NEHRP Recommended Provisions:
Design Examples
FEMA 451  August 2006
Building Seismic Safety Council of the National Institute of Building Sciences
NEHRP Recommended Provisions:
Design Examples
FEMA 451  August 2006
Prepared by the
Building Seismic Safety Council
for the
Federal Emergency Management Agency
of the Department of Homeland Security
National Institute of Building Sciences
Washington, D.C.
ii
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 EMW1998CO0419 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.
iii
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 2003 edition of the NEHRP Recommended Provisions
for Seismic Regulation of New Buildings and Other Structures and its Commentary affirmed
FEMA’s ongoing support to improve the seismic safety of construction in this country. The
NEHRP Recommended Provisions serves as the basis for the seismic requirements in the ASCE
7 Standard Minimum Design Loads for Buildings and Other Structures as well as both the
International Building Code 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 user of the NEHRP
Recommended Provisions. This material will also be of assistance to those using the ASCE 7
standard and the models codes that reference the standard.
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 be of significant assistance for a
significant number of users of the nation’s seismic building codes and their reference documents.
Department of Homeland Security/
Federal Emergency Management Agency
iv
v
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 application of seismic
requirements of the NEHRP (National Earthquake Hazards Reduction Program) Recommended
Provisions for Seismic Regulations for New Buildings and Other Structures and, by extension, the
model codes and standards because the Provisions are the source of seismic design requirements
in most of those documents including ASCE 7, Standard Minimum Design Loads for Buildings
and Other Structures; the International Building Code; and the NFPA 5000 Building
Construction and Safety Code.
This compilation of design examples is an expanded version of an earlier document (entitled
Guide to Application of the NEHRP Recommended Provisions, FEMA 140) and reflects 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 signals the
success of the Federal Emergency Management Agency and Building Seismic Safety Council
efforts 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 first 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 impact the NEHRP Recommended
Provisions has on frequently encountered design problems. Some of the examples draw heavily
on a BSSC trial design project conducted prior to the publication of the first edition of the
NEHRP Recommended Provisions in 1985 but most were created by the authors to illustrate
issues not covered in the trial design program. Further, the authors have made adjustments to
those examples drawn from the trial design program as necessary to reflect the 2000 Edition of
the NEHRP Recommended Provisions. Finally, 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.
The BSSC is grateful to all those individuals and organizations whose assistance made this set of
design examples a reality:
• James Robert Harris, J. R. Harris and Company, Denver, Colorado, who served as the
project manager, and Michael T. Valley, Magnusson Klemencic Associates, Seattle,
Washington, who served as the technical editor of this volume
• The chapter authors – Robert Bachman, Finley A. Charney, Richard Drake, Charles A. Kircher,
Teymour Manzouri, Frederick R. Rutz, Peter W. Somers, Harold O. Sprague, Jr., and Gene R.
Stevens – for there unstinting efforts
vi
• Greg Deierlein, J. Daniel Dolan, S. K. Ghosh, Robert D. Hanson, Neil Hawkins, and Thomas
Murray for their insightful reviews
• William Edmands and Cambria Lambertson for their hard work behind the scenes preparing figures
Special thanks go to Mike Valley and Peter Somers for their work annotating the design examples to
reflect the 2003 edition of the Provisions and updated versions of other standards referenced in the 2003
version. The BSSC Board is also grateful to FEMA Project Officer Michael Mahoney for his support and
guidance and to Claret Heider and Carita Tanner of the BSSC staff for their efforts preparing this volume
for publication and issuance as a CDROM.
Jim. W. Sealy, Chairman, BSSC Board of Direction
vii
TABLE OF CONTENTS
1. FUNDAMENTALS by James Robert Harris, P.E., Ph.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1 Earthquake Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Structural Response to Ground Shaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Engineering Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
1.4 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
1.5 Nonstructural Elements of Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
1.6 Quality Assurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2. GUIDE TO USE OF THE PROVISIONS by Michael Valley, P.E. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 STRUCTURAL ANALYSIS by Finley A. Charney, Ph.D., P.E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Irregular 12Story Steel Frame Building, Stockton, California . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 Description of Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.3 Provisions Analysis Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.4 Dynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.5 Equivalent Lateral Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
3.1.6 ModalResponseSpectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
3.1.7 ModalTimeHistory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
3.1.8 Comparison of Results from Various Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . 343
3.2 SixStory Steel Frame Building, Seattle, Washington . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
3.2.1 Description of Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
3.2.2 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Preliminaries to Main Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Description of Model Used for Detailed Structural Analysis . . . . . . . . . . . . . . . . . . . . . 365
3.2.5 Static Pushover Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
3.2.6 TimeHistory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115
3.2.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3143
4. FOUNDATION ANALYSIS AND DESIGN by Michael Valley, P.E. . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Shallow Foundations for a Seven Story Office Building, Los Angeles, California . . . . . . 44
4.1.1 Basic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.2 Design for Gravity Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.3 Design for MomentResisting Frame System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
4.1.4 Design for Concentrically Braced Frame System . . . . . . . . . . . . . . . . . . . . . . . . . . 417
4.1.5 Cost Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
4.2 Deep Foundations for a 12Story Building, Seismic Design Category D . . . . . . . . . . . . . 426
4.2.1 Basic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
4.2.2 Pile Analysis, Design, and Detailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
4.2.3 Other Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
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5. STRUCTURAL STEEL DESIGN by James R. Harris, P.E., Ph.D.,
Frederick R. Rutz, P.E.,Ph.D., and Teymour Manzouri, P.E., Ph.D. . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1 Industrial HighClearance Building, Astoria, Oregon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.2 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1.3 Structural Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.5 Proportioning and Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
5.2 SevenStory Office Building, Los Angeles, California . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
5.2.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
5.2.2 Basic Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
5.2.3 Structural Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
5.2.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
5.2.5 Cost Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
5.3 TwoStory Building, Oakland, California . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
5.3.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
5.3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
5.3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
5.3.4 Design of Eccentric Bracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
6. REINFORCED CONCRETE by Finley A. Charney, Ph.D., P.E. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.1 Development of Seismic Loads and Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.1.1 Seismicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.1.2 Structural Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.1.3 Structural Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2 Determination of Seismic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2.1 Approximate Period of Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2.2 Building Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
6.2.3 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
6.2.4 Accurate Periods from Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
6.2.5 Seismic Design Base Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
6.2.6 Development of Equivalent Lateral Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
6.3 Drift and Pdelta Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
6.3.1 Direct Drift and Pdelta Check for the Berkeley Building . . . . . . . . . . . . . . . . . . . . . . . . 619
6.3.2 Test for Torsional Irregularity for Berkeley Building . . . . . . . . . . . . . . . . . . . . . . . . . . 623
6.3.3 Direct Drift and Pdelta Check for the Honolulu Building . . . . . . . . . . . . . . . . . . . . . . . 624
6.3.4 Test for Torsional Irregularity for the Honolulu Building . . . . . . . . . . . . . . . . . . . . . . . 628
6.4 Structural Design of the Berkeley Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
6.4.1 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
6.4.2 Combination of Load Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
6.4.3 Comments on the Structure’s Behavior Under EW Loading . . . . . . . . . . . . . . . . . . . . . 631
6.4.4 Analysis of FrameOnly Structure for 25 Percent of Lateral Load . . . . . . . . . . . . . . . . . 632
6.4.5 Design of Frame Members for the Berkeley Building . . . . . . . . . . . . . . . . . . . . . . . . . . 636
6.5 Structural Design of the Honolulu Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
6.5.1 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
6.5.2 Combination of Load Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
6.5.3 Accidental Torsion and Orthogonal Loading (Seismic Versus Wind) . . . . . . . . . . . . . . 678
6.5.4 Design and Detailing of Members of Frame 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
ix
6.5.5 Design of Members of Frame 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692
7. PRECAST CONCRETE DESIGN by Gene R. Stevens, P.E. and James Robert Harris, P.E., Ph.D. 71
7.1 Horizontal Diaphragms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.1.1 Untopped Precast Concrete Units for FiveStory Masonry Buildings Located in
Birmingham, Alabama, and New York, New York . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.1.2 Topped Precast Concrete Units for FiveStory Masonry Building, Los Angeles, California
(See Guide Sec. 9.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
7.2 ThreeStory Office Building With Precast Concrete Shear Walls . . . . . . . . . . . . . . . . . . . . . . 730
7.2.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730
7.2.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
7.2.3 Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
7.2.4 Seismic Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
7.2.5 Proportioning and Detailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737
7.3 OneStory Precast Shear Wall Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749
7.3.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749
7.3.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751
7.3.3 Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
7.3.4 Seismic Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
7.3.5 Proportioning and Detailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756
8. COMPOSITE STEEL AND CONCRETE byJames Robert Harris, P.E., Ph.D. and
Frederick R. Rutz, P.E., Ph.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.2 Summary of Design Procedure for Composite Partially Restrained
Moment Frame System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8.3 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.3.1 Provisions Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.3.2 Structural Design Considerations Per the Provisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.3.3 Building Weight and Base Shear Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.4 Details of the PRC Connection and System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.4.1 Connection Mq Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.4.2 Connection Design and Connection Stiffness Analysis . . . . . . . . . . . . . . . . . . . . . . . . 811
8.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819
8.5.1 Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819
8.5.2 Drift and Pdelta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821
8.5.3 Required and Provided Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822
8.6 Details of the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
8.6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
8.6.2 WidthThickness Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824
8.6.3 Column Axial Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824
8.6.4 Details of the Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825
9 MASONRY by James Robert Harris, P.E., Ph.D., Frederick R. Rutz, P.E., Ph.D. and
Teymour Manzouri, P.E., Ph.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9.1 Warehouse with Masonry Walls and Wood Roof, Los Angeles, California . . . . . . . . . . . . . . . . 93
9.1.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.1.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
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9.1.3 Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
9.1.4 Seismic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.1.5 Longitudinal Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910
9.1.6 Transverse Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925
9.1.7 Bond Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940
9.1.8 InPlane Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940
9.2 Fivestory Masonry Residential Buildings in Birmingham, Alabama; New York, New York;
and Los Angeles, California . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
9.2.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
9.2.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944
9.2.3 Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947
9.2.4 Seismic Design for Birmingham 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949
9.2.5 Seismic Design for New York City . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965
9.2.6 Birmingham 2 Seismic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977
9.2.7 Seismic Design for Los Angeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984
9.3.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996
9.3.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997
9.3.3 Seismic Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9100
9.3.4 Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9126
9.3.5 OutofPlane Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9131
9.3.6 Orthogonal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9132
9.3.7 Anchorage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9132
9.3.8 Diaphragm Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9132
10 WOOD DESIGN by Peter W. Somers, P.E. and Michael Valley, P.E. . . . . . . . . . . . . . . . . . . . . . . 101
10.1 Threestory Wood Apartment Building; Seattle, Washington . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.1.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.1.2 Basic Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.1.3 Seismic Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010
10.1.4 Basic Proportioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016
10.2 Warehouse with Masonry Walls and Wood Roof, Los Angeles, California . . . . . . . . . . . . . 1044
10.2.1 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044
10.2.2 Basic Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
10.2.3 Seismic Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046
10.2.4 Basic Proportioning of Diaphragm Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047
11 SEISMICALLY ISOLATED STRUCTURES by Charles A. Kircher, P.E., Ph.D. . . . . . . . . . . . . 111
11.1 Background and Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
11.1.1 Types of Isolation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
11.1.2 Definition of Elements of an Isolated Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
11.1.3 Design Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
11.1.4 Effective Stiffness and Effective Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11.2 Criteria Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11.3 Equivalent Lateral Force Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
11.3.1 Isolation System Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
11.3.2 Design Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111
11.4 Dynamic Lateral Response Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113
11.4.1 Minimum Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113
xi
11.4.2 Modeling Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114
11.4.3 Response Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116
11.4.4 Time History Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116
11.5 Emergency Operations Center Using Elastomeric Bearings, San Francisco, California . . . 1117
11.5.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117
11.5.2 Basic Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1120
11.5.3 Seismic Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127
11.5.4 Preliminary Design Based on the ELF Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129
11.5.5 Design Verification Using Nonlinear Time History Analysis . . . . . . . . . . . . . . . . . . 1141
11.5.6 Design and Testing Criteria for Isolator Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145
12 NONBUILDING STRUCTURE DESIGN by Harold O. Sprague Jr., P.E. . . . . . . . . . . . . . . . . . . 121
12.1 Nonbuilding Structures Versus Nonstructural Components . . . . . . . . . . . . . . . . . . . . . . . . . . 122
12.1.1 Nonbuilding Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
12.1.2 Nonstructural Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
12.2 Pipe Rack, Oxford, Mississippi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
12.2.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
12.2.2 Provisions Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
12.2.3 Design in the Transverse Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
12.2.4 Design in the Longitudinal Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
12.3 Steel Storage Rack, Oxford, Mississippi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1211
12.3.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1211
12.3.2 Provisions Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212
12.3.3 Design of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212
12.4 Electric Generating Power Plant, Merna, Wyoming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216
12.4.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216
12.4.2 Provisions Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217
12.4.3 Design in the NorthSouth Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218
12.4.4 Design in the EastWest Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219
12.5 Pier/Wharf Design, Long Beach, California . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219
12.5.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219
12.5.2 Provisions Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1220
12.5.3 Design of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221
12.6 Tanks and Vessels, Everett, Washington . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1222
12.6.1 FlatBottom Water Storage Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223
12.6.2 Flatbottom Gasoline Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226
12.7 Emergency Electric Power Substation Structure, Ashport, Tennessee . . . . . . . . . . . . . . . . . 1228
12.7.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229
12.7.2 Provisions Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1230
12.7.3 Design of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1230
13 DESIGN FOR NONSTRUCTURAL COMPONENTS by Robert Bachman, P.E., and
Richard Drake, P.E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
13.1 Development and Background of the Provisions for Nonstructural Components . . . . . . . . . . 132
13.1.1 Approach to Nonstructural Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
13.1.2 Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
13.1.3 Load Combinations and Acceptance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
13.1.4 Component Amplification Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
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13.1.5 Seismic Coefficient at Grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
13.1.6 Relative Location Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
13.1.7 Component Response Modification Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
13.1.8 Component Importance Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
13.1.9 Accommodation of Seismic Relative Displacements . . . . . . . . . . . . . . . . . . . . . . . . . 136
13.1.10 Component Anchorage Factors and Acceptance Criteria . . . . . . . . . . . . . . . . . . . . . 137
13.1.11 Construction Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
13.2 Architectural Concrete Wall Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
13.2.1 Example Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
13.2.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
13.2.3 Spandrel Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311
13.2.4 Column Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317
13.2.5 Additional Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318
13.3 HVAC Fan Unit Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319
13.3.1 Example Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319
13.3.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1320
13.3.3 Direct Attachment to Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1321
13.3.4 Support on Vibration Isolation Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324
13.3.5 Additional Considerations for Support on Vibration Isolators . . . . . . . . . . . . . . . . . 1329
13.4 Analysis of Piping Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1330
13.4.1 ASME Code Allowable Stress Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1330
13.4.2 Allowable Stress Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1332
13.4.3 Application of the Provisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333
Appendix A THE BUILDING SEISMIC SAFETY COUNCIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1
LIST OF CHARTS, FIGURES, AND TABLES
Figure 1.21 Earthquake ground acceleration in epicentral regions . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Figure 1.22 Holiday Inn ground and building roof motion during the M6.4 1971 San Fernando
earthquake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Figure 1.23 Response spectrum of NorthSouth ground acceleration recorded at the Holiday Inn,
approximately 5 miles from the causative fault in the 1971 San Fernando earthquake . 16
Figure 1.24 Averaged spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Figure 1.25 Force controlled resistance versus displacement controlled resistance . . . . . . . . . . . . . 19
Figure 1.26 Initial yield load and failure load for a ductile portal frame . . . . . . . . . . . . . . . . . . . . 110
Chart 2.1 Overall Summary of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Chart 2.2 Scope of Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Chart 2.3 Application to Existing Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chart 2.4 Basic Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Chart 2.5 Structural Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Chart 2.6 Equivalent Lateral Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chart 2.7 SoilStructure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Chart 2.8 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Chart 2.9 Response History Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Chart 2.10 Seismically Isolated Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
xiii
Chart 2.11 Strength Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Chart 2.12 Deformation Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Chart 2.13 Design and Detailing Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Chart 2.14 Steel Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Chart 2.15 Concrete Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Chart 2.16 Precast Concrete Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Chart 2.17 Composite Steel and Concrete Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Chart 2.18 Masonry Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Chart 2.19 Wood Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Chart 2.20 Nonbuilding Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Chart 2.21 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Chart 2.22 Architectural, Mechanical, and Electrical Components . . . . . . . . . . . . . . . . . . . . . . . . 224
Chart 2.23 Quality Assurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Table 21 Navigating Among the 2000 and 2003 NEHRP Recommended Provisions
and ASCE 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
Figure 3.11 Various floor plans of 12story Stockton building . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Figure 3.12 Sections through Stockton building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Figure 3.13 Threedimensional wireframe model of Stockton building . . . . . . . . . . . . . . . . . . . . . 36
Table 3.11 Area Masses on Floor Diaphragms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Table 3.12 Line Masses on Floor Diaphragms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 3.14 Key Diagram for Computation of Floor Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Table 3.13 Floor Mass, Mass Moment of Inertia, and Center Mass Locations . . . . . . . . . . . . . . . 310
Figure 3.15 Computed ELF total acceleration response system spectrum . . . . . . . . . . . . . . . . . . . 312
Figure 3.16 Computed ELF relative displacement response spectrum . . . . . . . . . . . . . . . . . . . . . . 313
Table 3.14 Equivalent Lateral Forces for Building Responses in X and Y Directions . . . . . . . . . 314
Figure 3.17 Amplification of accidental torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
Table 3.15 Computation for Torsional Irregularity with ELF Loads Acting in X Direction . . . . . 316
Table 3.16 Computation for Torsional Irregularity with ELF Loads Acting in Y Direction . . . . . 316
Table 3.17 ELF Drift for Building Responding in X Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Table 3.18 ELF Drift for Building Responding in Y Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
Table 3.19 Rayleigh Analysis for XDirection Period of Vibration . . . . . . . . . . . . . . . . . . . . . . . 319
Table 3.110 Rayleigh Analysis for YDirection Period of Vibration . . . . . . . . . . . . . . . . . . . . . . . 320
Table 3.111 Computation of Pdelta Effects for XDirection Response . . . . . . . . . . . . . . . . . . . . . 321
Figure 3.18 Basic load causes used in ELF analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
Table 3.112 Seismic and Gravity Load Combinations as Run on SAP 2000 . . . . . . . . . . . . . . . . . 325
Figure 3.19 Seismic shears in girders (kips) as computed using ELF analysis . . . . . . . . . . . . . . . . 326
Figure 3.110 Mode shapes as computed using SAP2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
Table 3.113 Computed Periods and Directions Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Table 3.114 Computed Periods and Effective Mass Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Table 3.115 Response Structure Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
Table 3.116 Story Shears from ModalResponseSpectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . 332
Table 3.117 Response Spectrum Drift for Building Responding in X Direction . . . . . . . . . . . . . . 333
Table 3.118 Spectrum Response Drift for Building Responding in Y Direction . . . . . . . . . . . . . . 333
Table 3.119 Computation of Pdelta Effects for XDirection Response . . . . . . . . . . . . . . . . . . . . . 334
Figure 3.112 Load combinations from responsespectrum analysis . . . . . . . . . . . . . . . . . . . . . . . . . 335
Figure 3.113 Seismic shears in girders (kips) as computed using responsespectrum analysis . . . . 336
Table 3.120 Seattle Ground Motion Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
xiv
Figure 3.114 Unscaled SRSS of spectra of ground motion pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
Figure 3.115 Ratio average scaled SRSS spectrum to Provisions spectrum . . . . . . . . . . . . . . . . . . . 339
Table 3.121 Result Maxima from TimeHistory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
Table 3.122 Result Maxima from TimeHistory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
Table 3.123 TimeHistory Drift for Building Responding in X Direction to Motion . . . . . . . . . . . 342
Table 3.124 Scaled Inertial Force and Story Shear Envelopes from Analysis A00X . . . . . . . . . . . 342
Table 3.125 Computation of Pdelta Effects for XDirection Response . . . . . . . . . . . . . . . . . . . . . 342
Figure 3.116 Combinations 1 and 2, beam shears (kips) as computed using timehistory analysis . 343
Figure 3.117 All combinations, beam shears (kips) as computed using time history analysis . . . . . 344
Table 3.126 Summary of Results from Various Methods of Analysis: Story Shear . . . . . . . . . . . . 345
Table 3.127 Summary of Results from Various Methods of Analysis: Story Drift . . . . . . . . . . . . . 346
Table 3.128 Summary of Results from Various Methods of Analysis: Beam Shear . . . . . . . . . . . . 347
Figure 3.21 Plan of structural system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
Figure 3.22 Elevation of structural system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Table 3.21 Member Sizes Used in NS Moment Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Table 3.22 Gravity Loads on Seattle Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Figure 3.23 Element loads used in analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
Table 3.23 Equivalent Lateral Forces for Seattle Building Responding in NS Directions . . . . . . 358
Figure 3.24 Simple wire frame model used for preliminary analysis . . . . . . . . . . . . . . . . . . . . . . . 360
Table 3.24 Results of Preliminary Analysis Using Pdelta Effects . . . . . . . . . . . . . . . . . . . . . . . . 362
Table 3.25 Results of Preliminary Analysis Including Pdelta Effects . . . . . . . . . . . . . . . . . . . . . 362
Table 3.26 Periods of Vibration From Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Figure 3.25 Demandtocapacity ratios for elements from analysis with Pdelta effects included . 363
Table 3.27 Lateral Strength on Basis of RigidPlastic Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 365
Figure 3.26 Plastic mechanism for computing lateral strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
Figure 3.27 Detailed analytical model of 6story frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
Figure 3.28 Model of girder and panel zone region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
Figure 3.29 A compound node and attached spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
Figure 3.210 Krawinkler beamcolumn joint model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Figure 3.211 Column flange component of panel zone resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Figure 3.212 Column web component of panel zone resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
Figure 3.213 Forcedeformation behavior of panel zone region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Figure 3.414 Transforming shear deformation to rotational deformation in the Krawinkler model . 374
Table 3.88 Properties for the Krawinkler BeamColumn Joint Model . . . . . . . . . . . . . . . . . . . . . 375
Figure 3.215 Assumed stressstrain curve for modeling girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
Figure 3.216 Moment curvature diagram for W27x94 girder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Figure 3.217 Developing momentdeflection diagrams for a typical girder . . . . . . . . . . . . . . . . . . . 380
Figure 3.218 Development of equations for deflection of momentdeflection curves . . . . . . . . . . . 382
Table 3.29 Girder Properties as Modeled in DRAIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
Figure 3.219 Momentdeflection curve for W27x94 girder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Figure 3.220 Development of plastic hinge properties for the W27x97 girder . . . . . . . . . . . . . . . . . 386
Figure 3.221 Yield surface used for modeling columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Table 3.210 Lateral Load Patterns Used in Nonlinear Static Pushover Analysis . . . . . . . . . . . . . . 389
Figure 3.222 Two base shear components of pushover response . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
Figure 3.223 Response of strong panel model to three load pattern, excluding Pdelta effects . . . . 391
Figure 3.224 Response of strong panel model to three loading patters, including Pdelta effects . . 392
Figure 3.225 Response of strong panel model to ML loads, with and without Pdelta effects . . . . . 392
Figure 3.226 Tangent stiffness history for structure under ML loads, with and without
xv
Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Figure 3.227 Patterns of plastic hinge formation: SP model under ML load, including
Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
Table 3.211 Strength Comparisons: Pushover vs. Rigid Plastic . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
Figure 3.228 Weak panel zone model under ML load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
Figure 3.229 Comparison of weak panel zone model with strong panel zone model,
excluding Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
Figure 3.230 Comparison of weak panel zone model with strong panel zone model,
including Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Figure 3.231 Tangent stiffness history for structure under ML loads, strong versus
weak panels, including Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Figure 3.232 Patterns of plastic hinge formation: weak panel zone model under ML load, including Pdelta
effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
Table 3.212 Modal Properties and Expected Inelastic Displacements for the
Strong and Weak Panel Models Subjected to the Modal Load Pattern . . . . . . . . . . . 3100
Figure 3.233 A simple capacity spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3102
Figure 3.234 A simple demand spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3102
Figure 3.235 Capacity and demand spectra plotted together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3103
Figure 3.236 Capacity spectrum showing control points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3104
Table 3.213 Damping Modification Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3105
Figure 3.237 Damping modification factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3106
Figure 3.238 Capacity spectrum used in iterative solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3107
Table 3.214 Results of Iteration for Maximum Expected Displacement . . . . . . . . . . . . . . . . . . . . 3109
Table 3.215 Points on Capacity Spectrum Corresponding to Chosen Damping Values . . . . . . . . 3110
Figure 3.239 Capacity spectrum with equivalent viscous damping points and
secant stiffnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111
Figure 3.240 Demand spectra for several equivalent viscous damping values . . . . . . . . . . . . . . . . 3112
Figure 3.241 Capacity and demand spectra on single plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113
Figure 3.242 Closeup view of portion of capacity and demand spectra . . . . . . . . . . . . . . . . . . . . 3114
Table 3.216 Summary of Results from Pushover Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115
Table 3.217 Structural Frequencies and Damping Factors Used in TimeHistory Analysis . . . . . 3117
Table 3.218 Seattle Ground Motion Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3118
Figure 3.243 Time histories and response spectra for Record A . . . . . . . . . . . . . . . . . . . . . . . . . . . 3119
Figure 3.244 Time histories and response for Record B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3120
Figure 3.245 Time histories and response spectra for Record C . . . . . . . . . . . . . . . . . . . . . . . . . . . 3121
Figure 3.246 Ground motion scaling parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3123
Table 3.219 Maximum Base Shear in Frame Analyzed with 5 Percent Damping,
Strong Panels, Excluding Pdelta Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3124
Table 3.220 Maximum Story Drifts from TimeHistory Analysis with 5 Percent Damping, Strong
Panels, Excluding Pdelta Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3124
Table 3.221 Maximum Base Shear in Frame analyzed with 5 Percent Damping,
Strong Panels, Excluding Pdelta Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3124
Table 3.222 Maximum Story Drifts from TimeHistory Analysis with 5 Percent Damping, Strong
Panels, Including Pdelta Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3125
Figure 3.247 Time history of roof and firststory displacement, ground Motion A00,
excluding Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3126
Figure 3.248 Time history of total base shear, Ground Motion A00, excluding Pdelta effects . . . 3126
Figure 3.249 Energy time history, Ground Motion A00, excluding Pdelta effects . . . . . . . . . . . . 3127
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Figure 3.250 Time history of roof and first story displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . 3127
Figure 3.251 Time history of total base shear, Ground Motion B00,
excluding Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3128
Figure 3.252 Energy time history, Ground Motion B00, excluding Pdelta effects . . . . . . . . . . . . 3128
Figure 3.253 Time history of roof and firststory displacement, Ground Motion C00,
excluding Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3129
Figure 3.254 Time history of total base shear, Ground Motion C00,
excluding Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3129
Figure 3.255 Energy time history, Ground Motion C00, excluding Pdelta effects . . . . . . . . . . . . 3130
Figure 3.256 Time history of roof and firststory displacement, Ground motion A00,
including Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3130
Figure 3.257 Time history of total base shear, Ground Motion A00,
including Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3131
Figure 3.258 Energy time history, Ground Motion A00, including Pdelta effects . . . . . . . . . . . . 3131
Figure 3.259 Time history of roof and firststory displacement, Ground Motion B00,
including Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3132
Figure 3.260 Time history of total base shear, Ground motion B00,
including Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3132
Figure 3.261 Energy time history, Ground Motion B00, including Pdelta effects . . . . . . . . . . . . 3133
Figure 3.262 Time history of roof and firststory displacement, Ground Motion C00,
including Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3133
Figure 3.263 Time history of total base shear, Ground Motion C00,
including Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3134
Figure 3.264 Energy time history, Ground Motion C00, including Pdelta effects . . . . . . . . . . . . 3134
Figure 3.265 Timehistory of roof displacement, Ground Motion A00,
with and without Pdelta effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3135
Figure 3.266 Time history of base shear, Ground Motion A00, 135
with and without Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3135
Figure 3.267 Yielding locations for structure with strong panels subjected to Ground Motion A00,
including Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3136
Table 3.223 Summary of All Analyses for Strong Panel Structure,
Including Pdelta Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3137
Figure 3.268 Comparison of inertial force patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3138
Table 3.224 Maximum Base Shear (kips) in Frame Analyzed Ground Motion A00,
Strong Panels, Including Pdelta Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3138
Table 3.225 Maximum Story Drifts (in.) from TimeHistory Analysis Ground Motion A00, Strong
Panels, Including Pdelta Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3139
Figure 3.269 Modeling a simple damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3140
Figure 3.270 Response of structure with discrete dampers and with
equivalent viscous damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3141
Figure 3.271 Base shear time histories obtained from column forces . . . . . . . . . . . . . . . . . . . . . . . 3141
Figure 3.272 Base shear time histories as obtained from inertial forces . . . . . . . . . . . . . . . . . . . . . 3142
Figure 3.273 Energy timehistory for structure with discrete added damping . . . . . . . . . . . . . . . . 3143
Figure 4.11 Typical framing plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Table 4.11 Geotechnical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 4.12 Critical sections for isolated footings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Figure 4.13 Soil pressure distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
xvii
Table 4.12 Footing Design for Gravity Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
Figure 4.14 Foundation plan for gravityloadresisting system . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
Figure 4.15 Framing plan for moment resisting frame system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
Table 4.13 Demands for MomentResisting Frame System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
Figure 4.16 Foundation plan for momentresisting frame system . . . . . . . . . . . . . . . . . . . . . . . . . . 416
Figure 4.17 Framing plan for concentrically braced frame system . . . . . . . . . . . . . . . . . . . . . . . . . 416
Figure 4.18 Soil Pressures for controlling bidirectional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
Figure 4.19 Foundation plan for concentrically braced frame system . . . . . . . . . . . . . . . . . . . . . . 419
Table 4.14 Mat Foundation Section Capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Figure 4.110 Envelope of mat foundation flexural demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
Figure 4.111 Mat foundation flexural reinforcement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
Figure 4.112 Section of mat foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Figure 4.113 Critical sections for shear and envelope of mat foundation shear demands . . . . . . . . 424
Table 4.15 Summary of Material Quantities and Cost Comparison . . . . . . . . . . . . . . . . . . . . . . . 425
Figure 4.21 Design condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Table 4.21 Geotechnical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
Table 4.22 Gravity and Seismic Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
Figure 4.22 Schematic model of deep foundation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Figure 4.23 Pile cap free body diagram curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Figure 4.24 Representative py curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Figure 4.25 Passive pressure mobilization curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
Figure 4.26 Calculated group effect factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
Figure 4.27 Results of pile analysis – sheer versus depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Figure 4.28 Results of pile analysis – moment versus depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Figure 4.29 Results of pile analysis – displacement versus depth . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Figure 4.210 Results of pile analysis – applied lateral load versus head moment . . . . . . . . . . . . . . 436
Figure 4.211 Results of pile – head displacement versus applied lateral load . . . . . . . . . . . . . . . . . 436
Figure 4.212 PM interaction diagram for Site Class C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
Figure 4.213 PM interaction diagram for Site Class E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Figure 4.214 Pile axial capacity as a function of length for Site Class C . . . . . . . . . . . . . . . . . . . . . 441
Figure 4.215 Pile axial capacity as a function of length for Site Class E . . . . . . . . . . . . . . . . . . . . . 442
Table 4.23 Pile Lengths Required for Axial Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
Table 4.24 Summary of Pile Size, Length, and Longitudinal Reinforcement . . . . . . . . . . . . . . . . 443
Figure 4.216 Pile detailing for Site Class C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
Figure 4.217 Pile detailing for Site Class E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Figure 4.218 Foundation tie section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
Figure 5.11 Framing elevation and sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 5.12 Roof framing and mezzanine framing plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Figure 5.13 Foundation plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Table 5.11 ELF Vertical Distribution for NS Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Table 5.12 ELF Analysis in NS Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
Figure 5.14 Design response spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
Table 5.13 Design Response Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
Table 5.14 Moments in Gable Frame Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Table 5.15 Axial Forces in Gable Frames Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Figure 5.15 Moment diagram for seismic load combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Figure 5.16 Gable frame schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
xviii
Table 5.16 Comparison of Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
Figure 5.17 Arrangement at knee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
Figure 5.18 Bolted stiffened connection at knee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
Figure 5.29 End plate connection at ridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
Figure 5.110 Mezzanine framing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
Figure 5.211 Shear force in roof deck diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Figure 5.21 Typical floor framing plan and building section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
Figure 5.22 Framing plan for special moment frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
Figure 5.23 Concentrically braced frame elevations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
Figure 5.24 Approximate effect of accidental of torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
Table 5.21 Alternative A, Moment Frame Seismic Forces and Moments by Level . . . . . . . . . . . 541
Table 5.22 Alternative B, Braced Frame Seismic Forces and Moments by Level . . . . . . . . . . . . 541
Table 5.23 Alternative C, Dual System Seismic Forces and Moments by Level . . . . . . . . . . . . . 542
Figure 5.25 SMRF frame in EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
Figure 5.26 SMRF frame in NS direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
Table 5.24 Alternative A (Moment Frame) Story Drifts under Seismic Loads . . . . . . . . . . . . . . . 545
Table 5.25 Alternative A Torsional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
Figure 5.27 Projection of expected moment strength of beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
Figure 5.28 Story height and clear height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
Figure 5.29 Free body diagram bounded by plastic hinges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
Table 5.24 ColumnBeam Moment Ratios for SevenBay Frame . . . . . . . . . . . . . . . . . . . . . . . . . 549
Table 5.25 ColumnBeam Moment Ratios for FiveBay Frame . . . . . . . . . . . . . . . . . . . . . . . . . . 549
Figure 5.210 Illustration of AISC Seismic vs. FEMA 350 Methods for panel zone shear . . . . . . . . 550
Figure 5.211 Column shears for EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
Figure 5.212 Column shears for NS direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
Figure 5.213 Forces at beam/column connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
Figure 5.214 WUFW connection, Second level, NSdirection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
Figure 5.215 WUFW weld detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
Figure 5.216 Braced frame in EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
Figure 5.217 Braced frame in NS direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
Table 5.26 Alternative B Amplification of Accidental Torison . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
Table 5.27 Alternative B Story Drifts under Seismic Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
Figure 5.218 Lateral force component in braces for NS direction . . . . . . . . . . . . . . . . . . . . . . . . . . 563
Figure 5.219 Bracing connection detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
Figure 5.220 Whitmore section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
Figure 5.221 Bracetobrace connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
Figure 5.222 Moment frame of dual system in EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
Figure 5.223 Moment frame of dual system in NS direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
Table 5.28 Alternative C Amplification of Accidental Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
Figure 5.224 Braced frame of dual system in EWdirection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
Figure 5.225 Braced frame of dual system in NS direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
Table 5.29 Alternative C Story Drifts under Seismic Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Figure 5.31 Main floor framing plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
Figure 5.32 Section on Grid F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
Table 5.31 Summary of Critical Member Design Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
Figure 5.33 Diagram of eccentric braced bream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
Figure 5.34 Typical eccentric braced frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
Figure 5.35 Link and upper brace connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
xix
Figure 5.36 Lower brace connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
Figure 61 Typical floor plan of the Berkeley building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Figure 62 typical elevations of the Berkeley building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Table 61 Response Modification, Overstrength, and Deflection Amplification
Coefficients for Structural Systems Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Table 62 Story Weights, Masses, and Moments of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
Table 63 Periods and Modal Response Characteristics for the Berkeley building . . . . . . . . . . . 613
Table 64 Periods and Modal Response for the Honolulu Building . . . . . . . . . . . . . . . . . . . . . . 613
Table 65 Comparison of Approximate and “Exact” Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
Table 66 Comparison of Periods, Seismic Shears Coefficients, and Base Shears for the Berkeley
and Honolulu buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
Table 67a Vertical Distribution of NS Seismic Forces for the Berkeley Building . . . . . . . . . . . 616
Table 67b Vertical Distribution of EW Seismic Forces for the Berkeley Building . . . . . . . . . . 616
Table 68 Vertical Distribution of NS and EW Seismic Forces
for the Honolulu Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
Figure 63 Comparison of wind and seismic story shears for the Berkeley building . . . . . . . . . . 618
Figure 64 Comparison of wind and seismic story shears for the Honolulu building . . . . . . . . . . 619
Figure 65 Drift profile for Berkeley building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
Table 69a Drift Computations for the Berkeley Building Loaded in the NS Direction . . . . . . . 621
Table 69b Drift Computations for the Berkeley Building Loaded in the EW Direction . . . . . . . 621
Table 610a Pdelta Computations for the Berkeley Building Loaded in the NS Direction . . . . . . 623
Table 610b Pdelta Computations for the Berkeley Building Loaded in the EW Direction . . . . . 623
Figure 66 Drift profile for the Honolulu building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
Table 611a Drift Computations for the Honolulu Building Loaded in the NS Direction . . . . . . . 626
Table 611b Drift Computations for the Honolulu Building Loaded in the EW Direction . . . . . . 626
Table 612a Pdelta Computations for the Honolulu Building Loaded in the NS Direction . . . . . 627
Table 612b Pdelta Computations for the Honolulu Building Loaded in the EW Direction . . . . . 628
Figure 67 Story forces in the EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
Figure 68 Story shears in the EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
Figure 69 Story overturning moments in the EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
Figure 610 25 percent story shears, Frame 1 EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634
Figure 611 25 percent story shears, Frame 2 EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
Figure 612 25 percent story shear, Frame 3 EW direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
Figure 613 Layout for beam reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
Table 613 Tension Development Length Requirements for Hooked Bars and
Straight Bars in 4,000 psi LW Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
Figure 614 Bending moments Frame 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
Figure 615 Preliminary rebar layout for Frame 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
Table 614 Design and Maximum Probable Flexural Strength for Beams in Frame 1 . . . . . . . . . 643
Figure 616 Diagram for computing column shears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
Figure 617 Computing joint shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Figure 618 Loading for determination of rebar cutoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
Figure 619 Free body diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
Figure 620 Development length for top bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
Figure 621 Final bar arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
Table 615 Design and Maximum Probable Flexural Strength For Beams in Frame 1 . . . . . . . . . 650
Figure 622 Shears forces for transverse reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
xx
Figure 623 Detailed shear force envelope in Span BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
Figure 624 Layout and loads on column of Frame A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
Figure 625 Design interaction diagram for column on Gridline A . . . . . . . . . . . . . . . . . . . . . . . . 656
Figure 626 Details of reinforcement for column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
Figure 627 Design forces and detailing of haunched girder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
Figure 628 Computing shear in haunched girder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
Table 616 Design of Shear Reinforcement for Haunched Girder . . . . . . . . . . . . . . . . . . . . . . . . . 664
Figure 629 Computation of column shears for use in joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665
Figure 630 Computing joint shear force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
Figure 631 Column loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
Figure 632 Interaction diagram and column design forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
Figure 633 Column detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
Table 617 Design Forces for Structural Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670
Table 618 Design of Structural Wall for Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671
Figure 634 Interaction diagram for structural wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673
Figure 635 Variation of neutral axis depth with compressive force . . . . . . . . . . . . . . . . . . . . . . . . 674
Figure 636 Details of structural wall boundary element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
Figure 637 Overall details of structural wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
Figure 638 With loading requirements from ASCE 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
Figure 639 Wind vs. seismic shears in exterior bay of Frame 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 681
Figure 640 Bending moment envelopes at Level 5 of Frame 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
Figure 641 Preliminary reinforcement layout for Level 5 of Frame 1 . . . . . . . . . . . . . . . . . . . . . . 683
Figure 642 Shear strength envelopes for Span AB of Frame 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
Figure 643 Isolated view of Column A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
Figure 644 Interaction diagram for column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
Figure 645 Column reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
Figure 646 Loads, moments, and reinforcement for haunched girder . . . . . . . . . . . . . . . . . . . . . . 692
Figure 647 Shear force envelope for haunched girder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
Figure 648 Loading for Column A, Frame 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
Figure 649 Interaction diagram for Column A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698
Figure 650 Details for Column A, Frame 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
Table 7.11 Design Parameters from Example 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Table 7.22 Shear Wall Overstrength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Table 7.13 Birmingham 1 Fpx Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Figure 7.11 Diagram force distribution and analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
Figure 7.12 Diagram plan and critical design regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712
Figure 7.13 Joint 3 chord reinforcement at the exterior edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
Figure 7.14 Interior joint reinforcement at the ends of plank and the collector
reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716
Figure 7.15 Anchorage region of shear reinforcement and collector reinforcement . . . . . . . . . . . . 717
Figure 7.16 Joint 2 transverse wall joint reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
Figure 7.17 Joint 4 exterior longitudinal walls to diaphragm reinforcement and outofplane anchor7ag2e0
Figure 7.18 Walltodiaphragm reinforcement along interior longitudinal walls . . . . . . . . . . . . . . 721
Table 7.14 Design Parameters form Sec. 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
Table 7.15 Fpx Calculations from Sec. 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723
Figure 7.19 Diaphragm plan and section cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
Figure 7.110 Boundary member and chord and collector reinforcement . . . . . . . . . . . . . . . . . . . . . 726
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Figure 7.111 Collector reinforcement at the end of the interior longitudinal walls . . . . . . . . . . . . . 726
Figure 7.112 Walltowall diaphragm reinforcement along interior longitudinal walls . . . . . . . . . . 727
Figure 7.113 Exterior longitudinal walltodiaphragm reinforcement and outofplane
anchorage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728
Figure 7.21 Threestory building plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730
Table 7.21 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
Figure 7.22 Forces on the longitudinal walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735
Figure 7.23 Forces on the transverse walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736
Figure 7.24 Freebody diaphragm for longitudinal walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737
Figure 7.25 Freebody diaphragm of the transverse walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
Figure 7.26 Overturning connection detail at the base of the walls . . . . . . . . . . . . . . . . . . . . . . . . 741
Figure 7.27 Shear connection at base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
Figure 7.28 Shear connections on each side of the wall at the second and third floors . . . . . . . . . 746
Figure 7.31 Singlestory industrial warehouse building plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749
Table 7.31 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750
Figure 7.32 Freebody diagram of a panel in the longitudinal direction . . . . . . . . . . . . . . . . . . . . . 754
Figure 7.33 Freebody diagram of a panel in the transverse direction . . . . . . . . . . . . . . . . . . . . . . 755
Figure 7.34 Cross section of the DT drypacked at the footing . . . . . . . . . . . . . . . . . . . . . . . . . . . 757
Figure 7.35 Cross section of one DT leg showing the location of the bonded prestressing tendons or
strand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
Figure 7.36 Freebody of the angle and the fillet weld connecting the embedded
plates in the DT and the footing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760
Figure 7.37 Freebody of angle with welds, top view, showing only shear forces and
resisting moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762
Figure 7.38 Section at the connection of the precast/prestressed shear wall panel and the footing 764
Figure 7.39 Details of the embedded plate in the DT at the base . . . . . . . . . . . . . . . . . . . . . . . . . . 766
Figure 7.311 Sketch of connection of loadbearing DT wall panel at the roof . . . . . . . . . . . . . . . . 766
Figure 7.310 Sketch of connection of nonloadbearing DT wall panel at the roof . . . . . . . . . . . . . 767
Figure 81 Typical floor plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Figure 82 Building and elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Figure 83 Building side elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Figure 84 Typical composite connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Table 81 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Figure 85 M. Curve for W18x35 connection with 6#5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810
Figure 86 M. Curve for W21x44 connection with 8#5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811
Table 82 Partially Restrained Composite Connection Design . . . . . . . . . . . . . . . . . . . . . . . . . . 812
Figure 87 Analysis of seatangle for tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815
Figure 88 Moment diagram for typical beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818
Figure 810 elevation of typical connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819
Figure 811 Illustration of input for load combination for 1.2D + 0.5L + 1.0QE + 0.2SDSD . . . . . 820
Figure 812 Illustration of input for load combination for 0.9D + 1.0QE 0.2DSD . . . . . . . . . . . . . 821
Table 83 Story Drift (in.) and Pdelta Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822
Table 84 Maximum Connection Moments and Capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822
Table 85 Column Strength Check, for W10x77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
Figure 813 Detail at column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825
Figure 814 Detail at spandrel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826
Figure 815 Detail at building corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826
xxii
Figure 816 Force transfer from deck to column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827
Figure 9.11 Roof plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Figure 9.12 End wall elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Table 9.11 Comparison of Em . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911
Figure 9.13 Trial design for 8in.thick CMU wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912
Figure 9.14 Investigation ofoutplane ductility for the 8in.thick CMU side wall . . . . . . . . . . . . 914
Table 9.12 Comparison of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915
Figure 9.16 Basis for interpolation of modulus of rupture, fx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917
Figure 9.17 Cracked moment of inertia (Icr) for 8in.thick CMU side walls . . . . . . . . . . . . . . . . . 917
Figure 9.18 Outofplane strength for 8in.thick CMU walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 920
Figure 9.19 Inplane ductility check for side walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922
Figure 9.110 Grout cells solid within 10 ft of each end of side walls . . . . . . . . . . . . . . . . . . . . . . . . 923
Table 9.13 Combined Loads for Shear in Side Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925
Figure 9.111 Trial design for piers on end walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926
Figure 9.112 Inplane loads on end walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926
Figure 9.113 Outofplane load diagram and resultant of lateral loads . . . . . . . . . . . . . . . . . . . . . . 927
Figure 9.114 Investigation for outofplane ductility for end walls . . . . . . . . . . . . . . . . . . . . . . . . . 928
Figure 9.115 Cracked moment of inertia for end walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929
Figure 9.116 Outofplane seismic strength of pier on end wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931
Figure 9.117 Input loads for inplane and end wall analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932
Figure 9.118 Inplane design condition for 8ft.wide pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932
Figure 9.119 Inplane ductility check for 8ft.wide pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933
Figure 9.120 Inplane seismic strength of pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935
Figure 9.121 Inplane FP11 – FM11 diagram for pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937
Table 9.14 Combined Loads for Flexure in End Pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937
Figure 9.122 Inplane shear on end wall and pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938
Table 9.15 Combined Loads for Shear in End Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940
Figure 9.123 Inplane deflection of end wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941
Figure 9.21 Typical floor plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
Figure 9.22 Building elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943
Figure 9.23 Plan of walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944
Table 9.21 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945
Table 9.22 Birmingham 1 Seismic Forces and moments by Level . . . . . . . . . . . . . . . . . . . . . . . . 951
Figure 9.24 Location of moments due to story shears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951
Table 9.23 Shear Strength Calculations for Birmingham 1 Wall D . . . . . . . . . . . . . . . . . . . . . . . . 954
Table 9.24 Demands for Birmingham 1 Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954
Figure 9.25 Strength of Birmingham 1 Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957
Figure 9.26 FP11 – FM11 diagram for Birmingham 1 Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960
Figure 9.27 Ductility check for Birmingham 1 Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961
Table 9.25 Birmingham 1 Cracked Wall Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963
Figure 9.28 Shear wall deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963
Table 9.26 Deflections, Birmingham 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964
Table 9.27 New York City Seismic Forces and Moments by Level . . . . . . . . . . . . . . . . . . . . . . . 966
Table 9.28 New York City Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
Table 9.29 Demands for New York City Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
Figure 9.29 Strength of New York City and Birmingham 2 Wall D . . . . . . . . . . . . . . . . . . . . . . . . 970
Figure 9.210 FP11 – FM11 Diagram for New York City and Birmingham 2 Wall D . . . . . . . . . . . 972
xxiii
Figure 9.211 Ductility check for New York City and Birmingham 2 Wall D . . . . . . . . . . . . . . . . . 974
Table 9.210 New York City Cracked Wall Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975
Table 9.211 New York City Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976
Table 9.212 Birmington 2 Periods, Mass Participation Factors, and Modal Base Shears . . . . . . . . 979
Table 9.213 Birmingham 2 Seismic Forces and Moments by Level . . . . . . . . . . . . . . . . . . . . . . . . 980
Table 9.214 Birmingham Periods, Mass Participation Factors, and Modal Base . . . . . . . . . . . . . . 981
Table 9.215 Shear Strength Calculations for Wall D, Birmingham 2 . . . . . . . . . . . . . . . . . . . . . . . 982
Table 9.216 Birmingham 2 Demands for Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982
Figure 9.212 Typical wall section fro the Los Angeles location . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985
Table 9.217 Los Angeles Seismic Forces and Moments by Level . . . . . . . . . . . . . . . . . . . . . . . . . 986
Table 9.218 Los Angeles Shear Strength Calculations for Wall D . . . . . . . . . . . . . . . . . . . . . . . . . 987
Table 9.219 Los Angeles Load Combinations for Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988
Figure 9.213 Los Angeles: Strength of wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989
Figure 9.214 FP11 – FM11 diagram for Los Angeles Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991
Figure 9.215 Ductility check for Los Angeles Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992
Table 9.220 Los Angeles Cracked Wall Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993
Table 9.221 Los Angeles Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994
Table 9.222 Variation in Reinforcement and Grout by Location . . . . . . . . . . . . . . . . . . . . . . . . . . 995
Figure 9.31 Floor plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996
Figure 9.32 Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997
Table 9.31 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998
Table 9.32 Periods, Mass Particiaptions Ratios, and Modal Base Shears . . . . . . . . . . . . . . . . . . 9103
Table 9.33 Seismic Forces and Moments by Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9104
Table 9.34 Relative Rigidities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9106
Figure 9.33 Wall dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9107
Table 9.35 Shear for Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9108
Table 9.36 Periods, Mass Participations Ratios, and Modal Base Shears . . . . . . . . . . . . . . . . . . 9109
Table 9.37 Load Combinations for Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110
Figure 9.34 Bulb reinforcement at lower levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9111
Figure 9.35 Strength of Wall D, Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9112
Figure 9.36 FP11 – FM11 Diagram for Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115
Figure 9.37 Ductility check for Wall D, Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9116
Figure 9.38 Bulb reinforcement at upper levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9118
Figure 9.39 Strength of Wall D at Level 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119
Figure 9.310 FP11 – FM11 Diagram for Level 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9122
Figure 9.311 Ductility check for Wall D, Level 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9123
Table 9.38 Shear Strength for Wall D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9125
Table 9.39 Cracked Wall Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127
Table 9.310 Deflection for ELF Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9129
Table 9.311 Displacements from Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9130
Figure 9.312 Floor anchorage to wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9132
Table 9.312 Diaphragm Seismic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9132
Figure 9.313 Shears and moments for diaphragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9134
Figure 10.11 Typical floor plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Figure 10.12 Longitudinal section and elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Figure 10.13 Foundation plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Figure 10.14 Load path and shear walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
xxiv
Figure 10.15 Vertical shear distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011
Table 10.11 Seismic Coefficients, Forces, and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012
Figure 10.16 Transverse section: end wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018
Table 10.12 Fastener slip equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1020
Table 10.13 Wall Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021
Figure 10.17 Force distribution for flexural deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021
Table 10.14 Wall Deflection (per story) Due to Bending and Anchorage Slip . . . . . . . . . . . . . . . 1022
Table 10.15 Total Elastic Deflection and Drift of End Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022
Figure 10.18 Shear wall tie down at suspended floor framing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025
Figure 10.19 Transverse wall: overturning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028
Figure10.110 Bearing wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029
Figure10.111 Nonbearing wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029
Figure10.112 Foundation wall detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1030
Figure10.113 Diaphragm chord splice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032
Table 10.16a Total Deflection and Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036
Table 10.16b Total Deflection and Drift (Structural I Plywood Shear Walls) . . . . . . . . . . . . . . . . 1036
Table 10.17 Pdelta Stability Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037
Figure10.114 Perforated shear wall at exterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039
Figure10.115 Perforated shear wall detail at foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043
Figure10.116 Perforated shear wall detail at floor framing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043
Figure 10.21 Building plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044
Table 10.21 Roof Diaphragm Framing and Nailing Requirements . . . . . . . . . . . . . . . . . . . . . . . . 1049
Figure 10.22 Diaphragm framing and nailing layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
Figure 10.23 Plywood layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050
Figure 10.24 Chord splice detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051
Figure 10.25 Adjustment for nonuniform nail spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052
Figure 10.26 Diaphragm at roof opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054
Figure 10.27 Chord forces and Element 1 freebody diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055
Figure 10.28 Freebody diagram for Element 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056
Figure 10.29 Anchorage of masonry wall perpendicular to joists . . . . . . . . . . . . . . . . . . . . . . . . . . 1058
Figure10.210 Chord tie at roof opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059
Figure10.211 Cross tie plan layout and subdiaphragm freebody diagram for side walls . . . . . . . . 1060
Figure10.212 Anchorage of masonry wall parallel to joists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
Figure10.213 Cross tie plan layout and subdiaphragm freebody diagram for end walls . . . . . . . . 1062
Figure 11.11 Isolation system terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Figure 11.12 Effective stiffness and effective damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Table 11.21 Acceptable Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Figure 11.31 Isolation system capacity and earthquake demand . . . . . . . . . . . . . . . . . . . . . . . . . . 1110
Figure 11.32 Design, maximum, and total maximum displacement . . . . . . . . . . . . . . . . . . . . . . . . 1111
Figure 11.33 Isolation system displacement and shear force as function of period . . . . . . . . . . . . 1113
Table 11.41 Summary of Minimum Design Criteria for Dynamic Analysis . . . . . . . . . . . . . . . . . 1114
Figure 11.41 Moments due to horizontal shear and Pdelta effects . . . . . . . . . . . . . . . . . . . . . . . . 1115
Figure 11.42 Bilinear idealization of isolator unit behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116
Figure 11.51 Threedimensional model of the structural system . . . . . . . . . . . . . . . . . . . . . . . . . . 1117
Figure 11.52 Typical floor framing plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118
Figure 11.53 Penthouse roof framing plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118
Figure 11.54 Longitudinal bracing elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119
xxv
Figure 11.55 Transverse bracing elevations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119
Table 11.51 Gravity Loads on Isolator Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121
Figure 11.56 Example design spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123
Table 11.52 Earthquake Time History Records and Scaling Factors . . . . . . . . . . . . . . . . . . . . . . 1124
Figures11.57 Comparison of design earthquake spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125
Table 11.53 Vertical Distribution of Reduced Design Earthquake Forces . . . . . . . . . . . . . . . . . . 1132
Table 11.54 Vertical Distribution of Unreduced DE and MCE Forces . . . . . . . . . . . . . . . . . . . . . 1132
Table 11.55 Maximum Downward Force for Isolator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133
Table 11.56 Minimum Downward Force for Isolator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133
Table 11.57 Maximum Downward Force on Isolator Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134
Table 11.58 Maximum Uplift Displacement of Isolator Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134
Figure 11.58 Elevation of framing on Column Line 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135
Figure 11.59 Elevation of framing on Column Line B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136
Figure11.510 First floor framing plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136
Table 11.59 Summary of Key Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138
Figure11.511 Typical detail of the isolation system at columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139
Figure11.512 Stiffness and damping properties of EOC isolator units . . . . . . . . . . . . . . . . . . . . . . 1142
Figure11.513 Comparison of modeled isolator properties to test data . . . . . . . . . . . . . . . . . . . . . . 1143
Table 11.510 Design Earthquake Response Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144
Table 11.511 Maximum Considered Earthquake Response Parameters . . . . . . . . . . . . . . . . . . . . . 1144
Table 11.512 Maximum Downward Force (kips) on Isolator Units . . . . . . . . . . . . . . . . . . . . . . . . 1145
Table 11.513 Maximum Uplift Displacement (in.) Of Isolator Units . . . . . . . . . . . . . . . . . . . . . . . 1145
Figure11.514 Isolator dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146
Table 11.514 Prototype Test Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147
Table 121 Applicability of the Chapters of the Provisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Figure 121 Combustion turbine building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Figure 122 Pipe rack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Figure 123 Steel storage rack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1211
Table 12.31 Seismic Forces, Shears and Overturning Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 1214
Figure 124 Boiler building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216
Figure 125 Pier plan and elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1220
Figure 126 Storage tank section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223
Figure 127 Platform for elevated transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229
Figure 13.21 Fivestory building evaluation showing panel location . . . . . . . . . . . . . . . . . . . . . . . 138
Figure 13.22 Detailed building elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Figure 13.23 Spandrel panel connection payout from interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311
Figure 13.24 Spandrel panel moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314
Figure 13.25 Spandrel panel connections forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316
Figure 13.26 Column cover connection layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317
Figure 13.31 Air handling fan unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1320
Figure 13.32 Freebody diagram for seismic force analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1322
Figure 13.33 Anchor for direct attachment to structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323
Figure 13.34 ASHRAE diagonal seismic force analysis for vibration isolation springs . . . . . . . . . 1325
Figure 13.35 Anchor and snubber loads for support on vibration isolation springs . . . . . . . . . . . . 1327
Figure 13.36 Lateral restraint required to resist seismic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329
11
1
FUNDAMENTALS
James Robert Harris, P.E., Ph.D.
In introducing their wellknown 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 earthquakeresistant 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 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 Provisions for Seismic Regulations for New Buildings and Other Structures are
mentioned and reference is made to particularly relevant portions of the document. Note that through
2000, the NEHRP Recommended Provisions has been composed of two volumes of text and a separate set
of maps. Part 1 (referred to herein as the Provisions) contains the actual requirements and Part 2 (referred
to herein as the Commentary) provides a discussion of various aspects of the requirements.
Although the set of design examples is based on the 2000 Provisions, it is annotated to reflect changes
made to the 2003 Provisions. Annotations within brackets, [ ], indicate both organizational changes (as a
result of a reformat of all of the chapters of the 2003 Provisions) and substantive technical changes to the
2003 Provisions and its primary reference documents. While the general concepts of the changes are
FEMA 451, NEHRP Recommended Provisions: Design Examples
12
described, the design examples and calculations in this book have not been revised to reflect the changes
to the 2003 Provisions. Where related to the discussion in this chapter, significant changes to the 2003
Provisions and primary reference documents are noted. However, some minor changes to the 2003
Provisions and the reference documents may not be noted.
1.1 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.
The Commentary includes a method for prediction of susceptibility to liquefaction as well as general
guidelines for locating potential fault rupture zones.
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 uses two sets of maps define the hazard of seismic ground
shaking – one is pertinent for higher frequency motion (the SS maps) and the other for lower frequencies
(the S1 maps). There is evidence that extreme motions near the fault in certain types of large earthquakes
Chapter 1, Fundamentals
13
are not captured by the maps, but interim adjustments to design requirements for such a possibility are
included in the Provisions.
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 are 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 hazard maps. Ground shaking, however, is known to vary considerably
over small distances and the Provisions maps do not attempt to capture all such local variations
(commonly called microzoning).
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. In particular, note the
description of the newest generation of maps introduced in 1997 and their close relationship to the
development of a new design criterion. There are extended discussions of these issues in the appendices
to the Commentary. 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 50year reference period. In the long term, the change from life safety
to structural collapse prevention as the limit state will create significant changes in procedures for design
analysis. In the present interim, the ground motions for use with present design procedures are simply
taken as being twothirds of the MCE ground motions.
1.2 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.
1.2.1 Response Spectra
Figure 1.21 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 are analyzed for specific response to a specific ground
motion.
FEMA 451, NEHRP Recommended Provisions: Design Examples
14
El Centro 1940
Imperial 6 (Hudson) 1979
Landers
(Joshua Tree) 1992
Kern Taft 1952
Kobe 1995
Loma Prieta
(Oakland Wharf) 1989
Mexico City 1985
Morgan Hill (Gilroy) 1984
North Palm Springs 1986
Northridge
(Sylmar 90°) 1994
San Fernando
(Pacoima Dam) 1971
San Fernando (Orion
Blvd.) 1971
Northridge
(Sylmar 360°) 1994
Tabas 1978
Figure 1.21 Earthquake ground acceleration in epicentral regions (all accelerograms are plotted to the
same scale for time and acceleration). Great earthquakes extend for much longer periods of time.
Figure 1.22 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
Chapter 1, Fundamentals
15
20
0
20
30
0
30
250
0
250
Acceleration,
cm\s\s
Velocity,
cm\s
Displacement,
cm
0 10 20 30 40
Time, s
(a) Ground acceleration, velocity, and displacement
0
30
0
10 20 30 40
30
70
0
70
500
0
500
Displacement,
cm
Velocity,
cm\s
Acceleration,
cm\s\s
Time, s
(b) Roof acceleration, velocity, and displacement
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 1.22(b) (the
vertical scales are different). 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 1.22 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). Note that the vertical scale of (b) is
different from (a). 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 predicted 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 1.22). The underlying theory is based entirely on the response of a singledegreeoffreedom
oscillator such as a simple onestory frame with the mass concentrated at the roof. The vibrational
characteristics of such a simple oscillator may be reduced to two: the natural frequency and the amount
of damping. By recalculating the record of response versus time to a specific ground motion for a wide
range of natural frequencies 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 frequency and damping.
Figure 1.23 shows such a result for the ground motion of Figure 1.22(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. Different earthquake ground motions
lead to response spectra with peaks and valleys at different points with respect to the natural frequency.
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.
FEMA 451, NEHRP Recommended Provisions: Design Examples
16
Figure 1.23 Response spectrum of northsouth ground acceleration (0, 0.02, 0.05, 0.10, 0.20 of
critical damping) recorded at the Holiday Inn, approximately 5 miles from the causative fault in
the 1971 San Fernando earthquake (Housner and Jennings 1982).
Figure 1.24 is an example of an averaged spectrum. Note that the horizontal axes of Figures 1.23 and
1.24 are different, one being for the known frequency (period) while the other is for the cyclic frequency.
Cyclic frequency is the inverse of period; therefore, Figure 1.24 should be rotated about the line f = 1 to
compare it with Figure 1.23. Note that acceleration, velocity, or displacement may be obtained from
Figure 1.23 or 1.24 for a structure with known frequency (period) and damping.
Chapter 1, Fundamentals
17
Figure 1.24 Averaged spectrum (Newmark, Blume, and Kapur 1973). Mean and mean plus one
standard deviation acceleration, horizontal components (2.0 percent of critical damping).
Reprinted with permission from the American Society of Civil Engineers.
Prior to the 1997 editions of the Provisions, the maps that characterized the ground shaking hazard were
plotted in terms of peak ground acceleration, 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 MCE ground motions, 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. 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 singledegreeoffreedom 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 (higher frequencies) to the total response is relatively minor.
FEMA 451, NEHRP Recommended Provisions: Design Examples
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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 MCE ground motion maps 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 contains a thorough explanation of this feature.
1.2.2 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 1.23 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 are resisting 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 some damage to the structure is accepted. Figure 1.25 illustrates
the large amount of strain energy that may be stored by a ductile system in a displacementcontrolled
event such as an earthquake. The 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 gravity weight or wind pressure, while part (c) is characteristic of induced
displacements such as foundation settlement or earthquake ground shaking. The figures should not be
interpreted as a horizontal beam and a vertical column. Figure 1.25(a) would represent a beam if the
load W were small and a column if W were large. 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 low natural frequencies, 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 high natural frequencies, 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 frequencies (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.
Chapter 1, Fundamentals
19
H
H H
Force
control
Displacement
control
HY HU . Y . U
.
.
H /H y 1
(b)
U Y >> 1
(c)
. U /. Y
.
(a)
W
HY
HU
. Y
. U
Figure 1.25 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. In part (c) the displacement is the independent variable. As the displacement is increased, the
base moment (FR) increases until the yield point is reached. As the displacement increases still more, the
base moment increases only a small amount. For a 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 1.26 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, 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.
FEMA 451, NEHRP Recommended Provisions: Design Examples
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2 1
4 3
5 10 10 5
H
d
(a) Structures (b) H  d curve
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
20
40
60
80
100
120
140
160
HY
HU
Figure 1.26 Initial yield load and failure load 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
frequency, 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 O0 – 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. O0 is
intended to deliver a reasonably high estimate of the peak force that would develop in the structure. Sets
of R, Cd, and O0 are specified in the Provisions for the most common structural materials and systems.
1.2.3 Building Materials
The following brief comments about building materials and systems are included as general guidelines
only, not for specific application.
1.2.3.1 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 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
Chapter 1, Fundamentals
111
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.
1.2.3.2 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. Other 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 demonstration has greatly increased knowledge of how to avoid
low ductility details in steel construction.
1.2.3.3 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 for design of structural concrete explain how controlling 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 lead to larger reductions
from the elastic response.
1.2.3.4 Masonry
Masonry is a more diverse material than those mentioned above, but 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 the 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 and the masonry unit create additional failure phenomena. Thus, the
reduction factors for 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.
1.2.3.5 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
FEMA 451, NEHRP Recommended Provisions: Design Examples
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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, for seismic design of precast structures.
1.2.3.6 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.
1.2.4 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 with small reduction factors.
Connection details often make development of ductility difficult in braced frames, and buckling of
compression members also limits their inelastic response. Eccentrically braced steel frames and new
proportioning and detailing rules for concentrically braced frames have been developed to overcome these
shortcomings. [The 2003 Provisions include proportioning and detailing rules for bucklingrestrained
braced frames. This new system has the advantages of a special steel concentrically braced frame, but
with performance that is superior as brace buckling is prevented. Design provisions appear in 2003
Provisions Sec. 8.6.] Walls that are not load bearing 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). 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
redundance has been introduced in recent editions of the Provisions in an attempt to prevent use of large
reductions from elastic response in structures that actually possess very little redundancy. As with many
new empirical measures, it is not universally accepted and is likely to change in the future. [In the 2003
Provisions, a radical change was made to the requirements related to redundancy. Only two values of the
redundancy factor, ., are defined: 1.0 and 1.3. Assignment of a value for . is based on explicit
consideration of the consequence of failure of a single element of the seismicforceresisting system. A
simple, deemedtocomply exception is provided for certain structures.]
1.3 ENGINEERING PHILOSOPHY
Chapter 1, Fundamentals
113
The Provisions, under “Purpose,” states:
The design earthquake ground motion levels specified herein could result in both structural
and nonstructural damage. For most structures designed and constructed according to the
Provisions, structural damage from the design earthquake ground motion would be repairable
although perhaps not economically so. For essential facilities, it is expected that the damage
from the design earthquake ground motion would not be so severe as to preclude continued
occupancy and function of the facility. . . . For ground motions larger than the design levels,
the intent of the Provisions is that there be low likelihood of structural collapse.
The two points to be emphasized are that damage is to be expected when an earthquake (equivalent to the
design earthquake) occurs and that the probability of collapse is not zero. The design earthquake ground
motion level mentioned is twothirds of the MCE ground motion.
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
strengths greater than those required by the O0 amplifier on 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 as 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, 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 Seismic
Use Group (SUG). A combined classification called the Seismic Design Category (SDC) incorporates
both the seismic hazard and the SUG. 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.
1.4 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 (often referred to as a timehistory or responsehistory analysis), and a
nonlinear method are also permitted, subject to certain limitations. These methods use real or synthetic
ground motion histories as input but require them to be scaled to the basic response spectrum at the site
FEMA 451, NEHRP Recommended Provisions: Design Examples
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for the range of periods of interest for the structure in question. Nonlinear analyses are very sensitive to
assumptions made in the analysis and a peer review is required. A nonlinear static method, also know as
a pushover analysis, is described in an appendix for trial use and comment. [In the 2003 Provisions,
substantial changes were made to the appendix for the nonlinear static procedure based, in part, on the
results of the Applied Technology Council’s Project 55.]
The two most common linear methods make use of the same design spectrum. The entire 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. 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.
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 resulting internal forces are combined with the effects of gravity loads and then compared to the full
strength of the members, which are not reduced 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 the 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 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 model represents 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
forces, and the resulting story shears and overturning moments, 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
Chapter 1, Fundamentals
115
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 forces applied at each story, 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 story forces will not match the moment from the
summation). 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. This may be very significant; yet, 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.
1.5 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 SUGs 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 time 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 a useful in
understanding the demands upon nonstructural components.
FEMA 451, NEHRP Recommended Provisions: Design Examples
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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.
1.6 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. 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 fact that the design is 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 ensure 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.
21
2
GUIDE TO USE OF THE PROVISIONS
Michael Valley, P.E.
The flow charts and table 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, the International Building Code, and NFPA 5000. The flow charts provide an
overview of the complete process for satisfying the Provisions, including the content of all technical
chapters. The table that concludes this chapter provides cross references for ASCE 7 and the 2000 and
2000 editions of the NEHRP Recommended Provisions.
The flow charts 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. Notes
indicate discrepancies and errors in the Provisions. Both editions of the Provisions can be obtained free
from the FEMA Publications Distribution Center by calling 18004802520. Order by FEMA
Publication number; the 2003 Provisions is available as FEMA 450 in CD form (only a limited number of
paper copies are available) and the 2000 Provisions are available as FEMA 368 and 369 (2 volumes and
maps).
Although the examples in this volume are based on the 2000 Provisions, they have been annotated to
reflect changes made to the 2003 Provisions. Annotations within brackets, [ ], indicate both
organizational changes (as a result of a reformat of all of the chapters of the 2003 Provisions) and
substantive technical changes to the 2003 Provisions and its primary reference documents. For those
readers coming from ASCE 705, see the cross reference table at the end of this chapter.
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.
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 2.1 provides an overall summary of the process which begins with consideration of the Scope of
Coverage and ends with Quality Assurance Requirements. All of the specific provisions pertaining to
nonbuilding structures are collected together on one page (Chart 2.20); application for nonbuilding
structures requires the use of various portions of the Provisions with appropriate modification.
Additions to, changes of use in, and alterations of existing structures are covered by the NEHRP
Recommended Provisions (see Chart 2.3), but evaluation and rehabilitation of existing structures is not.
FEMA 451, NEHRP Recommended Provisions: Design Examples
22
In recent years FEMA has sponsored several coordinated efforts dealing with seismic safety in existing
buildings. A Handbook for Seismic Evaluation of Buildings (FEMA 310) was developed as an update to
the original FEMA 178, although this document has since been replaced by the ASCE 31 Standard
(Seismic Evaluation of Existing Buildings). Guidelines for the Seismic Rehabilitation of Buildings
(FEMA 273) and a corresponding Commentary (FEMA 274) have also been developed. A prestandard
(FEMA 356, Prestandard and Commentary for the Seismic Rehabilitation of Buildings) based on FEMA
273 has been developed and is in balloting as ASCE 41. In addition, specific recommendations have been
developed for the evaluation, repair, and rehabilitation of earthquakedamaged concrete and masonry wall
buildings (FEMA 306, 307, and 308) and for the evaluation, rehabilitation, postearthquake assessment,
and repair of steel moment frame structures (FEMA 351 and 352).
Chapter 2, Guide to Use of the Provisions
23
Chart 2.2
Scope of Coverage
Chart 2.3
Application to Existing
Structures
Chart 2.4
Basic Requirements
Chart 2.22
Architectural, Mechanical,
Electrical Requirements
Chart 2.11
Strength Requirements
Chart 2.12
Deformation Requirements
Chart 2.13
Design and Detailing
Requirements
Chart 2.21
Foundations
Chart 2.5
Structural Design
Chart 2.23
Quality Assurance
Requirements
Chart 2.8
Modal Analysis
Chart 2.7
SoilStructure
Interaction
Chart 2.6
ELF Analysis
Chart 2.10
Seismically
Isolated
Chart 2.14
Steel
Chart 2.15
Concrete
Chart 2.16
Precast
Chart 2.17
Composite
Chart 2.18
Masonry
Chart 2.19
Wood
Chart 2.20
Nonbuilding
Structures
Chart 2.9
Response History
Analysis
Chart 2.1
Overall Summary of Flow
FEMA 451, NEHRP Recommended Provisions: Design Examples
24
No
No
No
No
Yes
Yes
Yes
Yes
Yes
No
No
Yes
Yes
Yes
No
No
Tank in SUG III?
Satisfy freeboard
requirement (Sec. 14.7.3.6.1.2)
[Sec. 14.4.7.5.3].
Yes No
*The Provisions has never defined clearly the scope of application for structures assigned to Seismic
Design Category A. Although the framers of the Provisions intended application of only a few simple
requirements in Seismic Design Category A, a strict reading of the 2000 Provisions would lead to a
substantial list of items that remain within the scope. [As a result of the complete rewrite of the Provisions
at the beginning of the 2003 update cycle, this situation is improved considerably as the requirements for
Seismic Design Category A all appear in Sec. 1.5.]
Determine if structure falls in scope
of the Provisions (Sec. 1.2 [1.1.2]).
Is it a selfsupporting structure
which carries gravity loads?
Is structure a bridge, nuclear
power generation plant, offshore
platform, or dam?
Is the use agricultural with only
incidental human occupancy?
Is it a detached 1 or
2family dwelling?
S
1 = 0.04 and
S
S = 0.15?
Is it an existing structure?
Provisions not
applicable.
No requirements.
No additional
requirements.
Satisfy Sec. 5.2.5 and 5.2.6.1
(i.e., consider as Seismic
Design Category A)*.
Go to Chart 2.3.
Go to Chart 2.4.
SDC A, B, or C?
Wood frame dwelling designed
and constructed in accordance
with Sec. 12.5?
Determine S
S and S
1
(Sec. 4.1.2 [3.3.1]).
Chart 2.2
Scope of Coverage
Chapter 2, Guide to Use of the Provisions
25
Addition to existing structure
(Sec. 1.2.2 [1.1.2.2]).
Change of use
(Sec. 1.2.3 [1.1.2.3]).
Is addition structurally
independent from existing
structure?
Is any element's seismic force
increased by more than 5% or its
seismic resistance decreased?
Do the affected elements still
comply with the Provisions?
Only addition or alteration
designed as new structure.
Go to Chart 2.4.
Entire structure designed
as new structure.
Go to Chart 2.4.
Change to higher
Seismic Use Group?
Change from Seismic Use
Group I to II and S
DS < 0.3?
No requirements.
Is seismic force on existing
structural elements increased
beyond their design strength?
Such alteration
not permitted.
Yes
No
No
Yes
Yes
No
* The Provisions applies to existing structures only in the cases of additions to, changes of use in, and
alterations of such structures.
No
No
Yes
Yes
*
Does alteration increase seismic
forces to or decrease design
strength of existing structural
elements by more than 5 percent?
Alteration of existing
structure (Sec. 1.2.4)
[Sec. 1.1.2.4].
Is the design strength of existing
structural elements required to
resist seismic forces reduced?
New structural elements and new
or relocated nonstructural elements
must be detailed and connected as
required by the Provisions.
No
No
No
Yes
Yes
*
Yes
*
Chart 2.3
Application to Existing Structures
FEMA 451, NEHRP Recommended Provisions: Design Examples
26
Determine Seismic Use Group
(Sec. 1.3 [1.2]) and Occupancy
Importance Factor (Sec. 1.4 [1.3]).
Using Spectral Acceleration Maps 1 through 24
(or CDROM) [Fig. 3.31 through 3.314],
determine the Maximum Considered Earthquake
(MCE) spectral response acceleration at short
periods (S
S) and at 1 second (S
1).
S
S = 0.15 and
S
1 = 0.04?
Soil properties known in sufficient
detail to determine Site Class?
Classify the site (Sec. 4.1.2.1
through 4.1.2.3) [Sec. 3.5].
Site Class E or F?
Use Site Class D unless authority having
jurisdiction determines that Site Class E
or F could be present at the site.
Assign structure to Seismic Design Category A.
[As a result of the rewrite of the Provisions during
the 2003 update cycle, all of the requirements for
Seismic Design Category A appear in Sec. 1.5 and
application is greatly simplified.]
Adjust MCE acceleration parameters
for site class (Sec. 4.1.2.4 [3.3.2]).
Calculate design earthquake acceleration
parameters S
DS and S
D1 (Sec. 4.1.2.5 [3.3.3]).
Design response spectrum required
for the analysis to be used?
Detached 1 or 2family
dwelling of lightframe
construction?
Site Class F or
base isolated?
Fulfill site limitation
(Sec. 4.2.2 [1.4.2]).
Perform sitespecific evaluation
to determine design response
spectrum (Sec. 4.1.3 [3.4]).
S
1 > 0.6 and
base isolated?
[For assignment of Seismic Design Category
and determination of values needed for other
calculations, compute S
DS, S
D1 , S
MS, and S
M1
as indicated in Sec. 3.4.5.]
Determine Seismic Design
Category (Sec. 4.2.1 [1.4.1]).
Calculate design
response spectrum
(Sec. 4.1.2.6 [3.3.4]).*
* [Sec. 3.3.4 of the 2003 Provisions defines reduced spectral ordinates for periods greater than T
L.]
Go to Chart 2.5 for
structural requirements.
Go to Chart 2.22 for architectural,
mechanical, and electrical requirements.
Go to Chart 2.23 for quality
assurance requirements.
Yes
Yes
No
No
Yes No
No
Yes
Yes
No Yes
No
Yes
No
Chart 2.4
Basic Requirements
Chapter 2, Guide to Use of the Provisions
27
Use load combinations and
nonearthquake loads from
ASCE 7 (Sec. 5.1 [4.2.2]).
Comply with the stated design basis
(Sec. 5.2.1 [4.2.1]).
Seismic Design
Category A?
Height limits in Table 5.2.2
[4.31] apply.
Seismic Design
Category B or C?
Requirements for building height, interaction
effects, deformational compatibility, and
special moment frame continuity
(Sec. 5.2.2.4 [4.3.1.4, 4.5.2, 4.5.3]).
Seismic Design
Category F?
More stringent height limit
(Sec. 5.2.2.5 [4.3.1.6]).
Classify diaphragm flexibility (Sec. 5.2.3.1
[4.3.2.1]). Examine plan and vertical
regularity and meet minimum requirements
for irregular structures (Tables 5.2.3.2 and
5.2.3.3 [4.32 and 4.33]).
Calculate reliability factor, ., and
satisfy limitations for special moment
frame systems (Sec. 5.2.4 [4.3.3]).
(Note that . = 1.0 for SDC B and C.)
Determine required level of seismic
force analysis (Sec. 5.2.5 [4.4]).
Analyze for minimum lateral
force, F
x = 0.01w
x (Sec. 5.2.5.1
[1.5.1]). Go to Chart 2.11.
Go to Chart 2.6
for ELF analysis.
Go to Chart 2.8 for
modal analysis.
No
No
Yes
No
Yes
Seismically isolated?
Go to Chart 2.10.
Yes
Classify the structural framing system and
note R, O
0, and C
d for later use (Sec.
5.2.2 [4.3.1] and Table 5.2.2 [4.31]).
Opt to perform
more involved
analysis?
Go to Chart 2.9 for
response history
analysis.
Yes
No
Yes
[A new "Simplified Design Procedure"
that appears in the Appendix to Chapter 4
may be used in lieu of 2003 Provisions
Chapters 4 and 5 for certain structures.]
Chart 2.5
Structural Design
FEMA 451, NEHRP Recommended Provisions: Design Examples
28
Determine total weight, W (Sec. 5.3 [5.2.1]).
Go to Chart 2.11.
Check the first order deformation for stability and
amplify the forces if necessary (Sec. 5.4.6.2 [5.2.6.2]).
Determine the story drifts. A reanalysis based upon
a period larger than the upper limit is permitted for
calculating deformations (Sec. 5.4.6.1 [5.2.6.1]).
To determine the internal forces, perform a linear elastic analysis with an
appropriate distribution of forces within stories due to the relative lateral stiffnesses
of vertical elements and diaphragms (Sec. 5.4.4 [5.2.4]). Include appropriately
amplified (Sec. 5.4.4.3 [5.2.4.3]) inherent (Sec. 5.4.4.1 [5.2.4.1]) and accidental
torsions (Sec. 5.4.4.2 [5.2.4.2]). Calculate the overturning moments for all stories
and apply the permitted reduction for foundations (Sec. 5.4.5 [5.2.5]).
Distribute the base shear to the stories
of the building (Sec. 5.4.3 [5.2.3]).
Determine the seismic response coefficient, C
s, (Sec. 5.4.1
[5.2.1.1]) and the total base shear (Eqn. 5.4.1 [5.22]).
Determine fundamental period of vibration for the
building (Sec. 5.4.2 [5.2.2]). Carefully note the upper
limit placed on periods calculated from analytical
models of the structure (Table 5.4.2 [5.21]).
Consider
soilstructureinteraction?
(Optional)
Go to Chart 2.7 to
calculate reduced
base shear.
Yes
No
Chart 2.6
Equivalent Lateral Force (ELF) Analysis
Chapter 2, Guide to Use of the Provisions
29
Modal Analysis: Follow SSI procedure
for ELF analysis (Sec. 5.8.2 [5.6.2]) with
these modifications (Sec. 5.8.3 [5.6.3]).
Calculate effective period
using Eq. 5.8.2.1.11 [5.63].
Point bearing piles? or
Uniform soft soils over a
stiff deposit?
Read foundation damping factor
from Figure 5.8.2.1.2 [5.61].
Calculate effective damping using Eq.
5.8.2.1.21 [5.69]. Effective damping
need not be less than 5 percent of critical.
Calculate reduced base shear, V, per Sec. 5.8.2.1
[5.61], which cannot be less than 0.7V.
Revise deflections to include foundation
rotation (Sec. 5.8.2.3 [5.6.2.3]).
Use Eq. 5.8.2.1.24 [5.610]
to modify foundation
damping factor.
This SSI procedure applies only to the
fundamental mode of vibration (Sec. 5.8.3.1
[5.6.3.1]). Therefore, substitute W
1 for W,
T
1 for T, V
1 for V, etc.
Use Eq. 5.5.42 [5.32] to calculate W
1.
Use Eq. 5.8.3.12 [5.613] to calculate h.
Calculate reduced base shear for
the first mode, V
1, per Sec.
5.8.3.1 [5.6.3.1], which cannot be
less than 0.7V
1. Use standard
modal combination techniques
(Sec. 5.8.3.2 [5.6.3.2]).
Yes
No
Return to Chart 2.6. Return to Chart 2.8.
ELF Analysis: Follow this
procedure (Sec. 5.8.2 [5.6.2]).
Calculate the foundation stiffnesses K
y
and K
. (Commentary) at the expected
strain level (Table 5.8.2.1.1 [5.61]).
Calculate effective gravity load, W (as a
fraction of W), effective height, h (as a
fraction of h), and effective stiffness, k,
of the fixed base structure.
Chart 2.7
SoilStructure Interaction (SSI)
FEMA 451, NEHRP Recommended Provisions: Design Examples
210
Combine modal quantities by either the SRSS or CQC technique*.
Compare base shear to lower limit based upon 85 percent of that
computed using Sec. 5.4 [5.2] with T = C
uT
a. Amplify all quantities
if necessary to increase the base shear (Sec. 5.5.7 [5.3.7]).
To determine the internal forces, perform a linear elastic
analysis. Include inherent and accidental torsions (Sec. 5.4.4
[5.2.4]). Amplify torsions (Sec. 5.4.4.3 [5.2.4.3]) that are not
in the dynamic model (Sec. 5.5.8 [5.3.8]). May reduce the
overturning moments at the foundationsoil interface by only
10 percent (Sec. 5.5.9 [5.3.9]).
Check the first order deformations for stability and
amplify the forces if necessary (Sec. 5.5.10 [5.3.10]).
Go to Chart 2.11.
Consider
soilstructureinteraction?
(Optional)
Go to Chart 2.7 to
calculate reduced
base shear.
*As indicated in the text, use of the CQC technique is required where closely spaced periods
in the translational and torsional modes will result in crosscorrelation of the modes.
Yes
No
Determine whether a threedimensional model is
required; identify the appropriate degrees of freedom,
possibly including diaphragm flexibility; and model
elements as directed (Sec. 5.5.1 [5.3.1]). Determine
the number of modes to consider (Sec. 5.5.2 [5.3.2]).
Use linear elastic analysis to determine
periods and mode shapes (Sec. 5.5.3 [5.3.3]).
Determine seismic response coefficient, C
sm,
effective gravity load, W
m, and base shear, V
m, for
each mode (Sec. 5.5.4 [5.3.4]).
Determine story forces, displacements, and drifts in each mode
(Sec. 5.5.5 [5.3.5]). Use statics to determine story shear and
overturning moments in each mode (Sec. 5.5.6 [5.3.6]).
Chart 2.8
Modal Analysis
Chapter 2, Guide to Use of the Provisions
211
Scale analysis results so that the
maximum base shear is
consistent with that from the ELF
procedure (Sec. 5.6.3 [5.4.3]).
Determine response parameters for
use in design as follows. If at least
seven ground motions are analyzed,
may use the average value. If fewer
than seven are analyzed, must use the
maximum value (Sec. 5.6.3 [5.4.3]).
Go to Chart 2.11.
Global modeling requirements are
similar to those for modal analysis.
Modeling of hysteretic behavior of
elements must be consistent with
laboratory test results and expected
material properties (Sec. 5.7.1 [5.5.1]).
Select and scale ground motion
as for linear response history
analysis (Sec. 5.7.2 [5.5.2]).
Analysis results need
not be scaled.
As for linear response history
analysis, use average or
maximum values depending on
number of ground motions
analyzed (Sec. 5.7.3 [5.5.3]).
Subsequent steps of the design process change. For
instance, typical load combinations and the
overstrength factor do not apply (Sec. 5.7.3.1
[5.5.3.1]), member deformations must be considered
explicitly (Sec. 5.7.3.2 [5.5.3.2]), and story drift limits
are increased (Sec. 5.7.3.3 [5.5.3.3]). The design must
be subjected to independent review (Sec. 5.7.4 [5.5.4]).
Nonlinear analyses must directly
include dead loads and not less than
25 percent of required live loads.
An appendix to Chapter 5
contains requirements for the
application of nonlinear static
(pushover) analysis to the design
of new structures.
Linear Response History
Analysis: Follow this
procedure (Sec. 5.6 [5.4]).
Nonlinear Response History
Analysis: Follow this procedure
(Sec. 5.7 [5.5]).
Model structure as
for modal analysis
(Sec. 5.5.1 [5.3.1]).
Select and scale ground motions based on
spectral values in the period range of interest
(Sec. 5.6.2 [5.4.2]), as follows. For 2D
analysis, the average is not less than the
design spectrum. For 3D analysis, the
average of the SRSS spectra computed for
each pair of ground motions is not less than
1.3 times the design spectrum.
Chart 2.9
Response History Analysis
FEMA 451, NEHRP Recommended Provisions: Design Examples
212
Go to Chart 2.11.
Yes
No
Yes
No
Yes Yes
No No
Satisfy detailed requirements for isolation
system (Sec. 13.6.2 [13.2.5]) and structural
system (Sec. 13.6.3 [13.2.6]). Satisfy
requirements for elements of structures and
nonstructural components (Sec. 13.5 [13.2.7]).
Perform design review
(Sec. 13.8 [13.5]).
Satisfy testing
requirements
(Sec. 13.9 [13.6]).
[In the 2003 Provisions, requirements for
structures with damping systems appear
in Chapter 15 (rather than in an appendix
to Chapter 13).]
Do the structure and
isolation system satisfy
the criteria of Section
13.2.5.2 [13.2.4.1]?
Site Class A, B, C, or D?
and
isolation system meets the
criteria of Sec. 13.2.5.2
[13.2.4.1], item 7?
Opt to perform
dynamic analysis?
Opt to perform
timehistory analysis?
Perform ELF analysis
(see Chart 2.6) and
satisfy the provisions of
Sec. 13.3.
Perform modal analysis
(see Chart 2.8) and satisfy
the appropriate provisions
of Sec. 13.4.
Perform timehistory
analysis as described
in Sec. 13.4.
Chart 2.10
Seismically Isolated Structures
Chapter 2, Guide to Use of the Provisions
213
Go to Chart 2.12.
Combine gravity loads and seismic forces as indicated in
ASCE 7, where the seismic load, E, is defined in Provisions
Sec. 5.2.7 [4.2.2]. Must consider critical loading direction for
each component (Sec. 5.2.5.2 [4.4.2]).
Minimum force effects for connections (Sec. 5.2.6.1.1 [4.6.1.1]) and anchorage
of concrete or masonry walls (Sec. 5.2.6.1.2 [4.6.1.2]). Special requirements
for strength of moment frames in dual systems (Sec. 5.2.2.1 [4.3.1.1]) and
combinations of framing systems (Sec. 5.2.2.2 [4.3.1.2]).
Seismic Design
Category A?
Consideration must be given to Pdelta effects (Sec. 5.2.6.2.1 [5.2.6.2]). Limits on
vertical discontinuities (Sec. 5.2.6.2.3 [4.6.1.6]). Minimum force effects for
diaphragms (Sec. 5.2.6.2.6 [4.6.1.9]), bearing walls (Sec. 5.2.6.2.7 [4.6.1.3]),
inverted pendulum structures (Sec. 5.2.6.2.8 [4.6.1.5]), and anchorage of
nonstructural systems (Sec. 5.2.6.2.9 [4.6.1.10]). Special load combinations of
Sec. 5.2.7.1 [4.2.2.2] apply to columns supporting discontinuous walls or frames
(Sec. 5.2.6.2.10 [4.6.1.7]).
Seismic Design
Category B?
Must consider orthogonal effects for some plan irregular structures (Sec. 5.2.5.2.2
[4.4.2.2]). Special load combinations of Sec. 5.2.7.1 [4.2.2.2] apply to collector
elements (Sec. 5.2.6.3.1 [4.6.2.2]). Minimum forces for anchorage of concrete or
masonry walls to flexible diaphragms (Sec. 5.2.6.3.2 [4.6.2.1]) and diaphragms that
are not flexible (Sec. 6.1.3 [6.2.2]).
Seismic Design
Category C?
Orthogonal effects must be considered (Sec. 5.2.5.2.3 [4.4.2.3]). Increased
forces for plan or vertical irregularities (Sec. 5.2.6.4.2 [4.6.3.2]). Vertical
seismic forces must be considered for some horizontal components (Sec.
5.2.6.4.3 [4.6.3.1]). Minimum forces for diaphragms (Sec. 5.2.6.4.4 [4.6.3.4]).
Yes
No
Yes
No
Yes
No
Go to Chart 2.13. [All requirements for
the structures assigned to Seismic Design
Category A now appear in Sec. 1.5.]
Chart 2.11
Strength Requirements
FEMA 451, NEHRP Recommended Provisions: Design Examples
214
Enter with story drifts from the analysis of seismic
force effects. These drifts must include the deflection
amplification factor, C
d, given in Table 5.2.2 [4.31]
(Sec. 5.2.8 [4.5.1]).
Go to Chart 2.13.
Compare with the limits established in Table 5.2.8 [4.51].
[In the 2003 Provisions the allowable drift is reduced by
the redundancy factor for systems with moment frames in
Seismic Design Category D, E, or F.]
Separations between adjacent buildings
(including at seismic joints) must be
sufficient to avoid damaging contact.
Chart 2.12
Deformation Requirements
Chapter 2, Guide to Use of the Provisions
215
Continuous diaphragm crossties required. Limit on subdiaphragm
aspect ratio. Special detailing for wood diaphragms, metal deck
diaphragms, and embedded straps (Sec. 5.2.6.3.2 [4.6.2.1]).
Consider effect of diaphragm displacement on attached elements
(Sec. 5.2.6.2.6 [4.5.2]).
Seismic Design
Category C?
Satisfy requirement for deformational
compatibility (Sec. 5.2.2.4.3 [4.5.3]).
Yes
No
Yes
No
Yes
No
Seismic Design
Category D?
No
Yes
Certain plan and vertical irregularities not
permitted (Sec. 5.2.6.5.1 [4.3.1.5.1]).
Chart 2.14
Chart 2.15
Chart 2.16
Chart 2.17
Chart 2.18
Chart 2.19
For nonbuilding structures,
go to Chart 2.20. For various
materials, go to these charts:
Steel
Concrete
Precast
Composite
Masonry
Wood
Seismic Design
Category A?
Openings in shear walls and diaphragms must be detailed
(Sec. 5.2.6.2.2 [4.6.1.4]). System redundancy must be considered
(Sec. 5.2.6.2.4 [4.2.1]). Requirements for diaphragm ties, struts,
and connections (Sec. 5.2.6.2.6 [4.6.1.5]) and interconnection of
wall elements (Sec. 5.2.6.2.7 [4.6.1.3]).
Seismic Design
Category B?
Chart 2.13
Design and Detailing Requirements
FEMA 451, NEHRP Recommended Provisions: Design Examples
216
Yes
No
No
Seismic Design Yes
Category A, B, or C?
Using a "structural steel system
not specifically designed for
seismic resistance?"
Select an R value from Table
5.2.2 [4.31] for the appropriate
steel system.
The system must be designed and
detailed in accordance with the AISC
Seismic as modified in Sec. 8.4 [8.3]
or
Sec. 8.6 [8.4.2] for lightframed,
coldformed steel wall systems.
From Table 5.2.2
[4.31], R = 3.
Any of the reference
documents in Sec. 8.1 [8.1.2]
may be used for design.
Sec. 8.5 [8.4.1] modifies the reference standards
for design of coldformed steel members.
Sec. 8.7 [8.4.4] applies to steel deck diaphragms.
Sec. 8.8 [8.5] applies to steel cables.
[In the 2003 Provisions requirements are added
for bucklingrestrained braced frames (Sec. 8.6)
and special steel plate shear walls (Sec. 8.7).]
Go to Chart 2.21.
Chart 2.14
Steel Structures
Chapter 2, Guide to Use of the Provisions
217
Modifications to ACI 318 for load combinations and
resistance factors, permitted reinforcement, axial strength
of columns, diaphragm connectors, structural walls, and
coupling beams (Sec. 9.1.1 [9.2.2]). [Many of the
requirements in this chapter of the 2003 Provisions are
different due to changes made in ACI 31802 and the
introduction of new systems.]
Design of anchors (Sec. 9.2)
[ACI 31802 Appendix D].
Classification of shear
walls (Sec. 9.3 [9.2.1]).
Seismic Design
Category A?
Limit on use of ordinary moment
frames (Sec. 9.5.1 [9.3.1]).
Seismic Design
Category B?
Limits on moment frame and shear wall systems,
discontinuous members, and plain concrete.
Requirements for anchor bolts in the tops of
columns (Sec. 9.6 [9.4]). [Some of these
requirements are removed in the 2003 Provisions
as they are now in ACI 31802.]
Seismic Design
Category C?
Moment frames to be "special"; shear walls to
be "special reinforced." Detailing for
deflection compatibility (Sec. 9.7 [9.5]). [All
of these requirements now appear in ACI
31802 or in the basic requirements of the
2003 Provisions.]
Design in accordance with ACI 318
(Sec. 9.4 [1.5]). [All requirements
for the structures assigned to Seismic
Design Category A now appear in
Sec. 1.5.]
Go to Chart 2.21.
Yes
Yes
Yes
No
No
No
Chart 2.15
Concrete Structures
FEMA 451, NEHRP Recommended Provisions: Design Examples
218
General modifications to ACI 318 to include additional notation and definitions
(Sec. 9.1.1.1 and 9.1.1.2) and new sections. [All of the requirements on this
chart now appear in ACI 31802. The 2003 Provisions add some requirements
for intermediate and special precast walls (Sec. 9.2.2.4 and 9.2.2.5).]
Requirements for layout of seismicforceresisting
system: diaphragm strength, aspect ratios, and number
of moment resisting frame bays. Requirements for
gravity beamtocolumn connections: design force and
connection characteristics (Sec. 9.1.1.4).
Requirements for strong
connections (Sec. 9.1.1.12):
location, anchorage and
splices, design forces,
columntocolumn and
columnface connections
(ACI 318 new Sec. 21.11.5).
System must comply with all
applicable requirements of
monolithic concrete
construction for resisting
seismic forces (Sec. 9.1.1.12)
and connections (either wet
or dry) must satisfy the
requirements of ACI 318
new Sec. 21.11.3.1.
Must demonstrate suitability
of system by analysis and
substantiating experimental
evidence based on cyclic,
inelastic testing (Sec.
9.1.1.12) as indicated in
ACI ITG/T1.1 and the
additional items in ACI 318
new Sec. 21.11.4.
Yes No
Yes
Yes No
No
Yes
No
No
Yes
Connections must satisfy ACI 318
new Sec. 21.11.6 (Sec. 9.1.1.12).
Precast gravity load
carrying system?
Diaphragm composed
of precast elements?
Go to Chart 2.21.
Topping slabs must satisfy ACI
318. An appendix is provided for
untopped precast diaphragms.
Precast
seismicforceresisting
system?
Emulates monolithic
reinforced concrete
construction?
Ductile
connections?
Chart 2.16
Precast Concrete Structures
Chapter 2, Guide to Use of the Provisions
219
Yes
No
Select an R value from Table 5.2.2 [4.31]
for the appropriate composite system.
Seismic Design
Category A, B, or C?
Go to Chart 2.21.
The system must be designed and detailed in
accordance with the AISC Seismic Parts I and II.
[The 2003 Provisions make extensive
modifications to Part II of AISC Seismic.]
Must provide "substantiating
evidence" based on cyclic testing
(Sec. 10.2 [10.4]).
Chart 2.17
Composite Steel and Concrete Structures
FEMA 451, NEHRP Recommended Provisions: Design Examples
220
Must construct in accordance
with ACI 530 and use materials
in conformance with ACI 530.1
Seismic Design
Category A?
Seismic Design
Category B?
Seismic Design
Category C?
Seismic Design
Category D?
Empirical design (per
Chapter 9 of ACI 530)
may be used.
Go to Chart 2.21.
Special requirements for screen
walls and cavity walls, limits on
certain materials, and rules for wall
reinforcement and connection to
masonry columns. Ordinary plain
masonry and ordinary reinforced
masonry shear walls not
permitted(Sec. 11.3.7).
Limits on materials and rules for
wall reinforcement and
concrete/masonry interface.
Minimum wall thickness, column
reinforcement, and column
minimum dimensions. Detailed
plain masonry and intermediate
reinforced masonry shear walls not
permitted (Sec. 11.3.8).
Requirements for grout,
hollow units, and stack bond
(Sec. 11.3.9).
Material properties (Sec. 11.3.10),
section properties (Sec. 11.3.11),
and anchor bolts (Sec. 11.3.12).
Reinforcement detailing
(Sec. 11.4).
Strength and deformation
requirements (Sec. 11.5), flexure
and axial loads (Sec. 11.6), and
shear (Sec. 11.7).
Special requirements for beams
(Sec. 11.8), columns (Sec. 11.9),
and shear walls (Sec. 11.10).
Special moment frames of masonry
(Sec. 11.11).
Glass unit masonry and masonry
veneer (Sec. 11.12).
Yes
No
No
Yes
Yes
Yes
No
No
[A significant portion of 2003 Provisions Chapter 11 has been
replaced by a reference to ACI 53002. The updated chapter,
however, does not result in significant technical changes, as ACI
53002 is in substantial agreement with the strength design
methodology contained in the 2000 Provisions.]
Chart 2.18
Masonry Structures
Chapter 2, Guide to Use of the Provisions
221
Yes
Yes
Yes
Yes
Yes
Yes
No No
No No
No No
Satisfy an exception of
Sec. 1.2.1 [1.1.2.1]?
Seismic Design
Category A?
Go to Chart 2.21
Seismic Design
Category B, C, or D?
Seismic Use
Group I?
Satisfy an exception of
Sec. 1.2.1 [1.1.2.1]?
Satisfy the Section 12.5
requirements for conventional
lightframe construction?
Unblocked diaphragms not
permitted. Sheathing applied
directly to framing. Shear wall
resistance reduced for structures
with concrete or masonry
walls(Sec. 12.8 [12.2.2]).
Design and construct using any
applicable materials and procedures
in the reference documents. If satisfy
Sec. 12.5, deemed to comply with
Sec. 5.2.6.1 [1.5].
Conform to requirements for engineered wood
construction (Sec. 12.3) and diaphragms and shear
walls (Sec. 12.4). [A significant portion of 2003
Provisions Chapter 12, including the diaphragm and
shear wall tables, has been replaced by a reference to
the AF&PA, ASD/LRFD Supplement, Special Design
Provisions for Wind and Seismic. The updated
chapter, however, does not result in significant
technical changes, as the Supplement is in substantial
agreement with the 2000 Provisions.]
Chart 2.19
Wood Structures
FEMA 451, NEHRP Recommended Provisions: Design Examples
222
Use applicable strength and other design criteria from
other sections of the Provisions.
or
Use approved standards. Reduced seismic forces for
use with allowable stress standards are defined.
Nonbuilding structure
supported by another
structure?
Select R value and
calculate design forces
per Sec. 14.4 [14.1.5].
Determine Importance Factor and Seismic Use Group
(Sec. 14.5.1.2 [14.2.1]). Calculate seismic weight (Sec.
14.5.3 [14.2.6]) and fundamental period (Sec. 14.5.4
[14.2.9]). May be exempted from drift limits on the basis
of rational analysis (Sec. 14.5.5 [14.2.11]).
Classify system, determine
importance factor, and calculate
design forces per Sec. 14.5 [14.2.4].
Dynamic response
similar to that of
building structures?
Go to Chart 2.21.
Structures Similar to Buildings
Specific provisions for: pipe racks; steel storage
racks; electrical power generating facilities;
structural towers for tanks and vessels; and piers
and wharves (Sec. 14.6 [14.3]).
Structures Not Similar to Buildings
Specific provisions for: earth retaining structures;
tanks and vessels; stacks and chimneys;
amusement structures; special hydraulic
structures; and secondary containment systems
(Sec. 14.7 [14.4]). (Appendix contains provisions
for: electrical transmission, substation, and
distribution structures; telecommunication towers;
and buried structures.) [In the 2003 Provisions
the requirements for electrical structures and
telecommunication towers have been removed
since the corresponding national standards have
been updated appropriately.]
Yes No
Yes
No
Chart 2.20
Nonbuilding Structures
Chapter 2, Guide to Use of the Provisions
223
Requirements for: report
concerning potential site hazards;
ties between spread footings; and
reinforcement of piles. Design
of piles must consider curvatures
due to both freefield soil strains
and structure response. (Sec. 7.5).
Go to Chart 2.22.
Strength and detailing of
foundation components must
satisfy material chapter
requirements (Sec. 7.2.1).
Yes
No
Yes
No
Requirement for soil capacity (Sec.
7.2.2). [In the 2003 Provisions Chapter 7
includes a strength design method for
foundations and guidance for the explicit
modeling of foundation loaddeformation
characteristics.]
Seismic Design
Category A or B?
Requirements for: report of seismic soil
investigation; pole type structures; ties
between piles or piers; and reinforcement
of piles (Sec. 7.4).
Seismic Design
Category C?
Chart 2.21
Foundations
FEMA 451, NEHRP Recommended Provisions: Design Examples
224
Seismic Design
Category A?
Go to Chart 2.23.
Satisfy requirements for
construction documents
(Sec. 6.1.7 [6.2.9]).
Note component
exemptions in
Sec. 6.1 [6.1.1].
Must consider both flexibility and strength for components and
support structures [Sec. 6.2.4]. Avoid collateral failures by
considering functional and physical interrelationship of
components (Sec. 6.1 [6.2.3]). Components require positive
attachment to the structure without reliance on gravityinduced
friction (Sec. 6.1.2 [6.2.5]).
Determine the periods of mechanical and electrical components (Sec. 6.3.3
[6.4.1]). Select a
p and R
p values from Tables 6.2.2 and 6.3.2 [6.31 and 6.41]
and component importance factors (Sec. 6.1.5 [6.2.2]). Calculate design
seismic forces per Sec. 6.1.3 [6.2.6]. Calculate vertical load effects per Sec.
5.2.7 [6.2.6]. (Don't forget to consider nonseismic horizontal loads.) Note
additional requirements for component anchorage (Sec. 6.1.6 [6.2.8]).
Compute seismic relative displacements (Sec. 6.1.4 [6.2.7])
and accommodate such displacements (Sec. 6.2.3 and 6.3.5
[6.3.1 and 6.4.4]).
Architectural Components
Specific provisions for: exterior nonstructural
wall elements and connections; outofplane
bending; suspended ceilings; access floors;
partitions; and steel storage racks
(Sec. 6.2 [6.3]).
Mechanical and Electrical Components
Specific provisions for: component
certification; utility and service lines; storage
tanks; HVAC ductwork; piping systems;
boilers and pressure vessels; mechanical and
electrical equipment, attachments, and
supports; alternative seismic qualification
methods; and elevator design (Sec. 6.3 [6.4]).
Yes
No
Chart 2.22
Architectural, Mechanical, and Electrical Components
Chapter 2, Guide to Use of the Provisions
225
Seismicforceresisting system
assigned to Seismic Design
Category C, D, E, or F? or
Designated seismic system in
structure assigned to Seismic
Design Category D, E, or F?
Satisfy exceptions
in Sec. 3.2 [2.2]?
QA plan not
required.
Registered design professional must prepare
QA plan (Sec. 3.2.1 [2.2.1]) and affected
contractors must submit statements of
responsibility (Sec. 3.2.2 [2.2.2]).
Satisfy testing and
inspection requirements
in the reference standards
(Ch. 8 through 14).
Reporting and compliance procedures Done.
are given (Sec. 3.6 [2.6]).
Registered design
professional must
perform structural
observations.
Seismic Use Group II or III? or
Height > 75 ft? or
Seismic Design Category E or F
and more than two stories?
Seismic Design
Category C?
Special inspection is required for some
aspects of the following: deep foundations,
reinforcing steel, concrete, masonry, steel
connections, wood connections, coldformed
steel connections, selected architectural
components, selected mechanical and
electrical components, isolator units, and
energy dissipation devices (Sec. 3.3 [2.3]).
Special testing is required for some aspects
of the following: reinforcing and prestressing
steel, welded steel, mechanical and electrical
components and mounting systems (Sec. 3.4
[2.4]), and seismic isolation systems
(Sec. 13.9 [13.6]).
No
Yes
Yes
No
Yes
No
No
Yes
Chart 2.23
Quality Assurance
Table 21 Navigating Among the 2000 and 2003 NEHRP Recommended Provisions
and ASCE 7
ASCE 7
Section
NEHRP 2000
Section
NEHRP 2003
Section Topic
Chapter 11 SEISMIC DESIGN CRITERIA
11.1 1.1, 1.2 1.1 General
11.2 2.1 1.1.4 Definitions
11.3 2.2 1.1.5 Notation
11.4 4.1 3.3 Seismic Ground Motion Values
11.5 1.3, 1.4 1.2, 1.3 Importance Factor and Occupancy Category
11.6 4.2 1.4 Seismic Design Category
11.7 5.2.6.1 1.5 Design Requirements for Seismic Design Category A
11.8 4.2, 7.4, 7.5 1.4.2, 7.4, 7.5 Geologic hazards and Geotechnical Investigation
Chapter 12 5 4, 5 SEISMIC DESIGN REQUIREMENTS FOR BUILDING
STRUCTURES
12.1 5.2 4.2.1 Structural Design Basis
12.2 5.2.2 4.3.1 Structural System Selection
12.3 5.2.3, 5.2.6,
5.2.4
4.3.2 Diaphragm Flexibility, Configuration Irregularities and
Redundancy
12.4 5.2.7, 5.2.6 4.2.2 Seismic Load Effects and Combinations
12.5 5.2.5 4.4.2 Direction of Loading
12.6 5.2.5 4.4.1 Analysis Procedure Selection
12.7 5.2, 5.6.2 Modeling Criteria
12.8 5.5 5.2 Equivalent Lateral Force Procedures
12.9 5.6 5.3 Modal Response Spectrum Analysis
12.10 5.2.6 4.6 Diaphragms, Chords and Collectors
12.11 5.2.6 4.6 Structural Walls and Their Anchorage
12.12 5.2.8 4.5 Drift and Deformation
12.13 7 7 Foundation Design
12.14 5.4 4 Alt. Simplified Alternative Structural Design Criteria for
Simple Bearing Wall of Building Frame System
Chapter 13 SEISMIC REQUIREMENTS FOR NONSTRUCTURAL
COMPONENTS
13.1 6.1 6.1 General
13.2 6.1 6.2 General Design Requirements
13.3 6.1.3, 6.1.4 6.2 Seismic Demands on Nonstructural Components
13.4 6.1.2 6.2 Nonstructural Component Anchorage
13.5 6.2 6.3 Architectural Components
13.6 6.3 6.4 Mechanical and Electrical Components
Chapter 14 MATERIAL SPECIFIC SEISMIC DESIGN AND
DETAILING REQUIREMENTS
14 Scope
14.1 8 8 Steel
14.2 9 9 Concrete
14.3 10 10 Composite Steel and Concrete Structures
14.4 11 11 Masonry
14.5 12 12 Wood
Chapter 2, Guide to Use of the Provisions
227
Chapter 15 14 14 SEISMIC DESIGN REQUIREMENTS FOR
NONBUILDING STRUCTURES
15.1 14.1 14.1 General
15.2 14.2,14.3 14.1.2 Reference Documents
15.3 14.4 14.1.5 Nonbuilding Structures Supported by Other Structures
15.4 14.5 14.2 Structural Design Requirements
15.5 14.6 14.3 Nonbuilding Structures Similar to Buildings
15.6 14.7 14.4 General Requirements for Nonbuilding Structures Not
Similar to Buildings
15.7 14.7.3 14.4.7 Tanks and Vessels
Chapter 16 SEISMIC RESPONSE HISTORY PROCEDURES
16.1 5.7 5.4 Linear Response History Analysis
16.2 5.8 5.5 Nonlinear Response History Procedure
Chapter 17 13 13 SEISMIC DESDIGN REQUIREMENTS FOR
SEISMICALLY ISOLATED STRUCTURES
17.1 13.1 13.1 General
17.2 13.5, 13.6 13.2 General design Requirements
17.3 13.4.4 13.2.3 Ground Motion for Isolated Systems
17.4 13.2.5 13.2.4 Analysis Procedure Selection
17.5 13.3 13.3 Equivalent Lateral Force Procedure
17.6 13.4 13.4 Dynamic Analysis Procedures
17.7 13.8 13.5 Design Review
17.8 13.9 13.6 Testing
Chapter 18 13A 15 SEISMIC DESIGN REQUIREMENTS FOR
STRUCTURES WITH DAMPING SYSTEMS
18.1 13A.1 15.1 General
18.2 13A.2, 13A.8 15.2 General Design Requirements
18.3 13A.6 15.3 Nonlinear Procedures
18.4 13A.5 15.4 Response Spectrum Procedure
18.5 13A.4 15.5 Equivalent Lateral Force Procedure
18.6 13A.3 15.6 Damped Response Modification
18.7 13A.7 15.7 Seismic Load Conditions and Acceptance
18.8 13A.9 15.8 Design Review
18.9 13A.10 15.9 Testing
Chapter 19 SOIL STRUCTURE INTERACTION FOR SEISMIC
DESIGN
19.1 5.8.1 5.6.1 General
19.2 5.8.2 5.6.2 Equivalent Lateral Force Procedure
19.3 5.8.3 5.6.3 Modal Analysis Procedure
Chapter 20 SITE CLASSIFICATION PROCEDURE FOR SEISMIC
DESIGN
20.1 4.1 3.5 Site Classification
20.2 4.1 3.5 Site Response Analysis for Site Class F Soil
20.3 4.1 3.5 Site Class Definitions
20.4 4.1 3.5 Definitions of Site Class Parameters
FEMA 451, NEHRP Recommended Provisions: Design Examples
228
Chapter 21 SITESPECIFIC GROUND MOTION PROCEDURES
FOR SEISMIC DESIGN
21.1 4.1 3.4 Site Response Analysis
21.2 4.1 3.4 Ground Motion Hazard Analysis
21.3 4.1 3.4 Design Response Spectrum
21.4 4.1 3.4 Design Acceleration Parameters
Chapter 22 4.1 3.3 SEISMIC GROUND MOTION AND LONG PERIOD
TRANSITION MAPS
Chapter 23 SEISMIC DESIGN REFERENCE DOCUMENTS
23.1 Consensus Standards and Other Reference Documents
11A 3 2 QUALITY ASSURANCE PROVISIONS
11A.1 3.1, 3.2, 3.3 2.1, 2.2, 2.3 Quality Assurance
11A.2 3.4 2.4 Testing
11A.3 3.5 2.5 Structural Observations
11A.4 3.6 2.6 Reporting and Compliance Procedures
11B EXISTING BUILDING PROVISIONS
11B.1 1.2.1 1.1.2 Scope
11B.2 1.2.2.1 1.1.2.2 Structurally Independent Additions
11B.3 1.2.2.2 1.1.2.2 Structurally Dependent Additions
11B.4 1.2.4 1.1.2.4 Alterations
11B.5 1.2.3 1.1.2.3 Change of Use
31
3
STRUCTURAL ANALYSIS
Finley A. Charney, Ph.D., P.E.
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 (ELF) analysis, modalresponsespectrum analysis,
and modal timehistory 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 limitations of available software, the analyses are performed
for only two dimensions. For the nonlinear analysis, two approaches are used: static pushover
analysis in association with the capacitydemand spectrum method and direct timehistory analysis.
In the nonlinear analysis, special attention is paid to the modeling of the beamcolumn joint regions
of the structure. The relative merits of pushover analysis versus timehistory analysis are discussed.
Although the Seattle building, as originally designed, responds reasonably well under the design ground
motions, a second set of timehistory 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.
Although this volume of design examples is based on the 2000 Provisions, it has been annotated to reflect
changes made to the 2003 Provisions. Annotations within brackets, [ ], indicate both organizational
changes (as a result of a reformat of all of the chapters of the 2003 Provisions) and substantive technical
changes to the 2003 Provisions and its primary reference documents. While the general concepts of the
changes are described, the design examples and calculations have not been revised to reflect the changes
to the 2003 Provisions.
A number of noteworthy changes were made to the analysis requirements of the 2003 Provisions. These
include elimination of the minimum base shear equation in areas without nearsource effects, a change in
the treatment of Pdelta effects, revision of the redundancy factor, and refinement of the pushover
analysis procedure. In addition to changes in analysis requirements, the basic earthquake hazard maps
were updated and an approach to defining longperiod ordinates for the design response spectrum was
developed. Where they affect the design examples in this chapter, significant changes to the 2003
Provisions and primary reference documents are noted. However, some minor changes to the 2003
Provisions and the reference documents may not be noted.
In addition to the 2000 NEHRP Recommended Provisions (herein, the Provisions), the following
documents are referenced:
FEMA 451, NEHRP Recommended Provisions: Design Examples
32
AISC Seismic American Institute of Steel Construction. 1997 [2002]. Seismic Provisions for
Structural Steel Buildings.
ATC40 Applied Technology Council. 1996. Seismic Evaluation and Retrofit of Concrete
Buildings.
Bertero Bertero, R. D., and V.V. Bertero. 2002. “Performance Based Seismic Engineering:
The Need for a Reliable Comprehensive Approach,” Earthquake Engineering and
Structural Dynamics 31, 3 (March).
Chopra 1999 Chopra, A. K., and R. K. Goel. 1999. CapacityDemandDiagram Methods for
Estimating Seismic Deformation of Inelastic Structures: SDF Systems.
PEER1999/02. Berkeley, California: Pacific Engineering Research Center, College
on Engineering, University of California, Berkeley.
