channel:
fc = (0.85f'c) = 0.65(0.85)(5 ksi) = 2.76 ksi
The C8 has the following properties:
tw = 0.487 in.
bf = 2.53 in.
tf = 0.39 in. (average)
The bearing will be controlled by bending in the web (because of the tapered flange, the critical
flange thickness is greater than the web thickness). Conservatively ignoring the concrete s
resistance to vertical deformation of the flange, compute the width (b) of flange loaded at 2.76 ksi
that develops the plastic moment in the web:
Mp = Fytw2/4 = (0.9)(50 ksi)(0.4872 in2)/4 = 2.67 inkip/in.
Mu = fc[(btw)2/2  (tw/2)2/2] = 2.76[(b  0.243 in.)2  (0.243 in.)2]/2
setting the two equal results in b = 1.65 inches.
Therefore, bearing on the channel is:
Vc = fc(2  tw)(l) = (2.76 ksi)[(2(1.65)  0.487 in.](6 in.) = 46.6 kip
To the bearing capacity on the channel is added the four #5 diagonal bars, which are effective in tension
and compression; = 0.75 for shear is used here:
Vs = fyAscosà = (0.75)(60 ksi)(4)(0.31 in2)(cos45ø) = 39.5 kip
Thus, the total capacity for transfer to concrete is:
Vn = Vc + Vs = 46.6 + 39.6 = 86.1 kip
The capacity of the plate in the foundation is governed by the headed anchor studs. ACI 318 Appendix D
has detailed information on calculating the strength of headed anchor studs. ACI 318 Section D3.3 has
additional requirements for anchors resisting seismic forces in Seismic Design Categories C through F.
Capacity in shear for anchors located far from an edge of concrete, such as these and with sufficient
embedment to avoid the pryout failure mode is governed by the capacity of the steel, which is required by
ACI 318 Section D3.3.4:
Vsa = n Ase futa = (0.65)(6 studs)(0.44 in2 per stud)(60 ksi) = 103 kip
In summary, the various shear capacities of the connection are as follows:
Shear in the two loose angles: 130.5 kip
Weld at toe of loose angles: 89.1 kip
Weld at face plates: 83.0 kip
Transfer to concrete: 86.1 kip
Headed anchor studs at foundation: 103 kip
The number of embedded plates (n) required for a panel is:
n =402/83.0 = 4.8
Use five connection assemblies, equally spaced along each side (4'0" on center works well to avoid the
end reinforcement). The plates are recessed to position the #5 bars within the thickness of the panel and
within the reinforcement of the panel.
It is instructive to consider how much moment capacity is added by the resistance of these connections to
vertical lift at the joint. The vertical force at the tip of the angle that will create the plastic moment in the
leg of the angle is:
T = Mp/x = Fylt2/4 / (lk) = (36 ksi)(8 in.)(0.31252 in2)/4]/(4 in.  0.69 in.) = 2.12 kips
There are five assemblies with two loose angles each, giving a total vertical force of 21 kips. The
moment resistance is this force times half the length of the panel, which yields 265 ftkips. The total
demand moment, for which the entire system is proportioned, is 5,320 ftkips. Thus, these connections
will add approximately 5 percent to the resistance and ignoring this contribution is reasonable. If a
straight plate measuring 1/4 inch by 8 inches (which would be sufficient) were used and if the welds and
foundation embedment did not fail first, the tensile capacity would be 72 kips each, a factor of 42 increase
over the angles and the shear connections would have the unintended effect of more than doubling the
flexural resistance, which would require a much higher shear force to develop a plastic hinge at the wall
base.
Using ACI 318 Section 11.10, check the shear strength of the precast panel at the first floor:
= 239 kips
Because Vc ò Vu = 190 kips, the wall is adequate for shear without even considering the reinforcement.
Note that the shear strength of the wall itself is not governed by the overstrength required for the
connection. However, since Vu ò 0.5 Vc = 120 kips, ACI 318 Section 11.9.8 requires minimum wall
reinforcement in accordance with ACI 318 Section 11.9.9 rather than Chapters 14 or 16. For the
minimum required h = 0.0025, the required reinforcement is:
Av = 0.0025(8)(12) = 0.24 in2/ft
As before, use two layers of welded wire reinforcement, WWF 4 4  W4.0 W4.0, one on each face. The
shear reinforcement provided is:
Av = 0.12(2) = 0.24 in2/ft
Next, compute the required connection capacity at Level 2. Even though the end reinforcing at the base
extends to the top of the shear wall, the connection still needs to be checked for flexure in accordance
with Provisions Section 21.4.3 (ACI 318 Sec. 21.4.4). At Level 2:
ME = (95 kips)(24 ft) + (63.5 kips)(12 ft) = 3,042 ftkips
There are two possible approaches to the design of the joint at Level 2.
First, if Type 2 couplers are used at the Level 2 flexural connection, then the connection can be
considered to have been designed to yield, and no overstrength is required for the design of the flexural
connection. In this case, the bars are designed for the moment demand at the Level 2 joint.
Alternately, if a nonyielding connection is used at the Level 2 connection, then to meet the requirements
of Provisions Section 21.4.4 (ACI 318 Sec. 21.4.3), the flexural strength of the connection at Level 2
must be 1.5Sy or:
Mu = 1.5(1.41)ME = 1.5(1.41)(3,042 ftkips) = 6,433 ftkips
At Level 2, the gravity loads on the wall are:
D = wall + exterior floors/roof + lobby floors + penthouse floor + penthouse roof
= (25 ft)(36 ft)(0.1 ksf) + (25 ft)(48 ft / 2)(0.070 ksf)(2) + (25 ft)(8 ft / 2)(0.070 ksf)(1) +
(25 ft)(8 ft / 2)(0.18 ksf) + (25 ft )(24 ft / 2)(0.02 ksf)
= 90 + 84 + 7 + 18 + 6 = 205 kips
L = (25 ft)(48 ft / 2)(0.05 ksf)(1) + (25 ft)(8 ft / 2)(0.1 ksf) = 30 + 10 = 40 kips
S = (25ft)(48 ft + 24 ft)(0.03 ksf)/2 = 27 kips
Pmax = 1.26(205) + 0.5(40) + 0.2(27) = 285 kips
Pmin = 0.843(205) = 173 kips
Note that since the maximum axial load was used to determine the maximum yield strength of the base
moment connection, the maximum axial load is used here to determine the nominal strength of the
Level 2 connection. For completeness, the base moment overstrength provided should be checked using
the minimum axial load as well and compared to the moment strength at Level 2 using the minimum axial
load.
= 5,552 ftkips
Therefore, the nonyielding flexural connection at Level 2 must be strengthened.
Provide:
= 131 kips
The required reinforcement is:
As = Tu/ fy = (131 kips)/[0.9(60 ksi)] = 2.43 in2
In addition to the two #9 bars that extend to the roof, provide one #6 bar developed into the wall panel
above and below the joint. Note that no increase on the development length for the #6 bar is required for
this connection since the connection itself has been designed for the loads to promote base yielding per
Provisions Section 21.4.4 (ACI 318 Sec. 21.4.3).
Since the Level 2 connection is prevented from yielding, shear friction can reasonably be used to resist
shear sliding at this location. Also, because of the lack of flexural yield at the joint, it is not necessary to
make the shear connection flexible with respect to vertical movement should an embedded plate detail be
desired.
The design shear for this location is:
Vu,Level 2 = 1.5(1.41)(95+63.5) = 335 kips
Using the same recessed embedded plate assemblies in the panel as at the base, but welded with a straight
plate, the number of plates, n, is 335/83.0 = 4.04. Use four plates, equally spaced along each side.
Figure 8.28 shows the shear connection at the second and third floors of the longitudinal precast concrete
shear wall panels.
Figure 8.28 Shear connections on each side of the wall at the second and third floors
(1.0 in = 25.4 mm)
8.3 ONESTORY PRECAST SHEAR WALL BUILDING
This example illustrates the design of a precast concrete shear wall for a singlestory building in a region
of high seismicity. For buildings assigned to Seismic Design Category D, ACI 318 Section 21.10
requires that special structural walls constructed of precast concrete meet the requirements of ACI 318
Section 21.9, in addition to the requirements for intermediate precast structural walls. Alternately, special
structural walls constructed using precast concrete are allowed if they satisfy the requirements of ACI
ITG5.1, Acceptance Criteria for Special Unbonded PostTensioned Precast Structural Walls Based on
Validation Testing (ACI ITG 5.107). Design requirements for one such type of wall have been
developed by ACI ITG 5 and have been published by ACI as Requirements for Design of a Special
Unbonded PostTensioned Precast Shear Wall Satisfying ACI ITG5.1 (ACI ITG 5.209). ITG 5.1 and
ITG 5.2 describe requirements for precast walls for which a selfcentering mechanism is provided by
posttensioning located concentrically within the wall. More general requirements for special precast
walls are contained in Provisions Section 14.2.4. Section 14.2.4 is an updated version of Section 9.6 of
the 2003 Provisions, which formed the basis for ITG 5.1 and ITG 5.2.
8.3.1 Building Description
The precast concrete building is a singlestory industrial warehouse building (Occupancy Category II)
located in the Los Angeles area on Site Class C soils. The structure has 8footwide by 12.5inchdeep
prestressed double tee (DT) wall panels. The roof is light gage metal decking spanning to bar joists that
are spaced at 4 feet on center to match the location of the DT legs. The center supports for the joists are
joist girders spanning 40 feet to steel tube columns. The vertical seismic forceresisting system is the
precast/prestressed DT wall panels located around the perimeter of the building. The average roof height
is 20 feet and there is a 3foot parapet. Figure 8.31 shows the plan of the building, which is regular.
Figure 8.31 Singlestory industrial warehouse building plan
(1.0 ft = 0.3048 m)
The precast wall panels used in this building are typical DT wall panels commonly found in many
locations but not normally used in southern California. For these wall panels, an extra 1/2 inch has been
added to the thickness of the deck (flange). This extra thickness is intended to reduce cracking of the
flanges and provide cover for the bars used in the deck at the base. The use of thicker flanges is
addressed later.
The wall panels are normalweight concrete with a 28day compressive strength of f'c = 5,000 psi.
Reinforcing bars used in the welded connections of the panels and footings are ASTM A706 (60 ksi).
The concrete for the foundations has a 28day compressive strength of f'c = 4,000 psi.
In Standard Table 12.21 the values for special reinforced concrete shear walls are for both castinplace
and precast walls. In Section 2.2, ACI 318 defines a special structural wall as a castinplace or precast
wall complying with the requirements of 21.1.3 through 21.1.7, 21.9 and 21.10, as applicable, in addition
to the requirements for ordinary reinforced concrete structural walls. ACI 318 Section 21.10 defines
requirements for special structural walls constructed using precast concrete, including that the wall must
satisfy all of the requirements of ACI 318 Section 21.9.
Unfortunately, several of the requirements of ACI 318 Section 21.9 are problematic for a shear wall
system constructed using DT wall panels. These include the following:
1. ACI 318 Section 21.9.2.1 requires reinforcement to be spaced no more than 18 inches on center
and be continuous. This would require splices to the foundation along the DT flange.
2. ACI 318 Section 21.9.2.2 requires two curtains of reinforcement for walls with shear stress
greater than 2 ûf'c. For low loads, this might not be a problem, but for high shear stresses,
placing two layers of reinforcing in a DT flange would be a challenge.
3. While ACI 318 Section 21.1.5.3 allows prestressing steel to be used in precast walls, ACI 318
Commentary R21.1.5 states that the capability of a structural member to develop inelastic
rotation capacity is a function of the length of the yield region along the axis of the member. In
interpreting experimental results, the length of the yield region has been related to the relative
magnitudes of nominal and yield moments. Since prestressing steel does not have a defined
yield plateau, the ratio of nominal to yield moment is undefined. This limits the ability of the
structural member to develop inelastic rotation capacity a key assumption in the definition of
the R value for a special reinforced concrete wall system.
Therefore, these walls will be designed using the ACI category of intermediate precast structural walls.
8.3.2 Design Requirements
8.3.2.1 Seismic Parameters of the Provisions. The basic parameters affecting the design and detailing
of the building are shown in Table 8.31.
Table 8.31 Design Parameters
Design Parameter
Value
Occupancy Category II
I = 1.0
SS
1.5
S1
0.60
Site Class
C
Fa
1.0
Fv
1.3
SMS = FaSS
1.5
SM1 = FvS1
0.78
SDS = 2/3 SMS
1.0
SD1 = 2/3 SM1
0.52
Seismic Design Category
D
Basic Seismic ForceResisting System
Bearing Wall System
Wall Type
Intermediate Precast Structural Wall
R
4
ê0
2.5
Cd
4
8.3.2.2 Structural Design Considerations
8.3.2.2.1 Intermediate Precast Structural Walls Constructed Using Precast Concrete. The intent of
the intermediate precast structural wall requirements is to provide yielding in a dry connection for
bending at the base of each precast shear wall panel while maintaining significant shear resistance in the
connection. The flexural connection for a wall panel at the base is located in one DT leg while the
connection at the other leg is used for compression. Per ACI 318 Section 21.4, these connections must
yield only in steel elements or reinforcement and all other elements of the connection (including shear
resistance) must be designed for 1.5 times the force associated with the flexural yield strength of the
connection.
Yielding will develop in the dry connection at the base by bending in the horizontal leg of the steel angle
welded between the embedded plates of the DT and footing. The horizontal leg of this angle is designed
in a manner to resist the seismic tension of the shear wall due to overturning and then yield and deform
inelastically. The connections on the two legs of the DT are each designed to resist 50 percent of the
shear. The anchorage of the connection into the concrete is designed to satisfy the 1.5Sy requirements of
ACI 318 Section 21.4.3. Careful attention to structural details of these connections is required to ensure
tension ductility and resistance to large shear forces that are applied to the embedded plates in the DT and
footing.
8.3.2.2.2 Building System. The height limit in Seismic Design Category D (Standard Table 12.21) is
40 feet.
The metal deck roof acts as a flexible horizontal diaphragm to distribute seismic inertia forces to the walls
parallel to the earthquake motion (Standard Sec. 12.3.1.1).
The building is regular both in plan and elevation.
The redundancy factor, , is determined in accordance with Standard Section 12.3.4.2. For this structure,
which is regular and has more than two perimeter wall panels (bays) on each side in each direction,
= 1.0.
The structural analysis to be used is the ELF procedure (Standard Sec. 12.8) as permitted by Standard
Table 12.61.
Orthogonal load combinations are not required for flexible diaphragms in Seismic Design Category D
(Standard Sec. 12.5.4).
This example does not include design of the foundation system, the metal deck diaphragm, or the
nonstructural elements.
Ties, continuity and anchorage (Standard 12.11) must be considered explicitly when detailing
connections between the roof and the wall panels. This example does not include the design of those
connections, but sketches of details are provided to guide the design engineer.
There are no drift limits for singlestory buildings as long as they are designed to accommodate predicted
lateral displacements (Standard Table 12.121, Footnote c).
8.3.3 Load Combinations
The basic load combinations (Standard Sec. 12.4.2.3) require that seismic forces and gravity loads be
combined in accordance with the following factored load combinations:
(1.2 + 0.2SDS)D ñ QE + 0.5L+ 0.2S
(0.9  0.2SDS)D ñ QE + 1.6H
At this flat site, both S and H equal 0. Note that roof live load need not be combined with seismic loads,
so the live load term, L, can be omitted from the equation. Therefore:
1.4D + QE
0.7D  QE
These load combinations are for the inplane direction of the shear walls.
8.3.4 Seismic Force Analysis
8.3.4.1 Weight Calculations. Compute the weight tributary to the roof diaphragm:
Roofing = 2.0 psf
Metal decking = 1.8 psf
Insulation = 1.5 psf
Lights, mechanical, sprinkler system, etc. = 3.2 psf
Bar joists = 2.7 psf
Joist girder and columns = 0.8 psf
Total = 12.0 psf
The total weight of the roof is computed as:
(120 ft 96 ft)(12 psf / 1,000) = 138 kips
The exterior DT wall weight tributary to the roof is:
(20 ft / 2 + 3 ft)[42 psf / 1,000](120 ft + 96 ft)2 = 236 kips
Total building weight for seismic lateral load, W = 138+236 = 374 kips
8.3.4.2 Base Shear. The seismic response coefficient (Cs) is computed using Standard Equation 12.82
as:
= 0.25
except that it need not exceed the value from Standard Equation 12.83, as follows:
= 0.69
where T is the fundamental period of the building computed using the approximate method of Standard
Equation 12.87:
= 0.189 sec
Therefore, use Cs = 0.25, which is larger than the minimum specified in Standard Equation 12.85:
Cs = 0.044(SDS)(I) ò 0.01 = 0.044(1.0)(1.0) = 0.044
The total seismic base shear is then calculated using Standard Equation 12.81, as:
V = CsW = (0.25)(374) = 93.5 kips
8.3.4.3 Horizontal Shear Distribution and Torsion. Torsion is not considered in the shear distribution
in buildings with flexible diaphragms. The shear along each side of the building will be equal, based on a
tributary area force distribution.
8.3.4.3.1 Longitudinal Direction. The total shear along each side of the building is V/2 = 46.75 kips.
The maximum shear on longitudinal panels (at the side with the openings) is:
Vlu = 46.75/11 = 4.25 kips
On each side, each longitudinal wall panel resists the same shear force as shown in the freebody diagram
of Figure 8.32, where D1 represents roof joist reactions and D2 is the panel weight.
Figure 8.32 Freebody diagram of a panel in the longitudinal direction
(1.0 ft = 0.3048 m)
8.3.4.3.2 Transverse Direction. Seismic forces on the transverse wall panels are all equal and are:
Vtu = 46.75/12 = 3.90 kips
Figure 8.33 shows the transverse wall panel freebody diagram.
Note the assumption of uniform distribution to the wall panels in a line requires that the roof diaphragm
be provided with a collector element along its edge. The chord designed for diaphragm action in the
perpendicular direction will normally be capable of fulfilling this function, but an explicit check should be
made in the design.
Figure 8.33 Freebody diagram of a panel in the transverse direction
(1.0 ft = 0.3048 m)
8.3.5 Proportioning and Detailing
The strength of members and components is determined using the strengths permitted and required in
ACI 318 including Chapter 21.
8.3.5.1 Tension and Shear Forces at the Panel Base. Design each precast shear panel to resist the
seismic overturning moment by means of a ductile tension connector at the base of the panel. A steel
angle connector will be provided at the connection of each leg of the DT panel to the concrete footing.
The horizontal leg of the angle is designed to yield in bending as needed in an earthquake. ACI 318
Section 21.4 requires that dry connections at locations of nonlinear action comply with applicable
requirements of monolithic concrete construction and satisfy both of the following:
1. Where the moment action on the connection is assumed equal to 1.5My, the coexisting forces on all
other components of the connection other than the yielding element shall not exceed their design
strength.
2. The nominal shear strength for the connection shall not be less than the shear associated with the
development of 1.5My at the connection.
8.3.5.1.1 Longitudinal Direction. Use the freebody diagram shown in Figure 8.32. The maximum
tension for the connection at the base of the precast panel to the concrete footing is governed by the
seismic overturning moment and the dead loads of the panel and the roof. The weight for the roof is
11.2 psf, which excludes the joist girders and columns.
