FEMA P440A/ June 2009
Effects of Strength and Stiffness Degradation on
Seismic Response
Prepared by
APPLIED TECHNOLOGY COUNCIL
201 Redwood Shores Parkway, Suite 240
Redwood City, California 94065
www.ATCouncil.org
Prepared for
FEDERAL EMERGENCY MANAGEMENT AGENCY
Department of Homeland Security (DHS)
Michael Mahoney, Project Officer
Robert D. Hanson, Technical Monitor
Washington, D.C.
ATC MANAGEMENT AND OVERSIGHT
Christopher Rojahn (Project Executive)
Jon A. Heintz (Project Quality Control Monitor)
William T. Holmes (Project Tech. Monitor)
PROJECT MANAGEMENT COMMITTEE
Craig Comartin (Project Technical Director)
Eduardo Miranda
Michael Valley
PROJECT REVIEW PANEL
Kenneth Elwood
Subhash Goel
Farzad Naeim
CONSULTANT
Dimitrios Vamvatsikos
Notice
Any opinions, findings, conclusions, or recommendations expressed in this publication do not necessarily
reflect the views of the Applied Technology Council (ATC), the Department of Homeland Security (DHS), or
the Federal Emergency Management Agency (FEMA). Additionally, neither ATC, DHS, FEMA, nor any of
their employees, makes any warranty, expressed or implied, nor assumes any legal liability or responsibility
for the accuracy, completeness, or usefulness of any information, product, or process included in this
publication. Users of information from this publication assume all liability arising from such use.
FEMA P440A Foreword iii
Foreword
One of the primary goals of the Federal Emergency Management Agency
(FEMA) and the National Earthquake Hazards Reduction Program (NEHRP)
is to encourage design and construction practices that address the earthquake
hazard and minimize the potential damage resulting from that hazard. This
document, Effects of Strength and Stiffness on Degradation on Seismic
Response (FEMA P440A), is a follow-on publication to Improvement of
Nonlinear Static Seismic Analysis Procedures (FEMA 440). It builds on
another FEMA publication addressing the seismic retrofit of existing
buildings, the Prestandard and Commentary for Seismic Rehabilitation of
Buildings (FEMA 356) and the subsequent publication, ASCE/SEI Standard
41-06 Seismic Rehabilitation of Existing Buildings (ASCE 41).
The goal of FEMA 440 was improvement of nonlinear static analysis
procedures, as depicted in FEMA 356 and ASCE 41, and development of
guidance on when and how such procedures should be used. It was a
resource guide for capturing the current state of the art in improved
understanding of nonlinear static procedures, and for generating future
improvements to those products. One of the recommendations to come out
of that work was to fund additional studies of cyclic and in-cycle strength
and stiffness degradation, and their impact on response and response
stability.
This publication provides information that will improve nonlinear analysis
for cyclic response, considering cyclic and in-cycle degradation of strength
and stiffness. Recent work has demonstrated that it is important to be able to
differentiate between cyclic and in-cycle degradation in order to more
accurately model degrading behavior, while current practice only recognizes
cyclic degradation, or does not distinguish between the two. The material
contained within this publication is expected to improve nonlinear modeling
of structural systems, and ultimately make the seismic retrofit of existing
hazardous buildings more cost-effective.
This publication reaffirms FEMA’s ongoing efforts to improve the seismic
safety of new and existing buildings nationwide. This project is an excellent
example of the interagency cooperation that is made possible through the
NEHRP. FEMA is proud to have sponsored the development of this resource
document through the Applied Technology Council (ATC), and is grateful
iv Foreword FEMA P440A
for work done by the Project Technical Director, Craig Comartin, the Project
Management Committee, the Project Review Panel, the Project Working
Group, and all other contributors who made this publication possible. All
those who participated are listed at the end of this document, and FEMA
appreciates their involvement.
Federal Emergency Management Agency
FEMA P440A Preface v
Preface
In September 2004 the Applied Technology Council (ATC) was awarded a
“Seismic and Multi-Hazard Technical Guidance Development and Support”
contract (HSFEHQ-04-D-0641) by the Federal Emergency Management
Agency (FEMA) to conduct a variety of tasks, including one entitled
“Advanced Seismic Analysis Methods – Resolution of Issues” (ATC-62
Project). The purpose of this project was to resolve a series of difficult
technical issues that were identified during the preparation of the FEMA 440
report, Improvement of Nonlinear Static Seismic Analysis Procedures
(FEMA, 2005).
FEMA 440 was funded by FEMA to develop improvements to nonlinear
static analysis procedures contained in the FEMA 356 Prestandard and
Commentary for the Seismic Rehabilitation of Buildings (FEMA, 2000), and
the ATC-40 Report, Seismic Evaluation and Retrofit of Concrete Buildings
(ATC, 1996). Unresolved technical issues identified in FEMA 440 included
the need for additional guidance and direction on: (1) component and global
modeling to consider nonlinear degrading response; (2) soil and foundationstructure
interaction modeling; and (3) simplified nonlinear multiple-degreeof-
freedom modeling.
Of these issues, this project has investigated nonlinear degrading response
and conducted limited initial studies on multiple-degree-of-freedom effects.
Work has included an extensive literature search and review of past studies
on nonlinear strength and stiffness degradation, and review of available
hysteretic models for capturing degrading strength and stiffness behavior. To
supplement the existing body of knowledge, focused analytical studies were
performed to explore the effects of nonlinear degradation on structural
response. This report presents the findings and recommendations resulting
from these efforts.
ATC is indebted to the members of the ATC-62 Project Team who
participated in the preparation of this report. Direction of technical activities,
review, and development of detailed recommendations were performed by
the Project Management Committee, consisting of Craig Comartin (Project
Technical Director), Eduardo Miranda, and Michael Valley. Literature
reviews and focused analytical studies were conducted by Dimitrios
Vamvatsikos. Technical review and comment at critical developmental
vi Preface FEMA P440A
stages were provided by the Project Review Panel, consisting of Kenneth
Elwood, Subhash Goel, and Farzad Naeim. A workshop of invited experts
was convened to obtain feedback on preliminary findings and
recommendations, and input from this group was instrumental in shaping the
final product. The names and affiliations individuals who contributed to this
work are included in the list of Project Participants provided at the end of this
report.
ATC also gratefully acknowledges Michael Mahoney (FEMA Project
Officer), Robert Hanson (FEMA Technical Monitor), and William Holmes
(ATC Project Technical Monitor) for their input and guidance in the
preparation of this report, Peter N. Mork for ATC report production services,
and David Hutchinson as ATC Board Contact.
Jon A. Heintz Christopher Rojahn
ATC Director of Projects ATC Executive Director
FEMA P440A Executive Summary vii
Executive Summary
Much of the nation’s work regarding performance-based seismic design has
been funded by the Federal Emergency Management Agency (FEMA), under
its role in the National Earthquake Hazards Reduction Program (NEHRP).
Prevailing practice for performance-based seismic design is based on FEMA
273, NEHRP Guidelines for the Seismic Rehabilitation of Buildings (FEMA,
1997) and its successor documents, FEMA 356, Prestandard and
Commentary for the Seismic Rehabilitation of Buildings (FEMA, 2000), and
ASCE/SEI Standard 41-06, Seismic Rehabilitation of Existing Buildings
(ASCE, 2006b). This series of documents has been under development for
over twenty years, and has been increasingly absorbed into engineering
practice over that period.
The FEMA 440 report, Improvement of Nonlinear Static Seismic Analysis
Procedures (FEMA, 2005), was commissioned to evaluate and develop
improvements to nonlinear static analysis procedures used in prevailing
practice. Recommendations contained within FEMA 440 resulted in
immediate improvement in nonlinear static analysis procedures, and were
incorporated in the development of ASCE/SEI 41-06. However, several
difficult technical issues remained unresolved.
1. Project Objectives
The Applied Technology Council (ATC) was commissioned by FEMA under
the ATC-62 Project to further investigate the issue of component and global
response to degradation of strength and stiffness. Using FEMA 440 as a
starting point, the objectives of the project were to advance the understanding
of degradation and dynamic instability by:
. Investigating and documenting currently available empirical and
theoretical knowledge on nonlinear cyclic and in-cycle strength and
stiffness degradation, and their affects on the stability of structural
systems
. Supplementing and refining the existing knowledge base with focused
analytical studies
viii Executive Summary FEMA P440A
. Developing practical suggestions, where possible, to account for
nonlinear degrading response in the context of current seismic analysis
procedures.
This report presents the findings and conclusions resulting from the literature
search and focused analytical studies, and provides recommendations that
can be used to improve both nonlinear static and nonlinear response history
analysis modeling of strength and stiffness degradation for use in
performance-based seismic design.
2. Literature Review
Past research has shown that in-cycle strength and stiffness degradation are
real phenomena, and recent investigations confirm that the effects of in-cycle
strength and stiffness degradation are critical in determining the possibility of
lateral dynamic instability.
The body of knowledge is dominated by studies conducted within the last 20
years; however, relevant data on this topic extends as far back as the 1940s.
A summary of background information taken from the literature is provided
in Chapter 2. A comprehensive collection technical references on this
subject is provided in Appendix A.
3. Focused Analytical Studies
To supplement the existing body of knowledge, focused analytical studies
were performed using a set of eight nonlinear springs representing different
types of inelastic hysteretic behavior. These basic spring types were used to
develop 160 single-spring systems and 600 multi-spring systems with
differing characteristics. Each system was subjected to incremental dynamic
analysis with 56 ground motion records scaled to multiple levels of
increasing intensity. The result is an extensive collection of data on
nonlinear degrading response from over 2.6 million nonlinear response
history analyses on single- and multi-spring systems.
Development of single- and multi-spring models is described in Chapter 3,
analytical results are summarized in Chapter 4, and sets of analytical data are
provided in the appendices. A Microsoft Excel visualization tool that was
developed to view all available data from multi-spring studies is included on
the CD accompanying this report.