Chopra 2001 Chopra, A. K., and R. K. Goel. 2001. A Modal Pushover Procedure to Estimate
Seismic Demands for Buildings: Theory and Preliminary Evaluation, PEER2001/03.
Berkeley, California: Pacific Engineering Research Center, College on Engineering,
University of California, Berkeley.
FEMA 356 American Society of Civil Engineers. 2000. Prestandard and Commentary for the
Seismic Rehabilitation of Buildings.
Krawinkler Krawinkler, Helmut. 1978. “Shear in BeamColumn Joints in Seismic Design of
Frames,” Engineering Journal, Third Quarter.
Chapter 3, Structural Analysis
33
3.1 IRREGULAR 12STORY STEEL FRAME BUILDING, STOCKTON, CALIFORNIA
3.1.1 Introduction
This example presents the analysis of a 12story steel frame building under seismic effects acting alone.
Gravity forces due to live and dead load are not computed. For this reason, member stress checks,
member design, and detailing are not discussed. For detailed examples of the seismicresistant design of
structural steel buildings, see Chapter 5 of this volume of design examples.
The analysis of the structure, shown in Figures 3.11 through 3.13, is performed using three methods:
1. Equivalent lateral force (ELF) procedure based on the requirements of Provisions Chapter 5,
2. Threedimensional, modalresponsespectrum analysis based on the requirements of Provisions
Chapter 5, and
3. Threedimensional, modal timehistory analysis using a suite of three different recorded ground
motions based on the requirements of Provisions Chapter 5.
In each case, special attention is given to applying the Provisions rules for orthogonal loading and
accidental torsion. All analyses were performed using the finite element analysis program SAP2000
(developed by Computers and Structures, Inc., Berkeley, California).
3.1.2 Description of Structure
The structure is a 12story special moment frame of structural steel. The building is laid out on a
rectangular grid with a maximum of seven 30ftwide bays in the X direction, and seven 25ft 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 ft except for the first story which is 18 ft high. The structure has
a full onestory basement that extends 18.0 ft below grade. Reinforced 1ftthick concrete walls form the
perimeter of the basement. The total height of the building above grade is 155.5 ft.
Gravity loads are resisted by composite beams and girders that support a normal weight concrete slab on
metal deck. The slab has an average thickness of 4.0 in. at all levels except Levels G, 5, and 9. The slabs
on Levels 5 and 9 have an average thickness of 6.0 in. for more effective shear transfer through the
diaphragm. The slab at Level G is 6.0 in. 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 piles. Interior columns
are supported by concrete caps over piles. All tie beams and pile caps are connected by a grid of
reinforced concrete grade beams.
FEMA 451, NEHRP Recommended Provisions: Design Examples
34
7 at 3 0 '0"
45'0"
62'6"
A
Y
X
(c ) L evel 2
(b ) L evel 6
(a) Level 10
45'0"
62'6"
B
A
O rig in for
center
o f m ass
A
B
Y
X
Y
X
Figure 3.11 Various floor plans of 12story Stockton building (1.0 ft = 0.3048 m).
Chapter 3, Structural Analysis
35
Figure 3.12 Sections through Stockton building (1.0 ft. = 0.3048 m).
B
G
2
3
4
5
6
7
8
9
10
11
12
R
Section BB
2 at 18'0" 11 at 12'6"
Moment
connections
Pinned
connections
B
G
2
3
4
5
6
7
8
9
10
11
12
R
2 at 18'0" 11 at 12'6"
7 at 30'0"
7 at 25'0"
All moment
connections
Section AA
FEMA 451, NEHRP Recommended Provisions: Design Examples
36
The lateralloadresisting system consists of special 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 lateralloadresisting girders terminate at Level 5 for Grid C and Level 9 for Grid F. Girders below
these levels are simply connected. Due to the fact that the momentresisting girders terminate in Frames
C and F, much of the Ydirection 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 outriggering action provided by 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 moment
resisting frames have a yield strength of 36 ksi, and floor members and interior columns that are sized
strictly for gravity forces are 50 ksi.
3.1.3 Provisions Analysis Parameters
Stockton, California, is in San Joaquin County approximately 60 miles east of Oakland. According to
Provisions Maps 7 and 8, the shortperiod and 1second mapped spectral acceleration parameters are:
Ss = 1.25
S1 = 0.40
[The 2003 Provisions have adopted the 2002 USGS probabilistic seismic hazard maps, and the maps have
been added to the body of the 2003 Provisions as figures in Chapter 3 (instead of being issued in a
separate map package).]
Y
Z
X
Figure 3.13 Threedimensional wireframe model of Stockton
building.
Chapter 3, Structural Analysis
37
Assuming Site Class C, the adjusted maximum considered 5percentdamped spectral accelerations are
obtained from Provisions Eq. 4.1.2.41 and Eq. 4.1.2.42 [3.31 and 3.32]:
SMS =FaSS =1.0(1.25)=1.25
SM1=FvS1=1.4(0.4)=0.56
where the coefficients Fa = 1.0 and Fv = 1.4 come from Provisions Tables 4.1.2.4(a) and 4.1.2.4(b) [3.31
and 3.32], respectively.
According to Provisions Eq. 4.1.2.51 and 4.1.2.52 [3.33 and 3.34], the design level spectral
acceleration parameters are 2/3 of the above values:
2 2(1.25) 0.833
SDS=3SMS=3 =
2 2(0.56) 0.373
SD1=3SM1=3 =
As the primary occupancy of the building is business offices, the Seismic Use Group (SUG) is I and,
according to Provisions Table 1.4 [1.31], the importance factor (I) is 1. According to Provisions Tables
4.2.1(a) and 4.2.1(b) [1.41 and 1.42], the Seismic Design Category (SDC) for this building is D.
The lateralloadresisting system of the building is a special momentresisting frame of structural steel.
For this type of system, Provisions Table 5.2.2 [4.31] gives a response modification coefficient (R) of 8
and a deflection amplification coefficient (Cd) of 5.5. Note that there is no height limit placed on special
moment frames.
According to Provisions Table 5.2.5.1 [4.41] if the building has certain types of irregularities or if the
computed building period exceeds 3.5 seconds where TS = SD1/SDS = 0.45 seconds, the minimum level of
analysis required for this structure is modalresponsespectrum analysis. This requirement is based on
apparent plan and vertical irregularities as described in Provisions Tables 5.2.3.2 and 5.2.3.3 [4.32 and
4.33]. The ELF procedure would not be allowed for a final design but, as explained later, certain aspects
of an ELF analysis are needed in the modalresponsespectrum analysis. For this reason, and for
comparison purposes, a complete ELF analysis is carried out and described herein.
3.1.4 Dynamic Properties
Before any analysis can be carried out, it is necessary to determine the dynamic properties of the
structure. These properties include mass, periods of vibration and their associated mode shapes, and
damping.
3.1.4.1 Mass
For twodimensional analysis, only the translational mass is required. To perform a threedimensional
modal or timehistory analysis, it is necessary to compute the mass moment of inertia for floor plates
rotating about the vertical axis and to find the location of the center of mass of each level of the structure.
This may be done two different ways:
1. The mass moments of inertia may be computed “automatically” by SAP2000 by modeling the floor
diaphragms as shell elements and entering the proper mass density of the elements. Line masses,
such as window walls and exterior cladding, may be modeled as point masses. The floor diaphragms
FEMA 451, NEHRP Recommended Provisions: Design Examples
38
may be modeled as rigid inplane by imposing displacement constraints or as flexible inplane by
allowing the shell elements to deform in their own plane. Modeling the diaphragms as flexible is not
necessary in most cases and may have the disadvantage of increasing solution time because of the
additional number of degrees of freedom required to model the diaphragm.
2. The floor is assumed to be rigid inplane but is modeled without explicit diaphragm elements.
Displacement constraints are used to represent the inplane rigidity of the diaphragm. In this case,
floor masses are computed by hand (or an auxiliary program) and entered at the “master node”
location of each floor diaphragm. The location of the master node should coincide with the center of
mass of the floor plate. (Note that this is the approach traditionally used in programs such as ETABS
which, by default, assumed rigid inplane diaphragms and modeled the diaphragms using constraints.)
In the analysis performed herein, both approaches are illustrated. Final analysis used Approach 1, but the
frequencies and mode shapes obtained from Approach 1 were verified with a separate model using
Approach 2. The computation of the floor masses using Approach 2 is described below.
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 associated with the various floor plates are given in Tables 3.11 and
3.12. The line masses are based on a cladding weight of 15.0 psf, story heights of 12.5 or 18.0 ft, and
parapets 4.0 ft high bordering each roof region. Figure 3.14 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
3.13. The reference point for center of mass location is the intersection of Grids A and 8. Note that the
dimensional units of mass moment of inertia (in.kipsec2/radian), when multiplied by angular
acceleration (radians/sec2), must yield units of torsional moment (in.kips).
Table 3.13 includes a mass computed for Level G of the building. This mass is associated with an
extremely stiff story (the basement level) and is not dynamically excited by the earthquake. As shown
later, this mass is not included in equivalent lateral force computations.
Table 3.11 Area Masses on Floor Diaphragms
Mass Type
Area Mass 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 3.14 for mass location.
1.0 psf = 47.9 N/m2.
Chapter 3, Structural Analysis
39
1
1
1
1
2
2
2
2
2
2
2 3 1
2
1
1
2
2
2
2
2
2
1 2
1 2
3
1
2 2
2
2 2
2
2 2
2
5 5
2
4
4
4
4
A
4
4
4
4
5
5
5
5
5
5
5
5
A
2
Area mass
A Line mass
B
B
A D B A
A B
D
C
Roof Levels 1012 Level 9
Levels 68 Level 5 Levels 34
Level 2 Level G
Figure 3.14 Key diagram for computation of floor mass.
Table 3.12 Line Masses on Floor Diaphragms
Mass Type
Line Mass 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
1350.0
1485.0
See Figure 3.14 for mass location.
1.0 plf = 14.6 N/m.
FEMA 451, NEHRP Recommended Provisions: Design Examples
1This requirements seems odd to the writer since the Commentary to the Provisions states that timehistory analysis is superior
to responsespectrum analysis. Nevertheless, the timehistory analysis performed later will be scaled as required by the Provisions.
310
Table 3.13 Floor Mass, Mass Moment of Inertia, and Center of Mass Locations
Level
Weight
(kips)
Mass
(kipsec2/in.)
Mass Moment of
Inertia (in.kipsec2//
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
S
1656.5
1595.8
1595.8
1595.8
3403.0
2330.8
2330.8
2330.8
4323.8
3066.1
3066.1
3097.0
6526.3
36918.6
4.287
4.130
4.130
4.130
8.807
6.032
6.032
6.032
11.190
7.935
7.935
8.015
16.890
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
1260
1260
1260
1260
1637
1551
1551
1551
1159
1260
1260
1260
1260
1050
1050
1050
1050
1175
1145
1145
1145
1212
1194
1194
1193
1187
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
3.1.4.2 Period of Vibration
3.1.4.2.1 Approximate Period of Vibration
The formula in Provisions Eq. 5.4.2.11 [5.26] is used to estimate the building period:
x
Ta=Crhn
where Cr = 0.028 and x = 0.8 for a steel moment frame from Provisions Table 5.4.2.1 [5.22]. Using hn =
the total building height (above grade) = 155.5 ft, Ta = 0.028(155.5)0.8 = 1.59 sec.
When the period is computed from a properly substantiated analysis, the Provisions requires that the
computed period not exceed CuTa where Cu = 1.4 (from Provisions Table 5.4.2 [5.21] using SD1 =
0.373g). For the structure under consideration, CuTa = 1.4(1.59) = 2.23 seconds. When a modalresponse
spectrum is used, Provisions Sec. 5.5.7 [5.3.7] requires that the displacements, drift, and member design
forces be scaled to a value consistent with 85 percent of the equivalent lateral force base shear computed
using the period CuTa = 2.23 sec. Provisions Sec. 5.6.3 [5.4.3] requires that timehistory analysis results
be scaled up to an ELF shear consistent with T = CuTa (without the 0.85 factor).1
Note that when the accurately computed period (such as from a Rayleigh analysis) is less than the
approximate value shown above, the computed period should be used. In no case, however, must a period
less than Ta = 1.59 seconds be used. 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.
Chapter 3, Structural Analysis
2For an explanation of the use of the virtual force technique, see “Economy of Steel Framed Structures Through Identification
of Structural Behavior” by F. Charney, Proceedings of the 1993 AISC Steel Construction Conference, Orlando, Florida, 1993.
311
3.1.4.3 Damping
When a modalresponsespectrum analysis is performed, the structure’s damping is included in the
response spectrum. A damping ratio of 0.05 (5 percent of critical) is appropriate for steel structures. This
is consistent with the level of damping assumed in the development of the mapped spectral acceleration
values.
When recombining the individual modal responses, 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, Provisions Sec. 5.5.7 [5.3.7] requires that the CQC approach be used when the modes
are closely spaced. When using CQC, the analyst must correctly specify a damping factor. This factor
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 timehistory 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.
3.1.5 Equivalent Lateral Force Analysis
Prior to performing modal or timehistory analysis, it is often necessary to perform an equivalent lateral
force (ELF) analysis of the structure. This analysis typically is used for preliminary design and for
assessing the threedimensional response characteristics of the structure. ELF analysis is also useful for
investigating the behavior of driftcontrolled structures, particularly when a virtual force analysis is used
for determining member displacement participation factors.2 The virtual force techniques cannot be used
for modalresponsespectrum analysis because signs are lost in the CQC combinations.
In anticipation of the “true” 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 3.13) is the total weight minus the grade level weight, or 36918.6  6526.3 = 30392.3 kips.
3.1.5.1 Base Shear and Vertical Distribution of Force
Using Provisions Eq. 5.4.1 [5.21], the total seismic shear is:
V=CSW
where W is the total weight of the structure. From Provisions Eq. 5.4.1.11 [5.22], the maximum
(constant acceleration region) spectral acceleration is:
0.833 0.104
max ( / ) (8/1)
DS
S
C S
R I
= = =
FEMA 451, NEHRP Recommended Provisions: Design Examples
312
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Period, sec
Spectral acceleration, g
Equation 5.4.1.13
Equation 5.4.1.12
T = 2.23 sec
Figure 3.15 Computed ELF total acceleration response spectrum.
Provisions Eq. 5.4.1.12 [5.23] controls in the constant velocity region:
0.373 0.021
( / ) 2.23(8/1)
D1
S
C S
T R I
= = =
However, the acceleration must not be less than that given by Provisions Eq. 5.4.1.13 [replaced by 0.010
in the 2003 Provisions]:
0.044 0.044(1)(0.833) 0.037 CSmin= ISDS= =
[With the change of this base shear equation, the result of Eq. 5.23 would control, reducing the design
base shear significantly. This change would also result in removal of the horizontal line in Figure 3.15
and the corresponding segment of Figure 3.16.]
The value from Eq. 5.4.1.13 [not applicable in the 2003 Provisions] controls for this building. Using W
= 30,392 kips, V = 0.037(30,392) = 1,124 kips. The acceleration response spectrum given by the above
equations is plotted in Figure 3.15.
Chapter 3, Structural Analysis
313
0
1
2
3
4
5
6
7
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Period, sec
Displacement, in.
Equation 5.4.4.13
Equation 5.4.4.12
T = 2.23 sec
Figure 3.16 Computed ELF relative displacement response spectrum (1.0 in. = 25.4 mm).
While it is certainly reasonable to enforce a minimum base shear, Provisions Sec. 5.4.6.1 has correctly
recognized that displacements predicted using Eq. 5.4.1.13 are not reasonable. Therefore, it is very
important to note that Provisions Eq. 5.4.1.13, when it controls, should be used for determining member
forces, but should not be used for computing drift. For drift calculations, forces computed according to
Eq. 5.4.1.12 [5.23]should be used. The effect of using Eq. 5.4.1.13 for drift is shown in Figure 3.16,
where it can be seen that the fine line, representing Eq. 5.4.1.13, will predict significantly larger
displacements than Eq. 5.4.1.12 [5.23].
[The minimum base shear is 1% of the weight in the 2003 Provisions (CS = 0.01). For this combination
of SD1 and R, the new minimum controls for periods larger than 4.66 second. The minimum base shear
equation for nearsource sites (now triggered in the Provisions by S1 greater than or equal to 0.6) has been
retained.]
In this example, all ELF analysis is performed using the forces obtained from Eq. 5.4.1.13, but for the
purposes of computing drift, the story deflections computed using the forces from Eq. 5.4.1.13 are
multiplied by the ratio (0.021/0.037 = 0.568).
The base shear computed according to Provisions Eq. 5.4.1.13 is distributed along the height of the
building using Provisions Eq. 5.4.3.1 and 5.4.3.2 [5.210 and 5.211]:
Fx =CvxV
and
1
k
x
vx n k
i i
i
C w h
wh
=
=
S
FEMA 451, NEHRP Recommended Provisions: Design Examples
314
where k = 0.75 + 0.5T = 0.75 + 0.5(2.23) = 1.86. The story forces, story shears, and story overturning
moments are summarized in Table 3.14.
Table 3.14 Equivalent Lateral Forces for Building Responding in X and Y Directions
Level
x
wx
(kips)
hx
(ft) wxhx
k Cvx
Fx
(kips)
Vx
(kips)
Mx
(ftkips)
R 1656.5 155.5 20266027 0.1662 186.9 186.9 2336
12 1595.8 143.0 16698604 0.1370 154.0 340.9 6597
11 1595.8 130.5 14079657 0.1155 129.9 470.8 12482
10 1595.8 118.0 11669128 0.0957 107.6 578.4 19712
9 3403.0 105.5 20194253 0.1656 186.3 764.7 29271
8 2330.8 93.0 10932657 0.0897 100.8 865.5 40090
7 2330.8 80.5 8352458 0.0685 77.0 942.5 51871
6 2330.8 68.0 6097272 0.0500 56.2 998.8 64356
5 4323.8 55.5 7744119 0.0635 71.4 1070.2 77733
4 3066.1 43.0 3411968 0.0280 31.5 1101.7 91505
3 3066.1 30.5 1798066 0.0147 16.6 1118.2 103372
2 3097.0 18.0 679242 0.0056 6.3 1124.5 120694
S 30392.3  121923430 1.00 1124.5
1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN.
3.1.5.2 Accidental Torsion and Orthogonal Loading Effects
When using the ELF method as the basis for structural design, two effects must be added to the direct
lateral forces shown in Table 3.14. The first of these effects accounts for the fact that the earthquake can
produce inertial forces that act in any direction. For SDC D, E, and F buildings, Provisions Sec. 5.2.5.2.3
[4.4.2.3] requires that the structure be investigated for forces that act in the direction that causes the
“critical load effect.” Since this direction is not easily defined, the Provisions 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 (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. This requirement, given in Provisions Sec. 5.4.4.2
[5.2.4.2], can be satisfied for torsionally regular buildings by applying the equivalent lateral force at an
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 structures in SDC C, D, E, or F, these accidental eccentricities (and inherent torsion) must be
amplified if the structure is classified as torsionally irregular. According to Provisions Table 5.2.3.2, a
torsional irregularity exists if:
max 1.2
avg
d
d
=
where, as shown in Figure 3.17, dmax is the maximum displacement at the edge of the floor diaphragm,
and davg is the average displacement of the diaphragm. If the ratio of displacements is greater than 1.4, the
torsional irregularity is referred to as “extreme.” In computing the displacements, the structure must be
loaded with the basic equivalent lateral forces applied at a 5 percent eccentricity.
Chapter 3, Structural Analysis
315
B
.
d average
d minimum
d maximum
Figure 3.17 Amplification of accidental
torsion.
The analysis of the structure for accidental torsion was performed on SAP2000. The same model was
used for ELF, modalresponsespectrum, and modaltimehistory analysis. The following approach was
used for the mathematical model of the structure:
1. The floor diaphragm was modeled as infinitely rigid inplane and infinitely flexible outofplane.
Shell elements were used to represent the diaphragm mass. Additional point masses were used to
represent cladding and other concentrated masses.
2. Flexural, shear, axial, and torsional deformations were included in all columns. Flexural, shear, and
torsional deformations were included in the beams. Due to the rigid diaphragm assumption, axial
deformation in beams was neglected.
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 in light of the fact that most of the girders are on the perimeter of the building
and are under reverse curvature.
5. Except for those lateralloadresisting columns that terminate at Levels 5 and 9, all columns were
assumed to be fixed at their base.
The results of the accidental torsion analysis are shown in Tables 3.15 and 3.16. As may be observed,
the largest ratio of maximum to average floor displacements is 1.16 at Level 5 of the building under Y
direction loading. Hence, this structure is not torsionally irregular and the story torsions do not need to be
amplified.
FEMA 451, NEHRP Recommended Provisions: Design Examples
316
Table 3.15 Computation for Torsional Irregularity with ELF Loads Acting in X Direction
Level d1 (in.) d2 (in.) davg (in.) dmax (in.) dmax/davg Irregularity
R 6.04 7.43 6.74 7.43 1.10 none
12 5.75 7.10 6.43 7.10 1.11 none
11 5.33 6.61 5.97 6.61 1.11 none
10 4.82 6.01 5.42 6.01 1.11 none
9 4.26 5.34 4.80 5.34 1.11 none
8 3.74 4.67 4.21 4.67 1.11 none
7 3.17 3.96 3.57 3.96 1.11 none
6 2.60 3.23 2.92 3.23 1.11 none
5 2.04 2.52 2.28 2.52 1.11 none
4 1.56 1.91 1.74 1.91 1.10 none
3 1.07 1.30 1.19 1.30 1.10 none
2 0.59 0.71 0.65 0.71 1.09 none
Tabulated displacements are not amplified by Cd. Analysis includes accidental torsion. 1.0 in. = 25.4 mm.
Table 3.16 Computation for Torsional Irregularity with ELF Loads Acting in Y Direction
Level d1 (in.) d2 (in.) davg (in.) dmax (in) dmax/davg Irregularity
R 5.88 5.96 5.92 5.96 1.01 none
12 5.68 5.73 5.71 5.73 1.00 none
11 5.34 5.35 5.35 5.35 1.00 none
10 4.92 4.87 4.90 4.92 1.01 none
9 4.39 4.29 4.34 4.39 1.01 none
8 3.83 3.88 3.86 3.88 1.01 none
7 3.19 3.40 3.30 3.40 1.03 none
6 2.54 2.91 2.73 2.91 1.07 none
5 1.72 2.83 2.05 2.38 1.16 none
4 1.34 1.83 1.59 1.83 1.15 none
3 0.93 1.27 1.10 1.27 1.15 none
2 0.52 0.71 0.62 0.71 1.15 none
Tabulated displacements are not amplified by Cd. Analysis includes accidental torsion. 1.0 in. = 25.4 mm.
3.1.5.3 Drift and PDelta Effects
Using the basic structural configuration shown in Figure 3.11 and the equivalent lateral forces shown in
Table 3.14, the total story deflections were computed as shown in the previous section. In this section,
story drifts are computed and compared to the allowable drifts specified by the Provisions.
The results of the analysis are shown in Tables 3.17 and 3.18. The tabulated drift values are somewhat
different from those shown in Table 3.15 because the analysis for drift did not include accidental torsion,
whereas the analysis for torsional irregularity did. In Tables 3.17 and 3.18, the values in the first
numbered column are the average story displacements computed by the SAP2000 program using the
lateral forces of Table 3.14. Average story drifts are used here instead of maximum story drifts because
this structure does not have a “significant torsional response.” If the torsional effect were significant, the
maximum drifts at the extreme edge of the diaphragm would need to be checked.
The values in column 2 of Tables 3.17 and 3.18 are the story drifts as reported by SAP2000. These drift
values, however, are much less than those that will actually occur because the structure will respond
inelastically to the earthquake. The true inelastic story drift, which by assumption is equal to Cd = 5.5
Chapter 3, Structural Analysis
317
times the SAP2000 drift, is shown in Column 3. As discussed above in Sec. 3.1.5.1, the values in column
4 are multiplied by 0.568 to scale the results to the base shear calculated ignoring Provisions Eq. 5.4.1.13
since that limit does not apply to drift checks. [Recall that the minimum base shear is different in the
2003 Provisions.] The allowable story drift of 2.0 percent of the story height per Provisions Table 5.28
is shown in column 5. (Recall that this building is assigned to Seismic Use Group I.) It is clear from
Tables 3.17 and 3.18 that the allowable drift is not exceeded at any level.
Table 3.17 ELF Drift for Building Responding in X Direction
Level
1
Total Drift
from SAP2000
(in.)
2
Story Drift from
SAP2000
(in.)
3
Inelastic Story
Drift
(in.)
4
Inelastic Drift
Times 0.568
(in.)
5
Allowable Drift
(in.)
R
12
11
10
9
8
7
6
5
4
3
2
6.71
6.40
5.95
5.39
4.77
4.19
3.55
2.90
2.27
1.73
1.18
0.65
0.32
0.45
0.56
5.39
0.59
0.64
0.65
0.63
0.55
0.55
0.54
0.65
1.73
2.48
3.08
3.38
3.22
3.52
3.58
3.44
3.00
3.00
2.94
3.55
0.982
1.41
1.75
1.92
1.83
2.00
2.03
1.95
1.70
1.70
1.67
2.02
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
4.32
Column 4 adjusts for Provisions Eq. 5.4.1.12 (for drift) vs 5.4.1.13 (for strength). [Such a modification is not
necessary when using the 2003 Provisions because the minimum base shear is different. Instead, the design forces
applied to the model, which produce the drifts in Columns 1 and 2, would be lower by a factor of 0.568.]
1.0 in. = 25.4 mm.
FEMA 451, NEHRP Recommended Provisions: Design Examples
318
Table 3.18 ELF Drift for Building Responding in Y Direction
Level
1
Total Drift
from SAP2000
(in.)
2
Story Drift from
SAP2000
(in.)
3
Inelastic Story
Drift
(in.)
4
Inelastic Drift
Times 0.568
(in.)
5
Allowable Drift
(in.)
R
12
11
10
9
8
7
6
5
4
3
2
6.01
5.79
5.43
4.98
4.32
3.83
3.26
2.68
2.05
1.59
1.10
0.61
0.22
0.36
0.45
0.67
0.49
0.57
0.58
0.64
0.46
0.49
0.49
0.61
1.21
1.98
2.48
3.66
2.70
3.11
3.19
3.49
2.53
2.67
2.70
3.36
0.687
1.12
1.41
2.08
1.53
1.77
1.81
1.98
1.43
1.52
1.53
1.91
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
4.32
Column 4 adjusts for Provisions Eq. 5.4.1.12 (for drift) vs 5.4.1.13 (for strength). [Such a modification is not
necessary when using the 2003 Provisions because the minimum base shear is different. Instead, the design forces
applied to the model, which produce the drifts in Columns 1 and 2, would be lower by a factor of 0.568.]
1.0 in. = 25.4 mm.
3.1.5.3.1 Using ELF Forces and Drift to Compute Accurate Period
Before continuing with the example, it is helpful 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 is usually 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:
Kf =.2Mf
where K is the structure stiffness matrix, M is the (diagonal) mass matrix, and f, is a vector containing
the components of the mode shape associated with ..
If an approximate mode shape d is used instead of f, where d is the deflected shape under the
equivalent lateral forces F, the frequency . can be closely approximated. Making the substitution of
d for f, premultiplying both sides of the above equation by dT (the transpose of the displacement
vector), noting that F = Kd, and M = (1/g)W, the following is obtained:
2
TF 2TM TW
g
.
d =. d d= d d
where W is a vector containing the story weights and g is the acceleration due to gravity (a scalar).
After rearranging terms, this gives:
T
T
g F
W
d
.
d d
=
Chapter 3, Structural Analysis
319
Using the relationship between period and frequency, T 2 .
p
.
=
Using F from Table 3.14 and d from Column 1 of Tables 3.17 and 3.18, the periods of vibration are
computed as shown in Tables 3.19 and 3.110 for the structure loaded in the X and Y directions,
respectively. As may be seen from the tables, the Xdirection period of 2.87 seconds and the Ydirection
period of 2.73 seconds are much greater than the approximate period of Ta = 1.59 seconds and also exceed
the upper limit on period of CuTa = 2.23 seconds.
Table 3.19 Rayleigh Analysis for XDirection Period of Vibration
Level Drift, d (in.) Force, F (kips) Weight, W (kips) dF (in.kips) d2W/g
(in.kipssec2)
R 6.71 186.9 1656 1259.71 194.69
12 6.40 154.0 1598 990.22 170.99
11 5.95 129.9 1598 775.50 147.40
10 5.39 107.6 1598 583.19 121.49
9 4.77 186.3 3403 894.24 202.91
8 4.19 100.8 2330 424.37 106.88
7 3.55 77.0 2330 274.89 76.85
6 2.90 56.2 2330 164.10 51.41
5 2.27 71.4 4323 162.79 58.16
4 1.73 31.5 3066 54.81 24.02
3 1.18 16.6 3066 19.75 11.24
2 0.65 6.3 3097 4.10 3.39
S 5607.64 1169.42
. = (5607/1169)0.5 = 2.19 rad/sec. T = 2p/. = 2.87 sec. 1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
FEMA 451, NEHRP Recommended Provisions: Design Examples
320
Table 3.110 Rayleigh Analysis for YDirection Period of Vibration
Level Drift, d (in.) Force, F (kips) Weight, W (kips) dF d2W/g
R 6.01 186.9 1656 1123.27 154.80
12 5.79 154.0 1598 891.66 138.64
11 5.43 129.9 1598 705.36 121.94
10 4.98 107.6 1598 535.85 102.56
9 4.32 186.3 3403 804.82 164.36
8 3.83 100.8 2330 386.06 88.45
7 3.26 77.0 2330 251.02 64.08
6 2.68 56.2 2330 150.62 43.31
5 2.05 71.4 4323 146.37 47.02
4 1.59 31.5 3066 50.09 20.06
3 1.10 16.6 3066 18.26 9.60
2 0.61 6.3 3097 3.84 2.98
S 5067.21 957.81
. = (5067/9589)0.5 = 2.30 rad/sec. T = 2p/. = 2.73 sec. 1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
3.1.5.3.2 PDelta Effects
Pdelta effects are computed for the Xdirection response in Table 3.111. The last column of the table
shows the story stability ratio computed according to Provisions Eq. 5.4.6.21 [5.216]:
x
x sx d
P
V h C
.
.
=
[In the 2003 Provisions, the equation for the story stability ratio was changed by introducing the
importance factor (I) to the numerator. As previously formulated, larger axial loads (Px) would be
permitted where the design shears (Vx) included an importance factor greater than 1.0; that effect was
unintended.]
Provisions Eq. 5.4.6.22 places an upper limit on .:
0.5
max
Cd
.
ß
=
where ß is the ratio of shear demand to shear capacity for the story. Conservatively taking ß = 1.0 and
using Cd = 5.5, .max = 0.091. [In the 2003 Provisions, this upper limit equation has been eliminated.
Instead, the Provisions require that where . > 0.10 a special analysis be performed in accordance with
Sec. A5.2.3. This example constitutes a borderline case as the maximum stability ratio (at Level 3, as
shown in Table 3.111) is 0.103.]
The . terms in Table 3.111 below are taken from Column 3 of Table 3.17 because these are consistent
with the ELF story shears of Table 3.14 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 as it would appear in both the numerator and denominator.)
Chapter 3, Structural Analysis
321
Table 3.111 Computation of PDelta Effects for XDirection Response
Level hsx (in.) . (in.) PD (kips) PL (kips) PT (kips) PX (kips) VX(kips) .X
R 150 1.73 1656.5 315.0 1971.5 1971.5 186.9 0.022
12 150 2.48 1595.8 315.0 1910.8 3882.3 340.9 0.034
11 150 3.08 1595.8 315.0 1910.8 5793.1 470.8 0.046
10 150 3.38 1595.8 315.0 1910.8 7703.9 578.4 0.055
9 150 3.22 3403.0 465.0 3868.0 11571.9 764.7 0.059
8 150 3.52 2330.8 465.0 2795.8 14367.7 865.8 0.071
7 150 3.58 2330.8 465.0 2795.8 17163.5 942.5 0.079
6 150 3.44 2330.8 465.0 2795.8 19959.3 998.8 0.083
5 150 3.00 4323.8 615.0 4938.8 24898.1 1070.2 0.085
4 150 3.00 3066.1 615.0 3681.1 28579.2 1101.7 0.094
3 150 2.94 3066.1 615.0 3681.1 32260.3 1118.2 0.103
2 216 3.55 3097.0 615.0 3712.0 35972.3 1124.5 0.096
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
The gravity force terms include a 20 psf uniform live load over 100 percent of the floor and roof area.
The stability ratio just exceeds 0.091 at Levels 2 through 4. However, ß was very conservatively taken as
1.0. Because a more refined analysis would most likely show a lower value of ß, we will proceed
assuming that Pdelta effects are not a problem for this structure. Calculations for the Y direction
produced similar results, but are not included herein.
3.1.5.4 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.
3.1.5.4.1 Orthogonal Loading Effects and Accidental Torsion
For a nonsymmetric structure such as the one being analyzed, four directions of seismic force (+X, X,
+Y, Y) must be considered and, for each direction of force, there are two possible directions for 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. When the 30 percent orthogonal loading
rule is applied, 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 3.18 illustrates the basic possibilities of application of load. Although this figure shows 16
different load combinations, it may be observed that eight of these combinations – 7, 8, 5, 6, 15, 16, 13,
and 14 – are negatives of one of Combinations 1, 2, 3, 4, 9, 10, 11, and 12, respectively.
FEMA 451, NEHRP Recommended Provisions: Design Examples
322
4 8 12 16
3
2
1 5 9 13
14
7 11 15
6 10
Figure 3.18 Basic load cases used in ELF analysis.
3.1.5.4.2 Load Combinations
The basic load combinations for this structure come from ASCE 7 with the earthquake loadings modified
according to Provisions Sec. 5.2.7 [4.2.2.1].
The basic ASCE 7 load conditions that include earthquake are:
1.4D + 1.2L + E + 0.2S
and
0.9D + E
From Provisions Eq. 5.2.71 and Eq. 5.2.72 [4.21 and 4.22]:
E = .QE + 0.2SDSQD
and
E = .QE  0.2SDSQD
Chapter 3, Structural Analysis
323
where . is a redundancy factor (explained later), QE is the earthquake load effect, QD is the dead load
effect, and SDS is the short period spectral design acceleration.
Using SDS = 0.833 and assuming the snow load is negligible in Stockton, California, the basic load
combinations become:
1.37D + 0.5L + .E
and
0.73D + .E
[The redundancy requirements have been changed substantially in the 2003 Provisions. Instead of
performing the calculations that follow, 2003 Provisions Sec. 4.3.3.2 would require that an analysis
determine the most severe effect on story strength and torsional response of loss of moment resistance at
the beamtocolumn connections at both ends of any single beam. Where the calculated effects fall within
permitted limits, or the system is configured so as to satisfy prescriptive requirements in the exception,
the redundancy factor is 1.0. Otherwise, . = 1.3. Although consideration of all possible single beam
failures would require substantial effort, in most cases an experienced analyst would be able to identify a
few critical elements that would be likely to produce the maximum effects and then explicitly consider
only those conditions.]
Based on Provisions Eq. 5.2.4.2, the redundancy factor (.) is the largest value of .x computed for each
story:
2 20
x
x
rmax Ax
. = 
In this equation, is a ratio of element shear to story shear, and Ax is the area of the floor diaphragm rmaxx
immediately above the story under consideration; .x need not be taken greater than 1.5, but it may not be
less than 1.0. [In the 2003 Provisions, . is either 1.0 or 1.3.]
For this structure, the check is illustrated for the lower level only where the area of the diaphragm is
30,750 ft2. Figure 3.11 shows that the structure has 18 columns resisting load in the X direction and 18
columns resisting load in the Y direction. If it is assumed that each of these columns equally resists base
shear and the check, as specified by the Provisions, is made for any two adjacent columns:
2/18 0.111 and . rmaxx = = 2 20 0.963
0.11 30750 . x =  =
Checks for upper levels will produce an even lower value of .x; therefore, .x may be taken a 1.0 for this
structure. Hence, the final load conditions to be used for design are:
1.37D + 0.5L +E
and
0.73D + E
FEMA 451, NEHRP Recommended Provisions: Design Examples
324
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 ASCE 7 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.
3.1.5.4.3 Setting up the Load Combinations in SAP2000
The load combinations required for the analysis are shown in Table 3.112.
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 responsespectrum 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.
Chapter 3, Structural Analysis
325
Table 3.112 Seismic and Gravity Load Combinations as Run on SAP 2000
Run Combination Lateral* Gravity
A B 1 (Dead) 2 (Live)
One 1 [1] 1.37 0.5
2 [1] 0.73 0.0
3 [1] 1.37 0.5
4 [1] 0.73 0.0
5 [2] 1.37 0.5
6 [2] 0.73 0.0
7 [2] 1.37 0.5
8 [2] 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 [3] 1. 37 0.5
6 [3] 0.73 0.0
7 [4] 1. 37 0.5
8 [4] 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 [9] 1. 37 0.5
6 [9] 0.73 0.0
7 [10] 1. 37 0.5
8 [10] 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 [11] 1. 37 0.5
6 [11] 0.73 0.0
7 [12] 1. 37 0.5
8 [12] 0.73 0.0
* Numbers in brackets [#] represent load conditions shown in Figure 3.18. A negative sign [#] indicates that
all lateral load effects act in the direction opposite that shown in the figure.
3.1.5.4.4 Member Forces
For this portion of the analysis, the earthquake shears in the girders along Gridline 1 are computed. This
analysis considers only 100 percent of the Xdirection forces applied in combination with 30 percent of
the (positive or negative) Ydirection forces. The Xdirection forces are applied with a 5 percent
accidental eccentricity to produces a clockwise rotation of the floor plates. The Ydirection forces are
applied without eccentricity.
The results of the member force analysis are shown in Figure 3.19. In a later part of this example, the
girder shears are compared to those obtained from modalresponsespectrum and modaltimehistory
analyses.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3It should be emphasized that, in general, the principal direction of structural response will not coincide with one of the axes
used to describe the structure in threedimensional space.
326
8.31 9.54 9.07
R12 16.1 17.6 17.1
1211 25.8 26.3 26.9
1110 31.2 31.0 32.9
109 32.7 32.7 30.4 28.9 12.5
98 34.5 34.1 32.3 36.0 22.4
87 39.1 38.1 36.5 39.2 24.2
76 40.4 38.4 37.2 39.6 24.8
65 13.1 30.0 31.7 34.3 33.1 34.9 22.2
54 22.1 33.6 29.1 31.0 30.1 31.6 20.4
43 22.0 33.0 30.5 31.7 31.1 32.2 21.4
32 20.9 33.0 30.9 31.8 31.1 32.4 20.4
2G
Figure 3.19 Seismic shears in girders (kips) as computed using ELF analysis. Analysis includes
orthogonal loading and accidental torsion. (1.0 kip = 4.45 kn)
3.1.6 ModalResponseSpectrum Analysis
The first step in the modalresponsespectrum analysis is the computation of the structural mode shapes
and associated periods of vibration. Using the Table 3.14 structural masses and the same mathematical
model as used for the ELF and the Rayleigh analyses, the mode shapes and frequencies are automatically
computed by SAP2000.
The computed periods of vibration for the first 10 modes are summarized in Table 3.113, which also
shows values called the modal direction factor for each mode. Note that the longest period, 2.867
seconds, is significantly greater than CuTa = 2.23 seconds. Therefore, displacements, drift, and member
forces as computed from the true modal properties may have to be scaled up to a value consistent with 85
percent of the ELF base shear using T = CuTa. The smallest period shown in Table 3.113 is 0.427
seconds.
The modal direction factors shown in Table 3.113 are indices that quantify the direction of the mode. A
direction factor of 100.0 in any particular direction would indicate that this mode responds entirely along
one of the orthogonal (X, Y or .Z axes) of the structure.3 As Table 3.113 shows, the first mode is
predominantly X translation, the second mode is primarily Y translation, and the third mode is largely
Chapter 3, Structural Analysis
327
torsional. Modes 4 and 5 also are nearly unidirectional, but Modes 6 through 10 have significant
lateraltorsional coupling. Plots showing the first eight mode shapes are given in Figure 3.110.
It is interesting to note that the Xdirection Rayleigh period (2.87 seconds) is virtually identical to the first
mode predominately Xdirection period (2.867 seconds) computed from the eigenvalue analysis.
Similarly, the Ydirection Rayleigh period (2.73 seconds) is very close to second mode predominantly
Ydirection period (2.744 seconds) from the eigenvalue analysis. The closeness of the Rayleigh and
eigenvalue periods of this building arises from the fact that the first and seconds 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.
In Table 3.114, the effective mass in Modes 1 through 10 is given as a percentage of total mass. The
values shown in parentheses in Table 3.114 are the accumulated effective masses and should total 100
percent of the total mass when all modes are considered. By Mode 10, the accumulated effective mass
value is approximately 80 percent of the total mass for the translational modes and 72 percent of the total
mass for the torsional mode. Provisions Sec. 5.5.2 [5.3.2] requires that a sufficient number of modes be
represented to capture at least 90 percent of the total mass of the structure. On first glance, it would seem
that the use of 10 modes as shown in Table 3.114 violates this rule. However, approximately 18 percent
of the total mass for this structure is located at grade level and, as this level is extremely stiff, this mass
does not show up as an effective mass until Modes 37, 38, and 39 are considered. In the case of the
building modeled as a 13story building with a very stiff first story, the accumulated 80 percent of
effective translational mass in Mode 10 actually represents almost 100 percent of the dynamically
excitable mass. In this sense, the Provisions requirements are clearly met when using only the first 10
modes in the response spectrum or timehistory analysis. For good measure, 14 modes were used in the
actual analysis.
FEMA 451, NEHRP Recommended Provisions: Design Examples
328
Mode 1 T = 2.87 sec Mode 2 T = 2.74 sec
Mode 3 T = 1.57 sec Mode 4 T = 1.15 sec
Mode 5 T = 1.07 sec Mode 6 T = 0.72 sec
Mode 7 T = 0.70 sec Mode 8 T = 0.63 sec
Y Z
X
Y Z
X
Y Z
X
Y Z
X
Y Z
X
Y Z
X
Y Z
X
Y Z
X
Figure 3.110 Mode shapes as computed using SAP2000.
Chapter 3, Structural Analysis
329
Table 3.113 Computed Periods and Direction Factors
Mode Period
(seconds)
Modal Direction Factor
X Translation Y Translation Z Torsion
1
2
3
4
5
6
7
8
9
10
2.867
2.745
1.565
1.149
1.074
0.724
0.697
0.631
0.434
0.427
99.2
0.8
1.7
98.2
0.4
7.9
91.7
0.3
30.0
70.3
0.7
99.0
9.6
0.8
92.1
44.4
5.23
50.0
5.7
2.0
0.1
0.2
88.7
1.0
7.5
47.7
3.12
49.7
64.3
27.7
Table 3.114 Computed Periods and Effective Mass Factors
Mode Period
(seconds)
Effective Mass Factor
X Translation Y Translation Z Torsion
1
2
3
4
5
6
7
8
9
10
2.867
2.744
1.565
1.149
1.074
0.724
0.697
0.631
0.434
0.427
64.04 (64.0)
0.51 (64.6)
0.34 (64.9)
10.78 (75.7)
0.04 (75.7)
0.23 (75.9)
2.94 (78.9)
0.01 (78.9)
0.38 (79.3)
1.37 (80.6)
0.46 (0.5)
64.25 (64.7)
0.93 (65.6)
0.07 (65.7)
10.64 (76.3)
1.08 (77.4)
0.15 (77.6)
1.43 (79.0)
0.00 (79.0)
0.01 (79.0)
0.04 (0.0)
0.02 (0.1)
51.06 (51.1)
0.46 (51.6)
5.30 (56.9)
2.96 (59.8)
0.03 (59.9)
8.93 (68.8)
3.32 (71.1)
1.15 (72.3)
3.1.6.1 Response Spectrum Coordinates and Computation of Modal Forces
The coordinates of the response spectrum are based on Provisions Eq. 4.1.2.61 and 4.1.2.62 [3.35 and
3.36]. [In the 2003 Provisions, the design response spectrum has reduced ordinates at very long periods
as indicated in Sec. 3.3.4. The new portion of the spectrum reflects a constant ground displacement at
periods greater than TL, the value of which is based on the magnitude of the source earthquake that
dominates the probabilistic ground motion at the site.]
For periods less than T0:
0.6 DS 0.4
a DS
0
S ST S
T
= +
and for periods greater than TS:
FEMA 451, NEHRP Recommended Provisions: Design Examples
330
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Period, sec
Acceleration, g
I=1, R=1
I=1, R=8
Figure 3.111 Total acceleration response spectrum used in analysis.
D1
a
S S
T
=
where T0= 0.2SDS/SD1andTS=SD1/SDS.
Using SDS = 0.833 and SD1 = 0.373, T0 = 0.089 seconds and TS = 0.448 seconds. The computed responsespectrum
coordinates for several period values are shown in Table 3.115 and the response spectrum,
shown with and without the I/R =1/8 modification, is plotted in Figure 3.111. The spectrum does not
include the high period limit on Cs (Cs, min = 0.044ISDS), which controlled the ELF base shear for this
structure and which ultimately will control the scaling of the results from the responsespectrum 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 3.115 Response Spectrum Coordinates
Tm (seconds) Csm Csm(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 = 8.7.
Chapter 3, Structural Analysis
331
Using the response spectrum coordinates of Table 3.115, the responsespectrum analysis was carried out
using SAP2000. As mentioned above, the first 14 modes of response were computed and superimposed
using complete quadratic combination (CQC). 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. The response spectrum used in the analysis did include I/R.
3.1.6.1.1 Dynamic Base Shear
After specifying member “groups,” SAP2000 automatically computes and prints the CQC story shears.
Groups were defined such that total shears would be printed for each story of the structure. The base
shears were printed as follows:
Xdirection base shear = 437.7 kips
Ydirection base shear = 454.6 kips
These values are much lower that the ELF base shear of 1124 kips. Recall that the ELF base shear was
controlled by Provisions Eq. 5.4.1.13. The modalresponsespectrum shears are less than the ELF shears
because the fundamental period of the structure used in the responsespectrum analysis is 2.87 seconds
(vs 2.23) and because the response spectrum of Figure 3.111 does not include the minimum base shear
limit imposed by Provisions Eq. 5.4.1.13. [Recall that the equation for minimum base shear coefficient
does not appear in the 2003 Provisions.]
According to Provisions Sec. 5.5.7 [5.3.7], the base shears from the modalresponsespectrum 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 and displacements 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:
Xdirection scale factor = 0.85(1124)/437.7 = 2.18
Ydirection scale factor = 0.85(1124)/454.6 = 2.10
The computed and scaled story shears are as shown in Table 3.116. 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 3.111 and Table 3.14. However, it is clear that the vertical distribution of forces is somewhat
similar when computed by ELF and modalresponse spectrum.
FEMA 451, NEHRP Recommended Provisions: Design Examples
332
Table 3.116 Story Shears from ModalResponseSpectrum Analysis
Story
X Direction (SF = 2.18) Y Direction (SF = 2.10)
Unscaled Shear
(kips)
Scaled Shear
(kips)
Unscaled Shear
(kips)
Scaled Shear
(kips)
R12
1211
1110
109
98
87
76
65
54
43
32
2G
82.5
131.0
163.7
191.1
239.6
268.4
292.5
315.2
358.6
383.9
409.4
437.7
180
286
358
417
523
586
638
688
783
838
894
956
79.2
127.6
163.5
195.0
247.6
277.2
302.1
326.0
371.8
400.5
426.2
454.6
167
268
344
410
521
583
635
686
782
843
897
956
1.0 kip = 4.45 kN.
3.1.6.2 Drift and PDelta Effects
According to Provisions Sec. 5.5.7 [5.3.7], the computed displacements and drift (as based on the
response spectrum of Figure 3.111) must also be scaled by the base shear factors (SF) of 2.18 and 2.10
for the structure loaded in the X and Y directions, respectively.
In Tables 3.117 and 3.118, the story displacement from the responsespectrum analysis, the scaled story
displacement, the scaled story drift, the amplified story drift (as multiplied by Cd = 5.5), and the allowable
story drift are listed. As may be observed from the tables, the allowable drift is not exceeded at any level.
Pdelta effects are computed for the Xdirection response as shown in Table 3.119. Note that the scaled
story shears from Table 3.116 are used in association with the scaled story drifts (including Cd) from
Table 3.117. The story stability factors are above the limit (.max = 0.091) only at the bottom two levels of
the structure and are only marginally above the limit. As the ß factor was conservatively set at 1.0 in
computing the limit, it is likely that a refined analysis for ß would indicate that Pdelta effects are not of
particular concern for this structure.
Chapter 3, Structural Analysis
333
Table 3.117 Response Spectrum Drift for Building Responding in X Direction
Level
1
Total Drift
from R.S.
Analysis
(in.)
2
Scaled Total
Drift
[Col1 × 2.18]
(in.)
3
Scaled Drift
(in.)
4
Scaled Story
Drift × Cd
(in.)
5
Allowable
Story Drift
(in.)
R
12
11
10
9
8
7
6
5
4
3
2
1.96
1.88
1.76
1.62
1.47
1.32
1.15
0.968
0.789
0.615
0.439
0.245
4.28
4.10
3.84
3.54
3.21
2.87
2.51
2.11
1.72
1.34
0.958
0.534
0.18
0.26
0.30
0.33
0.34
0.36
0.40
0.39
0.38
0.38
0.42
0.53
0.99
1.43
1.65
1.82
1.87
1.98
2.20
2.14
2.09
2.09
2.31
2.91
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
4.32
1.0 in. = 25.4 mm.
Table 3.118 Spectrum Response Drift for Building Responding in Y Direction
Level
1
Total Drift
from R.S.
Analysis
(in.)
2
Scaled Total
Drift
[Col1 × 2.18]
(in.)
3
Scaled Drift
(in.)
4
Scaled Story
Drift × Cd
(in.)
5
Allowable Story
Drift
(in.)
R
12
11
10
9
8
7
6
5
4
3
2
1.84
1.79
1.69
1.58
1.40
1.26
1.10
0.938
0.757
0.605
0.432
0.247
3.87
3.75
3.55
3.31
2.94
2.65
2.32
1.97
1.59
1.27
0.908
0.518
0.12
0.20
0.24
0.37
0.29
0.33
0.35
0.38
0.32
0.36
0.39
0.52
0.66
1.10
1.32
2.04
1.60
1.82
1.93
2.09
1.76
2.00
2.14
2.86
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
4.32
1.0 in. = 25.4 mm.
FEMA 451, NEHRP Recommended Provisions: Design Examples
334
Table 3.119 Computation of PDelta Effects for XDirection Response
Level hsx (in.) . (in.) PD (kips) PL (kips) PT (kips) PX (kips) VX (kips) 2X
R 150 0.99 1656.5 315.0 1971.5 1971.5 180 0.013
12 150 1.43 1595.8 315.0 1910.8 3882.3 286 0.024
11 150 1.65 1595.8 315.0 1910.8 5793.1 358 0.032
10 150 1.82 1595.8 315.0 1910.8 7703.9 417 0.041
9 150 1.87 3403.0 465.0 3868.0 11571.9 523 0.050
8 150 1.98 2330.8 465.0 2795.8 14367.7 586 0.059
7 150 2.20 2330.8 465.0 2795.8 17163.5 638 0.072
6 150 2.14 2330.8 465.0 2795.8 19959.3 688 0.075
5 150 2.09 4323.8 615.0 4938.8 24898.1 783 0.081
4 150 2.09 3066.1 615.0 3681.1 28579.2 838 0.086
3 150 2.31 3066.1 615.0 3681.1 32260.3 894 0.101
2 216 2.91 3097.0 615.0 3712.0 35972.3 956 0.092
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
3.1.6.3 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 modalresponsespectrum analysis, there
are generally two approaches that can be taken:
1. 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.
2. Compute the effects of accidental torsion by creating a load condition with the story torques applied
as static forces. Member forces created by the accidental torsion are then added directly to the results
of the responsespectrum analysis. Since the sign of member forces in the responsespectrum analysis
is lost as a result of SRSS or CQC combinations, the absolute value of the member forces resulting
from accidental torsion should be used. 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 or Y
directions. Because of the required scaling, the static torsional forces should be based on 85 percent
of the ELF forces.
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 when the different runs incorporate a different set of mode shapes and
frequencies. For structures that are torsionally regular and will not require amplification of torsion, the
second approach is preferred. For torsionally flexible structures, the first approach may be preferred
because the dynamic analysis will automatically amplify the torsional effects. In the analysis that
follows, the second approach has been used because the structure has essentially rigid diaphragms and
high torsional rigidity and amplification of accidental torsion is not required.
Chapter 3, Structural Analysis
4This method has been forwarded in the unpublished paper A Seismic Analysis Method Which Satisfies the 1988 UBC Lateral
Force Requirements, written in 1989 by Wilson, Suharwardy, and Habibullah. The paper also suggests the use of a single scale
factor, where the scale factor is based on the total base shear developed along the principal axes of the structure. As stated in the
paper, the major advantage of the method is that one set of dynamic design forces, including the effect of accidental torsion, is
produced in one computer run. In addition, the resulting structural design has equal resistance to seismic motions in all possible
directions.
335
0.3RS Y
RSX
T
0.3RS X
RSY
T
Figure 3.112 Load combinations from responsespectrum analysis.
There are three possible methods for applying the orthogonal loading rule:
1. Run the responsespectrum analysis with 100 percent of the scaled X spectrum acting in one
direction, concurrent with the application of 30 percent of the scaled Y spectrum acting in the
orthogonal direction. Use CQC for combining modal maxima. Perform a similar analysis for the
larger seismic forces acting in the Y direction.
2. Run two separate responsespectrum 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 Xdirection results with 30 percent of the scaled Ydirection results. Perform a similar
analysis for the larger loads acting in the Y direction.
3. Run two separate responsespectrum analyses, one in the X direction and one in the Ydirection, with
CQC being used for modal combinations in each analysis. Using SRSS, combine 100 percent of the
scaled Xdirection results with 100 percent of the scaled Ydirection results.4
All seismic effects can be considered in only two load cases by using Approach 2 for accidental torsion
and Approach 2 for orthogonal loading. These are shown in Figure 3.112. When the load combinations
required by Provisions Sec. 5.2.7 [4.2.2.1] are included, the total number of load combinations will
double to four.
3.1.6.4 Member Design Forces
Earthquake shear forces in the beams of Frame 1 are given in Figure 3.113 for the X direction of
response. These forces include 100 percent of the scaled Xdirection spectrum added to the 30 percent of
the scaled Ydirection spectrum. Accidental torsion is then added to the combined spectral loading. The
design force for the Level 12 beam in Bay 3 (shown in bold type in Figure 3.113) was computed as
follows:
FEMA 451, NEHRP Recommended Provisions: Design Examples
336
Force from 100 percent Xdirection spectrum = 6.94 kips (as based on CQC combination for
structure loaded with X spectrum only).
Force from 100 percent Ydirection spectrum = 1.26 kips (as based on CQC combination for structure
loaded with Y spectrum only).
Force from accidental torsion = 1.25 kips.
Scale factor for Xdirection response = 2.18.
Scale factor for Ydirection response = 2.10.
Earthquake shear force = (2.18 × 6.94) + (2.10 × 0.30 × 1.26) + (0.85 × 1.25) = 17.0 kips
9.4 9.7 9.9
R12 17.0 17.7 17.8
1211 25.0 24.9 26.0
1110 28.2 27.7 29.8
109 26.6 26.5 24.8 22.9 10.2
98 27.2 26.7 25.5 28.0 18.0
87 30.9 28.8 28.8 30.5 19.4
76 32.3 30.4 29.8 31.1 20.1
65 11.1 24.4 26.0 27.7 27.1 27.9 18.6
54 19.0 28.8 25.7 27.0 26.6 27.1 18.6
43 20.1 29.7 28.0 28.8 28.4 29.0 20.2
32 20.0 31.5 30.1 30.6 30.4 31.1 20.1
2G
Figure 3.113 Seismic shears in girders (kips) as computed using responsespectrum analysis. Analysis
includes orthogonal loading and accidental torsion (1.0 kip = 4.45 kN).
3.1.7 ModalTimeHistory Analysis
In modaltimehistory analysis, the response in each mode is computed using stepbystep integration of
the equations of motion, the modal responses are transformed to the structural coordinate system, linearly
superimposed, and then used to compute structural displacements and member forces. The displacement
and member forces for each time step in the analysis or minimum and maximum quantities (response
envelopes) may be printed.
Requirements for timehistory analysis are provided in Provisions Sec. 5.6 [5.4]. The same mathematical
model of the structure used for the ELF and responsespectrum analysis is used for the timehistory
analysis.
Chapter 3, Structural Analysis
5See Sec. 3.2.6.2 of this volume of design examples for a detailed discussion of the selected ground motions.
337
As allowed by Provisions Sec. 5.6.2 [5.4.2], the structure will be analyzed using three different pairs of
ground motion timehistories. The development of a proper suite of ground motions is one of the most
critical and difficult aspects of timehistory 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.
For the purposes of this example, however, the emphasis is on the implementation of the timehistory
approach rather than on selection of realistic ground motions. For this reason, the motion suite developed
for Example 3.2 is also used for the present example.5 The structure for Example 3.2 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 Provisions. However, a realistic comparison may still be made between the ELF, response
spectrum, and timehistory approaches.
3.1.7.1 The Seattle Ground Motion Suite
It is beneficial to provide some basic information on the Seattle motion suites in Table 3.120 below.
Refer to Figures 3.240 through 3.242 for additional information, including plots of the ground motion
time histories and 5percentdamped response spectra for each motion.
Table 3.120 Seattle Ground Motion Parameters (Unscaled)
Record Name Orientation Number of Points and
Time Increment
Peak Ground
Acceleration (g) Source Motion
Record A00 NS 8192 @ 0.005 seconds 0.443 Lucern (Landers)
Record A90 EW 8192 @ 0.005 seconds 0.454 Lucern (Landers)
Record B00 NS 4096 @ 0.005 seconds 0.460 USC Lick (Loma Prieta)
Record B90 EW 4096 @ 0.005 seconds 0.435 USC Lick (Loma Prieta)
Record C00 NS 1024 @ 0.02 seconds 0.460 Dayhook (Tabas, Iran)
Record C90 EW 1024 @ 0.02 seconds 0.407 Dayhook (Tabas, Iran)
Before the ground motions may be used in the timehistory analysis, they must be scaled using the
procedure described in Provisions Sec. 5.6.2.2 [5.4.2.2]. One scale factor will be determined for each pair
of ground motions. The scale factors for record sets A, B, and C will be called SA, SB, and SC,
respectively.
The scaling process proceeds as follows:
1. For each pair of motions (A, B, and C):
a Assume an initial scale factor (SA, SB, SC),
b. Compute the 5percentdamped elastic response spectrum for each component in the pair,
c. Compute the SRSS of the spectra for the two components, and
d. Scale the SRSS using the factor from (a) above.
2. Adjust scale factors (SA, SB, and SC) such that the average of the three scaled SRSS spectra over the
period range 0.2T1 to 1.5 T1 is not less than 1.3 times the 5percentdamped spectrum determined in
accordance with Provisions Sec. 4.1.2.6 [3.3.4]. T1 is the fundamental mode period of vibration of
FEMA 451, NEHRP Recommended Provisions: Design Examples
6The “degree of freedom” in selecting the scaling factors may be used to reduce the effect of a particularly demanding motion.
7NONLIN, developed by Finley Charney, may be downloaded at no cost at www.fema.gov/emi. To find the latest version, do
a search for NONLIN.
338
0
100
200
300
400
500
600
0 1 2 3 4 5
Period, sec
Acceleration, in./sec
Average of SRSS
NEHRP Spectrum
1.5 T 1 = 4.30 sec
T1 = 2.87 sec
0.2T 1 = 0.57 sec
2
Figure 3.114 Unscaled SRSS of spectra of ground motion pairs together with Provisions spectrum (1.0 in. =
25.4 mm).
the structure. (The factor of 1.3 more than compensates for the fact that taking the SRSS of the two
components of a ground motion effectively increases their magnitude by a factor of 1.414.)
Note that the scale factors so determined are not unique. An infinite number of different scale factors will
satisfy the above requirements, and it is up to the engineer to make sure that the selected scale factors are
reasonable.6 Because the original ground motions are similar in terms of peak ground acceleration, the
same scale factor will be used for each motion; hence, SA = SB = SC. This equality in scale factors would
not necessarily be appropriate for other suites of motions.
Given the 5percentdamped spectra of the ground motions, this process is best carried out using an Excel
spreadsheet. The spectra themselves were computed using the program NONLIN.7 The results of the
analysis are shown in Figures 3.114 and 3.115. Figure 3.114 shows the average of the SRSS of the
unscaled spectra together with the Provisions response spectrum using SDS = 0.833g (322 in./sec2) and SD1
= 0.373g (144 in./sec2). Figure 3.115 shows the ratio of the average SRSS spectrum to the Provisions
spectrum over the period range 0.573 seconds to 4.30 seconds, where a scale factor SA = SB = SC = 0.922
has been applied to each original spectrum. As can be seen, the minimum ratio of 1.3 occurs at a period
of approximately 3.8 seconds.
At all other periods, the effect of using the 0.922 scale factor to provide a minimum ratio of 1.3 over the
target period range is to have a relatively higher scale factor at all other periods if those periods
significantly contribute to the response. For example, at the structure’s fundamental mode, with T =
2.867 sec, the ratio of the scaled average SRSS to the Provisions spectrum is 1.38, not 1.30. At the higher
modes, the effect is even more pronounced. For example, at the second translational X mode, T = 1.149
Chapter 3, Structural Analysis
The Provisions is not particularly clear regarding the scaling of displacements in timehistory analysis. The first paragraph of
Sec. 5.6.3 states that member forces should be scaled, but displacements are not mentioned. The second paragraph states that
member forces and displacements should be scaled. In this example, the displacements will be scaled, mainly to be consistent
with the response spectrum procedure which, in Provisions Sec. 5.5.7, explicitly states that forces and displacements should be
scaled. See Sec. 3.1.8 of this volume of design examples for more discussion of this apparent inconsistency in the Provisions.
339
seconds and the computed ratio is 1.62. This, of course, is an inherent difficulty of using a single scale
factor to scale ground motion spectra to a target code spectrum.
When performing lineartimehistory analysis, the ground motions also should be scaled by the factor I/R.
In this case, I = 1 and R = 8, so the actual scale factor applied to each ground motion will be 0.922(1/8) =
0.115.
If the maximum base shear from any of the analyses is less than that computed from Provisions
Eq. 5.4.1.13 (Cs = 0.044ISDS), all forces and displacements8 computed from the timehistory analysis
must again be scaled such that peak base shear from the timehistory analysis is equal to the minimum
shear computed from Eq. 5.4.1.13. This is stated in Provisions Sec. 5.6.3 [5.4.3]. Recall that the base
shear controlled by Eq. 5.4.1.13 is 1124 kips in each direction. [In the 2003 Provisions base shear
scaling is still required, but recall that the minimum base shear has been revised.]
The second paragraph of Provisions Sec. 5.6.3 [5.4.3] states that if fewer than seven ground motion pairs
are used in the analysis, the design of the structure should be based on the maximum scaled quantity
among all analyses.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 1 2 3 4 5
Period, sec
Ratio
Ratio of average SRSS to NEHRP
1.3 Target
Figure 3.115 Ratio of average scaled SRSS spectrum to Provisions spectrum.
FEMA 451, NEHRP Recommended Provisions: Design Examples
340
Twelve individual timehistory analyses were carried out using SAP2000: one for each NS ground
motion acting in the X direction, one for each NS motion acting in the Y direction, one for each EW
motion acting in the X direction, and one for each EW motion acting in the Y direction. As with the
responsespectrum analysis, 14 modes were used in the analysis. Five percent of critical damping was
used in each mode. The integration timestep used in all analyses was 0.005 seconds. The results from
the analyses are summarized Tables 3.121 and 3.122.
As may be observed from Table 3.121, the maximum scaled base shears computed from the timehistory
analysis are significantly less than the ELF minimum of 1124 kips. This is expected because the ELF
base shear was controlled by Provisions Eq. 5.4.1.13. Hence, each of the analyses will need to be scaled
up. The required scale factors are shown in Table 3.122. Also shown in that table are the scaled
maximum deflections with and without Cd = 5.5.
Table 3.121 Result Maxima from TimeHistory Analysis (Unscaled)
Analysis
Maximum Base
Shear
(S.F. = 0.115)
(kips)
Time of Maximum
Shear
(sec)
Maximum Roof
Displacement
(S.F. = 0.115)
(in.)
Time of
Maximum
Displacement
(sec)
A00X 394.5 12.73 2.28 11.39
A00Y 398.2 11.84 2.11 11.36
A90X 473.8 15.42 2.13 12.77
A90Y 523.9 15.12 1.91 10.90
B00X 393.5 15.35 2.11 14.17
B00Y 475.1 14.29 1.91 19.43
B90X 399.6 13.31 1.77 16.27
B90Y 454.2 12.83 1.68 12.80
C00X 403.1 6.96 1.86 7.02
C00Y 519.2 6.96 1.70 7.02
C90X 381.5 19.40 1.95 19.38
C90Y 388.5 19.38 1.85 19.30
1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
Table 3.122 Result Maxima from TimeHistory Analysis (Scaled)
Analysis
Maximum Base
Shear
(SF = 0.115)
(kips)
Required Additional
Scale Factor for
V = 1124 kips
Adjusted
Maximum Roof
Displacement
(in.)
Adjusted Max
Roof Disp. × Cd
(in.)
A00X 394.5 2.85 6.51 35.7
A00Y 398.2 2.82 5.95 32.7
A90X 473.8 2.37 5.05 27.8
A90Y 523.9 2.15 4.11 22.6
B00X 393.5 2.86 6.03 33.2
B00Y 475.1 2.37 4.53 24.9
B90X 399.6 2.81 4.97 27.4
B90Y 454.2 2.48 4.17 22.9
C00X 403.1 2.79 5.19 28.5
C00Y 519.2 2.16 3.67 20.2
C90X 381.5 2.95 5.75 31.6
C90Y 388.5 2.89 5.35 29.4
Scaled base shear = 1124 kips for all cases. 1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.
Chapter 3, Structural Analysis
341
3.1.7.2 Drift and PDelta Effects
In this section, drift and Pdelta effects are checked only for the structure subjected to Motion A00 acting
in the X direction of the building. As can be seen from Table 3.122, this analysis produced the largest
roof displacement, but not necessarily the maximum story drift. To be sure that the maximum drift has
been determined, it would be necessary to compute the scaled drifts histories from each analysis and then
find the maximum drift among all analyses.
As may be observed from Table 3.123, the allowable drift has been exceeded at several levels. For
example, at Level 11, the computed drift is 4.14 in. compared to the limit of 3.00 inches.
Before computing Pdelta effects, it is necessary to determine the story shears that exist at the time of
maximum displacement. These shears, together with the inertial story forces, are shown in the first two
columns of Table 3.124. The maximum base shear at the time of maximum displacement is only 668.9
kips, significantly less that the peak base shear of 1124 kips. For comparison purposes, Table 3.124 also
shows the story shears and inertial forces that occur at the time of peak base shear.
As may be seen from Table 3.125, the Pdelta effects are marginally exceeded at the lower three levels of
the structure, as the maximum allowable stability ratio for the structure is 0.091 (see Sec. 3.1.5.3 of this
example). As mentioned earlier, the fact that the limit has been exceeded is probably of no concern
because the factor ß was conservatively taken as 1.0.
Table 3.123 TimeHistory Drift for Building Responding in X Direction to Motion A00X
Level
1
Elastic Total
Drift (in.)
2
Elastic Story
Drift (in.)
3
Inelastic Story
Drift (in.)
4
Allowable Drift
(in.)
R
12
11
10
9
8
7
6
5
4
3
2
6.51
6.05
5.39
4.63
3.88
3.27
2.66
2.08
1.54
1.12
0.74
0.39
0.47
0.66
0.75
0.75
0.62
0.61
0.58
0.54
0.42
0.39
0.34
0.39
2.57
3.63
4.14
4.12
3.40
3.34
3.20
2.95
2.32
2.12
1.89
2.13
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
4.32
Computations are at time of maximum roof displacement from analysis A00X. 1.0 in. = 25.4 mm.
FEMA 451, NEHRP Recommended Provisions: Design Examples
342
Table 3.124 Scaled Inertial Force and Story Shear Envelopes from Analysis A00X
Level
At Time of Maximum Roof
Displacement
(T = 11.39 sec)
At Time of Maximum Base Shear
(T = 12.73 sec)
Story Shear
(kips)
Inertial Force
(kips)
Story Shear
(kips)
Inertial Force
(kips)
R
12
11
10
9
8
7
6
5
4
3
2
307.4
529.7
664.9
730.5
787.9
817.5
843.8
855.0
828.7
778.7
716.1
668.9
307.4
222.3
135.2
65.6
57.4
29.6
26.3
11.2
26.3
50.0
62.6
47.2
40.2
44.3
45.7
95.6
319.0
468.1
559.2
596.5
662.7
785.5
971.7
1124.0
40.2
4.1
1.4
49.9
223.4
149.1
91.1
37.3
66.2
122.8
186.2
148.3
1.0 kip = 4.45 kN.
Table 3.125 Computation of PDelta Effects for XDirection Response
Level hsx (in.) . (in.) PD (kips) PL (kips) PT (kips) PX (kips) VX(kips) 2X
R 150 2.57 1656.5 315.0 1971.5 1971.5 307.4 0.020
12 150 3.63 1595.8 315.0 1910.8 3882.3 529.7 0.032
11 150 4.14 1595.8 315.0 1910.8 5793.1 664.9 0.044
10 150 4.12 1595.8 315.0 1910.8 7703.9 730.5 0.053
9 150 3.40 3403.0 465.0 3868.0 11571.9 787.9 0.061
8 150 3.34 2330.8 465.0 2795.8 14367.7 817.5 0.071
7 150 3.20 2330.8 465.0 2795.8 17163.5 843.8 0.079
6 150 2.95 2330.8 465.0 2795.8 19959.3 855.0 0.083
5 150 2.32 4323.8 615.0 4938.8 24898.1 828.7 0.084
4 150 2.12 3066.1 615.0 3681.1 28579.2 778.7 0.094
3 150 1.89 3066.1 615.0 3681.1 32260.3 716.1 0.103
2 216 2.13 3097.0 615.0 3712.0 35972.3 668.9 0.096
Computations are at time of maximum roof displacement from analysis A00X. 1.0 in. = 25.4 mm, 1.0 kip = 4.45
kN.
3.1.7.3 Torsion, Orthogonal Loading, and Load Combinations
As with ELF or responsespectrum analysis, it is necessary to add the effects of accidental torsion and
orthogonal loading into the analysis. Accidental torsion is applied in exactly the same manner as done for
the response spectrum approach, except that the factor 0.85 is not used. Orthogonal loading is
automatically accounted for by concurrently running one ground motion in one principal direction with
Chapter 3, Structural Analysis
343
30 percent of the companion motion being applied in the orthogonal direction. Because the signs of the
ground motions are arbitrary, it is appropriate to add the absolute values of the responses from the two
directions. Six dynamic load combinations result:
Combination 1: A00X + 0.3 A90Y + Torsion
Combination 2: A90X + 0.3 A00Y + Torsion
Combination 3: B00X + 0.3 B90Y + Torsion
Combination 4: B90X + 0.3 B00Y + Torsion
Combination 5: C00X + 0.3 C90Y + Torsion
Combination 6: C90X + 0.3 C00Y + Torsion
3.1.7.4 Member Design Forces
Using the method outlined above, the individual beam shear maxima developed in Fame 1 were computed
for each load combination. The envelope values from only the first two combinations are shown in
Figure 3.116. Envelope values from all combinations are shown in Figure 3.117. Note that some of the
other combinations (Combinations 3 through 8) control the member shears at the lower levels of the
building. These forces are compared to the forces obtained using ELF and modalresponsespectrum
analysis in the following discussion.
16.2 17.5 17.6
R12 30.4 32.3 32.3
1211 45.5 45.6 47.6
1110 50.0 49.3 52.8
109 43.9 44.4 40.9 37.8 16.3
98 42.0 41.9 39.6 44.4 28.4
87 44.9 44.3 42.1 45.4 28.9
76 43.4 42.0 40.2 43.7 27.9
65 13.7 30.3 32.3 34.2 33.5 34.9 23.0
54 23.2 35.6 31.1 32.9 32.4 33.2 22.8
43 23.7 35.6 32.6 34.0 33.6 34.4 24.0
32 23.1 36.2 35.1 35.3 35.4 35.8 23.4
2G
Figure 3.116 For Combinations 1 and 2, beam shears (kips) as computed using timehistory analysis; analysis
includes orthogonal loading and accidental torsion (1.0 kip = 4.45 kN).
FEMA 451, NEHRP Recommended Provisions: Design Examples
344
16.2 17.5 17.6
R12 30.4 32.3 32.3
1211 45.5 45.6 47.6
1110 50.0 49.3 52.8
109 44.7 44.5 41.7 38.5 17.3
98 43.9 43.5 41.3 45.8 29.6
87 46.6 45.4 43.6 46.7 29.6
76 45.2 42.9 41.8 44.1 28.5
65 14.9 32.4 34.4 36.4 35.6 36.7 24.2
54 24.9 37.9 33.5 35.3 34.8 35.6 24.2
43 25.3 37.1 35.6 36.1 36.0 36.2 25.3
32 24.6 38.2 36.9 37.3 37.3 37.8 24.6
2G
Figure 3.117 For all combinations, beam shears (kips) as computed using timehistory analysis; analysis includes
orthogonal loading and accidental torsion (1.0 kip = 4.45 kN).
3.1.8 Comparison of Results from Various Methods of Analysis
A summary of the results from all of the analyses is provided in Tables 3.126 through 3.128.
3.1.8.1 Comparison of Base Shear and Story Shear
The maximum story shears are shown In Table 3.126. For the timehistory analysis, the shears computed
at the time of maximum displacement and time of maximum base shear (from analysis A00X only) are
provided. Also shown from the timehistory analysis is the envelope of story shears computed among all
analyses. As may be observed, the shears from ELF and responsespectrum analysis seem to differ
primarily on the basis of the factor 0.85 used in scaling the response spectrum results. ELF does,
however, produce relatively larger forces at Levels 6 through 10.
The difference between ELF shears and timehistory envelope shears is much more pronounced,
particularly at the upper levels where timehistory analysis gives larger forces. One reason for the
difference is that the scaling of the ground motions has greatly increased the contribution of the higher
modes of response.
The timehistory analysis also gives shears larger than those computed using the response spectrum
procedure, particularly for the upper levels.
Chapter 3, Structural Analysis
345
3.1.8.2 Comparison of Drift
Table 3.127 summarizes the drifts computed from each of the analyses. The timehistory drifts are from
a single analysis, A00X; envelope values would be somewhat greater. As with shear, the ELF and modalresponse
spectrum approaches appear to produce similar results, but the drifts from timehistory analysis
are significantly greater. Aside from the fact that the 0.85 factor is not applied to timehistory response, it
is not clear why the timehistory drifts are as high as they are. One possible explanation is that the drifts
are dominated by one particular pulse in one particular ground motion. As mentioned above, it is also
possible that the effect of scaling has been to artificially enhance the higher mode response.
3.1.8.3 Comparison Member Forces
The shears developed in Bay DE of Frame 1 are compared in Table 3.128. The shears from the timehistory
(TH) analysis are envelope values among all analyses, including torsion and orthogonal load
effects. The timehistory approach produced larger beam shears than the ELF and response spectrum
(RS) approaches, particularly in the upper levels of the building. The effect of higher modes on the
response is again the likely explanation for the noted differences.
Table 3.126 Summary of Results from Various Methods of Analysis: Story Shear
Level
Story Shear (kips)
ELF RS TH at Time of
Maximum
Displacement
TH at Time of
Maximum Base
Shear
TH. at Envelope
R
12
11
10
9
8
7
6
5
4
3
2
187
341
471
578
765
866
943
999
1070
1102
1118
1124
180
286
358
417
523
586
638
688
783
838
894
956
307
530
664
731
788
818
844
856
829
779
718
669
40.2
44.3
45.7
95.6
319
468
559
596
663
786
972
1124
325
551
683
743
930
975
964
957
1083
1091
1045
1124
1.0 kip = 4.45 kN.
FEMA 451, NEHRP Recommended Provisions: Design Examples
346
Table 3.127 Summary of Results from Various Methods of Analysis: Story Drift
Level
XDirection Drift (in.)
ELF RS TH
R
12
11
10
9
8
7
6
5
4
3
2
0.982
1.41
1.75
1.92
1.83
2.00
2.03
1.95
1.70
1.70
1.67
2.02
0.99
1.43
1.65
1.82
1.87
1.98
2.20
2.14
2.09
2.09
2.31
2.91
2.57
3.63
4.14
4.12
3.40
3.34
3.20
2.95
2.32
2.12
1.89
2.13
1.0 in. = 25.4 mm.
Table 3.128 Summary of Results from Various Methods of Analysis: Beam Shear
Level
Beam Shear Force in Bay DE of Frame 1 (kips)
ELF RS TH
R
12
11
10
9
8
7
6
5
4
3
2
9.54
17.6
26.3
31.0
32.7
34.1
38.1
38.4
34.3
31.0
31.7
31.8
9.70
17.7
24.9
27.7
26.5
26.7
28.8
30.4
27.7
27.0
28.8
30.6
17.5
32.3
45.6
49.3
44.5
43.5
45.4
42.9
36.4
35.3
36.1
37.3
1.0 kip = 4.45 kN.
3.1.8.4 A Commentary on the Provisions Requirements for Analysis
From the writer’s perspective, there are two principal inconsistencies between the requirements for ELF,
modalresponsespectrum, and modaltimehistory analyses:
1. In ELF analysis, the Provisions allows displacements to be computed using base shears consistent
with Eq. 5.4.1.42 [5.23] (Cs = SD1/T(R/I) when Eq. 5.4.1.43 (CS = 0.044ISDS) controls for strength.
For both modalresponsespectrum analysis and modal timehistory analysis, however, the computed
Chapter 3, Structural Analysis
347
shears and displacements must be scaled if the computed base shear falls below the ELF shear
computed using Eq. 5.1.1.13. [Because the minimum base shear has been revised in the 2003
Provisions, this inconsistency would not affect this example.]
2. The factor of 0.85 is allowed when scaling modalresponsespectrum analysis, but not when scaling
timehistory results. This penalty for timehistory analysis is in addition to the penalty imposed by
selecting a scale factor that is controlled by the response at one particular period (and thus exceeding
the target at other periods). [In the 2003 Provisions these inconsistencies are partially resolved. The
minimum base shear has been revised, but timehistory analysis results are still scaled to a higher base
shear than are modal response spectrum analysis results.]
The effect of these inconsistencies is evident in the results shown in Tables 3.126 through 3.1
28 and should be addressed prior to finalizing the 2003 edition of the Provisions.
3.1.8.5 Which Method Is Best?
In this example, an analysis of an irregular steel moment frame was performed using three
different techniques: equivalentlateralforce, modalresponsespectrum, and modaltimehistory
analyses. 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
the Provisions, the structure will likely survive an earthquake consistent with the maximum
considered ground motion. The exception would be if a highly irregular structure were analyzed
using the ELF procedure. Fortunately, the Provisions 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, when computer programs
(e.g., SAP2000 and ETABS) that can perform modalresponsespectrum analysis with only
marginally increased effort over that required for ELF are available, 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 or importance of ELF analysis because
such an analysis is useful for preliminary design and components of the ELF analysis are
necessary for application of accidental torsion.
The use of timehistory analysis is limited when applied to a linear elastic model of the structure.
The amount of additional effort required to select and scale the ground motions, perform the
timehistory analysis, scale the results, and determine envelope values for use in design is simply
not warranted when compared to the effort required for modalresponsespectrum analysis. This
might change in the future when “standard” suites of ground motions are developed and are
made available to the earthquake engineering community. Also, significant improvement is
FEMA 451, NEHRP Recommended Provisions: Design Examples
348
needed in the software available for the preprocessing and particularly, for the postprocessing of
the huge amounts of information that arise from the analysis.
Scaling ground motions used for timehistory analysis is also an issue. The Provisions requires
that the selected motions be consistent with the magnitude, distance, and source mechanism of a
maximum considered earthquake expected at the site. If the ground motions satisfy this criteria,
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 Provisions 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 Provisions scaling requirements compensates for this effect, but only for very
knowledgeable users.
The main benefit of timehistory 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 3.2.
Chapter 3, Structural Analysis
349
3.2 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:
1. Linear static analysis,
2. Plastic strength analysis (using virtual work),
3. Nonlinear static (pushover) analysis,
4. Linear dynamic analysis, and
5. Nonlinear dynamic analysis.
The primary purpose of this example is to highlight some of the more advanced analytical techniques;
hence, more detail is provided on the last three analytical techniques. The Provisions provides some
guidance and requirements for the advanced analysis techniques. Nonlinear static analysis is covered in
the Appendix to Chapter 5, nonlinear dynamic analysis is covered in Sec. 5.7 [5.5], and analysis of
structures with added damping is prescribed in the Appendix to Chapter 13 [new Chapter 15].
3.2.1 Description of Structure
The structure analyzed for this example is a 6story office building in Seattle, Washington. According to
the descriptions in Provisions Sec. 1.3 [1.2], the building is assigned to Seismic Use Group I. From
Provisions Table 1.4 [1.31], the occupancy importance factor (I) is 1.0. A plan and elevation of the
building are shown in Figures 3.21 and 3.22, respectively. The lateralloadresisting system consists of
steel momentresisting frames on the perimeter of the building. There are five bays at 28 ft on center in
the NS direction and six bays at 30 ft on center in the EW direction. The typical story height is 12 ft6
in. with the exception of the first story, which has a height of 15 ft. There are a 5fttall perimeter parapet
at the roof and one basement level that extends 15 ft below grade. For this example, it is assumed that the
columns of the momentresisting frames are embedded into pilasters formed into the 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. It is
assumed that these and all other fully welded connections are constructed and inspected according to
postNorthridge protocol. Only the demand side of the required behavior of these connections is
addressed in this example.
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
FEMA 451, NEHRP Recommended Provisions: Design Examples
350
Moment
connection
(typical)
W E
S
N
28'0" 28'0" 28'0" 28'0" 28'0"
1'6" 30'0" 30'0" 30'0" 30'0" 30'0" 30'0"
(typical)
Figure 3.21 Plan of structural system.
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 will be for lateral loads acting in
the NS direction. Analysis for lateral loads acting in the EW direction would be performed in a similar
manner.
Chapter 3, Structural Analysis
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. A 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.
351
5 at 28'0" = 140'0"
Basement
wall
15'0" 15'0" 5 at 12'6" = 62'6" 5'0"
Figure 3.22 Elevation of structural system.
Prior to analyzing the structure, a preliminary design was performed in accordance with the AISC
Seismic. All members, including miscellaneous plates, were designed using steel with a nominal yield
stress of 50 ksi. Detailed calculations for the design are beyond the scope of this example. Table 3.21
summarizes the members selected for the preliminary design.1
Table 3.21 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
FEMA 451, NEHRP Recommended Provisions: Design Examples
352
The sections shown in Table 3.21 meet the widthtothickness requirements for special moment frames,
and the size of the column relative to the girders should ensure that plastic hinges will form in the girders.
Doubler plates 0.875 in. thick are used at each of the interior columns at Levels 2 and 3, and 1.00 in. thick
plates are used at the interior columns at Levels 4, 5, 6, and R. Doubler plates were not used in the
exterior columns.
3.2.2 Loads
3.2.2.1 Gravity Loads
It is assumed that the floor system of the building consists of a normal weight composite concrete slab on
formed metal deck. The slab is supported by floor beams that span in the NS direction. These floor
beams have a span of 28 ft and are spaced 10 ft on center.
The dead weight of the structural floor system is estimated at 70 psf. Adding 15 psf for ceiling and
mechanical, 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 over the full floor. Twentyfive percent of this load, or 12.5 psf, is assumed to act concurrent with
seismic forces. A similar reduced live load is used for the roof.
Based on these loads, the total dead load, live load, and dead plus live load applied to each level are given
in Table 3.22. The slight difference in 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 3.23. These are based on a total dead load of 95 psf, a cladding weight of 15 psf,
and a live load of 0.25(50) = 12.5 psf.
Table 3.22 Gravity Loads on Seattle Building
Dead Load (kips) Reduced Live Load (kips) Total Load (kips)
Level Story Accumulated Story Accumulated Story Accumulated
R 2,549 2,549 321 321 2,870 2,870
6 2,561 5,110 321 642 2,882 5,752
5 2,561 7,671 321 963 2,882 8,634
4 2,561 10,232 321 1,284 2,882 11,516
3 2,561 12,792 321 1,605 2,882 14,398
2 2,573 15,366 321 1,926 2,894 17,292
3.2.2.2 Earthquake Loads
Although the main analysis in this example is nonlinear, equivalent static forces are computed in
accordance with the Provisions. These forces are used in a preliminary static analysis to determine
whether the structure, as designed, conforms to the drift requirements of the Provisions.
The structure is situated in Seattle, Washington. The short period and the 1second mapped spectral
Chapter 3, Structural Analysis
353
acceleration parameters for the site are:
SS = 1.63
S1 = 0.57
The structure is situated on Site Class C materials. From Provisions Tables 4.1.2.4(a) and 4.1.2.4(b)
[Tables 3.31 and 3.32]:
Fa = 1.00
Fv = 1.30
From Provisions Eq. 4.1.2.41 and 4.1.2.42 [3.31 and 3.32], the maximum considered spectral
acceleration parameters are:
SMS = FaSS = 1.00(1.63)
= 1.63
SM1 = FvS1 = 1.30(0.57)
= 0.741
And from Provisions Eq. 4.1.2.51 and Eq. 4.1.2.52 [3.33 and 3.34], the design acceleration parameters
are:
SDS = (2/3)SM1 = (2/3)1.63
= 1.09
SD1 = (2/3)SM1 = (2/3)0.741
= 0.494
Based on the above coefficients and on Provisions Tables 4.2.1a and 4.2.1b [1.41 and 1.42], 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 Provisions
Table 5.2.2 [4.31], has R = 8, Cd = 5.5, and O0 = 3.0.
FEMA 451, NEHRP Recommended Provisions: Design Examples
354
A B
(a) Tributary area for columns
(b) Tributary area for girders
C C
(c) Element and nodal loads
R
6
5
PA  RC PB  2R C PB  2R C
1'6" 28'0" 28'0"
15'0" 1'6"
30'0"
1'6" 28'0" 28'0"
5'0"
Figure 3.23 Element loads used in analysis.
Chapter 3, Structural Analysis
355
3.2.2.2.1 Approximate Period of Vibration
Provisions Eq. 5.4.2.11 [5.26] is used to estimate the building period:
x
Ta=Crhn
where, from Provisions Table 5.4.2.1 [5.52], Cr = 0.028 and x = 0.8 for a steel moment frame. Using hn
(the total building height above grade) = 77.5 ft, Ta = 0.028(77.5)0.8 = 0.91 sec.
When the period is determined from a properly substantiated analysis, the Provisions requires that the
period used for computing base shear not exceed CuTa where, from Provisions Table 5.4.2 [5.21] (using
SD1 = 0.494), Cu = 1.4. For the structure under consideration, CuTa = 1.4(0.91) = 1.27 sec.
3.2.2.2.2 Computation of Base Shear
Using Provisions Eq. 5.4.1 [5.21], the total seismic shear is:
V=CSW
where W is the total weight of the structure. From Provisions Eq. 5.4.1.11 [5.22], the maximum
(constant acceleration region) seismic response coefficient is:
1.09 0.136
max ( / ) (8/1)
DS
S
C S
R I
= = =
Provisions Eq. 5.4.1.12 [5.23] controls in the constant velocity region:
0.494 0.0485
( / ) 1.27(8/1)
D1
S
C S
T R I
= = =
The seismic response coefficient, however, must not be less than that given by Eq. 5.4.1.13 [revised for
the 2003 Provisions]:
0.044 0.044(1)(1.09) 0.0480 . CSmin= ISDS= =
[In the 2003 Provisions, this equation for minimum base shear coefficient has been revised. The results
of this example problem would not be affected by the change.]
Thus, the value from Eq. 5.4.1.12 [5.23] controls for this building. Using W = 15,366 kips, V =
0.0485(15,366) = 745 kips.
FEMA 451, NEHRP Recommended Provisions: Design Examples
356
3.2.2.2.3 Vertical Distribution of Forces
The Provisions Eq. 5.4.1.12 [5.23] base shear is distributed along the height of the building using
Provisions Eq. 5.4.3.1 and 5.4.3.2 [5.210 and 5.211]:
Fx=CvxV
and
1
k
x
vx n k
i i
i
C w h
wh
=
=
S
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 and story overturning moments acting at the bottom of the story below the indicated level are
summarized in Table 3.23. These are the forces acting on the whole building. For analysis of a single
frame, onehalf of the tabulated values are used.
Table 3.23 Equivalent Lateral Forces for Seattle Building Responding in NS Direction
Level x wx
(kips)
hx
(ft) wxhx
k Cvx
Fx
(kips)
Vx
(kips)
Mx
(ftkips)
R 2,549 77.5 1,060,663 0.321 239.2 239.2 2,990
6 2,561 65.0 835,094 0.253 188.3 427.5 8,334
5 2,561 52.5 621,077 0.188 140.1 567.6 15,429
4 2,561 40.0 426,009 0.129 96.1 663.7 23,725
3 2,561 27.5 253,408 0.077 57.1 720.8 32,735
2 2,561 15.0 109,882 0.033 24.8 745.6 43,919
S 15,366 3,306,133 1.000 745.6
3.2.3 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.
3.2.3.1 The Computer Program DRAIN2Dx
The computer program DRAIN2Dx (henceforth called DRAIN) was used for all of the analyses
described in this example. DRAIN allows linear and nonlinear static and dynamic analysis of twodimensional
(planar) structures only.
Chapter 3, Structural Analysis
357
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:
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, the braces
in the damped system, 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 and
columns 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 considered 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.
3.2.3.2 Description of Preliminary Model and Summary of Preliminary Results
The preliminary DRAIN model is shown in Figure 3.24. Important characteristics of the model are as
follows:
1. Only a single frame was modeled. Hence onehalf of the loads shown in Tables 3.22 and 3.23 were
applied.
2. Columns were fixed at their base.
3. Each beam or column element was modeled using a Type 2 element. For the columns, axial, flexural,
and shear deformations were included. For the girders, flexural and shear deformations were
included but, because of diaphragm slaving, axial deformation was not included. Composite action in
the floor slab was ignored for all analysis.
4. Members were modeled using centerline dimensions without rigid end offsets. This allows, in an
approximate but reasonably accurate manner, deformations to occur in the beamcolumn joint region.
Note that this model does not provide any increase in beamcolumn joint stiffness due to the presence
of doubler plates.
5. Pdelta effects were modeled using the leaner column shown in Figure 3.24 at the right of the main
frame. This column was modeled with an axially rigid Type 1 (truss) element. Pdelta effects were
activated for this column only (Pdelta effects were turned off for the columns of the main frame).
The lateral degree of freedom at each level of the Pdelta column was slaved to the floor diaphragm at
FEMA 451, NEHRP Recommended Provisions: Design Examples
358
R
Y
X
6
5
4
3
2
Frame A or G P. column
Figure 3.24 Simple wire frame model used for preliminary analysis.
the matching elevation. When Pdelta effects were included in the analysis, a special initial load case
was created and executed. This special load case consisted of a vertical force equal to onehalf of the
total story weight (dead load plus fully reduced live load) applied to the appropriate node of the
Pdelta column. Pdelta effects were modeled in this manner to avoid the inconsistency of needing
true column axial forces for assessing strength and requiring total story forces for assessing stability.
3.2.3.2.1 Results of Preliminary Analysis: Drift and Period of Vibration
The results of the preliminary analysis for drift are shown in Tables 3.24 and 3.25 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 1.25 times the limit provided
by Provisions Table 5.2.8. This is in accordance with Provisions Sec. 5.7.3.3 [5.5.3.3] which allows such
an increase in drift when a nonlinear analysis is performed. This increased limit is used here for
consistency with the results from the following nonlinear timehistory analysis.
When 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 3.45 in. in Story 2. If
the 1.25 multiplier were not used, the allowable story drift would reduce to 3.00 in., and the computed
story drift at Levels 3 and 4 would exceed the limit.
As a preliminary estimate of the importance of Pdelta effects, story stability coefficients (.) were
computed in accordance with Provisions Sec. 5.4.6.2 [5.2.6.2]. At Story 2, the stability coefficient is
0.0839. Provisions Sec. 5.4.6.2 [5.2.6.2] allows Pdelta effects to be ignored when the stability
coefficient is less than 0.10. For this example, however, analyses are performed with and without Pdelta
effects. [In the 2003 Provisions, the stability coefficient equation has been revised to include the
importance factor in the numerator and the calculated result is used simply to determine whether a special
Chapter 3, Structural Analysis
2The story drifts including Pdelta effects can be estimated as the drifts without Pdelta times the quantity 1/(1.) , where . is
the stability coefficient for the story.
359
analysis (in accordance with Sec. A5.2.3) is required.]
When Pdelta effects are included, the drifts at the lower stories increase by about 10 percent as expected
from the previously computed stability ratios. (Hence, the stability ratios provide a useful check.2)
Recall that this analysis ignored the stiffening effect of doubler plates.
Table 3.24 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 3.14 0.33 1.82 3.75 0.0264
5 2.81 0.50 2.75 3.75 0.0448
4 2.31 0.54 2.97 3.75 0.0548
3 1.77 0.61 3.36 3.75 0.0706
2 1.16 0.63 3.45 3.75 0.0839
1 0.53 0.53 2.91 4.50 0.0683
Table 3.25 Results of Preliminary Analysis Including Pdelta Effects
Story Total Drift
(in.)
Story Drift
(in.)
Magnified
Story Drift (in.)
Drift Limit
(in.)
6 3.35 0.34 1.87 3.75
5 3.01 0.53 2.91 3.75
4 2.48 0.57 3.15 3.75
3 1.91 0.66 3.63 3.75
2 1.25 0.68 3.74 3.75
1 0.57 0.57 3.14 4.50
The computed periods for the first three natural modes of vibration are shown in Table 3.26. 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 Provisions Eq. 5.4.2.11 [5.26]. Recall from previous calculations that this period (Ta) is
0.91 seconds, and the upper limit on computed period CuTa is 1.4(0.91) = 1.27 seconds. When doubler
plate effects are included in the analysis, the period will decrease slightly, but it remains obvious that the
structure is quite flexible.
FEMA 451, NEHRP Recommended Provisions: Design Examples
360
Table 3.26 Periods of Vibration From Preliminary Analysis (sec)
Mode Pdelta Excluded Pdelta Included
1 1.985 2.055
2 0.664 0.679
3 0.361 0.367
3.2.3.2.2 Results of Preliminary Analysis: DemandtoCapacity Ratios
To determine the likelihood of and possible order of yielding, demandtocapacity ratios were computed
for each element. The results are shown in Figure 3.25. For this analysis, the structure was subjected to
full dead load plus 25 percent of live load followed by the equivalent lateral forces of Table 3.23.
Pdelta effects were included.
For girders, the demandtocapacity ratio is simply the maximum moment in the member divided by the
member’s plastic moment capacity where the plastic capacity is ZgirderFy. For columns, the ratio is similar
except that the plastic flexural capacity is estimated to be Zcol(Fy  Pu/Acol) where Pu is the total axial force
in the column. The ratios were computed at the end of the member, not at the face of the column or
girder. This results in slightly conservative ratios, particularly for the columns, because the columns have
a smaller ratio of clear span to total span than do the girders.
Level R 0.176 0.177 0.169 0.172 0.164
0.066 0.182 0.177 0.177 0.170 0.135
Level 6 0.282 0.281 0.277 0.282 0.280
0.148 0.257 0.255 0.255 0.253 0.189
Level 5 0.344 0.333 0.333 0.333 0.354
0.133 0.274 0.269 0.269 0.269 0.175
Level 4 0.407 0.394 0.394 0.394 0.420
0.165 0.314 0.308 0.308 0.309 0.211
Level 3 0.452 0.435 0.435 0.434 0.470
0.162 0.344 0.333 0.333 0.340 0.223
Level 2 0.451 0.425 0.430 0.424 0.474
0.413 0.492 0.485 0.485 0.487 0.492
Figure 3.25 Demandtocapacity ratios for elements from analysis with Pdelta effects included.
Chapter 3, Structural Analysis
3To determine the demandtocapacity ratio on the basis of an elastic analysis, multiply all the values listed in Table 3.26 by
R = 8. With this modification, the ratios are an approximation of the ductility demand for the individual elements.
361
It is very important to note that the ratios shown in Figure 3.25 are based on the inelastic seismic forces
(using R = 8). Hence, a ratio of 1.0 means that the element is just at yield, a value less than 1.0 means the
element is still elastic, and a ratio greater than 1.0 indicates yielding.3
Several observations are made regarding the likely inelastic behavior of the frame:
1. The structure has considerable overstrength, particularly at the upper levels.
2. The sequence of yielding will progress from the lower level girders to the upper level girders.
Because of the uniform demandtocapacity ratios in the girders of each level, all the hinges in the
girders in a level will form almost simultaneously.
3. With the possible exception of the first level, the girders should yield before the columns. While not
shown in the table, it should be noted that the demandtocapacity ratios for the lower story columns
were controlled by the moment at the base of the column. It is usually very difficult to prevent
yielding of the base of the first story 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.
3.2.3.2.3 Results of Preliminary Analysis: Overall System Strength
The last step in the preliminary analysis was 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 ZgirderFy and the hinges form 5.0 in. from the face of the column. Columns hinge only at
the base, and the plastic moment capacity is assumed to be Zcol(Fy  Pu/Acol). The fully plastic mechanism
for the system is illustrated in Figure 3.26. The inset to the figure shows how the angle modification
term s was computed. The strength (V) for the total structure is computed from the following
relationships (see Figure 3.26 for nomenclature):
Internal Work = External Work
Internal Work = 2[20s.MPA + 40s.MPB + .(MPC + 4MPD + MPE)]
External Work = where
1
nLevels
i i
i
V. FH
=
... S ... 1
1.0
nLevels
i
i
F
=
S =
Three lateral force patterns were used: uniform, upper triangular, and Provisions where the Provisions
pattern is consistent with the vertical force distribution of Table 3.23 in this volume of design examples.
The results of the analysis are shown in Table 3.27. As expected, the strength under uniform load is
significantly greater than under triangular or Provisions load. The closeness of the Provisions and
triangular load strengths is due to the fact that the verticalloaddistributing parameter (k) was 1.385,
which is close to 1.0. The difference between the uniform and the triangular or Provisions patterns is an
FEMA 451, NEHRP Recommended Provisions: Design Examples
362
indicator that the results of a capacityspectrum analysis of the system will be quite sensitive to the lateral
force pattern applied to the structure when performing the pushover analysis.
The equivalentlateralforce (ELF) base shear, 746 kips (see Table 3.23), when divided by the Provisions
pattern capacity, 2886 kips, is 0.26. This is reasonably consistent with the demand to capacity ratios
shown in Figure 3.25.
Before proceeding, three important points should be made:
1. The rigidplastic analysis did not include strain hardening, which is an additional source of
overstrength.
2. The rigidplastic analysis did 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.
3. Slightly more than 10 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 “error” in total strength is less than 1 percent.
Table 3.27 Lateral Strength on Basis of RigidPlastic Mechanism
Lateral Load Pattern Lateral Strength (kips)
Entire Structure
Lateral Strength (kips)
Single Frame
Uniform 3,850 1,925
Upper Triangular 3,046 1,523
Provisions 2,886 1,443
3.2.4 Description of Model Used for Detailed Structural Analysis
Nonlinearstatic and 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 3.27. A detail of a girder and its
connection to two interior columns is shown in Figure 3.28. The detail illustrates the two main
features of the model: an explicit representation of the panel zone region and the use of
concentrated (Type 4 element) plastic hinges in the girders.
Chapter 3, Structural Analysis
363
Figure 3.26 Plastic mechanism for computing lateral strength.
In Figure 3.27, the column shown to the right of the structure is used to represent Pdelta effects. See
Sec. 3.2.3.2 of this example for details.
Y
X
M
M
(c)
(a)
s.
s.
.
(c)
c
b
c
(b) (d)
.
.'
PA
PA
MPB
MPB
MPB
MPB
MPC MPD MPD MPD MPD MPE
d
e = 0.5 (d )+ 5"
d
e
e L2e e
e . 2e .
FEMA 451, NEHRP Recommended Provisions: Design Examples
364
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.
3.2.4.1 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, through a simple transformation process, deformations in the panel zone
region of beamcolumn joints.
See Figure 3.28
28'0"
Typical
15'0" 5 at 12'6"
Figure 3.27 Detailed analytical model of 6story frame.
Panel zone
flange spring
(Typical)
Panel zone
panel spring
(Typical)
Girder
plastic hinge
Figure 3.28 Model of girder and panel zone region.
Chapter 3, Structural Analysis
365
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 3.29. The figure also shows the assumed bilinear, inelastic
momentrotation behavior for the spring.
Figure 3.29 A compound node and attached spring.
Rotational spring
.Slave
.Master
Master node d. = . M a s t e r  .Slave
Slave node
(a)
(b)
a.
1
.
My
My
d.
1
(c)
Master
Slave
Rotational spring
FEMA 451, NEHRP Recommended Provisions: Design Examples
4The author of this example is completing research at Virginia Tech to determine whether the scissors model is adequate to
model steel moment frames. Preliminary results indicate that the kinematics error is not significant and that very good results may
be obtained by a properly formulated scissors model.
366
Figure 3.210 Krawinkler beamcolumn joint model.
3.2.4.2 Modeling of BeamColumn Joint Regions
A very significant portion of the total story drift of a momentresisting frame may be due to deformations
that occur in the panel zone region of the beamcolumn joint. In this example, panel zones are modeled
using an approach developed by Krawinkler (1978). This model, illustrated in Figure 3.210, has the
advantage of being conceptually simple, yet robust. The disadvantage of the approach is that the number
of degrees of freedom required to model a structure is significantly increased.
A simpler model, often referred to as the scissors model, also has been developed to represent panel zone
behavior. The scissors model has the advantage of using fewer degrees of freedom. However, due to its
simplicity, it is generally considered to inadequately represent the kinematics of the problem.4 For this
reason, the scissors model is not used here.
The Krawinkler model assumes that the panel zone area has two resistance mechanisms acting in parallel:
1. Shear resistance of the web of the column, including doubler plates and
2. Flexural resistance of the flanges of the column.
These two resistance mechanisms are apparent in AISC Seismic Eq. (91), which is used for determining
panel zone shear strength:
Chapter 3, Structural Analysis
367
.
3 2
0.6 1 cf cf
v ycp
b c p
b t
R Fdt
d d t
. .
= .+ .
.. ..
The equation can be rewritten as:
2
0.6 1.8 y cf cf 1.8
v ycp Panel Flanges
b
F b t
R Fdt V V
d
= + = +
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:
Fy = 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, and
db = total depth of girder.
Additional terms used in the subsequent discussion are:
tbf = girder flange thickness and
G = shear modulus of steel.
FEMA 451, NEHRP Recommended Provisions: Design Examples
368
(a)
b
V
M
V
.
4M . = VF l a n g e s d b .
V = 4M
db
(b)
p
p
Yielding of
column flange
Flanges
Flanges
p
d
Figure 3.211 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 0.6Fy. (The 0.6 factor is a simplification of the Von Mises yield
criterion that gives the yield stress in shear as 1/ 3 = 0.577 times the strength in tension.)
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 3.211(a). If it can be
assumed that the kink is represented by a plastic hinge with a plastic moment capacity of Mp = FyZ =
Fybcftcf
2/4, it follows from virtual work (see Figure 3.211b) that the equivalent shear strength of the
column flanges is:
Chapter 3, Structural Analysis
369
b
VPanel
.=.
Thickness = t
c
p
d
d
d
Figure 3.212 Column web component of
panel zone resistance.
4 p
Flanges
b
M
V
d
=
and by simple substitution for Mp:
2
y cf cf
Flanges
b
F b t
V
d
=
This value does not include the 1.8 multiplier that appears in the AISC equation. This multiplier is based
on 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 3.212:
K V V
Panel d
Panel Panel
b
,. . d
= =
noting that the displacement d can be written as:
d
.
=
=
.
. ..
.
. ..
=
V d
Gt d
K V
V d
Gt d d
Gt d
Panel b
p c
Panel
Panel
Panel b
p c b
p c
,
,
1
FEMA 451, NEHRP Recommended Provisions: Design Examples
370
Shear
Panel V
Panel
Total resistance
1
Flanges V Flanges .
.y 4.y
Shear
., flanges
. ., panel .
Shear strain, .
Figure 3.213 Forcedeformation behavior of panel zone region.
Krawinkler assumes that the column flange component yields at four times the yield deformation of the
panel component, where the panel yield deformation is:
.
,
Panel 0.6 y c p 0.6 y
y
Panel c p
V F d t F
K . Gd t G
. = = =
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 3.213. This figure is applicable 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).
The actual Krawinkler model is shown in Figure 3.210. 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) do not have
rotational springs and thereby 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 3.214. 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:
Chapter 3, Structural Analysis
371
MPanel=VPaneldb=0.6Fydcdbtp
KPanel,.=KPanel,.db=Gdcdbtp
It is interesting to note that the shear strength in terms of the rotation spring is simply 0.6Fy 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:
MFlanges=1.8VFlangesdb=1.8Fybcftc2f
Shear = V
V
.
Moment = Vd b
(a) (b)
(c)
Panel spring
Web spring
Note . = .
d d
.
db
Figure 3.214 Transforming shear deformation to rotational deformation in the
Krawinkler model.
FEMA 451, NEHRP Recommended Provisions: Design Examples
372
Because of the equivalence of rotation and shear deformation, the yield rotation of the panel is the same as
the yield strain in shear:
.
,
Panel 0.6 y
y y
Panel
M F
K . G
. =. = =
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,
2 .
, 0.75
4
Flanges
Flanges cf cf
y
M
K. . Gbt
= =
The complete resistance mechanism, in terms of rotational spring properties, is shown in Figure 3.213.
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. Krawinkler suggests using a strainhardening
stiffness equal to 3 percent of the initial stiffness of the joint. In this analysis, the strain hardening
component was simply added to both the panel and the flange components:
KSH,.=0.03(KPanel,.+KFlanges,.) .
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:
dc=dc,nomtcf
where the nom part of the subscript indicates the property listed as the total depth in the AISC Manual of
Steel Construction.
The Krawinkler properties are now computed for a typical interior subassembly of the 6story frame. A
summary of the properties used for all connections is shown in Table 3.28.
Chapter 3, Structural Analysis
373
Table 3.28 Properties for the Krawinkler BeamColumn Joint Model
Connection Girder Column Doubler Plate
(in.)
Mpanel,.
(in.k)
Kpanel,.
(in.k/rad)
Mflanges,.
(in.k/rad)
Kflanges,q
(in.k/rad)
A W24x84 W21x122 – 8,701 3,480,000 1,028 102,800
B W24x84 W21x122 1.00 23,203 9,281,000 1,028 102,800
C W27x94 W21x147 – 11,822 4,729,000 1,489 148,900
D W27x94 W21x147 1.00 28,248 11,298,000 1,489 148,900
E W27x94 W21x201 – 15,292 6,117,000 3,006 300,600
F W27x94 W21x201 0.875 29,900 11,998,000 3,006 300,600
Example calculations shown for row in bold type.
The sample calculations below are for Connection D in Table 3.28.
Material Properties:
Fy = 50.0 ksi (girder, column, and doubler plate)
G = 12,000 ksi
Girder:
W27x94
db,nom 26.92 in.
tf 0.745 in.
db 26.18 in.
Column:
W21x147
dc,nom 22.06 in.
tw 0.72 in.
tcf 1.150 in.
dc 20.91 in.
bcf 12.51 in.
Doubler plate: 1.00 in.
Total panel zone thickness = tp = 0.72 + 1.00 = 1.72 in.
VPanel=0.6Fydctp=0.6(50)(20.91)(1.72)=1,079 kips
1.8 2 1.8 50(12.51)(1.152 ) 56.9 kips
26.18
y cf cf
Flanges
b
F b t
V
d
= = =
FEMA 451, NEHRP Recommended Provisions: Design Examples
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.
374
KPanel,. =Gtpdc=12,000(1.72)(20.91)=431,582 kips/unit shear strain
0.6 0.6(50,000) 0.0025
12,000
y
y y
F
G
. =. = = =
MPanel=VPaneldb=1,079(26.18)=28,248 in.kips
KPanel,.=KPanel,.db=431,582(26.18)=11,298,000 in.kips/radian
MFlanges=VFlangesdb=56.9(26.18)=1,489 in.kips
,
1,489 148,900 in.kips/radian
4 4(0.0025)
Flanges
Flanges
y
M
K . .
= = =
3.2.4.3 Modeling Girders
Because this structure is designed in accordance with the strongcolumn/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 3.28. 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 mid span 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:
1. The initial stiffness (moment per unit rotation),
2. The effective yield moment,
3. The secondary stiffness, and
4. The location of the hinge with respect to the face of the column.
Determination of the above properties, particularly the location of the hinge, is complicated by the fact that
the plastic hinge grows in length during increasing story drift. Unfortunately, there is no effective way to
represent a changing hinge length in DRAIN, so one must make do with a fixed hinge length and location.
Fortunately, the behavior of the structure is relatively insensitive to the location of the hinges.
Chapter 3, Structural Analysis
375
50
40
30
20
10
0
1
Eo
1 ESH
0.002 0.004 0.006
Strain
Stress, ksi
0
Figure 3.215 Assumed stressstrain curve for modeling girders.
To determine the hinge properties, it is necessary to perform a momentcurvature analysis of the cross
section, and this, in turn, is a function of the stressstrain curve of the material. In this example, a
relatively simple stressstrain curve is used to represent the 50 ksi steel in the girders. This curve does not
display a yield plateau, which is consistent with the assumption that the section has yielded in previous
cycles, with the Baushinger effect erasing any trace of the yield plateau. The idealized stressstrain curve is
shown in Figure 3.215.
To compute the momentcurvature relationship, the girder cross section was divided into 50 horizontal
slices, with 10 slices in each flange and 30 slices in the web. The girder cross section was then subjected
to gradually increasing rotation. For each value of rotation, strain compatibility (plane sections remain
plane) was used to determine fiber strain. Fiber stress was obtained from the stressstrain law and stresses
were multiplied by fiber area to determine fiber force. The forces were then multiplied by the distance to
the neutral axis to determine that fiber’s contribution to the section’s resisting moment. The fiber
contributions were summed to determine the total resisting moment. Analysis was performed using a
Microsoft Excel worksheet. Curves were computed for an assumed strain hardening ratio of 1, 3, and 5
percent of the initial stiffness. The resulting momentcurvature relationship is shown for the W27x94
girder in Figure 3.216. Because of the assumed bilinear stressstrain curve, the momentcurvature
relationships are essentially bilinear. Residual stresses due to section rolling were ignored, and it was
assumed that local buckling of the flanges or the web would not occur.
FEMA 451, NEHRP Recommended Provisions: Design Examples
376
To determine the parameters for the plastic hinge in the DRAIN model, a separate analysis was performed
on the structure shown in Figure 3.217(a). This structure represents half of the clear span of the girder
supported as a cantilever. The purpose of the special analysis was to determine a momentdeflection
relationship for the cantilever loaded at the tip with a vertical force V. A similar momentdeflection
relationship was determined for the structure shown in Figure 3.217(b), which consists of a cantilever
with a compound node used to represent the inelastic rotation in the plastic hinge. Two Type4 DRAIN
elements were used at each compound node. The first of these is rigidperfectly plastic and the second is
bilinear. The resulting behavior is illustrated in Figure 3.217(c).
If the momentcurvature relationship is idealized as bilinear, it is a straightforward matter to compute the
deflections of the structure of Figure 3.217(a). The method is developed in Figure 3.218. Figure 3.2
18(a) is a bilinear momentcurvature diagram. The girder is loaded to some moment M, which is greater
than the yield moment. The moment diagram for the member is shown in Figure 3.218(b). At some
distance x the moment is equal to the yield moment:
MyL
x
M
=
0
5,000
10,000
15,000
20,000
25,000
0 0.0005 0.001 0.0015 0.002 0.0025
Curvature, radians/in.
Stress, ksi
5 percent strain hardening
3 percent strain hardening
1 percent strain hardening
Figure 3.216 Moment curvature diagram for W27x94 girder.
Chapter 3, Structural Analysis
377
(a)
V
V
(b)
Component 1
Component 2
c
c
e
L' = (L  d )/2
d
L/2
Tip deflection
End moment
Component 1
Component 2
Combined
(c)
Figure 3.217 Developing momentdeflection
diagrams for a typical girder.
The curvature along the length of the member is shown in Figure 3.218(c). At the distance x, the
curvature is the yield curvature (fy), and at the support, the curvature ( fM) is the curvature corresponding
to the Point M on the momentcurvature diagram. The deflection is computed using the momentarea
method, and consists of three parts:
2
1
2
2 3 3
fyx x fyx
. = · =
( ) 2
( )( )
2 2
y
y
L x L x L x L x
f
f
. = '  ' + = ' '+
FEMA 451, NEHRP Recommended Provisions: Design Examples
378
( )( ) ( )
( )( )( )
. 3 2
2
3
2
6
=
 ' 
· +
. ' 
. ..
.
. ..
=
 '  ' +
f f
f f
M y
M y
L x
x
L x
L x L x
The first two parts of the deflection are for elastic response and the third is for inelastic response. The
elastic part of the deflection is handled by the Type2 elements in Figure 3.217(b). The inelastic part is
represented by the two Type4 elements at the compound node of the structure.
The development of the momentdeflection relationship for the W27x94 girder is illustrated in Figure
3.219. Part (a) of the figure is the idealized bilinear momentcurvature relationship for 3 percent strain
hardening. Displacements were computed for 11 points on the structure. The resulting momentdeflection
diagram is shown in Figure 3.219(b), where the total deflection (.1+.2+.3) is indicated. The inelastic part
of the deflection (.3 only) is shown separately in Figure 3.219(c), where the moment axis has been
truncated below 12,000 in.kips.
Finally, the simple DRAIN cantilever model of Figure 3.217(b) is analyzed. The compound node has
arbitrarily been placed a distance e = 5 in. from the face of the support. (The analysis is relatively
insensitive to the assumed hinge location.)
Chapter 3, Structural Analysis
379
M
y
f
M
Moment
y M
f
Curvature
(a)
My
M
(b)
(c)
f M
f y
3
2
1
x
L'
x
L'
Figure 3.218 Development of equations for deflection of momentdeflection curves.
FEMA 451, NEHRP Recommended Provisions: Design Examples
380
The moment diagram is shown in Figure 3.220(a) for the model subjected to a load producing a support
moment, MS, greater than the yield moment. The corresponding curvature diagram is shown in Figure
3.220(b). At the location of the plastic hinge, the moment is:
( )
H S
M M L e
L
' 
=
'
and all inelastic curvature is concentrated into a plastic hinge with rotation .H. The tip deflection of the
structure of Figure 3.220(c) consists of two parts:
2
3
Support
E
M L
EI
'
. =
Chapter 3, Structural Analysis
381
(a)
0
5,000
10,000
15,000
20,000
25,000
0.0000 0.0005 0.0010 0.0015 0.0020 0.0025
Curvature, radians/in.
Moment, in.kips
0
5,000
10,000
15,000
20,000
25,000
0 1 2 3 4 5 6
Total tip deflection, in.
End moment, in.kips
12 000
13,000
14,000
15,000
16,000
17,000
18,000
19,000
20,000
End moment, in.kips
From cantilever analysis
Idealized for drain
(b)
(c)
Figure 3.219 Momentdeflection curve for W27x94 girder with 3 percent strain
hardening.
FEMA 451, NEHRP Recommended Provisions: Design Examples
382
.I=.H(L'e).
The first part is the elastic deflection and the second part is the inelastic deflection. Note that .E and
(.1 + .2) are not quite equal because the shapes of the curvature diagram used to generate the deflections
are not the same. For the small values of strain hardening assumed in this analysis, however, there is little
error in assuming that the two deflections are equal. As .E is simply the elastic displacement of a simple
cantilever beam, it is possible to model the main portion of the girder using its nominal moment of inertia.
The challenge is to determine the properties of the two Type4 elements such that the deflections predicted
using .I are close to those produced using .3. This is a trialanderror procedure, which is difficult to
reproduce in this example. However, the development of the hinge properties is greatly facilitated by the
fact that one component of the hinge must be rigidplastic, with the second component being bilinear. The
resulting “fit” for the W27x94 girder is shown in Figure 3.219. The resulting properties for the model are
shown in Table 3.29. The properties for the W24x84 girder are also shown in the table. Note that the
first yield of the model will be the yield moment from Component 1, and that this moment is roughly equal
to the fully plastic moment of the section.
Chapter 3, Structural Analysis
383
Table 3.29 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
(see note below)
Yield Moment (in.kip) 11,025 13,538
Initial Stiffness (in.kip/radian) 10E10 10E10
S.H. Ratio 0.0 0.0
Inelastic Component 2 Yield Moment (in.kip) 1,196 1,494
Initial Stiffness (in.kip/radian) 326,000 450,192
S.H. Ratio 0.284 0.295
Comparative Property Yield Moment = SxFy 9,800 12,150
Plastic Moment = ZxFy 11,200 13,900
In some versions of DRAIN the strain hardening stiffness of the Type4 springs is set to some small value (e.g. 0.001) if a zero
value is entered in the appropriate data field. This may cause very large artificial strain hardening moments to develop in the
hinge after it yields. It is recommended, therefore, to input a strain hardening value of 1020 to prevent this from happening.
FEMA 451, NEHRP Recommended Provisions: Design Examples
384
My
M
(a)
Inelastic
Elastic
Moment
Deflection
(c)
(b)
H H
Elastic part
Plastic part
e
L'
f = .
L'
1"
Figure 3.220 Development of plastic hinge properties for the
W27x97 girder.
Chapter 3, Structural Analysis
385
30,000
20,000
10,000
0
10,000
20,000
30,000
4,000 3,000 2,000 1,000 0 1,000 2,000 3,000 4,000
Moment, in.kips
Axial force, kips
W21x201
W21x147
W21x122
Figure 3.221 Yield surface used for modeling columns.
3.2.4.4 Modeling Columns
All columns in the analysis were modeled as Type2 elements. Preliminary analysis indicated that
columns should not yield, except at the base of the first story. Subsequent analysis showed that the
columns will yield in the upper portion of the structure as well. For this reason, column yielding had to be
activated in all of the Type2 column elements. The columns were 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 surface used by DRAIN is shown in Figure 3.221.
The rules employed by DRAIN to model column yielding are adequate for eventtoevent nonlinear static
pushover analysis, but leave much to be desired when 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.
3.2.5 Static Pushover Analysis
Nonlinear static analysis is covered for the first time in the Appendix to Chapter 5 of the 2000 Provisions.
Inclusion of these requirements in an appendix rather than the main body indicates that pushover analysis
is in the developmental stage and may not be “ready for prime time.” For this reason, some liberties are
taken in this example; however, for the most part, the example follows the appendix. [In the 2003
FEMA 451, NEHRP Recommended Provisions: Design Examples
6The mathematical model does not represent strength loss due to premature fracture of welded connections. If such fracture is
likely, the mathematical model must be adjusted accordingly.
386
Provisions, a number of substantive technical changes have been made to the appendix, largely as a result
of work performed by the Applied Technology Council in Project 55, Evaluation and Improvement of
Inelastic Seismic Analysis Procedures).]
Nonlinear static pushover analysis, in itself, provides the location and sequence of expected yielding in a
structure. Additional analysis is required to estimate the amount of inelastic deformation that may occur
during an earthquake. These inelastic deformations may then be compared to the deformations that have
been deemed acceptable under the ground motion parameters that have been selected. Provisions Sec.
5A.1.3 [Appendix to Chapter 5] provides a simple methodology for estimating the inelastic deformations
but does not provide specific acceptance criteria.
Another wellknown method for determining maximum inelastic displacement is based on the capacity
spectrum approach. This method is described in some detail in ATC 40 (Applied Technology Council,
1996). The capacity spectrum method is somewhat controversial and, in some cases may produce
unreliable results (Chopra and Goel, 1999). However, as the method is still very popular and is
incorporated in several commercial computer programs, it will be utilized here, and the results obtained
will be compared to those computed using the simple approach.
Provisions Sec. 5A1.1 [A5.2.1] discusses modeling requirements for the pushover analysis in relatively
vague terms, possibly reflecting the newness of the approach. However, it is felt that the model of the
structure described earlier in this example is consistent with the spirit of the Provisions.6
The pushover curve obtained from a nonlinear static analysis is a function of the way the structure is both
modeled and loaded. In the analysis reported herein, the structure was first subjected to the full dead load
plus 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, four different load patterns were initially
considered:
UL = uniform load (equal force at each level)
TL = triangular (loads proportional to height)
ML = modal load (lateral loads proportional to first mode shape)
BL = Provisions load distribution (using the forces indicated in Table 3.23)
Relative values of these load patterns are summarized in Table 3.210. The loads have been normalized to
a value of 15 kips at Level 2. Because of the similarity between the TL and ML distributions, the results
from the TL distribution are not presented.
DRAIN analyses were run with Pdelta effects included and, for comparison purposes, with such effects
excluded. The Provisions requires “the influence of axial loads” to be considered when the axial load in
the column exceeds 15 percent of the buckling load but presents no guidance on exactly how the buckling
load is to be determined nor on what is meant by “influence.” In this analysis the influence was taken as
inclusion of the storylevel Pdelta effect. This effect may be easily represented through linearized
geometric stiffness, which is the basis of the outrigger column shown in Figure 3.24. Consistent
Chapter 3, Structural Analysis
7If Pdelta effects have been included, this procedure needs to be used when recovering base shear from column shear forces.
This is true for displacement controlled static analysis, force controlled static analysis, and dynamic timehistory analysis.
387
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. That approach was not used here.
Table 3.210 Lateral Load Patterns Used in Nonlinear Static Pushover Analysis
Level
Uniform Load
UL
(kips)
Triangular Load
TL
(kips)
Modal Load
ML
(kips)
BSSC Load
BL
(kips)
R65432
15.0
15.0
15.0
15.0
15.0
15.0
77.5
65.0
52.5
40.0
27.5
15.0
88.4
80.4
67.8
50.3
32.0
15.0
150.0
118.0
88.0
60.0
36.0
15.0
As described later, the pushover analysis indicated all yielding in the structure occurred in the clear span of
the girders and columns. Panel zone hinging did not occur. For this reason, the ML analysis was repeated
for a structure with thinner doubler plates and without doubler plates. Because the behavior of the
structure with thin doubler plates was not significantly different from the behavior with the thicker plates,
the only comparison made here will be between the structures with and without doubler plates. These
structures are referred to as the strong panel (SP) and weak panel (WP) structures, respectively.
The analyses were 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. Of course, the computed response of the structure after strength loss is
completely fictitious in the context of a static loading environment. Under a dynamic loading, however,
structures can display strength loss and be incrementally stable. It is for this reason that the poststrength
loss realm of the pushover response is of interest.
When 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:
FEMA 451, NEHRP Recommended Provisions: Design Examples
388
Roof displacement, in.
0 5 10 15 20 25 30 35 40 45
0
500
1000
500
1000
1500
2000
Total Base Shear
Shear, kips
PDelta Forces
Column Shear Forces
Figure 3.222 Two base shear components of pushover response.
1 1
,
1 1
n
C i
i
V V P
= h
.
=S 
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 were
included through the use of the outrigger column shown at the right of Figure 3.24. Figure 3.222 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 results because these
forces are based on firststory displacement, which, for an inelastic system, will not generally be
proportional to the roof displacement.
For all of the pushover analyses reported for this example, the maximum displacement at the roof is 42.0
in. This value is slightly greater than 1.5 times the total drift limit for the structure where the total drift
limit is taken as 1.25 times 2 percent of the total height. The drift limit is taken from Provisions Table
5.2.8 [4.51] and the 1.25 factor is taken from Provisions Sec. 5A.1.4.3. [In the 2003 Provisions, Sec.
A5.2.6 requires multiplication by 0.85R/Cd rather than by 1.25.] As discussed below in Sec. 3.2.5.3, the
Appendix to Chapter 5 of the Provisions requires only that the pushover analysis be run to a maximum
displacement of 1.5 times the expected inelastic displacement. If this limit were used, the pushover
analysis of this structure would only be run to a total displacement of about 13.5 in.
3.2.5.1 Pushover Response of Strong Panel Structure
Figure 3.223 shows the pushover response of the SP structure to all three lateral load patterns when
Pdelta effects are excluded. In each case, gravity loads were applied first and then the lateral loads were
applied using the displacement control algorithm. Figure 3.224 shows the response curves if Pdelta
effects are included. In Figure 3.225, the response of the structure under ML loading with and without
Chapter 3, Structural Analysis
389
Roof displacement, in.
0
Base shear, kips
0
10 20 30 40 50
200
400
600
800
1,000
1,200
1,400
1,600
1,800
2,000
BL Loading
ML Loading
UL Loading
Figure 3.223 Response of strong panel model to three load pattern, excluding
Pdelta effects.
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 Provisions, which ignore Pdelta effects in elastic analysis if the maximum stability ratio is less
than 0.10 (see Provisions Sec. 5.4.6.2 [5.2.6.2]). For this structure, the maximum computed stability ratio
was 0.0839 (see Table 3.24), which is less than 0.10 and is also less than the upper limit of 0.0901. The
upper limit is computed according to Provisions Eq. 5.4.6.22 and is based on the very conservative
assumption that ß = 1.0. While the Provisions allow the analyst to exclude Pdelta effects in an elastic
analysis, this clearly should not be done in the pushover analysis (or in timehistory analysis). [In the 2003
Provisions, the upper limit for the stability ratio is eliminated. Where the calculated . is greater than 0.10
a special analysis must be performed in accordance with Sec. A5.2.3. Sec. A5.2.1 requires that Pdelta
effects be considered for all pushover analyses.]
FEMA 451, NEHRP Recommended Provisions: Design Examples
390
Roof displacement, in.
0
Base shear, kips
0
5 10 15 20 25 30 35 40 45
400
800
1,200
1,600
2,000
Including PDelta
Excluding PDelta
Figure 3.225 Response of strong panel model to ML loads, with and wthout Pdelta
effects.
BL Loading
ML Loading
UL Loading
Roof displacement, in.
0
Base shear, kips
0
5 10 15 20 25 30 35 40 45
200
400
600
800
1,000
1,200
1,400
1,600
Figure 3.224 Response of strong panel model to three load patterns, including
Pdelta effects.
Chapter 3, Structural Analysis
391
Roof displacement, in.
0
"Tangent stiffness", kips/in.
20
5 10 15 20 25 30 35 40 45
Including PDelta
Excluding PDelta
0
20
40
60
80
100
120
140
Figure 3.226 Tangent stiffness history for structure under ML loads, with and
without Pdelta effects.
In Figure 3.226, 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 3.226 illustrates, the first significant yield occurs at a roof displacement of
approximately 6.5 in. and that most of the structure’s original stiffness is exhausted by the time the roof
drift reaches 10 in.
For the case with Pdelta effects excluded, the final stiffness shown in Figure 3.226 is approximately 10
kips/in., compared to an original value of 133 kips/in. Hence, the strainhardening stiffness of the structure
is 0.075 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.
When Pdelta effects are included, the final stiffness is 1.6 kips per in. The structure attains this negative
residual stiffness at a displacement of approximately 23 in.
3.2.5.1.1 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 3.227. 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.
FEMA 451, NEHRP Recommended Provisions: Design Examples
392
Several important observations are made from Figure 3.227:
1. There was no hinging in Levels 6 and R,
2. There was no hinging in any of the panel zones,
3. Hinges formed at the base of all the firststory columns,
4. All columns on Story 3 and all the interior columns on Story 4 formed plastic hinges, and
5. Both ends of all the girders at Levels 2 through 5 yielded.
It appears the structure is somewhat weak in the middle two stories and is too strong at the upper stories.
The doubler plates added to the interior columns prevented panel zone yielding (even at the extreme roof
displacement of 42 in.).
The presence of column hinging at Levels 3 and 4 is a bit troublesome because the structure was 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 did
form at the bottom of each column in the third story, hinges did not form at the top of these columns, and a
complete story mechanism was 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 Sec. 3.2.5.3.
Chapter 3, Structural Analysis
393
20 22 22 22 22
19 18 18 18 17
24 24 24 24
12 16 16 16 15
14 12 12 12 12 13
3
3 3 3 1
6 8 8 8 8
4 5 5 5
7 7 7 7
26 21 21 21 21 25
4
10 11 11 11 9
1
4
8
10
16 17 19 20 23 24 26
(b)
0
0
5 10 15 20 25 30 35 40 45
Drift, in.
200
400
600
800
1,000
1,200
Total shear, kips
2
(a)
Figure 3.227 Patterns of plastic hinge formation: SP model under ML load, including Pdelta effects.
FEMA 451, NEHRP Recommended Provisions: Design Examples
394
3.2.5.1.2 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 3.211. The
strength from the case with Pdelta excluded was estimated from the curves shown in Figure 3.223 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 significantly 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 3.211 Strength Comparisons: Pushover vs Rigid Plastic
Pattern
Lateral Strength (kips)
Pdelta Excluded Pdelta Included RigidPlastic
Uniform
Modal (Triangular)
BSSC
1220
1137
1108
1223
1101
1069
1925
1523
1443
3.2.5.2 Pushover Response of Weak Panel Structure
Before continuing, the structure should be reanalyzed without panel zone reinforcing and the behavior
compared with that determined from the analysis described above. For this exercise, only the modal load
pattern d 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
3.228. Curves for the analyses run with and without Pdelta effects are included. Figures 3.229 and 3.2
30 are more informative because they compare the response of the structures with and without panel zone
reinforcement. Figure 3.231 shows the tangent stiffness history comparison for the structures with and
without doubler plates. In both cases Pdelta effects have been included.
From Figures 3.228 through 3.231 it may be seen that the doubler plates, which represent approximately
2.0 percent of the volume of the structure, increase the strength by approximately 12 percent and increase
the initial stiffness by about 10 percent.
Chapter 3, Structural Analysis
395
Roof displacement, in.
0
Base shear, kips
5 10 15 20 25 30 35 40 45
Including PDelta
Excluding PDelta
0
200
400
600
800
1,000
1,200
1,400
1,600
1,800
2,000
Figure 3.228 Weak panel zone model under ML load.
Roof displacement, in.
0
Base shear, kips
0 5 10 15 20 25 30 35 40 45
200
400
600
800
1,000
1,200
1,400
1,600
1,800
Strong Panels
Weak Panels
Figure 3.229 Comparison of weak panel zone model with strong panel zone model,
excluding Pdelta effects.
FEMA 451, NEHRP Recommended Provisions: Design Examples
396
Roof displacement, in.
0
0 5 10 15 20 25 30 35 40 45
Base shear, kips
200
400
600
800
1,000
1,200
Strong Panels
Weak Panels
Figure 3.230 Comparison of weak panel zone model with strong
panel zone model, including Pdelta effects.
Roof displacement, in.
0 5 10 15 20 25 30 35 40 45
"Tangent stiffness", kips/in.
Strong Panels
Weak Panels
0
20
20
40
60
80
100
110
120
Figure 3.231 Tangent stiffness history for structure under ML loads, strong versus
weak panels, including Pdelta effects.
Chapter 3, Structural Analysis
397
26 21 21 21 20 25
11 15 14 34 18 33 5
27 7 28 6 26
33 34
8 6 29 7
6
1
1 1 19
13 30
21
31
21 2
31
2
30 12
22
10
24
52
48
33
54
39
63
38
59
68
23
61
52 12
2
9
16
40
18
55
42
10
67 35
51
58
46 28
4
36
57
47 29
4
10
18
62 69
41
17
49
3
37
66 17
31
64
44
65
50 51 51
23
60
60
32
9
54 56
39
69
43
41 41
1
11
22 31 36 37 39 44 47 49 56 59 61 65 69
0
0
5 10 15 20 25 30 35 40 45
Drift, in.
200
400
600
800
1,000
1,200
Total shear, kips
(b)
(a)
Figure 3.232 Patterns of plastic hinge formation: weak panel zone model under ML load, including
Pdelta effects.
FEMA 451, NEHRP Recommended Provisions: Design Examples
398
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 3.232. 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 second level of the structure.
It should be noted that 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 is behaving 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.
3.2.5.3 Predictions of Total Displacement and Story Drift from Pushover Analysis
In the following discussion, the only loading pattern considered is the modal load pattern discussed earlier.
This is consistent with the requirements of Provisions Sec. 5A.1.2 [A5.2.2]. The structure with both strong
and weak panel zones is analyzed, and separate analyses are performed including and excluding Pdelta
effects.
3.2.5.3.1 Expected Inelastic Displacements Computed According to the Provisions
The expected inelastic displacement was computed using the procedures of Provisions Sec. 5.5 [5.3]. In
the Provisions, the displacement is computed using responsespectrum analysis with only the first mode
included. The expected roof displacement will be equal to the displacement computed from the 5percentdamped
response spectrum multiplied by the modal participation factor which is multiplied by the first
mode displacement at the roof level of the structure. In the present analysis, the roof level first mode
displacement is 1.0.
Details of the calculations are not provided herein. The relevant modal quantities and the expected
inelastic displacements are provided in Table 3.212. Note that only those values associated with the ML
lateral load pattern were used.
Chapter 3, Structural Analysis
399
Table 3.212 Modal Properties and Expected Inelastic Displacements for the Strong and Weak Panel
Models Subjected to the Modal Load Pattern
Computed Quantity Strong Panel
w/o PDelta
Strong Panel
with PDelta
Weak Panel
w/o PDelta
Weak Panel
with PDelta
Period (seconds)
Modal Participation Factor
Effective Modal Mass (%)
Expected Inelastic Disp. (in.)
Base Shear Demand (kips)
6th Story Drift (in.)
5th Story Drift (in.)
4th Story Drift (in.)
3rd Story Drift (in.)
2nd Story Drift (in.)
1st Story Drift (in.)
1.950
1.308
82.6
12.31
1168
1.09
1.74
2.28
2.10
2.54
2.18
2.015
1.305
82.8
12.70
1051
1.02
1.77
2.34
2.73
2.73
2.23
2.028
1.315
82.1
12.78
1099
1.12
1.84
2.44
2.74
2.56
2.09
2.102
1.311
82.2
13.33
987
1.11
1.88
2.53
2.90
2.71
2.18
As the table indicates, the modal quantities are only slightly influenced by Pdelta effects and the inclusion
or exclusion of doubler plates. The maximum inelastic displacements are in the range of 12.2 to 13.3 in.
The information provided in Figures 3.223 through 3.232 indicates that at a target displacement of, for
example, 13.0 in., some yielding has occurred but the displacements are not of such a magnitude that the
slope of the pushover curve is negative when Pdelta effects are included.
It should be noted that FEMA 356, Prestandard and Commentary for the Seismic Rehabilitation of
Buildings, provides a simplified methodology for computing the target displacement that is similar to but
somewhat more detailed than the approach illustrated above. See Sec. 3.3.3.3.2 of FEMA 356 for details.
3.2.5.3.2 Inelastic Displacements Computed According to the Capacity Spectrum Method
In the capacity spectrum method, the pushover curve is transformed to a capacity curve that represents the
first mode inelastic response of the full structure. Figure 3.233 shows a bilinear capacity curve. The
horizontal axis of the capacity curve measures the first mode displacement of the simplified system. The
vertical axis is a measure of simplified system strength to system weight. When multiplied by the
acceleration due to gravity (g), the vertical axis represents the acceleration of the mass of the simple
system.
Point E on the horizontal axis is the value of interest, the expected inelastic displacement of the simplified
system. This displacement is often called the target displacement. The point on the capacity curve directly
above Point E is marked with a small circle, and the line passing from the origin through this point
represents the secant stiffness of the simplified system. If the values on the vertical axis are multiplied by
the acceleration due to gravity, the slope of the line passing through the small circle is equal to the
acceleration divided by the displacement. This value is the same as the square of the circular frequency of
the simplified system. Thus, the sloped line is also a measure of the secant period of the simplified
structure. As will be shown later, an equivalent viscous damping value (.E) can be computed for the
simple structure deformed to Point E.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3100
Spectral displacement, in.
Spectral pseudoacceleration, g
. E
E 2
.
1
E
Figure 3.233 A simple capacity spectrum.
Figure 3.234 shows a response spectrum with the vertical axis representing spectral acceleration as a ratio
of the acceleration due to gravity and the horizontal axis representing displacement. This spectrum, called
a demand spectrum, is somewhat different from the traditional spectrum that uses period of vibration as the
horizontal axis. The demand spectrum is drawn for a particular damping value (.). Using the demand
spectrum, the displacement of a SDOF system may be determined if its period of vibration is known and
the system’s damping matches the damping used in the development of the demand spectrum. If the
system’s damping is equal to .E, and its stiffness is the same as that represented by the sloped line in
Figure 3.233, the displacement computed from the demand spectrum will be the same as the expected
inelastic displacement shown in Figure 3.233.
The capacity spectrum and demand spectrum are shown together in Figure 3.235. The demand spectrum
is drawn for a damping value exactly equal to .E, but .E is not known a priori and must be determined by
the analyst. There are several ways to determine .E. In this example, two different methods will be
demonstrated: an iterative approach and a semigraphical approach.
Chapter 3, Structural Analysis
3101
Spectral displacement, in.
Spectral pseudoacceleration, g
. E
E 2
.
1
Demand spectrum
for damping
E
Figure 3.234 A simple demand spectrum.
Spectral displacement, in. Spectral pseudoacceleration, g
. E
E 2
.
1
Demand Spectrum
for damping
. E
. E
Capacity spectrum
for damping
E
Figure 3.235 Capacity and demand spectra plotted together.
FEMA 451, NEHRP Recommended Provisions: Design Examples
8Expressions in this section are taken from ATC40 but have been modified to conform to the nomenclature used herein.
3102
The first step in either approach is to convert the pushover curve into a capacity spectrum curve. This is
done using the following two transformations:8
1. To obtain spectral displacement, multiply each displacement value in the original pushover curve by
the quantity:
1 ,1
1
PFf Roof
where PF1 is the modal participation factor for the fundamental mode and fRoof,1 is the value of the first
mode shape at the top level of the structure. The modal participation factor and the modal
displacement must be computed using a consistent normalization of the mode shapes. One must be
particularly careful when using DRAIN because the printed mode shapes and the printed modal
participation factors use inconsistent normalizations – the mode shapes are normalized to a maximum
value of 1.0 and the modal participation factors are based on a normalization that produces a unit
generalized mass matrix. For most frametype structures, the first mode participation factor will be in
the range of 1.3 to 1.4 if the mode shapes are normalized for a maximum value of 1.0.
2. To obtain spectral pseudoacceleration, divide each force value in the pushover curve by the total
weight of the structure, and then multiply by the quantity:
1
1
a
where a1 is the ratio of the effective mass in the first mode to the total mass in the structure. For frame
structures, a1 will be in the range of 0.8 to 0.85. Note that a1 is not a function of mode shape
normalization.
After performing the transformation, convert the smooth capacity curve into a simple bilinear capacity
curve. This step is somewhat subjective in terms of defining the effective yield point, but the results are
typically insensitive to different values that could be assumed for the yield point. Figure 3.236 shows a
typical capacity spectrum in which the yield point is represented by points aY and dY. The displacement
and acceleration at the expected inelastic displacement are dE and aE, respectively. The two slopes of the
demand spectrum are K1 and K2, and the intercept on the vertical axis is aI.
Chapter 3, Structural Analysis
3103
Spectral displacement, in.
Spectral pseudoacceleration, g
K 1
1
d Y d E
a E
Y
I
a 2
a
1
K
Figure 3.236 Capacity spectrum showing control points.
At this point the iterative method and the direct method diverge somewhat. The iterative method will be
presented first, followed by the direct method.
Given the capacity spectrum, the iterative approach is as follows:
I1. Guess the expected inelastic displacement dE. The displacement computed from the simplified
procedure of the Provisions is a good starting point.
I2. Compute the equivalent viscous damping value at the above displacement. This damping value, in
terms of percent critical, may be estimated as:
5 63.7( Y E Y E)
E
E E
a d d a
a d
.