At the base:
ME = (4.25 kips)(20 ft) = 85.0 ftkips
Dead loads:
= 1.08 kips
D2 = 0.042(23)(8) = 7.73 kips
D = 2(1.08) + 7.73 = 9.89 kips
1.4D = 13.8 kips
0.7D = 6.92 kips
Compute the tension force due to net overturning based on an effective moment arm, d, of 4.0 feet (the
distance between the DT legs). The maximum is found when combined with 0.7D:
Tu = ME/d  0.7D/2 = 85.0/4  6.92/2 = 17.8 kips
8.3.5.1.2 Transverse Direction. For the transverse direction, use the freebody diagram of Figure 8.33.
The maximum tension for connection at the base of the precast panel to the concrete footing is governed
by the seismic overturning moment and the dead loads of just the panel. No load from the roof is
included, since it is negligible.
At the base:
ME = (3.90 kips)(20 ft) = 78.0 ftkips
The dead load of the panel (as computed above) is D2 = 7.73 kips and 0.7D = 5.41.
The tension force is computed as above for d = 4.0 feet (the distance between the DT legs):
Tu = 78.0/4  5.41/2 = 16.8 kips
This tension force is less than that at the longitudinal wall panels. Use the tension force of the
longitudinal wall panels for the design of the angle connections.
8.3.5.2 Size the Yielding Angle. The angle, which is the ductile element of the connection, is welded
between the plates embedded in the DT leg and the footing. This angle is an L5 31/2 3/4 0'61/2"
with the long leg vertical. The steel for the angle and embedded plates will be ASTM A572, Grade 50.
The horizontal leg of the angle needs to be long enough to provide significant displacement at the roof,
although this is not stated as a requirement in either the Provisions or ACI 318. This will be examined
briefly here. The angle and its welds are shown in Figure 8.34.
Figure 8.34 Freebody of the angle and the fillet weld connecting the embedded plates
in the DT and the footing (elevation and section)
(1.0 in = 25.4 mm)
The location of the plastic hinge in the angle is at the toe of the fillet (at a distance, k, from the heel of the
angle.) The bending moment at this location is:
Mu = Tu(3.5  k) = 17.8(3.5  1.1875) = 41.2 in.kips
= 41.1 inkips
Providing a stronger angle (e.g., a shorter horizontal leg) will simply increase the demands on the
remainder of the assembly. Using ACI 318 Section 21.4.3, the tension force for the remainder of this
connection and the balance of the wall design are based upon a probable strength equal to 150 percent of
the yield strength. Thus:
= 27.0 kips
The amplifier, required for the design of the balance of the connection, is:
= 1.52
The shear on the connection associated with this force in the angle is:
= 6.46 kips
Check the welds for the tension force of 27.0 kips and a shear force 6.46 kips.
The Provisions Section 21.4.4 (ACI 318 21.4.3) requires that connections that are designed to yield be
capable of maintaining 80 percent of their design strength at the deformation induced by the design
displacement. For yielding of a flat bar (angle leg), this can be checked by calculating the ductility
capacity of the bar and comparing it to Cd. Note that the element ductility demand (to be calculated
below for the yielding angle) and the system ductility, Cd, are only equal if the balance of the system is
rigid. This is a reasonable assumption for the intermediate precast structural wall system described in this
example.
The idealized yield deformation of the angle can be calculated as follows:
= 19.8 kips
= 0.012 in.
It is conservative to limit the maximum strain in the bar to sh = 15 y. At this strain, a flat bar would be
expected to retain all its strength and thus meet the requirement of maintaining 80 percent of its strength.
Assuming a plastic hinge length equal to the section thickness:
= 0.06897
= 0.112 in.
Since the ductility capacity at strain hardening is 0.112/0.012 = 9.3 is larger than Cd = 4 for this system,
the requirement of Provision Section 21.4.4 (ACI 318 Sec. 21.4.3) is met.
8.3.5.3 Welds to Connection Angle. Welds will be fillet welds using E70 electrodes.
For the base metal, Rn = (Fy)ABM.
For which the limiting stress is Fy = 0.9(50) = 45.0 ksi.
For the weld metal, Rn = (Fy)Aw = 0.75(0.6)70(0.707)Aw.
For which the limiting stress is 22.3 ksi.
Size a fillet weld, 6.5 inches long at the angle to the embedded plate in the footing. Using an elastic
approach:
Resultant force = = 27.8 kips
Aw = 27.8/22.3 = 1.24 in2
t = Aw/l =1.24 in2 / 6.5 in. = 0.19 in.
For a 3/4 inch angle leg, use a 5/16 inch fillet weld. Given the importance of this weld, increasing the
size to 3/8 inch would be a reasonable step. With ordinary quality control to avoid flaws, increasing the
strength of this weld by such an amount should not have a detrimental effect elsewhere in the connection.
Now size the weld to the plate in the DT. Continue to use the conservative elastic method to calculate
weld stresses. Try a fillet weld 6.5 inches long across the top and 4 inches long on each vertical leg of the
angle. Using the freebody diagram of Figure 8.34 for tension and Figure 8.35 for shear, the weld
moments and stresses are:
Mx = Tpr(3.5) = 27.0(3.5) = 94.5 inkips
My = Vpr(3.5) = (6.46)(3.5) = 22.6 inkips
Mz = Vpr(yb + 1.0)
= 6.46(2.77 + 1.0) = 24.4 inkips
Figure 8.35 Freebody of angle with welds, top view, showing only
shear forces and resisting moments
For the weld between the angle and the embedded plate in the DT as shown in Figure 8.35, the section
properties for a weld leg (t) are:
A = 14.5t in2
Ix = 25.0t in4
Iy = 107.4t in4
Ip = Ix + Iy = 132.4t in4
yb = 2.90 in.
xL = 3.25 in.
To check the weld, stresses are computed at all four ends (and corners). The maximum stress is at the
lower right end of the inverted U shown in Figure 8.34.
Thus, t = 11.9/22.3 = 0.53 inch, which can be taken as 9/16 inch. Field welds are conservatively sized
with the elastic method for simplicity and to minimize construction issues.
8.3.5.4 Panel Reinforcement. Check the maximum compressive stress in the DT leg. Note that for an
intermediate precast structural wall, ACI 318 Section 21.9.6 does not apply and transverse boundary
element reinforcing is not required. However, the cross section must be designed for the loads associated
with 1.5 times the moment that yields the base connectors.
Figure 8.36 shows the cross section used. The section is limited by the area of drypack under the DT at
the footing.
Figure 8.36 Cross section of the DT drypacked at the footing
(1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)
The reason to limit the area of drypack at the footing is to locate the boundary elements in the legs of the
DT, at least at the bottom of the panel. The flange between the legs of the DT is not as susceptible to
cracking during transportation as are the corners of DT flanges outside the confines of the legs. The
compressive stress due to the overturning moment at the top of the footing and dead load is:
A = 227 in2
S = 3240 in3
= 539 ksi
Roof live loads need not be included as a factored axial load in the compressive stress check, but the force
from the prestress steel will be added to the compression stress above because the prestress force will be
effective a few feet above the base and will add compression to the DT leg. Each leg of the DT will be
reinforced with one 1/2inchdiameter strand and one 3/8inchdiameter strand. Figure 8.37 shows the
location of these prestressed strands.
Figure 8.37 Cross section of one DT leg showing the location of
the bonded prestressing tendons or strand
(1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)
Next, compute the compressive stress resulting from these strands. Note that the moment at the height of
strand development above the footing, about 26 inches for the effective stress (fse), is less than at the top
of footing. This reduces the compressive stress by:
= 34 psi
In each leg, use:
P = 0.58fpu Aps = 0.58(270 ksi)[0.153 + 0.085] = 37.3 kips
A = 168 in2
e = yb  CGStrand = 9.48  8.57 = 0.91 in.
Sb = 189 in3
= 402 psi
Therefore, the total compressive stress is approximately 539 + 402  34 = 907 psi.
Since yielding is restricted to the steel angle and the DT is designed to be 1.5 times stronger than the yield
force in the steel angle, the full strength of the strand can be used to resist axial forces in the DT stem,
without concern for yielding in the strand.
D2 = (0.042)(20.83)(8) = 7.0 kips
Pmin = 0.7(7.0 + 2(1.08)) = 6.41 kips
ME = (1.52)(4.25)(17.83) = 115.2 ftkips
Tu,stem = ME/d  Pmin/2= 25.5 kips
The area of tension reinforcement required is:
Aps = Tu,stem/ fpy = (25.5 kips)/[0.9(270 ksi)] = 0.10 in2
The area of one 1/2inchdiameter strand and one 3/8inchdiameter strand is 0.153 in2 + 0.085 in2 =
0.236 in2. The mesh in the legs is available for tension resistance but is not required in this check.
To determine the nominal shear strength of the concrete for the connection design, complete the shear
calculation for the panel in accordance with ACI 318 Section 11.9. The demand on each panel is:
Vu = Vpr = 6.46 kips
Only the deck between the DT legs is used to resist the inplane shear (the legs act like flanges, meaning
that the area effective for shear is the deck between the legs). First, determine the minimum required
shear reinforcement based on ACI 318 Section 11.9.
= 12.7 kips
Since Vu of 6.46 kips exceeds Vu/2 of 6.36 kips, provide minimum reinforcement per ACI 318
Section 11.9.9.2. Using welded wire reinforcement, the required areas of reinforcement are:
Av = Avh = (0.0025)(2.5)(12) = 0.075 in2/ft
Provide 6 6 W4.0 W4.0 welded wire reinforcement.
Asv = Ash = 0.08 in2/ft
The prestress force and the area of the DT legs are excluded from the calculation of the nominal shear
strength of the DT wall panel. The prestress force is not effective at the base, where the connection is and
the legs are like the flanges of a channel, which are not effective in shear.
8.3.5.5 Tension and Shear at the Footing Embedment. Reinforcement to anchor the embedded plates
is sized for the same tension and shear. Reinforcement in the DT leg and in the footing will be welded to
embedded plates as shown in Figure 8.38.
The welded reinforcement is sloped to provide concrete cover and to embed the bars in the central region
of the DT leg and footing. The tension reinforcement area required in the footing is:
= 0.56 in2
Use two #5 bars (As = 0.62 in2) at each embedded plate in the footing.
The shear bars in the footing will be two #4 bars placed on an angle of twotoone. The resultant shear
resistance is:
Vn = 0.75(0.2)(2)(60)(cos26.5ø) = 16.1 kips
Figure 8.38 Section at the connection of the precast/prestressed shear wall panel
and the footing
(1.0 in = 25.4 mm)
8.3.5.6 Tension and Shear at the DT Embedment. The area of reinforcement for the welded bars of the
embedded plate in the DT, which develops tension as the angle bends through cycles, is:
= 0.503 in2
Two #5 bars are adequate. Note that the bars in the DT leg are required to extend upward the
development length of the bar, which would be 22 inches. In this case, they will be extended 22 inches
past the point of development of the effective stress in the strand, which totals approximately 48 inches.
The same embedded plate used for tension will also be used to resist onehalf the nominal shear. This
shear force is 6.46 kips. The transfer of direct shear to the concrete is easily accomplished with bearing
on the sides of the reinforcing bars welded to the plate. Two #5 and two #4 bars (explained later) are
welded to the plate. The available bearing area is approximately Abr = 4(0.5 in.)(5 in.[available]) = 10 in2
and the bearing capacity of the concrete is Vn = (0.65)(0.85)(5 ksi)(10 in2) = 27.6 kips, which is greater
than the 6.46 kip demand.
The weld of these bars to the plate must develop both the tensile demand and this shear force. The weld
is a flare bevel weld, with an effective throat of 0.2 times the bar diameter along each side of the bar.
(Refer to the PCI Handbook.) Using the weld capacity for the #5 bar:
Vn = (0.75)(0.6)(70 ksi)(0.2)(0.625 in.)(2) = 7.9 kips/in
The shear demand is prorated among the four bars as (6.46 kip)/4 = 1.6 kips. The tension demand is
Tu,stem/2(13.5 kips). The vectorial sum of shear and tension demand is 13.6 kips. Thus, the minimum
length of weld is 13.6/7.9 = 1.7 inches.
8.3.5.7 Resolution of Eccentricities at the DT Embedment. Check the twisting of the embedded plate
in the DT for Mz. Use Mz = 24.4 inkips.
= 0.05 in2
Use one #4 bar on each side of the vertical embedded plate in the DT as shown in Figure 8.39. This is
the same bar used to transfer direct shear in bearing.
Check the DT embedded plate for My (equal to 22.6 inkips) and Mx (equal to 94.5 inkips) using the two
#4 bars welded to the back side of the plate near the corners of the weld on the loose angle and the two #3
bars welded to the back side of the plate near the bottom of the DT leg (as shown in Figure 8.39). It is
relatively straightforward to compute the resultant moment magnitude and direction, assume a triangular
compression block in the concrete and then compute the resisting moment. It is quicker to make a
reasonable assumption as to the bars that are effective and then compute resisting moments about the X
and Y axes. That approximate method is demonstrated here. The #5 bars are effective in resisting Mx and
one each of the #3 and #5 bars are effective in resisting My. For My assume that the effective depth
extends 1 inch beyond the edge of the angle (equal to twice the thickness of the plate). Begin by
assigning onehalf of the corner #5 bar to each component.
With Asx = 0.31 + 0.31/2 = 0.47 in2:
Mnx = As fy jd = (0.9)(0.47 in2)(60 ksi)(0.95)(5 in.) = 120 inkips (> 94.5 inkips)
With Asy = 0.11 + 0.31/2 = 0.27 in2:
Mny = As fy jd = (0.9)(0.27 in2)(60 ksi)(0.95)(5 in.) = 69 inkips (> 22.6 inkips)
Each component is strong enough, so the proposed bars are satisfactory.
Figure 8.39 Details of the embedded plate in the DT at the base
(1.0 in = 25.4 mm)
Figure 8.310 Sketch of connection of nonloadbearing DT wall panel at the roof
(1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)
Figure 8.311 Sketch of connection of loadbearing DT wall panel at the roof
(1.0 in = 25.4 mm)
8.3.5.8 Other Connections. This design assumes that there is no inplane shear transmitted from panel to
panel. Therefore, if connections are installed along the vertical joints between DT panels to control the
outofplane alignment, they should not constrain relative movement inplane. In a practical sense, this
means the chord for the roof diaphragm should not be a part of the panels. Figures 8.310 and 8.311
show the connections at the roof and DT wall panels. These connections are not designed here. Note that
the continuous steel angle would be expected to undergo vertical deformations as the panels deform
laterally.
Because the diaphragm supports concrete walls out of their plane, Standard Section 12.11.2.1 requires
specific force minimums for the connection and requires continuous ties across the diaphragm. Also, it
specifically prohibits use of the metal deck as the ties in the direction perpendicular to the deck span. In
that direction, the designer may wish to use the top chord of the bar joists, with an appropriate connection
at the joist girder, as the continuous cross ties. In the direction parallel to the deck span, the deck may be
used, but the laps should be detailed accordingly.
In precast DT shear wall panels with flanges thicker than 21/2 inches, consideration may be given to
using vertical connections between the wall panels to transfer vertical forces resulting from overturning
moments and thereby reduce the overturning moment demand. These types of connections are not
considered here, since the uplift force is small relative to the shear force and cyclic loading of bars in thin
concrete flanges is not always reliable in earthquakes.
8.4 SPECIAL MOMENT FRAMES CONSTRUCTED USING PRECAST CONCRETE
As for special concrete walls, the Standard does not distinguish between a castinplace and a precast
concrete special moment frame in Table 12.21. However, ACI 318 Section 21.8 provides requirements
for special moment frames constructed using precast concrete. That section provides requirements for
designing special precast concrete frame systems using either ductile connections (ACI 318 Sec. 21.8.2)
or strong connections (ACI 318 Sec. 21.8.3.) ACI 318 Section 21.8.4 also explicitly allows precast
moment frame systems that meet the requirements of ACI 374.1, Acceptance Criteria for Moment
Frames based on Structural Testing.
8.4.1 Ductile Connections
For moment frames constructed using ductile connections, ACI 318 requires that plastic hinges be able to
form in the connection region. All of the requirements for special moment frames must still be met, plus
there is an increased factor that must be used in developing the shear demand at the joint.
It is interesting to note that while Type 2 connectors can be used anywhere (including in a plastic hinge
region) in a castinplace frame, these same connectors cannot be used closer than h/2 from the joint face
in a ductile connection. The rationale behind this requirement is that in a jointed system, a concentrated
crack occurs at the joint between precast elements in a ductile connection. Thus the rotation is
concentrated at this location. Type 2 connectors are actually strong connections, relative to the bar, as
they are designed to develop the tensile strength of the bar. The objective of Type 2 connectors is that
they relocate the yielding away from the connector, into the bar itself.
If a Type 2 connector is used at the face of a column as shown in Figure 8.41 and the bar size is the same
in both the column and the beam, yielding will occur at the joint at the face of the column but not be able
to spread into the beam to develop a plastic hinge, due to the strength of the connector. This concentrates
the yielding in the bar to the left of the connector and likely will fracture the bar when significant rotation
is imposed on the beam.
Figure 8.41 Type 2 coupler location in a strong connection
(1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)
In a ductile connection, frame yielding takes place at the connection. This is most easily accomplished by
extending the reinforcement out of the precast column element and coupling this rebar at the end of the
precast beam. Since the couplers have to be located a minimum distance of h/2 from the joint face (i.e.,
column face) the resulting gap between the precast beam and precast column is filled with castinplace
concrete as shown in Figure 8.42.
Figure 8.42 Type 2 coupler location in a ductile connection
(1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)
8.4.2 Strong Connections
ACI 318 also provides design rules for strong connections used in special moment frames. The concept is
to provide connections that are strong enough to remain elastic when a plastic hinge forms in the beam.
Thus the frame behavior is the same as would occur if the connection were monolithic.
Using the frame in Figure 8.43 (ignoring gravity forces for simplicity), design forces for the plastic hinge
region and the associated forces on the precast connection are computed. Assuming inflection points at
midheight of the columns and a seismic shear force of Vcol on each column:
Figure 8.43 Moment frame geometry
(1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)
Under seismic loads alone, the shear is constant along the beam length. Therefore, the moment at the
joint between the end of the beam and the column is:
The plastic hinge, however, will be relocated to the side of the Type 2 coupler away from the column.
With a coupler length of lcoupler, the moment at the end of the coupler is:
In order to ensure that the hinge forms at the intended location (away from the precast connection), the
connection needs to be designed to be stronger than the moment associated with the development of the
plastic hinge. This is done by upsizing the bar that is anchored into the column.
8.4.2.1 Strong Connection Example. In the following numerical example, a singlebay frame is
designed to meet the requirements of a precast frame using strong connections at the beamcolumn
interface. Using Figure 8.43 and the following geometry:
Hcol = 12 ft
hcol = 36 in.
Lb = 30 ft (column centerline to column centerline)
lcoupler = 18 in.
Lclr = Lb  hcol  2lcoupler = 24 ft (distance between plastic hinge locations)
hbeam = 42 in.
Reinforcing the beam with three #10 bars top and bottom, the nominal moment strength of the beam is:
= 3.0 in.
= 540 ftkips
This is the moment strength at the plastic hinge location. The strong precast connection must be designed
for the loads that occur at the connection when the beam at the plastic hinge location develops its
probable strength.