FEMA P440A Executive Summary ix
4. Comparison with FEMA 440 Limitations on
Strength for Lateral Dynamic Instability
In FEMA 440, a minimum strength requirement (Rmax) was developed as an
approximate measure of the need to further investigate the potential for
lateral dynamic instability caused by in-cycle strength degradation and Pdelta
effects. To further investigate correlation between Rmax and lateral
dynamic instability, the results of this equation were compared to quantile
incremental dynamic analysis (IDA) curves for selected multi-spring systems
included in this investigation. Results indicate that values predicted by the
FEMA 440 equation for Rmax are variable, but generally plot between the
median and 84th percentile results for lateral dynamic instability of the
systems investigated. Observed trends indicate that an improved equation, in
a form similar to Rmax, could be developed as a more accurate (less variable)
predictor of lateral dynamic instability for use in current nonlinear static
analysis procedures.
5. Findings, Conclusions, and Recommendations
Findings, conclusions, and recommendations resulting from the literature
review and focused analytical studies of this investigation are collected and
summarized in Chapter 5, grouped into the following categories:
. Findings related to improved understanding of nonlinear degrading
response and judgment in implementation of nonlinear analysis results in
engineering practice.
. Recommended improvements to current nonlinear
analysis procedures
. Suggestions for further study
6. Findings Related to Improved Understanding and
Judgment
Results from focused analytical studies were used to identify predominant
characteristics of median incremental dynamic analysis (IDA) curves and
determine the effects of different degrading behaviors on the dynamic
stability of structural systems. Observed practical ramifications from these
studies are summarized below:
. Behavior of real structures can include loss of vertical-load-carrying
capacity at lateral displacements that are significantly smaller than those
associated with sidesway collapse. Use of the findings of this
investigation with regard to lateral dynamic instability (sidesway
x Executive Summary FEMA P440A
collapse) in engineering practice should include consideration of possible
vertical collapse modes that could be present in the structure under
consideration.
. Historically, the term “backbone curve” has referred to many different
things. For this reason, two new terms have been introduced to
distinguish between different aspects of hysteretic behavior. These are
the force-displacement capacity boundary, and cyclic envelope.
. Nonlinear component parameters should be based on a forcedisplacement
capacity boundary, rather than a cyclic envelope.
Determining the force-displacement capacity boundary from test results
using a single cyclic loading protocol can result in overly conservative
predictions of maximum displacement.
. Observed relationships between selected features of the forcedisplacement
capacity boundary and the resulting characteristics of
median IDA curves support the conclusion that the nonlinear dynamic
response of a system can be correlated to the parameters of the forcedisplacement
capacity boundary of that system. Of particular interest is
the relationship between global deformation demand and the intensity of
the ground motion at lateral dynamic instability (collapse). Results
indicate that it is possible to use nonlinear static procedures to estimate
the potential for lateral dynamic instability of systems exhibiting in-cycle
degradation.
. It is important to consider the dependence on period of vibration in
conjunction with the effects of other parameters identified in this
investigation. The generalized effect of any one single parameter can be
misleading.
. It is important to recognize the level of uncertainty that is inherent in
nonlinear analysis, particularly regarding variability in response due to
ground motion uncertainty.
. In most cases the effects of in-cycle strength degradation dominate the
nonlinear dynamic behavior of a system. This suggests that in many
cases the effects of cyclic degradation can be neglected.
. Two situations in which the effects of cyclic degradation were observed
to be important include: (1) short period systems; and (2) systems with
very strong in-cycle strength degradation effects (very steep negative
slopes and very large drops in lateral strength).
FEMA P440A Executive Summary xi
7. Improved Equation for Evaluating Lateral
Dynamic Instability
An improved estimate for the strength ratio at which lateral dynamic
instability might occur (Rdi) was developed based on nonlinear regression of
the extensive volume of data generated during this investigation. In
performing this regression, results were calibrated to the median response of
the SDOF spring systems studied in this investigation. Since the proposed
equation for Rdi has been calibrated to median response, use of this equation
could eliminate some of the conservatism inherent in the current Rmax
limitation on use of nonlinear static procedures. Calibrated using the
extensive volume of data generated during this investigation, use of this
equation could improve the reliability of current nonlinear static procedures
with regard to cyclic and in-cycle degradation.
Median response, however, implies a fifty percent chance of being above or
below the specified value. Use of Rdi in engineering practice should consider
whether or not a median predictor represents an appropriate level of safety
against the potential for lateral dynamic instability. If needed, a reduction
factor could be applied to the proposed equation for Rdi to achieve a higher
level of safety on the prediction of lateral dynamic instability.
8. Simplified Nonlinear Dynamic Analysis Procedure
Focused analytical studies comparing force-displacement capacity
boundaries to incremental dynamic analysis results led to the concept of a
simplified nonlinear dynamic analysis procedure. In this procedure, a
nonlinear static analysis is used to generate an idealized force-deformation
curve (i.e., static pushover curve), which is then used as a force-displacement
capacity boundary to constrain the hysteretic behavior of an equivalent
SDOF oscillator. This SDOF oscillator is then subjected to incremental
dynamic analysis, or approximate IDA results are obtained using the open
source software tool, Static Pushover 2 Incremental Dynamic Analysis,
SPO2IDA (Vamvatsikos and Cornell 2006). A Microsoft Excel version of
the SPO2IDA application is included on the CD accompanying this report.
The procedure is simplified because only a SDOF oscillator is subjected to
nonlinear dynamic analysis. Further simplification is achieved through the
use of SPO2IDA, which avoids the computational effort associated with
incremental dynamic analysis. This simplified procedure is shown to have
several advantages over nonlinear static analysis procedures. Use of the
procedure is explained in more detail in the example application contained in
Appendix F.
xii Executive Summary FEMA P440A
9. Application of Results to Multiple-Degree-of-
Freedom Systems
Multi-story buildings are more complex dynamic systems whose seismic
response is more difficult to estimate than that of SDOF systems. Recent
studies have suggested that it may be possible to estimate the collapse
capacity of multiple-degree-of-freedom (MDOF) systems through dynamic
analysis of equivalent SDOF systems. As part of the focused analytical
work, preliminary studies of MDOF systems were performed. Results
indicate that many of the findings for SDOF systems in this investigation
(e.g., the relationship between force-displacement capacity boundary and
IDA curves; the equation for Rdi) may be applicable to MDOF systems.
Results of MDOF investigations are summarized in Appendix G. More
detailed study of the application of these results to MDOF systems is
recommended, and additional investigations are planned under a project
funded by the National Institute of Standards and Technology (NIST).
10. Concluding Remarks
Using FEMA 440 as a starting point, this investigation has advanced the
understanding of degradation and dynamic instability by:
. Investigating and documenting currently available empirical and
theoretical knowledge on nonlinear cyclic and in-cycle strength and
stiffness degradation, and their affects on the stability of structural
systems
. Supplementing and refining the existing knowledge base with focused
analytical studies
Results from this investigation have confirmed conclusions regarding
degradation and dynamic instability presented in FEMA 440, provided
updated information on modeling to differentiate between cyclic and in-cycle
strength and stiffness degradation, and linked nonlinear dynamic response to
major characteristics of component and system degrading behavior. This
information will ultimately improve nonlinear modeling of structural
components, improve the characterization of lateral dynamic instability, and
reduce conservatism in current analysis procedures making it more costeffective
to strengthen existing buildings for improved seismic resistance in
the future.
FEMA P440A Contents xiii
Table of Contents
Foreword ........................................................................................................ iii
Preface .............................................................................................................v
Executive Summary ...................................................................................... vii
List of Figures .............................................................................................. xix
List of Tables .......................................................................................... xxxvii
1. Introduction ........................................................................................... 1-1
1.1. Project Objectives .......................................................................... 1-2
1.2. Scope of Investigation ................................................................... 1-3
1.2.1. Literature Review ............................................................. 1-3
1.2.2. Focused Analytical Studies .............................................. 1-4
1.3. Report Organization and Content .................................................. 1-5
2. Background Concepts ............................................................................ 2-1
2.1. Effects of Hysteretic Behavior on Seismic Response .................... 2-1
2.1.1. Elasto-Plastic Behavior .................................................... 2-2
2.1.2. Strength-Hardening Behavior .......................................... 2-3
2.1.3. Stiffness-Degrading Behavior .......................................... 2-4
2.1.4. Pinching Behavior ............................................................ 2-6
2.1.5. Cyclic Strength Degradation ............................................ 2-7
2.1.6. Combined Stiffness Degradation and Cyclic Strength
Degradation ...................................................................... 2-9
2.1.7. In-Cycle Strength Degradation ......................................... 2-9
2.1.8. Differences Between Cyclic and In-Cycle Strength
Degradation .................................................................... 2-10
2.2. Concepts and Terminology .......................................................... 2-13
2.2.1. Force-Displacement Capacity Boundary........................ 2-13
2.2.2. Cyclic Envelope ............................................................. 2-16
2.2.3. Influence of Loading Protocol on the Cyclic
Envelope ......................................................................... 2-17
2.2.4. Relationship between Loading Protocol, Cyclic
Envelope, and Force-Displacement Capacity
Boundary ........................................................................ 2-19
3. Development of Single-Degree-of-Freedom Models for Focused
Analytical Studies .................................................................................. 3-1
3.1. Overview of Focused Analytical Studies ...................................... 3-1
3.1.1. Purpose ............................................................................. 3-1
3.1.2. Process .............................................................................. 3-1
3.1.3. Incremental Dynamic Analysis Procedure ....................... 3-3
3.1.4. Ground Motion Records ................................................... 3-6
3.1.5. Analytical Models ............................................................ 3-8
3.2. Single-Spring Models .................................................................... 3-9
xiv Contents FEMA P440A
3.2.1. Springs 1a and 1b – Typical Gravity Frame Systems ..... 3-13
3.2.2. Springs 2a and 2b – Non-Ductile Moment Frame
Systems ........................................................................... 3-15