= +
I3. Compute the secant period of vibration:
2
E
E
E
T
g a
d
p
=
×
where g is the acceleration due to gravity.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3104
I4. An estimated displacement must now be determined from the demand spectrum. A damping value
of .E will be assumed in the development of the spectrum. The demand spectrum at this damping
value is adapted from the response spectrum given by Provisions Sec. 4.1.2.6 [3.3.4]. This spectrum
is based on 5 percent of critical damping; therefore, it must be modified for the higher equivalent
damping represented by .E. For the example presented here, the modification factors for systems
with higher damping values are obtained from Provisions Table 13.3.3.1 [13.31], which is
reproduced in a somewhat different form as Table 3.213 below. In Table 3.213, the modification
factors are shown as multiplying factors instead of dividing factors as is done in the Provisions. The
use of the table can be explained by a simple example: the spectral ordinate for a system with 10
percent of critical damping is obtained by multiplying the 5percentdamped value by 0.833.
The values in Table 3.213 are intended for use only for ductile systems without significant strength
loss. They are also to be used only in the longer period constant velocity region of the response
spectrum. This will be adequate for our needs because the initial period of vibration of our structure
is in the neighborhood of 2.0 seconds. See ATC 40 for conditions where the structure does have
strength loss or where the period of vibration is such that the constant acceleration region of the
spectrum controls. During iteration it may be more convenient to use the information from Table
3.213 in graphic form as shown in Figure 3.237.
Table 3.213 Damping Modification Factors
Effective Damping (% critical) Damping Modification Factor
5 1.000
10 0.833
20 0.667
30 0.588
40 0.526
50 or greater 0.500
Chapter 3, Structural Analysis
3105
Damping, % critical
0
0.0
10 20 30 40 50 60
0.2
0.4
0.6
0.8
1.0
1.2
Spectral modification factor
Figure 3.237 Damping modification factors.
I5. Using the period of vibration computed in Step 3 and the damping computed in Step 4, compute the
updated estimate of spectral acceleration new and convert to displacement using the following
aE
expression:
[ ]2 2 /
new
new E
E
E
d g a
p T
×
=
If this displacement is the same as that estimated in Step 1, the iteration is complete. If not, set the
displacement in Step 1 to new and perform another cycle. Continue iterating until the desired level
dE
of accuracy is achieved.
I6. Convert the displacement for the simple system to the expected inelastic displacement for the
complete structure by multiplying by the product of the modal participation factor and the first mode
roof displacement.
The procedure will now be demonstrated for the strong panel structure subjected to the ML load pattern.
Pdelta effects are excluded.
For this structure, the modal participation factor and effective modal mass factor for the first mode are:
FEMA 451, NEHRP Recommended Provisions: Design Examples
3106
f1 = 1.308 and a1 = 0.826
The original pushover curve is shown in Figure 3.223. The capacity spectrum version of the curve is
shown in Figure 3.238 as is a bilinear representation of the capacity curve.
The control values for the bilinear curve are:
dY = 6.592 in.
aY = 0.1750 g
aI = 0.1544 g
K1 = 0.0265 g/in.
K2 = 0.00311 g/in.
The initial period of the structure (from DRAIN) is 1.95 sec. The same period may be recovered from the
demand curve as follows:
Spectral displacement, in.
0
0.00
5 10 15 20 25 30 35
Actual
Simplified
6.59
0.05
0.10
0.15
0.20
0.25
0.30
Spectral pseudoacceleration, g
0.175
Figure 3.238 Capacity spectrum used in iterative solution.
Chapter 3, Structural Analysis
3107
2 2 1.95 sec.
386.1 0.175
6.659
Y
Y
T
g a
d
p p
= = .
× ×
The 5percentdamped demand spectrum for this example is based on Provisions Figure 4.1.2.6 [3.315].
Since the initial period is nearly 2.0 seconds, the only pertinent part of the spectrum is the part that is
inversely proportional to period. Using a value of SD1 of 0.494 (see Sec. 3.2.2.2), the spectral acceleration
as a function of period T is a = 0.494/T where a is in terms of the acceleration due to gravity. For higher
damping values, the acceleration will be multiplied by the appropriate value from Table 3.213 of this
example.
At this point the iteration may commence. Assume an initial displacement dE of 8.5 in. This is the value
computed earlier (see Table 3.212) from the simplified procedure in the Provisions. At this displacement,
the acceleration aE is:
aE=aI+K2dE=0.1544+0.00311(8.5)=0.1808 g .
At this acceleration and displacement, the equivalent damping is:
.
5 63.7( ) 5 63.7(0.175 8.5 6.592 0.1808) 17.2% critical
0.1808 8.5
Y E Y E
E
E E
a d d a
a d
.
 ×  ×
= + = + =
×
The updated secant period of vibration is:
2 2 2.19sec.
386.4 0.1808
8.5
E
E
T
g a
d
p p
= = =
× ×
From Table 3.213 (or Figure 3.237), the damping modification factor for .E = 17.2 percent is 0.71.
Therefore, the updated acceleration is:
new 0.71(0.494) / 2.19 0.160 g .
aE = =
Using this acceleration, the updated displacement for the next iteration is:
[ ]2 [ ]2
386.4 0.160 7.52 in.
2 / 2 / 2.19
new
new E
E
E
d g a
p T p
× ×
= = =
The complete iteration is summarized in Table 3.214, where the final displacement from the iteration is
7.82 in. This must be multiplied by the modal participation factor, 1.308, to obtain the actual roof
displacement. This value is 7.82(1.308) = 10.2 in. and is somewhat greater than the value of 8.5 in.
predicted from the simplified method of the Provisions.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3108
This example converged even though some of the accelerations from the demand spectrum were less than
the yield value in the development of the capacity spectrum (e.g., 0.161 in iteration 1 is less than 0.175).
This particular example predicts displacements very close to the yield displacement dY; consequently, there
may be some influence of the choice of aY and dY on the computed displacement.
Table 3.214 Results of Iteration for Maximum Expected Displacement
Iteration
a*
(g)
dE
(in.)
aE
(g)
Damping
(%)
Damping
Mod. Factor
TE
(sec.)
8.50 0.181 17.2 0.71 2.19
123456789
10
0.161
0.189
0.173
0.183
0.176
0.180
0.178
0.179
0.178
0.179
7.52
8.01
7.70
7.88
7.77
7.84
7.80
7.82
7.81
7.82
0.178
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
11.8
14.7
12.9
14.0
13.4
13.7
13.5
13.6
13.6
13.6
0.80
0.75
0.78
0.76
0.77
0.76
0.77
0.76
0.76
0.76
2.08
2.14
2.10
2.12
2.12
2.12
2.11
2.11
2.11
2.11
Note: a* is from demand spectrum at period TE.
In the direct approach, a family of demand spectra are plotted together with the capacity spectrum and the
desired displacement is found graphically. The steps in the procedure are as follows:
D1. Develop a bilinear capacity spectrum for the structure.
D2. Find the points on the capacity spectrum that represent 5, 10, 15, 20, 25, and 30 percent damping.
D3. Draw a series of secant stiffness lines, one for each damping value listed above.
D4. Develop demand spectra for damping values of 5, 10, 15, 20, 25, and 30 percent of critical.
D5. Draw the demand spectra on the same plot as the capacity spectrum.
D6. Find the points where the secant stiffness lines (from Step 3) for each damping value cross the
demand spectrum line for the same damping value.
D7. Draw a curve connecting the points found in Step 6.
D8. Find the point where the curve from Step 7 intersects the capacity spectrum. This is the target
displacement, but it is still in SDOF spectrum space.
D9. Convert the target displacement to structural space.
Chapter 3, Structural Analysis
3109
This procedure is now illustrated for the strong panel structure subjected to the modal load pattern. For
this example, Pdelta effects are excluded.
1. The original pushover curve for this structure is shown in Figure 3.223. The effective mass in the
first mode is 0.826 times the total mass, and the first mode participation factor is 1.308. The first
mode displacement at the roof of the building is 1.0. Half of the dead weight of the structure was
used in the conversion because the pushover curve represents the response of one of the two frames.
The resulting capacity curve and its bilinear equivalent are shown in Figure 3.238. For this
example, the yield displacement (dy) is taken as 6.59 in. and the corresponding yield strength (ay) is
0.175g. The secant stiffness through the yield point is 0.0263g/in. or 10.2 (rad/sec)2. Note that the
secant stiffness through this point is mathematically equivalent to the circular frequency squared of
the structure; therefore, the frequency is 3.19 rad/sec and the period is 1.96 seconds. This period, as
required, is the same as that obtained from DRAIN. (The main purpose of computing the period
from the initial stiffness of the capacity spectrum is to perform an intermediate check on the
analysis.)
23. The points on the capacity curve representing ßeff values of 5, 10, 15, 20, 25, and 30 percent critical
damping are shown in Table 3.215. The points are also shown as small diamonds on the capacity
spectrum of Figure 3.239. The secant lines through the points are also shown.
Table 3.215 Points on Capacity Spectrum Corresponding to Chosen Damping Values
Effective Damping
(% critical)
Displacement dpi
(in.)
Spectral Acceleration api
(g)
5 6.59 0.175
10 7.25 0.177
15 8.07 0.180
20 9.15 0.183
25 10.7 0.188
30 13.1 0.195
FEMA 451, NEHRP Recommended Provisions: Design Examples
3110
Spectral displacement, in.
0
0.00
Spectral pseudoacceleration, g
5 10 15 20 25
0.10
0.20
0.30
0.40
0.50
0.60
20%
25%
15%
10%
5%
Figure 3.239 Capacity spectrum with equivalent viscous damping points and secant
stiffnesses.
Chapter 3, Structural Analysis
3111
Spectral displacement, in.
0
0.00
Spectral pseudoacceleration, g
5 10 15 20 25
0.20
0.40
0.60
0.80
1.00
1.20
5%
10%
15%
20%
25%
Figure 3.240 Demand spectra for several equivalent viscous damping values.
45. The demand spectra are based on the short period and 1second period accelerations obtained in
Sec. 3.2.2.2e. These values are SDS = 1.09 and SD1 = 0.494. Plots for these spectra are shown
individually in Figure 3.237. The damping modification factors used to obtain the curves were
taken directly or by interpolation from Table 3.213. The demand spectra are shown on the same
plot as the capacity spectrum in Figure 3.241.
68. The final steps of the analysis are facilitated by Figure 3.242, which is a closeup of the relevant
portion of Figure 3.241. The expected inelastic roof displacement, still in spectral space, is
approximately 7.8 in. This is the same as that found from the iterative solution.
9. The expected inelastic roof displacement for the actual structure is 1.308(7.8) or 10.2 in. This is 20
percent greater than the value of 8.5 in. obtained from the first mode elastic responsespectrum
analysis.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3112
Spectral displacement, in.
0
0.00
Spectral pseudoacceleration, g
5 10 15 20 25
0.20
0.40
0.60
0.80
1.00
1.20
5%
10%
15%
20%
25%
5%
10%
15%
20%
25%
Figure 3.241 Capacity and demand spectra on single plot.
Chapter 3, Structural Analysis
3113
5.0
0.00
Spectral pseudoacceleration, g
Spectral displacement, in.
6.0 7.0 8.0 9.0 10.0
0.05
0.30
0.25
0.20
0.15
0.10
10%
15%
20%
25%
5%
5%
10%
15%
20%
25%
Figure 3.242 Closeup view of portion of capacity and demand spectra.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3114
Results for all the strong and weak panel structures under modal load are summarized in Table 3.216. All
drifts and rotations are consistent with the expected inelastic roof displacement shown at the top of the
table.
Table 3.216 Summary of Results from Pushover Analysis
Computed Quantity Strong Panel
w/o PDelta
Strong Panel
with PDelta
Weak Panel
w/o PDelta
Weak Panel
with PDelta
Expected Inelastic Disp. (in.)
Base Shear Demand (kips)
6th Story Drift (in.)
5th Story Drift (in.)
4th Story Drift (in.)
3rd Story Drift (in.)
2nd Story Drift (in.)
1st Story Drift (in.)
Max beam plastic hinge rot. (rad)
Max column plastic hinge rot. (rad)
Max panel zone hinge rot. (rad)
10.2
1125
0.81
1.35
1.82
2.19
2.20
1.83
0.00522
0.0
0.0
10.3
1031
0.78
1.31
1.81
2.23
2.27
1.90
0.00564
0.0
0.0
10.2
1033
0.87
1.55
1.96
2.21
2.06
1.64
0.00511
0.0
0.00421
10.4
953
0.84
1.45
2.00
2.29
2.14
1.68
0.00524
0.0
0.00437
3.2.5.4 Summary and Observations from Pushover Analysis
1. The simplified approach from the Provisions predicts maximum expected displacements about 8 to
10 percent lower than the much more complicated capacity spectrum method. Conclusions cannot
be drawn from this comparison, however, as only one structure has been analyzed.
2. Pdelta effects had a small but significant effect on the response of the system. In particular, base
shears for the structure with Pdelta effects included were about 8 percent lower than for the
structure without Pdelta effects. If the maximum expected displacement was larger, the differences
between response with and without Pdelta effects would have been much more significant.
3. The inelastic deformation demands in the hinging regions of the beams and in the panel zones of the
beamcolumn joints were small and are certainly within acceptable limits. The small inelastic
deformations are attributed to the considerable overstrength provided when preliminary member
sizes were adjusted to satisfy story drift limits.
4. The structure without panel zone reinforcement appears to perform as well as the structure with such
reinforcement. This is again attributed to the overstrength provided.
3.2.6 TimeHistory Analysis
Because of the many assumptions and uncertainties inherent in the capacity spectrum method, it is
reasonable to consider the use of timehistory analysis for the computation of global and local deformation
demands. A timehistory analysis, while by no means perfect, does eliminate two of the main problems
with static pushover analysis: selection of the appropriate lateral load pattern and use of equivalent linear
Chapter 3, Structural Analysis
9See Ray W. Clough and Joseph Penzien, Dynamics of Structures, 2nd Edition.
3115
viscous damping in the demand spectrum to represent inelastic hysteretic energy dissipation. However,
timehistory analysis does introduce its own problems, most particularly selection and scaling of ground
motions, choice of hysteretic model, and inclusion of inherent (viscous) damping.
The timehistory analysis of Example 2 is used to estimate the deformation demands for the structure
shown in Figures 3.21 and 3.22. The analysis, conducted only for the structure with panel zone
reinforcement, is carried out for a suite of three ground motions specifically prepared for the site.
Analyses included and excluded Pdelta effects.
3.2.6.1 Modeling and Analysis Procedure
The DRAIN2Dx program was used for each of the timehistory analyses. The structural model was
identical to that used in the static pushover analysis. Second order effects were included through the use of
the outrigger element shown to the right of the actual frame in Figure 3.24.
Inelastic hysteretic behavior was 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 for this structure when inelastic rotations are less
than about 0.02 radians. (Previous analysis has indicated a low likelihood of rotations significantly greater
than 0.02 radians.) At inelastic rotations greater than 0.02 radians it is possible for local inelastic buckling
to reduce the apparent strength and stiffness.
Rayleigh proportional damping was used to represent viscous energy dissipation in the structure. The
mass and stiffness proportional damping factors were set to produce 5 percent damping in the first and
third modes. This was done primarily for consistency with the pushover analysis, which use a baseline
damping of 5 percent of critical. Some analysts would use a lower damping, say 2.5 percent, to
compensate for the fact that bilinear hysteretic models tend to overestimate energy dissipation in plastic
hinges.
In Rayleigh proportional damping, the damping matrix (D) is a linear combination of the mass matrix M
and the initial stiffness matrix K:
D=aM+ßK
where a and ß 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:9
1 3
1 3
2
1
a . . .
ß . .
. . . .
. . = . .
. . + . .
FEMA 451, NEHRP Recommended Provisions: Design Examples
3116
Note that a and ß are directly proportional to .. To increase . from 5 percent to 10 percent of critical
requires only that a and ß be increased by a factor of 2.0. The structural frequencies and damping
proportionality factors are shown in Table 3.217 for the models analyzed by the timehistory method.
Table 3.217 Structural Frequencies and Damping Factors Used in TimeHistory Analysis.
(Damping Factors that Produce 5 Percent Damping in Modes 1 and 3)
Model/Damping Parameters .1
(Hz.)
.3
(Hz.)
a ß
Strong Panel with PDelta
Strong Panel without PDelta
3.118
3.223
18.65
18.92
0.267
0.275
0.00459
0.00451
It is very important to note that the stiffness proportional damping factor must not be included in the
Type4 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, effecting 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.
The structure was subjected to dead load and full reduced live load, followed by ground acceleration. The
incremental differential equations of motion were solved in a stepbystep manner using the Newmark
constant average acceleration approach. Time steps and other integration parameters were carefully
controlled to minimize errors. The minium time step used for analysis was 0.00025 seconds. Later
analyses used time steps as large as 0.001 seconds.
3.2.6.2 Development of Ground Motion Records
The ground motion time histories used in the analysis were developed specifically for the site. Basic
information for the records was shown previously in Table 3.120 and is repeated as Table 3.218.
Chapter 3, Structural Analysis
3117
Table 3.218 Seattle Ground Motion Parameters (Unscaled)
Record Name Orientation Number of Points and
Time Increment
Peak Ground
Acceleration (g) Source Motion
Record A00 NS 8192 @ 0.005 seconds 0.443 Lucern (Landers)
Record A90 EW 8192 @ 0.005 seconds 0.454 Lucern (Landers)
Record B00 NS 4096 @ 0.005 seconds 0.460 USC Lick (Loma Prieta)
Record B90 EW 4096 @ 0.005 seconds 0.435 USC Lick (Loma Prieta)
Record C00 NS 1024 @ 0.02 seconds 0.460 Dayhook (Tabas, Iran)
Record C90 EW 1024 @ 0.02 seconds 0.407 Dayhook (Tabas, Iran)
Time histories and 5percentdamped response spectra for each of the motions are shown in Figures 3.243
through 3.245.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3118
0.01
0.00
Pseudoacceleration, g
Period, sec
0.10 1.00 10.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
0.01
Period, sec
0.10 1.00 10.00
0.00
Pseudoacceleration, g
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
Record A00 Record A90
Time, sec
0
0.60
Acceleration, g
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
0.40
0.20
0.00
0.20
0.40
0.60
Record A90
Time, sec
0
0.60
Acceleration, g
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
0.40
0.20
0.00
0.20
0.40
0.60
Record A00
Figure 3.243 Time histories and response spectra for Record A.
Chapter 3, Structural Analysis
3119
FEMA 451, NEHRP Recommended Provisions: Design Examples
3120
0.01
Period, sec
0.10 1.00 10.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
Record B00 Record B90
Time, sec
0
0.60
Acceleration, g
2 4 6
Record B90
0.01
Period, sec
0.10 1.00 10.00
0.00
Pseudoacceleration, g
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
8 10 12 14 16 18 20
0.40
0.20
0.00
0.20
0.40
0.60
Time, sec
0
0.60
Acceleration, g
2 4 6
Record B00
8 10 12 14 16 18 20
0.40
0.20
0.00
0.20
0.40
0.60
Figure 3.244 Time histories and response spectra for Record B.
Chapter 3, Structural Analysis
3121
Record C90
Time, sec
0
0.60
Acceleration, g
2 4 6
Record C90
0.01
Period, sec
0.10 1.00 10.00
0.00
Pseudoacceleration, g
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
8 10 12 14 16 18 20
0.40
0.20
0.00
0.20
0.40
0.60
Record C00
0.01
Period, sec
0.10 1.00 10.00
0.00
Pseudoacceleration, g
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
Time, sec
0
0.60
Acceleration, g
2 4 6
Record C00
8 10 12 14 16 18 20
0.40
0.20
0.00
0.20
0.40
0.60
Figure 3.245 Time histories and response spectra for Record C.
FEMA 451, NEHRP Recommended Provisions: Design Examples
102.00 seconds is approximately the average of the period of the strong panel model with and without Pdelta effects. See Table
3.212.
3122
Because only a twodimensional analysis of the structure is performed using DRAIN, only a single
component of ground motion is applied at one time. For the analyses reported herein, only the NS (00)
records of each ground motion were utilized. A complete analysis would require consideration of both sets
of ground motions.
When analyzing structures in two dimensions, Provisions Sec. 5.6.2.1 [5.4.2.1] gives the following
instructions for scaling:
1. For each pair of motions:
a. Assume an initial scale factor for each motion pair (for example, SA for ground motion A00).
b. Compute the 5percentdamped elastic response spectrum for each component in the pair.
2. Adjust scale factors SA, SB, and SC such that the average of the scaled response spectra over the period
range 0.2T1 to 1.5 T1 is not less than the 5percentdamped spectrum determined in accordance with
Provisions Sec. 4.1.3. T1 is the fundamental mode period of vibration of the structure.
As with the threedimensional timehistory analysis for the first example in this chapter, it will be assumed
that the scale factors for the three earthquakes are to be the same. If a scale factor of 1.51 is used, Figure
3.246 indicates that the criteria specified by the Provisions have been met for all periods in the range
0.2(2.00) = 0.40 sec to 1.5(2.00) = 3.0 seconds.10 The scale factor of 1.51 is probably conservative
because it is controlled by the period at 0.47 seconds, which will clearly be in the higher modes of
response of the structure. If the Provisions had called for a cutoff of 0.25T instead of the (somewhat
arbitrary) value of 0.2T, the required scale factor would be reduced to 1.26.
3.2.6.3 Results of TimeHistory Analysis
Timehistory analyses were performed for the structure subjected to the first 20 seconds of the three
different ground motions described earlier. The 20second cutoff was based on a series of preliminary
analyses that used the full duration.
The following parameters were varied to determine the sensitivity of the response to the particular
variation:
1. Analysis was run with and without Pdelta effects for all three ground motions.
2. Analysis was run with 2.5, 5, 10, and 20 percent damping (Ground Motion A00, including Pdelta
effects). These analyses were performed to assess the potential benefit of added viscous fluid damping
devices.
3.2.6.3.1 Response of Structure with 5 Percent of Critical Damping
Chapter 3, Structural Analysis
3123
Pseudoacceleration, in./sec
Period, sec
0.4
0.0
Ratio, scaled average/NEHRP
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Period, sec
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
0
2
100
200
300
400
500
600
700
NEHRP Spectrum
Average of scaled EQ Windows
(b) Ratio of Average of Scaled Spectra to NEHRP Spectrum (S.F.=1.51)
(a) Comparison of Average of Scaled Spectra and NEHRP Spectrum (S.F.=1.51)
Figure 3.246 Ground motion scaling parameters.
The results from the first series of analyses, all run with 5 percent of critical damping, are summarized in
Tables 3.219 through 3.222. Selected timehistory traces are shown in Figures 3.247 through 3.264.
Energy time histories are included for each analysis.
The tabulated shears in Tables 3.219 and 3.221 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 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
FEMA 451, NEHRP Recommended Provisions: Design Examples
3124
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 drift and the peak story drifts listed in Tables 3.220 and 3.222 are peak (envelope) values
at each story and are not necessarily concurrent.
Tables 3.219 and 3.220 summarize the global response of the structure with excluding Pdelta effects.
Timehistory traces are shown in Figures 3.247 through 3.255. Significant yielding occurred in the
girders, columns, and panel zone regions for each of the ground motions. Local quantification of such
effects is provided later for the structure responding to Ground Motion A00.
Table 3.219 Maximum Base Shear (kips) in Frame Analyzed with 5 Percent
Damping, Strong Panels, Excluding PDelta Effects
Level Motion A00 Motion B00 Motion C00
Column Forces 1559 1567 1636
Inertial Forces 1307 1370 1464
Table 3.220 Maximum Story Drifts (in.) from TimeHistory Analysis with 5 percent Damping,
Strong Panels, Excluding PDelta Effects
Level Motion A00 Motion B00 Motion C00 Limit
Total Roof
R6
65
54
43
32
2G
16.7
1.78
3.15
3.41
3.37
3.98
4.81
13.0
1.60
2.52
2.67
2.75
2.88
3.04
11.4
1.82
2.63
2.65
2.33
2.51
3.13
NA
3.75
3.75
3.75
3.75
3.75
4.50
Table 3.221 Maximum Base Shear (kips) in Frame Analyzed with 5 Percent
Damping, Strong Panels, Including PDelta Effects
Level Motion A00 Motion B00 Motion C00
Column Forces 1426 1449 1474
Inertial Forces 1282 1354 1441
Chapter 3, Structural Analysis
3125
Table 3.222 Maximum Story Drifts (in.) from TimeHistory Analysis with 5 Percent Damping,
Strong Panels, Including PDelta Effects
Level Motion A00 Motion B00 Motion C00 Limit
Total Roof
R6
65
54
43
32
2G
17.4
1.90
3.31
3.48
3.60
4.08
4.84
14.2
1.59
2.48
2.66
2.89
3.08
3.11
10.9
1.78
2.61
2.47
2.31
2.78
3.75
NA
3.75
3.75
3.75
3.75
3.75
4.50
The peak base shears (for a single frame), taken from the sum of column forces, are very similar for each
of the ground motions and range from 1307 kips to 1464 kips. There is, however, a pronounced difference
in the recorded peak displacements. For Ground Motion A00 the roof displacement reached a maximum
value of 16.7 in., while the peak roof displacement from Ground Motion C00 was only 11.4 in. Similar
differences occurred for the firststory displacement. For Ground Motion A00, the maximum story drift
was 4.81 in. for Level 1 and 3.98 in. for Levels 2 through 6. The firststory drift of 4.81 in. exceeds the
allowable drift of 4.50 in. Recall that the allowable drift includes a factor of 1.25 that is permitted when
nonlinear analysis is performed.
As shown in Figure 3.247, the larger displacements observed in Ground Motion A00 are due to a
permanent inelastic displacement offset that occurs at about 10.5 seconds into the earthquake. The sharp
increase in energy at this time is evident in Figure 3.249. Responses for the other two ground motions
shown in Figures 3.250 and 3.253 do not have a significant residual displacement. The reason for the
differences in response to the three ground motions is not evident from their ground acceleration
timehistory traces (see Figures 3.243 through 3.245).
The response of the structure including Pdelta effects is summarized in Tables 3.221 and 3.222. Timehistory
traces are shown in Figures 3.256 through 3.264. Pdelta effects have a significant influence on
the response of the structure to each of the ground motions. This is illustrated in Figures 3.265 and
3.266, which are history traces of roof displacement and base shear, respectively, in response to Ground
Motion A00. Responses for analysis with and without Pdelta effects are shown in the same figure. The
Pdelta effect is most evident after the structure has yielded.
Table 3.221 summarizes the base shear response and indicates that the maximum base shear from the
column forces, 1441 kips, occurs during Ground Motion C00. This shear is somewhat less than the shear
of 1464 kips which occurs under the same ground motion when Pdelta effects are excluded. A reduction
in base shear is to be expected for yielding structures when Pdelta effects are included.
Table 3.222 shows that inclusion of Pdelta effects led to a general increase in displacements with the
peak roof displacement of 17.4 in. occurring during ground motion A00. The story drift at the lower level
of the structure is 4.84 in. when Pdelta effects are included and this exceeds the limit of 4.5 in. The larger
drifts recorded during Ground Motion A00 are again associated with residual inelastic deformations. This
may be seen clearly in the timehistory trace of roof and firststory displacement shown in Figure 3.256.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3126
2000
1500
1000
500
0
500
1000
1500
2000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Base shear, kips
Figure 3.248 Time history of total base shear, Ground Motion A00, excluding Pdelta effects.
20
15
10
5
0
5
10
15
20
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Displacement, in.
Total (Roof)
First Story
Figure 3.247 Time history of roof and firststory displacement, Ground Motion A00, excluding Pdelta
effects.
Chapter 3, Structural Analysis
3127
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 4 8 12 16 20
Time, Seconds
Energy, InchKips
Strain + Hysteretic
Strain + Hysteretic + Viscous
Total
Figure 3.249 Energy time history, Ground MotionA00, excluding Pdelta
effects.
20
15
10
5
0
5
10
15
20
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Displacement, in.
Total (Roof)
First Story
Figure 3.250 Time history of roof and firststory displacement. Ground Motion B00, excluding
Pdelta effects.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3128
2000
1500
1000
500
0
500
1000
1500
2000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Base shear, kips
Figure 3.251 Time history of total base shear, Ground Motion B00, excluding Pdelta effects.
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Energy, in.kips
Total
Strain + Hysteretic + Viscous
Strain + Hysteretic
Figure 3.252 Energy time history, Ground Motion B00, excluding Pdelta effects.
Chapter 3, Structural Analysis
3129
20
15
10
5
0
5
10
15
20
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Displacement, in.
Total (Roof)
First Story
Figure 3.253 Time history of roof and firststory displacement, Ground Motion C00, excluding Pdelta
effects.
2000
1500
1000
500
0
500
1000
1500
2000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Base shear, kips
Figure 3.254 Time history of total base shear, Ground Motion C00, excluding Pdelta effects.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3130
20
15
10
5
0
5
10
15
20
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Displacement, in.
Total (Roof)
First Story
Figure 3.256 Time history of roof and firststory displacement, Ground Motion A00, including Pdelta
effects.
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 4 8 12 16 20
Time, Seconds
Energy, InchKips
Strain + Hysteretic
Strain + Hysteretic + Viscous
Total
Figure 3.255 Energy time history, Ground Motion C00, excluding Pdelta effects.
Chapter 3, Structural Analysis
3131
2000
1500
1000
500
0
500
1000
1500
2000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Base shear, kips
Figure 3.257 Time history of total base shear, Ground Motion A00, including Pdelta effects.
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Energy, in.kips
Total
Strain + Hysteretic + Viscous
Strain + Hysteretic
Figure 3.258 Energy time history, Ground Motion A00, including Pdelta effects.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3132
20
15
10
5
0
5
10
15
20
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Displacement, in.
Total (Roof)
First Story
Figure 3.259 Time history of roof and firststory displacement, Ground Motion B00, including Pdelta
effects.
2000
1500
1000
500
0
500
1000
1500
2000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Base shear, kips
Figure 3.260 Time history of total base shear, Ground Motion B00, including Pdelta effects.
Chapter 3, Structural Analysis
3133
20
15
10
5
0
5
10
15
20
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Displacement, in.
Total (Roof)
First Story
Figure 3.262 Time history of roof and firststory displacement, Ground Motion C00, including Pdelta
effects.
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Energy, in.kips
Total
Strain + Hysteretic + Viscous
Strain + Hysteretic
Figure 3.261 Energy time history, Ground Motion B00, including Pdelta effects.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3134
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Energy, in.kips
Total
Strain + Hysteretic + Viscous
Strain + Hysteretic
Figure 3.264 Energy time history, Ground Motion C00, including Pdelta effects.
20.0
15.0
10.0
5.0
0.0
5.0
10.0
15.0
20.0
0 2 4 6 8 10 12 14 16 18 20
Time, Seconds
Displacement, Inches
Roof Level 1
Figure 3.263 Time history of total base shear, Ground Motion C00, including Pdelta effects.
Chapter 3, Structural Analysis
3135
2000
1500
1000
500
0
500
1000
1500
2000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Base shear, kips
Including PDelta
Excluding PDelta
Figure 3.266 Time history of base shear, Ground Motion A00, with and without Pdelta effects.
20
15
10
5
0
5
10
15
20
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Displacement, in.
Including PDelta
Excluding PDelta
Figure 3.265 Timehistory of roof displacement, Ground Motion A00,
with and without Pdelta effect.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3136
Panel zone, max
= 0.0102 rad
Column, max
= 0.0246 rad
Girder, max
= 0.00121 rad
Figure 3.267 Yielding locations for structure with strong panels subjected to
Ground Motion A00, including Pdelta effects.
Figure 3.267 shows the pattern of yielding in the structure subjected to Gound Motion A00 including
Pdelta effects. Recall that the model analyzed incorporated panel zone reinforcement at the interior
beamcolumn joints. Yielding patterns for the other ground motions and for analyses run with and without
Pdelta effects were similar but are not shown here. The circles on the figure represent yielding at any
time during the response; consequently, yielding does not necessarily occur at all locations simultaneously.
Circles shown at the upper left corner of the beamcolumn joint region indicate yielding in the rotational
spring that represents the web component of panel zone behavior. Circles at the lower right corner of the
panel zone represent yielding of the flange component.
Figure 3.267 shows that yielding occurred at both ends of each of the girders at Levels 2, 3, 4, 5, and 6,
and in the columns at Stories 1 and 5. The panels zones at the exterior joints of Levels 2 and 6 also
yielded. The maximum plastic hinge rotations are shown at the locations they occur for the columns,
girders, and panel zones. Tabulated values are shown in Table 3.223. The maximum plastic shear strain
in the web of the panel zone is identical to the computed hinge rotation in the panel zone spring.
3.2.6.3.2 Comparison with Results from Other Analyses
Table 3.223 compares the results obtained from the timehistory analysis with those obtained from the
ELF and the nonlinear static pushover analyses. Recall that the base shears in the table represent half of
the total shear in the building. The differences shown in the results are quite striking:
1. The base shear from nonlinear dynamic analysis is more than four times the value computed from the
ELF analysis, but the predicted displacements and story drifts are similar. 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.
2. The nonlinear static pushover analysis predicts base shears and story displacements that are
significantly less than those obtained from timehistory analysis. It is also very interesting to note that
Chapter 3, Structural Analysis
3137
the pushover analysis indicates no yielding in the panel zones, even at an applied roof displacement of
42 in.
While part of the difference in the pushover and timehistory response is due to the scale factor of 1.51 that
was required for the timehistory analysis, the most significant reason for the difference is the use of the
firstmode lateral loading pattern in the nonlinear static pushover response. Figure 3.268 illustrates this
by plotting the inertial forces that occur in the structure at the time of peak base shear and comparing this
pattern to the force system applied for nonlinear static analysis. The differences are quite remarkable. The
higher mode effects shown in the Figure 3.268 are the likely cause of the different hinging patterns and
are certainly the reason for the very high base shear developed in the timehistory 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 1100 kips. See, for example, Figure 3.224.)
Table 3.223 Summary of All Analyses for Strong Panel Structure, Including PDelta Effects
Response Quantity
Analysis Method
Equivalent
Lateral Forces
Static Pushover
Provisions
Method
Static Pushover
Capacity
Spectrum
Nonlinear Dynamic
Base Shear (kips)
Roof Disp. (in.)
Drift R6 (in.)
Drift 65 (in.)
Drift 54 (in.)
Drift 43 (in.)
Drift 32 (in.)
Drift 21 (in.)
Girder Hinge Rot. (rad)
Column Hinge Rot. (rad)
Panel Hinge Rot. (rad)
Panel Plastic Shear Strain
373
18.4
1.87
2.91
3.15
3.63
3.74
3.14
NA
NA
NA
NA
1051
12.7
1.02
1.77
2.34
2.73
2.73
2.23
0.0065
0.00130
No Yielding
No Yielding
1031
10.3
0.78
1.31
1.81
2.23
2.27
1.90
0.00732
0.00131
No Yielding
No Yielding
1474
17.4
1.90
3.31
3.48
3.60
4.08
4.84
0.0140
0.0192
0.00624
0.00624
Note: Shears are for half of total structure.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3138
First Mode
Pattern
Time History
Analysis
288k
209k
19k
590k
640k
75k
Figure 3.268 Comparison of inertial
force patterns.
3.2.6.3.3 Effect of Increased Damping on Response
The timehistory analysis of the structure with panel zone reinforcement indicates that excessive drift may
occur in the first story. The most cost effective measure to enhance the performance of the structure would
probably be to provide additional strength and/or stiffness at this story. However, added damping is also a
viable approach.
To determine the effect of added damping on the behavior of the structure, preliminary analysis was
performed by simply increasing the damping ratio from 5 percent to 20 percent of critical in 5percent
increments. For comparison purposes, an additional analysis was performed for a system with only 2.5
percent damping. In each case, the structure was subjected to Ground Motion A00, the panel zones were
reinforced, and Pdelta effects were included. A summary of the results is shown in Tables 3.224 and
3.225. As may be seen, an increase in damping from 5 to 10 percent of critical eliminates the drift
problem. Even greater improvement is obtained by increasing damping to 20 percent of critical. In is
interesting to note, however, that an increase in damping had little effect on the inertial base shear, which
is the true shear in the system.
Table 3.224 Maximum Base Shear (kips) in Frame Analyzed Ground Motion A00, Strong
Panels, Including PDelta Effects
Item
Damping Ratio
2.5% 5% 10% 20% 28%
Column Forces 1354 1284 1250 1150 1132
Inertial Forces 1440 1426 1520 1421 1872
Chapter 3, Structural Analysis
3139
Table 3.225 Maximum Story Drifts (in.) from TimeHistory Analysis Ground Motion A00,
Strong Panels, Including PDelta Effects
Level
Damping Ratio
2.5% 5% 10% 20% 28%
Total Roof
R6
65
54
43
32
2G
18.1
1.81
3.72
3.87
3.70
4.11
4.93
17.4
1.90
3.31
3.48
3.60
4.08
4.84
15.8
1.74
2.71
3.00
3.33
3.69
4.21
12.9
1.43
2.08
2.42
2.77
2.86
2.90
11.4
1.21
1.79
2.13
2.40
2.37
2.18
If added damping were a viable option, additional analysis that treats the individual dampers explicitly
would be required. This is easily accomplished in DRAIN by use of the stiffness proportional component
of Rayleigh damping; however, only linear damping is possible in DRAIN. In practice, added damping
systems usually employ devices with a “softening” nonlinear relationship between the deformational
velocity in the device and the force in the device.
If a linear viscous fluid damping device (Figure 3.269) were to be used in a particular story, it could be
modeled through the use of a Type1 (truss bar) element. If a damping constant Cdevice were required, it
would be obtained as follows:
Let the length of the Type1 damper element be Ldevice, the cross sectional area Adevice, and modulus of
elasticity Edevice.
The elastic stiffness of the damper element is simply:
device device
device
device
A E
k
L
=
As stiffness proportional damping is used, the damping constant for the element is:
Cdevice = ß devicekdevice
The damper elastic stiffness should be negligible so set kD = 0.001 kips/in. Thus:
1000
0.001
device
device device
C
ß = = C
When modeling added dampers in this manner, the author typically sets Edevice = 0.001 and Adevice = the
damper length Ldevice.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3140
This value of ßdevice is for the added damper element only. Different dampers may require different values.
Also, a different (global) value of ß will be required to model the stiffness proportional component of
damping in the remaining nondamper elements.
Modeling the dynamic response using Type1 elements is exact within the typical limitations of finite
element analysis. Using the modal strain energy approach, DRAIN will report a damping value in each
mode. These modal damping values are approximate and may be poor estimates of actual modal damping,
particularly when there is excessive flexibility in the mechanism that connects the damper to the structure.
In order to compare the response of the structure with fictitiously high Rayleigh damping to the response
with actual discrete dampers, dampers were added in a chevron configuration along column lines C and D,
between Bays 3 and 4 (see Figure 3.21). As before, the structure is subjected to Ground Motion A00, has
strong panels, and has Pdelta effects included.
Devices with a damping constant (C) of 80 kipsec/in. were added in Stories 1 and 2, devices with C = 70
kipsec/in. were added in Stories 3 and 4, and dampers with C = 60 kipsec/in. were added at Stories 5 and
6. The chevron braces used to connect the devices to the main structure had sufficient stiffness to
eliminate any loss of efficiency of the devices. Using these devices, an equivalent viscous damping of
approximately 28 percent of critical was obtained in the first mode, 55 percent of critical damping was
obtained in the second mode, and in excess of 70 percent of critical damping was obtained in modes three
through six..
The analysis was repeated using Rayleigh damping wherein the above stated modal damping ratios were
approximately obtained. The peak shears and displacements obtained from the analysis with Rayleigh
damping are shown at the extreme right of Tables 3.224 and 3.225. As may be observed, the trend of
decreased displacements and increased inertial shears with higher damping is continued.
Figure 3.270 shows the time history of roof displacements for the structure without added damping, with
true viscous dampers, and with equivalent Rayleigh damping. As may be seen, there is a dramatic
L
Brace Brace
Damper
i
j
Figure 3.269 Modeling a simple damper.
Chapter 3, Structural Analysis
3141
20
15
10
5
0
5
10
15
20
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Roof displacement, in.
No added damping
With discrete added damping
With Rayleigh added damping
Figure 3.270 Response of structure with discrete dampers and with equivalent viscous damping (1.0 in. =
25.4 mm).
2000
1500
1000
500
0
500
1000
1500
2000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Base shear, kips
No added damping
With discrete added damping
With Rayleigh added damping
Figure 1Figure 3.271 Base shear time histories obtained from column forces (1.0 kip = 4.45 kN).
decrease in roof displacement. It is also clear that the discrete dampers and the equivalent Rayleigh
damping produce very similar results.
Figure 3.271 shows the time history of base shears for the structure without added damping, with discrete
dampers, and with equivalent viscous damping. These base shears were obtained from the summation of
column forces, including Pdelta effects. For the discrete damper case, the base shears include the
horizontal component of the forces in the chevron braces. The base shears for the discretely damped
system are greater than the shears for the system without added damping. The peak base shear for the
system with equivalent viscous damping is less than the shear in the system without added damping.
FEMA 451, NEHRP Recommended Provisions: Design Examples
3142
2000
1500
1000
500
0
500
1000
1500
2000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Base shear, kips
Discrete damping
Rayleigh damping
Figure 3.272 Base shear time histories as obtained from inertial forces (1.0 kip = 4.45 kN).
The inertial base shears in the system with discrete damping and with equivalent viscous damping are
shown in Figure 3.272. As may be observed, the responses are almost identical. The inertial forces
represent the true base shear in the structure, and should always be used in lieu of the sum of column
forces.
As might be expected, the use of added discrete damping reduces the hysteretic energy demand on the
structure. This effect is shown in Figure 3.273, which is an energy time history for the structure with
added discrete damping (which yields equivalent viscous damping of 28 percent of critical). This figure
should be compared to Figure 3.258, which is the energy history for the structure without added damping.
The reduction in hysteretic energy demand for the system with added damping will reduce the damage in
the structure.
Chapter 3, Structural Analysis
11Improved methods are becoming available for pushover analysis (see, for example, Chopra and Goel 2001).
3143
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
0 2 4 6 8 10 12 14 16 18 20
Time, sec
Energy, in.kips
Total
Strain + Hysteretic + Viscous
Strain + Hysteretic
Figure 3.273 Energy timehistory for structure with discrete added damping (1.0 in.kip = 0.113 kNm).
3.2.7 Summary and Conclusions
In this example, five different analytical approaches were used to estimate the deformation demands in a
simple unbraced steel 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 analysis
5. Nonlinear dynamic timehistory analysis
Approaches 1, 3, and 5 were carried to a point that allowed comparison of results. In modeling the
structure, particular attention was paid to representing possible inelastic behavior in the panelzone regions
of the beamcolumn joints.
The results obtained from the three different analytical approaches were quite dissimilar. Because of the
influence of the higher mode effects on the response, pushover analysis, when used alone, is inadequate.11
[In the 2003 Provisions, a number of substantive technical changes have been made to the appendix,
FEMA 451, NEHRP Recommended Provisions: Design Examples
3144
largely as a result of work performed in the development of ATC 55. That report outlines numerous other
technical modifications that could be considered in application of nonlinear static analysis methods.]
Except for preliminary design, the ELF approach should not be used in explicit performance evaluation as
it has no mechanism for determining location and extent of yielding in the structure.
This leaves timehistory analysis as the most viable approach. Given the speed and memory capacity of
personal computers, it is expected that timehistory analysis will eventually play a more dominant role in
the seismic analysis of buildings. However, significant shortcomings, limitations, and uncertainties in
timehistory 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 tools and scaling methodologies to make
the motions represent the site. The scaling techniques that are currently recommended in the Provisions
are a start but need improving.
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 author believes 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 seriously considered.
The ideas presented above are certainly not original. They have been presented by many academics and
practicing engineers. What is still lacking is a comprehensive approach for seismicresistant design based
on these principles. Bertero and Bertero (2002) have presented valuable discussions in these regards.
41
4
FOUNDATION ANALYSIS AND DESIGN
Michael Valley, P.E.
This chapter illustrates application of the 2000 Edition of the NEHRP Recommended Provisions to the
design of foundation elements. Example 4.1 completes the analysis and design of shallow foundations for
two of the alternate framing arrangements considered for the building featured in Example 5.2. Example
4.2 illustrates the analysis and design of deep foundations for a building similar to the one highlighted in
Chapter 6 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.
Although this chapter is based on the 2000 Provisions, it has been annotated to reflect changes made to
the 2003 Provisions. Annotations within brackets, [ ], indicate both organizational changes (as a result of
a reformat of all of the chapters of the 2003 Provisions) and substantive technical changes to the 2003
Provisions and its primary reference documents. While the general concepts of the changes are
described, the design examples and calculations have not been revised to reflect the changes to the 2003
Provisions. The most significant change to the foundation chapter in the 2003 Provisions is the addition
of a strength design method for foundations. Another change was made to introduce guidance for the
explicit modeling of foundation loaddeformation characteristics. Where they affect the design examples
in this chapter, other significant changes to the 2003 Provisions and primary reference documents are
noted. However, some minor changes to the 2003 Provisions and the reference documents may not be
noted.
FEMA 451, NEHRP Recommended Provisions: Design Examples
42
In addition to the 2000 NEHRP Recommended Provisions and Commentary (referred to herein as
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. 1999 [2002]. Building Code Requirements and
Commentary for Structural Concrete.
ASCE 7 American Society of Civil Engineers. 1998 [2002]. Minimum Design Loads for
Buildings and Other Structures.
Bowles Bowles, J. E. 1988. Foundation Analysis and Design. McGrawHill.
Brown 1987 Brown, D. A., L. C. Reese, and M. W. O’Neill. 1987. “Cyclic Lateral Loading
of a LargeScale Pile Group,” Journal of Geotechnical Engineering, Vol. 113,
No. 11 (November). ASCE.
Brown 1988 Brown, D. A., C. Morrison, and L. C. Reese. 1988. “Lateral Load Behavior of
Pile Group in Sand.” Journal of Geotechnical Engineering, Vol 114, No. 11,
(November). ASCE.
CRSI Concrete Reinforcing Steel Institute. 1996. CRSI Design Handbook. Concrete
Reinforcing Steel Institute.
FEMA 356 ASCE. 2000. Prestandard and Commentary for the Seismic Rehabilitation of
Buildings, FEMA 356, prepared by the American Society of Civil Engineers for
the Federal Emergency Management Agency.
GROUP Reese, L. C., and S. T. Wang. 1996. Manual for GROUP 4.0 for Windows.
Ensoft.
Kramer Kramer, S. L. 1996. Geotechnical Earthquake Engineering. Prentice Hall.
LPILE Reese, L. C., and S. T. Wang. 1997. Technical Manual for LPILE Plus 3.0 for
Windows. Ensoft.
Martin Martin, G. R., and I. PoLam. 1995. “Seismic Design of Pile Foundations:
Structural and Geotechnical Issues.” Proceedings: Third International
Conference on Recent Advances in Geotechnical Earthquake Engineering and
Soil Dynamics.
Pender Pender, M. J. 1993. “Aseismic Pile Foundation Design Analysis.” Bulletin of
the New Zealand National Society for Earthquake Engineering, Vol. 26, No. 1
(March).
PoLam PoLam, I., M. Kapuskar, and D. Chaudhuri. 1998. Modeling of Pile Footings
and Drilled Shafts for Seismic Design, MCEER980018. Multidisciplinary
Center for Earthquake Engineering Research.
Wang & Salmon Wang, C.K., and C. G. Salmon. 1992. Reinforced Concrete Design .
HarperCollins.
Chapter 4, Foundation Analysis and Design
43
Youd Youd, T. L., Idriss, I. M., and et al. 2001. “Liquefaction Resistance of Soils:
Summary Report from the 1996 NCEER and 1998 NCEER/NSF Workshops on
Evaluation of Liquefaction Resistance of Soils.” Journal of Geotechnical and
Geoenvironmental Engineering (October). ASCE.
Several commercially available programs were used to perform the calculations described in this chapter.
RISA: 3D is used to determine the shears and moments in a concrete mat foundation; LPILE, in the
analysis of laterally loaded single piles; and PCACOL, to determine concrete pile section capacities.
FEMA 451, NEHRP Recommended Provisions: Design Examples
44
127'4"
25'0" 1'2" 25'0" 25'0" 25'0" 25'0" 1'2"
1'2" 25'0" 25'0" 25'0" 25'0" 25'0" 25'0" 25'0" 1'2"
177'4"
N
Figure 4.11 Typical framing plan.
4.1 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 Sec. 5.2 of this volume of design
examples. Refer to that example for more detailed building information and for the design of the
superstructure; because Chapter 4 was completed after Chapter 5, some values may differ slightly
between the two chapters.
4.1.1 Basic Information
4.1.1.1 Description
The framing plan in Figure 4.11 shows the gravityloadresisting system for a representative level of the
building. The site soils, consisting of medium dense sands, are suitable for shallow foundations. Table
4.11 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.
Chapter 4, Foundation Analysis and Design
45
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.
[The 2003 Provisions discuss the settlement and strength limit states in Sec. 7.2.2.2 using slightly
different nomenclature.]
Table 4.11 Geotechnical Parameters
Parameter Value
Basic soil properties Medium dense sand
(SPT) N = 20
. = 125 pcf
angle of internal friction = 33 deg
Net bearing pressure
(to control settlement due to
sustained loads)
# 4000 psf for B # 20 ft
# 2000 psf for B $ 40 ft
(may interpolate for intermediate dimensions)
Bearing capacity
(for plastic equilibrium strength
checks with factored loads)
2000 B psf for concentrically loaded square footings
3000 B' 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, f = 0.6
[In the 2003 Provisions, the f factor for cohesionless soil
is explicitly defined; 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, f = 0.8
[In the 2003 Provisions, the f factor for cohesionless soil
is explicitly defined; the value is set at 0.7 for vertical,
lateral, and rocking resistance.]
The structural material properties assumed for this example are:
f'c = 4,000 psi
fy = 60,000 psi
4.1.1.2 Provisions Parameters
The complete set of parameters used in applying the Provisions to design of the superstructure is
described in Sec. 5.2.2.1 of this volume of design examples. The following parameters, which are used
during foundation design, are duplicated here.
Site Class = D
SDS = 1.0
FEMA 451, NEHRP Recommended Provisions: Design Examples
46
Seismic Design Category = D
4.1.1.3 Design Approach
4.1.1.3.1 Selecting Footing Size and Reinforcement
Most foundation failures are related to excessive movement rather than loss of loadcarrying capacity.
Settlement control should be addressed first. 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 load bearing pressures for all of the individual
footings is encouraged as it will tend to reduce differential settlements, which are usually of more concern
than are total settlements.
When 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 4.12 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 the pile caps), these critical
section definitions become less meaningful and other approaches (e.g., 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 Sec. 15.4
defines how flexural reinforcement is to be distributed for footings of various shapes.
Sec. 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.” Sec. 10.5.4 relaxes
the minimum reinforcement requirement for footings of uniform thickness. Such elements need only
satisfy the shrinkage reinforcement requirements of Sec. 7.12. Sec. 10.5.4 also imposes limits on the
maximum spacing of bars.
4.1.1.3.2 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
Chapter 4, Foundation Analysis and Design
47
d/2
(all sides)
(c)
Critical section
for twoway shear
(b)
Critical section
for oneway shear
(a)
Critical section
for flexure
Outside face of concrete
column or line midway
between face of steel
column and edge of
steel base plate (typical)
extent of footing
(typical)
d
Figure 4.12 Critical sections for isolated
footings.
(a)
Loading
(b)
Elastic, no uplift
(c)
Elastic, at uplift
(d)
Elastic, after uplift
(e)
Some plastification
(f)
Plastic limit
M
P
Figure 4.13 Soil pressure distributions.
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 4.13 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 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 FEMA 356
provides a useful discussion of foundation compliance, rocking, and other advanced considerations.)
4.1.2 Design for Gravity Loads
FEMA 451, NEHRP Recommended Provisions: Design Examples
48
Although most of the examples in the 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 Sec. 4.1.5 below meaningful.
Detailed calculations are shown for a typical interior footing. The results for all three footing types are
summarized in Sec. 4.1.2.5.
4.1.2.1 Demands
Dead and live load reactions are determined as part of the threedimensional analysis described in Sec. 5.2
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 Sec. 2.3.2 of ASCE 7, the controlling gravity load
combination is 1.2D + 1.6L. Because ASCE 7 load combinations are employed, the alternate strength
reduction factors found in ACI 318 Appendix C must be used. [The 2003 Provisions refer to ACI 318
02, in which the basic resistance factors have been revised to be consistent with the load combinations in
ASCE 7. These new resistance factors (not those found in the ACI 318 Appendix) are used for seismic
design. This change would affect slightly the results of the example calculations in this chapter .]
4.1.2.2 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 ft, the allowable bearing pressure (see Table 4.11) is
4000 psf. Therefore, the required footing area is 487,000 lb/4000 psf = 121.25 ft2.
Check a footing that is 11'0" by 11'0":
Pallow = 11 ft(11 ft)(4000 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 4.11, the bearing capacity (qc) is 2000 B = 2000 × 11 = 22000 psf = 22 ksf.
Chapter 4, Foundation Analysis and Design
49
The design capacity for the foundation is:
fPn = fqcB2 = 0.6(22 ksf)(11 ft)2 = 1597 kips o 621 kips. OK
For use in subsequent calculations, the factored bearing pressure qu = 621 kips/(11 ft)2 = 5.13 ksf.
4.1.2.3 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. Because
the demands are calculated at critical sections (see Figure 4.12) that depend on the footing thickness,
iteration is required.
Check a footing that is 26 in. 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 about 16 in. Accounting for cover and
expected bar sizes, d = 26  (3 + 1.5(1)) = 21.5 in.
Oneway shear:
( ) .
16
12 11 11 21.5 5.13 172kips
Vu 2 12
.  .
= .  . =
. .
( ) ( )( )( 1 ) > 172 kips. OK
1000 fVn=fVc= 0.75 2 4000 11×12 21.5 =269kips
Twoway shear:
( ) ( ) . 16 21.5 2
12 Vu =621 + 5.13 =571kips
( ) ( ) ( )( 1 ) > 571 kips. OK
1000 fVn=fVc= 0.75 4 4000..4× 16+21.5 ..21.5 =612kips
4.1.2.4 Footing Reinforcement
Footing reinforcement is selected considering both flexural demands and minimum reinforcement
requirements. The following calculations treat flexure first because it usually controls:
( ) ( ) .
16 2
12 1 11 11 5.13 659 ftkips
Mu 2 2
.  .
= . . =
. .
Try 10 #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 = 10(0.79)(60) = 474 kips.
Noting that C = T and solving the expression C = 0.85 f'c b a for a produces a = 1.06 in.
( ) ( )( 1.06 )( 1 ) > 659 ftkips. OK
2 212 a 0.80 474 21.5 663ftkips
fMn=fTd =  =
FEMA 451, NEHRP Recommended Provisions: Design Examples
410
The ratio of reinforcement provided . = 10(0.79)/[(11)(12)(21.5)] = 0.00278. The distance between bars
spaced uniformly across the width of the footing s = [(11)(12)2(3+0.5)]/(101) = 13.9 in.
According to ACI 318 Sec. 7.12, the minimum reinforcement ratio = 0.0018 < 0.00278. OK
and the maximum spacing is the lesser of 3 × 26 in. or 18 = 18 in. > 13.9 in. OK
4.1.2.5 Design Results
The calculations performed in Sec. 4.1.2.2 through 4.1.2.4 are repeated for typical perimeter and corner
footings. The footing design for gravity loads is summarized in Table 4.12; Figure 4.14 depicts the
resulting foundation plan.
Table 4.12 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
10#8 bars each way
Pallow = 484 kip
fPn = 1597 kip
Oneway shear: Vu = 172 kip
fVn = 269 kip
Twoway shear: Vu = 571 kip
fVn = 612 kip
Flexure: Mu = 659 ftkip
fMn = 663 ftkip
Perimeter D = 206 kip
L = 45 kip
P = 251 kip
Pu = 319 kip
8'0" × 8'0" × 1'6" deep
10#6 bars each way
Pallow = 256 kip
fPn = 614 kip
Oneway shear: Vu = 88.1 kip
fVn = 123 kip
Twoway shear: Vu = 289 kip
fVn = 302 kip
Flexure: Mu = 222 ftkip
fMn = 230 ftkip
Corner D = 104 kip
L = 23 kip
P = 127 kip
Pu = 162 kip
6'0" × 6'0" × 1'2" deep
7#5 bars each way
Pallow = 144 kip
fPn = 259 kip
Oneway shear: Vu = 41.5 kip
fVn = 64.9 kip
Twoway shear: Vu = 141 kip
fVn = 184 kip
Flexure: Mu = 73.3 ftkip
fMn = 80.2 ftkip
[Use of the new resistance factors in ACI 31802 would change these results.]
Chapter 4, Foundation Analysis and Design
411
Corner:
6'x6'x1'2" thick
Perimeter:
8'x8'x1'6" thick
Interior:
11'x11'x2'2" thick
Figure 4.14 Foundation plan for gravityloadresisting system.
4.1.3 Design for MomentResisting Frame System
Framing Alternate A in Sec. 5.2 of this volume of design examples includes a perimeter moment resisting
frame as the seismicforceresisting system. A framing plan for the system is shown in Figure 4.15.
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 Sec. 4.1.2. The results for all footing
types are summarized in Sec. 4.1.3.4.
FEMA 451, NEHRP Recommended Provisions: Design Examples
412
5 at 25'0"
7 at 25'0"
N
Figure 4.15 Framing plan for moment resisting frame system.
4.1.3.1 Demands
A threedimensional analysis of the superstructure, in accordance with the requirements for the equivalent
lateral force (ELF) procedure, is performed using the RAMFRAME program. Foundation reactions at
selected grids are reported in Table 4.13.
Table 4.13 Demands from MomentResisting Frame System
Location Load Rx Ry Rz 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.
Sec. 5.2.3.5 of this volume of design examples outlines the design load combinations, which include the
redundancy factor as appropriate. Considering two senses of accidental torsion for loading in each
direction and including orthogonal effects results in a large number of load cases. 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 + 0.32Ex + 1.11Ey) and the upward case
Chapter 4, Foundation Analysis and Design
413
(0.7D + 0.32Ex + 1.11Ey). [Because the redundancy factor is changed substantially in the 2003
Provisions, the factors in these load combinations would change.]
Before loads can be computed, attention must be given to Provisions Sec. 5.4.5 [5.2.5]. That section
permits “foundations of structures . . . to be designed for threefourths of the foundation overturning
design moment, Mf.” Because the overturning moment in question is the global overturning moment for
the system, judgment must be used in determining which design actions may be reduced. If the seismicforce
resisting 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 (Rz) from load cases Ex
and Ey will be multiplied by 0.75 and all other load effects will remain unreduced.
4.1.3.2 Downward Case (1.4D + 0.5L + 0.32Ex + 1.11Ey)
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 10 ft wide by 40 ft long by 5 ft thick. Further, assume that the top of the footing is
2 ft 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 Provisions. 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[0.32(3.8 + 51.8) + 1.11(21.3 + 281)]
 1.2[10(40)(5)(0.15) + 10(40)(2)(0.125)]
= 714 kips.
Mxx = direct moments + moment due to eccentricity of applied axial loads
= 0.32(53.6 + 47.7) + 1.11(1011.5  891.0)
+ [1.4(203.8) + 0.5(43.8) + 0.75(0.32)(3.8) + 0.75(1.11)(21.3)](12.5)
+ [1.4(103.5) + 0.5(22.3) + 0.75(0.32)(51.8) + 0.75(1.11)(281)](12.5)
= 7258 ftkips.
Myy = 0.32(243.1  246.9) + 1.11(8.1 + 13.4)
= 133 ftkips. (The resulting eccentricity is small enough to neglect here, which simplifies the
problem considerably.)
Vx = 0.32(13.8  14.1) + 1.11(0.5 + 0.8)
= 7.49 kips.
Vy = 0.32(4.6 + 3.7) + 1.11(85.1 68.2)
= 167.5 kips.
Note that the above load combination does not yield the maximum downward load. Reversing the
direction of the seismic load results in P = 1173 kips and Mxx = 3490 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.
FEMA 451, NEHRP Recommended Provisions: Design Examples
414
The soil calculations that follow use a different sign convention than that in the analysis results noted
above; compression is positive for the soil calculations. The eccentricity is:
e = M/P = 7258/714 = 10.17 ft.
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 ft, uplift occurs and
the maximum bearing pressure is:
max
2 2(714) 4.84ksf
3 3(10) 40 10.17
2 2
q P
B L e
= = =
.. .. ..  ..
. . . .
and the length of the footing in contact with the soil is:
3 3 40 10.17 29.5ft .
2 2
L'= ..Le..= ..  ..=
. . . .
The bearing capacity qc = 3000 B' = 3000 × min(B, L'/2) = 3000 × min(10, 29.5/2) = 30,000 psf = 30 ksf.
(L'/2 is used as an adjustment to account for the gradient in the bearing pressure in that dimension.)
The design bearing capacity fqc = 0.6(30 ksf) = 18 ksf > 4.84 ksf. OK
The foundation satisfies overturning and bearing capacity checks. The upward case, which follows, will
control the sliding check.
4.1.3.3 Upward Case (0.7D + 0.32Ex + 1.11Ey)
For the upward case the loads are:
P = 346 kips
Mxx = 6240 ftkips
Myy = 133 ftkips (negligible)
Vx = 7.5 kips
Vy = 167 kips
The eccentricity is:
e = M/P = 6240/346 = 18.0 ft.
Again, e is greater than L/6, so uplift occurs and the maximum bearing pressure is:
max
2(346) 11.5ksf
3(10) 40 18.0
2
q = =
..  ..
. .
and the length of the footing in contact with the soil is:
3 40 18.0 6.0ft .
2
L'= ..  ..=
. .
Chapter 4, Foundation Analysis and Design
415
The bearing capacity qc = 3000 × min(10, 6/2) = 9,000 psf = 9.0 ksf.
The design bearing capacity fqc = 0.6(9.0 ksf) = 5.4 ksf < 11.5 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 L' = 4.39 ft.
The bearing capacity qc = 3000 × min(10, 4.39) = 13,170 psf = 13.2 ksf. (No adjustment to L' is needed
as the pressure is uniform.)
The design bearing capacity fqc = 0.6(13.2 ksf) = 7.92 ksf.
(7.92)(4.39)(10) = 348 kips . 346 kips, so equilibrium is satisfied; the difference is rounded off.
The resisting moment, MR = P (L/2L'/2) = 346 (40/2  4.39/2) = 6160 ftkip . 6240 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:
V= Vx2+Vy2= (7.49)2+(167)2=167.2kips .
As calculated previously, the total compression force at the bottom of the foundation is 346 kips. The
design sliding resistance is:
fVc = f × friction coefficient × P = 0.8(0.65)(346 kips) = 180 kips > 167.2 kips. OK
If base traction alone had been insufficient, resistance due to passive pressure on the leading face could be
included. Sec. 4.2.2.2 below illustrates passive pressure calculations for a pile cap.
4.1.3.4 Design Results
The calculations performed in Sec. 4.1.3.2 and 4.1.3.3 are repeated for combined footings at middle and
side locations. Figure 4.16 shows the results.
FEMA 451, NEHRP Recommended Provisions: Design Examples
416
Corner:
10'x40'x5'0" w/
top of footing
2'0" below grade
Middle:
5'x30'x4'0"
Side:
8'x32'x4'0"
Figure 4.16 Foundation plan for momentresisting frame system.
Figure 4.17 Framing plan for concentrically
braced 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.
4.1.4 Design for Concentrically Braced Frame System
Chapter 4, Foundation Analysis and Design
417
Framing Alternate B in Sec. 5.2 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 4.17.
4.1.4.1 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 ft by 95 ft by 7 ft 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 4.18 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 RISA
3D 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 ft and the maximum soil
pressure is 16.9 ksf.
The bearing capacity qc = 3000 × min(95, 11.1/2) = 16,650 psf = 16.7 ksf.
The design bearing capacity fqc = 0.6(16.7 ksf) = 10.0 ksf < 16.9 ksf. NG
FEMA 451, NEHRP Recommended Provisions: Design Examples
418
(a)
Plastic
solution
(b)
Elastic solution
pressures (ksf)
0
4
8
12
16
12.2 ksf
~
Figure 4.18 Soil pressures for controlling bidirectional case.
As was done in Sec. 4.1.3.3 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 4.18 is determined. The length of the
contact area is 4.46 ft at the left side and 9.10 ft at the right side. The average contact length, for use in
determining the bearing capacity, is (4.46 + 9.10)/2 = 6.78 ft. The distances from the center of the mat to
the centroid of the contact area are
5.42 ft
18.98 ft
x
y
=
=
The bearing capacity qc = 3000 × min(95, 6.78) = 20,340 psf = 20.3 ksf.
The design bearing capacity fqc = 0.6(20.3 ksf) = 12.2 ksf.
(12.2)(6.78)(95) = 7,858 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, MR,xx =Py=7849(18.98)=148,974ftkips >148,439 ftkips. OK
So, the checks of stability and bearing capacity are satisfied. The mat dimensions are shown in Figure
4.19.
Chapter 4, Foundation Analysis and Design
419
Mat:
45'x95'x7'0"
with top of mat
3'6" below grade
Figure 4.19 Foundation plan for concentrically braced frame system.
4.1.4.2 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
12.2 ksf).
[Note that Sec. 7.2.3 of the 2003 Provisions 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 Sec. 10.5 of ACI 318 were
discussed in Sec. 4.1.1.3 above. Although all of the reinforcement provided to satisfy Sec. 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
in. for this 7ftthick mat and assuming one or two layers of bars, the section capacities indicate in Table
4.14 (presented in order of decreasing strength) may be precomputed for use in design. The amount of
FEMA 451, NEHRP Recommended Provisions: Design Examples
420
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 Sec. 7.12.
Table 4.14 Mat Foundation Section Capacities
Mark Reinforcement As (in.2 per ft) fMn (ftkip/ft) 3/4fMn (ftkip/ft)
A 2 layers of #10 bars
at 10 in. o.c.
3.05 899 not used
B 2 layers of #9 bars
at 10 in. o.c.
2.40 not used 534
C 2 layers of #8 bars
at 10 in. o.c.
1.90 not used 424
D #8 bars
at 10 in. o.c.
0.95 not used 215
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 fMn 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 4.110) using contours that correspond to the capacities shown for the
reinforcement patterns noted in Table 4.14.
Chapter 4, Foundation Analysis and Design
421
(a)
M x positive
CL
(b)
M x negative
(c)
M y positive
(d)
M y negative
CL
CL CL
B
C
D B
B
D C C D
B
C
D
B
C
C
C B
B +
669
+
881
B
B
B B
C
C
C
D
D D
D
D
D
+
884
444
+
Figure 4.110 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 4.111, the selected reinforcement is superimposed on the demand contours. Figure 4.112 shows
a section of the mat along Gridline C.
FEMA 451, NEHRP Recommended Provisions: Design Examples
422
CL CL
(a)
EW bottom
reinforcement
(b)
EW top
reinforcement
(c)
NS bottom
reinforcement
(d)
NS top
reinforcement
CL CL
A B A B C D C D
A B
A
B
A
B
8'4" 10'0" 3'4" 5'0" 7'6" 2'6"
4'2" 4'2" 4'2" 4'2"
Figure 4.111 Mat foundation flexural reinforcement.
Chapter 4, Foundation Analysis and Design
423
3" clear
(typical)
8" 8"
Figure 4.112 Section of mat foundation.
Figure 4.113 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 f 4 fc' and in the shaded areas it exceeds f 2 fc' . The critical sections for twoway shear (as
discussed in Sec. 4.1.1.3 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 f 2 fc' 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 created 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, bars may be used both to chair the upper decks of reinforcement and provide resistance to shear
in which case they may be bent thus: .
FEMA 451, NEHRP Recommended Provisions: Design Examples
424
(a) Vx
(b) Vy
Critical section
(typical)
Figure 4.113 Critical sections for shear and envelope of
mat foundation shear demands.
4.1.5 COST COMPARISON
Table 4.15 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
Chapter 4, Foundation Analysis and Design
425
foundations. In this case the added cost amounts to about $0.80/ft2, which is an increase of perhaps 4 or 5
percent to the cost of the structural system.
Table 4.15 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 4.14)
310 cy at $150/cy
= $46,500
310 cy at $15/cy
= $4,650
$ 51,150
Moment frame
(see Figure 4.16)
233 cy at $150/cy
= $34,950
537 cy at $180/cy
= $96,660
800 cy at $15/cy
= $12,000
$143,610
Braced frame
(see Figure 4.19)
233 cy at $150/cy
= $34,950
1108 cy at $180/cy
= $199,440
1895 cy at $15/cy
= $28,425
$262,815
FEMA 451, NEHRP Recommended Provisions: Design Examples
426
Figure 4.21 Design condition: column of concrete moment resisting frame
supported by pile cap and castinplace piles.
4.2 DEEP FOUNDATIONS FOR A 12STORY BUILDING, SEISMIC DESIGN
CATEGORY D
This example features the analysis and design of deep foundations for a 12tory reinforced concrete
momentresisting frame building similar to that described in Chapter 6 of this volume of design examples.
4.2.1 Basic Information
4.2.1.1 Description
Figure 4.21 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 4.21.
Chapter 4, Foundation Analysis and Design
427
Table 4.21 Geotechnical Parameters
Depth Class E Site Class C Site
0 to 3 ft Loose sand/fill
. = 110 pcf
angle of internal friction = 28 deg
soil modulus parameter, k = 25 pci
neglect skin friction
neglect end bearing
Loose sand/fill
. = 110 pcf
angle of internal friction = 30 deg
soil modulus parameter, k = 50 pci
neglect skin friction
neglect end bearing
3 to 30 ft Soft clay
. = 110 pcf
undrained shear strength = 430 psf
soil modulus parameter, k = 25 pci
strain at 50 percent of maximum stress,
e50 = 0.01
skin friction (ksf) = 0.3
neglect end bearing
Dense sand (one layer: 3 to 100 ft depth)
. = 130 pcf
angle of internal friction = 42 deg
soil modulus parameter, k = 125 pci
skin friction (ksf)* = 0.3 + 0.03/ft # 2
end bearing (ksf)* = 65 + 0.6/ft # 150
30 to 100 ft Medium dense sand
. = 120 pcf
angle of internal friction = 36 deg
soil modulus parameter, k = 50 pci
skin friction (ksf)* = 0.9 + 0.025/ft # 2
end bearing (ksf)* = 40 + 0.5/ft # 100
Pile cap
resistance
300 pcf, ultimate passive pressure 575 pcf, ultimate passive pressure
Resistance factor for capacity checks (f) = 0.75.
Safety factor for settlement checks = 2.5.
[In the 2003 Provisions, f factors for
cohesive and cohesionless soils are
explicitly defined; for vertical, lateral and
rocking resistance, the values would be 0.8
for the clay layer and 0.7 for the sand
layers.]
*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
4.2.1.2 Provisions Parameters
Site Class = C and E (both conditions considered in this example)
SDS = 0.9
Seismic Design Category = D (for both conditions)
FEMA 451, NEHRP Recommended Provisions: Design Examples
428
4.2.1.3 Demands
The unfactored demands from the moment frame system are shown in Table 4.22.
Table 4.22 Gravity and Seismic Demands
Location Load Rx Ry Rz Mxx Myy
Corner D 351.0
L 36.0
Vx 40.7 0.6 142.5 4.8 439.0
Vy 0.8 46.9 305.6 489.0 7.0
ATx 1.2 2.6 12.0 27.4 12.9
ATy 3.1 6.7 31.9 70.2 33.0
Side D 702.0
L 72.0
Vx 29.1 0.5 163.4 3.5 276.6
Vy 0.8 59.3 18.9 567.4 6.5
ATx 0.1 3.3 8.7 31.6 1.3
ATy 0.4 8.4 22.2 80.8 3.4
Note: Units are kips and feet. Load Vx is for loads applied toward the east. ATx is the corresponding
accidental torsion case. Load Vy is for loads applied toward the north. ATy is the corresponding accidental
torsion case.
Using ASCE 7 Load Combinations 5 and 7, E as defined in Provisions Sec. 5.2.7 [4.2.2] (with 0.2SDSD =
0.18D 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. [Although the
redundancy factor is changed substantially in the 2003 Provisions, it is expected that this system would
still satisfy the conditions needed for . = 1.0, so these load combinations would not change.]
1.38D + 0.5L ± 1.0Vx ± 0.3Vy ± max(1.0ATx, 0.3ATy)
1.38D + 0.5L ± 0.3Vx ± 1.0Vy ± max(0.3ATx, 1.0ATy)
0.72D ± 1.0Vx ± 0.3Vy ± max(1.0ATx, 0.3ATy)
0.72D ± 0.3Vx ± 1.0Vy ± max(0.3ATx, 1.0ATy)
4.2.1.4 Design Approach
For typical deep foundation systems resistance to lateral loads is provided by both piles and pile cap.
Figure 4.22 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.
Chapter 4, Foundation Analysis and Design
429
Passive resistance
(see Figure 4.25)
py springs
(see Figure 4.24)
Pile
cap
Pile
Figure 4.22 Schematic model of deep foundation system.
Pgroup
Pp Pp
= +
Pgroup
Pot Pot
M
Mgroup
Vgroup
Mgroup
Vgroup
Vpassive
M
V
O
Figure 4.23 Pile cap free body diagram.
4.2.1.4.1 Pile Group Mechanics
With reference to the free body diagram (of a 2×2 pile group) shown in Figure 4.23, demands on
individual piles as a result of loads applied to the group may be determined as follows:
and M = V × R, where R is a characteristic length determined from analysis of a
4
Vgroup Vpassive
V