Therefore, the moment strength at the beamcolumn interface (which also is the precast joint location)
must be at least:
Where:
= 750 ftkips
Therefore, the design strength of the connection must be at least:
= 843 ftkips
Using #14 bars in the column side of the Type 2 coupler:
= 5.3 in.
= 921 ftkips
which is greater than the load at the connection (843 ftkips) when the plastic hinge develops.
If columntocolumn connections are required, ACI 318 Section 21.8.3(d) requires a 1.4 amplification
factor, in addition to loads associated with the development of the plastic hinge in the beam. Locating the
column splice near the point of inflection, while difficult for construction, can help to make these forces
manageable.
The beam shear, when the plastic hinge location reaches its nominal strength, is:
= 20 kips
Assuming inflection points at the midspan of the beam and midheight of the column, the column shear
is:
= 50 kips
However, the column shear must be amplified to account for the development of the plastic hinge.
= 69 kips
The column design moment is:
= 293 ftkips
At the connection, this moment is amplified by 1.4 for a strong connection design moment of 410 ftkips.
This moment must be combined with the axial load on the connection from both gravity loads and
amplified seismic forces.
The balance of the design is the same as for a castinplace special moment frame.
9
Composite Steel and Concrete
Clinton O. Rex, P.E., PhD
Originally developed by
James Robert Harris, P.E., PhD and Frederick R. Rutz, P.E., PhD
Contents
9.1 BUILDING DESCRIPTION 3
9.2 PARTIALLY RESTRAINED COMPOSITE CONNECTIONS 7
9.2.1 Connection Details 7
9.2.2 Connection MomentRotation Curves 10
9.2.3 Connection Design 13
9.3 LOADS AND LOAD COMBINATIONS 17
9.3.1 Gravity Loads and Seismic Weight 17
9.3.2 Seismic Loads 18
9.3.3 Wind Loads 19
9.3.4 Notional Loads 19
9.3.5 Load Combinations 20
9.4 DESIGN OF CPRMF SYSTEM 21
9.4.1 Preliminary Design 21
9.4.2 Application of Loading 22
9.4.3 Beam and Column Moment of Inertia 23
9.4.4 Connection Behavior Modeling 24
9.4.5 Building Drift and Pdelta Checks 24
9.4.6 Beam Design 26
9.4.7 Column Design 27
9.4.8 Connection Design 28
9.4.9 Column Splices 29
9.4.10 Column Base Design 29
The 2009 NEHRP Recommended Provisions for the design of a composite building using a Composite
Partially Restrained Moment Frame (CPRMF) as the lateral forceresisting system is illustrated in this
chapter by means of an example design. The CPRMF lateral forceresisting system is recognized in
Standard Section 12.2 and in AISC 341 Part II Section 8; and it is an appropriate choice for buildings in
low to moderate Seismic Design Categories (SDC A to D). There are other composite lateral force
resisting systems recognized by the Standard and AISC 341; however, the CPRMF is the only one
illustrated in this set of design examples.
The design of a CPRMF is different from the design of a more traditional steel moment frame in three
important ways. First, the design of a Partially Restrained Composite Connection (PRCC) differs in that
the connection itself is not designed to be stronger than the beam it is connecting. Consequently, the
lateral system typically will hinge within the connections and not within the associated beams or columns.
Second, because the connections are neither simple nor rigid, their stiffness must be accounted for in the
frame analysis. Third, because the connections are weaker than fully restrained moment connections, the
lateral forceresisting system requires more frames with more connections, resulting in a highly redundant
system.
In addition to the 2009 NEHRP Recommended Provisions (referred to herein as the Provisions), the
following documents are referenced throughout the example:
ACI 318 American Concrete Institute. 2008. Building Code Requirements for Structural
Concrete.
AISC 341 American Institute of Steel Construction. 2005. Seismic Provisions for Structural
Steel Buildings, including Supplement No. 1.
AISC 360 American Institute of Steel Construction. 2005. Specification for Structural Steel
Buildings.
AISC Manual American Institute of Steel Construction. 2005. Steel Construction Manual.
Thirteenth Edition.
AISC SDGS8 American Institute of Steel Construction. 1996. Partially Restrained Composite
Connections, Steel Design Guide Series 8. Chicago: AISC.
AISC SDM American Institute of Steel Construction. 2006. Seismic Design Manual. First
Edition.
Arum (1996) Mayangarum, Arum, 1251996. Design, Analysis and Application of Bolted Semi
Rigid Connections for Moment Resisting Frames, MS Thesis, Lehigh University.
ASCE TC American Society of Civil Engineers Task Committee on Design Criteria for
Composite Structures in Steel and Concrete. October 1998. Design Guide for
Partially Restrained Composite Connections, Journal of Structural Engineering
124(10).
RCSC Research Council on Structural Connections. 2004. Specification for Structural Joints
Using ASTM A325 or A490 Bolts.
Standard American Society of Civil Engineers, 2005, ASCE/SEI 705 Minimum Design Loads
for Buildings and other Structures
Yura (2006) Yura, Joseph A and Helwig, Todd A. (282006) Notes from SSRC/AISC Short
Course 2 on Beam Buckling and Bracing The shortform designations presented
above for each citation are used throughout.
The PRCC used in the example has been subjected to extensive laboratory testing, resulting in the
recommendations of AISC SDGS8 and ASCE TC. ASCE TC is the newest of the two guidance
documents and is referenced here more often; however, AISC SDGS8 provides information not in ASCE
TC, which is still pertinent to the design of this type of frame. While both of these documents provide
guidance for design of PRCC, the method presented in this design example deviates from that guidance
based on more recent code requirements for stability and on years of experience in designing CPRMF
systems.
The structure is analyzed using threedimensional, static, nonlinear methods. The SAP 2000 analysis
program, v. 14.0 (Computers and Structures, Inc., Berkeley, California) is used in the example.
The symbols used in this chapter are from Chapter 2 of the Standard or the above referenced documents,
or are as defined in the text. U.S. Customary units are used.
9.1 BUILDING DESCRIPTION
The example building is a fourstory steel framed medical office building located in Denver, Colorado
(see Figures 9.11 through 9.13). The building is free of plan and vertical irregularities. Floor and roof
slabs are 4.5inch normalweight reinforced concrete on 0.6inch form deck (total slab depth of
4.5 inches.). Typically slabs are supported by open web steel joists which are supported by composite
steel girders. Composite steel beams replace the joists at the spandrel locations to help control cladding
deflections. The lateral loadresisting system is a CPRMF in accordance with Standard Table 12.21 and
AISC 341 Part II Section 8. The CPRMF uses PRCCs at almost all beamtocolumn connections. A
conceptual detail of a PRCC is presented in Figure 9.14. The key advantage of this type of moment
connection is that it requires no welding. The lack of field welding results in erection that is quicker and
easier than that for more traditional moment connections with CJP welding and the associated
inspections.
Figure 9.11 Typical floor and roof plan
Figure 9.12 Building end elevation
Figure 9.13 Building side elevation
The building is located in a relatively low seismic hazard region, but localized internal storage loading
and Site Class E are used in this example to provide somewhat higher seismic design forces for purposes
of illustration and to push the example building into Seismic Design Category C.
Figure 9.14 Conceptual partially restrained composite connection (PRCC)
There are no foundations designed in this example. For this location and system, the typical foundation
would be a drilled pier and voided grade beam system, which would provide flexural restraint for the
strong axis of the columns at their base (very similar to the foundation for a conventional steel moment
frame). The main purpose here is to illustrate the procedures for the PRCCs. The floor and roof slabs
serve as horizontal diaphragms distributing the seismic forces and by inspection they are stiff enough to
be considered as rigid.
The typical bay spacing is 25 feet. Architectural considerations allowed an extra column at the end bay of
each side in the northsouth direction, which is useful in what is the naturally weaker direction. The
exterior frames in the northsouth direction have momentresisting connections at all columns. The
frames in each bay in the eastwest direction have momentresisting connections at all columns except the
end columns. Composite connections to the weak axis of the column are feasible, but they are not used
for this design. The PRCC connection locations are illustrated in Figure 9.11.
Material properties in this example are as follows:
Structural steel beams and columns (ASTM A992): Fy = 50 ksi
Structural steel connection angles and plates (ASTM A36): Fy = 36 ksi
Concrete slab (4.5 inches thick on form deck, normal weight): fc' = 3,000 psi
Steel reinforcing bars (ASTM A615): Fy = 60 ksi
9.2 PARTIALLY RESTRAINED COMPOSITE CONNECTIONS
9.2.1 Connection Details
The type of PRCC used for this example building consists of a reinforced composite slab, a doubleangle
bolted web connection and a bolted seat angle. In real partially restrained building design, it is
advantageous to select and design the complete PRCC simply based on beam depth and element
capacities. Generally it is impractical to tune connections to beam plastic moment capacities and/or
lateral load demands. This allows the designer to develop an inhouse suite of PRCC details and
associated behavior curves for each nominal beam depth ahead of time. Slight adjustments can be made
later to account for real versus nominal beam depth.
It is considered good practice (particularly for capacitybased seismic design) to provide substantial
rotation capacity at connections while avoiding nonductile failure modes. This requirement for ductile
rotation capacity is expressed in AISC 341 Part II Section 8.4 as a requirement for story drift of
0.04 radians. Because much of the drift in a partially restrained building comes from connection rotation,
this story drift requirement implies a connection rotation ductility requirement. In short, connections
must be detailed to allow ductile modes to dominate over nonductile failure modes.
Practical detailing is limited by commonly available components. For instance, the largest angle leg
commonly available is 8 inches, which can reasonably accommodate four 1inchdiameter bolts. As a
result, the maximum shear that can be delivered from the beam flange to the seat angle is limited by shear
in four A490X bolts. Bolt shear failure is generally considered to be nonductile, so the rest of the
connection design and detailing aims to maximize moment capacity of the connection while avoiding this
limit state.
The connection details chosen for this example are illustrated in Figures 9.21, 9.22, 9.23 and 9.24.
Figure 9.21 Typical interior W18x35 PRCC
Figure 9.22 Typical spandrel W21x44 PRCC
Figure 9.23 Typical corner PRCC
Figure 9.24 Typical PRCC reinforcing plan
9.2.2 Connection MomentRotation Curves
Two connection momentrotation curves are required for the design of partially restrained buildings: the
nominal momentrotation curve and the modified momentrotation curve.
The nominal momentrotation curve, obtained from connection test data or from published moment
rotation prediction models, is used for servicelevel load design. For this example, the published
momentrotation prediction model given in ASCE TC is used to define the momentrotation curve for the
PRCC.
Negative momentrotation behavior (slab in tension):
(ASCE TC, Eq. 4)
Where:
C1 = 0.18(4 ArbFyrb + 0.857AsaFya)(d + Y3), kipin.
C2 = 0.775
C3 = 0.007(Asa + Awa)Fya (d + Y3), kipin.
= connection rotation (mrad = radians 1,000)
d = beam depth, in.
Y3 = distance from top of beam to the centroid of the longitudinal slab reinforcement, in.
Arb = area of longitudinal slab reinforcement, in2
Asa = gross area of seat angle leg, in2
(For use in these equations, Asa is limited to a maximum of 1.5Arb)
Awa = gross area of double web angles for shear calculations, in2
(For use in these equations, Awa is limited to a maximum of 2.0Asa)
Fyrb = yield stress of reinforcing, ksi
Fya = yield stress of seat and web angles, ksi
Positive momentrotation behavior (slab in compression):
(ASCE TC, Eq. 3)
Where:
C1 = 0.2400[(0.48Awa) + Asa](d + Y3)Fya, kipin.
C2 = 0.0210(d + Y3/2)
C3 = 0.0100(AwL + AL)(d + Y3)Fya, kipin.
C4 = 0.0065 AwL(d + Y3)Fya, kipin.
The modified momentrotation curve is used for strength level load design. The Direct Analysis Method
requires two modifications to the nominal momentrotation curve: an elastic stiffness reduction and a
strength reduction. AISC 360 Section 7.3(3) requires an elastic stiffness reduction of 0.8, which is
accomplished by translating the connection rotation by an elastic stiffness reduction offset. This
translation can be shown as follows:
Where:
Mc = connection moment from the nominal momentrotation curve, kipin.
Kci = connection initial stiffness, kipin./mrad; because the momentrotation curve is nonlinear, it is
necessary to define how the initial stiffness will be measured. For this example, the initial
stiffness will be taken as the secant stiffness to the momentrotation curve at = 2.5 mrad as
suggested in ASCE TC. Note that this will be different values for the positive and negative
momentrotation portions of the connection behavior.
The second modification to the nominal momentrotation curve is a strength reduction associated with .
ASCE TC recommends using equal to 0.85. The associated connection strength is given by:
McDAM = 0.85 Mc
From these equations, curves for M can be developed for a particular connection. The momentrotation
curves for the typical connections associated with the W18x35 girder and the W21x44 spandrel beam are
presented in Figures 9.25 and 9.26, respectively.
Figure 9.25 Typical interior W18x35 PRCC M curves
Figure 9.26 Typical spandrel W21x44 PRCC M curves
Important key values from the above connection curves are summarized in Table 9.21 for reference in
later parts of the example design.
Table 9.21 Key Connection Values From MomentRotation Curves
W18x35 PRCC
W21x44 PRCC
Kci (kipin/rad) (nominal)
704,497
1,115,253
Kci+ (kipin/rad) (nominal)
338,910
554,498
Mc @ 20 mrad (kipft)
(nominal/modified)
232/206
367/326
Mc+ @ 10 mrad (kipft)
(nominal/modified)
151/127
240/202
These curves and the corresponding equations do not reproduce the results of any single test. Rather, they
are averages fitted to real test data using numerical methods and they smear out the slip of bolts into
bearing. Articles in the AISC Engineering Journal (Vol. 24, No.2; Vol. 24, No.4; Vol. 27, No.1; Vol. 27,
No. 2; and Vol. 31, No. 2) describe actual test results. Those tests demonstrate clearly the ability of the
connection to satisfy the rotation requirements of AISC 341 Part II Section 8.4.
9.2.3 Connection Design
This section illustrates the detailed design decisions and checks associated with the typical W21x44
spandrel beam connection. A complete design would require similar checks for each different connection
type in the building. Design typically involves iteration on some of the chosen details until all the design
checks are within acceptable limits.
9.2.3.1 Longitudinal Reinforcing Steel. The primary negative moment resistance derives from tensile
yielding of slab reinforcing steel. Since ductile response of the connection requires that the reinforcing
steel yield and elongate prior to failure of other connection components, providing too much reinforcing
is not a good thing. The following recommendations are from ASCE TC.
A minimum of six bars (three bars each side of column), #6 or smaller, should be used (eight #5 bars have
been used in this example). The bars should be distributed symmetrically within a total effective width of
seven column flange widths (36 inches at each side of the column has been used in this example). For
edge beams, the steel should be distributed as symmetrically as possible, with at least onethird (minimum
three bars) of the total reinforcing on the exterior side of the column. Bars should extend a minimum of
onefourth of the beam length or 24 bar diameters past the assumed inflection point at each side of the
column. For seismic design a minimum of 50 percent of the reinforcing steel should be detailed
continuously. Continuous reinforcing should be spliced with a Class B tension lap splice and minimum
cover should be in accordance with ACI 318.
9.2.3.2 Transverse Reinforcing Steel. The purpose of the transverse reinforcing steel is to help promote
the force transfer from the tension reinforcing to the column and to prevent potential shear splitting of the
slab over the beams, thus allowing the beam studs to transfer the reinforcing tension force into the beam.
ASCE TC recommends the following.
Provide transverse reinforcement, consistent with a strutandtie model as shown in Figure 9.27. In the
limit (maximum), this amount will be equal to the longitudinal reinforcement. The transverse reinforcing
should be placed below the top of the studs to prevent a conetype failure over the studs. The transverse
bars should extend at least 12 bar diameters or 12 inches, whichever is larger, on either side of the outside
longitudinal bars.
Figure 9.27 Force transfer mechanism from slab to column
Concrete bearing stresses on the column flange should be limited to per the ASCE TC
recommendations. For the W21x44 PRCC, the sum of the positive and negative moment capacity is
607 kipft. The moment arm is approximately 22.95 inches (20.7 + 4.5/2). So the maximum possible
transfer of force from the slab to the column, if each connection is at maximum and opposite strengths on
each side of the column, is 607 ftkip/22.95 inches = 317 kips. A W10x88 column has a 10.3inchwide
flange. Assuming uniform bearing of the concrete on each flange, the bearing stress would be 317 kips /
2 flanges / 4.5inchthick slab / 10.3inchwide flange = 3.42 ksi, which is less than the recommended
limit of . It is also necessary to check this force against the flange local bending and web local
yielding limit states given in Chapter J of AISC 360. It is important to have concrete filling the gap
between column flanges; otherwise, the force must be transferred by a single column flange.
9.2.3.3 Connection Moment Capacity Limits. AISC 341 Part II Section 8.4 requires that the PRCC
have a nominal strength that is at least equal to 50 percent of the nominal Mp for the connected beam
ignoring composite action. ASCE TC recommends 75 percent as a good target, with 50 percent as a
lower limit and 100 percent as an upper limit. ASCE TC also recommends using the moment capacity at
20 mrad for negative moment and 10 mrad for positive moment to determine the nominal connection
moment capacity. From the W21x44 PRCC connection curve, the negative moment capacity at 20 mrad
is 367 kipft and the positive moment capacity at 10 mrad is 240 kipft. With Mp of the beam being
398 kipft, the ratio of connectiontobeam moment capacity is 0.922 and 0.603 for negative and positive
moments, respectively.
9.2.3.4 Seat Angle. The typical gage for the bolts attaching the seat angle to the column is 5.5 inches to
allow sufficient room for bolt tightening on the inside of the column. For a 1inch bolt diameter and a
1.75inch minimum edge distance to a sheared edge, the minimum angle length is 9 inches. Per ASCE
TC, the minimum area of the outstanding angle leg should be:
Asamin = 1.33 Fyrd Arb / Fya = 5.497 in2
A 5/8inch thick angle with the 9inch angle length results in Asa equal to 5.625 in2.
The outstanding angle dimension is controlled by the number of bolts attaching the angle to the beam
flange. As previously discussed, a minimum 8inch dimension is desired here to allow room for four
1inchdiameter bolts.
The vertical angle dimension has to be sufficient both to allow room for bolts to the column flange and to
permit yielding when the seat angle is in tension. The ductility of the connection, when in positive
bending, is derived from angle hinging, as shown in Figure 9.28.
Figure 9.28 Typical angle tension hinging mechanism
This mechanism is based on research by Arum (1996). The following equations can be used to determine
the associated angle tension, prying forces and bolt forces associated with the angle hinging mechanism.
a = Lvsa gsa + dbsa / 2 = 2.500 in.
c = (Wsa dbsa) / 2 = 0.313 in.
b = Lvsa a c ksa = 2.062 in.
Mpsa = Fya tsa2 Lsa / 4 = 31.641 kipin
Tsa = 2 Mpsa / b = 30.682 kips
Qsa = Mpsa / a (1 + 2 c / b ) = 16.491 kips
Bsa = Tsa + Qsa = 47.173 kips
The above equations were derived in the same fashion as the prying action equations currently given in
Section 9 of the AISC Manual with the same limitations applied to a . The nomenclature in the above
equations is shown in Figure 9.29.