3.2.3. Springs 3a and 3b – Ductile Moment Frame Systems .... 3-17
3.2.4. Springs 4a and 4b – Stiff, Non-Ductile Systems ............ 3-19
3.2.5. Springs 5a and 5b – Stiff, Highly-Pinched Non-Ductile
Systems ........................................................................... 3-21
3.2.6. Springs 6a and 6b – Elastic-Perfectly-Plastic
Systems ........................................................................... 3-23
3.2.7. Springs 7a and 7b – Limited-Ductility Moment Frame
Systems ........................................................................... 3-24
3.2.8. Springs 8a and 8b – Non-Ductile Gravity Frame
Systems ........................................................................... 3-26
3.3. Multiple Spring Models ............................................................... 3-27
3.3.1. Multi-Spring Combinations of Single-Spring
Systems ........................................................................... 3-28
4. Results from Single-Degree-of-Freedom Focused Analytical
Studies.................................................................................................... 4-1
4.1. Summary of Analytical Results ..................................................... 4-1
4.2. Observations from Single-Spring Studies ...................................... 4-1
4.3. Characteristics of Median IDA Curves .......................................... 4-2
4.3.1. Dependence on Period of Vibration ................................. 4-4
4.3.2. Dispersion in Response .................................................... 4-4
4.4. Influence of the Force-Displacement Capacity
Boundary ........................................................................................ 4-5
4.4.1. Post-Yield Behavior and Onset of Degradation ............... 4-9
4.4.2. Slope of Degradation ...................................................... 4-10
4.4.3. Ultimate Deformation Capacity ...................................... 4-11
4.4.4. Degradation of the Force-Displacement Capacity
Boundary (Cyclic Degradation) ...................................... 4-12
4.5. Observations from Multi-Spring Studies ..................................... 4-14
4.5.1. Normalized versus Non-Normalized Results ................. 4-14
4.5.2. Comparison of Multi-Spring Force-Displacement
Capacity Boundaries ....................................................... 4-15
4.5.3. Influence of the Combined Force-Displacement
Capacity Boundary in Multi-Spring Systems ................. 4-16
4.5.4. Effects of the Lateral Strength of Multi-Spring
Systems ........................................................................... 4-19
4.5.5. Effects of Secondary System Characteristics ................. 4-21
4.6. Comparison with FEMA 440 Limitations on Strength for
Lateral Dynamic Instability ......................................................... 4-23
4.6.1. Improved Equation for Evaluating Lateral Dynamic
Instability ........................................................................ 4-25
5. Conclusions and Recommendations ...................................................... 5-1
5.1. Findings Related to Improved Understanding and
Judgment ........................................................................................ 5-2
5.1.1. Sidesway Collapse versus Vertical Collapse .................... 5-2
FEMA P440A Contents xv
5.1.2. Relationship between Loading Protocol, Cyclic
Envelope, and Force-Displacement Capacity
Boundary .......................................................................... 5-2
5.1.3. Characteristics of Median IDA Curves ............................ 5-5
5.1.4. Dependence on Period of Vibration ................................. 5-6
5.1.5. Dispersion in Response .................................................... 5-7
5.1.6. Influence of the Force-Displacement Capacity
Boundary .......................................................................... 5-8
5.1.7. Cyclic Degradation of the Force-Displacement
Capacity Boundary ......................................................... 5-10
5.1.8. Effects of Secondary System Characteristics ................. 5-11
5.1.9. Effects of Lateral Strength ............................................. 5-12
5.2. Recommended Improvements to Current Nonlinear Analysis
Procedures ................................................................................... 5-13
5.2.1. Current Nonlinear Static Procedures .............................. 5-13
5.2.2. Clarification of Terminology and Use of the Force-
Displacement Capacity Boundary for Component
Modeling ........................................................................ 5-14
5.2.3. Improved Equation for Evaluating Lateral Dynamic
Instability ........................................................................ 5-16
5.2.4. Simplified Nonlinear Dynamic Analysis Procedure ...... 5-18
5.3. Suggestions for Further Study ..................................................... 5-20
5.3.1. Application of Results to Multiple-Degree-of-Freedom
Systems .......................................................................... 5-20
5.3.2. Development of Physical Testing Protocols for
Determination of Force-Displacement Capacity
Boundaries ...................................................................... 5-20
5.3.3. Development and Refinement of Tools for
Approximate Nonlinear Dynamic Analysis ................... 5-21
5.4. Concluding Remarks ................................................................... 5-21
Appendix A: Detailed Summary of Previous Research .............................. A-1
A.1. Summary of the Development of Hysteretic Models ................... A-1
A.1.1. Non-Deteriorating Models .............................................. A-1
A.1.2. Piecewise Linear Deteriorating Models .......................... A-2
A.1.3. Smooth Deteriorating Hysteretic Models ........................ A-8
A.1.4. Hysteretic Models for Steel Braces ................................. A-9
A.2. Detailed Summaries of Relevant Publications ........................... A-14
A.2.1. Instability of Buildings During Seismic Response ....... A-16
A.2.2. Seismic Analysis of Older Reinforced Concrete
Columns ........................................................................ A-18
A.2.3. Spectral Displacement Demands of Stiffness- and
Strength-Degrading Systems ......................................... A-22
A.2.4. Dynamic Instability of Simple Structural Systems ....... A-24
A.2.5. Tests to Structural Collapse of Single-Degree-of-
Freedom Frames Subjected to Earthquake Excitations . A-27
A.2.6. Methods to Evaluate the Dynamic Stability of
Structures – Shake Table Tests and Nonlinear
Dynamic Analyses ........................................................ A-30
xvi Contents FEMA P440A
A.2.7. Seismic Performance, Capacity and Reliability of
Structures as Seen Through Incremental Dynamic
Analysis ......................................................................... A-32
A.2.8. Hysteretic Models that Incorporate Strength and
Stiffness Deterioration ................................................... A-36
A.2.9. Global Collapse of Frame Structures Under Seismic
Excitations ..................................................................... A-39
A.2.10. Object-Oriented Development of Strength and Stiffness
Degrading Models for Reinforced Concrete
Structures ....................................................................... A-43
A.2.11. Shake Table Tests and Analytical Studies on the
Gravity Load Collapse of Reinforced Concrete
Frames ........................................................................... A-47
A.2.12. Determination of Ductility Factor Considering
Different Hysteretic Models .......................................... A-50
A.2.13. Effects of Hysteresis Type on the Seismic Response
of Buildings ................................................................... A-53
A.2.14. Performance-Based Assessment of Existing
Structures Accounting For Residual Displacements ..... A-56
A.2.15. Inelastic Spectra for Infilled Reinforced Concrete
Frames ........................................................................... A-61
Appendix B: Quantile IDA Curves for Single-Spring Systems .................. B-1
Appendix C: Median IDA Curves for Multi-Spring Systems versus
Normalized Intensity Measures .................................................... C-1
C.1. Visualization Tool ......................................................................... C-1
Appendix D: Median IDA Curves for Multi-Spring Systems versus
Non-Normalized Intensity Measures ............................................ D-1
D.1. Visualization Tool ......................................................................... D-1
Appendix E: Uncertainty, Fragility, and Probability................................... E-1
E.1. Conversion of IDA Results to Fragilities ...................................... E-1
E.2. Calculation of Annualized Probability .......................................... E-3
Appendix F: Example Application ............................................................... F-1
F.1. Simplified Nonlinear Dynamic Analysis Procedure ...................... F-1
F.2. Example Building .......................................................................... F-2
F.3. Structural Analysis Model ............................................................. F-2
F.4. Nonlinear Static Pushover Analysis ............................................... F-5
F.5. Evaluation of Limit States of Interest ............................................ F-6
F.6. Incremental Dynamic Analysis ...................................................... F-9
F.7. Determination of Probabilities Associated with Limit States
of Interest ..................................................................................... F-10
F.8. Retrofit Strategies ........................................................................ F-12
F.8.1. Addition of a Secondary Lateral System .......................... F-12
F.8.2. Improvement of Primary System Strength and
Ductility............................................................................ F-13
Appendix G: Preliminary Multiple-Degree-of-Freedom System Studies ... G-1
G.1. Four-Story Code-Compliant Reinforced Concrete Building ........ G-2
FEMA P440A Contents xvii
G.2. Eight-Story Code-Compliant Reinforced Concrete Building ....... G-5
G.3. Twelve-Story Code-Compliant Reinforced Concrete
Building ........................................................................................ G-8
G.4. Twenty-Story Code-Compliant Reinforced Concrete
Building ...................................................................................... G-10
G.5. Nine-Story Pre-Northridge Steel Moment-Resisting Frame
Building ........................................................................................ G13
G.6. Twenty-Story Pre-Northridge Steel Moment-Resisting Frame
Building ...................................................................................... G-15
G.7. Summary and Recommendations ............................................... G-17
References and Bibliography ...................................................................... H-1
Project Participants ....................................................................................... I-1
FEMA P440A List of Figures xix
List of Figures
Figure 1-1 Types of degradation defined in FEMA 440 ..................... 1-2
Figure 2-1 Elasto-plastic non-degrading piecewise linear hysteretic
model ................................................................................. 2-2
Figure 2-2 Strength-hardening non-degrading piecewise linear
hysteretic model ................................................................. 2-3
Figure 2-3 Three examples of stiffness-degrading piecewise linear
hysteretic models ............................................................... 2-5
Figure 2-4 Examples of hysteretic models with: (a) moderate
pinching behavior; and (b) severe pinching behavior ........ 2-6
Figure 2-5 Examples of cyclic strength degradation: (a) due to
increasing inelastic displacement; and (b) due to
repeated cyclic displacement ............................................. 2-7
Figure 2-6 Hysteretic models combining stiffness degradation and
cyclic strength degradation: (a) moderate stiffness and
cyclic strength degradation; and (b) severe stiffness and
cyclic strength degradation ................................................ 2-9
Figure 2-7 In-cycle strength degradation ........................................... 2-10
Figure 2-8 Hysteretic behavior for models subjected to Loading
Protocol 1 with: (a) cyclic strength degradation; and
(b) in-cycle degradation ................................................... 2-11
Figure 2-9 Loading Protocol 1 used to illustrate the effects of cyclic
and in-cycle strength degradation .................................... 2-11
Figure 2-10 Loading Protocol 2 used to illustrate the effects of
cyclic and in-cycle strength degradation .......................... 2-11