=
laterally loaded single pile.
, where s is the pile spacing, h is the height of the pile cap,
4
2
group group p passive
ot
V h M M hV
P
s
+ + 
=
and hp is the height of Vpassive above Point O.
and P = Pot + Pp 4
group
p
P
P =
FEMA 451, NEHRP Recommended Provisions: Design Examples
430
Site Class E, depth = 10 ft
Site Class E, depth = 30 ft
Site Class C, depth = 10 ft
Site Class C, depth = 30 ft
0.0
1
Soil resistance, p (lb/in.)
10
100
1,000
10,000
0.1 0.2 0.3
Pile deflection, y (in.)
0.4 0.5 0.6
0.7
100,000
0.8 0.9 1.0
Figure 4.24 Representative py curves (note that a logarithmic scale is used
on the vertical axis).
4.2.1.4.2 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 22 in. diameter pile) are shown in Figure 4.24. 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 is 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.
4.2.1.4.3 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
4.25 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).
Chapter 4, Foundation Analysis and Design
431
0.05
0.6
0.2
0
0.1
0.0
0.01
P/P
ult
0.5
0.4
0.3
1.0
0.9
0.8
0.7
0.02 0.03 0.04
d/H
0.06
Figure 4.25 Passive pressure mobilization curve (after FEMA 356).
4.2.1.4.4 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.) A useful discussion of this “group effect” may be
found in PoLam Sec. 2.6.4, from which the following observations are taken:
The pile group effect has been a popular research topic within the geotechnical community for
almost 50 years. At present, there is no common consensus on the approach for group effects.
Fullsize and model tests by a number of authors show that in general, 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 experimental research reported in Brown 1987, Brown 1988, and other publications] . . .
yielded information that largely corroborated each other on the following aspects:
(1) Most of these experiments first used the single pile data to verify the validity of the widely used
Reese’s and Matlock’s benchmark py criteria and all concluded that the Reese and Matlock py criteria
provide reasonable solutions.
(2) The observed group effects appeared to be associated with shadowing effects and the various
researchers found relatively consistent pile group behavior in that the leading piles would be loaded more
heavily than the trailing piles when all piles are loaded to the same deflection. ... All referenced researchers
recommended to modify the single pile py curves by adjusting the resistance value on the single pile py
curves (i.e. pmultiplier). . . .
The experiments reported by McVay also included data for pile centertocenter spacing of 5D
which showed pmultipliers of 1.0, 0.85, and 0.7 for the front, middle and back row piles,
respectively. For such multipliers, the group stiffness efficiency would be about 95% and group
effects would be practically negligible.
FEMA 451, NEHRP Recommended Provisions: Design Examples
432
The basis of the calculation procedure for group effect factors that is shown below is described in Chapter
6 of GROUP. In these expressions, D is the pile diameter and s is the centertocenter spacing between
the piles in question. In the equation for each efficiency factor, where s/D equals or exceeds the noted
upper limit, the corresponding value of ß is 1.0.
For piles that are side by side with respect to the applied load, a factor to reflect the reduction in
efficiency, ßa
, may be calculated as:
.
0.5659
a 0.5292 for 1 3.28
s s
D D
ß= .. .. = <
. .
For piles that are inline with respect to the applied load, a factor to reflect the reduction in efficiency ( ßb
)
may be calculated as follows:
Leading piles: .
0.2579
bL 0.7309 for 1 3.37
s s
D D
ß= .. .. = <
. .
Trailing piles: .
0.3251
bT 0.5791 for 1 5.37
s s
D D
ß = .. .. = <
. .
For piles that are skewed (neither in line nor side by side) with respect to the applied load, a factor to
reflect the reduction in efficiency ( ßs
) may be calculated as:
ßs= ßa2cos2.+ßb2sin2.
where ßa
and ßb
are calculated as defined above using s equal to the centertocenter distance along the
skew and setting . equal to the angle between the direction of loading and a line connecting the two piles.
If a group contains more than two piles, the effect of each pile on each other pile must be considered. If
the effect of pile j on pile i is called ßji and it is noted that ßji = 1.0 when j = i (as this is a single pile
condition), the preduction factor for any given pile i is
.
1
n
mi ji
j
f ß
=
= .
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:
.
1 1
1 n n
m ji
i j
f
n
ß
= =
= S.
For a 2×2 pile group thus with s = 3D, the group effect factor is calculated as:
3 1
4 2
ß11 = 1.0,
Chapter 4, Foundation Analysis and Design
433
1
0.0
0.2
2
Group effect factor
0.4
0.6
0.8
1.0
3 5
Group size (piles per side)
4
s = 1.5 D
s = 2 D
s = 3 D
s = 4 D
Figure 4.26 Calculated group effect factors.
,
0.5659
21
0.5292 3 1.0 0.985
ß =ßaßb= .. 1 .. × =
. .
, and
0.2579
31
1.0 0.7309 3 0.970
ß =ßaßb= × .. 1 .. =
. .
ß41 = ßa
ßb
= (1.0)(1.0) = 1.0 (because s/D = 4.24).
Thus, fm1 = ß11 × ß21 × ß31 × ß41 = (1.00)(0.985)(0.970)(1.00) = 0.955 . 0.96.
By similar calculations, fm2 = 0.96, fm3 = 0.79, and fm4 = 0.79.
And finally, . 0.96 0.96 0.79 0.79 0.87
fm 4
+ + +
= =
Figure 4.26 shows the group effect factors that are calculated for square pile groups of various sizes with
piles at several different spacings.
4.2.2 Pile Analysis, Design, and Detailing
4.2.2.1 Pile Analysis
For this design example it is assumed that all piles will be fixedhead, 22in.diameter, castinplace piles
arranged in 2×2 pile groups with piles spaced at 66 inches centertocenter. The computer program
LPILE Plus 3.0 is used to analyze single piles for both soil conditions shown in Table 4.21 assuming a
FEMA 451, NEHRP Recommended Provisions: Design Examples
434
30
25
20
15
10
5
0
5 0 5 10 15
Shear, V (kip)
Depth (ft)
Site Class C
Site Class E
Figure 4.27 Results of pile analysis – shear versus
depth (applied lateral load is 15 kips).
30
25
20
15
10
5
0
1000 500 0 500
Moment, M (in.kips)
Depth (ft)
Site Class C
Site Class E
Figure 4.28 Results of pile analysis – moment versus
depth (applied lateral load is 15 kips).
length of 50 ft. 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 0.87 for group effects (as computed at the end of Sec. 4.2.1.4) is
used in all cases. Figures 4.27, 4.28, and 4.29 show the variation of shear, moment, and displacement
with depth (within the top 30 ft) 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 ft for the Class E site (or
about 25 ft 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.
Chapter 4, Foundation Analysis and Design
435
30
25
20
15
10
5
0
0.1 0.0 0.1 0.2 0.3
Displacement (in.)
Depth (ft)
Site Class C
Site Class E
Figure 4.29 Results of pile analysis – displacement
versus depth (applied lateral load is 15 kips)
The analyses performed to develop Figures 4.27 through 4.29 are repeated for different levels of applied
lateral load. Figures 4.210 and 4.211 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 4.210 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:
R = 46 in. for the Class C site
R = 70 in. for the Class E site
FEMA 451, NEHRP Recommended Provisions: Design Examples
436
0
0 5 20 25
Applied lateral load, V (kip)
Head moment, M (inkip)
Site Class C
Site Class E
10 15 30
400
800
1200
1600
Figure 4.210 Results of pile analysis – applied
lateral load versus head moment.
Site Class C
Site Class E
0.0
0
0.2 0.4 0.6 0.8
Head displacement, . (inch)
Applied lateral load, V (kip)
5
10
15
20
25
30
Figure 4.211 Results of pile analysis –
head displacement versus applied lateral
load.
A similar examination of Figure 4.211 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
4.2.2.2 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 Sec. 4.2.1.4 for each of the 32 load combinations discussed in
Sec. 4.2.1.3. 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 4.23, the demands on the group are:
P = 1097 kip
Myy = 93 ftkips
Vx = 10 kips
Myy = 659 ftkips
Vy = 69 kips
From preliminary checks, assume that the displacements in the x and y directions are sufficient to
mobilize 15 percent and 30 percent, respectively, of the ultimate passive pressure:
( 1 )
, 1000
0.15(575) 18 48 48 110 11.0kips
Vpassive x 12 2(12) 12 12
= .. + .... .... .. =
. .. .. .
and
( 1 )
, 1000
0.30(575) 18 48 48 110 22.1kips
Vpassive y 12 2(12) 12 12
= .. + .... .... .. =
. .. .. .
Chapter 4, Foundation Analysis and Design
437
and conservatively take hp = h/3 = 16 in.
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 1.5 kips
to each pile in the x direction. In the y direction the shear in each pile is:
69 22.1 11.7 kips .
4
V