Figure 9.29 Seat angle nomenclature
The author recommends that the ratio of tsa/b be limited to no more than 0.5, so that the angle can
properly develop the assumed hinges. For the example detail, the ratio is 0.303.
9.2.3.5 Bolts in Vertical Seat Angle Leg. The bolts in the vertical seat angle leg are designed primarily
to resist tension in the case of connection positive moment. To protect against premature tension failure,
the bolt force calculated in the previous section should be magnified by Ry from AISC 341 Table I61.
Ry Bsa = 1.5 47.173 kips = 70.76 kips
The tension capacity for two 1inchdiameter A490 bolts is 133 kips.
9.2.3.6 Bolts in Outstanding Seat Angle Leg. The bolts in the outstanding leg of the seat angle must be
designed for the shear transfer between the beam flange and the seat angle. For positive moments, this
force is limited by tension hinging of the seat angle as calculated previously. For negative moments, this
force is the sum of tension from the reinforcing steel and tension developed from hinging of the web
angles. In general, the later will be significantly more than the former. The tension hinging capacity of
the web angles, Twa, is calculated in the same way as the tension hinging of the seat angle. Again, to
protect against premature shear failure of bolts, the tension capacity of the web angle and the reinforcing
steel is magnified by an appropriate Ry. ASCE TC recommends Ry = 1.25 for the reinforcing steel.
Ry Twa + Ry Fyrd Arb = 1.5 22.5 kips + 1.25 60 ksi 2.48 in2 = 220 kips
The published shear capacity for four 1inchdiameter A490X bolts is 177 kips; however, this capacity
includes a 0.8 reduction to account for joint lengths up to 50 inches per the RCSC. The RCSC further
states that this reduction does not apply in cases where the distribution of force is essentially uniform
along the joint. When one increases the published shear capacity by 1/0.8, the revised shear capacity is
221 kips. Bolt bearing at the beam flange and at the seat angle should also be checked.
9.2.3.7 Double Angle Web Connection. The primary purpose of the double angle web connection is to
resist shear. Therefore, it can be selected directly from the AISC Manual; the specific design limits will
not be addressed here. The required shear force is determined by adding the seismic demand to the
gravity demand. The seismic demand for the W21x44 PRCC is the sum of the positive and negative
moment capacity (607 kipft) divided by the appropriate beam length. For the typical 25foot beam
length, the seismic shear is approximately 25 kips.
9.3 LOADS AND LOAD COMBINATIONS
9.3.1 Gravity Loads and Seismic Weight
The design gravity loads and the associated seismic weights for the example building are summarized in
Table 9.31. The seismic weight of the storage live load is taken as 50 percent of the design gravity load
(a minimum of 25 percent is required by Standard Section 12.7.2). To simplify this design example, the
roof design is assumed to be the same as the floor design and floor loads are used rather than considering
special roof and snow loads.
Table 9.31 Gravity Load and Seismic Weight
Gravity
Load
Seismic
Weight
NonComposite Dead Loads (Dnc)
4.5in. Slab on 0.6in. Form Deck (4.5in. total thickness)
plus Concrete Ponding
58 psf
58 psf
Joist and Beam Framing
6 psf
6 psf
Columns
2 psf
2 psf
Total:
66 psf
66 psf
Composite Dead Loads (Dc)
Fire Insulation
4 psf
4 psf
Mechanical and Electrical
6 psf
6 psf
Ceiling
2 psf
2 psf
Total:
12 psf
12 psf
Precast Cladding System
800 plf
800 plf
Live Loads (L)
Typical Area Live and Partitions (Reducible)
70 psf
10 psf
Records Storage Area Live (NonReducible)
200 psf
100 psf
The reason for categorizing dead loads as noncomposite and composite is explained in Section 9.4.2.
Live loads are applied to beams in the analytical model, with corresponding live load reductions
appropriate for beam design. Column live loads are adjusted to account for different live load reduction
factors, including the 20 percent reduction on storage loads for columns supporting two or more floors per
Standard Section 4.8.2.
9.3.2 Seismic Loads
The basic seismic design parameters are summarized in Table 9.32
Table 9.32 Seismic Design Parameters
Parameter
Value
Ss
0.20
S1
0.06
Site Class
E
Fa
2.5
Fv
3.5
SMS = FaSs
0.50
SM1 = FvS1
0.21
SDS = 2/3SMS
0.33
SD1 = 2/3SM1
0.14
Occupancy Category
II
Importance Factor
1.0
Seismic Design Category (SDC)
C
Frame Type per Standard Table 12.21
Composite Partially Restrained
Moment Frame
R
6
0
3
Cd
5.5
For Seismic Design Category C, the height limit is 160 feet, so the selected system is permitted for this
52foottall example building. The building is regular in both plan and elevation; consequently, the
Equivalent Lateral Force Procedure of Section 12.8 is permitted in accordance with Standard
Table 12.61. The seismic weight, W, totals 7,978 kips. The approximate period is determined to be
0.66 seconds using Equation 12.87 and the steel momentresisting frame parameters of Table 12.82.
The coefficient for upper limit on calculated period, Cu, from Table 12.81 is 1.62, resulting in Tmax of
1.07 seconds for purposes of determining strengthlevel seismic forces.
A specific value for PRCC stiffness must be selected in order to conduct a dynamic analysis to determine
the building period. It is recommended that the designer use Kci of the negative momentrotation behavior
given in Section 9.2.2 above for this analysis. This should result in the shortest possible analytical
building period and thus the largest seismic design forces. For the example building, the computed
periods of vibration in the first modes are 2.13 and 1.95 seconds in the northsouth and eastwest
directions, respectively. These values exceed Tmax, so strengthlevel seismic forces must be computed
using Tmax for the period. The seismic response coefficient is then given by:
The total seismic forces or base shear is then calculated as:
V = Cs W = (0.022)(7,978) = 174 kips (Standard Eq. 12.81)
The distribution of the base shear to each floor (by methods similar to those used elsewhere in this
volume of design examples) is:
Roof (Level 4): 77 kips
Story 4 (Level 3): 53 kips
Story 3 (Level 2): 31 kips
Story 2 (Level 1): 13 kips
: 174 kips
For Seismic Design Category C, the value of is permitted to be taken as 1.0 per Standard
Section 12.3.4.1, so the above story shears are applied as Eh without any additional magnification.
9.3.3 Wind Loads
From calculations not illustrated here, the gross servicelevel wind force following ASCE 7 is 83 kips
(assuming 90 mph, 3secondgust wind speed). Including the directionality effect and the strength load
factor, the design wind force is less than the design seismic base shear. The wind force is not distributed
in the same fashion as the seismic force, thus the story shears and the overturning moments for wind are
considerably less than for seismic. The distribution of the wind base shear to each floor is:
Roof (Level 4): 13 kips
Story 4 (Level 3): 25 kips
Story 3 (Level 2): 23 kips
Story 2 (Level 1): 22 kips
: 83 kips
Because the wind loads are substantially below the seismic loads, they are not considered in subsequent
strength design calculations; however, wind drift is considered in the design.
9.3.4 Notional Loads
AISC 360 now requires that notional loads be included in the building analysis. As shown later, the
example building qualifies for application of notional loads to gravityonly load combinations. The
notional load at level i is Ni = 0.002Yi, where Yi is the gravity load applied at level i. For our example
building, these values are as follows:
NDnc = 4,258 kips 0.002 = 8.516 kips / 4 floors = 2.13 kips/floor
NDc = 2,393 kips 0.002 = 4.786 kips / 4 floors = 1.20 kips/floor
NL = 4,469 kips 0.002 = 8.938 kips / 4 floors = 2.23 kips/floor
The notional loads are applied in the same manner as the seismic and wind loads in each orthogonal
direction of the building and they are factored by the same load factors that are applied to their
corresponding source (such as 1.2 or 1.4 for dead loads). It is important to note that, in general, notional
loads should be determined, at a minimum, on a columnbycolumn basis rather than for an entire floor as
done above. This will allow the design to capture the effect of gravity loads that are not symmetric about
the center of the building. The example building happens to have gravity loads that are concentric with
the center of the building, so it does not matter in this case.
9.3.5 Load Combinations
Three load combinations (from Standard Section 2.3.2) are considered in this design example.
Load Combination 2: 1.2D + 1.6L
Load Combination 5: 1.2D + 0.5L + 1.0E
Load Combination 7: 0.9D + 1.0E
Expanding the combinations for vertical and horizontal earthquake effects, breaking D into Dnc and Dc
(defined in Section 9.3.1) and including notional loads, results in:
Load Combination 2: 1.2(Dnc + NDnc) + 1.2(Dc + NDc) + 1.6(L + NL)
Load Combination 5: 1.2Dnc + 1.2Dc + 0.5L + 1.0Eh +1.0Ev
Ev = 0.2SDS (Dnc + Dc) = 0.2(0.33)(Dnc + Dc) = 0.067(Dnc + Dc)
1.267Dnc + 1.267Dc + 0.5L + 1.0Eh
Load Combination 7: 0.9 Dnc + 0.9 Dc + 1.0 Eh 1.0 Ev
0.833 Dnc + 0.833 Dc + 1.0 Eh
Dnc has to be applied separately to the columns and beams because of the twostage connection behavior
(discussed later). Dncc is for column loading and Dncb is for beam loading. This breakout of the loading
results in the following combinations:
Stage 1 Analysis:
Load Combinations 2 and 5: 1.2 Dncb
Load Combination 7: 0.9 Dncb
Stage 2 Analysis:
Load Combination 2: 1.2(Dncc + NDnc) + 1.2(Dc + NDc) + 1.6(L + NL)
Load Combination 5: 1.2Dncc + 0.067Dncb + 1.267Dc + 0.5L + 1.0Eh
Load Combination 7: 0.9Dncc  0.067Dncb + 0.833Dc + 1.0Eh
The columns are designed from the Stage 2 Analysis and the beams are designed from the linear
combination of the Stage 1 and Stage 2 Analyses.
Because partially restrained connection behavior is nonlinear, seismic and wind drift analyses must be
carried out for each complete load combination, rather than for horizontal loads by themselves. Note that
Standard Section 12.8.6.2 allows drifts to be checked using seismic loads based on the analytical building
period.
Seismic Drift: 1.0Dncc + 0.067Dncb + 1.0Dc + 0.5L + 1.0Eh
Wind Drift: 1.0Dncc + 1.0Dc + 0.5L + 1.0W
The typical permeations of the above combinations have to be generated for each orthogonal direction of
the building; however, orthogonal effects need not be considered for Seismic Design Category C provided
the structure does not have a horizontal structural irregularity (Standard Sec. 12.5.3).
9.4 DESIGN OF CPRMF SYSTEM
9.4.1 Preliminary Design
The goal of an efficient partially restrained building design is to have a sufficient number of beams,
columns and connections participating in the lateral system so that the forces developed in any of these
elements from lateral loads is relatively small compared to the gravity design. In other words, design for
gravity as if the connections are pinned; add the connections and check to see if any beams or columns
must be upsized to handle the lateral loads. The author cautions designers against trying to reduce beam
sizes below the initial gravity sizes unless a full inelastic, pathdependent analysis accounting for
potential shakedown of the connections is conducted. At this time, such an analysis typically is relegated
to academic study and is not applied in real building design. The analysis methods described below do
not go to that level of detail.
Once the building has been designed for gravity, a preliminary lateral analysis can be made to assess
whether the proposed steel framing sizes may be suitable for lateral loads in combination with gravity
loads. Typically this is done assuming all the PRCCs are rigid connections. Two basic checks can be
based on this preliminary analysis. First, review connection moments that come from the lateral load
cases alone (earthquake moments and wind moments) without gravity. If these moments (at strength
levels) exceed approximately 75 percent of the negative moment capacity of the PRCC then either
additional beams, columns and connections need to be added to the lateral system or existing beams need
to be upsized to provide larger PRCCs with higher capacities. Second, perform a preliminary assessment
of the building drift. While there is no simple, reliable relationship between rigid frame drift and
CPRMF drift, the author typically assumes that the partially restrained system will drift approximately
twice as much as a fully rigid analysis indicates. Keep in mind that these preliminary checks are made to
establish basic system proportions before extensive modeling efforts are made to include the real partially
restrained behavior of the building.
Using this preliminary design method, initial floor framing was selected. In accordance with the
ASCE TC, the beams are designed to be 100 percent composite; no partial composite design is used.
The W18x35 typical interior girder is determined from a simple beam design. This typical size would
work for all locations with the exception of the girders that support storage load on both sides (Grids 4
and 5 between Grids C and D). For simplicity, the example design was not further refined. The W18x35
size would also work as the Grid Line A and F spandrel beams; however, a W21x44 spandrel beam is
used to help control drift in the northsouth direction and help equalize the building periods in both
directions. Note that the W21x44 improves drift more due to the increase in beam depth, which increases
PRCC momentrotation stiffness, rather than because of the increase in the moment of inertia of the steel
beam section.
9.4.2 Application of Loading
PRCC do not develop substantial beam end restraint until after the concrete has hardened (since the
reinforcing steel cannot be mobilized without the concrete). At the time of concrete casting, the bare steel
elements of the connection are all that are present to resist rotation at the beam ends. The degree of
restraint provided by the bare steel connection varies depending on the details; however, for purposes of
design, the connection stiffness prior to concrete hardening typically is assumed to be zero (a pinned
beam end). Consequently, the connection actually has two stages of behavior that need to be accounted
for in the analysis. These two stages are the precomposite stage, when the connection is assumed to
behave as a pin and the postcomposite stage, when the connection is assumed to have the full moment
rotation behavior determined in Section 9.2.2. In a building where the complete lateral system is
provided by PRCCs, temporary bracing may be required to provide lateral stability prior to concrete
hardening.
The above twostage connection behavior requires separation of dead load into portions consistent with
each stage. This is why the dead loads in Section 9.3.1 are separated into Dnc and Dc. The Dnc load is
placed on the beams during the Stage 1 analysis (when the connections are pins) but is not placed on the
beams (other than the seismic fraction) during the Stage 2 analysis (when the connections have PRCC
stiffness). In Stage 2 analysis, the Dnc loads are placed directly on the columns so that their destabilizing
effects are accounted for properly in the nonlinear Pdelta analysis. That is why Dnc loads are further
broken down into Dncc and Dncb. The Stage 2 load combinations are presented graphically in Figures 9.41
and 9.42.
Figure 9.41 Stage 2 Load Combination 5
Figure 9.42 Stage 2 Load Combination 7
9.4.3 Beam and Column Moment of Inertia
ASCE TC recommends that the beam moment of inertia used for frame analysis be increased to account
for the stiffening effect that the composite slab has on the beam moment of inertia. The use of the
increased moment of inertia is also required by AISC 341 Part II Section 8.3. The following equivalent
moment of inertia is recommended:
Ieq = 0.6ILB+ + 0.4ILB (Eq. 5, ASCE TC)
ILB+ and ILB are the lower bound moments of inertia in positive and negative bending, respectively. ILB+
can be determined from Table 320 in the AISC Manual as 1,594 in4 for the W18x35 interior girder and
1,570 in4 for the W21x44 spandrel beam once composite beam design values are known. Note that the
W21x44 spandrel 100 percent composite design is limited by the effective slab capacity, which is why its
composite moment of inertia is so close to that of the W18x35 interior girder. ILB can be assumed as the
bare steel moment of inertia, as 510 in4 for the W18x35 interior girder and 843 in4 for the W21x44
spandrel beam. It is permitted to account for the transformed area of the reinforcing steel in calculating
ILB, but the bare steel beam property has been used in this example. The equivalent moment of inertia is
then calculated as:
W18x35 Interior Girder: Ieq = 0.6(1,594) + 0.4(510) = 1,160 in4
W21x44 Spandrel Beam: Ieq = 0.6(1,570) + 0.4(843) = 1,279 in4
The bare steel moment of inertia values in the building analysis are revised to these values, which are
suitable for servicelevel limit state checks. Use of a 0.8 reduction factor on the beam moment of inertia
is required by AISC 360 Section 7.3(3) for strengthlevel checks from direct analysis.
The bare steel moment of inertia for the columns is appropriate for servicelevel checks. For strength
level checks, the same 0.8 reduction factor on the moment of inertia used on beams would apply to the
columns. A further reduction on the column moment of inertia for strengthlevel checks is required if
Pr/Py exceeds 0.5. A quick scan of the column loads from the building analysis results indicates that the
only columns that exceed this value are the firststory columns at Grids C4, C5, D4 and D5 for Load
Combination 2 only. The adjustment factor is calculated to be:
b = 4[Pr/Py(1Pr/Py)] = 4[612 kips/1130 kips (1 612 kips/1130 kips)] = 0.99
In the author s judgment, the above reduction on so few columns will have little or no effect on the
building analysis results and it is ignored for this example.
9.4.4 Connection Behavior Modeling
For each connection type (such as W18 PRCC or W21 PRCC), there are four different connection
behavior models used, as developed in Section 9.2.2. First, the connection is modeled as a linear spring
with nominal stiffness Kci. This is done for the dynamic analysis of the building needed to determine the
building period. Second, a servicelevel analysis is conducted using the full nonlinear nominal service
momentrotation behavior. Third, a connection Stage 1 building analysis is done with the connections
having no moment resistance (analytical pins) so the beam precomposite loads can be applied. Finally, a
Stage 2 building strength analysis under factored loads is performed with the full modified nonlinear
momentrotation behavior.
The multilinear elastic link option provided in SAP2000 is used to model the connection springs for all
stages. This nonlinear spring model allows userdefined behavior for two types of analysis, linear and
nonlinear, for each spring type. This is helpful to handle the various connection behaviors because the
dynamic analysis and the Stage 1 precomposite beam load analysis can both be linear analysis which
automatically switches the connection spring to the defined linear behavior. Another important point is
that this particular spring model stays on the defined connection curve in a nonlinearelastic manner.
That is, the analysis simply rides up and down always converging at momentrotation points on the
connection backbone curve. This allows what is known as a path independent analysis; the order of the
loading does not matter. This is in contrast with a spring model with different connection unloading
behavior, such as might be used to model the full hysteric connection behavior. If the connection
unloading behavior is considered, the analysis is no longer path independent because the answer will
depend on the sequence of loads that are applied. This pathdependent analysis is more accurate and
allows consideration of connection shakedown to be captured in the model; however, it is also much more
complicated when compared to the pathindependent analysis. Since the simpler, pathindependent
connection modeling approach does not capture connection shakedown behavior, the author does not
recommend reducing beam sizes from the pure simple pinned gravity design discussed in Section 9.4.1.
9.4.5 Building Drift and Pdelta Checks
Drifts should be checked using the service momentrotation curves along with the full moment of inertias
for the beams and columns (no 0.8 reduction). Because of the nonlinear connection behavior, the analysis
is nonlinear. Though optional, the author recommends including Pdelta effects in the service drift checks
for partially restrained building designs. Drifts are computed for the nonlinear load combinations
developed in Section 9.3.5.
9.4.5.1 Torsional Irregularity Check. Standard Table 12.31 defines torsional irregularities. The story
drift values at the each end of the example building and their average story drift values including Pdelta
are presented in Table 9.41. Since the ratio of maximum drift to average drift does not exceed 1.2, no
torsional irregularity exists, accidental torsion need not be amplified and drift may be checked at the
center of the building (rather than at the corners).