Figure 2-11 Hysteretic behavior for models subjected to Loading
Protocol 2 with: (a) cyclic strength degradation; and
(b) in-cycle degradation ................................................... 2-12
Figure 2-12 Displacement time histories for models subjected to the
1992 Landers Earthquake with: (a) cyclic strength
degradation; and (b) in-cycle strength degradation .......... 2-13
Figure 2-13 Examples of commonly used force-displacement
capacity boundaries .......................................................... 2-14
Figure 2-14 Interaction between the cyclic load path and the
force-displacement capacity boundary ............................ 2-14
Figure 2-15 Degradation of the force-displacement capacity
boundary .......................................................................... 2-15
Figure 2-16 Example of a cyclic envelope .......................................... 2-16
xx List of Figures FEMA P440A
Figure 2-17 Loading protocols and resulting hysteretic plots for
identical reinforced concrete bridge pier specimens:
(a) Loading Protocol TP01; and (b) Loading Protocol
TP02 ................................................................................. 2-17
Figure 2-18 Loading protocols and resulting hysteretic plots for
identical reinforced concrete bridge pier specimens:
(a) Loading Protocol TP03; and (b) Loading Protocol
TP04 ................................................................................. 2-18
Figure 2-19 Loading protocols and resulting hysteretic plots for
identical reinforced concrete bridge pier specimens:
(a) Loading Protocol TP05; and (b) Loading Protocol
TP06 ................................................................................. 2-18
Figure 2-20 Comparison of cyclic envelopes of reinforced concrete
bridge pier specimens subjected to six different loading
protocols ........................................................................... 2-19
Figure 2-21 Examples of a force-displacement capacity boundary
that is (a) equal to the cyclic envelope, and (b) extends
beyond the cyclic envelope .............................................. 2-20
Figure 2-22 Comparison of hysteretic behavior when the forcedisplacement
capacity boundary is: (a) equal to the
cyclic envelope, and (b) extends beyond the cyclic
envelope ........................................................................... 2-20
Figure 3-1 Features of the force-displacement capacity boundary
varied in focused analytical studies .................................... 3-2
Figure 3-2 Different collapse behaviors: (a) vertical collapse due
to loss of vertical-load-carrying capacity; and
(b) incipient sidesway collapse due to loss of lateralforce-
resisting capacity ...................................................... 3-4
Figure 3-3 Examples depicting incremental dynamic analysis
results; (a) suite of individual IDA curves from 30
different ground motion records; and (b) statistically
derived quantile curves given . or R .................................. 3-5
Figure 3-4 Hysteretic model confined by a force-displacement
capacity boundary ............................................................ 3-10
Figure 3-5 Generic force-displacement capacity boundary used for
all single-spring system models ....................................... 3-11
Figure 3-6 Comparison of eight basic single-spring system
models .............................................................................. 3-11
Figure 3-7 Force-displacement capacity boundaries for Spring 1a
and Spring 1b ................................................................... 3-13
Figure 3-8 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 1a: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-13
Figure 3-9 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 1b: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-14
FEMA P440A List of Figures xxi
Figure 3-10 Hysteretic behavior from experimental tests on beamto-
column shear tab connections ...................................... 3-14
Figure 3-11 Force-displacement capacity boundaries for Spring 2a
and Spring 2b ................................................................... 3-15
Figure 3-12 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 2a: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-15
Figure 3-13 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 2b: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-16
Figure 3-14 Hysteretic behavior from experimental tests on:
(a) pre-Northridge welded steel beam-column
connections; and (b) shear-critical reinforced concrete
columns ............................................................................ 3-16
Figure 3-15 Force-displacement capacity boundaries for Spring 3a
and Spring 3b ................................................................... 3-17
Figure 3-16 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 3a: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-18
Figure 3-17 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 3b: (a) without
cyclic degradation and (b) with cyclic degradation ......... 3-18
Figure 3-18 Hysteretic behavior from experimental tests on post-
Northridge reduced-beam steel moment connections ...... 3-18
Figure 3-19 Force-displacement capacity boundaries for Spring 4a
and Spring 4b ................................................................... 3-19
Figure 3-20 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 4a: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-20
Figure 3-21 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 4b: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-20
Figure 3-22 Hysteretic behavior from experimental tests on steel
concentric braced frames ................................................. 3-20
Figure 3-23 Force-displacement capacity boundaries for Spring 5a
and Spring 5b ................................................................... 3-21
Figure 3-24 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 5a: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-22
Figure 3-25 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 5b: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-22
Figure 3-26 Hysteretic behavior from experimental tests on:
(a) unreinforced masonry walls; and (b) concrete
frames with masonry infill ............................................... 3-22
xxii List of Figures FEMA P440A
Figure 3-27 Force-displacement capacity boundaries for Spring 6a
and Spring 6b ................................................................... 3-23
Figure 3-28 Force-displacement capacity boundary overlaid onto
hysteretic behaviors for: (a) Spring 6a without cyclic
degradation; and (b) Spring 6b without cyclic
degradation ....................................................................... 3-23
Figure 3-29 Force-displacement capacity boundaries for Spring 7a
and Spring 7b ................................................................... 3-24
Figure 3-30 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 7a: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-25
Figure 3-31 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 7b: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-25
Figure 3-32 Hysteretic behavior from experimental tests on lightly
reinforced concrete columns ............................................ 3-25
Figure 3-33 Force-displacement capacity boundaries for Spring 8a
and Spring 8b ................................................................... 3-26
Figure 3-34 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 8a: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-27
Figure 3-35 Initial force-displacement capacity boundary overlaid
onto hysteretic behaviors for Spring 8b: (a) without
cyclic degradation; and (b) with cyclic degradation ........ 3-27
Figure 3-36 Combined force-displacement capacity boundary for
spring 2a +1a (normalized by the strength of
Spring 1a) ......................................................................... 3-29
Figure 3-37 Combined force-displacement capacity boundary for
spring 3a +1a (normalized by the strength of
Spring 1a) ......................................................................... 3-29
Figure 3-38 Combined force-displacement capacity boundary for
spring 4a +1a (normalized by the strength of
Spring 1a) ......................................................................... 3-30
Figure 3-39 Combined force-displacement capacity boundary for
spring 5a +1a (normalized by the strength of
Spring 1a) ......................................................................... 3-30
Figure 3-40 Combined force-displacement capacity boundary for
spring 6a +1a (normalized by the strength of
Spring 1a) ......................................................................... 3-30
Figure 3-41 Combined force-displacement capacity boundary for
spring 7a +1a (normalized by the strength of
Spring 1a) ......................................................................... 3-31
Figure 3-42 Initial force-displacement capacity boundary overlaid
onto hysteretic behavior for: (a) Spring 1x2a+1a; and
(b) Spring 2x2a+1a; both with cyclic degradation ........... 3-31
FEMA P440A List of Figures xxiii
Figure 3-43 Initial force-displacement capacity boundary overlaid
onto hysteretic behavior for: (a) Spring 1x3a+1a; and
(b) Spring 5x2a+1a; both with cyclic degradation ........... 3-32
Figure 3-44 Initial force-displacement capacity boundary overlaid
onto hysteretic behavior for: (a) Spring 9x2a+1a; and
(b) individual Spring 2a; both with cyclic degradation .... 3-32
Figure 4-1 Characteristic segments of a median IDA curve ................ 4-3
Figure 4-2 Characteristic segments of a median IDA curve with a
pseudo-linear segment ....................................................... 4-3
Figure 4-3 Force-displacement capacity boundary and median IDA
curves for Spring 3a with various periods of vibration ...... 4-4
Figure 4-4 Force-displacement capacity boundary and 16th, 50th and
84th percentile IDA curves for Spring 3b with a period of
vibration T=2.0s ................................................................. 4-5
Figure 4-5 Force-displacement capacity boundary and median IDA
curve for Spring 3b with a period of vibration T=2.0s ...... 4-6
Figure 4-6 Force-displacement capacity boundary and median IDA
curve for Spring 2a with a period of vibration T=2.0s ....... 4-6
Figure 4-7 Force-displacement capacity boundary and median IDA
curve for Spring 6a with a period of vibration T=2.0s ....... 4-7
Figure 4-8 Force-displacement capacity boundary and median IDA
curve for Spring 8a with a period of vibration T=2.0s ....... 4-7
Figure 4-9 Relationship between IDA curves and the features of a
typical force-displacement capacity boundary ................... 4-8
Figure 4-10 Effect of post-yield behavior on the collapse capacity
of a system (Springs 2a, 3a and 6a with T=2.0s). .............. 4-9
Figure 4-11 Effect of slope of degradation on the collapse capacity
of a system (Springs 2a and 2b with T=1.0s) ................... 4-10
Figure 4-12 Effect of slope of degradation on the collapse capacity
of a system (Springs 5a and 5b with T=1.0s) ................... 4-11
Figure 4-13 Effect of ultimate deformation capacity on the collapse
capacity of a system (Springs 1a and 1b with T=1.0s). ... 4-11
Figure 4-14 Effect of ultimate deformation capacity on the collapse
capacity of a system (Springs 6a and 6b with T=1.0s). ... 4-12
Figure 4-15 Effect of degradation of the force-displacement
capacity boundary on the collapse capacity of a system
(Spring 3b, T=2.0s, with and without cyclic
degradation). .................................................................... 4-13
Figure 4-16 Effect of degradation of the force-displacement
capacity boundary on the collapse capacity of a system
(Spring 2b, T=0.2s, with and without cyclic
degradation). .................................................................... 4-13
Figure 4-17 Force-displacement capacity boundaries for multi-spring
systems Nx2a+1a and Nx3a+1a, normalized by the yield
strength, Fy, of the combined system .............................. 4-16
xxiv List of Figures FEMA P440A
Figure 4-18 Force-displacement capacity boundaries for multi-spring
systems Nx2a+1a and Nx3a+1a, normalized by the yield
strength of the weakest system ......................................... 4-16
Figure 4-19 Median IDA curves plotted versus the normalized
intensity measure Sa(T,5%)/Say(T,5%) for systems
Nx2a+1a and Nx3a+1a with a mass of 8.87 tons ............. 4-17
Figure 4-20 Median IDA curves plotted versus the normalized
intensity measure Sa(T,5%)/Say(T,5%) for systems
Nx2a+1a and Nx3a+1a with a mass of 35.46 tons ........... 4-18
Figure 4-21 Median IDA curves plotted versus the normalized
intensity measure Sa(T,5%)/Say(T,5%) for systems
Nx3a+1a and Nx3b+1a with a mass of 8.87 tons ............. 4-19
Figure 4-22 Median IDA curves plotted versus the common
intensity measure Sa(1s,5%) for systems Nx2a+1a