= =
The corresponding pile moments are:
M = 1.5(46) = 69 in.kips for xdirection loading
and
M = 11.7(46) = 538 in.kips for ydirection loading.
The maximum axial load due to overturning for xdirection loading is:
10(48) 93(12) 4(69) 16(4) 13.7 kips
Pot 2(66)
+ + 
= =
and for ydirection loading (determined similarly) Pot = 98.6 kips.
The axial load due to direct loading is Pp = 1097/4 = 274 kips.
Therefore the maximum load effects on the most heavily loaded pile are:
Pu = 13.7 + 98.6 + 274 = 386 kips
Mu= (69)2+(538)2=542in.kips .
The expected displacement in the y direction is computed as:
d = V/k = 11.7/175 = 0.067 in., which is 0.14% of the pile cap height (h).
Reading Figure 4.25 with d/H = 0.0014, P/Pult . 0.34, so the assumption that 30 percent of Pult would be
mobilized was reasonable.
4.2.2.3 Design of Pile Section
The calculations shown in Sec. 4.2.2.2 are repeated for each of the 32 load combinations under each of
the four design conditions. The results are shown in Figures 4.212 and 4.213. In these figures, circles
indicate demands on piles under side columns and squares indicate demands on piles under corner
columns. Also plotted are the fPfM design strengths for the 22in.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 4.24, following calculation of the
required pile length.
FEMA 451, NEHRP Recommended Provisions: Design Examples
438
300
200
100
0
100
200
300
400
500
600
700
800
0 500 1000 1500 2000 2500
Moment, M (inkip)
Axial load, P (kip)
8#7
8#6
6#6
6#5
Side
Corner
Figure 4.212 PM interaction diagram for Site Class C.
300
200
100
0
100
200
300
400
500
600
700
800
0 500 1000 1500 2000 2500
Moment, M (inkip)
Axial Load, P (kip)
8#7
8#6
6#6
6#5
Side
Corner
Figure 4.213 PM interaction diagram for Site Class E.
Chapter 4, Foundation Analysis and Design
439
4.2.2.4 Pile Length for Axial Loads
For the calculations that follow, recall that skin friction and end bearing are neglected for the top three
feet in this example. (In these calculations, the pile cap depth is ignored – effectively assuming that piles
begin at the ground surface. Because the soil capacity increases with depth and the resulting pile lengths
are applied below the bottom of the pile cap, the results are slightly conservative.)
4.2.2.4.1 Length for Settlement
Service loads per pile are calculated as P = (PD + PL)/4.
Check pile group under side column in Site Class C, assuming L = 47 ft:
P = (702 + 72)/4 = 194 kips.
Pskin = average friction capacity × pile perimeter × pile length for friction
= 0.5[0.3 + 0.3 + 44(0.03)]p(22/12)(44) = 243 kips.
Pend = end bearing capacity at depth × end bearing area
= [65 + 44(0.6)](p/4)(22/12)2 = 241 kips.
Pallow = (Pskin + Pend)/S.F. = (243 + 241)/2.5 = 194 kips = 194 kips (demand). OK
Check pile group under corner column in Site Class E, assuming L = 43 ft:
P = (351 + 36)/4 = 97 kips.
Pskin = [friction capacity in first layer + average friction capacity in second layer] × pile perimeter
= [27(0.3) + (13/2)(0.9 + 0.9 + 13[0.025])]p(22/12) = 126 kips.
Pend = [40 + 13(0.5)](p/4)(22/12)2 = 123 kips.
Pallow = (126 + 123)/2.5 = 100 kips > 97 kips. OK
4.2.2.4.2 Length for Compression Capacity
All of the strengthlevel load combinations (discussed in Sec. 4.2.1.3) must be considered.
Check pile group under side column in Site Class C, assuming L = 50 ft:
As seen in Figure 4.112, the maximum compression demand for this condition is Pu = 390 kips.
Pskin = 0.5[0.3 + 0.3 + 47(0.03)]p(22/12)(47) = 272 kips.
Pend = [65 + 47(0.6)](p/4)(22/12)2 = 246 kips.
fPn = f(Pskin + Pend) = 0.75(272 + 246) = 389 kips . 390 kips. OK
Check pile group under corner column in Site Class E, assuming L = 64 ft:
As seen in Figure 4.213, the maximum compression demand for this condition is Pu = 340 kips.
FEMA 451, NEHRP Recommended Provisions: Design Examples
440
Pskin = [27(0.3) + (34/2)(0.9 + 0.9 + 34[0.025])]p(22/12) = 306 kips.
Pend = [40 + 34(0.5)](p/4)(22/12)2 = 150 kips.
fPn = f(Pskin + Pend) = 0.75(306 + 150) = 342 kips > 340 kips. OK
4.2.2.4.3 Length for Uplift Capacity
Again, all of the strengthlevel load combinations (discussed in Sec. 4.2.1.3) must be considered.
Check pile group under side column in Site Class C, assuming L = 5 ft:
As seen in Figure 4.212, the maximum tension demand for this condition is Pu = 1.9 kips.
Pskin = 0.5[0.3 + 0.3 + 2(0.03)]p(22/12)(2) = 3.8 kips.
fPn = f(Pskin) = 0.75(3.8) = 2.9 kips > 1.9 kips. OK
Check pile group under corner column in Site Class E, assuming L = 52 ft.
As seen in Figure 4.213, the maximum tension demand for this condition is Pu = 144 kips.
Pskin = [27(0.3) + (22/2)(0.9 + 0.9 + 22[0.025])]p(22/12) = 196 kips.
fPn = f(Pskin) = 0.75(196) = 147 kips > 144 kips. OK
4.2.2.4.4 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 precalculate 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 4.214 and 4.215.
Chapter 4, Foundation Analysis and Design
441
80
70
60
50
40
30
20
10
0
0 100 200 300 400 500 600 700
Design resistance (kip)
Pile depth (ft)
Compression
Tension
Figure 4.214 Pile axial capacity as a function of length for Site Class C.
80
70
60
50
40
30
20
10
0
0 100 200 300 400 500 600 700
Design resistance (kip)
Pile depth (ft)
Compression
Tension
Figure 4.215 Pile axial capacity as a function of length for Site Class E.
4.2.2.4.5 Results of Pile Length Calculations
FEMA 451, NEHRP Recommended Provisions: Design Examples
442
Detailed calculations for the required pile lengths are provided above for two of the design conditions.
Table 4.23 summarizes the lengths required to satisfy strength and serviceability requirements for all
four design conditions.
Table 4.23 Pile Lengths Required for Axial Loads
Piles Under Corner Column Piles Under Side Column
Condition Load Min Length Condition Load Min Length
Site Class C
Compression 331 kip 43 ft Compression 390 kip 50 ft
Uplift 133 kip 40 ft Uplift 1.9 kip 5 ft
Settlement 97 kip 19 ft Settlement 194 kip 47 ft
Site Class E
Compression 340 kip 64 ft Compression 400 kip 71 ft
Uplift 144 kip 52 ft Uplift 14.7 kip 14 ft
Settlement 97 kip 43 ft Settlement 194 kip 67 ft
4.2.2.5 Design Results
The design results for all four pile conditions are shown in Table 4.24. 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 4.24 Summary of Pile Size, Length, and Longitudinal Reinforcement
Piles Under Corner Column Piles Under Side Column
Site Class C
22 in. diameter by 43 ft long 22 in. diameter by 50 ft long
8#6 bars 6#5 bars
Site Class E
22 in. diameter by 64 ft long 22 in. diameter by 71 ft long
8#7 bars 6#6 bars
4.2.2.6 Pile Detailing
Provisions Sec. 7.4.4 and 7.5.4, respectively, 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:
1. All pile reinforcement must be developed in the pile cap (Provisions Sec. 7.4.4).
2. 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.
3. Continuous longitudinal reinforcement must be provided over the entire length resisting design
tension forces (ACI 318 Sec. 21.8.4.2 [21.10.4.2]).
Chapter 4, Foundation Analysis and Design
443
The discussion that follows refers to the detailing shown in Figures 4.216 and 4.217.
4.2.2.6.1 Development at the Pile Cap
Where neither uplift nor flexural restraint are required, the development length is the full development
length for compression (Provisions Sec. 7.4.4). 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 (Provisions Sec. 7.5.4). 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 Provisions, 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
Provisions Sec. 7.4.4), 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 Provisions.
FEMA 451, NEHRP Recommended Provisions: Design Examples
444
(4) #5
#4 spiral at
9 inch pitch
(6) #5
#4 spiral at
9 inch pitch
(6) #5
#4 spiral at
4.5 inch pitch
4" pile
embedment
Section A
Section B
Section C
C
B
A
21'0" 23'0" 6'4"
Figure 4.216 Pile detailing for Site Class C (under side column).
Chapter 4, Foundation Analysis and Design
445
(4) #7
#4 spiral at
9 inch pitch
(6) #7
#5 spiral at
3.5 inch pitch
(8) #7
#5 spiral at
3.5 inch pitch
4" pile
embedment
Section A
Section B
Section C
C
B
A
32'0" 20'0" 12'4"
Figure 4.217 Pile detailing for Site Class E (under corner column).
FEMA 451, NEHRP Recommended Provisions: Design Examples
446
4.2.2.6.2 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 ft 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 Sec. 9.5.2.3) exceeds the calculated flexural demand at
that point. [A change made in Sec. 7.4.4.1 of the 2003 Provisions makes it clear that the longitudinal
reinforcement must be developed beyond this 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 6 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 in. in diameter, the transverse reinforcement may not be smaller than 0.5
in. diameter. For the piles shown in Figures 4.216 and 4.217 the spacing of the transverse reinforcement
in the top half of the pile length may not exceed the least of: 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 ASCE 318 Sec. 21.4.4.1(a)
of ACI 318 (since the site is not Class E, F, or liquefiable) and the requirements of Sec. 21.4.4.2 and
21.4.4.3 must be satisfied. Note that Sec. 21.4.4.1(a) refers to Eq. (106) [105], which often will govern.
In this case, the minimum volumetric ratio of spiral reinforcement is onehalf that determined using ACI
318 Eq. (106) [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 in., but the maximum spacing permitted by Sec. 21.4.4.2 is 22/4 =
5.5 in. or 6db = 3.75 in., so a #4 spiral at 3.75 in. pitch is used.
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” (Provisions Sec. 7.5.4). 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 Provisions does not indicate the extent of such
detailing into the firmer material. Taking into account the soil layering shown in Table 4.21 and the pile
cap depth and thickness, the tightly spaced transverse reinforcement shown in Figure 4.217 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 Sec. 21.4.4.1
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 in. The maximum spacing permitted by Sec. 21.4.4.2 is 22/4 = 5.5 in. or
6db = 5.25 in., so a #5 spiral at 3.5 in. pitch is used.
4.2.2.6.3 Continuous Longitudinal Reinforcement for Tension
Table 4.23 shows the pile lengths required for resistance to uplift demands. For the Site Class E
condition under a corner column (Figure 4.217), longitudinal reinforcement must resist tension for at
least the top 52 ft (being developed at that point). Extending four longitudinal bars for the full length and
providing widely spaced spirals at such bars reflect the designer’s judgment (not specific requirements of
the Provisions). For the Site Class C condition under a side column (Figure 4.216), design tension due
to uplift extends only about 5 ft below the bottom of the pile cap. Therefore, a design with Section C of
Figure 4.216 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.
Chapter 4, Foundation Analysis and Design
447
(2) #6 top bars
(3) #6 bottom bars
#4 ties at 7" o.c.
2" clear
at sides
3" clear at
top and bottom
Figure 4.218 Foundation tie section.
4.2.3 Other Considerations
4.2.3.1 Foundation Tie Design and Detailing
Provisions Sec. 7.4.3 requires that individual pile caps be connected by ties. Such ties are often grade
beams, but the Provisions 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 will be designed. The resulting
section is shown in Figure 4.218.
For pile caps with an assumed centertocenter spacing of 32 ft in each direction, and given Pgroup = 1121
kips under a side column and Pgroup = 812 kips under a corner column, the tie is designed as follows.
As indicated in Provisions Sec. 7.4.3, the minimum tie force in tension or compression equals the product
of the larger column load times SDS divided by 10 = 1121(0.90)/10 = 101 kips.
The design strength for five # 6 bars is fAs fy = 0.8(5)(0.44)(60) = 106 kips > 101 kips. OK
According to ACI 318 Sec. 21.8.3.2 [21.10.3.2], 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 in.
Use a tie beam that is 14 in. wide and 16 in. deep. ACI 318 Sec. 21.8.3.2 [21.10.3.2] further indicates that
closed ties must be provided at a spacing of not more than onehalf the minimum dimension = 14/2 = 7 in.
Assuming that the surrounding soil provides restraint against buckling, the design strength of the tie beam
concentrically loaded in compression is:
fPn = 0.8f[0.85f'c(Ag  Ast) + fyAst]
= 0.8(0.65)[0.85(3)(16)(14) + 60(5)(0.44)] = 366 kips > 101 kips. OK
4.2.3.2 Liquefaction
For Seismic Design Categories C, D, E and F, Provisions Sec. 7.4.1 requires that the geotechnical report
address potential hazards due to liquefaction. For Seismic Design Categories D, E and F, Provisions Sec.
7.5.1 and 7.5.3 [7.5.1 and 7.5.2] further require that the geotechnical report describe the likelihood and
potential consequences of liquefaction and soil strength loss (including estimates of differential
settlement, lateral movement, and reduction in foundation soilbearing capacity) and discuss mitigation
measures. [In the 2003 Provisions, Sec. 7.5.2 also requires that the geotechnical report describe lateral
loads on foundations, increases in lateral pressures on retaining walls, and flotation of embedded
FEMA 451, NEHRP Recommended Provisions: Design Examples
448
structures.] 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 in the 2003 Provisions]) must be considered.
Commentary Section 7.4.1 contains a calculation procedure that can be used to evaluate the liquefaction
hazard, but readers should refer to Youd for an update of the methods described in the Commentary.
[Sec. 7.4.1 of the 2003 Commentary has been updated to reflect Youd and other recent references.]
4.2.3.3 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, the first two sentences of Provisions Sec. 7.5.4 require 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 Provisions would be as follows:
1. The geotechnical consultant performs appropriate kinematic interaction analyses considering
freefield ground motions and the stiffness of the piles to be used in design.
2. The resulting pile demands, which generally are greatest at the interface between stiff and soft
strata, are reported to the structural engineer.
3. 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 would be 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. The third sentence of Provisions
Sec. 7.5.4, which defines a specific instance in which this second method is to be employed to define
areas requiring additional transverse reinforcement, helps to make an argument for general application of
this practical approach.
4.2.3.4 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)
Chapter 4, Foundation Analysis and Design
449
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 as 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 the
2002 edition 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.
4.2.3.5 Foundation Flexibility and Its Impact on Performance
4.2.3.5.1 Discussion
Most engineers routinely use fixedbase models. Nothing in the Provisions prohibits that common
practice; the consideration of soilstructure interaction effects (Provisions Sec. 5.8 [5.6]) 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 Provisions 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 Provisions
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
Commentary Sec. 5.2.8 [4.5.1]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
Provisions 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 Seismic Use Group II or III in the hope that those measures will improve
performance without requiring explicit consideration of such performance. Although foundation
flexibility can affect structural performance significantly, the fact that all consideration of performance in
the context of the Provisions is approximate and judgmentbased has made it 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 are often resisted by
the user community.
The engineering framework established in FEMA 356 is more conducive to explicit use of performance
measures. In that document (Sec. 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
FEMA 451, NEHRP Recommended Provisions: Design Examples
450
sensitive to base rotations or other types of foundation movement.” In this case the focus is on damage
control rather than structural stability.
4.2.3.5.2 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 Sec. 4.2.2.1 and consider the
northsouth response of the concrete moment frame building located in Berkeley (Sec. 6.2) as
representative for this building.
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 62 shows that the total weight
of the structure is 36,462 kips. Table 65 shows that the calculated period of the fixedbase structure is
2.50 seconds, and Table 64 indicates that 80.2 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:
2 2 ( )
2 2
4 4 (0.802)36, 462 386.1 478 kip/in.
2.50
K M
T
p p
= = =
Foundation Stiffness. As seen in Figure 61 there are 36 moment frame columns. Assume that a 2×2 pile
group supports each column. As shown in Sec. 4.2.2.1, 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 = 5760 kip/in.
Effect of Foundation Flexibility. Because the foundation stiffness is more than 10 times 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:
111 111 441 kip/in.
478 5760
combined
structure fdn
K
K K
= = =
+ +
Assume that the weight of the foundation system is 4000 kips and that 100 percent of the corresponding
mass participates in the new fundamental mode of vibration. The period of the combined system is
[(0.802)(36, 462) (1.0)(4000)]386.1
2 2 2.78 sec
441
T M
K
p p
+
= = =
which is an 11percent increase 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 10 percent smaller and displacements that are 11 percent
larger. In the context of earthquake engineering, those differences are not significant.
51
5
STRUCTURAL STEEL DESIGN
James R. Harris, P.E., Ph.D., Frederick R. Rutz,
P.E., Ph.D., and Teymour Manzouri, P.E., Ph.D.
This chapter illustrates how the 2000 NEHRP Recommended Provisions (hereafter the Provisions) is
applied to the design of steel framed buildings. The three examples include:
1. An industrial warehouse structure in Astoria, Oregon;
2. A multistory office building in Los Angeles, California; and
3. A lowrise hospital facility in the San Francisco Bay area of California.
The discussion examines the following types of structural framing for resisting horizontal forces:
1. Concentrically braced frames,
2. Intermediate moment frames,
3. Special moment frames,
4. A dual system consisting of moment frames and concentrically braced frames, and
5. Eccentrically braced frames.
For determining the strength of steel members and connections, the 1993 [1999] Load and Resistance
Factor Design Specification for Structural Steel Buildings, published by the American Institute of Steel
Construction, is used throughout. In addition, the requirements of the 1997 [2002] AISC Seismic
Provisions for Structural Steel Buildings are followed where applicable.
The examples only cover design for seismic forces in combination with gravity, and 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, and detailing.
All structures are analyzed using threedimensional static or dynamic methods. The SAP2000 Building
Analysis Program (Computers & Structures, Inc., Berkeley, California, v.6.11, 1997) is used in Example
5.1, and the RAMFRAME Analysis Program (RAM International, Carlsbad, California, v. 5.04, 1997 ) is
used in Examples 5.2 and 5.3.
In addition to the 2000 NEHRP Recommended Provisions, the following documents are referenced:
AISC LRFD American Institute of Steel Construction. 1999. Load and Resistance Factor Design
Specification for Structural Steel Buildings.
FEMA 451, NEHRP Recommended Provisions: Design Examples
52
AISC Manual American Institute of Steel Construction. 2001. Manual of Steel Construction, Load
and Resistance Factor Design, 3rd Edition.
AISC Seismic American Institute of Steel Construction. 2000. [2002] Seismic Provisions for
Structural Steel Buildings, 1997, including Supplement No. 2.
IBC International Code Council, Inc. 2000. 2000 International Building Code.
FEMA 350 SAC Joint Venture. 2000. Recommended Seismic Design Criteria for New Steel
MomentFrame Buildings.
AISC SDGS4 AISC Steel Design Guide Series 4. 1990. Extended EndPlate Moment Connections,
1990.
SDI Luttrell, Larry D. 1981. Steel Deck Institute Diaphragm Design Manual. Steel
Deck Institute.
The symbols used in this chapter are from Chapter 2 of the Provisions, the above referenced documents,
or are as defined in the text. Customary U.S. units are used.
Although the these design examples are based on the 2000 Provisions, it is annotated to reflect changes
made to the 2003 Provisions. Annotations within brackets, [ ], indicate both organizational changes (as a
result of a reformat of all of the chapters of the 2003 Provisions) and substantive technical changes to the
2003 Provisions and its primary reference documents. While the general concepts of the changes are
described, the design examples and calculations have not been revised to reflect the changes to the 2003
Provisions.
The most significant change to the steel chapter in the 2003 Provisions is the addition of two new lateral
systems, buckling restrained braced frames and steel plate shear walls, neither of which are covered in
this set of design examples. Other changes are generally related to maintaining compatibility between the
Provisions and the 2002 edition of AISC Seismic. Updates to the reference documents, in particular
AISC Seismic, have some effects on the calculations illustrated herein.
Some general technical changes in the 2003 Provisions that relate to the calculations and/or design in this
chapter include updated seismic hazard maps, changes to Seismic Design Category classification for short
period structures and revisions to the redundancy requirements, new Simplified Design Procedure would
not be applicable to the examples in this chapter.
Where they affect the design examples in this chapter, other significant changes to the 2003 Provisions
and primary reference documents are noted. However, some minor changes to the 2003 Provisions and
the reference documents may not be noted.
It is worth noting that the 2002 edition of AISC Seismic has incorporated many of the design provisions
for steel moment frames contained in FEMA 350. The design provisions incorporated into AISC Seismic
are similar in substance to FEMA 350, but the organization and format are significantly different.
Therefore, due to the difficulty in crossreferencing, the references to FEMA 350 sections, tables, and
equations in this chapter have not been annotated. The design professional is encouraged to review AISC
Seismic for updated moment frame design provisions related to the examples in this chapter.
Chapter 5, Structural Steel Design
53
5.1 INDUSTRIAL HIGHCLEARANCE BUILDING, ASTORIA, OREGON
This example features a transverse steel moment frame and a longitudinal steel braced frame. The
following features of seismic design of steel buildings are illustrated:
1. Seismic design parameters,
2. Equivalent lateral force analysis,
3. Threedimension (3D) modal analysis,
4. Drift check,
5. Check of compactness and brace spacing for moment frame,
6. Moment frame connection design, and
7. Proportioning of concentric diagonal bracing.
5.1.1 Building Description
This industrial building has plan dimensions of 180 ft by 90 ft and a clear height of approximately 30 ft.
It includes a 12fthigh, 40ftwide mezzanine area at the east end of the building. The structure consists
of 10 gable frames spanning 90 ft in the transverse (NS) direction. Spaced at 20 ft o.c., these frames are
braced in the longitudinal (EW) 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 5.11. Longitudinal struts at
the eaves and 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 5.12. The framing consists of a steel roof
deck supported by joists between transverse gable frames. Because the frames resist lateral loading at
each frame position, 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 10 ft
o.c. The floor beams are supported on girders continuous over two intermediate columns spaced
approximately 30 ft 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 in. and lateral sway due to wind was limited to 2 in. (which did not
control). These serviceability limits are not considered to control any aspect of the seismicresistant
design. However, many aspects of seismic design are driven by actual capacities so, in that sense, the
serviceability limits do affect the seismic design.
FEMA 451, NEHRP Recommended Provisions: Design Examples
54
(b)
(a)
(c)
East
30'6" 3'9"
9'0"
32'0"
West
Siding: 6" concrete
insulated sandwich
panels.
Eave
Ridge
Eave
strut
Collector
Concrete slab
on grade
Momentresisting
steel frame.
3'0"
Ceiling
34'3"
35'0"
Braces
Mezzanine
Figure 5.11 Framing elevation and sections (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
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
avoid shear resistance by the panels. (This building arrangement has been intentionally contrived to
illustrate what can happen to a taperedmoment frame building if high seismic demands are placed on it.
More likely, if this were a real building, the concrete panels would be connected directly to the steel
frame, and the building would tend to act as a shear wall building. But for this example, the connections
have been arranged to permit the steel frame to move at the point of attachment in the inplane direction
of the concrete panels. This was done to cause the steel frame to resist lateral forces and, thus, shearwall
action of the panels does not influence the frames.)
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 5.13. 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.
Chapter 5, Structural Steel Design
55
182'0"
Mezzanine
90'0"
1200 MJ12
Cjoist at
4'0" o.c.
W12x62
W14x43
11
2" type "B"
22 gage
metal deck
3" embossed
20 gage deck
N
Figure 5.12 Roof framing and mezzanine framing plan (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
N
90'0"
9 bays at 20'0"=180'0" 30'0" 30'0" 30'0"
Building is
symmetrical
about center
line
6" concrete slab
with 6x6W1.4x
W1.4 wwf over
6" gravel
Typical 3'4"x
3'4"x1'0"
footings
11
4" dia. tie rod
(or equal) at each
frame. Embed in
thickened slab
Mezzanine
5'6"x5'6"x
1'4" footings
Mezzanine
6'6"x6'8"x
1'4" footings
40'0"
Mezzanine
20'0"
(typical)
Figure 5.13 Foundation plan (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
FEMA 451, NEHRP Recommended Provisions: Design Examples
1R must be taken as 4.5 in this direction, due to Provisions Sec. 5.2.2.2.1 [4.3.1.2], which states that if the value of R in
either direction is less than 5, the smaller value of R must be used in both directions. If the ordinary steel moment frame were
chosen for the NS direction, this R factor would change to 3.5.
56
5.1.2 Design Parameters
5.1.2.1 Provisions Parameters
Site Class = D (Provisions Sec. 4.1.2.1[3.5])
SS = 1.5 (Provisions Map 9 [Figure 3.31])
S1 = 0.6 (Provisions Map 10 [Figure 3.32])
Fa = 1.0 (Provisions Table 4.1.2.4a [3.31])
Fv = 1.5 (Provisions Table 4.1.2.4b [3.32])
SMS = FaSS = 1.5 (Provisions Eq. 4.1.2.41 [3.31])
SM1 = FvS1 = 0.9 (Provisions Eq. 4.1.2.42 [3.32])
SDS = 2/3 SMS = 1.0 (Provisions Eq. 4.1.2.51 [3.33])
SD1 = 2/3 SM1 = 0.6 (Provisions Eq. 4.1.2.52 [3.34])
Seismic Use Group = I (Provisions Sec. 1.3 [1.2])
Seismic Design Category = D (Provisions Sec. 4.2.1 [1.4])
[The 2003 Provisions have adopted the 2002 USGS probabilistic seismic hazard maps, and the maps have
been added to the body of the 2003 Provisions as figures in Chapter 3 (instead of the previously used
separate map package).]
Note that Provisions Table 5.2.2 [4.31] permits an ordinary momentresisting steel frame for buildings
that do not exceed one story and 60 feet tall with a roof dead load not exceeding 15 psf. This building
would fall within that restriction, but the intermediate steel moment frame with stiffened bolted end plates
is chosen to illustrate the connection design issues.
[The height and tributary weight limitations for ordinary momentresisting frames have been revised in
the 2003 Provisions. In Seismic Design Category D, these frames are permitted only in singlestory
structures up to 65 feet in height, with fieldbolted end plate moment connections, and roof dead load not
exceeding 20 psf. Refer to 2003 Provisions Table 4.31, footnote h. The building in this example seems
to fit these criteria, but the presence of the mezzanine could be questionable. Similarly, the limitations on
intermediate momentresisting frames in Seismic Design Category D have been revised. The same
singlestory height and weight limits apply, but the type of connection is not limited.]
NS direction:
Momentresisting frame system = intermediate steel moment frame
R = 4.5 (Provisions Table 5.2.2 [4.31])
O0
= 3 (Provisions Table 5.2.2 [4.31])
Cd = 4 (Provisions Table 5.2.2 [4.31])
EW direction:
Braced frame system = ordinary steel concentrically braced frame
(Provisions Table 5.2.2 [4.31])
R = 5 (Provisions Table 5.2.2 [4.31])1
O0
= 2 (Provisions Table 5.2.2 [4.31])
Cd = 4.5 (Provisions Table 5.2.2 [4.31])
Chapter 5, Structural Steel Design
57
5.1.2.2 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, and 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 Provisions Sec. 5.3 [5.2.1].) Therefore, the
mezzanine seismic weight is 69 + 0.25(125) = 100 psf.
5.1.2.3 Materials
Concrete for footings fc' = 2.5 ksi
Slabsongrade fc' = 4.5 ksi
Mezzanine concreteon metal deck fc' = 3.0 ksi
Reinforcing bars ASTM A615, Grade 60
Structural steel (wide flange sections) ASTM A992, Grade 50
Plates ASTM A36
Bolts ASTM A325
5.1.3 Structural Design Criteria
5.1.3.1 Building Configuration
Because there is a mezzanine at one end, the building might be considered vertically irregular (Provisions
Sec. 5.2.3.3 [4.3.2.3]). However, the upper level is a roof, and the Provisions exempts roofs from weight
irregularities. There also are plan irregularities in this building in the transverse direction, again because
of the mezzanine (Provisions Sec. 5.2.3.2 [4.3.2.2]).
5.1.3.2 Redundancy
For a structure in Seismic Design Category D, Provisions Eq. 5.2.4.2 [not applicable in the 2003
Provisions] defines the reliability factor (.) as:
2 20
rmaxxAx
. = 
where the roof area (Ax) = 16,200 sq ft.
To checking . in an approximate manner. In the NS (transverse) direction, there are (2 adjacent
columns)/(2 x 9 bays) so:
0.11 and =0.57 < 1.00 . rmaxx = .
Therefore, use . = 1.00.
FEMA 451, NEHRP Recommended Provisions: Design Examples
58
In the EW (longitudinal) direction, the braces are equally loaded (ignoring accidental torsion), so there is
(1 brace)/(4 braces) so
0.25 and = 1.37 .
maxx r = .
Thus, the reliability multiplier is 1.00 in the transverse direction and 1.37 in the longitudinal direction.
The reliability factor applies only to the determination of forces, not to deflection calculations.
[The redundancy requirements have been substantially changed in the 2003 Provisions. For a building
assigned to Seismic Design Category D, . = 1.0 as long as it can be shown that failure of beamtocolumn
connections at both ends of a single beam (moment frame system) or failure of an individual brace
(braced frame system) would not result in more than a 33 percent reduction in story strength or create an
extreme torsional irregularity. Therefore, the redundancy factor would have to be investigated in both
directions based on the new criteria in the 2003 Provisions.]
5.1.3.3 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 structures in Seismic Design Category D (Provisions Sec.
5.2.5.2.3 and 5.2.5.2.2 [4.4.2.3 and 4.4.2.2, respectively]).
5.1.3.4 Structural Component Load Effects
The effect of seismic load (Provisions Eq. 5.2.71 and 5.2.72 [4.21 and 4.22, respectively]) is:
E = .QE ± 0.2SDSD.
Recall that SDS = 1.0 for this example. The seismic load is combined with the gravity loads as follows:
1.2D + 1.0L + 0.2S + E = 1.2D +1.0L +.QE + 0.2D = 1.4D + 1.0L + 0.2S +.QE.
Note 1.0L is for the storage load on the mezzanine; the coefficient on L is 0.5 for many common live
loads:
0.9D + E = 0.9D + .QE 0.2D = 0.7D + .QE.
5.1.3.5 Drift Limits
For a building in Seismic Use Group I, the allowable story drift (Provisions 5.2.8 [4.51]) is:
.a
= 0.025 hsx.
At the roof ridge, hsx = 34 ft3 in. and .a = 10.28 in.
At the hip (columnroof intersection), hsx = 30 ft6 in. and .a
= 9.15 in.
At the mezzanine floor, hsx = 12 ft and .a
= 3.60 in.
Footnote b in Provisions Table 5.2.8 [4.51, footnote c] permits unlimited drift for singlestory buildings
with interior walls, partitions, etc., that have been designed to accommodate the story drifts. See Sec.
5.1.4.3 for further discussion. The main frame of the building can be considered to be a onestory
Chapter 5, Structural Steel Design
59
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 the footprint of the building.)
5.1.3.6 Seismic Weight
The weights that contribute to seismic forces are:
EW direction NS direction
Roof D and L = (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 = (0.100)(90)(40) = 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 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.
5.1.4 Analysis
Base shear will be determined using an equivalent lateral force (ELF) analysis; a modal analysis then will
examine the torsional irregularity of the building. The base shear as computed by the ELF analysis will
be needed later when evaluating the base shear as computed by the modal analysis (see Provisions Sec.
5.5.7 [5.3.7]).
5.1.4.1 Equivalent Lateral Force Procedure
In the longitudinal direction where stiffness is provided only by the diagonal bracing, the approximate
period is computed using Provisions Eq. 5.4.2.11 [5.26]:
Ta = Crhn
x = (0.02)(34.250.75) = 0.28 sec
In accordance with Provisions Sec. 5.4.2 [5.2.2], the computed period of the structure must not exceed:
Tmax = CuTa = (1.4)(0.28) = 0.39 sec.
The subsequent 3D modal analysis finds the computed period to be 0.54 seconds.
In the transverse direction where stiffness is provided by momentresisting frames (Provisions Eq.
5.4.2.11 [5.26]):
Ta = Crhn
x = (0.028)(34.250.8) = 0.47 sec
and
Tmax = CuTa = (1.4)(0.47) = 0.66 sec.
FEMA 451, NEHRP Recommended Provisions: Design Examples
510
Also note that the dynamic analysis found a computed period of 1.03 seconds.
The seismic response coefficient (Cs) is computed in accordance with Provisions Sec. 5.4.1.1 [5.2.1.1]. In
the longitudinal direction:
1.0 0.222
4.5 1
DS
s
C S
R I
= = =
but need not exceed
0.6 0.342
( / ) (0.39)(4.5/1)
D1
s
C S
T R I
= = =
Therefore, use Cs = 0.222 for the longitudinal direction.
In the transverse direction (Provisions Eq. 5.4.1.11 and 5.4.1.12 [5.22 and 5.23, respectively]):
1.0 0.222
4.5 1
DS
s
C S
R I
= = =
but need not exceed
0.6 0.202
( / ) (0.66)(4.5/1)
D1
s
C S
T R I
= = =
Therefore, use Cs = 0.202 for the transverse direction.
In both directions the value of Cs exceeds the minimum value (Provisions Eq. 5.4.1.13 [not applicable in
the 2003 Provisions]) computed as:
Cs = 0.044I SDS = (0.044)(1)(1.0) = 0.044
[This minimum Cs value has been removed in the 2003 Provisions. In its place is a minimum Cs value for
longperiod structures, which is not applicable to this example.]
The seismic base shear in the longitudinal direction (Provisions Eq. 5.4.1 [5.21]) is:
V = CsW = (0.222)889 kips) = 197 kips.
The seismic base shear in the transverse direction is:
V = CsW = (0.202)(1,102 kips) = 223 kips.
The seismic force must be increased by the reliability factor as indicated previously. Although this is not
applicable to the determination of deflections, it is applicable in the determination of required strengths.
The reliability multiplier ( .) will enter the calculation later as the modal analysis is developed. If the
ELF method was used exclusively, the seismic base shear in the longitudinal direction would be increased
by . now:
V = . (197)
V = (1.37)(197) = 270 kips
Chapter 5, Structural Steel Design
511
[See Sec. 5.1.3.2 for discussion of the changes to the redundancy requirements in the 2003 Provisions.]
Provisions Sec. 5.4.3 [5.2.3] 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 Sec. 5.1.3.6 of this example and interpolating the exponent k as 1.08 for the
period of 0.66 sec, the distribution of forces for the NS analysis is shown in Table 5.11.
Table 5.11 ELF Vertical Distribution for NS Analysis
Level Weight (wx) Height (hx) wxhx
k Cvx Fx
Roof 707 kips 30.5 ft. 28340 0.83 185 kips
Mezzanine 395 kips 12 ft. 5780 0.17 38 kips
Total 1102 kips 34120 223 kips
It is not immediately clear as to whether the roof (a 22gauge steel deck with conventional roofing over it)
will behave as a flexible or rigid diaphragm. If one were to assume that the roof were a flexible
diaphragm while the mezzanine were rigid, the following forces would be applied to the frames:
Typical frame at roof (tributary basis) = 185 kips / 9 bays = 20.6 kips
End frame at roof = 20.6/2 = 10.3 kips
Mezzanine frame at mezzanine = 38 kips/3 frames = 12.7 kips
If one were to assume the roof were rigid, it would be necessary to compute the stiffness for each of the
two types of frames and for the braced frames. For this example, a 3D model was created in SAP 2000.
5.1.4.2 ThreeDimension Static and Modal Response Spectrum Analyses
The 3D analysis is performed for this example to account for:
1. The significance of differing stiffness of the gable frames with and without the mezzanine level,
2. The significance of the different centers of mass for the roof and the mezzanine,
3. The relative stiffness of the roof deck with respect to the gable frames, and
4. 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 3D beamcolumn elements. The collector at the
hip 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 3D model in the transverse direction yields two important results: the roof
diaphragm behaves as a rigid diaphragm and the displacements result in the building being classified as
torsionally irregular. The forces at the roof are distributed to each frame line in a fashion that offsets the
center of force 5 percent of 180 ft (9 ft) to the west of the center of the roof. The forces at the mezzanine
are similarly distributed to offset the center of the mezzanine force 5 percent of 40 ft to the west of the
FEMA 451, NEHRP Recommended Provisions: Design Examples
512
center of the mezzanine. Using grid locations numbered from west to east, the applied forces and the
resulting displacements are shown in Table 5.12.
Table 5.12 ELF Analysis in NS Direction
Grid Roof Force,
kips
Mezzanine
Force, kips
Roof Displacement,
in.
1 13.19 4.56
2 25.35 4.45
3 23.98 4.29
4 22.61 4.08
5 21.24 3.82
6 19.87 3.53
7 18.50 3.21
8 17.13 14.57 2.86
9 15.76 12.67 2.60
10 7.36 10.77 2.42
Totals 184.99 38.01
The average of the extreme displacements is 3.49 in. The displacement at the centroid of the roof is 3.67
in. Thus, the deviation of the diaphragm from a straight line is 0.18 in. whereas the average frame
displacement is about 20 times that. Clearly then, the behavior is as a rigid diaphragm. The ratio of
maximum to average displacement is 1.31, which exceeds the 1.2 limit given in Provisions Table 5.2.3.2
[4.32] and places the structure in the category “torsionally irregular.” Provisions Table 5.2.5.1 [4.41]
then requires that the seismic force analysis be any one of several types of dynamic analysis. The
simplest of these is the modal response spectrum (MRS) analysis.
The MRS is an easy next step once the 3D model has been assembled. A 3D dynamic design response
spectrum analysis is performed per Provisions Sec. 5.5 [5.3] using the SAP 2000 program. The design
response spectrum is based on Provisions Sec. 4.1.2.6 [3.3.4] and is shown in Figure 5.14. [Although it
has no affect on this example, the design response spectrum has been changed for long periods in the
2003 Provisions. See the discussion in Chapter 3 of this volume of design examples.]
Chapter 5, Structural Steel Design
513
0.12
0.2 0.4 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Period, T (sec.)
Spectral response acceleration, S a (g)
0.2
0.4
0.6
0.8
1.0
0.6
S a = S D 1
T
S D 1 =
S D S =
0
0
Figure 5.14 Design response spectrum.
The modal seismic response coefficient (Provisions Eq. 5.5.43 [5.33]) is am . The design
sm
C S
R I
=
response spectra expressed in units of g and ft/sec2 are shown in Table 5.13.
Table 5.13 Design Response Spectra
T
(sec)
Sa (g)
Sam = Sa (g)
( / )
am
sm
S
C
R I
=
R = 4.5
Csm (ft/sec2)
0.0
0.12
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
0.4
0.9
1.0
0.857
0.750
0.666
0.600
0.545
0.500
0.461
0.429
0.089
0.222
0.222
0.190
0.167
0.148
0.133
0.121
0.111
0.102
0.095
2.862
7.155
7.155
6.132
5.367
4.766
4.293
3.900
3.578
3.299
3.070
1.0 ft = 0.3048 m.
With this model, the first 24 periods of vibration and mode shapes of the structure were computed using
the SAP2000 program. The first mode had a period of vibration of 1.03 seconds with predominantly
transverse participation. The third mode period was 0.54 seconds with a predominantly longitudinal
participation. The first 24 modes accounted for approximately 98 percent of the total mass of the
FEMA 451, NEHRP Recommended Provisions: Design Examples
514
structure in the transverse direction and approximately 93 percent in the longitudinal direction, both of
which are is greater than the 90 percent requirement of Provisions Sec. 5.5.2 [5.3.2].
The design value for modal base shear (Vt) is determined by combining the modal values for base shear.
The SAP 2000 program uses the complete quadratic combination (CQC) of the modal values, which
accounts for coupling of closely spaced modes. In the absence of damping, the CQC is simply the square
root of the sum of the squares (SRSS) of each modal value. Base shears thus obtained are:
Longitudinal Vt = 159.5 kips
Transverse Vt = 137.2 kips
In accordance with Provisions Sec. 5.5.7 [5.3.7], compare the design values of modal base shear 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, a factor to bring the modal base shear up
to this comparison ELF value must be applied to the modal story shears, moments, drifts, and floor
deflections. According to Provisions Eq. 5.5.7.1 [5.310]:
Modification factor = 0.85 (V/Vt)
EW modification factor = 0.85(V/Vt) = (0.85)(197 kips/159.5 kips) = 1.05
NS modification factor = 0.85(V/Vt) = (0.85)(223 kips/137.2 kips) = 1.38
The response spectra for the 3D modal analysis is then revised by the above modification factors:
EW (1.0)(1.05)(xdirection spectrum)
NS (1.0)(1.38)(ydirection spectrum)
The model is then run again.
The maximum lateral displacements at the ridge due to seismic loads (i.e., design response spectra as
increased by the modification factors above) from the second analysis are:
EW deflection dxe = 0.84 in.
NS deflection dye = 2.99 in. at the first frame in from the west end
where dxe and dye are deflections determined by the elastic modal analysis. Those frames closer to the
mezzanine had smaller NS lateral deflections in much the same fashion as was shown for the ELF
analysis. Before going further, the deflections should be checked as discussed in Sec. 5.1.4.3 below.
The response spectra for the 3D modal analysis are combined to meet the orthogonality requirement of
Provisions Sec. 5.2.5.2.2a [4.4.2.3]:
EW (1.0)(EW direction spectrum) + (0.3)(NS direction spectrum)
NS (0.3)(EW direction spectrum) + (1.0)(NS direction spectrum)
Finally, the design response spectra for the 3D modal analysis is again revised by increasing the EW
direction response by the reliability factor, . = 1.37. Note that . is equal to unity in the NS direction.
Thus, the factors on the basic spectrum for the load combinations become:
EW (1.0)(1.05)(1.37)(EW direction spectrum) + (0.3)(1.38)(1.00)(NS direction spectrum)
NS (0.3)(1.05)(1.37)(EW direction spectrum) + (1.0)(1.38)(1.00)(NS direction spectrum)
Chapter 5, Structural Steel Design
515
and the model is run once again to obtain the final result for design forces, shears, and moments. From
this third analysis, the final design base shears are obtained. Applying the . factor (1.37) is equivalent to
increasing the EW base shear from (0.85 x 197 kips) = 167.5 kips to 230 kips.
5.1.4.3 Drift
The lateral deflection cited previously must be multiplied by Cd = 4 to find the transverse drift:
. 4.0(2.99) 12.0 in
1.0
d xe
x
C
I
d = d = =
This exceeds the limit of 10.28 in. 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. (The heavy wall panels were selected to make an interesting example
problem, and the high transverse drift is a consequence of this. Some real buildings, such as refrigerated
warehouses, have heavy wall panels and would be expected to have high seismic drifts. Special attention
to detailing the connections of such features is necessary.)
In the longitudinal direction, the lateral deflection was much smaller and obviously is within the limits.
Recall that the deflection computations do not consider the reliability factor. This value must be
multiplied by a Cd factor to find the transverse drift. The tabulated value of Cd is 4.5, but this is for use
when design is based upon R = 5. The Provisions does not give guidance for Cd when the system R factor
is overridden by the limitation of Provisions Sec. 5.2.2.1 [4.3.1.2]. The authors suggest adjusting by a
ratio of R factors.
5.1.4.4 Pdelta
The AISC LRFD Specification requires Pdelta analyses for frames. This was investigated by a 3D Pdelta
analysis, which determined that secondary Pdelta effect on the frame in the transverse direction was
less than 1 percent of the primary demand. As such, for this example, Pdelta was considered to be
insignificant and was not investigated further. (Pdelta may be significant for a different structure, say
one with higher mass at the roof. Pdelta should always be investigated for unbraced frames.)
5.1.4.5 Force Summary
The maximum moments and axial forces caused by dead, live, and earthquake loads on the gable frames
are listed in Tables 5.12 and 5.13. 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 5.14 and the axial forces
are given in Table 5.15. The moment diagram for the combined load condition is shown in Figure 5.15.
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 and design of the braces are discussed in Sec. 5.1.5.5 of this chapter.
FEMA 451, NEHRP Recommended Provisions: Design Examples
516
104 ft  kips
447 ft  kips
53 ft  kips
447 ft  kips