Table 9.41 Torsional Irregularity and Seismic Drift Checks
Northsouth Direction (in.)
Eastwest Direction (in.)
Story
Displacement
Story Drift
Displacement
Story Drift
A1
F1
A1
F1
avg
max/avg
F1
F8
F1
F8
avg
max/avg
1
0.40
0.45
0.40
0.45
0.43
1.06
0.31
0.37
0.31
0.37
0.34
1.08
2
0.91
1.03
0.51
0.58
0.55
1.06
0.72
0.84
0.41
0.47
0.44
1.07
3
1.32
1.49
0.41
0.46
0.43
1.06
1.05
1.22
0.33
0.38
0.35
1.08
4
1.55
1.76
0.23
0.27
0.25
1.06
1.23
1.44
0.19
0.22
0.20
1.08
9.4.5.2 Seismic Drift and Pdelta Effect. The allowable seismic story drift is taken from Standard Table
12.121 as 0.025hsx = (0.025)(13 ft 12 in./ft) = 3.9 in. With Cd of 5.5 and I of 1.0, this corresponds to a
story drift limit of 0.71 inch under the equivalent elastic forces (see Standard Section 12.8.6 for story drift
determination). Review of the average drift values in Table 9.41 shows that all drifts are within the
0.71inch limit.
Table 9.42 Pdelta Effect Checks
Northsouth Direction (in.)
Eastwest Direction (in.)
Story
Displacement
Story Drift
Displacement
Story Drift
w/o
w/
w/o
w/
w/o
w/
w/o
w/
P
P
P
P
P amp
P
P
P
P
P amp
1
0.38
0.43
0.38
0.43
1.14
0.12
0.30
0.34
0.30
0.34
1.12
0.10
2
0.86
0.97
0.48
0.55
1.14
0.12
0.70
0.78
0.40
0.44
1.12
0.10
3
1.25
1.41
0.39
0.43
1.10
0.09
1.02
1.13
0.32
0.35
1.09
0.08
4
1.48
1.66
0.24
0.25
1.06
0.06
1.22
1.33
0.19
0.20
1.05
0.04
Separate analyses are conducted to determine seismic drifts with and without Pdelta effects. Due to the
nonlinear connection behavior, all of the analyses are nonlinear. The ratio of these two drifts (P amp) is
compared to the 1.5 limit for ratio of secondorder drift to firstorder drift set forth in AISC 360
Section 7.3(2). Because the ratios are all below the 1.5 limit, it is permissible to apply the notional loads
as a minimum lateral load for the gravityonly combination and not in combination with other lateral
loads. The results of these analyses are given in Table 9.42.
Provisions Section 12.8.7 now defines the stability coefficient ( ) as follows:
The story drift ( ) is defined in Standard Section 12.8.6 as:
Replacing in the stability coefficient equation results in:
This value of can also be calculated from the Pdelta amplifier presented in Table 9.42 by the
following:
The stability coefficients presented in Table 9.42 were calculated in this manner. Review of the values
shows that varies from 0.04 to 0.12. Provisions Section 12.8.7 now requires that not exceed 0.10
unless the building satisfies certain criteria when subjected to either nonlinear static (pushover) analysis
or nonlinear response history analysis. Because for the building in the northsouth direction exceeds
0.10 in the lower stories, the designer would have to either increase the building stiffness in that direction
or conduct an approved nonlinear analysis. Such nonlinear analysis is beyond the scope of this example.
9.4.5.3 Wind Drift. A wind drift limit of hsx/400 was chosen based on typical office practice for this type
of building. This gives a story drift limit of 13 12 / 400 = 0.39 inch. The wind drift values presented in
Table 9.43 were determined for the 50year return interval wind loads previously determined in
Section 9.3.3 above. Review of the drift values indicates that all drifts are within the 0.39inch limit.
Table 9.43 Wind Drift Results
Northsouth Direction (in.)
Eastwest Direction (in.)
Story
Displacement
Story Drift
Displacement
Story Drift
1
0.19
0.19
0.15
0.15
2
0.39
0.20
0.32
0.17
3
0.52
0.13
0.42
0.11
4
0.57
0.05
0.47
0.04
9.4.6 Beam Design
AISC 341 Part II Section 8.3 requires that composite beams be designed in accordance with AISC 360
Chapter I. The beams are designed for 100 percent composite action and sufficient shear studs to develop
100 percent composite action are provided between the end and midspan. They do not develop
100 percent composite action between the column and the inflection point, but it may be easily
demonstrated that they are more than capable of developing the full force in the reinforcing steel within
that distance. Composite beam design is not unique to this example; however, composite beams acting as
part of the lateral loadresisting system is unique and deserves further attention.
As a result of connection restraint, negative moments will develop at beam ends. These moments must be
considered when checking beam strength. The inflection point cannot be counted on as a brace point, so
it may be necessary to consider the full beam length as unbraced for checking lateraltorsional buckling
and comparing that capacity to the negative end moments. Note that there are Cb equations in the
literature that do a better job (as compared to the standard Cb equation in AISC 360) of predicting the
lateraltorsional buckling strength of beams that are continuously attached to a composite slab floor
system (Yura, 2006)
AISC 341 Part II Section 8 does not specifically address compactness criteria for beams; however, given
that the beams are not being required to develop Mp, other than possibly under gravity loads, it is unlikely
they would need to be seismically compact. The author recommends that they meet the compactness
criteria of AISC 360. A quick check in Table 11 of the AISC Manual indicates that both W18x35 and
W21x44 are compact for flexure.
9.4.7 Column Design
Requirements for column design are found in AISC 341 and AISC 360. AISC 341 Part II Section 8.2
requires that columns meet the requirements of AISC 341 Part I Sections 6 and 8. W10 columns of A992
steel meet all Section 6 material requirements.
AISC 341 Part I Section 8.3 requires a special load combination if Pu/ cPn exceeds 0.4 for a column in a
seismic load combination. The only columns that exceed this limit are the interior columns on Grid
Lines C and D under the storage load. Because they are so close to the center of the building, the seismic
axial force in these columns is very small. Consequently, including the overstrength factor of 3.0 on the
seismic axial portion of the column load will not have a meaningful effect on the column loads and can be
ignored in this example.
The nominal strength of the columns is determined using K = 1.0 in accordance with AISC 360
Section 7.1. The associated column strength unity checks are presented in Table 9.44. The unity checks
presented are for the first story of the center four columns in the building.
Table 9.44 Column Strength Check for W10x77
Seismic Load Combination
Gravity Load Combination
Axial force, Pu
370 kip
612 kip
Moment, Mu
55 ftkip
35 ftkip
Interaction
0.606
0.866
Part II Section 8 does not specifically address the required compactness criteria; however, given the high
R value for the lateral loadresisting system, the author has assumed that the columns would need to meet
the seismically compact criteria given in Part I Table I91. A W10x77 column from the lower level of an
interior bay with storage load is illustrated (the axial load from the seismic load combination is used):
Column Flange:
= 7.22
= 5.86 (AISC Manual) < 7.22
Column Web:
= 0.364 > 0.125
= 14.8 (AISC Manual) < 53.03
As an alternative to calculating the compactness criteria by hand, the designer can use the AISC SDM
Table 12. A quick review of this table indicates that the W10x77 is compact for flexure (beam) and for
axial loads (column). The dash in the table indicates that applied axial loads as large as Py still result in
the column meeting the seismically compact criteria.
The equivalent of the weakbeam strongcolumn concept for the CPRMF lateral system is a weak
connectionstrong column. This is not specifically addressed in AISC 341; however, ASCE TC
recommends the following check:
For the same lower level interior W10x77 one gets:
9.4.8 Connection Design
There is really little to do with the connection design at this stage because the full nonlinear connection
behavior is being used in the analysis. This means that the connection moments will never exceed the
connection capacity during the analysis. This is in contrast to any analysis method that models the
connections with linear behavior. When the connections are modeled with linear behavior, it is up to the
designer to confirm that the final connection results are consistent with the expected connection behavior.
This might be very easy for building designs where connection moments are small; however, when the
connections are being pushed close to their capacity, that sort of independent connection check by the
designer can be problematic.
Although not entirely necessary, it is useful to check where the connections are along the expected
behavior curves for any given analysis so one can see just how hard the connections are being pushed.
The connection moment demand versus design capacities (including ) are presented in Table 9.45. The
demand values are from different load combinations. A quick check of this table indicates that this
building design is not being pushed particularly hard and that there is likely significant reserve capacity in
the lateral system.
Table 9.45 Connection Moment Demand vs. Capacity (kipft)
W21 PRCC
W18 PRCC
() M
(+) M
() M
(+) M
Demand
136
87.0
126
37.0
Capacity
312
204
197
128
Ratio
0.44
0.43
0.64
0.29
9.4.9 Column Splices
Column splice design would be in accordance with AISC 341 Part I Section 8.4 but is not illustrated in
this example.
9.4.10 Column Base Design
Column base design would be in accordance with AISC 341 Part I Section 8.5 but is not illustrated in this
example.
10
Masonry
James Robert Harris, PE, PhD and
Frederick R. Rutz, PE, PhD
Contents
10.1 WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF, LOS ANGELES,
CALIFORNIA 3
10.1.1 Building Description 3
10.1.2 Design Requirements 4
10.1.3 Load Combinations 6
10.1.4 Seismic Forces 8
10.1.5 Side Walls 9
10.1.6 End Walls 25
10.1.7 InPlane Deflection End Walls 44
10.1.8 Bond Beam Side Walls (and End Walls) 45
10.2 FIVESTORY MASONRY RESIDENTIAL BUILDINGS IN BIRMINGHAM, ALABAMA;
ALBUQUERQUE, NEW MEXICO; AND SAN RAFAEL, CALIFORNIA 45
10.2.1 Building Description 45
10.2.2 Design Requirements 48
10.2.3 Load Combinations 50
10.2.4 Seismic Design for Birmingham 1 51
10.2.5 Seismic Design for Albuquerque 69
10.2.6 Birmingham 2 Seismic Design 81
10.2.7 Seismic Design for San Rafael 89
10.2.8 Summary of Wall D Design for All Four Locations 101
This chapter illustrates application of the 2009 NEHRP Recommended Provisions (the Provisions) to the
design of a variety of reinforced masonry structures in regions with different levels of seismicity.
Example 10.1 features a singlestory masonry warehouse building with tall, slender walls, and
Example 10.2 presents a fivestory masonry hotel building with a bearing wall system designed in areas
with different seismicities. Selected portions of each building are designed to demonstrate specific
aspects of the design provisions.
Masonry is a discontinuous and heterogeneous material. The design philosophy of reinforced grouted
masonry approaches that of reinforced concrete; however, there are significant differences between
masonry and concrete in terms of restrictions on the placement of reinforcement and the effects of the
joints. These physical differences create significant differences in the design criteria.
All structures were analyzed using twodimensional (2D) static methods using the RISA 2D program,
V.5.5 (Risa Technologies, Foothill Ranch, California). Example 10.2 also uses the SAP 2000 program,
V6.11 (Computers and Structures, Berkeley, California) for dynamic analyses to determine the structural
periods.
All examples are for buildings of concrete masonry units (CMU); neither prestressed masonry shear walls
nor autoclaved aerated concrete masonry shear walls are included.
In addition to the Provisions and the Standard, the following documents are referenced in this chapter:
ACI 318 American Concrete Institute. 2008. Building Code Requirements for Structural
Concrete.
TMS 402 The Masonry Society. 2008. Building Code Requirements for Masonry Structures,
TMS 402/ACI 530/ASCE 5.
IBC International Code Council. 2009. International Building Code.
NCMA National Concrete Masonry Association. A Manual of Facts on Concrete Masonry,
NCMATEK is an information series from the National Concrete Masonry Association,
various dates. NCMATEK 141BA, Section Properties of Concrete Masonry Walls and
NCMATEK 1411B, Strength Design of Concrete Masonry Walls for Axial Load &
Flexure, are referenced here.
USGS United States Geological Survey. Seismic Design Maps web application.
The short form designations for each citation are used throughout. The citation to the IBC is because one
of the designs employees a tall, slender wall that is partially governed by wind loads and the IBC
provisions are used for that design.
Regarding TMS 402:
The 2005 edition of the Standard, in its Supplement 1, refers to the 2005 edition of TMS 402.
The 2010 edition of the Standard refers to the 2008 edition of TMS 402.
The examples herein are prepared according to the 2008 edition of TMS 402.
10.1 WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF, LOS ANGELES,
CALIFORNIA
This example features a onestory building with reinforced masonry bearing walls and shear walls.
10.1.1 Building Description
This simple rectangular warehouse is 100 feet by 200 feet in plan (Figure 10.11). The masonry walls are
30 feet high on all sides, with the upper 2 feet being a parapet. The wood roof structure slopes slightly
higher towards the center of the building for drainage. The walls are 8 inches thick on the long side of the
building, for which the slender wall design method is adopted, and 12 inches thick on both ends. The
masonry is grouted in the cells containing reinforcement, but it is not grouted solid. The specified
strength of masonry is 2,000 psi. Normalweight CMU with Type S mortar are assumed.
Figure 10.11 Roof plan
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
The long side walls are solid (no openings). The end walls are penetrated by several large doors, which
results in more highly stressed piers between the doors (Figure 9.12); thus, the greater thickness for the
end walls.
Figure 10.12 End wall elevation
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
The floor is concrete slabongrade construction. Conventional spread footings are used to support the
interior steel columns. The soil at the site is a dense, gravelly sand.
The roof structure is wood and acts as a diaphragm to carry lateral loads in its plane from and to the
exterior walls. The roofing is ballasted, yielding a total roof dead load of 20 psf. There are no interior
walls for seismic resistance. This design results in a highly stressed diaphragm with large calculated
deflections. The design of the wood roof diaphragm and the masonry walltodiaphragm connections is
illustrated in Sec. 11.2.
In this example, the following aspects of the structural design are considered:
Design of reinforced masonry walls for seismic loads
Computation of Pdelta effects.
10.1.2 Design Requirements
This building could qualify for the simplified approach in Standard Section 12.14, although the long
method per Standard Section 11.411.6 has been followed.
10.1.2.1 Seismic parameters. The ground motion response coefficients are found from USGS based
upon latitude and longitude. The site class is taken from a sitespecific geotechnical report and is typical
for dense sands and gravels. The warehouse is not designated for hazardous materials and does not house
any essential facility, thus the occupancy category is all other .
Site Class = C
SS = 2.14
S1 = 0.74
Occupancy Category (Standard Sec. 1.5.1) = II
The remaining basic parameters depend on the ground motion adjusted for site conditions.
10.1.2.2 Response parameter determination. The mapped spectral response factors must be adjusted
for site class in accordance with Standard Section 11.4.3. The adjusted spectral response acceleration
parameters are computed according to Standard Equations 11.41 and 11.42 for the short period and one
second period, respectively, as follows:
SMS = FaSS = 1.0(2.14) = 2.14
SM1 = FvS1 = 1.3(0.74) = 0.96
Where Fa and Fv are site coefficients defined in Standard Tables 11.41 and 11.42, respectively. The
design spectral response acceleration parameters (Standard Sec. 11.4.4) are determined in accordance
with Standard Equations 11.43 and 11.44 for the shortperiod and onesecond period, respectively:
The Seismic Design Category may be determined by the design spectral acceleration parameters
combined with the Occupancy Category. For buildings assigned to Seismic Design Category D, masonry
shear walls must satisfy the requirements for special reinforced masonry shear walls in accordance with
Standard Section 12.2. A summary of the seismic design parameters follows:
Seismic Design Category (Standard Sec. 11.6): D
Seismic ForceResisting System (Standard Table 12.21) : Special Reinforced Masonry Shear
Wall
Response Modification Factor, R: 5
Deflection Amplification Factor, Cd: 3.5
System Overstrength Factor, ê0: 2.5
Redundancy Factor, (Standard Sec. 12.3.4.2): 1.0
(Determination of is discussed in Section 10.1.3 below.)
10.1.2.3 Structural design considerations. With respect to the lateral load path, the roof diaphragm
supports approximately the upper 16 feet of the masonry walls (half the clear span plus the parapet) in the
outofplane direction, transferring the lateral force to inplane masonry shear walls. This is more
precisely calculated in Section 10.1.4.1.
Soil structure interaction is not considered.
The building is of bearing wall construction.
Other than the opening in the roof, the building is symmetric about both principal axes, and the vertical
elements of the seismic forceresisting system are arrayed entirely at the perimeter. The opening is not
large enough to be considered an irregularity (per Standard Table 12.31); thus, the building is regular,
both horizontally and vertically. Standard Table 12.61 permits several analytical procedures to be used;
the equivalent lateral force (ELF) procedure (Standard Sec. 12.8) is selected for use in this example. The
direction of loading requirements of Standard Section 12.5 are for walls that act in both principal
directions, which is not the case for this structure, as will be discussed in more detail.
There is no inherent torsion because the building is symmetric. The effects of accidental torsion and its
potential amplification, need not be included because the roof diaphragm is flexible (Standard
Sec. 12.8.4.2).
The masonry bearing walls also must be designed for forces perpendicular to their plane (Standard
Sec. 12.11.1).
For inplane loading, the walls are treated as cantilevered shear walls. For outofplane loading, the walls
are treated as simply supported at top and bottom. The assumption of a pinned connection at the base is
deemed appropriate because the foundation is shallow and narrow, which permits rotation near the base of
the wall.
10.1.3 Load Combinations
The basic load combinations are the same as specified in Standard Section 2.3.2. The seismic load effect,
E, is defined by Standard Equations 12.41, 12.43 and 12.44, as follows:
E = Eh + Ev = QE ñ 0.2SDSD = (1.0)QE ñ 0.2(1.43)D = QE ñ 0.286D
This assumes = 1.0 as will be confirmed in the following section.
10.1.3.1 Redundancy Factor. In accordance with Standard Section 12.3.4.2, the redundancy factor, ,
applies to the inplane load direction.
In order to achieve the two conditions in Standard Section 12.3.4.2 must be met. In the long
direction there are no walls with heightto length ratios exceeding 1.0; thus = 1.0 in the long direction.
In the short direction the pier heights do exceed the length; thus their conditions must be checked. For
our case, both are met.
Although the calculation is not shown here, note that a single 8footlong pier carries approximately
23 percent (determined by considering the relative rigidities of the piers) of the inplane load for each end
wall. Thus, failure of a single pier results in less than 33 percent reduction in base shear resistance.
Loss of a single pier will not result in extreme torsional irregularity because the diaphragm is flexible.
Even if the diaphragm were rigid, an extreme torsional irregularity would not be created. The lateral
deflection of end wall with all piers in place is approximately 0.018 inch (determined by RISA analysis).
Lateral deflection of end wall with one pier removed is 0.024 inch. The larger deflection divided by the
average of both deflections is less than 1.4:
Therefore, even if the diaphragm were rigid, there is no extreme torsional irregularity as per Standard
Table 12.31.
10.1.3.2 Combination of load effects. Load combinations for the inplane loading direction from
Standard Section 2.3.2 are:
1.2D + 1.0E + 0.5L + 0.2S
and
0.9D + 1.0E + 1.6H
L, S and H do not apply for this example (roof live load, Lr, is not floor live load, L) so the load
combinations become:
1.2D + 1.0E
and
0.9D + 1.0E
For this case, E = Eh ñ Ev = QE ñ 0.2 SDSD = (1.0)QE ñ (0.2)(1.43)D = QE ñ 0.286D
Where the effect of the earthquake determined above, 1.2D + 1.0(QE ñ 0.2D), is inserted in each of the
load combinations, the controlling cases are 1.486D + QE when gravity and seismic are additive and
0.614D  QE when gravity and seismic counteract.