and Nx3a+1a with a mass of 8.87 tons ............................. 4-20
Figure 4-23 Median IDA curves plotted versus the common
intensity measure Sa(2s,5%) for systems Nx2a+1a
and Nx3a+1a with a mass of 35.46 tons ........................... 4-20
Figure 4-24 Median IDA curves plotted versus the normalized
intensity measure Sa(T,5%)/Say(T,5%) for systems
Nx2a+1a and Nx2a+1b with a mass of 8.87 tons ............. 4-22
Figure 4-25 Median IDA curves plotted versus the normalized
intensity measure Sa(T,5%)/Say(T,5%) for systems
Nx2a+1a and Nx3a+1a with a mass of 8.87 tons ............. 4-23
Figure 4-26 Idealized force-displacement curve for nonlinear static
analysis (from FEMA 440) .............................................. 4-24
Figure 4-27 Idealization of multi-spring force-displacement capacity
boundaries to estimate effective negative stiffness for
use in the FEMA 440 equation for Rmax. ........................... 4-25
Figure 4-28 Simplified force-displacement boundary for estimating
the median collapse capacity associated with lateral
dynamic instability ........................................................... 4-26
Figure 4-29 Relationship between Equation 4-4 and the segments
of a typical IDA curve ...................................................... 4-26
Figure 4-30 Comparison of Rdi with FEMA 440 Rmax and IDA results
for system 2x2a+1a with T=1.18s .................................... 4-27
Figure 4-31 Comparison of Rdi with FEMA 440 Rmax and IDA results
for system 3x3b+1b with T=1.0s ..................................... 4-28
Figure 4-32 Comparison of Rdi with FEMA 440 Rmax and IDA results
for system 9x3b+1b with T=0.61s ................................... 4-28
Figure 4-33 Comparison of Rdi with FEMA 440 Rmax and IDA results
for system 5x5a+1a with T=1.15s .................................... 4-29
Figure 4-34 Comparison of Rdi with FEMA 440 Rmax and IDA results
for system 5x5a+1a with T=0.58s .................................... 4-29
FEMA P440A List of Figures xxv
Figure 4-35 Comparison of Rdi with FEMA 440 Rmax and IDA results
for system 9x5a+1a with T=0.34s .................................... 4-30
Figure 5-1 Example of a force-displacement capacity boundary ........ 5-3
Figure 5-2 Example of a cyclic envelope ............................................ 5-3
Figure 5-3 Comparison of hysteretic behavior when the forcedisplacement
capacity boundary is: (a) equal to the
cyclic envelope, and (b) extends beyond the cyclic
envelope ............................................................................. 5-4
Figure 5-4 Relationship between IDA curves and the features of a
typical force-displacement capacity boundary ................... 5-5
Figure 5-5 Force-displacement capacity boundary and median IDA
curves for Spring 3a with various periods of vibration ...... 5-7
Figure 5-6 Force-displacement capacity boundary and 16th, 50th and
84th percentile IDA curves for Spring 3b with a period of
vibration T=2.0s ................................................................. 5-7
Figure 5-7 Effect of post-yield behavior on the collapse capacity
of a system (Springs 2a, 3a and 6a with T=2.0s). .............. 5-8
Figure 5-8 Effect of slope of degradation on the collapse capacity
of a system (Springs 2a and 2b with T=1.0s). .................... 5-9
Figure 5-9 Effect of ultimate deformation capacity on the collapse
capacity of a system (Springs 1a and 1b with T=1.0s) ...... 5-9
Figure 5-10 Effect of degradation of the force-displacement
capacity boundary on the collapse capacity of a system
(Spring 3b, T=2.0s, with and without cyclic
degradation) ..................................................................... 5-10
Figure 5-11 Median IDA curves plotted versus the normalized
intensity measure Sa(T,5%)/Say(T,5%) for systems
Nx2a+1a and Nx2a+1b with a mass of 8.87 tons ............ 5-11
Figure 5-12 Force-displacement capacity boundaries and median
IDA curves plotted versus the common intensity
measure Sa(2s,5%) for system Nx3a+1a with a mass of
35.46 tons ......................................................................... 5-12
Figure 5-13 Conceptual force-displacement relationship
(“backbone”) used in ASCE/SEI 41-06 (adapted
from FEMA 356) ............................................................. 5-15
Figure 5-14 Simplified force-displacement boundary for estimating
the median collapse capacity associated with dynamic
instability ......................................................................... 5-17
Figure A-1 Effect of mechanism shape on the monotonic work vs.
amplitude relationship ..................................................... A-17
Figure A-2 Illustration of the definition of stability coefficient:
(a) general load deformation relationship,
(b) elasto-plastic system .................................................. A-17
Figure A-3 The RC column element formulation ............................. A-19
Figure A-4 The hysteretic laws for shear and moment springs ......... A-20
xxvi List of Figures FEMA P440A
Figure A-5 The force (or moment) unbalance is subtracted after an
arbitrary positive stiffness step towards the “correct”
displacement. Very small load steps are needed for
accuracy, even at the SDOF level ................................... A-20
Figure A-6 Observed versus calculated response for a column
specimen SC3 (shear critical). ......................................... A-21
Figure A-7 Observed versus calculated response for a column
specimen 2CLH18 (fails in shear after considerable
flexural deformation). ..................................................... A-21
Figure A-8 (a) Hysteresis law used for the SDOF system and
(b) ratio of degrading to non-degrading displacement
amplification factors for the post-peak stiffness equal
to -1% or -3% of the elastic stiffness .............................. A-23
Figure A-9 Force-displacement characteristics of bilinear systems
considered ....................................................................... A-25
Figure A-10 Effect of period of vibration and post-yield stiffness
on the mean strength ratio at which dynamic instability
is produced ...................................................................... A-26
Figure A-11 (a) Schematic of test setup .............................................. A-28
Figure A-12 Simplified bilinear force deformation model .................. A-28
Figure A-13 Comparison of experimental (left) and analytical
(right) results ................................................................... A-29
Figure A-14 (a) Static pushover curves for the two frames and (b)
modeling of the column plastic hinges in OpenSEES ..... A-31
Figure A-15 The backbone of the studied oscillator ............................ A-33
Figure A-16 The interface of the SPO2IDA tool for moderate
periods ............................................................................. A-34
Figure A-17 Influence of (a) the post-peak and (b) post-yield
stiffness on the median dynamic response of the
oscillator. When the negative segment is the same
then the hardening slope has a negligible effect .............. A-35
Figure A-18 (a) Influence of the load pattern on the pushover curve
shape and (b) the predicted versus actual dynamic
response for various intensity levels using SPO2IDA
and the worst-case pushover for a 9-story steel moment
frame ............................................................................... A-35
Figure A-19 The backbone of the proposed hysteretic model ............. A-37
Figure A-20 Basic rules for peak-oriented hysteretic model ............... A-38
Figure A-21 Pinching hysteretic model: (a) basic model rules; and
(b) modification if reloading deformation is to the
right of break point .......................................................... A-38
Figure A-22 Examples of comparisons between experimental and
analytical results for (a) non-ductile reinforced
concrete column; and (b) plywood shear wall ................. A-38
FEMA P440A List of Figures xxvii
Figure A-23 (a) Backbone curve used for the investigations and
(b) post-peak stiffness cyclic deterioration considered ... A-41
Figure A-24 (a) Effect of the post-peak stiffness to the median
collapse capacity spectra for a peak-oriented model
and (b) the ratio of collapse capacities for different
hysteretic models ............................................................ A-41
Figure A-25 Effect of (a) post-yield slope and (b) reloading
stiffness cyclic deterioration on the collapse capacity .... A-42
Figure A-26 (a) Effect of the beam-hinge hysteretic model on the
median MDOF collapse capacity and (b) the generation
of an equivalent SDOF system by using an auxiliary
backbone curve to incorporate P-.. ................................ A-42
Figure A-27 Idealization of the (a) flexure spring and (b) shear
spring backbones ............................................................. A-45
Figure A-28 (a) Full and (b) half cycle pinching hysteresis for the
shear spring ..................................................................... A-45
Figure A-29 Comparison of calculated versus experimental results
for (a) a moment-critical column and (b) a shear-critical
column ............................................................................ A-46
Figure A-30 Use of Sezen model to estimate (a) shear capacity and
(b) displacement ductility capacity ................................. A-48
Figure A-31 Comparison of the Sezen shear strength model and
the proposed drift capacity model ................................... A-48
Figure A-32 Redefinition of backbone in Elwood’s model after
shear failure is detected ................................................... A-48
Figure A-33 Comparison of calculated versus experimental results
for two shear-critical columns ........................................ A-49
Figure A-34 The parameters investigated: (a) backbone hardening
ratio; (b) unloading/reloading cyclic stiffness
degradation; (c) strength degradation; and (d) degree
of pinching. ..................................................................... A-52
Figure A-35 The effect of (a) cyclic strength degradation and
(b) degree of pinching on the mean R-factor for a
given ductility ................................................................. A-52
Figure A-36 Ratio of maximum displacement for all buildings and
hinge types versus the hinge type 1 (kinematic
hardening, no degradation). ............................................ A-54
Figure A-37 The hysteresis types considered for the beam-hinges ..... A-55
Figure A-38 Hysteretic models used in this investigation that only
have stiffness degradation. (a) Modified-Clough (MC);
(b) Takeda model (TK); and (c) Origin-Oriented model
(O-O) ............................................................................... A-57
Figure A-39 Hysteretic models used in this investigation with
stiffness and cyclic strength degradation. (b) Moderate
Degrading (MSD); and (c) Severely Degrading (SSD). . A-58
xxviii List of Figures FEMA P440A
Figure A-40 Mean ratios of maximum deformation of bilinear to
elastoplastic systems: (a).. = 3%; and (b).. = 5% ........ A-58
Figure A-41 Mean ratio of inelastic displacement demands in
structural degrading and bilinear systems:
(a) SSD-1 model; and (b) SSD-2 model .......................... A-58
Figure A-42 Influence of hysteretic behavior on maximum
deformation for three types of stiffness-degrading
systems: (a) Modified-Clough model; (b) Takeda model;
and (c) Origin-oriented model ......................................... A-59
Figure A-43 The SDOF system: (a) force-displacement envelope;
and (b) mathematic model ............................................... A-62
Figure A-44 The hysteretic behavior of the equivalent SDOF
system .............................................................................. A-62
Figure A-45 The influence of negative slope and residual plateau
on the mean ductility for given R-factor ......................... A-63
Figure B-1 Quantile IDA curves plotted versus Sa(T,5%) for
Spring 1a and Spring 1b with a period of T = 0.5s .......... B-2
Figure B-2 Quantile IDA curves plotted versus Sa(T,5%) for
Spring 1a and Spring 1b with a period of T = 1.0s. .......... B-2
Figure B-3 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 1a and Spring 1b with a period of T = 2.0s. .......... B-2
Figure B-4 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 2a and Spring 2b with a period of T = 0.5s. .......... B-3
Figure B-5 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 2a and Spring 2b with a period of T = 1.0s. .......... B-3
Figure B-6 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 2a and Spring 2b with a period of T = 2.0s. .......... B-3
Figure B-7 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 3a and Spring 3b with a period of T = 0.5s. .......... B-4
Figure B-8 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 3a and Spring 3b with a period of T = 1.0s. .......... B-4
Figure B-9 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 3a and Spring 3b with a period of T = 2.0s. .......... B-4