40 ft  kips
1.4D + 0.25 + . Q E 0.7D  . Q E
53 ft  kips
Figure 5.15 Moment diagram for seismic load combinations (1.0 ftkip = 1.36 kNm).
Table 5.14 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 maximums 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 5.15 Axial Forces in Gable Frames 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 maximums 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.
5.1.5 Proportioning and Details
Chapter 5, Structural Steel Design
517
Mezzanine (2 end bays)
Tapered column
12'0"
30'6"
Tapered roof beam
Figure 5.16 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 5.18 (1.0 in. = 25.4+ mm).
The gable frame is shown schematically in Figure 5.16. Using load combinations presented in Sec.
5.1.3.4 and the loads from Tables 5.12 and 5.13, 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 5.11,
was proportioned similarly. The diagonal bracing, shown in Figure 5.11 at the east end of the building,
is proportioned using tension forces determined from the 3D modal analysis.
5.1.5.1 Frame Compactness and Brace Spacing
According to Provisions Sec. 8.4 [8.2.2], steel structures assigned to Seismic Design Categories D, E, and
F must be designed and detailed (with a few exceptions) per AISC Seismic. For an intermediate moment
frame (IMF), AISC Seismic Part I, Section 1, “Scope,” stipulates that those requirements are to be applied
in conjunction with AISC LRFD. Part I, Section 10 of AISC Seismic itemizes a few exceptions from
AISC LRFD for intermediate moment frames, but otherwise the intermediate moment frames are to be
designed per the AISC LRFD Specification.
Terminology for momentresisting frames varies among the several standards; Table 5.16 is intended to
assist the reader in keeping track of the terminology.
FEMA 451, NEHRP Recommended Provisions: Design Examples
518
Table 5.16 Comparison of Standards
Total Rotation
(story drift
angle)
Plastic
Rotation
AISC Seismic
(1997) FEMA 350 AISC Seismic
(Supplement No. 2) Provisions
0.04 0.03 SMF SMF SMF SMF
0.03 0.02 IMF Not used Not used Not used
0.02 0.01* OMF OMF IMF IMF
Not defined Minimal Not used Not used OMF OMF
*This is called “limited inelastic deformations” in AISC Seismic.
SMF = special moment frame.
IMF = intermediate moment frame.
OMF = ordinary moment frame.
For this example, IMF per the Provisions corresponds to IMF per AISC Seismic.
[The terminology in the 2002 edition of AISC Seismic is the same as Supplement No. 2 to the 1997
edition as listed in Table 5.16. Therefore, the terminology is unchanged from the 2000 Provisions.]
Because AISC Seismic does not impose more restrictive widththickness ratios for IMF, the widththickness
ratios of AISC LRFD, Table B5.1, will be used for our IMF example. (If the frame were an
SMF, then AISC Seismic would impose more restrictive requirements.)
The tapered members are approximated as short prismatic segments; thus, the adjustments of AISC LRFD
Specification for webtapered members will not affect the results of the 3D SAP 2000 analysis.
All widththickness ratios are less than the limiting .p from AISC LRFD Table B5.1. All PM ratios
(combined compression and flexure) were less than 1.00. This is based on proper spacing of lateral
bracing.
Lateral bracing is provided by the roof joists and wall girts. The spacing of lateral bracing is illustrated
for the high moment area of the tapered beam near the knee. The maximum moment at the face of the
column under factored load combinations is less than the plastic moment, but under the design seismic
ground motion the plastic moment will be reached. At that point the moment gradient will be higher than
under the design load combinations (the shear will be higher), so the moment gradient at design
conditions will be used to compute the maximum spacing of bracing. The moment at the face of the
column is 659 ft.kip, and 4.0 ft away the moment is 427 ft.kip. The member is in single curvature here,
so the sign on the ratio in the design equation is negative (AISC LRFD Eq. F117):
1
2
pd 0.12 0.076 y
y
L M Er
M F
. . ... .
.. .. ...... ..
= +
0.12 0.076 488 29,000 (1.35) 49.9 in. > 48 in. OK
Lpd 659 50 . . ... .
. . ... .
. . ... .
+  =
Chapter 5, Structural Steel Design
519
Section "A"
Elevation
Filler pad
L3x3
Section "B"
MC8 girt
11
8" dia. A325
(typical)
L3x3
Gusset plate
2x2
Xbrace
Figure 5.17 Arrangement at knee (1.0 in. = 25.4 mm).
Also, per AISC LRFD Eq. F14:
Lp = 300ry/Fyf
Lp=(300)(1.35) / 50=57 in.>48 in. OK
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 5.17).
FEMA 451, NEHRP Recommended Provisions: Design Examples
520
Bolts: 1" dia.
A490
g = 4"
p
Typical
Tapered
beam
Tapered
column
Plate: 1
2"x7x1'05
8"
t = 1
2"
Plate: 2"x7x1'05
8"
t = 2"
b = 9"
3'0"
s
1
2"
1
2"
1"
L st
p = 11
2" f
p = 3" b
1"
Varies
18" to 36"
Varies
12" to 36"
3
4" 2"
1
2"
8"
8"
2"
7
16"
1
2" 1
2"
Detail "1"
2"
3
4"
2.5
1
Weld per
AWS D1.1
t =
p
p
Detail "1"
30°
d = 37.25" 0
d = 30.75" 1
d = 36" b
3
4"
3
4"
Figure 5.18 Bolted stiffened connection at knee (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
5.1.5.2 Knee of the Frame
The knee detail is shown in Figures 5.17 and 5.18. The vertical plate shown near the upper left corner
in Figure 5.17 is a gusset providing connection for Xbracing in the longitudinal direction. The beam to
column connection requires special consideration. The method of FEMA 350 for bolted, stiffened end
plate connections is used for a design guide here. (FEMA 350 has design criteria for specific connection
details. The connection for our moment frame, which has a tapered column and a tapered beam is not one
of the specific details per FEMA 350. However, FEMA 350 is used as a guide for this example because it
is the closest design method developed to date for such a connection.) Refer to Figure 5.18 for
configuration. Highlights from this method are shown for this portion of the example Refer to FEMA
350 for a discussion of the entire procedure. AISC SDGS4 is also useful.
Chapter 5, Structural Steel Design
521
The FEMA 350 method for bolted stiffened end plate connection requires the determination of the
maximum moment that can be developed by the beam. The steps in FEMA 350 for bolted stiffened end
plates follow:
Step 1. The location of the plastic hinge is distance x from the face of the column. 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 (Mpr) at the plastic
hinge is computed (FEMA 350 Eq. 31) as follows:
Mpr = CprRyZeFy.
Per FEMA 350 Eq. 32:
.
+ (50 65) 1.15
2 (2)(50)
y u
pr
u
F F
C
F
= = + =
AISC Seismic Table I61 indicates:
Ry = 1.1
Ze = 267 in.3 at d/2 from the end plate (the plastic hinge location)
Fy = 50 ksi
Therefore, Mpr = (1.15)(1.1)(267)(50) = 16,888 in.kips. = 1,407 ftkips.
The moment at the column flange, Mf , which drives the connection design, is determined from
FEMA 350 Figure 34 as:
Mf = Mpr + Vpx
where
Vp = Shear at location of plastic hinge, assuming the frame has formed two hinges, one near each
column.
1 2 (0.52 klf) 81 ft 1407 1407 ftk 55.8 kips
2 2 81ft
pr pr
p g
V w l M M
l
= + + = .. ..+ + =
. .
l = 81 ft comes from the 90 ft outtoout dimension of the frame, less the column depth and
distance to the hinge at each end. Where the gravity moments are a large fraction of the
section capacity, the second hinge to form, which will be in positive moment, may be away
from the column face, which will reduce l and usually increase Vp. That is not the
circumstance for this frame.
x = db /2 = 18 in. = 1.5 ft
Thus, Mf = 1407 + (55.8)(1.5) = 1491 ftkips
In a like manner, the moment at the column centerline is found:
FEMA 451, NEHRP Recommended Provisions: Design Examples
522
1407 55.8(1.5 1.5) 1574 ftkips
2
c
c pr p
M M V.x d .
. .
. .
= + + = + + =
Step 2. Find bolt size for end plates. For a connection with two rows of two bolts inside and outside the
flange, FEMA 350 Eq. 331 indicates:
Mf < 3.4 Tub(do + di)
(1491)(12) < 3.4 Tub(37.25 in. + 30.75 in.)
77.38 < Tub
77.38 < 113 Ab (for A490 bolts)
0.685 in.2 < Ab
Use 1 in. Diameter A490 bolts.
Now confirm that Tub satisfies FEMA 350 Eq. 332:
0.591 2.583
0.895 1.909 0.327 0.965
0.00002305 f fu
ub b
p bt s p
p F
T T
t d t b
= +
where:
pf = dimension from top of flange to top of first bolt = 1.5 in.
tp = end plate thickness = 2 in. (Trial tp)
dbt = bolt diameter = 1 in.
ts = thickness of stiffener plate = 0.44 in.
bp = width of end plate = 9 in.
Tb = bolt pretension per AISC LRFD Table J3.1
Tub = 113 Ab = (113)(0.785) = 88.7 kips
0.591 2.583
0.895 1.909 0.327 0.965
(0.00002305)(1.5) (504) 64
(2) (1) (0.44) (9) Tub= +
Tub = 88.7 kips > 87.5 kips OK
Therefore, a 2in.thick end plate is acceptable.
Step 3. Check the bolt size to preclude shear failure. This step is skipped here because 16 bolts will
obviously carry the shear for our example.
Step 4. Determine the minimum end plate thickness necessary to preclude flexural yielding by comparing
the thickness determined above against FEMA 350 Eq. 334:
0.9 0.6 0.9
0.9 0.1 0.7
0.00609f fu
p
bt s p
p g F
t
d t b
=
0.9 0.6 0.9
0.9 0.1 0.7
(0.00609)(1.5) (4) (504)
(1) (0.44) (9) tp =
2 in. > 1.27 in. OK
Chapter 5, Structural Steel Design
523
and against FEMA 350 Eq. 335:
0.25 0.15
0.7 0.15 0.3
0.00413f fu
p
bt s p
p g F
t
d t b
=
0.25 0.15
0.7 0.15 0.3
(0.00413)(1.5) (4) (504)
tp = (1) (0.44) (9)
2 in. > 1.66 in. OK
Therefore, use a 2in.thick end plate.
Step 5. Determine the minimum column flange thickness required to resist beam flange tension using
FEMA 350 Eq. 337:
3
0.9 (3.5 )
m fu
cf
yc b
F C
t
F p c
a
>
+
where
3 1
= 4 1 0.75 1.00 in.
2 bt 24
C= g dk   =
(For purposes of this example, k1 is taken to be the thickness of the column web, 0.5 in. and
an assumed 0.25 in. fillet weld for a total of 0.75 in.).
Using FEMA 350 Eq. 338:
1 1
3 3 3
1 0.25
4
(2)(8)(0.5) 1
(1.48) 1.19
(35)(0.44) (1)
( )
f
m a
w
bt
A C
C
A
d
a= = = . . . .
.. .. .. ..
(1.19)(504)(1.00) 0.95 in.
tcf>(0.9)(50)[(3.5)(3) + (3.5)]=
Minimum tcf = 0.95 in. but this will be revised in Step 7.
Step 6. Check column web thickness for adequacy for beam flange compression. This is a check on web
crippling using FEMA 350 Eq. 340:
(1491)(12) 1.44 in.
( )(6 2 ) (36 0.5)[(6)(0.75) (2)(2) (0.5)](50)
f
wc
b fb p fb yc
M
t
d t k t t F
= = =
 + +  + +
twc reqd = 1.44 in. > 0.5 in. = twc OK
Therefore, a continuity plate is needed at the compression flange. See FEMA 350 Sec. 3.3.3.1 for
continuity plate sizing. For onesided connections, the necessary thickness of the continuity
plate is 0.5(tbf + tbf) = 0.5 in.
Step 7. Because continuity plates are required, tcf must be at least as thick as the end plate thickness tp.
Therefore, tcf = 2 in. For this column, the 2in.thick flange does not need to be full height but
must continue well away from the region of beam flange compression and the high moment
FEMA 451, NEHRP Recommended Provisions: Design Examples
524
1
12
Tappered
roof beam
Unstiffened bolted
end plate
Figure 5.19 End plate connection at ridge.
portion of the column knee area. Some judgment is necessary here. For this case, the 2in. flange
is continued 36 in. down from the bottom of the beam, where it is welded to the 0.75in. thick
flange. This weld needs to be carefully detailed.
Step 8. Check the panel zone shear in accordance with FEMA 350, Sec. 3.3.3.2. For purposes of this
check, use db = 35.5 + 1.5 + 3 + 1.5 = 41.5 in. Per FEMA 350 Eq. 37:
(0.9)(0.6 ) ( )
b
y c
y yc c b fb
C M h d
t h
F R d d t
. .
. .
. .