These load combinations are for the inplane direction of loading. Load combinations for the outofplane
direction of loading are similar except that the redundancy factor, is not applicable. Thus, for this
example (where = 1.0), the load combinations for both the inplane and the outofplane directions are
1.486D + QE and 0.614D  QE.
The combination of earthquake motion (and corresponding loading) in two orthogonal directions as per
Standard Section 12.5.3.a need not be considered. Standard Section 12.5.4 for Seismic Design
Category D refers to Section 12.5.3 for Category C, which requires consideration of direction to produce
maximum effect where horizontal irregularity Type 5 exists ( nonparallel systems ); this building does
not have that irregularity. Standard Section 12.5.4 also requires consideration of direction for maximum
effect for elements that are part of intersecting systems if those elements receive an axial load from
seismic action that exceeds 20 percent of their axial strength; axial loads are less than that for this
building.
If a masonry control joint is provided at the corner, there are no elements acting in two directions. The
short pier at the corner can be designed as an L shaped element, which means that it does participate in
both directions. The vertical seismic force in that pier, generated by frame action, is small and easily less
than 20 percent of its capacity. Therefore, no element of the seismic forceresisting system is required to
be checked for the direction of load that produces the maximum effect. Although it is not required, the
typical pier in the end wall will be checked using the method of Standard Section 12.5.3.a. to illustrate
the Standard s method for design to account for orthogonal effects.
10.1.4 Seismic Forces
Seismic base shear, diaphragm force and wall forces are discussed below.
10.1.4.1 Base Shear. Base shear is computed using the parameters determined previously. The
Standard does not recognize the effect of long, flexible diaphragms on the fundamental period of
vibration. The approximate period equations, which limit the computed period, are based only on the
height. Since the structure is relatively short and stiff, shortperiod response will govern the design
equations. According to Standard Section 12.8 (for shortperiod structures):
The seismic weight for forces in the long direction is as follows:
Roof = (20 psf)(100 ft)(200 ft) = 400 kips
End walls = (103 psf) (2 walls)[(30 ft)(100 ft) (5)(12 ft)(12 ft)](17.8 ft/28 ft)* = 299 kips
Side walls = (65 psf) (30ft)(200ft)(2 walls) = 780 kips
Total = 1,479 kips
*Only the portion of the end walls that is distributed to the roof contributes to seismic weight in the long
direction.
(The initial estimates of 65 psf for 8inch CMU and 103 psf for 12inch CMU are slightly higher than
normalweight CMU with grouted cells at 24 inches on center. However, grouted bond beams at 4 feet on
center will be included, as will certain additional grouted cells.)
Note that the centroid of the end walls is determined to be 17.8 feet above the base, so the portion of the
weight distributed to the roof is approximately the total weight multiplied by 17.8 feet/28 feet (weights
and section properties of the walls are described subsequently).
Therefore, the base shear to each of the long walls is as follows:
Vu = (0.286)(1,479 kips)/2 = 211 kips
The seismic weight for forces in the short direction is:
Roof = (20 psf)(100 ft)(200 ft) = 400 kips
Side walls = (65 psf)(2 walls)(30ft)(200ft)(15ft/28ft)* = 418 kips
End walls = (103 psf)(2 walls)[(30ft)(100ft)5(12ft)(12ft)] = 470 kips
Total = 1,288 kips
*Only the portion of the side walls that is distributed to the roof contributes to seismic weight in the short
direction.
The base shear to each of the short walls is as follows:
Vu = (0.286)(1,288 kips)/2 = 184 kips
10.1.4.2 Diaphragm force. See Section 11.2 for diaphragm forces and design.
10.1.4.3 Wall forces. Because the diaphragm is flexible with respect to the walls, shear is distributed to
the walls on the basis of beam theory, ignoring walls perpendicular to the motion (this is the "tributary"
basis).
The building is symmetric. Given the previously explained assumption that accidental torsion need not
be applied, the force to each wall becomes half the force on the diaphragm.
All exterior walls are bearing walls and, according to Standard Section 12.11.1, must be designed for a
normal (outofplane) force of 0.4SDSIWw where Ww = weight of wall. The outofplane design is shown
in Section 10.1.5.2.
10.1.5 Side Walls
The total base shear is the design force. Standard Section 14.4, which cites TMS 402, is the reference for
design strengths. The compressive strength of the masonry, fm', is 2,000 psi. TMS 402 Section 1.8.2.2
gives Em = 900fm' = (900)(2 ksi) = 1,800 ksi.
For 8inchthick CMU with vertical cells grouted at 24 inches on center and horizontal bond beams at
48 inches on center, the weight is conservatively taken as 65 psf (recall the CMU are normal weight) and
the net bedded area is 51.3 in.2/ft based on tabulations in NCMATEK 141B.
10.1.5.1 Horizontal reinforcement side walls. As determined in Section 10.1.4.1, the design base
shear tributary to each longitudinal wall is 211 kips. Based on TMS 402 Section 1.17.3.6.2.1, the design
shear strength must exceed either the shear corresponding to the development of 1.25 times the nominal
flexure strength of the wall (which is very unlikely in this example due to the length of wall) or 2.5 times
Vu , which in this case is 2.5(211) = 528 kips.
From TMS 402 Section 3.3.4.1.2.1, the masonry component of the shear strength capacity for reinforced
masonry is as follows:
For a singlestory cantilever wall, Mu/Vudv = h/d, which is (28/200) = 0.14 for this case. For the long
walls and conservatively treating P as 0.614 times the weight of the wall only, without considering the
roof weight contribution:
and
Vm = 0.8(1,783) = 1,426 kips > 528 kips OK
where = 0.8 is the strength reduction factor for shear from TMS 402 Section 3.1.4.3.
Horizontal reinforcement therefore is not required for shear strength but is required if the wall is to
qualify as a Special Reinforced Masonry Wall (TMS 402 Sec. 1.17.3.2.6.b). Standard Table 12.21 does
not permit lower quality masonry walls in Seismic Design Category D.
According to TMS 402 Section 1.17.3.2.6.c, minimum horizontal reinforcement is 0.0007Ag =
(0.0007)(7.625 in.)(8 in.) = 0.043 in.2 per course, but the authors believe it prudent to use more horizontal
reinforcement for shrinkage in this very long wall and then use minimum reinforcement in the vertical
direction [this concept applies even though this wall requires far more than the minimum reinforcement
(also 0.0007Ag) in the vertical direction due to its large heighttothickness ratio]. Two #5 bars at
48 inches on center provide 0.103 in.2 per course. This amounts to 0.4 percent of the area of masonry
plus the grout in the bond beams. The actual shrinkage properties of the masonry and the grout as well as
local experience should be considered in deciding how much reinforcement to provide. For long walls
that have no control joints, as in this example, providing more than minimum horizontal reinforcement is
appropriate.
10.1.5.2 Outofplane flexure side walls. The design demand for seismic outofplane flexure is
0.4SDSIww (Standard Sec. 12.11.1). For a wall weight of 65 psf for the 8inchthick CMU side walls, this
demand is 0.4(1.43)(1)(65 psf) = 37 psf.
Calculations for outofplane flexure become somewhat involved and include the following:
1. Select a trial design.
2. Investigate to ensure ductility (i.e., check maximum reinforcement limit).
3. Make sure the trial design is suitable for wind (or other nonseismic) lateral loadings using the wind
provisions of the Standard (which satisfies the IBC). (This is not illustrated in this example).
4. Calculate midheight deflection due to wind by TMS 402. (Not illustrated in this example). (Note
that while the Standard has story drift requirements, it does not impose a midheight deflection limit
for walls).
5. Calculate seismic demand. This computation requires consideration of Pdelta effects because of the
wall slenderness. (Seismic demand is greater than wind for this wall.)
6. Determine seismic resistance and compare to the demand determined in Step 5.
Proceed with these steps as follows:
10.1.5.2.1 Trial design. A trial design of #7 bars at 24 inches on center is selected. See Figure 10.13.
Figure 10.13 Trial design for 8inchthick CMU wall
(1.0 in. = 25.4 mm)
10.1.5.2.2 Investigate to ensure ductility. The critical strain condition corresponds to a strain in the
extreme tension reinforcement (which is a single #7 centered in the wall in this example) equal to times
the strain at yield stress. is the tension reinforcement strain factor (equal to 1.5 for outofplane flexure
due to wind; see TMS 402 Commentary 3.3.3.5).
Based on TMS 402 Section 3.3.3.5.1.a and Commentary 3.3.3.5 (where is defined) for this case:
t = 7.63 in.
d = t/2 = 3.81 in.
îm = 0.0025
îs = 1.5îy = 1.5(fy/Es) = 1.5(60 ksi / 29,000 ksi) = 0.0031
a = 0.8c = 1.36 in.
The Whitney compression stress block, a = 1.36 inches for this strain distribution, is greater than the
1.25inch face shell width. Thus, the compression stress block is broken into two components: one for
full compression against solid masonry (the face shell), and another for compression against the webs and
grouted cells but accounting for the open cells. These are shown as C1 and C2 in Figure 10.14:
C1 = 0.80fm' (1.25 in.)b = (0.80)(2 ksi)(1.25)(24) = 48 kips (for a 24inch length)
C2 = 0.80 fm' (a1.25 in.)(8 in.) = (0.80)(2 ksi)(1.361.25)(8) = 1.41 kips (for a 24inch length)
The 8inch dimension in the C2 calculation is for the combined width of the grouted cell and the adjacent
mortared webs over a 24inch length of wall. The actual width of one cell plus the two adjacent webs will
vary with various block manufacturers and may be larger or smaller than 8 inches. The 8inch value has
the benefit of simplicity (and is correct for solidly grouted walls).
Figure 10.14 Investigation of outofplane ductility for the 8inchthick CMU side walls
(1.0 in. = 25.4 mm)
T is based on FyAs (TMS 402 Sec. 3.3.3.5.1.c):
T = FyAs = (1.0)(60 ksi)(0.60 in.2) = 36 kips (for a 24inch length)
P is based on the load combination of D + 0.75L +0.525QE (TMS 402 Sec. 3.3.3.5.1.d).
QE is the effect of horizontal seismic motions, and P is a vertical force. QE produces overturning forces,
but because this is such a long wall, the vertical force due to horizontal seismic motion is not significant,
so the net total vertical force is taken as zero here. Therefore QE is zero in determining P for this wall.
Conservatively neglecting the roof weight:
Check C1 + C2 > T + P (all for 24inch length):
T + P = 36 + 2.08 = 38.1 kips
C1 + C2 = 49.4 kips > 38.1 kips OK
The compression capacity is greater than the tension capacity; therefore, the ductile failure mode criterion
is satisfied.
10.1.5.2.3 Check for wind load. The wind design check is beyond the scope of this seismic example, so
it is not presented here. Both strength and deflection need to be ascertained in accordance with a building
code; most are based on the Standard, which we are using. For our example, a check on strength to resist
wind was found to conform to the Standard and is not shown here.
10.1.5.2.4 Calculate midheight deflection due to wind by Standard. Deflection due to wind was
found to conform to the Standard and is not shown here.
Figure 10.15 Basis for outofplane deflection calculation
10.1.5.2.5 Calculate seismic demand. For this case, the two load factors for dead load apply: 0.614D
and 1.486D. Conventional wisdom holds that the lower dead load will result in lower momentresisting
capacity of the wall, so the 0.614D load factor would be expected to govern. However, the lower dead
load also results in lower Pdelta, so both cases should be checked. (As it turns out, the higher factor of
1.486D controls).
wu = 37 psf (from Sec. 10.1.5.3)
Check moment capacity for 0.614D: TMS 402 Section 3.3.5.3 requires consideration of the secondary
moment from the axial force acting through the deflection. TMS 402 Section 3.3.5.4 gives an equation
that is essentially bilinear (two straight lines joined at the point of cracking). NCMA TEK 141B
illustrates that determining the final moment by this method requires iteration.
Roof load, Pf = (20 psf)(10 ft) = 200 plf
Wall load (at midheight), Pw = (65 psf)(16 ft) = 1,040 plf
P = Pf + Pw = 1,240 plf
Puf = (0.614)(200 plf) = 123 plf
Puw = (0.614)(1,040 plf) = 638 plf
Pu = Puf + Puw = 761 plf
Eccentricity, e = 7.32 in. (distance from wall centerline to roof reaction centerline)
Modulus of elasticity, Em = 1,800,000 psi
fm' = 2000 psi
Modular ratio,
The modulus of rupture, fr, is found from TMS 402 Table 3.1.8.2. The values given in the table are for
either hollow CMU or fully grouted CMU. Values for partially grouted CMU are not given; Footnote a
indicates that interpolation between these values must be performed. As illustrated in Figure 9.16, and
shown below, the interpolated value for this example is based on relative areas of hollow and grouted
cells:
Another method for making the interpolation, while approximate, is simpler. It is based on 2/3 of the
cells being hollow and 1/3 of the cells being grouted for the case of grouted cells at 24 inches on center:
For this example, the value of fr = 98 psi will be used.
From mechanics:
Ig = 443 in.4/ft
Sg = 116 in.3/ft
From NCMA TEK 141B:
In= 355 in.4/ft
Sn = 93.2 in.3/ft
An = 51.3 in.2/ft
Mcr = Sn(fr + P/An) = 93.2(98 + 1240/51.3) = 11,386 inlb/ft. Use Mcr = 11,400 in.lb/ft. Note: this
equation for Mcr is not in TMS 402; however, it is valid based on mechanics.
Figure 10.16 Basis for interpolation of modulus of rupture, fr
(1.0 in. = 25.4 mm, 1.0 psi = 6.89 kPa).
Refer to Figure 10.17 for determining Icr. The neutral axis shown on the figure is not the conventional
neutral axis by linear analysis; instead, it is the plastic centroid, which is simpler to locate, especially
where the neutral axis position results in a T beam crosssection. (For this wall, the neutral axis does not
produce a T section, but for the end wall in this building, a T section does result.) Cracked moments of
inertia computed by this procedure are less than those computed by linear analysis but generally not so
much less that the difference is significant. (This is the method used for computing the cracked section
moment of inertia for slender walls in the standard for concrete structures, ACI 318.) Axial load does
enter the computation of the plastic neutral axis and the effective area of reinforcement. Thus:
T = (0.60 in.2/ 2 ft)(60 ksi) = 18.0 klf
C = T + P = 19.24 klf
a = C/(0.8 f'mb) = (19.24 klf)/(0.8(2.0 ksi)(12 in./ft) = 1.002 in.
c = a/0.8 = 1.25 in.
Ase = As + P/fy = 0.30 in.2/ft + 1.240 klf/60 ksi = 0.32 in.2/ft
Note: TMS 402 uses the term As to mean the same thing as effective area of reinforcement (TMS 402
Sec. 1.5 and Commentary 3.3.5.4). Ase is used here to distinguish effective area from actual area, As.
Icr = nAse(dc)2 + bc3/3
= 16.1(0.32 in.2/ft)(3.81 in.  1.25 in.)2 + (12 in./ft)(1.25 in.)3/3
= 42.1 in.4/ft
Note that Icr could be recomputed for Pu = 0.614D and Pu = 1.486D, but that refinement is not pursued in
this example. In the opinion of the authors, the most correct method for computing the cracked section
properties is to use Pu. This will necessitate two sets of cracked section properties for this example. For
purposes of illustration, one set of cracked section properties, with P = 1.0D, is computed.
Figure 10.17 Cracked moment of inertia (Icr) for 8inchthick CMU side walls
(1.0 in. = 25.4 mm)
The computation of the secondary moment in an iterative fashion is shown below:
First iteration:
Mu1 = 43,512 + 450 + 0 = 43,962 in.lb/ft > Mcr = 11,400 in.lb/ft
Second iteration:
Third iteration
Mu3 = 43,512 + 450 + (761)(5.82) = 48,391 in.lb/ft
Convergence check:
Mu = 48,391 in.lb/ft (for the 0.614D load case)
Using the same procedure, find Mu for the 1.486D load case. The results are summarized below:
First iteration:
Pu = 1.486 (Pf + Pw) = 1.486(200 + 1,040) =297 + 1,545 = 1,843 plf
Mu1 = 44,811 in.lb/ft
u1 = 5.39 in.
Second iteration:
Mu2 = 54,739 in.lb/ft
u2 = 6.93 in.
Third iteration:
Mu3 = 57,579 in.lb/ft
3 = 7.37 in.
Fourth iteration:
Mu4 = 58,392 in.lb/ft
u4 = 7.50 in.
Check convergence:
Mu = 58,392 in.lb/ft (for the 1.486D load case)
The iterative method described above is consistent with NCMA TEK 1411B. The authors note that
ACI 318, the standard for concrete structures, includes provisions for the design of slender walls that are
somewhat different. For the computation of deflection at nominal strength, 75 percent of the cracked
stiffness is used. The 0.75 factor represents a margin for safety, because the required strength, Mu,
depends on the computed deflection. The absence of the bilinear relation is much closer to deflection
computations by other methods, such as given in TMS 402, Section 1.13.3.2. The absence of bilinear
relations allows direct computation of the final deflection and moment, rather than iteration. For
illustration, the method that predicts the secondary moment directly and upon which the ACI 318 slender
wall direct calculation is based, is shown here:
Where:
Therefore, for the 1.486D case:
which is approximately 6 percent larger than Mu = 58.4 in.k/ft by the iterative method above. In this
calculation, the 0.75 factor on Pe used in ACI 318 has not been included.
10.1.5.2.6 Determine flexural strength of wall. Refer to Figure 10.18. As in the case for the ductility
check, a strain diagram is drawn. Unlike the ductility check, the strain in the steel is not predetermined.
Instead, as in conventional strength design of reinforced concrete, a rectangular stress block is computed
first and then the flexural capacity is checked.
T = Asfy = (0.30 in.2/ft)60 ksi = 18.0 klf
The results for the two axial load cases are shown in Table 10.12 below.
Table 10.12 Flexural Strength of Side Wall
Load Case
0.614D + E
1.486D + E
Pu, klf
0.761
1.843
C = T + Pu, klf
18.76
19.84
a = C / (0.8f'mb), in.
0.978
1.03
Mn= C (d  a/2), in.kip/ft
62.3
65.35
Mn= 0.9Mn, in.kip/ft
56.1
58.8
Mu, in.kip/ft
48.4
58.4*
Acceptance
OK
OK
*The Mu from the alternative direct computation is approximately 5% higher than the
design strength.
Figure 10.18 Outofplane strength for 8inchthick CMU walls
(1.0 in. = 25.4 mm)
Note that either wind or earthquake may control the stiffness and strength outofplane; earthquake
controls for this example. A careful reading of Standard Section 12.5 should be made to see if the
orthogonal loading combination will be called for; as discussed earlier, the orthogonal combination is not
required for this example (although an orthogonal combination check will be made for illustration
purposes later).