Figure B-10 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 4a and Spring 4b with a period of T = 0.5s. .......... B-5
Figure B-11 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 4a and Spring 4b with a period of T = 1.0s. .......... B-5
Figure B-12 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 4a and Spring 4b with a period of T = 2.0s. .......... B-5
Figure B-13 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 5a and Spring 5b with a period of T = 0.5s. .......... B-6
Figure B-14 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 5a and Spring 5b with a period of T = 1.0s. .......... B-6
FEMA P440A List of Figures xxix
Figure B-15 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 5a and Spring 5b with a period of T = 2.0s. ........... B-6
Figure B-16 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 6a and Spring 6b with a period of T = 0.5s. ........... B-7
Figure B-17 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 6a and Spring 6b with a period of T = 1.0s. ........... B-7
Figure B-18 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 6a and Spring 6b with a period of T = 2.0s. ........... B-7
Figure B-19 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 7a and Spring 7b with a period of T = 0.5s. ........... B-8
Figure B-20 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 7a and Spring 7b with a period of T = 1.0s. ........... B-8
Figure B-21 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 7a and Spring 7b with a period of T = 2.0s. ........... B-8
Figure B-22 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 8a and Spring 8b with a period of T = 0.5s. ........... B-9
Figure B-23 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 8a and Spring 8b with a period of T = 1.0s. ........... B-9
Figure B-24 Quantile IDA curves plotted versus Sa (T,5%) for
Spring 8a and Spring 8b with a period of T = 2.0s. ........... B-9
Figure C-1 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx2a and Nx2b with mass M=8.87ton. ............................ C-2
Figure C-2 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx2a+1a and Nx2b+1a with mass M=8.87ton. ................ C-2
Figure C-3 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx2a+1b and Nx2b+1b with mass M=8.87ton. ................ C-2
Figure C-4 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx3a and Nx3b with mass M=8.87ton. ............................ C-3
Figure C-5 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx3a+1a and Nx3b+1a with mass M=8.87ton. ................. C-3
Figure C-6 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx3a+1b and Nx3b+1b with mass M=8.87ton. ................. C-3
Figure C-7 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx4a and Nx4b with mass M=8.87ton. ............................ C-4
Figure C-8 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx4a+1a and Nx4b+1a with mass M=8.87ton. ................ C-4
xxx List of Figures FEMA P440A
Figure C-9 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx4a+1b and Nx4b+1b with mass M=8.87ton. ............... C-4
Figure C-10 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx5a and Nx5b with mass M=8.87ton. ............................ C-5
Figure C-11 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx5a+1a and Nx5b+1a with mass M=8.87ton. ................ C-5
Figure C-12 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx5a+1b and Nx5b+1b with mass M=8.87ton. ............... C-5
Figure C-13 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx6a and Nx6b with mass M=8.87ton. ............................ C-6
Figure C-14 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx6a+1a and Nx6b+1a with mass M=8.87ton. ................ C-6
Figure C-15 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx6a+1b and Nx6b+1b with mass M=8.87ton. ............... C-6
Figure C-16 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx7a and Nx7b with mass M=8.87ton. ............................ C-7
Figure C-17 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx7a+1a and Nx7b+1a with mass M=8.87ton. ................ C-7
Figure C-18 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx7a+1b and Nx7b+1b with mass M=8.87ton. ............... C-7
Figure C-19 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx2a and Nx2b with mass M=35.46ton. .......................... C-8
Figure C-20 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx2a+1a and Nx2b+1a with mass M=35.46ton. .............. C-8
Figure C-21 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx2a+1b and Nx2b+1b with mass M=35.46ton. ............. C-8
Figure C-22 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx3a and Nx3b with mass M=35.46ton. .......................... C-9
Figure C-23 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx3a+1a and Nx3b+1a with mass M=35.46ton. .............. C-9
FEMA P440A List of Figures xxxi
Figure C-24 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx3a+1b and Nx3b+1b with mass M=35.46ton. ............. C-9
Figure C-25 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx4a and Nx4b with mass M=35.46ton. ........................ C-10
Figure C-26 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx4a+1a and Nx4b+1a with mass M=35.46ton. ............ C-10
Figure C-27 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx4a+1b and Nx4b+1b with mass M=35.46ton. ............ C-10
Figure C-28 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx5a and Nx5b with mass M=35.46ton. ........................ C-11
Figure C-29 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx5a+1a and Nx5b+1a with mass M=35.46ton. ............ C-11
Figure C-30 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx5a+1b and Nx5b+1b with mass M=35.46ton. ............ C-11
Figure C-31 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx6a and Nx6b with mass M=35.46ton. ........................ C-12
Figure C-32 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx6a+1a and Nx6b+1a with mass M=35.46ton. ............ C-12
Figure C-33 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx6a+1b and Nx6b+1b with mass M=35.46ton. ............ C-12
Figure C-34 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx7a and Nx7b with mass M=35.46ton. ........................ C-13
Figure C-35 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx7a+1a and Nx7b+1a with mass M=35.46ton. ............ C-13
Figure C-36 Median IDA curves plotted versus the normalized
intensity measure R = Sa(T1,5%)/Say(T1,5%) for systems
Nx7a+1b and Nx7b+1b with mass M=35.46ton. ............ C-13
Figure D-1 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx2a and Nx2b with mass
M=8.87ton. ...................................................................... D-2
Figure D-2 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx2a+1a and Nx2b+1a with mass
M=8.87ton. ...................................................................... D-2
xxxii List of Figures FEMA P440A
Figure D-3 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx2a+1b and Nx2b+1b with mass
M=8.87ton. ....................................................................... D-2
Figure D-4 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx3a and Nx3b
with mass M=8.87ton. ...................................................... D-3
Figure D-5 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx3a+1a and Nx3b+1a with mass
M=8.87ton. ........................................................................ D-3
Figure D-6 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx3a+1b and Nx3b+1b with mass
M=8.87ton. ........................................................................ D-3
Figure D-7 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx4a and Nx4b with mass
M=8.87ton. ....................................................................... D-4
Figure D-8 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx4a+1a and Nx4b+1a with mass
M=8.87ton. ....................................................................... D-4
Figure D-9 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx4a+1b and Nx4b+1b with mass
M=8.87ton. ....................................................................... D-4
Figure D-10 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx5a and Nx5b
with mass M=8.87ton. ...................................................... D-5
Figure D-11 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx5a+1a and Nx5b+1a with mass
M=8.87ton. ....................................................................... D-5
Figure D-12 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx5a+1b and Nx5b+1b with mass
M=8.87ton. ....................................................................... D-5
Figure D-13 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx6a and Nx6b
with mass M=8.87ton. ...................................................... D-6
Figure D-14 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx6a+1a and Nx6b+1a with mass
M=8.87ton. ....................................................................... D-6
Figure D-15 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx6a+1b and Nx6b+1b with mass
M=8.87ton. ....................................................................... D-6
Figure D-16 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx7a and Nx7b
with mass M=8.87ton. ...................................................... D-7
Figure D-17 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx7a+1a and Nx7b+1a with mass
M=8.87ton. ....................................................................... D-7
FEMA P440A List of Figures xxxiii
Figure D-18 Median IDA curves plotted versus the intensity measure
Sa(1s,5%) for systems Nx7a+1b and Nx7b+1b with mass
M=8.87ton. ...................................................................... D-7
Figure D-19 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx2a and Nx2b
with mass M=35.46ton. ................................................... D-8
Figure D-20 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx2a+1a and Nx2b+1a with mass
M=35.46ton. .................................................................... D-8
Figure D-21 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx2a+1b and Nx2b+1b with mass
M=35.46ton. .................................................................... D-8
Figure D-22 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx3a and Nx3b
with mass M=35.46ton. .................................................... D-9
Figure D-23 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx3a+1a and Nx3b+1a with mass
M=35.46ton ...................................................................... D-9
Figure D-24 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx3a+1b and Nx3b+1b with mass
M=35.46ton. ..................................................................... D-9
Figure D-25 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx4a and Nx4b
with mass M=35.46ton. ................................................. D-10
Figure D-26 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for (systems Nx4a+1a and Nx4b+1a with mass
M=35.46ton. .................................................................. D-10
Figure D-27 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx4a+1b and Nx4b+1b with mass
M=35.46ton. .................................................................. D-10
Figure D-28 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx5a and Nx5b with mass
M=35.46ton. ................................................................... D-11
Figure D-29 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx5a+1a and Nx5b+1a with mass
M=35.46ton. .................................................................. D-11
Figure D-30 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx5a+1b and Nx5b+1b with mass
M=35.46ton. .................................................................. D-11
Figure D-31 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx6a and Nx6b with mass
M=35.46ton. ................................................................... D-12
Figure D-32 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx6a+1a and Nx6b+1a with mass
M=35.46ton. .................................................................. D-12
xxxiv List of Figures FEMA P440A
Figure D-33 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx6a+1b and Nx6b+1b with mass
M=35.46ton. ................................................................... D-12
Figure D-34 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx7a and Nx7b
with mass M=35.46ton. ................................................... D-13
Figure D-35 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx7a+1a and Nx7b+1a with mass
M=35.46ton. ................................................................... D-13
Figure D-36 Median IDA curves plotted versus the intensity measure
Sa(2s,5%) for systems Nx7a+1b and Nx7b+1b with mass
M=35.46ton. ................................................................... D-13