=

where, according to FEMA 350 Eq. 34:
1 1 0.71
1.15 267
218
y
be
pr
b
C C Z
S
. .
. .
. .
= = =
(0.71)(1574 x 12) 366 41.5
366 0.31 in.
tcw (0.9)(0.6)(50)(1.1)(36)(36 0.5)
. .
. .
. .

= =

tcw required = 0.31 in. < 0.50 in. = tcw OK
5.1.5.3 Frame at the Ridge
The ridge joint detail is shown in Figure 5.19. An unstiffened bolted connection plate is selected.
This is an AISC LRFD designed connection, not a FEMA 350 designed connection because there should
not be a plastic hinge forming in this vicinity. Lateral seismic force produces no moment at the ridge
until yielding takes place at one of the knees. Vertical accelerations on the dead load do produce a
Chapter 5, Structural Steel Design
525
MC8x18.7
3" concrete slab
3" embossed 20 ga. deck
W14x43
Split W27x84
(b)
L3x3 strut
W21x62
Figure 5.110 Mezzanine framing (1.0 in. = 25.4 mm).
moment at this point; however, the value is small compared to all other moments and does not appear to
be a concern. Once lateral seismic loads produce yielding at one knee, further lateral displacement
produces some positive moment at the ridge. Under the condition on which the FEMA 350 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. If one uses
1.2D + 0.2S as the load for this scenario, 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.
5.1.5.4 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 concrete filled 3in., 20gauge steel deck of the mezzanine floor is supported on steel beams
spaced at 10 ft and spanning 20 ft (Figure 5.12). The steel beams rest on threespan girders connected at
each end to the portal frames and supported on two intermediate columns (Figure 5.11). The girder
spans are approximately 30 ft each. The design of the mezzanine framing is largely conventional as
seismic loads do not predominate. 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 5.110. The design of the end plate connection is similar to that at
the knee, but simpler because the beam is horizontal and not tapered.
5.1.5.5 Braced Frame Diagonal Bracing
FEMA 451, NEHRP Recommended Provisions: Design Examples
526
Although the force in the diagonal X braces 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 Seismic Sec. 14.2 (November 2000 Supplement) for requirements on braces for OCBFs. The
strength of the members and connections, including the columns in this area but excluding the brace
connections, shall be based on AISC Seismic Eq. 41.
1.2D + 0.5L + 0.2S + O0
QE
Recall that a 1.0 factor is applied to L when the live load is greater than 100 psf (AISC Seismic Sec. 4.1).
For the case discussed here, the “tension only” brace does not carry any live load so the load factor does
not matter. For the braced design, O0 = 2.
However, Provisions Sec. 5.2.7.1, Eq. 5.2.7.11 and 2 [4.23 and 4.24, respectively] requires that the
design seismic force on components sensitive to overstrength shall be defined by:
E = O0
QE ± 0.2SDSD
Given that the Provisions is being following, the AISC Seismic equation will be used but E will be
substituted for QE. Thus, the load combination for design of the brace members reduces to:
1.4D + 0.5L + 0.2S + O0
QE
[The special load combinations have been removed from the 2002 edition of AISC Seismic to eliminate
inconsistencies with other building codes and standards but the design of ordinary braced frames is not
really changed because there is a reference to the load combinations including “simplified seismic loads.”
Therefore, 2003 Provisions Eq. 4.23 and 4.24 should be used in conjunction with the load combinations
in ASCE 7 as is done here.]
From analysis using this load combination, the maximum axial force in the X brace located at the east end
of the building is 66 kips computed from the combined orthogonal earthquake loads (longitudinal
direction predominates). With the O0
factor, the required strength becomes 132 kips. All braces will
have the same design. Using A36 steel for angles:
Tn = fFyAg
132 4.07 in.2
(0.9)(36)
n
g
y
A P
fF
= = =
Try (2) L4 ×3 × 3/8:
Ag = (2)(2.49) = 4.98 in.2 > 4.07 in.2 OK
AISC Seismic Sec. 14.2 requires the design strength of the brace connections to be based on the expected
tensile strength:
RyFyAg = (1.5)(36 ksi)(4.98 in.2) = 269 kips.
Also be sure to check the eave strut at the roof. The eave strut, part of the braced frame, has to carry
compression and that compression is determined using the overstrength factor.
The kl/r requirement of AISC Seismic Sec. 14.2 does not apply because this is not a V or an inverted V
Chapter 5, Structural Steel Design
527
configuration.
5.1.5.6 Roof Deck Diaphragm
Figure 5.111 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 230 kips ( Sec. 5.1.4.2) with 83 percent or 191 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 (191,000/2)/180 ft. = 530 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 3D 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 might be expected given the significant change in
stiffness. There does not appear to be any particularly good simple approximation to estimate the shear
here without a 3D model. The shear is also high at the longitudinal braced bays because they tend to
resist the horizontal torsion. The shear at the braced bays is lower than observed for the EW motion,
however.
FEMA 451, NEHRP Recommended Provisions: Design Examples
528
Roof diaphragm shear, EastWest motion, pound per foot.
Roof diaphragm shear, NorthSouth motion, pound per foot.
Figure 5.111 Shear force in roof deck diaphragm; upper diagram is for EW motion and lower is for NS motion
(1.0 lb. /ft. = 14.59 N/M).
Chapter 5, Structural Steel Design
529
5.2 SEVENSTORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA
Three alternative framing arrangements for a sevenstory office building are illustrated.
5.2.1 Building Description
5.2.1.1 General Description
This sevenstory office building of rectangular plan configuration is 177 ft, 4 in. long in the EW
direction and 127 ft, 4 in. wide in the NS direction (Figure 5.21). The building has a penthouse. It
extends a total of 118 ft, 4 in. above grade. It is framed in structural steel with 25ft bays in each
direction. The story height is 13 ft, 4 in. except for the first story which is 22 ft, 4 in. high. The
penthouse extends 16 ft above the roof level of the building and covers the area bounded by gridlines C,
F, 2, and 5 in Figure 5.21. Floors consist of 31/4 in. lightweight concrete placed on composite metal
deck. The elevators and stairs are located in the central three bays. The building is planned for heavy
filing systems (350 psf) covering approximately four bays on each floor.
5.2.1.2 Alternatives
This example features three alternatives – a steel momentresisting frame, concentrically braced frame,
and a dual system with a momentresisting frame at the perimeter and a concentrically braced frame at the
core area – as follows:
1. Alternative A – Seismic force resistance is provided by special moment frames located on the
perimeter of the building (on lines A, H, 1, and 6 in Figure 5.21, also illustrated in Figure 5.22).
2. Alternative B – Seismic force resistance is provided by four special concentrically braced frames in
each direction. They are located in the elevator core walls between columns 3C and 3D, 3E and 3F,
4C and 4D, and 4E and 4F in the EW direction and between columns 3C and 4C, 3D and 4D, 3E
and 4E, and 3F and 4F in the NS direction (Figure 5.21). The braced frames in an X configuration
are designed for both diagonals being effective in tension and compression. The braced frames are
not identical, but are arranged to accommodate elevator door openings. Braced frame elevations are
shown in Figure 5.23.
3. Alternative C – Seismic force resistance is provided by a dual system with the special moment frames
at the perimeter of the building and a special concentrically braced frames at the core. The moment
frames are shown in Figure 5.22 and the braced frames are shown in Figure 5.23.
5.2.1.3 Scope
The example covers:
1. Seismic design parameters
2. Analysis of perimeter moment frames
3. Beam and column proportioning
4. Analysis of concentrically braced frames
5. Proportioning of braces
6. Analysis and proportioning of the dual system
FEMA 451, NEHRP Recommended Provisions: Design Examples
530
127'4"
25'0" 25'1'2" 0" 25'0" 25'0" 25'0" 1'2"
1'2" 25'0" 25'0" 25'0" 25'0" 25'0" 25'0" 25'0" 1'2"
177'4"
N
PH roof
Roof
7
6
5
4
3
2
22'4" 6 at 13'4" 16'0"
Figure 5.21 Typical floor framing plan and building section (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
Chapter 5, Structural Steel Design
531
5 at 25'0"
N
7 at 25'0"
Figure 5.22 Framing plan for special moment frame (1.0 in. = 25.4 mm, 1.0 ft = 0.3048
m).
Roof
PH roof
PH floor PH floor
25'0" 25'0" 25'0"
16'0"
22'4" 13'4" 13'4" 13'4" 13'4" 13'4" 13'4"
102'4"
102'4" 16'0"
22'4" 13'4" 13'4" 13'4" 13'4" 13'4" 13'4"
PH roof
Figure 5.23 Concentrically braced frame elevations (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
FEMA 451, NEHRP Recommended Provisions: Design Examples
532
5.2.2 Basic Requirements
5.2.2.1 Provisions Parameters
Site Class = D (Provisions Sec. 4.1.2.1 [3.5])
SS = 1.5 (Provisions Map 9 [Figure 3.33])
S1 =0.6 (Provisions Map 10 [Figure 3.34])
Fa = 1.0 (Provisions Table 4.1.2.4a [3.31])
Fv = 1.5 (Provisions Table 4.1.2.4b [3.32])
SMS = FaSS = 1.5 (Provisions Eq. 4.1.2.41 [3.31])
SM1 = FvS1 = 0.9 (Provisions Eq. 4.1.2.42 [3.32])
SDS = 2/3 SMS = 1.0 (Provisions Eq. 4.1.2.51 [3.33])
SD1 = 2/3 SM1 = 0.6 (Provisions Eq. 4.1.2.52 [3.34])
Seismic Use Group = I (Provisions Sec. 1.3 [1.2])
Seismic Design Category = D (Provisions Sec. 4.2.1 [1.4])
Alternative A, Special Steel Moment Frame (Provisions Table 5.2.2 [4.31])
R = 8
O0 = 3
Cd = 5.5
Alternative B, Special Steel Concentrically Braced Frame (Provisions Table 5.2.2 [4.31])
R = 6
O0
= 2
Cd = 5
Alternative C, Dual System of Special Steel Moment Frame Combined with Special Steel Concentrically
Braced Frame (Provisions Table 5.2.2 [4.31])
R = 8
O0
= 2.5
Cd = 6.5
5.2.2.2 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, M&E) = 55 psf
Exterior wall cladding = 25 psf of wall
Penthouse floor D = 65 psf
Floor L = 50 psf
Floor D (deck, beams, girders,
fireproofing, ceiling, M&E, partitions) = 62.5 psf
Floor L reductions per the IBC
5.2.2.3 Materials
Concrete for drilled piers fc' = 5 ksi, normal weight (NW)
Concrete for floors fc' = 3 ksi, lightweight (LW)
Chapter 5, Structural Steel Design
533
All other concrete fc' = 4 ksi, NW
Structural steel
Wide flange sections ASTM A992, Grade 50
HSS ASTM A500, Grade B
Plates ASTM A36
5.2.3 Structural Design Criteria
5.2.3.1 Building Configuration
The building is considered vertically regular despite the relatively tall height of the first story. The
exception of Provisions Sec. 5.2.3.3 [4.3.2.3]is taken in which the drift ratio of adjacent stories are
compared rather than the stiffness of the stories. In the 3D analysis, it will be shown that the first story
drift ratio is less than 130 percent of the story above. Because the building is symmetrical in plan, plan
irregularities would not be expected. Analysis reveals that Alternatives B and C are torsionally irregular,
which is not uncommon for corebraced buildings.
5.2.3.2 Redundancy
According to Provisions Sec. 5.2.4.2 [not applicable in the 2003 Provisions], the reliability factor, (.) for
a Seismic Design Category D structure is:
20
2
maxx r Ax
. = 
In a typical story, the floor area, Ax = 22,579 ft.2
The ratio of the design story shear resisted by the single element carrying the most shear force in the story
to the total story shear is as defined in Provisions Sec. 5.2.4.2. max x r
Preliminary . factors will be determined for use as multipliers on design force effects. These preliminary
. factors will be verified by subsequent analyses.
[The redundancy requirements have been substantially changed in the 2003 Provisions. For a building
assigned to Seismic Design Category D, . = 1.0 as long as it can be shown that failure of beamtocolumn
connections at both ends of a single beam (moment frame system) or failure of an individual brace
(braced frame system) would not result in more than a 33 percent reduction in story strength or create an
extreme torsional irregularity. Alternatively, if the structure is regular in plan and there are at least two
bays of perimeter framing on each side of the structure in each orthogonal direction, it is permitted to use
. = 1.0. Per 2003 Provisions Sec. 4.3.1.4.3, special moment frames in Seismic Design Category D must
be configured such that the structure satisfies the criteria for . = 1.0. There are no reductions in the
redundancy factor for dual systems. Based on the preliminary design, . = 1.0 for Alternative A because it
has a perimeter moment frame and is regular. The determination of . for Alternatives B and C (which are
torsionally irregular) requires the evaluation of connection and brace failures per 2003 Provisions Sec.
4.3.3.2.]
5.2.3.2.1 Alternative A (moment frame)
FEMA 451, NEHRP Recommended Provisions: Design Examples
534
For a momentresisting frame, max x is taken as the maximum of the sum of the shears in any two adjacent r
columns divided by the total story shear. The final calculation of . will be deferred until the building
frame analysis, which will include the effects of accidental torsion, is completed. At that point, we will
know the total shear in each story and the shear being carried by each column at every story. See Sec.
5.2.4.3.1.
Provisions Sec. 5.2.4.2 requires that the configuration be such that . shall not exceed 1.25 for special
moment frames. [1.0 in the 2003 Provisions] (There is no limit for other structures, although . need not
be taken larger than 1.50 in the design.) Therefore, it is a good idea to make a preliminary estimate of .,
which was done here. In this case, . was found to be 1.11 and 1.08 in the NS and EW directions,
respectively. A method for a preliminary estimate is explained in Alternative B.
Note that . is a multiplier that applies only to the force effects (strength of the members and connections),
not to displacements. As will be seen for this momentresisting frame, drift, and not strength, will govern
the design.
5.2.3.2.2 Alternative B (concentrically braced frame)
Again, the following preliminary analysis must be refined by the final calculation. For the braced frame
system, there are four bracedbay braces subject to shear at each story, so the direct shear on each line of
braces is equal to Vx/4. The effects of accidental torsion will be estimated as:
The torsional moment Mta = (0.05)(175)(Vx) = 8.75Vx.
The torsional force applied to either grid line C or F is Vt = MtaKd / SKd2.
Assuming all frame rigidity factors (K) are equal:
2 2
(37.5) 0.01
(2)(37.5) (6)(12.5)
ta
t ta
V M M
.. ..
= =
+
Vt = (0.01)(8.75 Vx) = 0.0875Vx
The amplification of torsional shear (Ax) must be considered in accordance with Provisions Sec. 5.4.4.1.3
[5.2.4.3]. Without dynamic amplification of torsion, the direct shear applied to each line of braces is Vx/4
and the torsional shear, Vt = 0.0875 Vx. Thus, the combined shear at Grid C is 0.25Vx  0.0875Vx =
0.1625Vx, and the combined shear at Grid F is 0.25Vx + 0.0875Vx = 0.3375Vx. As the torsional deflections
will be proportional to the shears and extrapolating to Grids A and H, the deflection at A can be seen to
be proportional to 0.250Vx + (0.0875Vx.)(87.5/37.5) = 0.454Vx. Likewise, the deflection at H is
proportional to 0.250Vx  (0.0875Vx)(87.5/37.5) = 0.046Vx. The average deflection is thus proportional to
[(0.454 + 0.046)/2]Vx = 0.250Vx. These torsional effects are illustrated in Figure 5.24.
Chapter 5, Structural Steel Design
535
0.046
0.250 =
0.163
0.338
0.454
daverage
= dmax
L
0.05 L x
x
Figure 5.24 Approximate effect of accidental of torsion (1.0 in. = 25.4 mm).
From the above estimation of deflections, the torsional amplification can be determined per Provisions
Eq. 5.4.4.1.3.1 [5.213] as:
2 2 0.454 2.29
1.2 (1.2)(0.250)
max
x
avg
A d
d
. . . .
... ... .. ..
= = =
The total shear in the NS direction on Gridlines C or F is the direct shear plus the amplified torsional
shear equal to:
Vx/4 + AxVt = [0.250 + (2.29)(0.0875)]Vx = 0.450Vx
As there are two braces in each braced bay (one in tension and the other in compression):
0.450 0.225
rmaxx= 2 =
and
2 20 2 20 1.41
(0.225) 22,579
maxx r Ax
. =  =  =
FEMA 451, NEHRP Recommended Provisions: Design Examples
536
Therefore, use . = 1.41 for the NS direction. In a like manner, the . factor for the EW direction is
determined to be . = 1.05. These preliminary values will be verified by the final calculations.
5.2.3.2.3 Alternative C (dual system)
For the dual system, the preliminary value for . is taken as 1.0. The reason for this decision is that, with
the dual system, the moment frame will substantially reduce the torsion at any story, so torsional
amplification will be low. The combined redundancy of the braced frame combined with the moment
frame (despite the fact that the moment frame is more flexible) will reduce . from either single system.
Finally, Provisions Sec. 5.2.4.2 [not applicable in the 2003 Provisions] calls for taking only 80 percent of
the calculated . value when a dual system is used. Thus, we expect the final value to fall below 1.0, for
which we will take . = 1.0. This will be verified by analysis later.
5.2.3.3 Orthogonal Load Effects
A combination of 100 percent of the seismic forces in one direction with 30 percent seismic forces in
orthogonal direction is required for structures in Seismic Design Category D (Provisions Sec. 5.2.5.2.3
and 5.2.5.2.2 [4.4.2.2]).
5.2.3.4 Structural Component Load Effects
The effect of seismic load is be defined by Provisions Eq, 5.2.71 [4.21] as:
E = .QE+ 0.2SDSD
Recall that SDS = 1.0. As stated above, . values are preliminary estimates to be checked and, if necessary,
refined later.
For Alternative A
NS direction E = (1.11)QE ± (0.2)D
EW direction E = (1.08)QE ± (0.2)D
Alternative B
NS direction E = (1.41)QE ± (0.2)D
EW direction E = (1.05)QE ± (0.2)D
Alternative. C
NS direction E = (1.00)QE ± (0.2)D
EW direction E = (1.00)QE ± (0.2)D
5.2.3.5 Load Combinations
Load combinations from ASCE 7 are:
1.2D + 1.0E + 0.5L + 0.2S
and
0.9D + 1.0E + 1.6H
Chapter 5, Structural Steel Design
537
To each of these load combinations, substitute E as determined above, showing the maximum additive
and minimum negative. Recall that QE acts both east and west (or north and south):
Alternative A
NS 1.4D + 1.11QE +0.5L and 0.7D + 1.11QE
EW 1.4D + 1.08QE +0.5L and 0.7D + 1.08QE
Alternative B
NS 1.4D + 1.41QE +0.5L and 0.7D + 1.41QE
EW 1.4D + 1.05QE +0.5L and 0.7D + 1.05QE
Alternative C
NS 1.4D + QE +0.5L and 0.7D + QE
EW 1.4D + QE +0.5L and 0.7D + QE
5.2.3.6 Drift Limits
The allowable story drift per Provisions Sec. 5.2.8 [4.51] 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.
Remember to adjust calculated story drifts by the appropriate Cd factor from Sec. 5.2.2.1.
Consider that the maximum story drifts summed to the roof of the sevenstory building, (102 ft4 in. main
roof/penthouse floor) is 24.56 in.
5.2.3.7 Basic Gravity Loads
Penthouse roof
Roof slab = (0.025)(75)(75) = 141 kips
Walls = (0.025)(8)(300) = 60 kips
Columns = (0.110)(8)(16) = 14 kips
Total = 215 kips
Lower roof
Roof slab = (0.055)[(127.33)(177.33)  (75)2] = 932 kips
Penthouse floor = (0.065)(75)(75) = 366 kips
Walls = 60 + (0.025)(609)(6.67) = 162 kips
Columns = 14 + (0.170)(6.67)(48) = 68 kips
Equipment (allowance for mechanical
equipment in penthouse) = 217 kips
Total = 1,745 kips
FEMA 451, NEHRP Recommended Provisions: Design Examples
538
Typical floor
Floor = (0.0625)(127.33)(177.33) = 1,412 kips
Walls = (0.025)(609)(13.33) = 203 kips
Columns = (0.285)(13.33)(48) = 182 kips
Heavy storage = (0.50)(4)(25 x 25)(350) = 438 kips
Total = 2,235 kips
Total weight of building = 215 + 1,745 + 6(2,235) = 15,370 kips
Note that this office building has heavy storage in the central bays of 280 psf over five bays. Use 50
percent of this weight as effective seismic mass. (This was done to add seismic mass to this example
thereby making it more interesting. It is not meant to imply that the authors believe such a step is
necessary for ordinary office buildings.)
5.2.4 Analysis
5.2.4.1 Equivalent Lateral Force Analysis
The equivalent lateral force (ELF) procedure will be used for each alternative building system. The
seismic base shear will be determined for each alternative in the following sections.
5.2.4.1.1 ELF Analysis for Alternative A, Moment Frame
First determine the building period (T) per Provisions Eq. 5.4.2.11 [5.26]:
0.8 (0.028)(102.3) 1.14 sec x
Ta Crhn = = =
where hn, the height to the main roof, is conservatively taken as 102.3 ft. The height of the penthouse (the
penthouse having a smaller contribution to seismic mass than the main roof or the floors) will be
neglected.
The seismic response coefficient (Cs,) is determined from Provisions Eq. 5.4.1.11 [5.22] as:
1 0.125
/ (8/1)
DS
S
S
C R I
= = =
However, Provisions Eq. 5.4.1.12 [5.23] indicates that the value for Cs need not exceed:
1 0.6 0.066
( / ) 1.14(8/1)
D
S
S
C T R I
= = =
and the minimum value for Cs per Provisions Eq. 5.4.1.13 [not applicable in the 2003 Provisions] is:
Cs=0.044ISDS=(0.044)(1)(1)=0.044
Therefore, use Cs = 0.066.
Chapter 5, Structural Steel Design
539
Seismic base shear is computed per Provisions Eq. 5.4.1 [5.21] as:
(V=CSW= 0.066)(15,370)=1014 kips
5.2.4.1.2 ELF Analysis for Alternative B, Braced Frame
As above, first find the building period (T) using Provisions Eq. 5.4.2.11 [5.26]:
x (0.02)(102.3)0.75 0.64 sec
Ta=Crhn= =
The seismic response coefficient (Cs) is determined from Provisions Eq. 5.4.1.11 [5.22] as:
1 0.167
/ (6/1)
DS
S
S
C R I
= = =
However, Provisions Eq. 5.4.1.12 [5.23] indicates that the value for Cs need not exceed:
1 0.6 0.156
( / ) (0.64)(6/1)
D
S
S
C T R I
= = =
and the minimum value for Cs per Provisions Eq. 5.4.1.13 [not applicable in 2003 Provisions] is:
Cs=0.044ISDS=(0.044)(1)(1)=0.044
Use Cs = 0.156.
Seismic base shear is computed using Provisions Eq. 5.4.1 [5.21] as:
V=CSW=(0.156)(15,370)=2,398 kips
5.2.4.1.3 ELF Analysis for Alternative C, Dual System
The building period (T) is the same as for the braced frame (Provisions Eq. 5.4.2.11 [5.26]):
x (0.02)(102.3)0.75 0.64 sec
a r n T =Ch = =
The seismic response coefficient (Cs) is determined as (Provisions Eq. 5.4.1.11 [5.22]):
1 0.125
/ (8/1)
DS
S
S
C R I
= = =
However, the value for Cs need not exceed (Provisions Eq. 5.4.1.12 [5.23]):
1 0.6 0.117
( / ) (0.64)(8/1)
D
S
S
C T R I
= = =
and the minimum value for Cs is (Provisions Eq. 5.4.1.13 [not applicable in the 2003 Provisions]):
Cs=0.044ISDS=(0.044)(1)(1)=0.044
Therefore, use Cs = 0.117.
FEMA 451, NEHRP Recommended Provisions: Design Examples
540
Seismic base shear is computed as (Provisions Eq. 5.4.1 [5.21]):
V = CsW = (0.117)(15,370) = 1,798 kips
5.2.4.2 Vertical Distribution of Seismic Forces
Provisions Sec. 5.4.3 [5.2.3] provides the procedure for determining the portion of the total seismic load
that goes to each floor level. The floor force Fx is calculated using Provisions Eq. 5.4.31 [5.210] as:
Fx = CvxV
where (Provisions Eq. 5.4.32 [5.211])
1
k
x x
vx n
k
i i i
C w h
wh
=
=
S
For Alternative A
T = 1.14 secs, thus k = 1.32
For Alternatives B and C
T = 0.64 sec, thus k = 1.07
Using Provisions Eq. 5.4.4 [5.212], the seismic design shear in any story is computed as:
n
x i x i
V = F = S
The story overturning moment is computed from Provisions Eq. 5.4.5 [5.214]:
( )
n
x i xi i x
M F h h
=
= S 
The application of these equations for the three alternative building frames is shown in Tables 5.21, 5.2
2, and 5.13.
Chapter 5, Structural Steel Design
541
Table 5.21 Alternative A, Moment Frame Seismic Forces and Moments by Level
Level (x)
Wx
(kips )
hx
(ft)
Wxhx
k
(ftkips)
Cvx Fx
(kips)
Vx
(kips)
Mx
(ftkips)
PH Roof 215 118.33 117,200 0.03 32 32 514
Main roof 1,745 102.33 785,200 0.21 215 247 3,810
Story 7 2,235 89.00 836,500 0.23 229 476 10,160
Story 6 2,235 75.67 675,200 0.18 185 661 18,980
Story 5 2,235 62.33 522,700 0.14 143 805 29,710
Story 4 2,235 49.00 380,500 0.10 104 909 41,830
Story 3 2,235 35.67 250,200 0.07 69 977 54,870
Story 2
S
2,235
15,370
22.33 134,800
3,702,500
0.04
1.00
37
1,014
1,014 77,520
1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m.
Table 5.22 Alternative B, Braced Frame Seismic Forces and Moments by Level
Level (x)
Wx
(kips )
hx
(ft)
Wxhx
k
(ftkips)
Cvx Fx
(kips)
Vx
(kips)
Mx
(ftkips)
PH Roof 215 118.33 35,500 0.03 67 67 1,070
Main roof 1,745 102.33 246,900 0.19 463 530 8,130
Story 7 2,235 89.00 272,300 0.21 511 1,041 22,010
Story 6 2,235 75.67 228,900 0.18 430 1,470 41,620
Story 5 2,235 62.33 186,000 0.15 349 1,819 65,870
Story 4 2,235 49.00 143,800 0.11 270 2,089 93,720
Story 3 2,235 35.67 102,400 0.08 192 2,281 124,160
Story 2
S
2,235
15,370
22.33 62,000
1,278,000
0.05
1.00
116
2,398
2,398 177,720
1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m.
FEMA 451, NEHRP Recommended Provisions: Design Examples
542
Table 5.23 Alternative C, Dual System Seismic Forces and Moments by Level
Level (x)
Wx
(kips )
hx
(ft)
Wxhx
k
(ftkips)
Cvx Fx
(kips)
Vx
(kips)
Mx
(ftkips)
PH Roof 215 118.33 35,500 0.03 50 50 800
Main roof 1,745 102.33 246,900 0.19 347 397 6,100
Story 7 2,235 89.00 272,350 0.21 383 781 16,500
Story 6 2,235 75.67 228,900 0.18 322 1,103 31,220
Story 5 2,235 62.33 186,000 0.15 262 1,365 49,400
Story 4 2,235 49.00 143,800 0.11 202 1,567 70,290
Story 3 2,235 35.67 102,386 0.08 144 1,711 93,120
Story 2
S
2,235
15,370
22.33 62,000
1,278,000
0.05
1.00
87
1,798
1,798 133,270
1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m.
Be sure to note that the seismic mass at any given level, which includes the lower half of the wall above
that level and the upper half of the wall below that level, produces the shear applied at that level and that
shear produces the moment which is applied at the top of the next level down. Resisting the overturning
moment is the weight of the building above that level combined with the moment resistance of the
framing at that level. Note that the story overturning moment is applied to the level below the level that
receives the story shear. (This is illustrated in Figure 9.24 in the masonry examples.)
5.2.4.3 Size Members
At this point we are ready to select the sizes of the framing members. The method for each alternative is
summarized below.
Alternative A, Special Moment Frame:
1. Select preliminary member sizes
2. Check deflection and drift (Provisions Sec. 5.2.8 [5.4.1])
3. Check torsional amplification (Provisions Sec. 5.4.4.1.3 [5.2.4.3])
4. Check the columnbeam moment ratio rule (AISC Seismic Sec. 9.6)
5. Check shear requirement at panelzone (AISC Seismic Sec. 9.3; FEMA 350
Sec. 3.3.3.2)
6. Check redundancy (Provisions Sec. 5.2.4.2 [5.3.3])
7. Check strength
Reproportion member sizes as necessary after each check. The most significant criteria for the design
are drift limits, relative strengths of columns and beams, and the panelzone shear.
Chapter 5, Structural Steel Design
543
Alternative B, Special Concentrically Braced Frame:
1. Select preliminary member sizes
2. Check strength
3. Check drift (Provisions Sec. 5.2.8 [4.5.1])
4. Check torsional amplification (Provisions Sec. 5.4.4.1 [5.2.4.3])
5. Check redundancy (Provisions Sec. 5.2.4.2 [4.3.3])
Reproportion member sizes as necessary after each check. The most significant criteria for this
design is torsional amplification.
Alternative C, Dual System:
1. Select preliminary member sizes
2. Check strength of moment frame for 25 percent of story shear (Provisions Sec. 5.2.2.1 [4.3.1.1])
3. Check strength of braced frames
4. Check drift for total building (Provisions Sec. 5.2.8 [4.5.1])
5. Check torsional amplification (Provisions Sec. 5.4.4.1 [5.2.4.3])
6. Check redundancy (Provisions Sec. 5.2.4.2 [4.3.3])
Reproportion member sizes as necessary after each check.
5.2.4.3.1 Size Members for Alternative A, Moment Frame
1. Select Preliminary Member Sizes – The preliminary member sizes are shown for the moment frame in
the Xdirection (7 bays) in Figure 5.25 and in the Y direction (5 bays) in Figure 5.26.
Check Local Stability – Check beam flange stability in accordance with AISC Seismic Table I91 [I
81] (same as FEMA 350 Sec. 3.3.1.1) and beam web stability in accordance with AISC Seismic
Table I91 [I81]. (FEMA 350 Sec.3.3.1.2.is more restrictive for cases with low Pu / fb
Py, such as in
this example.) Beam flange slenderness ratios are limited to 52/ and beam web heightto y F
thickness ratios are limited to 418/ . y F
[The terminology for local stability has been revised in the 2002 edition of AISC Seismic. The
limiting slenderness ratios in AISC Seismic use the notation .ps (“seismically compact”) to
differentiate them from .p in AISC LRFD. In addition, the formulas appear different because the
elastic modulus, Es, has been added as a variable. Both of these changes are essentially editorial, but
Table I81 in the 2002 edition of AISC Seismic has also been expanded to include more elements
than in the 1997 edition.]
Be careful because certain shapes found in the AISC Manual will not be permitted for Grade 50 steel
(but may have been permitted for Grade 36 steel) because of these restrictions. For Grade 50, b/t is
limited to 7.35.
Further note that for columns in special steel moment frames such as this example, AISC Seismic
9.4b [I81] requires that when the column moment strength to beam moment strength ratio is less
than or equal to 2.0, the more stringent .p
requirements apply for b/t, and when Pu/ fb
Py is less than or
equal to 0.125, the more stringent h/t requirements apply.
FEMA 451, NEHRP Recommended Provisions: Design Examples
544
W21x44
W14x145
W 14x145
W14x398 W14x283 W14x233
W14x370 W 14x257 W 14x233
W14x370
W14x370
W14x370
W14x370
W14x370
W14x398
W24x62
W27x94
W27x102
W30x108
W30x108 W30x108
W30x108
W30x108
W30x108
W30x108
W30x108
W30x108
W30x108
W30x108
W30x108
W30x108
W30x108
W33x141
W 14x257 W 14x145
W 14x257 W 14x145
W 14x257 W 14x145
W 14x257 W 14x145
W 14x257 W 14x145
W 14x283 W 14x233 W 14x145
W33x141
W27x102
W27x94
W24x62
W21x44
W33x141
W27x102
W27x94
W24x62
W21x44
W33x141
W27x102
W27x94
W24x62
W21x44
W33x141
W27x102
W27x94
W24x62
W21x44
W33x141
W27x102
W27x94
W24x62
W21x44
W33x141
W27x102
W27x94
W24x62
W21x44
W 14x233
W 14x233
W 14x233
W 14x233
W 14x233
Figure 5.25 SMRF frame in EW direction (penthouse not shown).
2. Check Drift – Check drift in accordance with Provisions Sec. 5.2.8 [4.5.1]. The building was
modeled in 3D using RAMFRAME. Displacements at the building centroid are used here because
the building is not torsionally irregular (see the next paragraph regarding torsional amplification).
Calculated story drifts and Cd amplification factors are summarized in Table 5.24. Pdelta effects are
included.
All story drifts are within the allowable story drift limit of 0.020hsx per Provisions Sec. 5.2.8 [4.5.1]
and Sec. 5.2.3.6 of this chapter.
As indicated below, the first story drift ratio is less than 130 percent of the story above (Provisions
Sec. 5.2.3.3 [4.3.2.3]):
story 2
story 3
5.17 in.
268 in. 0.98 1.30
3.14 in.
160 in.
d x
d x
C
C
.
= = <
.
. .
. .
. .
. .
. .
. .
Therefore, there is no vertical irregularity.
Chapter 5, Structural Steel Design
545
W33x141
W30x116
W30x108
W30x108
W27x94
W21x44
W14x398 W14x283 W14x233
W14x145
W14x145
W14x145
W14x145
W14x145
W14x145
W24x76
W14x233
W14x233
W14x233
W14x233
W14x233
W14x398 W14x283
W14x398 W14x283
W14x398 W14x283
W14x398 W14x283
W14x398 W14x283
W30x116
W30x108
W30x108
W27x94
W24x76
W21x44
W30x116
W30x108
W30x108
W27x94
W24x76
W21x44
W30x116
W30x108
W30x108
W27x94
W24x76
W21x44
W30x116
W30x108
W30x108
W27x94
W24x76
W21x44
W33x141 W33x141 W33x141 W33x141
Figure 5.26 SMRF frame in NS direction (penthouse not shown).
Table 5.24 Alternative A (Moment Frame) Story Drifts under Seismic Loads
Total Displacement
at Building Centroid
(86.5, 62.5)
Story Drift from
3D Elastic Analysis
at Building Centroid
Cd (Cd ) x
(Elastic Story Drift)
Allowable
Story Drift
dEW
(in.)
dNS
(in.)
.EW
(in.)
.NS
(in.)
5.5 .EW
(in.)
.NS
(in.)
. (in.)
Roof 4.24 4.24 0.48 0.47 5.5 2.64 2.59 3.20
Floor 7 3.76 3.77 0.57 0.58 5.5 3.14 3.19 3.20
Floor 6 3.19 3.19 0.54 0.53 5.5 2.97 2.92 3.20
Floor 5 2.65 2.66 0.57 0.58 5.5 3.14 3.19 3.20
Floor 4 2.08 2.08 0.57 0.58 5.5 3.14 3.19 3.20
Floor 3 1.51 1.50 0.57 0.57 5.5 3.14 3.14 3.20
Floor 2 0.94 0.93 0.94 0.93 5.5 5.17 5.12 5.36
1.0 in. = 25.4 mm.
FEMA 451, NEHRP Recommended Provisions: Design Examples
546
3. Check Torsional Amplification – The torsional amplification factor per Provisions Eq. 5.4.4.1.31
[5.213] is:
2
1.2
max
x
avg
A
d
d
. .
=.. ..
. .
If Ax < 1.0, then torsional amplification need not be considered. It is readily seen that if the ratio of
dmax/davg is less that 1.2, then torsional amplification will not be necessary.
The 3D analysis provided the story deflections listed in Table 5.25. Because none of the ratios for
dmax/davg exceed 1.2, torsional amplification of forces is not necessary for the moment frame
alternative.
Table 5.25 Alternative A Torsional Analysis
Torsion Checks
(in.) EWmax d
(175,0)
(in.) NSmax d
(125,0)
/ EWmax EWavg d d / NSmax NSavg d d
Roof 4.39 4.54 1.04 1.07
Story 7 3.89 4.04 1.04 1.07
Story 6 3.30 3.42 1.04 1.07
Story 5 2.75 2.85 1.03 1.07
Story 4 2.16 2.23 1.04 1.07
Story 3 1.57 1.62 1.04 1.08
Story 2 0.98 1.00 1.04 1.08
1.0 in. = 25.4 mm.
Member Design Considerations – Because Pu/fPn is typically less than 0.4 for the columns (re: AISC
Seismic Sec. 8.2 [8.3]), combinations involving O0
factors do not come into play for the special steel
moment frames. In sizing columns (and beams) for strength we will satisfy the most severe value
from interaction equations. However, this frame is controlled by drift. So, with both strength and
drift requirements satisfied, we will check the columnbeam moment ratio and the panel zone shear.
4. Check the ColumnBeam Moment Ratio – Check the columnbeam moment ratio per AISC Seismic
Sec. 9.6. For purposes of this check, the plastic hinge was taken to occur at 0.5db from the face or the
column in accordance with FEMA 350 for WUFW connections (see below for description of these
connections). The expected moment strength of the beams were projected from the plastic hinge
location to the column centerline per the requirements of AISC Seismic Sec. 9.6. This is illustrated in
Figure 5.27. For the columns, the moments at the location of the beam flanges is projected to the
columnbeam intersection as shown in Figure 5.28.
Chapter 5, Structural Steel Design
547
Assumed plastic
hinge location
MP MV
M*Pb
S
25'0"
Center line
of column
Center line
of column
Center line
of beam
Sh
h
Figure 5.27 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 G (W14×370 column and W33×141 beam). For the columns:
* uc
pc c yc
g
P M Z FA
. .
S =S ..  ..
. .
* 3
2
2 736 in. 50 ksi 500 kips 66,850 ftkips
pc 109 in. M S =... ...  ......=
Adjust this by the ratio of average story height to average clear height between beams, or (268 +
160)/ (251.35 + 128.44) = 1.13. Therefore, SM*pc = (1.13)(66,850) = 75,300 ftkips. For the beams,
* (1.1 ) pb y p v SM =S R M +M
where
Ry = 1.1 for Grade 50 steel
Mp = Fy Z = (50) (514) = 25,700 in.kips
Mv = VpSh
Sh = Distance from column centerline to plastic hinge = dc/2 + db/2 = 25.61 in.
Vp = Shear at plastic hinge location
FEMA 451, NEHRP Recommended Provisions: Design Examples
548
c
Center line
of column
Center line
of beam
h
clear height
h s
Center line
of beam
Figure 5.28 Story height and clear height.
Mp
V
Mp p
Vp
W
L' Plastic
hinges
Figure 5.29 Free body diagram bounded by
plastic hinges.
The shear at the plastic hinge (Figure 5.29) is computed as:
Vp=[2Mp+(wL' 2 /2] / L'
where
L' = Distance between plastic hinges = 248.8 in.
w = Factored uniform gravity load along beam
w = 1.4D + 0.5L = 1.4(0.0625 ksf)(12.5 ft)
+ 0.5(0.050 ksf)(12.5 ft) = 1.406 klf
Therefore,
2 2 2 (2)(25,700) (1.14206) (2482.8)2 221.2 kips
248.8
p
p
M wL
V
L
+ ' +.. ..
= = . .=
'
and
Mv = VpSh = (221.2)(25.61) = 5,665 in.kips
Finally, SM*
pb = S(1.1RyMp + Mv) = 2[(1.1)(1.1)(25,700) + 5,665] = 73,500 in.kips.
The ratio of column moment strengths to beam moment strengths is computed as:
Chapter 5, Structural Steel Design
549
OK
*
*
Ratio 76,900 1.05 1.0
73,500
pc
pb
M
M
S
= = = >
S
The columnbeam strength ratio for all the other stories is determined in a similar manner. They are
summarized in Table 5.24 for the EW direction (sevenbay) frame and in Table 5.25 for the NS
direction (fivebay) frame. All cases are acceptable because the columnbeam moment ratios are all
greater than 1.00.
Table 5.24 ColumnBeam Moment Ratios for SevenBay Frame (NS Direction)
Story Member SM*pc
(in.kips)
SM*pb
(in.kips)
Column
Beam Ratio
7 column W14×145
beam W24×62
29,000 21,300 1.36
5 column W14×233
beam W27×102
40,000 42,600 1.15
3 column W14×257
beam W30×108
53,900 48,800 1.11
2 column W14×370
beam W33×141
75,300 73,500 1.02
For levels with the same size column, the one with the larger beam size will govern; only these
are shown. 1.0 in.kip = 0.113 kNm.
Table 5.25 ColumnBeam Moment Ratios for FiveBay Frame (NS Direction)
Story Member SM*pc
(in.kips)
SM*pb
(in.kips)
Column
Beam Ratio
7 column W14×145
beam W24×76
29,400 27,700 1.06
5 column W14×233
beam W30×108
50,700 48,700 1.04
3 column W14×283
beam W30×116
63,100 53,900 1.17
2 column W14×398
beam W33×141
85,900 74,100 1.16
For levels with the same size column, the one with the larger beam size will govern; only these
are shown. 1.0 in.kip = 0.113 kNm.
5. Check Panel Zone – The Provisions defers to AISC Seismic for the panel zone shear calculation.
Because the two methods for calculating panel zone shear (AISC Seismic and FEMA 350) differ,
both are illustrated below.
FEMA 451, NEHRP Recommended Provisions: Design Examples
550
x L' x
dc
2 2
dc
l
Mpe
Mf
Mf
Mpe
c
M = Expected moment at plastic hinge
projected to face of column (AISC Seismic method)
f
dc dc
M = Expected moment at plastic hinge
projected to column centerline (FEMA 350 method)
L'
L
c
Column
center line
Mc Mpr
2 x
Column
center line
x 2
Mpr
Mc
Column
center line
Column
center line
Figure 5.210 Illustration of AISC Seismic vs. FEMA 350 methods for panel zone shear.
AISC Seismic Method
Check the shear requirement at the panel zone in accordance with AISC Seismic Sec. 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
f
u
b fb
M
R
d t
= S

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 5.210).
Chapter 5, Structural Steel Design
551
For a column with equal beams of equal spans framing into opposite faces (such as on Grids C, D, E,
F, 2, 3, 4, and 5), the effect of gravity loads offset, and
2
2
c
f y y x
c
M RFZ l
l x
. .
. .
. .
S =

where lc = the clear span and x = distance from column face to plastic hinge location.
For Grids 1 and 6, only one beam frames into the column; at Grids B and G, the distance x is different
on one side; at Grids A and H, there is no moment because the beams are pinconnected to the corner
columns. For all these cases, the above relationship needs to be modified accordingly.
For W33×141 beams framing into each side of a W14×370 column (such as Level 2 at Grid F):
(2)(1.1)(50)(514) 282.1 64,056 in.kips
Mf 282.1 (2)(16.55) . .
. .
. .
S = =

64,056 1,981 kips
33.30 0.96 u R= =

The shear transmitted to the joint from the story above opposes the direction of Ru and may be used to
reduce the demand. From analysis, this is 98 kips at this location. Thus the frame Ru = 1,981  98 =
1,883 kips.
The panel zone shear calculation for Story 2 of the frame in the EW direction at Grid F (column:
W14×370; beam: W33×141) is from AISC Seismic Eq. 91:
3 2
0.6 1 cf cf
v ycp
b c p
b t
R Fdt
d d t
. .
. .
.. ..
= +
(0.6)(50)(17.92)( ) 1 (3)(16.475)(2.660)2
v p (33.30)(17.92)( )
p
R t
t
. .
. .
.. ..
= +
v 537.6p1 0.586
p
R t
t
. .
. .
.. ..
= +
Rv=537.6tp+315
The required total (web plus doubler plate) thickness is determined by:
Rv = fRu
Therefore, 537.6tp + 315 = (1.0)(1883) and tp = 2.91 in.
Because the column web thickness is 1.655 in., the required doubler plate thickness is 1.26 in. Use a
plate thickness of 11/4 in.
FEMA 451, NEHRP Recommended Provisions: Design Examples
552
Both the column web thickness and the doubler plate thickness are checked for shear buckling during
inelastic deformations by AISC Seismic Eq. 92. If necessary, the doubler plate may be plugwelded
to the column web as indicated by AISC Seismic Commentary Figure C9.2. For this case, the
minimum individual thickness as limited by local buckling is:
t=(dz+wz)/90
(31.38 12.6) 0.49 in.
90
t= + =
Because both the column web thickness and the doubler plate thicknesses are greater than 0.49 in.,
plug welding of the doubler plate to the column web is not necessary.
In the case of thick doubler plates, to avoid thick welds, two doubler plates (each of half the required
thickness) may be used, one on each side of the column web. For such cases, buckling also must be
checked using AISC Seismic Eq. 92 as doubler plate buckling would be a greater concern. Also, the
detailing of connections that may be attached to the (thinner) doubler plate on the side of the weld
needs to be carefully reviewed for secondary effects such as undesirable outofplane bending or
prying.
FEMA 350 Method
For the FEMA 350 method, see FEMA 350 Sec. 3.3.3.2, “Panel Zone Strength,” to determine the
required total panel zone thickness (t):
(0.9)(0.6) ( )
b
y c
yc yc c b fb
h d
C M
t h
F R d d t

=

. .
S .. ..
(Please note the S; its omission from FEMA 350 Eq. 37 is an inadvertent typographical error.)
The term Mc refers to the expected beam moment projected to the centerline of the column; whereas
AISC Seismic uses the expected beam moment projected to the face of the column flange. (This
difference is illustrated in Figure 5.210.) The term is an adjustment similar to reducing Ru
b h d
h
.  .
.. ..
by the direct shear in the column, where h is the average story height. Cy is a factor that adjusts the
force on the panel down to the level at which the beam begins to yield in flexure (see FEMA 350 Sec.
3.2.7) and is computed from FEMA 350 Eq. 34:
1
y
be
pr
b
C C Z
S
=
Cpr, a factor accounting for the peak connection strength, includes the effects of strain hardening and
local restraint, among others (see FEMA 350 Sec. 3.2.4) and is computed from FEMA 350 Eq. 32:
( )
2
y u
pr
y
F F
C
F
+
=
For the case of a W33×141 beam and W14×370 column (same as used for the above AISC Seismic
method), values for the variables are:
Chapter 5, Structural Steel Design
553
Distance from column centerline to plastic hinge, Sh = dc/2 + db/2 = 17.92/2 + 33.30/2 = 25.61 in.
Span between plastic hinges, L' = 25 ft  2(25.61 in.)/12 = 20.73 ft
Mpr = CprRyZeFy (FEMA 350 Figure 34)
Mpr = (1.2)(1.1)(514)(50) = 33,924 in.kips (FEMA 350, Figure 34)
2
2
2
'
pr
p
M wL
V
L
. . ' ..
. +. ..
=. . ..
(2)(33,924) (1.266)(20.73)2
(12)(2)
273 kips
Vp (20.73)(12)
. . ..
. . ..
.. . ...
+
= =
Mc = Mpr + Vp(x + dc/2) (FEMA 350 Figure 34)
Mc = 33,924 + (273)(17.92/2 + 25.61/2) = 40,916 in.kips
1 (1.2)1514 0.73
448
y
be
pr
b
C C Z
S
= = =
Therefore,
= 2.93 in.
(0.73)(40,916) (214) (33.30)
(214)
2
(0.9)(0.6)(50)(1.1)(17.92)(33.30 0.96)
t
. . ..
. . ..
. . ..
. .
. .
.. ..