10.1.5.3 Inplane flexure side walls. Inplane calculations for flexure in masonry walls include two
items per the Provisions:
Ductility check
Strength check
It is recognized that this wall is very strong and stiff in the inplane direction. Many engineers would not
even consider these checks as necessary in ordinary design. The ductility check is illustrated here to show
a method of implementing the requirement.
10.1.5.3.1 Ductility check. For this case, with Mu/Vudv < 1 and R > 5, TMS 402 Section 3.3.3.5.4 refers
to Section 3.3.3.5.1, which stipulates that the critical strain condition corresponds to a strain in the
extreme tension reinforcement equal to 1.5 times the strain associated with Fy. This calculation uses
unfactored gravity loads. (See Figure 10.19.)
a = 0.8c = 71.43 ft
Cm = 0.8f mabavg = (0.8) (2 ksi)(71.43 ft)(51.3 in.2/lf) = 5,862 kips
Where bavg is taken from the average area used earlier, 51.3 in.2/ft results; see Figure 10.19 for locations
of tension steel and compression steel (the rebar in the compression zone will act as compression steel).
From this it can be seen that:
Figure 10.19 Inplane ductility check for side walls
(1.0 in. = 25.4 mm, 1.0 ksi = 6.89 MPa)
Some authorities would not consider the compression resistance of reinforcing steel that is not confined
within ties. In one location (Section 3.1.8.3) TMS 402 clearly requires transverse reinforcement (ties) for
any steel used in compression, while in another place (Section 3.3.3.5.1.e) it explicitly permits inclusion
of compression reinforcement with or without lateral restraining reinforcement for checks on maximum
flexural tensile reinforcement (i.e., ductility checks). TMS 402 Commentary 3.3.3.5 explains that
confinement reinforcement is not required because the maximum masonry compressive strain will be less
than ultimate values. This inconsistency does not usually have a significant effect on computed results.
The authors have taken credit for unconfined compression reinforcement for strength and included it in
ductility checks (but there is no objection to the practice of neglecting unconfined compression
reinforcement used by some engineers).
In the authors opinion, there are two approaches to the determination of P, one following TMS 402 and
the other following the Standard:
P is at the base of the wall rather than at the midheight:
TMS 402 Section 3.3.3.5.1.d:
D + 0.75L + 0.525 QE
Since QE represents the effect of horizontal seismic forces, which equals zero for our case, and
roof live load is not combined with seismic loads, this reduces to D:
P = Pw + Pf = [(0.065 ksf) (30 ft) + (0.02 ksf )(10 ft)](200 ft) = 430 kips
Standard Section 12.4.2.3:
(1.2 + 0.2 SDS)D + QE + L + 0.2S
which reduces to:
[1.2 + (0.2)(1.43)]D + 0 + 0 + 0
Pu = (1.486)(Pf + Pw) = (1.486)(480 kips) = 713 kips
Continuing with the inplane ductility check:
äC > P + äT
Cm + Cs1 + Cs2 > P + Ts1 + Ts2
And conservatively using the higher of the two values for P,
5,862 + 278 + 665 > 713 + 664 + 664 6,805 > 2,041 OK
Therefore, there is enough compression capacity to ensure ductile failure. Note that either of the two
values for P brings us to the same conclusion for this case.
It should also be noted that even if the compression reinforcement were neglected, there would still be
enough compression capacity to ensure ductile failure.
In the opinion of the authors, flexural yield is feasible for walls with Mu/Vudv in excess of 1.0; this
criterion limits the compressive strain in the masonry, which leads to good performance in strong ground
shaking. For walls with Mu/Vudv substantially less than 1.0, the wall will fail in shear before a flexural
yield is possible. Therefore, the criterion does not affect performance. Well distributed and well
developed reinforcement to control the shear cracks is the most important ductility attribute for such
walls.
10.1.5.4.2 Strength check. The wall is so long with respect to its height that inplane strength for
flexure is acceptable by inspection.
10.1.5.5 Shear side walls.
10.1.5.5.1 Outofplane shear in side walls. Compute outofplane shear at the base of a wall in
accordance with Standard Section 12.11.1:
Fp = 0.4SDSIww = (0.4)(1.43)(1.0)(65 psf)(28 ft/2) = (37 psf)(14 ft) = 521 plf.
The capacity design requirement in TMS 402 Section 1.17.3.2.6.1 applies to behavior inplane, not out
ofplane.
The capacity computed per TMS 402 Section 3.3.4.1.2.1 is as follows:
Mu/Vudv need not be taken larger than 1.0 (and Mu/Vudv does exceed 1.0 for a short distance above the
base). An, as determined earlier, is taken as 51.3 in.2/ft. Note: If the dimensions from Figure 109.7 are
used, An is taken as bwd = (8.32 in.)(3.81 in.) + (24  8.32 in.)(1.25 in.) = 51.3 in.2/ft, a similar value.
The authors note that traditional practice in reinforced masonry has been to compute shear stress on the
basis of areas equaling width times depth to reinforcement (bd). For the inplane shear strength, the
difference between bd and An is not too great, but for the outofplane shear strength of walls with one
layer of reinforcement in their centers, the difference is very substantial. Therefore, the authors have
substituted bwd (= 8 in. 3.81in. = 30.5 in.2 ; see Figure 10.17) for An in the equation below.
Because shear exists at both the bottom and the top of the wall, conservatively neglect the effect of P:
= 1.53 klf
Vm = (0.8)(1.53) = 1.23 klf > 0.52 klf OK
10.1.5.5.2 Inplane shear in side walls. As indicated in Sections 10.1.4.1 and 10.1.5.1, the inplane
demand at the base of the wall, Vu = 2.5(211 kips) = 528 kips and the shear capacity, Vm, is larger than
(4.13 klf)(200 ft) = 826 kips.
For the purpose of understanding likely behavior of the building somewhat better, Vn is estimated more
accurately than simply limiting Mu/Vudv to 1 for these long walls:
Mu/Vudv = h/l = 28/200 = 0.14
Pu = 0.614D = (0.614)(430 kips) = 264 kips
Vm = [4.0  1.75(0.14)][200(51.3)](0.045) + 0.25(264) = 1,733 + 66 = 1,799 kips
Vns = 0.5(Av/s)fyd = 0.5(0.62/4.0)(60)(200) = 930 kips (for 2#5 in bond beams at 4 ft o.c.)
Vn = 1,799 + 930 = 2,729 kips
Maximum = 6(0.045 ksi)(10,260 in.2) = 2,770 > 2,729 kips
Vn = 0.8(2,729) = 2,183 kip > 528 kips = Vu
The calculated seismic force, VE = 211kips (from Sec. 10.1.4.1)
Vn/VE = 10.3 >> R used in design
In other words, it is unlikely that the long masonry walls will yield in either inplane shear or flexure at
the design seismic ground motion. The walls will likely yield in outofplane response, and the roof
diaphragm may also yield. The roof diaphragm for this building is illustrated in Section 11.2.
10.1.6 End Walls
The transverse walls are designed in a manner similar to the longitudinal walls. Complicating the design
of the transverse walls are the door openings, which leave a series of masonry piers between the doors.
10.1.6.1 Horizontal reinforcement end walls. The minimum reinforcement, per TMS 402
Section 1.17.3.2.6, is 0.0007Ag = (0.0007)(11.625 in.)(8 in.) = 0.065 in.2 per course. The maximum
spacing of horizontal reinforcement is 48 inches, for which the minimum reinforcement is 0.39 in.2. Two
#4 in bond beams at 48 inches on center would satisfy the requirement. The large amount of vertical
reinforcement would combine to satisfy the minimum total reinforcement requirement. However, given
the 100foot length of the wall, a larger amount is desired for control of restrained shrinkage as discussed
in Section 10.1.5.1. Two #5 at 48 inches on center will be used.
10.1.6.2 Vertical reinforcement end walls. The area for each bay subject to outofplane wind is 20
feet wide by 30 feet high because wind load applied to the doors is transferred to the masonry piers.
However, the area per bay subject to both inplane and outofplane seismic forces is reduced by the area
of the doors. This is because the doors are relatively light compared to the masonry. See Figures 10.111
and 10.112.
10.1.6.3 Outofplane flexure end walls. Outofplane flexure is considered in a manner similar to
that illustrated in Section 10.1.5.2. The design of this wall must account for the effect of door openings
between a row of piers. The steps are the same as identified previously and are summarized here for
convenience:
1. Select a trial design.
2. Investigate to ensure ductility.
3. Make sure the trial design is suitable for wind (or other nonseismic) lateral loadings using the
wind provisions of the Standard.
4. If wind controls over seismic (it does not in this example), then calculate the midheight
deflection due to wind by TMS 402.
5. Calculate the seismic demand.
6. Determine the seismic resistance and compare to the demand determined in Step 5.
10.1.6.3.1 Trial design. A trial design of 12inchthick CMU reinforced with two #6 bars at 24 inches
on center is selected. The selfweight of the wall, accounting for horizontal bond beams at 4 feet on
center, is taken as 103 psf. Adjacent to each door jamb, the vertical reinforcement is placed into two
cells. See Figure 10.110.
Figure 10.110 Trial design for piers on end walls
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
Next, determine the design load locations. The centroid for seismic loads, outofplane, is the centroid of
the mass of the wall and, accounting for the door openings, is determined to be 17.8 feet above the base.
See Figures 10.111 and 10.112.
10.1.6.3.2 Investigate to ensure ductility. The critical strain condition corresponds to a strain in the
extreme tension reinforcement (which is a pair of #6 bars in the end cell in this example) equal to times
the strain at yield stress. As for the side walls, = 1.5 for outofplane flexure due to wind (TMS 402
Section 3.3.3.5 and Commentary 3.3.3.5). See Figure 10.113.
For this case:
t = 11.63 in.
d = 11.63  2.38 = 9.25 in.
îm = 0.0025 (TMS Sec. 402 3.3.2.c)
îs = 1.5 îy = 1.5 (fy/Es) = 1.5 (60 ksi /29,000 ksi) = 0.0031 (TMS 402 Sec. 3.3.3.5.1.a and
Commentary 3.3.3.5)
a = 0.8c = 3.30 in. (TMS 402 Sec. 3.3.3.5.1.b)
Figure 10.111 Inplane loads on end walls
(1.0 ft = 0.3048 m)
Figure 10.112 Outofplane load diagram and resultant of lateral loads
(1.0 ft = 0.3048 m, 1.0 lb = 4.45 N, 1.0 kip = 4.45 kN)
Figure 10.113 Investigation of outofplane ductility for end wall
(1.0 in. = 25.4 mm, 1.0 ksi = 6.89 MPa)
Note that the Whitney compression stress block, a = 3.30 inches deep, is greater than the 1.50inch face
shell thickness. Thus, the compression stress block is broken into two components: one for full
compression against solid masonry (the face shell) and another for compression against the webs and
grouted cells but accounting for the open cells. These are shown as C1 and C2 in Figure 10.114. The
values are computed using TMS 402 Section 3.3.2.g:
C1 = 0.80fm' (1.50 in.)b = (0.80)(2 ksi)(1.50)(96) = 230 kips (for full length of pier)
C2 = 0.80fm' (a  1.50 in.)(6(8 in.)) = (0.80)(2 ksi)(3.30  1.50)(48) = 138 kips
The 48inch dimension in the C2 calculation is the combined width of grouted cell and adjacent mortared
webs over the 96inch length of the pier.
T = FyAs = (60 ksi)(6 0.44 in.2) = 158 kips/pier
P is computed at the head of the doors. The dead load component of P is:
P = (Pf + Pw) = (0.020 ksf)(20 ft)(20 ft) + (0.103 ksf)(18 ft)(20 ft) = 8.0 + 37.1
P = 45.1 kips/pier
From TMS 402 Section 3.3.3.5.1.d, axial forces are taken from the load combination of the following:
P = D + 0.75L + 0.525QE with QE = Fp = 0.2SDSD = (0.2)(1.43)(45.1) = 12.9 kips/pier
P = 45.1 kips/pier + (0.75)(0) + (0.525)(12.9 kips/pier)
P = 51.9 kips/pier
C1 + C2 > P + T
368 kip > 210 kips
The compression capacity is greater than the tension capacity, so the ductility criterion is satisfied.
10.1.6.3.3 Check for wind loading. Wind pressure per bay is over the full 20footwide by 30foothigh
bay, as discussed above, and is based on the Standard. While both strength and deflection need to be
ascertained per a building code (the IBC was used), the calculations are not presented here.
Figure 10.114 Cracked moment of inertia (Icr) for end walls.
Dimension c depends on calculations shown for Figure 10.116.
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
10.1.6.3.4 Calculate outofplane seismic demand. For this example, the load combination 0.614D has
been used, and for this calculation, forces and moments over a single pier (width = 96 in.) are used. This
does not violate the b > 6t rule (TMS 402 Sec. 3.3.4.3.3.d) because the pier is reinforced at 24 inches on
center. The use of the full width of the pier instead of a 24inch width is simply for calculation
convenience.
For this example, a Pdelta analysis using RISA2D was run, resulting in the following:
Maximum moment, Mu = 95.6 ftkips/bay = 95.6/20 ft = 4.78 klf (does not control)
Moment at top of pier, Mu = 89.3 ftkips/pier = 89.3 / 8 ft = 11.2 klf (controls)
Shear at bottom of pier, Vu = 9.61 kips/pier
Reaction at roof, Vu = 17.5 kips/bay
Axial force at base, Ru = 31.2 kips/pier (includes load factor on D of 0.614)
10.1.6.3.6 Determine moment resistance at the top of the pier. See Figure 10.115.
As = 6#6 = 2.64 in.2/pier
d = 9.25 in.
T = 2.64(60) = 158.4 kip/pier
C = T + P = 184.1 kip/pier (P is based on D of (0.614)(37.1 + 8 kip) = 27.7 kip/pier at top of pier)
a = C / (0.8f'mb) = 184.1 / [(0.8)(2)96] = 1.20 in.
Because a is less than the face shell thickness (1.50 in.), compute as for a rectangular beam. Moments are
computed about the centerline of the wall.
Mn = C (t/2  a/2) + P (0) + T (d  t/2)
= 184.1(5.81  1.20/2) + 158.4(9.255.81) = 1,504 in.kip = 125.4 ftkip
íMn = 0.9(125.4) = 112.8 ftkip
Because moment capacity at the top of the pier, Mn = 112.8 ftkips, exceeds the maximum moment
demand at top of pier, Mu = 89.3ftkips, the condition is acceptable.
Figure 10.115 Outofplane seismic strength of pier on end wall
(1.0 in. = 25.4 mm, 1.0 ksi = 6.89 MPa)
10.1.6.4 Inplane flexure end walls. There are several possible methods to compute the shears and
moments in the individual piers of the end wall. For this example, the end wall was modeled using RISA
2D. The horizontal beam was modeled at the top of the opening, rather than at its midheight. The in
plane lateral loads (see Figure 10.111) were applied at the 12foot elevation and combined with joint
moments representing transfer of the horizontal forces from their point of action down to the 12foot
elevation. Vertical load due to roof beams and the selfweight of the end wall were included. The input
loads are shown in Figure 10.116. For this example:
w = (18 ft)(103 psf) + (20 ft)(20 psf) = 2.254 klf
H = (184 kip)/5 = 36.8 kip
M = Cs[(Vf long + Vw long)hlong + (Vw short)(hshort)] (refer to Fig. 10.111).
M = 0.286[(400 + 418)(28 ft 12 ft) + 470(17.8 ft 12 ft)] = 452 ftkip
Figure 10.116 Input loads for inplane end wall analysis
(1.0 ft = 0.3048 m)
The input forces at the end wall are distributed over all the piers to simulate actual conditions. The RISA
2D frame analysis accounts for the relative stiffnesses of the 4foot and 8footwide piers (continuity of
the 4footwide piers at the corners was not considered). The final distribution of forces, shears and
moments for an interior pier is shown on Figure 10.117.
Figure 10.117 Inplane design condition for 8footwide pier
(1.0 ft = 0.3048 m)
Continuing with the trial design for inplane pier design, use two #6 bars at 24 inches on center
supplemented by adding two #6 bars in the cells adjacent to the door jambs (see Figures 10.110 and
10.118).
Figure 10.118 Inplane ductility check for 8footwide pier
(1.0 in. = 25.4 mm, 1.0 ksi = 6.89 MPa)
The design values for inplane design at the top of the pier are:
Table 10.13 Inplane Design Values at Pier Top
Unfactored
0.614D + 1.0E
1.486D + 1.0E
P = 45.1 kips
Pu = 41.2 kips
Pu = 67.0 kips
V = 43.6 kips
Vu = 43.6 kips
Vu = 43.6 kips
M = 523 ftkips
Mu = 523 ftkips
Mu = 523 ftkips
Mu/Vudv
1.50
1.50
The ductility check is illustrated in Figure 10.118. Because Mu/Vudv > 1for this special reinforced
masonry shear wall subject to inplane loads, = 4:
îm = 0.0025
îs = 4îy = (4 )(60/29,000) = 0.0083
d = 92 in.
From the strain diagram (Fig. 10.118), the strains at the rebar locations from left to right are:
î = 0.0020
î = 0.0011
î = 0.0017
î = 0.0045
î = 0.0073
î = 0.0083
To check ductility, use unfactored loads (from Section 10.1.6.3.2):
P = Pf + Pw = 8 kips + 37.1 kips = 45.1 kips
a = 0.8c = 17.0 in.
Cm = (0.8fm' )ab = (1.6 ksi)(17.0 in.)(11.63 in.) = 315.5 kips
Ts1 = Ts2 = FyAs = (60 ksi)(2 0.44 in.2) = 52.8 kips
Ts3 = As = (0.0017)(29,000 ksi)(2 0.44 in.2) = 43.4 kip
Cs1 = As = (0.0021)(29,000 ksi)(2 0.44 in.2) = 53.6 kip
Cs2 = As = (0.0011)(29,000 ksi)(2 0.44 in.2) = 28.1 kip
C > T + P
Cm + Cs1 + Cs2 > Ts1 + Ts2 + Ts3 + P
315.5 + 53.6 + 28.1 > 52.8 + 52.8 + 43.4 + 45.1
397 kips > 194 kips OK
Because compression capacity exceeds tension capacity, the requirement for ductile behavior is OK.
Note that maximum P for the wall to remain ductile is Pmax = C  T = 248 kips. Thus, Pmax = 223 kips
in order to assure ductility.
For the strength check, see Figure 10.119.
Figure 10.119 Inplane seismic strength of pier.
Strain diagram superimposed on strength diagram for both cases.
Note that locations with low force in reinforcement, marked by *, are neglected.
(1.0 in. = 25.4 mm)
To ascertain the strength of the pier, a Pn  Mn curve is developed. Only the portion below the balance
point is examined since that portion is sufficient for the purposes of this example. (Ductile failures
occur only at points on the curve that are below the balance point, so this is consistent with the overall
approach).
For the P = 0 case, assume all bars in tension reach their yield stress and neglect compression steel (a
conservative assumption):
Ts1 = Ts2 = Ts3 = Ts4 = (2)(0.44 in.2)(60 ksi) = 52.8 kips
Cm = Ts = (4)(52.8) = 211.2 kips
Cm = 0.8f mab = (0.8)(2 ksi)a(11.63 in.) = 18.6a
Thus, a = 11.3 inches and c = a/ = 11.3 / 0.8 = 14.2 inches.