Figure E-1 IDA results for a single structure subjected to a suite
of ground motions of varying intensities ........................... E-1
Figure E-2 Cumulative distribution plot obtained by fitting a
lognormal distribution to collapse data from IDA
results ................................................................................ E-2
Figure E-3 Conceptual collapse fragility curves for sidesway
(lateral) collapse, vertical collapse, and a combination
of both ............................................................................... E-3
Figure F-1 Example building exterior elevation .................................. F-3
Figure F-2 Example building first floor plan ....................................... F-3
Figure F-3 Force-displacement modeling parameters for:
(a) column components; and (b) wall-like column
components ........................................................................ F-4
Figure F-4 Structural analysis model showing: (a) assumptions;
and (b) distortions .............................................................. F-5
Figure F-5 Pushover curve from nonlinear static analysis ................... F-5
Figure F-6 Uniform hazard spectrum for intensity corresponding to
10% chance of exceedance in 50 years .............................. F-6
Figure F-7 Uniform hazard spectrum for intensities corresponding
to 2% chance of exceedance in 50 years ............................ F-7
Figure F-8 Pushover curve from nonlinear static analysis and two
idealized system force-displacement capacity
boundaries .......................................................................... F-8
Figure F-9 Results of approximate incremental dynamic analysis
using SPO2IDA ................................................................ F-10
Figure F-10 Example building collapse fragilities for loss of verticalload-
carrying capacity (LVCC), lateral dynamic
instability (LDI), and a combination of both .................... F-11
Figure F-11 Hazard curve for representative soft site .......................... F-11
Figure F-12 Revised pushover curve for the example building after
retrofit with a secondary lateral system ............................ F-12
FEMA P440A List of Figures xxxv
Figure F-13 Revised pushover curve for the example building after
retrofit for improved strength and ductility of columns ... F-13
Figure G-1 (a) Monotonic pushover force-deformation curve and
(b) story drifts at a roof drift ratio of the 0.06 in a fourstory
concrete frame building ........................................... G-2
Figure G-2 Tri-linear capacity boundary selected for approximate
analysis ............................................................................. G-3
Figure G-3 Comparison of median collapse capacity for a fourstory
code-compliant concrete frame building
computed using incremental dynamic analysis and
approximate procedures .................................................... G-3
Figure G-4 Effect of selecting an alternate force-displacement
capacity boundary on estimates of median collapse
capacity for a four-story code-compliant concrete
frame building ................................................................... G-4
Figure G-5 Effect of selecting an alternate force-displacement
capacity boundary on estimates of median collapse
capacity for a four-story code-compliant concrete
frame building. .................................................................. G-4
Figure G-6 Incremental dynamic analysis results for a four-story
code-compliant concrete frame building subjected to
80 ground motions ............................................................ G-5
Figure G-7 (a) Monotonic pushover force-deformation curve and
(b) distribution of story drift demands at a roof drift
ratio of 2.6% in an eight-story concrete frame building ... G-6
Figure G-8 Tri-linear capacity boundary selected for approximate
analyses using SPO2IDA .................................................. G-6
Figure G-9 Comparison of median collapse capacity for an eightstory
code-compliant concrete frame building
computed using incremental dynamic analysis and
approximate procedures .................................................... G-7
Figure G-10 Incremental dynamic analysis results for an eight-story
code-compliant concrete frame building subjected to
80 ground motions ............................................................ G-7
Figure G-11 (a) Monotonic pushover force-deformation curve and
(b) distribution of story drift demands at a roof drift
ratio of 2.7% in a twelve-story concrete frame
building ............................................................................. G-8
Figure G-12 Tri-linear capacity boundary selected for approximate
analyses using SPO2IDA .................................................. G-9
Figure G-13 Comparison of median collapse capacity for a twelvestory
code-compliant concrete frame building computed
using incremental dynamic analysis and approximate
procedures ......................................................................... G-9
Figure G-14 Incremental dynamic analysis results for a twelvestory
code-compliant concrete frame building subjected
to 80 ground motions ...................................................... G-10
xxxvi List of Figures FEMA P440A
Figure G-15 (a) Monotonic pushover force-deformation curve and
(b) distribution of story drift demands at a roof drift
ratio of 1.8% in a twelve-story concrete frame
building ........................................................................... G-11
Figure G-16 Tri-linear capacity boundary selected for approximate
analyses using SPO2IDA ................................................. G-11
Figure G-17 Comparison of median collapse capacity for a twentystory
code-compliant concrete frame building
computed using incremental dynamic analysis and
approximate procedures .................................................. G-12
Figure G-18 Incremental dynamic analysis results for a twentystory
code-compliant concrete frame building subjected
to 80 ground motions ...................................................... G-12
Figure G-19 Monotonic pushover force-deformation curve, and
tri-linear approximation, for a nine-story pre-Northridge
steel moment frame building ........................................... G-14
Figure G-20 Comparison of median collapse capacity for a ninestory
pre-Northridge steel moment frame building
computed using incremental dynamic analysis and
approximate procedures .................................................. G-14
Figure G-21 Monotonic pushover force-deformation curve, and
tri-linear approximation, for a twenty-story pre-
Northridge steel moment frame building ........................ G-16
Figure G-22 Comparison of median collapse capacity for a twentystory
pre-Northridge steel moment frame building
computed using incremental dynamic analysis and
approximate procedures .................................................. G-16
FEMA P440A List of Tables xxxvii
List of Tables
Table 3-1 Earthquake Records Used in Focused Analytical Studies
(Both Horizontal Components) .......................................... 3-7
Table 3-2 Force-Displacement Capacity Boundary Control Points
for Single-Spring System Models .................................... 3-12
Table F-1 Mean Annual Frequencies for Collapse Limit States ...... F-10
FEMA P440A 1: Introduction 1-1
Chapter 1
Introduction
Much of the nation’s work regarding performance-based seismic design has
been funded by the Federal Emergency Management Agency (FEMA), under
its role in the National Earthquake Hazards Reduction Program (NEHRP).
Prevailing practice for performance-based seismic design is based on FEMA
273, NEHRP Guidelines for the Seismic Rehabilitation of Buildings (FEMA,
1997) and its successor documents, FEMA 356, Prestandard and
Commentary for the Seismic Rehabilitation of Buildings (FEMA, 2000), and
ASCE/SEI Standard 41-06, Seismic Rehabilitation of Existing Buildings
(ASCE, 2006b). This series of documents has been under development for
over twenty years, and has been increasingly absorbed into engineering
practice over that period.
The FEMA 440 report, Improvement of Nonlinear Static Seismic Analysis
Procedures (FEMA, 2005), was commissioned to evaluate and develop
improvements to nonlinear static analysis procedures used in prevailing
practice. In FEMA 440, deviation between nonlinear static and nonlinear
response history analyses was attributed to a number of factors including: (1)
inaccuracies in the “equal displacement approximation” in the short period
range; (2) dynamic P-delta effects and instability; (3) static load vector
assumptions; (4) strength and stiffness degradation; (5) multi-degree of
freedom effects; and (6) soil-structure interaction effects.
FEMA 440 identified and defined two types of degradation in inelastic
single-degree-of-freedom oscillators. These included cyclic degradation and
in-cycle degradation, as illustrated in Figure 1-1. Cyclic degradation was
characterized by loss of strength and stiffness occurring in subsequent cycles.
In-cycle degradation was characterized by loss of strength and negative
stiffness occurring within a single cycle. This distinction was necessary
because the consequences of cyclic degradation and in-cycle degradation
were observed to be vastly different. In general, systems with cyclic
degradation were shown to have stable dynamic response, while systems
with severe in-cycle degradation were prone to dynamic instability,
potentially leading to collapse.
Recommendations contained within FEMA 440 resulted in immediate
improvement in nonlinear static analysis procedures, and were incorporated
1-2 1: Introduction FEMA P440A
in the development of ASCE/SEI 41-06. However, several difficult technical
issues remained unresolved. These included the need for additional guidance
and direction on: (1) expansion of component and global modeling to include
nonlinear degradation of strength and stiffness; (2) improvement of
simplified nonlinear modeling to include multi-degree of freedom effects;
and (3) improvement of modeling to include soil and foundation structure
interaction effects.
Figure 1-1 Types of degradation defined in FEMA 440.
1.1 Project Objectives
The Applied Technology Council (ATC) was commissioned by FEMA under
the ATC-62 Project to further investigate the issue of component and global
response to degradation of strength and stiffness. Using FEMA 440 as a
starting point, the objectives of the project were to advance the understanding
of degradation and dynamic instability by:
. Investigating and documenting currently available empirical and
theoretical knowledge on nonlinear cyclic and in-cycle strength and
stiffness degradation, and their affects on the stability of structural
systems
. Supplementing and refining the existing knowledge base with focused
analytical studies
. Developing practical suggestions, where possible, to account for
nonlinear degrading response in the context of current seismic analysis
procedures.
The result is an extensive collection of available research on component
modeling of degradation, and a database of analytical results documenting
the effects of a variety of parameters on the overall response of single-
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Interstory Drift
Force
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Interstory Drift
Force
Cyclic strength degradation In-cycle strength degradation
Strength loss occurs during
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Story Drift Ratio
Force
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Story Drift Ratio
Force
Cyclic strength degradation In-cycle strength degradation
Strength loss occurs in subsequent cycles;
not in the same loaing cycle the loading cycle
FEMA P440A 1: Introduction 1-3
degree-of-freedom systems with degrading components. This report presents
the findings and conclusions resulting from focused analytical studies, and
provides recommendations that can be used to improve both nonlinear static
and nonlinear response history analysis modeling of strength and stiffness
degradation for use in performance-based seismic design.
1.2 Scope of Investigation
The scope of the investigative effort included two primary activities. The
first was to assemble and review currently available research on the effects of
degrading nonlinear component properties on structural system response.
The second was to augment this information with supplemental analytical
data, where needed.
1.2.1 Literature Review
Work included an extensive review of existing research on hysteretic models
that have been developed and used for modeling nonlinear response of
structures, with an emphasis on those that have incorporated degradation of
stiffness and strength. The review included theoretical and empirical
investigations that have studied the effect of hysteretic behavior on seismic
response. Interviews with selected researchers were also conducted.