=

The required doubler plate thickness is equal to t  tcw = 2.93 in.  1.655 in. = 1.27 in. Thus, the
doubler plate thickness for 1.27 in. by FEMA 350 is close to the thickness of 1.26 by AISC Seismic.
6. Check Redundancy – Return to the calculation of rx for the moment frame. In accordance with
Provisions Sec. 5.2.4.2 [not applicable in the 2003 Provisions], is taken as the maximum of the max x r
sum of the shears in any two adjacent columns in the plane of a moment frame divided by the story
shear. For columns common to two bays with moment resisting connections on opposite sides of the
column at the level under consideration, 70 percent of the shear in that column may be used in the
column shear summation (Figures 5.211 and 5.212).
FEMA 451, NEHRP Recommended Provisions: Design Examples
554
76.2 kips
x
14.1 kips 70.7 kips 76.7 kips
r = (0.7)(76.7) + (0.7)(76.2)
1,014 = 0.105
r = (1.0)(14.1) + (0.7)(70.7)
x 1,014 = 0.063
76.2 kips 76.7 kips 70.7 kips 14.1 kips
Figure 5.211 Column shears for EW direction (partial elevation, Level 2) (1.0 kip = 4.45 kN).
56.1 kips 113.3 kips 110.1 kips 110.1 kips 113.3 kips 56.1 kips
r = (1.0)(56.1) + (0.7)(113.3)
977 = 0.139
r = (0.7)(113.3) + (0.7)(110.1)
977 = 0.160
x
x
Figure 5.212 Column shears for NS direction (partial elevation, Level 3) (1.0 kip = 4.45 kN).
For this example, rx was computed for every column pair at every level in both directions. The shear
carried by each column comes from the RAMFRAME analysis, which includes the effect of
accidental torsion. Selected results are illustrated in the figures. The maximum value of in the max x r
NS direction is 0.160, and . is now determined using Provisions Eq.5.2.4.2 [not applicable in the
2003 Provisions]:
2 20
x x rmax A
. = 
2
2 20 1.15
0.160 21,875 ft
. =  =
Because 1.15 is less than the limit of 1.25 for special moment frames per the exception in the
Provisions Sec. 5.2.4.2 [not applicable in the 2003 Provisions], use . =1.15. (If . > 1.25, then the
framing would have to be reconfigured until . < 1.25.)
Chapter 5, Structural Steel Design
555
Sh Sh
Vp Vp
Mpr
Mpr
Figure 5.213 Forces at beam/column connection.
In the EW direction, = 0.105 and . = 0.71, which is less than 1.00, so use . = 1.00. All design max x r
force effects (axial force, shear, moment) obtained from analysis must be increased by the . factors.
(However, drift controls the design in this example. Drift and deflections are not subject to the .
factor.)
7. Connection Design – One beamtocolumn connection for the momentresisting frame is now
designed to illustrate the FEMA 350 method for a prequalified connection. The welded unreinforced
flangeswelded web (WUFW) connection is selected because it is prequalified for special moment
frames with members of the size used in this example. FEMA 350 Sec. 3.5.2 notes that the WUFW
connection can perform reliably provided all the limitations are met and the quality assurance
requirements are satisfied. While the discussion of the design procedure below considers design
requirements, remember that the quality assurance requirements are a vital part of the total
requirements and must be enforced.
Figure 5.213 illustrates the forces at the beamtocolumn connection.
First review FEMA 350 Table 33 for prequalification data. Our case of a W36×135 beam connected
to a W14×398 column meets all of these. (Of course, here the panel zone strength requirement is
from FEMA 350, not the AISC Seismic method.)
The connection, shown in Figure 5.214, is based on the general design shown in FEMA 350
Figure 38. The design procedure outlined in FEMA 350 Sec. 3.5.2.1 for this application is reviewed
below. All other beamtocolumn connections in the moment frame will be similar.
The procedure outlined above for the FEMA 350 method for panel zone shear is repeated here to
determine Sh, Mpr, Vp, Mc , Cy and the required panel zone thickness.
Continuity plates are required in accordance with FEMA 350 Sec. 3.3.3.1:
0.4 1.8 yb yb
cf f f
yc yc
F R
t bt
F R
<
FEMA 451, NEHRP Recommended Provisions: Design Examples
556
1
2" R
See Figure 5.215
Figure 5.214 WUFW connection, Second level, NSdirection (1.0 in. = 25.4 mm).
= 1.65 in. required
0.4 (1.8)(11.950)(0.790) (50)(1.1)
cf (50)(1.1) t <
tcf = 2.845 in. > actual OK
Therefore, continuity plates are not necessary at this connection because the column flange is so
thick. But we will provide them anyway to illustrate continuity plates in the example. At a
minimum, continuity plates should be at least as thick as the beam flanges. Provide continuity plates
of 7/8 in. thickness, which is thicker than the beam flange of 0.79 in.
Chapter 5, Structural Steel Design
557
PP 4
5
23
8"
1" 2"
3 4"
3
3
8" R
Erection
bolt
1
4"  3
8" 6
Shear tab:
5
8" x 3"
2 5
16
Backing
bar 1
1"
1
Figure 5.215 WUFW weld detail (1.0 in. = 25.4 mm).
Check AISC LRFD K1.9:
Width of stiffener +
2 3
cw bf t = b
OK 5 1.77 5.88 in. 3.98 in. 11.950
2 3
.. + ..= > =
. .
2
f
stiffener
b
t =
OK
0.875 in. > 0.395 in. = 0.79
2
95
y
stiffener stiffener
F
t >w
FEMA 451, NEHRP Recommended Provisions: Design Examples
558
OK
0.875 in. > 0.37 in. = (5) 50
95
. .
.. ..
. .
The details shown in Figures 5.214 and 5.215 conform to the requirements of FEMA 350 for a
WUFW connection in a special moment frame.
Notes for Figure 5.215 (indicated by circles in the figure) are:
1. CJP groove weld at top and bottom flanges, made with backing bar.
2. Remove backing bar, backgouge, and add fillet weld.
3. Fillet weld shear tab to beam web. Weld size shall be equal to thickness of shear tab minus 1/16
in. Weld shall extend over the top and bottom third of the shear tab height and extend across the
top and bottom of the shear tab.
4. Full depth partial penetration weld from far side. Then fillet weld from near side. These are shop
welds of shear tab to column.
5. CJP groove weld full length between weld access holes. Provide nonfusible weld tabs, which
shall be removed after welding. Grind end of weld smooth at weld access holes.
6. Root opening between beam web and column prior to starting weld 5.
See also FEMA 350 Figure 38 for more elaboration on the welds.
5.2.4.3.2 Size Members for Alternative B, Braced Frame
1. Select Preliminary Member Sizes – The preliminary member sizes are shown for the braced frame in
the EW direction (seven bays) in Figure 5.216 and in the NS direction (five bays) in Figure 5.217.
The arrangement is dictated by architectural considerations regarding doorways into the stairwells.
2. Check Strength – First, check slenderness and widthtothickness ratios – the geometrical
requirements for local stability. In accordance with AISC Seismic Sec. 13.2, bracing members must
satisfy
1000 1000 141
50 y
kl
r F
= = =
The columns are all relatively heavy shapes, so kl/r is assumed to be acceptable and is not examined
in this example.
Wide flange members and channels must comply with the widthtothickness ratios contained in
AISC Seismic Table I91 [I81]. Flanges must satisfy:
52 52 7.35
2 50 y
b
t F
= = =
Webs in combined flexural and axial compression (where Pu/ fb
Py < 0.125, which is the case in this
example) must satisfy:
Chapter 5, Structural Steel Design
559
W14x665
W14x61
HSS12x12x5
8
W14x53
W14x48
W14x43
W14x38
W14x34
W14x34
W14x34
HSS10x10x5
8
HSS10x10x5
8
W14x61
W14x53
W14x48
W14x43
W14x38
W14x34
W14x34
W14x34
HSS10x10x5
8
HSS10x10x5
8
W14x665
W14x455
W14x455
W14x211
W14x211
W14x109
W14x109
W14x38
W14x38
W14x665
W14x665
W14x455
W14x455
HSS10x10x5
8
HSS10x10x5
8
HSS10x10x5
8
HSS10x10x5
8
W14x211
W14x211
W14x109
W14x109
HSS8x8x1
2
HSS8x8x1
2
W14x38
W14x38
HSS4x4x1
4
HSS4x4x1
4
HSS12x12x5
8
PH
R
7
6
5
4
3
2
HSS12x12x5
8
HSS12x12x5
8
HSS12x12x5
8
HSS12x12x5
8 HSS12x12x5
8
HSS12x12x5
8
HSS12x12x5
8
HSS12x12x5
8
HSS12x12x5
8
HSS12x12x5
8
HSS8x8x1
2
HSS8x8x1
2
HSS8x8x1
2
HSS8x8x1
2
HSS8x8x1
2
HSS8x8x1
2
HSS4x4x1
4
HSS4x4x1
4
Figure 5.216 Braced frame in EW direction.
HSS12x12x5
8
W14x665
W14x38
W14x38
W14x38
W14x38
W14x38
W14x38
W14x38
W14x34
W14x665
HSS12x12x5
8
W14x455
W14x455
W14x211
W14x211
W14x109
W14x109
W14x38
W14x38
HSS4x4x1
4
HSS4x4x1
4
HSS10x10x5
8
HSS10x10x5
8
HSS12x12x5
8
HSS12x12x5
8
HSS10x10x5
8
HSS10x10x5
8
HSS12x12x5
8
HSS12x12x5
8
PH
R
7
6
5
4
3
2
HSS12x12x5
8
HSS12x12x5
8
HSS12x12x5
8
HSS12x12x5
8
Figure 5.217 Braced frame
in NS direction.
c520 1 1.54u
w y by
h P
t F fP
. .
= . .
.. ..
Rectangular HSS members must satisfy:
110 110 16.2
46 y
b
t F
= = =
FEMA 451, NEHRP Recommended Provisions: Design Examples
560
Selected members are checked below:
W14×38: b/2t = 6.6 < 7.35 OK
W14×34: b/2t = 7.4 > 7.35, but is acceptable for this example. Note that the W14×34 is at the
penthouse roof, which is barely significant for this braced frame.
HSS12×12×5/8: OK
(1) 28.33 12
2 36.8 < 141
4.62
kl
r
. × .
. .
= . .=
OK
9.4 16.17 16.2
0.581
b
t
= = <
Also note that t for the HSS is actual, not nominal. The corner radius of HSS varies somewhat, which
affects the dimension b. The value of b used here, 9.40 in., depends on a corner radius slightly larger
than 2t, and it would have to be specified for this tube to meet the b/t limit.
3. Check Drift – Check drift in accordance with Provisions Sec. 5.2.8 [4.5]. The building was modeled
in 3D using RAMFRAME. Maximum displacements at the building corners are used here because
the building is torsionally irregular. Displacements at the building centroid are also calculated
because these will be the average between the maximum at one corner and the minimum at the
diagonally opposite corner. Use of the displacements at the centroid as the average displacements is
valid for a symmetrical building. Calculated story displacements are used to determine Ax, the
torsional amplification factor. This is summarized in Table 5.26. Pdelta effects are included.
Chapter 5, Structural Steel Design
561
Table 5.26 Alternative B Amplification of Accidental Torsion
Average Elastic
Displacement =
Displacement at
Building Centroid
(in.)
Maximum Elastic
Displacement at
Building Corner*
(in.)
** Torsional max
avg
d
d Amplification
Factor =
2
1.2
max
x
avg
A
d
d
=
. .
. .
. .
Amplified
Eccentricity =
Ax(0.05 L)***
(ft)
EW NS EW NS EW NS EW NS EW NS
R 2.38 2.08 3.03 3.37 1.28 1.62 1.13 1.82 7.08 15.95
7 2.04 1.79 2.62 2.93 1.29 1.64 1.15 1.88 7.20 16.41
6 1.65 1.47 2.15 2.44 1.30 1.67 1.18 1.93 7.37 16.86
5 1.30 1.16 1.70 1.96 1.32 1.69 1.2 1.99 7.52 17.41
4 0.95 0.86 1.27 1.48 1.33 1.72 1.23 2.06 7.71 17.99
3 0.66 0.59 0.89 1.03 1.34 1.75 1.25 2.14 7.80 18.70
2 0.39 0.34 0.53 0.60 1.35 1.79 1.26 2.23 7.89 19.57
* These values are taken directly from the analysis. Accidental torsion is not amplified here.
** Amplification of accidental torsion is required because this term is greater than 1.2 (Provisions Table 5.2.3.2
Item 1a [4.32, Item 1a]). The building is torsionally irregular in plan. Provisions Table 5.2.5.1 [4.41] indicates
that an ELF analysis is “not permitted” for torsionally irregular structures. However, given rigid diaphragms and
symmetry about both axes, a modal analysis will not give any difference in results than an ELF analysis insofar as
accidental torsion is concerned unless one arbitrarily offsets the center of mass. The Provisions does not require
an arbitrary offset for center of mass in dynamic analysis nor is it common practice to do so. One reason for this
is that the computed period of vibration would lengthen, which, in turn, would reduce the overall seismic demand.
See Sec. 9.2 and 9.3 of this volume of design examples for a more detailed examination of this issue.
*** The initial eccentricities of 0.05 in the EW and NS directions are multiplied by Ax to determine the
amplified eccentricities. These will be used in the next round of analysis.
1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m.
4. Check Torsional Amplification – A second RAMFRAME 3D analysis was made, using the
amplified eccentricity for accidental torsion instead of merely 0.05L for accidental torsion. The
results are summarized in Table 5.27.
FEMA 451, NEHRP Recommended Provisions: Design Examples
562
Table 5.27 Alternative B Story Drifts under Seismic Load
Max. Elastic
Displacement at
Building Corners
(in.)
Elastic Story Drift at
Location of Max.
Displacement (at
corners) (in.)
Cd (Cd) × (Elastic
Story Drift) (in.)
Allowable
Story Drift
(in.)
EW NS .EW .NS .EW .NS .
R 3.14 4.50 0.42 0.55 5 2.10 2.75 3.20
7 2.72 3.95 0.49 0.64 5 2.46 3.19 3.20
6 2.23 3.32 0.45 0.63 5 2.27 3.16 3.20
5 1.77 2.68 0.45 0.64 5 2.25 3.18 3.20
4 1.32 2.05 0.40 0.61 5 1.98 3.07 3.20
3 0.93 1.43 0.38 0.59 5 1.89 2.93 3.20
2 0.55 0.85 0.55 0.85 5 2.75 4.24 5.36
1.0 in. = 25.4 mm
All story drifts are within the allowable story drift limit of 0.020hsx in accordance with Provisions
Sec. 5.2.8 [4.51] and the allowable deflections for this building from Sec. 5.2.3.6 above. This a good
point to reflect on the impact of accidental torsion and its amplification on the design of this corebraced
structure. The sizes of members were increased substantially to bring the drift within the
limits (note how close the NS direction drifts are). For the final structure, the elastic displacements
at the main roof are:
At the centroid = 2.08 in.
At the corner with accidental torsion = 3.37 in.
At the corner with amplified accidental torsion = 4.50 in.
The two effects of torsional irregularity (in this case, it would more properly be called torsional
flexibility) of amplifying the accidental torsion and checking the drift limits at the corners combine to
create a demand for substantially more stiffness in the structure. Even though many braced frames
are controlled by strength, this is an example of how the Provisions places significant stiffness
demands on some braced structures.
5. Check Redundancy – Now return to the calculation of rx for the braced frame. Per Provisions Sec.
5.2.4.2 [not applicable in the 2003 Provisions], max x for braced frames is taken as the lateral force r
component in the most heavily loaded brace element divided by the story shear (Figure 5.218).
A value for rx was determined for every brace element at every level in both directions. The lateral
component carried by each brace element comes from the RAMFRAME analysis, which includes the
effect of amplified accidental torsion. Selected results are illustrated in the figures. The maximum rx
was found to be 0.223 below Level 7 in the NSdirection. The reliability factor (.) is now determined
using Provisions Eq. 5.2.4.2 [not applicable in the 2003 Provisions]:
Chapter 5, Structural Steel Design
563
7
r = 118.2
x 530 = 0.223
530 kips = story shear
118.2 kips
Figure 5.218 Lateral force component in braces for NS
direction – partial elevation, Level 7 (1.0 kip = 4.45 kN).
2
2 20 2 20 1.39
rmax xAx 0.223 21,875 ft
. =  =  =
In the NS direction, all design force effects (axial forces, shears, moments) obtained from analysis
must be increased by the . factor of 1.39. Similarly, for the EWdirection, and . are found to max x r
be 0.192 and 1.26, respectively. (However drift controls the design for this problem. Drift and
deflection are not subject to the . factor.)
[See Sec. 5.2.3.2 for a discussion of the significant changes to the redundancy requirements in the
2003 Provisions.]
6. Braced Frame Member Design Considerations – The design of members in the special concentrically
braced frame (SCBF) needs to satisfy AISC Seismic Sec. 13 and columns also need to satisfy AISC
Seismic Sec. 8. When Pu/fPn is greater than 0.4, as is the predominant case here, the required axial
strength needs to be determined from AISC Seismic Eq. 41 and 42 [Provisions Eq. 4.23 and 4.24].
These equations are for load combinations that include the O0
, or overstrength, factors. Moments are
generally small for the braced frame so load combinations with O0
can control column size for
strength considerations but, for this building, drift controls because of amplified accidental torsion.
Note that . is not used where O0
is used (see Provisions Sec. 5.2.7 [4.2.2.2]).
Bracing members have special requirements as well, although O0
factors do not apply to braces in a
SCBF. Note in particular AISC Seismic Sec. 13.2c, which requires that both the compression brace
and the tension brace share the force at each level (as opposed to the “tension only” braces of
Example 5.1). AISC Seismic Sec. 13.2 also stipulates a kl/r limitation and local buckling (widththickness)
ratio limits.
Beams in many configurations of braced frames have small moments and forces, which is the case
here. V and inverted V (chevron) configurations are an exception to this. There is a panel of chevron
bracing at the top story of one of the braced frames (Figure 5.216). The requirements of AISC
Seismic Sec. 13.4 should be checked although, in this case, certain limitations of AISC Seismic do
not apply because the beam is at the top story of a building. (The level above in Figure 5.216 is a
minor penthouse that is not considered to be a story.) If the chevron bay were not at the top story, the
size of the braces must be known in order to design the beam. The load combination for the beam is
modified using a Qb factor defined in AISC Seismic Sec. 13.4a. Basically, the beam must be able to
FEMA 451, NEHRP Recommended Provisions: Design Examples
564
Plate: 11
8" (A36)
W14x38
W14x211
HSS10x10x5
8
HSS12x12x5
8
Plastic hinge zone = 2t
3'1"
Leff
Plastic hinge zone = 2t
2'7"
2'61
2"
2'31
2"
7'11"
30°
(Typ.)
Figure 5.219 Bracing connection detail (1.0 in. = 25.4 mm, 1.0 ft =
0.3048 m).
carry a concentrated load equal to the difference in vertical force between the postbuckling strength
of the compression brace and the yield strength of the tension brace (i.e., the compression brace has
buckled, but the tension brace has not yet yielded). The prescribed load effect is to use 0.3 fc
Pn for
the compression brace and Py for the tension brace in order to determine a design vertical force to be
applied to the beam.
7. Connection Design – Figure 5.219 illustrates a typical connection design at a column per AISC
Seismic Sec. 13. First, check the slenderness and widthtothickness ratios (see above). The bracing
members satisfy these checks.
Chapter 5, Structural Steel Design
565
HSS12x12x5
8
30°
30°
37"
543
4"
12"
W14x211
30.5"
Figure 5.220 Whitmore section (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
Next, design the connections. The required strength of the connection is to be the nominal axial
tensile strength of the bracing member. For an HSS12×12×5/8, the nominal axial tensile strength is
computed using AISC Seismic Sec. 13.3a:
Pn = RyFyAg = (1.3)(46 ksi)(27.4 in.2) = 1,639 kips
The area of the gusset is determined using the plate thickness and the Whitmore section for effective
width. See Figure 5.220 for the determination of this dimension.
For tension yielding of the gusset plate:
fTn = fFyAg = (0.90)(36 ksi)(1.125 in. × 54.7 in.) = 1,993 kips > 1,639 kips OK
For fracture in the net section:
fTn = fFuAn = (0.75)(58 ksi )(1.125 in. × 54.7 in.) = 2,677 kips > 1,639 kips OK
Since 1,933 kips is less than 2,677 kips, yielding (ductile behavior) governs over fracture.
FEMA 451, NEHRP Recommended Provisions: Design Examples
566
For a tube slotted to fit over a connection plate, there will be four welds. The demand in each weld
will be 1,639 kips/4 = 410 kips. The design strength for a fillet weld per AISC LRFD Table J2.5 is:
fFw = f(0.6Fexx) = (0.75)(0.6)(70 ksi) = 31.5 ksi
For a 1/2 in. fillet weld, the required length of weld is determined to be:
. 410 kips 37 in
Lw=(0.707)(0.5 in.)(31.5 ksi)=
In accordance with the exception of AISC Seismic Sec. 13.3c, the design of brace connections need
not consider flexure if the connections meet the following criteria:
a. Inelastic rotation associated with brace postbuckling deformations: 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 Seismic C13.3c. With such a plastic
hinge, the compression brace may buckle outofplane when the tension braces are loaded.
Remember that during the earthquake, there will be alternating cycles of compression to 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.
b. The connection design strength must be at least equal to the nominal compressive strength of the
brace.
Therefore, the connection will be designed in accordance with these criteria. First, determine the
nominal compressive strength of the brace member. The effective brace length (Leff) is the distance
between the plastic hinges on the gusset plates at each end of the brace member. For the brace being
considered, Leff = 169 in. and the nominal compressive strength is determined using AISC LRFD Eq.
E24:
(1)(169) 46 0.466
(4.60) 29,000
y
c
kl F
r E
.
p p
= = =
Since .c
< 1.5, use AISC LRFD Eq. E22:
(0.658 c2 ) (0.6580.217 )(46) 42.0 ksi
cr y F= .F= =
Pcr = AgFcr = (27.4)(42.0) = 1,151 kips
Now, using a design compressive load from the brace of 1,151 kips, determine the buckling capacity
of the gusset plate using the Whitmore section method. By this method, illustrated by Figure 5.220,
the compressive force per unit length of gusset plate is (1,151 kips/54.7 in.) = 21.04 kips/in.
Try a plate thickness of 1.125 in.
fa = P/A = 21.04 kips/(1 in.× 1.25 in.) = 18.7 ksi
Chapter 5, Structural Steel Design
567
The gusset plate is modeled as a 1 in. wide by 1.125 in. deep rectangular section, pinned at one end
(the plastic hinge) and fixed at the other end where welded to column (see Whitmore section
diagram). The effective length factor (k) for this “column” is 0.8.
Per AISC LRFD Eq. E24:
(0.8)(30.5) 36 0.51
(0.54) 29,000
y
c
kl F
r E
.
p p
= = =
Since .c < 1.5, use AISC LRFD Eq. E22:
(0.658 c2 ) (0.6580.257 )(36) 32.3 ksi
cr y F= .F= =
fFcr = (0.85)(32.3) = 27.4 ksi
fFcr = 27.4 ksi > 18.7 ksi OK
Now consider the bracetobrace connection shown in Figure 5.221. The gusset plate will experience
the same tension force as the plate above, and the Whitmore section is the same. However, the
compression length is much less, so a thinner plate may be adequate.
Try a 15/16 in. plate. Again, the effective width is shown in Figure 5.220. For tension yielding of
the gusset plate:
fTn = fFyAg = (0.90)(36 ksi)(0.9375 in. × 54.7 in.) = 1,662 kips > 1,639 kips OK
For fracture in the net section:
fTn = fFyAg = (0.75)(58 ksi)(0.9375 in. × 54.7 in.) = 2,231 kips > 1,639 kips OK
Since 1,662 kips is less than 2,231 kips, yielding (ductile behavior) governs over fracture.
For compression loads, the plate must be detailed to develop a plastic hinge over a distance of 2t from
the end of the brace. The effective length for buckling of this plate will be k[12" + (2)(2t + weld
length)]. For this case, the effective length is 0.65[12 + (2)(2 × 15/16 + 5/16)] = 9.2 in. Compression
in the plate over this effective length is acceptable by inspection and will not be computed here.
Next, check the reduced section of thetube, which has a 1 1/4 in. wide slot for the gusset plate (at the
column). The reduction in HSS12×12×5/8 section due to the slot is (0.581 × 1.25 × 2) = 1.45 in.2,
and the net section, Anet = (25.7  1.45) = 24.25 in.2
Compare yield in the gross section with fracture in the net section:
Yield = FyAg =(46 ksi)(25.7 in.2) = 1,182 kips OK
Fracture = FuAn =(58 ksi)(24.25 in.2) = 1,406 kips OK
AISC Seismic 13.3b could be used to require design fracture strength (0.75 x 1,406 = 1,055 kips) to
exceed probable tensile yield (1,639 kips), but this is clearly impossible, even if the net area equaled
the gross area. This design is considered satisfactory.
FEMA 451, NEHRP Recommended Provisions: Design Examples
568
W14x38
HSS12x12x5
8
HSS12x12x5
8
Leff
Plastic hinge
zone = 2t + 5
16"
Figure 5.221 Bracetobrace connection (1.0 in. = 25.4 mm).
5.2.4.3.3 Size Members for Alternative C, Dual System
1. Select Preliminary Member Sizes – A dual system is a combination of a momentresisting frame with
either a shear wall or a braced frame. In accordance with the building systems listed in Provisions
Table 5.2.2 [4.31], a dual system consisting of special moment frames at the perimeter and special
concentrically braced frames at the core will be used.
2. Check Strength of Moment Frame – The moment frame is required to have sufficient strength to resist
25 percent of the design forces by itself (Provisions Sec. 5.2.2.1 [4.3.1]). This is a good place to start
a design. Preliminary sizes for the perimeter moment frames are shown in Figures 5.222 and 5.223.
It is designed for strength using 25 percent of the design lateral forces. All the design requirements
for special moment frames still apply (flange and web widthtothickness ratios, columnbeam
moment ratio, panel zone shear, drift, and redundancy) and all must be checked; however, it may be
prudent to defer some of the checks until the design has progressed a bit further. The methodology
for the analysis and these checks is covered in Sec. 5.2.4.3.1, so they will not be repeated here.
For some buildings this may present an opportunity to design the columns without doubler plates
because the strength requirement is only 25 percent of the total. However, for the members used in
this example, doubler plates will still be necessary. The increase in column size to avoid doubler
plates is substantial, but feasible. The sequence of column sizes that is shown is W 14×132  82  68 
Chapter 5, Structural Steel Design
569
W16x31
W14x132 W14x82 W14x68 W14x53
W 14x74 W 14x68
W16x31
W16x31
W18x35
W18x40
W21x44
W21x50
W14x132
W14x132
W14x132
W14x132
W14x132
W14x132
W14x132
W21x50
W21x44
W18x40
W18x35
W16x31
W16x31
W16x31
W21x50
W21x44
W18x40
W18x35
W16x31
W16x31
W16x31
W21x50
W21x44
W18x40
W18x35
W16x31
W16x31
W16x31
W21x50
W21x44
W18x40
W18x35
W16x31
W16x31
W16x31
W21x50
W21x44
W18x40
W18x35
W16x31
W16x31
W16x31
W21x50
W21x44
W18x40
W18x35
W16x31
W16x31
W16x31
W 14x53
W 14x74 W 14x68 W 14x53
W 14x74 W 14x68 W 14x53
W 14x74 W 14x68 W 14x53
W 14x74 W 14x68 W 14x53
W 14x74 W 14x68 W 14x53
W 14x82 W 14x68 W 14x53
Figure 5.222 Moment frame of dual system in EW direction.
53 and would become W14×257  233  211  176 to avoid doubler plates. The beams in Figures 5.2
22 and 5.223 are controlled by strength because drift is not a criterion.
Note that Pu /fPn > 0.4 for a few of the columns when analyzed without the braced frame so the
overstrength requirements of AISC Seismic Sec. 8.2 [8.3] apply to these columns. Because the check
using O0
E is for axial capacity only and the moment frame columns are dominated by bending
moment, the sizes are not controlled by the check using O0
E.
3. Check the Strength of the Braced Frames – The next step is to select the member sizes for the braced
frame. Because of the presence of the moment frame, the accidental torsion on the building will be
reduced as compared to a building with only a braced core. In combination with the larger R factor
(smaller design forces), this should help to realize significant savings in the braced frame member
sizes. A trial design is selected, followed by analysis of the entire dual system. All members need to
be checked for widththickness ratios and the braces need to be checked for slenderness. Note that
columns in the braced frame will need to satisfy the overstrength requirements of AISC Seismic Sec.
8.2 [8.3] because Pu/fPn > 0.4. This last requirement causes a significant increase in column sizes,
except in the upper few stories.
FEMA 451, NEHRP Recommended Provisions: Design Examples
570
W24x55
W21x50
W21x50
W21x44
W18x40
W16x31
W14x132 W14x82 W14x68 W14x53
W16x31
W24x55
W21x50
W21x50
W21x44
W18x40
W16x31
W16x31
W24x55
W21x50
W21x50
W21x44
W18x40
W16x31
W16x31
W24x55
W21x50
W21x50
W21x44
W18x40
W16x31
W16x31
W24x55
W21x50
W21x50
W21x44
W18x40
W16x31
W16x31
W14x132 W14x82 W14x68 W14x53
W14x132 W14x82 W14x68 W14x53
W14x132 W14x82 W14x68 W14x53
W14x132 W14x82 W14x68 W14x53
W14x132 W14x82 W14x68 W14x53
Figure 5.223 Moment frame of dual system in NS direction.
4. Check Drift – Check drift in accordance with Provisions Sec. 5.2.8 [4.5]. The building was modeled
in three dimensions using RAMFRAME. Maximum displacements at the building corners are used
here because the building is torsionally irregular. Displacements at the building centroid are also
calculated because these will be the average between the maximum at one corner and the minimum at
the diagonally opposite corner. Use of the displacements at the centroid as the average displacements
is valid for a symmetrical building.
5. Check Torsional Amplification – Calculated story drifts are used to determine Ax, the torsional
amplification factor (Table 5.28). Pdelta effects are included.
Chapter 5, Structural Steel Design
571
Table 5.28 Alternative C Amplification of Accidental Torsion
Average Elastic
Displacement =
Displacement at
Building
Centroid (in.)
Maximum
Elastic
Displacement at
Building
Corner* (in.)
max ** Torsional
avg
d
d Amplification
Factor =
2
max
avg 1.2 x A
d
d
=
. .
. .
. .
Amplified
Eccentricity
= Ax(0.05 L)***
(ft.)
EW NS EW NS EW NS EW NS EW NS
R 2.77 2.69 3.57 3.37 1.29 1.25 1.15 1.09 7.19 9.54
7 2.45 2.34 3.15 3.00 1.28 1.28 1.14 1.14 7.14 10.01
6 2.05 1.91 2.63 2.50 1.28 1.31 1.13 1.20 7.07 10.46
5 1.64 1.51 2.10 2.01 1.28 1.33 1.13 1.23 7.08 10.8
4 1.22 1.11 1.56 1.50 1.28 1.35 1.14 1.27 7.13 11.15
3 0.81 0.75 1.05 1.03 1.29 1.38 1.16 1.31 7.25 11.50
2 0.43 0.41 0.57 0.57 1.32 1.40 1.20 1.37 7.52 11.98
* These values are directly from the analysis. Accidental torsion is not amplified here.
** Amplification of accidental torsion is required because this term is greater than 1.2 (Provisions Table 5.2.3.2,
Item 1a [4.32, Item 1a). The building is torsionally irregular in plan. See discussion in footnote ** of
Table 5.2.6.
*** The initial eccentricities of 0.05L in the EW and NS directions are multiplied by Ax to determine the
amplified eccentricities. These will be used in the next round of analysis.
1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m.
The design that yielded the displacements shown in Table 5.28 does not quite satisfy the drift limits,
even without amplifying the accidental torsion. That design was revised by increasing various brace
and column sizes and then reanalyzing using the amplified eccentricity instead of merely 0.05L for
accidental torsion. After a few iterations, a design that satisfied the drift limits was achieved. These
member sizes are shown in Figures 5.224 and 5.225. That structure was then checked for its
response using the standard 0.05L accidental eccentricity in order to validate the amplifiers used in
design. The amplifier increased for the EW direction but decreased for the NS direction, which was
the controlling direction for torsion. The results are summarized in Table 5.29.
FEMA 451, NEHRP Recommended Provisions: Design Examples
572
W14x426
W14x48
W14x48
W14x48
W14x48
W14x48
W14x48
W14x48
W14x48
HSS8x8x1
2
W14x311 W14x176 W14x68 W14x48
HSS31
2x31
2x1
4
HSS6x6x1
2
PH
R
7
6
5
4
3
2
HSS6x6x1
2
HSS7x7x1
2
HSS7x7x1
2
HSS8x8x1
2
HSS8x8x1
2
HSS8x8x1
2
HSS31
2x31
2x1
4
HSS6x6x1
2
HSS6x6x1
2
HSS7x7x1
2
HSS7x7x1
2
HSS8x8x1
2
HSS8x8x1
2
W14x426 W14x311 W14x176 W14x68 W14x48
Figure 5.225 Braced frame
of dual system in NS
direction.
W14x426
W14x48
W14x48
W14x48
W14x48
W14x48
W14x48
W14x48
W14x34
W14x48
W14x48
W14x48
W14x48
W14x38
W14x48
W14x48
W14x34
HSS7x7x5
8
HSS7x7x5
8
W14x311 W14x176 W14x68 W14x48
PH
R
7
6
5
4
3
2
HSS10x10x5
8
HSS8x8x5
8
HSS8x8x5
8
HSS8x8x5
8
HSS6x6x1
2
HSS6x6x1
2
W14x426 W14x311 W14x176 W14x68 W14x48
W14x426 W14x311 W14x176 W14x68 W14x48
W14x426 W14x311 W14x176 W14x68 W14x48
HSS10x10x5
8
HSS8x8x5
8
HSS8x8x5
8
HSS8x8x5
8 HSS8x8x5
8
HSS8x8x5
8
HSS8x8x5
8
HSS8x8x5
8
HSS7x7x5
8 HSS7x7x5
8
HSS7x7x5
8
HSS7x7x5
8
HSS7x7x5
8 HSS7x7x5
8
HSS31
2x31
2x1
4
HSS31
2x31
2x1
4
HSS6x6x1
2
HSS6x6x1
2
HSS6x6x1
2
HSS6x6x1
2
HSS6x6x1
2
HSS6x6x1
2
Figure 5.224 Braced frame of dual system in EWdirection.
Chapter 5, Structural Steel Design
573
Table 5.29 Alternative C Story Drifts under Seismic Load
Max. Elastic
Displacement at
Building Corners
(in.)
Elastic Story Drift at
Location of Max.
Displacement (at
corners) (in.)
Cd (Cd ) x (Elastic Story
Drift)
(in.)
Allowable
Story
Drift
(in.)
EW NS .EW .NS .EW .NS .
R 3.06 3.42 0.37 0.37 6.5 2.43 2.42 3.20
7 2.69 3.05 0.45 0.47 6.5 2.94 3.05 3.20
6 2.24 2.58 0.45 0.49 6.5 2.89 3.17 3.20
5 1.79 2.09 0.45 0.51 6.5 2.93 3.30 3.20
4 1.34 1.58 0.41 0.48 6.5 2.66 3.09 3.20
3 0.93 1.11 0.39 0.46 6.5 2.55 3.01 3.20
2 0.54 0.64 0.54 0.64 6.5 3.52 4.17 5.36
1.0 in. = 25.4 mm
The story drifts are within the allowable story drift limit of 0.020hsx as per Provisions Sec. 5.2.8
[4.5.1]. Level 5 has a drift of 3.30 in. > 3.20 in. but the difference of only 0.1 in. is considered close
enough for this example.
6. Check Redundancy – Now return to the calculation of rx for the braced frame. In accordance with
Provisions Sec. 5.2.4.2 [not applicable in the 2003 Provisions], max x for braced frames is taken as the r
lateral force component in the most heavily loaded brace element divided by the story shear. This is
illustrated in Figure 5.218 for Alternative B.
For this design, rx was determined for every brace element at every level in both directions. The
lateral component carried by each brace element comes from the RAMFRAME analysis, which
includes the effect of amplified accidental torsion. The maximum value was found to be 0.1762 at the
base level in the NS direction. Thus, . is now determined to be (see Sec. 5.2.4.2):
2
0.8 2 20 0.8 2 20 0.986
rmax xAx 0.1762 21,875 ft.
.
. . . .
= . .= . .=
.. .. .. ..
The 0.8 factor comes from Provisions Sec. 5.2.4.2 [not applicable in the 2003 Provisions]. As . is
less than 1.0, . = 1.0 for this example.
In the EW direction, rmax is less; therefore, . will be less, so use . = 1.0 for both directions.
[See Sec. 5.2.3.2 for a discussion of the significant changes to the redundancy requirements in the
2003 Provisions.]
FEMA 451, NEHRP Recommended Provisions: Design Examples
574
7. Connection Design – Connections for both the moment frame and braced frames may be designed in
accordance with the methods illustrated in Sec. 5.2.4.3.1 and 5.2.4.3.2.
5.2.5 Cost Comparison
Material takeoffs were made for the three alternatives. In 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:
Alternative A, Special Steel Moment Resisting Frame 593 tons
Alternative B, Special Steel Concentrically Braced Frame 640 tons
Alternative C, Dual System 668 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.
Chapter 5, Structural Steel Design
575
Basement walls
N
120'0"
20'0" 20'0" 20'0" 20'0"
140'0"
20'0" 20'0" 20'0"
Brace (typical)
20'0" 20'0"
20'0" 20'0" 20'0" 20'0"
Figure 5.31 Main floor framing plan (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
5.3 TWOSTORY BUILDING, OAKLAND, CALIFORNIA
This example features an eccentrically braced frame (EBF) building. The following items of seismic
design of steelframed buildings are illustrated:
1. Analysis of eccentrically braced frames
2. Design of bracing members
3. Brace connections
5.3.1 Building Description
This twostory hospital, 120 ft by 140 ft in plan, is shown in Figure 5.31. The building has a basement
and two floors. It has an unusually high roof load because of a plaza with heavy planters on the roof.
The verticalloadcarrying system consists of concrete fill on steel deck floors supported by steel beams
and girders that span to steel columns and to the perimeter basement walls. The bay spacing is 20 ft each
way. Floor beams are spaced three to a bay. The beams and girders on the column lines are tied to the
slabs with stud connectors.
The building is founded on a thick mat. The foundation soils are deep stable deposits of sands, gravels,
and stiff clays overlying rock.
The lateralforceresisting system for Stories 1 and 2 consists of EBFs on Gridlines 1, 8, B, and F as
FEMA 451, NEHRP Recommended Provisions: Design Examples
576
3'0" 15'6" 12'8" 12'8"
Figure 5.34
Roof
2nd floor
1st floor
Basement
Figure 5.32 Section on Grid F (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
shown in Figure 5.31. A typical bracing elevation is shown in Figure 5.32. These EBFs transfer lateral
loads to the main floor diaphragm. The braced frames are designed for 100 percent of lateral load and
their share of vertical loads. EBFs have been selected for this building because they provide high
stiffness and a high degree of ductility while permitting limited storytostory height.
The structure illustrates a common situation for lowrise buildings with basements. The combination of
the basement walls and the first floor diaphragm is so much stiffer that the superstructure that the base
(see Provisions Chapter 2 [4.1.3] for definition) of the building is the first floor, not the foundation.
Therefore, the diagonal braces do not extend into the basement because the horizontal force is in the
basement walls (both in shear parallel to the motion considered and in direct pressure on perpendicular
walls). This has a similarity to the irregularity Type 4 “outofplane offsets” defined in Provisions Table
5.2.3.2 [4.32], but because it is below the base that classification does not apply. However, the columns
in the basement that are part of the EBFs must be designed and detailed as being the extension of the EBF
that they are. This affects widththickness ratios, overstrength checks, splice requirements, and so on.
Column design for an EBF is illustrated later in this example.
5.3.2 Method
The method for determining basic parameters is similar to that for other examples. It will not be repeated
here; rather the focus will be on the design of a specific example of an EBF starting with the forces in the
frame as obtained from a linear analysis. Keep in mind that the load path is from the floor diaphragm to
the beam to the braces. The fundamental concept behind the eccentric braced frame is that seismic energy
is absorbed by yielding of the link. Yielding in shear is more efficient than yielding in flexure, although
either is permitted. A summary of the method is as follows:
1. Select member preliminary sizes.
2. Perform an elastic analysis of the building frame. Compute elastic drift (de) and forces in the
members.
3. Compute the inelastic displacement as the product of Cd times de. The inelastic displacement must be
within the allowable story drift from Provisions Table 5.2.8 [4.51].
Chapter 5, Structural Steel Design
577
4. Compute the link rotation angle (a) and verify that it is less than 0.08 radians for yielding dominated
by shear in the link or 0.02 radians for yielding dominated by flexure in the link. (See Figure 5.34
for illustration of a). The criteria is based on the relationship between Mp and Vp as related to the
length of the link.
5. To meet the link rotation angle requirement, it may be necessary to modify member sizes, but the
more efficient approach is to increase the link length. (The tradeoff to increasing the link length is
that the moment in the link will increase. Should the moment become high enough to govern over
shear yielding, then a will have to be limited to 0.02 radians instead of 0.08 radians.)
6. The braces and building columns are to remain elastic. The portions of the beam outside the link are
to remain elastic; only the link portion of the beam yields.
7. For this case, there are momentresisting connections at the columns. Therefore from Provisions
Table 5.2.2 [4.31], R = 8, Cd = 4, and O0
=2. (Neither the Provisions nor AISC Seismic offer very
much detailed information about requirements for momentresisting connections for the beam to
column connection in an EBF. There are explicit requirements for the connection from a link to a
column. The EBF system will not impose large rotational demands on a beam to column connection;
the inelastic deformations are confined to the link. Therefore, without further detail, it is the authors’
interpretation that an ordinary moment resisting frame connection will be adequate).
5.3.3 Analysis
Because the determination of basic provisions and analysis are so similar to those of other examples, they
will not be presented here. An ELF analysis was used.
5.3.3.1 Member Design Forces
The critical forces for the design of individual structural elements, determined from computer analysis,
are summarized in Table 5.31.
Table 5.31 Summary of Critical Member Design Forces
Member Force Designation Magnitude
Link Plink
Vlink
Mleft
Mright
5.7 kips
85.2 kips
127.9 ftkips
121.3 ftkips
Brace Pbrace
Mtop
Mbot
120.0 kips
15.5 ftkips
9.5 ftkips
1.0 kip = 4.45 kN, 1.0 ftkip = 1.36 kNm.
The axial load in the link at Level 2 may be computed directly from the secondfloor forces. The force
from the braces coming down from the roof level has a direct pass to the braces below without affecting
the link. The axial forces in the link and brace may be determined as follows:
Total secondstory shear (determined elsewhere) = 535.6 kips
FEMA 451, NEHRP Recommended Provisions: Design Examples
578
Secondstory shear per braced line = 535.6/2 = 267.8 kips
Secondstory shear per individual EBF = 267.8/2 = 133.9 kips
Secondstory shear per brace = 133.9/2 = 66.95 kips
Axial force per brace = 66.95 (15.25 ft/8.5 ft) = 120.0 kips
Secondstory shear per braced line = 267.8 kips
Secondstory shear per linear foot = 267.8 kips/140 ft = 1.91 klf
Axial force in link = (1.91 klf)(3 ft) = 5.7 kips
5.3.3.2 Drift
From the linear computer analysis, the elastic drift was determined to be 0.247 inches. The total inelastic
drift is computed as:
Cddc = (4)(0.247) = 0.99 in.
The link rotation angle is computed for a span length, L = 20 ft, and a link length, e = 3 ft as follows:
20 ft 0.99 in. 0.043 radians
3 ft (12.67 ft)(12)
L
e
a .
=.. .. =.. .... ..=
. . . .. .
The design is satisfactory if we assume that shear yielding governs because the maximum permissible
rotation is 0.08 radians (AISC Seismic Sec. 15.2g [15.2]). For now, we will assume that shear yielding of
the link governs and will verify this later.
5.3.4 Design of Eccentric Bracing
Eccentric bracing adds two elements to the frame: braces and links. As can be seen in Figure 5.33, two
eccentric braces located in one story of the same bay intersect the upper beam a short distance apart, thus
creating a link subject to high shear. In a severe earthquake, energy is dissipated through shear yielding
of the links while diagonal braces and columns remain essentially elastic.
The criteria for the design of eccentric bracing are given in AISC Seismic Sec. 15. All section sizes and
connection details are made similar for all braced bays. The following sections have been selected as a
preliminary design:
Typical girders W16×57
Typical columns W14×132
Typical braces HSS 8×8×5/8
Since all members of the braced frames are to be essentially the same, further calculations deal with the
braced frames on Line F, shown in Figures 5.33 and 5.34.
Chapter 5, Structural Steel Design
579
Roof
Basement
Figure 5.34
2nd floor
1st floor
Figure 5.33 Diagram of eccentric braced frames on Grid F.
Drift
8'6" 3'0" 8'6"
12'8"
a
.
15'3"
Figure 5.34 Typical eccentric braced frame
(1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
5.3.4.1 Link Design
The firststory eccentric braced frame (identified in Figure 5.32) is examined first. The shear force and
end moments in the link (W16x57 beam section) are listed in Table 5.31 and repeated below:
Plink = 5.7 kips
Vlink = 85.2 kips
Mleft = 127.9 ftkips
Mright = 121.3 ftkips
FEMA 451, NEHRP Recommended Provisions: Design Examples
580
5.3.4.1.1 WidthThickness Ratio
The links are first verified to conform to AISC Seismic Sec. 15.2a [15.2], which refers to AISC Seismic
Table I91 [I81].
First, check the beam flange widththickness ratio. For the selected section, b/t = 4.98, which is less than
the permitted b/t ratio of :
OK
52 52
7.35
Fy 50
= =
The permitted web slenderness is dependent on the level of axial stress. The level of axial stress is
determined as:
5.8 0.008
(0.9)(16.8 50)
u
b y
P
f P
= =
×
It is less than 0.125; therefore, the ratio tw/hc = 33.0 for the selected section is less than the limiting widthto
thickness ratio computed as:
OK
253 253 35.7
50 y F
= =
5.3.4.1.2 Link Shear Strength
The forces Vlink, Mleft, and Mright must not exceed member strength computed from AISC Seismic Sec.
15.2d [15.2]. That section specifies that the required shear strength of the link (Vu) must not exceed the
design shear strength fVn, where Vu = Vlink = 85.2 kips and Vn is the nominal shear strength of link. The
nominal shear strength of the link is defined as the lesser of:
Vp = (0.60Fy)(d2tf)tw
and
2Mp
e
For the W16×57 section selected for the preliminary design:
Vp = (0.60)(50)[16.43  (2)(0.715)](0.430) = 193.5 kips
and
Mp = fMn = 0.9Fy Zx = (0.9)(50)(105) = 4725 in.kips
2 (2)(4725) 262.5 kips
(3 12)
Mp
e
= =
×
Chapter 5, Structural Steel Design
581
Therefore,
Vn = 193.5 kips
fVn = (0.9)(193.5) = 174.2 ftkips > 85.2 kips OK
5.3.4.1.3 Link Axial Strength
In accordance with AISC Seismic Sec. 15.2e [15.2], the link axial strength is examined:
Py of the link = FyAg = (50 ksi)(16.8 in) = 840 kips
0.15Py of the link = (0.15)(840) = 126 kips
Since the axial demand of 5.7 kips is less than 126 kips, the effect of axial force on the link design shear
strength need not be considered. Further, because Pu < Py, the additional requirements of AISC Seismic
Sec. 15.f [15.2] do not need to be invoked.
5.3.4.1.4 Link Rotation Angle
In accordance with AISC Seismic Sec. 15.2g [15.2], the link rotation angle is not permitted to exceed 0.08
radians for links 1.6Mp/Vp long or less. Therefore, the maximum link length is determined as:
1.6Mp/Vp = (1.6)(4725)/(193.5) = 39.1 in.
Since the link length (e) of 36 in. is less than 1.6Mp/Vp, the link rotation angle is permitted up to 0.08
radians. From Sec. 5.3.3.2, the link rotation angle, a, was determined to be 0.043 radians, which is
acceptable.
5.3.4.1.5 Link Stiffeners
AISC Seismic Sec. 15.3a [15.3] requires fulldepth web stiffeners on both sides of the link web at the
diagonal brace ends of the link. These serve to transfer the link shear forces to the reacting elements (the
braces) as well as restrain the link web against buckling.
Because the link length (e) is less than 1.6Mp/Vp, intermediate stiffeners are necessary in accordance with
AISC Seismic Sec. 15.3b [15.3]. Interpolation of the stiffener spacing based on the two equations
presented in AISC Seismic Sec. 15.3b.1 [15.3] will be necessary. For a link rotation angle of 0.08
radians:
Spacing = (30tw  d/5) = (30 × 0.430  16.43/5) = 9.6 in.
For link rotation angle of 0.02 radians:
Spacing = (52tw  d/5) = (52 × 0.430  16.43/5) = 19.1 in.
For our case the link rotation angle is 0.043 radians, and interpolation results in a spacing requirement of
15.4 in. Therefore, use a stiffener spacing of 12 in. because it conforms to the 15.4 in. requirement and
also fits nicely within the link length of 36 in.
In accordance with AISC Seismic Sec. 15.3a [5.3], full depth stiffeners must be provided on both sides of
the link, and the stiffeners must be sized as follows:
FEMA 451, NEHRP Recommended Provisions: Design Examples
582
Combined width at least (bf  2tw) = (7.120  2 × 0.430) = 6.26 in. Use 3.25 in. each.
Thickness at least 0.75tw or 3/8 in. Use 3/8 in.
5.3.4.1.6 Lateral Support of Link
The spacing of the lateral bracing of the link must not exceed the requirement of AISC LRFD Eq. F117,
which specifies a maximum unbraced length of:
[3,600 + 2,200( 1 / 2 )] [3,600 + 2,200(121.3/127.9)](1.60)
50
y 182 in.
pd
y
M M r
L
F
= = =
Accordingly, lateral bracing of beams with one brace at each end of the link (which is required for the
link design per AISC Seismic Sec. 15.5) is sufficient.
In accordance with AISC Seismic Sec. 15.5, the end lateral supports must have a design strength
computed as:
0.06RyFybftf = (0.06)(1.1)(50)(7.120)(0.715) = 16.8 kips
While shear studs on the top flange are expected to accommodate the transfer of this load into the
concrete deck, the brace at the bottom flange will need to be designed for this condition. Figure 5.35
shows angle braces attached to the lower flange of the link. Such angles will need to be designed for 16.8
kips tension or compression.
5.3.4.2 Brace Design
For the design equations used below, see Chapter E. of the AISC LRFD Specification. The braces,
determined to be 8×8×5/8 in. tubes with Fy = 46 ksi in the preliminary design, are subjected to a
calculated axial seismic load of 120 kips (from elastic analysis in Table 5.31). Taking the length of the
brace conservatively as the distance between panel points, the length is 15.26 feet. The slenderness ratio
is
(1)(15.26)(12) 61.9
2.96
kl
r
= =
(k has been conservatively taken as 1.0, but is actually lower because of restraint at the ends.)
Using AISC LRFD E24 for Fy = 46 ksi:
(1)(125..9266 12) 295,0000 0.817
y
c
kl F
r E
.
p p
= = × =
(0.658 c2 ) (0.6580.8172 )(46) 34.8 ksi
cr y F= .F= =
The design strength of the brace as an axial compression element is:
Pbr = fc AgFcr = (0.85)(17.4)(34.8) = 514 kips
AISC Seismic Sec. 15.6a [15.6] requires that the design axial and flexural strength of the braces be those
resulting from the expected nominal shear strength of the link (Vn) increased by Ry and a factor of 1.25.
Chapter 5, Structural Steel Design
583
Thus, the factored Vn is equal to (193.5 kips)(1.1)(1.25) = 266 kips. The shear in the link, determined
from elastic analysis, is 85.2 kips. Thus, the increase is 266/85.2 = 3.12. Let us now determine the
design values for brace axial force and moments by increasing the values determined from the elastic
analysis by the same factor:
Design Pbrace = (3.12)(120) = 374 kips
Design Mtop = (3.12)(15.5) = 48.4 ftkips
Design Mbot = (3.12)(9.5) = 29.6 ftkips
The design strength of the brace, 514 kips, exceeds the design demand of 374 kips, so the brace is
adequate for axial loading. However, the brace must also be checked for combined axial and flexure
using AISC LRFD Chapter H. For axial demandtocapacity ratio greater than 0.20, axial and flexure
interaction is governed by AISC LRFD H11a:
8 1.0
9
u u
n b n
P M
fP fM
+ =
. .
. .
. .
where
Pu = 374 kips
Pn = 514 kips
Mn = ZFy = (105)(50) = 5250 in.kips
The flexural demand, Mu, is computed in accordance with AISC LRFD Chapter C and must account for
second order effects. For a braced frame only two stories high and having several bays, the required
flexural strength in the brace to resist lateral translation of the frame only (Mlt) is negligible. Therefore,
the required flexural strength is computed from AISC LRFD C11 as:
Mu= B1Mnt
where Mnt = 48.4 ftkips as determined above and, per AISC LRFD C12:
1 1
m 1.0
u e
B C
P P
=

=
2 2 s (17.4)(46) 1,199 kip
(0.817)
g y
e
c
A F
P
.
= = =
1
2
0.6 0.4 0.6 0.4 29.6 0.36
48.4
Cm M
M
=  .. ..=  .. ..=
. . . .
Therefore,
FEMA 451, NEHRP Recommended Provisions: Design Examples
584
1
(0.36)(1 1 34784.4) (0.52)(48.4) 25.3 ftkips
1,199
nt
nt
m
u
u
e
M BM C MP
P
= = = = =
 
and
OK
98 (0.8357)(4514)89(0(2.95).(34),(81320) ) 0.92 < 1.00
u u
n bn
P M
fP fM
. . . .
+. . +. .
. . . .
= =
The design of the brace is satisfactory.
5.3.4.3 Brace Connections at Top of Brace
AISC Seismic Sec. 15.6 requires that, like the brace itself, the connection of the brace to the girder be
designed to remain elastic at yield of the link. The required strength of the bracetobeam connection
must be at least as much as the required strength of the brace. Because there is a moment at the top of the
brace, the connections must also be designed as a fully restrained moment connection. The beam, link,
and brace centerlines intersect at a common work point, and no part of this connection shall extend over
the link length.
The tube may be attached to the girder with a gusset plate welded to the bottom flange of the girder and to
the tube with fillet welds. The design of the gusset and connecting welds is conventional except that
cutting the gusset short of the link may require adding a flange. (Such a flange is shown ine Figure 5.3
5.) Adding a similar flange on the other side of the brace will keep the joint compact. In such a case, it
may be required, or at least desirable, to add another stiffener to the beam opposite the flange on the
gusset. It also should be remembered that the axial force in the brace may be either tension or
compression reflecting the reversal in seismic motions.
In addition to the design of the gusset and the connecting welds, a check should be made of stiffener
requirements on the beam web opposite the gusset flanges (if any) and the panel zone in the beam web
above the connection. All of these calculations are conventional and need no explanation here. Details of
the link and adjacent upper brace connection are shown in Figure 5.35.
Chapter 5, Structural Steel Design
585
C.P. C.P.
Plate 3
8"x31
4"
each side
Plate 1"x
both sides
2nd floor
or roof
(not shown) Equal Equal Equal
Link: 3'0"
W16x57
Angle brace
(typical)
Plate 3
8"x31
4"
stiffener
each side
Plate
1"x7"
TS 8x8x5 Gusset plate 1"
8
Gussets shall not
influence link
in this zone
Figure 5.35 Link and upper brace connection (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).
5.3.4.4 Brace Connections at Bottom of Brace
These braces are concentric at their lower end, framing into the columngirder intersection in a
conventional manner.
The design of the gusset plate and welds is conventional. Details of a lower brace connection are shown
in Figure 5.36. In order to be able to use R = 8, moment connections are required at the ends of the link
beams (at the roof and second floor levels). Moment connections could be used, but are not required,
outside of the EBF (e.g., the left beam in Figure 5.36) or at the bottom of the brace at the first floor (e.g.,
the right beam in Figure 5.36 if it is at the first floor level) . The beam on the left in Figure 5.36 could
be a collector. If so, the connection must carry the axial load (force from floor deck to collector) that is
being transferred through the beam to column connection to the link beam on the right side, as well as
beam vertical loads.
FEMA 451, NEHRP Recommended Provisions: Design Examples
586
1" Gusset plate
TS 8x8x5
8"
W.P.
W16x57
Beam framing to column web
(not shown)
Floor beam
with pinned
connection
2nd floor or
1st floor
(not shown)
CJP
Figure 5.36 Lower brace connections (1.0 in. = 25.4 mm).
5.3.4.5 Beam and Column Design
Refer to AISC Seismic Sec. 15.6 for design of the beam outside the link and AISC Seismic Sec. 15.8 for
design of the columns. The philosophy is very similar to that illustrated for the brace: the demand
becomes the forces associated with expected yield of the link.. Although the moment and shear are less in
the beam than in the link, the axial load is substantially higher.