Mcl = 0
Mn = 42.35 Cm + 44Ts1 + 36Ts2 + 12Ts3  12Ts4 = 13,168 in.kips
Mn = (0.9)(13,168) = 11,851 in.kips = 988 ftkips
For the balanced case:
d = 92 in.
î = 0.0025
îy = 60/29,000 = 0.00207
a = 0.8c = 40.3 in.
Compression values are determined from the Whitney compression block adjusted for fully grouted cells
or ungrouted cells:
Cm1 = (1.6 ksi)(16 in.)(11.63 in.) = 297.8 kips
Cm2 = (1.6 ksi)(16 in.)(2 1.50 in.) = 76.8 kips
Cm3 = (1.6 ksi)(8.3 in.)(11.63 in.) = 154.4 kips
Cs1 = (0.88 in.2)(60 ksi) = 52.8 kips
Cs2 = (0.88 in.2)(60 ksi)(0.0019 / 0.00207) = 48.5 kips
Ts1 = (0.88 in.2)(60 ksi) = 52.8 kips
Ts2 = (0.88 in.2)(60 ksi)(0.0017 / 0.00207) = 43.4 kips
Fy = 0:
Pn = C  T = 297.8 + 76.8 + 154.4 + 52.8 + 48.5 52.8  43.4 = 534 kips
Pn = (0.9)(534) = 481 kips
Mcl = 0:
Mn = 40Cm1 + 24Cm2 + 11.85Cm3 + 44Cs1 + 36Cs2 + 44Ts1 + 36Ts2 = 23,540 in.kips
Mn = (0.9)(23,540) = 21,186 in.kips = 1,765 ftkips
The two cases are plotted in Figure 10.120 to develop the Pn  Mn curve on the pier. The demand
(Pu,  Mu) also is plotted. As can be seen, the pier design is acceptable because the demand is within the
Pn  Mn curve. (See the Birmingham 1 example in Section 10.2 for additional discussion of Pn  Mn
curves.) By linear interpolation, Mn at the minimum axial load is 1,096 ft kip.
The authors note that the use of = 0.9 on Pn at the balance point is consistent with TMS 402, but,
because of the ductility requirement, the balance point will never be reached. The maximum Pn for this
pier, as per the ductility requirement (from Sec. 10.1.6.4), would be (397 kips  149 kips) = 248 kips (as
discussed above), well below the 481 kips at Pb. To illustrate the point, this maximum expressed as
Pnmax = 223 kips, is illustrated in Figure 10.120.
Figure 10.120 Inplane Pn  Mn diagram for pier
(1.0 kip = 4.45 kN, 1.0 ftkip = 1.36 kNm)
10.1.6.5 Combined loads. Although it is not required by the Standard, it is educational to illustrate the
orthogonal combination of seismic loads for this pier (as if Standard Section 12.5.3.a were required),
shown in Table 10.14:
Table 10.14 Combined Loads for Flexure in End Wall Pier
0.614D
OutofPlane
InPlane
Total
Case 1
1.0(87.7/112.8) +
0.3(523/1026) =
0.93 < 1.00 OK
Case 2
0.3(87.7/112.8) +
1.0(523/1026) =
0.74 < 1.00 OK
Values are in kips; 1.0 kip = 4.45 kN.
10.1.6.6 Shear end walls.
10.1.6.6.1 Inplane shear at end wall piers. The inplane shear at the base of the pier is 43.6 kips per
bay. At the head of the opening where the moment demand is highest, the inplane shear is slightly less
(based on the weight of the pier). There, V = 43.6 kips  (0.286)(8 ft)(12 ft)(0.103 ksf) = 40.8 kips. Per
TMS 402 Section 1.17.3.2.6.1.1, the design shear strength, Vn, must exceed the shear corresponding to
the development of 1.25 times the nominal flexural strength, Mn, or 2.5Vu, whichever is smaller. Using
the results in Figure 10.120, the 125 percent implies a factor on shear by analysis of:
But 2.91Vu > 2.5Vu; therefore, 2.5Vu controls (TMS 402 Sec. 1.17.3.2.6.1.1).
Therefore, the required shear capacities at the base and head of the pier are (2.5)(43.6 kips) = 109 kips
and (2.5)(40.8) = 102 kips, respectively.
The inplane shear capacity is computed as follows where the net area, An, of the pier is the area of face
shells plus the area of grouted cells and adjacent webs:
As discussed previously, Mu/Vudv need not exceed 1.0 in the above equation.
An = (96 in. 1.50 in. 2) + (6 cells 8 in. 8.63 in.) = 702 in.2 / bay
Recall that horizontal reinforcement is 2#5 at 48 inches in bond beams:
At the base of the pier:
Vm = [4.0  1.75(0)](702 in.2)(0.0447 ksi) + (0.25)(0.614 55.0 kips)
Vm = 79.0 kips/bay
Vn = (0.8)(79.0 + 37.2) = 116.2 kips/bay > 109 kips/bay = 2.5 Vu OK
At the head of the pier:
Vm = [4.0  1.75(1.0)](702 in.2)(0.0447 ksi) + (0.25)(0.614 45.1 kips) = 77.5 kips/bay
Vn = (0.8)(77.5 + 37.2) = 91.8 kips/bay < 102 kips/bay = 2.5 Vu N.G.
This nonductile situation can be addressed by increasing the compression capacity. For this case, the
other cells in the pier will be grouted, resulting in An = bwd = (11.63 in.)(92 in.) = 1070 in.2. (Note that
while TMS 402 permits An = bwdv, the authors have elected to use the slightly more conservative bwd in
the determination of area.) This results in Vm = 114.5 kips and Vn = 121 kips > 102 kips = 2.5 Vu which
is OK.
Note: The design of the piers in the end walls of this example will remain the same without iteration to
reflect the additional grouted cells. Note also that there is no additional vertical reinforcement; only grout
has been added to the cells.
Figure 10.121 Inplane shear at end wall
10.1.6.6.1 OutofPlane Shear at End Wall Piers. For outofplane shear, see Figure 10.112. Shear at
the top of wall is 15.3 kips/bay, and shear at the base of the pier is 10.3 kips/bay. From the RISA2D
analysis, which included Pdelta, the shear at the head of the opening is 4.57 kips. The same multiplier of
2.5 for development of 125 percent of flexural capacity will be applied to outofplane shear resulting in
38.25 kips at the top of the wall, 11.4 kips at the head of the opening (top of pier) and 25.8 kips at the
base of the pier.
Outofplane shear capacity is computed using the same equation. äbwd is taken as the net area An. Note
that Mu/Vudv is zero at the support because the moment is assumed to be zero; however, a few inches into
the vertical span, Mu/Vudv will exceed 1.0, so the limiting value of 1.0 is used here. This is typically the
case where considering outofplane loads on a wall.
For computing shear capacity at the top of the wall:
An = bwd = ((8 in./2 ft) 20 ft)(9.25 in.) = 740 in.2
Vm = [4.0  1.75(1)](740 in.2)(0.0447 ksi) + (0.25)(0.614 8.0) = 75.7 kips/bay
Vm = (0.8)(75.7) = 60.5 kips/bay
For computing shear capacity in the pier:
An = (8 in./cell)(12 cells)(9.25 in.) = 888 in.2
Vm = [4.0  1.75(1)](888 in.2)(0.0447 ksi) + (0.25)(0.614 41.67) = 95.7 kips/bay
Vm = (0.8)(95.7) = 76.6 kips/bay
At the top of wall:
Vn = Vm = 60.5 kips/bay > 15.3 kips/bay = Vu OK
At the pier:
Vn = Vm = 76.6 kips/bay > 10.3 kips/bay = Vu OK
10.1.7 InPlane Deflection End Walls
Deflection of the end wall (short wall) has two components as illustrated in Figure 10.122.
Figure 10.122 Inplane deflection of end wall
(1.0 ft = 0.3048 m)
As obtained from the RISA2D analysis of the piers, 1 = 0.047 in.:
where is the form factor equal to 6/5 and:
G = Em/2(1 + æ) = 1,800 ksi / 2(1 + 0.15) = 782 ksi
A = An = area of face shells + area of grouted cells
= (100 ft 12 in./ft 2 1.50 in.2) +(50)(8 in.)(8.63 in.) = 7,050 in.2
Note: Contribution to base shear of end walls (above the doors) is Cs (end wall weight) =
(0.286)[(470 kips/2)  (103 psf)(5)(8 ft)(12 ft)] = 53 kips. Contribution to base shear of long walls plus
roof is Cs (long wall + roof weight) = (0.286)[(400+418)/2] = 117 kips.
Therefore:
= 0.0008 + 0.0049 = 0.006 in.
and
total = Cd(0.047 + 0.006) = 3.5(0.053 in.) = 0.19 in. < 2.35 in. OK
where (2.35 = 0.007hn = 0.01hsx) (TMS 402 Sec. 3.3.5.4).
Note that the drift limits for masonry structures are smaller than for other types of structure. It is possible
to interpret Standards Table 12.121 to give a limit of 0.007hn for this structure, but that limit also is
easily satisfied. The real displacement in this structure is in the roof diaphragm; see Sec. 11.2.4.2.3.
10.1.8 Bond Beam Side Walls (and End Walls)
Reinforcement for the bond beam located at the elevation of the roof diaphragm can be used for the
diaphragm chord. The uniform lateral load for the design of the chord is the lateral load from the long
wall plus the lateral load from the roof and is equal to 1.17 klf. The maximum tension in rebar is equal to
the maximum moment divided by the diaphragm depth:
M/d = 5,850 ftkips/100 ft = 58.5 kips
The seismic load factor is 1.0. The required reinforcement is:
Areqd = T/ Fy = 58.5/(0.9)(60) = 1.081 in.2
This will be satisfied by two #7 bars, As = (2 0.60 in.2) = 1.20 in.2
In Sec. 11.2.4.2.2, the diaphragm chord is designed as a wood member utilizing the wood ledger member.
Using either the wood ledger or the bond beam is considered acceptable.
10.2 FIVESTORY MASONRY RESIDENTIAL BUILDINGS IN BIRMINGHAM,
ALABAMA; ALBUQUERQUE, NEW MEXICO; AND SAN RAFAEL, CALIFORNIA
10.2.1 Building Description
In plan, this fivestory residential building has bearing walls at 24 feet on center (see Figures 10.21 and
10.22). All structural walls are of 8inchthick concrete masonry units (CMU). The floor is of 8inch
thick hollow core precast, prestressed concrete planks. To demonstrate the incremental seismic
requirements for masonry structures, the building is partially designed for four locations: two sites in
Birmingham, Alabama; a site in Albuquerque, New Mexico; and a site in San Rafael, California. The two
sites in Birmingham have been selected to illustrate the influence of different soil profiles at the same
location. The building is designed for Site Classes C and E in Birmingham.
Figure 10.21 Typical floor plan
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
Figure 10.22 Building elevation
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)
For the Albuquerque and both Birmingham sites, it is assumed that shear friction reinforcement in the
joints of the diaphragm planks is sufficient to resist seismic forces, so no topping is used. For the San
Rafael site, a castinplace 2«inchthick reinforced lightweight concrete topping is applied to all floors.
The structure is free of irregularities both in plan and elevation. ACI 318, Sections 21.1.1.6 and 21.11.1,
require reinforced castinplace toppings as diaphragms in Seismic Design Category D and higher. Thus,
the Birmingham example in Site Class E / Seismic Design Category D would require a topping, although
that is not included in this example.
The design of an untopped diaphragm (for Seismic Design Categories A, B and C) is not addressed
explicitly in ACI 318. The designs of both untopped and topped diaphragms for these buildings are
described in Chapter 8 of this volume using ACI 318 for the topped diaphragm in the San Rafael building.
The Provisions provide guidance for the design of untopped precast plank diaphragms in Part 3, RP10.
For the purpose of determining the site class coefficient (Standard Sec. 11.4.2 and 20.3), a stiff soil
profile with standard penetration test results of 15 < N < 50 is assumed for the San Rafael site resulting in
a Site Class D for this location. The Birmingham 1 and Albuquerque sites have soft rock with N > 50,
resulting in Site Class C. The Birmingham 2 site has soft clay with N < 15, which results in Site Class E.
The two Birmingham sites are presented to illustrate how different soil conditions at the same location
(same seismicity) can result in different Seismic Design Categories. No foundations are designed in this
example. The foundation systems are assumed to be able to carry the superstructure loads including the
overturning moments.
The masonry walls in two perpendicular directions act as bearing and shear walls with different levels of
axial loads. The geometry of the building in plan and elevation results in nearly equal lateral resistance in
both directions. The walls are constructed of CMU and typically are minimally reinforced in all
locations. Figure 10.23 illustrates the wall layout.
Figure 10.23 Plan of walls
(1.0 ft = 0.3048 m)
The floors serve as horizontal diaphragms distributing the seismic forces to the walls and are assumed to
be stiff enough to be considered rigid. There is little information about the stiffness of untopped precast
diaphragms. The design procedure in Section RP10 of Part 3 of the Provisions results in a diaphragm
intended to remain below the elastic limit until the walls reach an upper bound estimate of strength;
therefore, it appears that the assumption is reasonable.
Material properties are as follows:
The compressive strength of masonry, f m, is taken as 2,000 psi, and the steel reinforcement has a
yield limit of 60 ksi.
The design snow load (on an exposed flat roof) is less than the roof live load for all locations.
This example covers the following aspects of a seismic design:
Determining the equivalent lateral forces
Design of selected masonry shear walls for their inplane loads
Computation of drifts
The story heights are small enough that the design of the masonry walls for outofplane forces is nearly
trivial. Inplane response governs both the reinforcement in the wall and the connections to the
diaphragms.
10.2.2 Design Requirements
10.2.2.1 Seismic parameters. The basic parameters affecting the design and detailing of the buildings
are shown in Table 10.21. The Seismic Design Category for Birmingham 2 deserves special comment.
The value of SDS would imply a Seismic Design Category of C, while the value of SD1 would imply
Seismic Design Category D, per Tables 11.61 and 11.62 of the Standard, where in Section 11.6 a
provision permits the use of Table 11.61 alone if T < 0.8 SD1/SDS and the floor diaphragm is considered
rigid or has a span of less than 40 feet. As will be shown for this building, Ta = 0.338 seconds and 0.8
SD1/SDS = 0.446. In the author s opinion, the untopped diaphragm may not be sufficiently rigid and thus
Table 11.62 is considered, resulting in Seismic Design Category D.
10.2.2.2 Structural design considerations. The floors act as horizontal diaphragms, and the walls
parallel to the motion act as shear walls for all four buildings.
The system is categorized as a bearing wall system (Standard Sec. 12.2). For Seismic Design
Category D, the bearing wall system has a height limit of 160 feet and must comply with the requirements
for special reinforced masonry shear walls. Note that the structural system is one of uncoupled shear
walls. Crossing beams over the interior doorways (their design is not included in this example) will need
to continue to support the gravity loads from the deck slabs above during the earthquake, but are not
designed to provide coupling between the shear walls.
The building is symmetric and appears to be regular both in plan and elevation. It will be shown,
however, that the building is torsionally irregular. Standard Table 12.61 permits use of the ELF
procedure in accordance with Standard Section 12.8 for Birmingham 1 and Albuquerque (Seismic Design
Categories B and C). By the same table, the Seismic Design Category D buildings (Birmingham 2 and
San Rafael) must use a dynamic analysis for design. A careful reading of Standard Table 12.61 for
Seismic Design Category D reveals that all of the rows do not apply to our building except the last, all
other structures ; thus, ELF analysis is not permitted, but modal analysis is permitted. For this particular
building arrangement, it will be shown that the modal response spectrum analysis does not identify any
particular effect of the horizontal torsional irregularity, as will be illustrated; thus it is the authors opinion
that ELF analysis would be sufficient.
Table 10.21 Design Parameters
Design Parameter
Value for
Birmingham 1
Value for
Birmingham 2
Value for
Albuquerque
Value for
San Rafael
Ss (Map 1)
0.266
0.266
0.456
1.5
S1 (Map 2)
0.105
0.105
0.137
0.6
Site Class
C
E
C
D
Fa
1.2
2.45
1.2
1
Fv
1.7
3.49
1.66
1.5
SMS = FaSs
0.32
0.65
0.55
1.5
SM1 = FvS1
0.18
0.37
0.23
0.9
SDS = 2/3 SMS
0.21
0.43
0.37
1
SD1 = 2/3 SM1
0.12
0.24
0.15
0.6
Seismic Design
Category
B
D
C
D
Diaphragm
Topping req d per
ACI 318
No
Yes*
No
Yes
Masonry Wall Type
Ordinary
Reinforced
Special
Reinforced
Intermediate
Reinforced
Special
Reinforced
Standard Design Coefficients (Table 12.21)
R
2.0
5
3.5
5
ê0
2.5
2.5
2.5
Cd
1.75
3.5
2.25
3.5
*For this masonry example, Birmingham 2 is designed without topping on the precast planks. It is assumed that the
precast planks at floors and roof have connections sufficiently rigid to permit the idealization of rigid horizontal
diaphragms.
The type of masonry shear wall is selected to illustrate the various requirements as well as to satisfy
Table 12.21 of the Standard. Note that Ordinary Reinforced Masonry Shear Walls could be used for
Seismic Design Category C at this height.
The orthogonal direction of loading combination requirement (Standard Sec. 12.5) needs to be considered
for structures assigned to Seismic Design Category D. However, the arrangement of this building is not
particularly susceptible to orthogonal effects; the walls are not subject to axial force from horizontal
seismic motions, only bending and shear.
The walls are all solid, and there are no significant discontinuities, as defined by Standard
Section 12.3.2.2, in the vertical elements of the seismic forceresisting system.
Ignoring the short walls at stairs and elevators, there are eight shear walls in each direction; therefore, the
system appears to have adequate redundancy (Standard Sec. 12.3.4.2). The redundancy factor, however,
will be computed.
Tie and continuity requirements (Standard Sec. 12.11) must be addressed when detailing connections
between floors and walls (see Chapter 8 of this volume).
Nonstructural elements (Standard Chapter 13) are not considered in this example.
Collector elements are required in the diaphragm for longitudinal response (Standard Sec. 12.10). Rebar
in the longitudinal direction, spliced into bond beams, is used for this purpose (see Chapter 8 of this
volume).
Diaphragms must be designed for the required forces (Standard Sec. 12.10 and Provisions Part 3,
Sec. RP10).
The structural walls must be designed for the required outofplane seismic forces (Standard Sec. 12.11)
in addition to outofplane wind on exterior walls and 5 psf differential air pressure on interior walls.
Each wall acts as a vertical cantilever in resisting inplane forces. The walls are classified as masonry
cantilever shear wall structures in Standard Table 12.121, which limits story drift to 0.01 times the story
height.
10.2.3 Load Combinations
The basic load combinations are those in Standard Section 2.3.2. The seismic load effect, E, is defined
by Standard Section 12.4, as follows:
E = Eh + Ev = QE ñ 0.2SDSD
10.2.3.1 Redundancy Factor. The Redundancy Factor, is a multiplier on design force effects and
applies only to the inplane direction of the shear walls. For structures in Seismic Design Categories A, B
and C, = 1.0 (Standard Sec. 12.3.4.1). For structures in Seismic Design Category D, is determined
per Standard Section 12.3.4.2.
For a shear wall building assigned to Seismic Design Category D, = 1.0 as long as it can be shown that
failure of a shear wall or pier with a heighttole