The body of knowledge is dominated by studies conducted within the last 20
years; however, relevant data on this topic extends as far back as the 1940s.
In summary, past research leads to the conclusion that in-cycle strength and
stiffness degradation are real phenomena, and recent investigations confirm
that the effects of in-cycle strength and stiffness degradation are critical in
determining the possibility of lateral dynamic instability.
Only a small number of analytical studies and experimental tests have
considered the dynamic loading effects of in-cycle strength and stiffness
degradation. Most experimental studies to date have only considered
individual components or individual subassemblies, and have not considered
larger systems of components with mixed hysteretic behavior. There are
only a few studies that have considered combined effects of strength,
stiffness, period of vibration together with in-cycle degradation.
A summary of background information taken from the literature is provided
in Chapter 2. A comprehensive collection and summary of technical
references on the development, evolution, and applicability of various
hysteretic models for use in nonlinear structural analysis is provided in
Appendix A.
1-4 1: Introduction FEMA P440A
1.2.2 Focused Analytical Studies
To supplement the current state of knowledge, a program of nonlinear
dynamic focused analytical studies was developed and implemented. The
purpose of this program was to investigate the response of systems
comprised of degrading components, test various characteristics of degrading
component behavior, and identify their effects on the dynamic stability of a
system.
The basis of the focused analytical studies is a set of eight nonlinear springs
representing different types of inelastic hysteretic behavior:
. Typical gravity frame (e.g., steel)
. Non-ductile moment frame (e.g., steel or concrete)
. Ductile moment frame (e.g., steel or concrete)
. Stiff non-ductile system (e.g., concentric braced frame)
. Stiff and highly pinched non-ductile system (e.g., infill wall)
. Idealized elastic-perfectly-plastic system (for comparison)
. Limited-ductility moment frame (e.g., concrete)
. Non-ductile gravity frame (e.g., concrete)
Each spring was defined with a hysteretic model based on information
available in the literature. While intended to be representative of realistic
degrading response that has been observed to occur in some structural
components, these idealized springs are not intended to be a detailed
characterization of the mechanical behavior exclusively associated with any
one specific structural component or structural assembly.
Individual springs were combined to approximate the behavior of more
complex systems consisting of a mixture of subassemblies having different
hysteretic characteristics. Combinations included gravity frame components
working with various different primary lateral-force resisting components to
approximate a range of possible building types encountered in practice. For
each such combined system, variations in the relative contribution of
individual springs to the initial stiffness and maximum lateral strength over a
range of periods were considered. Development of single-degree-of-freedom
(SDOF) models used in focused analytical studies is described in Chapter 3.
Extensive parametric studies varying the strength, stiffness, period, and postelastic
properties were conducted on each component spring and combined
system using Incremental Dynamic Analysis (IDA). Results of over 2.6
million nonlinear response history analyses are summarized in Chapter 4.
FEMA P440A 1: Introduction 1-5
A limited study of multiple-degree-of-freedom (MDOF) systems was also
conducted. This effort compared the results of nonlinear dynamic analyses
of MDOF buildings performed by others to analytical results for SDOF
representations of the same systems. The purpose was to investigate the
extent to which results from nonlinear static analyses might be combined
with dynamic analyses of SDOF systems to estimate the global response of
MDOF systems. Preliminary MDOF investigations are described in
Appendix G. Additional MDOF investigations are planned under a project
funded by the National Institute of Standards and Technology (NIST).
1.3 Report Organization and Content
Chapter 1 introduces the project context, objectives, and scope of the
investigation.
Chapter 2 provides background information related to modeling of
component hysteretic behavior, summarizes results of past studies, and
introduces new terminology.
Chapter 3 describes the development of SDOF models, and explains the
analytical procedures used in the conduct of focused analytical studies.
Chapter 4 summarizes the results of focused analytical studies on singlespring
and multi-spring systems, compares results to recommendations
contained in FEMA 440, and explains the development of a new equation
measuring the potential for lateral dynamic instability.
Chapter 5 collects and summarizes the findings, conclusions, and
recommendations resulting from this investigation related to improved
understanding of nonlinear degrading response and judgment in
implementation of nonlinear analysis results in engineering practice,
improvements to current nonlinear analysis procedures, and suggestions for
further study.
Appendix A provides a comprehensive collection and summary of technical
references on the development, evolution, and applicability of various
hysteretic models for use in nonlinear structural analysis.
Appendix B contains plots of selected incremental dynamic analysis results
for single-spring systems.
Appendix C contains normalized plots of selected incremental dynamic
analysis results for multi-spring systems.
1-6 1: Introduction FEMA P440A
Appendix D contains non-normalized plots of selected incremental dynamic
analysis results for multi-spring systems.
Appendix E explains the concepts of uncertainty and fragility, how
incremental dynamic analysis results can be converted into fragilities, and
how to use this information to calculate estimates of annualized probability
for limit states of interest.
Appendix F provides an example application of a simplified nonlinear
dynamic analysis procedure, including quantitative evaluation of alternative
retrofit strategies and development of probabilistic estimates of performance
using the concepts outlined in Appendix E.
Appendix G describes a set of preliminary studies of MDOF systems
comparing results of MDOF analyses with results from equivalent SDOF
representations of the systems, and provides recommendations for additional
MDOF studies.
A compact disc (CD) accompanying this report provides electronic files of
the report and appendices in PDF format, an electronic visualization tool in
Microsoft Excel format that can be used to view the entire collection of
multi-spring incremental dynamic analysis results, and the Static Pushover 2
Incremental Dynamic Analysis (SPO2IDA) software tool in Microsoft Excel
format (Vamvatsikos and Cornell, 2006) that can be used to estimate the
dynamic response of systems based on idealized force-displacement (static
pushover) curves.
FEMA P440A 2: Background Concepts 2-1
Chapter 2
Background Concepts
This chapter provides background information on modeling of component
hysteretic behavior. It summarizes how various types of hysteretic behavior
have been investigated in past studies, and explains how these behaviors
have been observed to affect seismic response. It introduces new
terminology, and explains how the new terms are related to observed
differences in nonlinear dynamic response.
2.1 Effects of Hysteretic Behavior on Seismic Response
Many hysteretic models have been proposed over the years with the purpose
of characterizing the mechanical nonlinear behavior of structural components
(e.g., members and connections) and estimating the seismic response of
structural systems (e.g., moment frames, braced frames, shear walls).
Available hysteretic models range from simple elasto-plastic models to
complex strength and stiffness degrading curvilinear hysteretic models. This
section presents a summary of the present state of knowledge on hysteretic
models, and their influence on the seismic response of structural systems. A
comprehensive summary of technical references on the development,
evolution, and applicability of various hysteretic models is presented in
Appendix A.
2.1.1 Elasto-Plastic Behavior
In the literature, most studies that have considered nonlinear behavior have
used non-degrading hysteretic models, or models in which the lateral
stiffness and the lateral yield strength remain constant throughout the
duration of loading. These models do not incorporate stiffness or strength
degradation when subjected to repeated cyclic load reversals. The simplest
and most commonly used non-deteriorating model is an elasto-plastic model
in which system behavior is linear-elastic until the yield strength is reached
(Figure 2-1). At yield, the stiffness switches from elastic stiffness to zero
stiffness. During unloading cycles, the stiffness is equal to the loading
(elastic) stiffness.
Early examples of the use of elasto-plastic models include studies by Berg
and Da Deppo (1960), Penzien (1960a, 1960b), and Veletsos and Newmark
(1960). The latter study was the first one to note that peak lateral
2-2 2: Background Concepts FEMA P440A
displacements of moderate and long-period single-degree-of-freedom
(SDOF) systems with elasto-plastic behavior were, on average, about the
same as that of linear elastic systems with the same period of vibration and
same damping ratio. Their observations formed the basis of what is now
known as the “equal displacement approximation.” This widely-used
approximation implies that the peak displacement of moderate and longperiod
non-degrading systems is proportional to the ground motion intensity,
meaning that if the ground motion intensity is doubled, the peak
displacement will be on average, approximately twice as large.
Displacement
Force
Figure 2-1 Elasto-plastic non-degrading piecewise linear hysteretic model.
Veletsos and Newmark also observed that peak lateral displacement of shortperiod
SDOF systems with elasto-plastic behavior were, on average, larger
than those of linear elastic systems, and increases in peak lateral
displacements were larger than the increment in ground motion intensity.
Thus, the equal displacement approximation was observed to be less
applicable to short-period structures.
Using many more ground motions, recent studies have corroborated some of
the early observations by Veletsos, identified some of the limitations in the
equal displacement approximation, and provided information on record-torecord
variability (Miranda, 1993, 2000; Ruiz-Garcia and Miranda, 2003;
Chopra and Chintanapakdee, 2004). These studies have shown that, in the
short-period range, peak inelastic system displacements increase with respect
to elastic system displacements as the period of vibration decreases and as
the lateral strength decreases. These observations formed the basis of the
improved displacement modification coefficient C1, which accounts for the
effects of inelastic behavior in the coefficient method of estimating peak
displacements, as documented in FEMA 440 Improvement of Nonlinear
Static Seismic Analysis Procedures (FEMA, 2005).
FEMA P440A 2: Background Concepts 2-3
2.1.2 Strength-Hardening Behavior
Another commonly used non-degrading hysteretic model is a strengthhardening
model, which is similar to the elasto-plastic model, except that the
post-yield stiffness is greater than zero (Figure 2-2). Early applications of
bilinear strength-hardening models include investigations by Caughey
(1960a, 1960b) and Iwan (1961). Positive post-yield stiffness is also referred
to as “strain hardening” because many materials exhibit gains in strength
(harden) when subjected to large strain levels after yield. Strength hardening
in components, connections, and systems after initial yield is also caused by
eventual mobilization of a full member crossection, or sequential yielding of
the remaining elements in a system. This is typically the most important
source of strength hardening observed in a structural system.
Displacement
Force
Figure 2-2 Strength-hardening non-degrading piecewise linear hysteretic
model.
Although many studies have considered elasto-plastic and bilinear strengthhardening
behavior, it was not until recently that comprehensive statistical
studies were conducted to systematically quantify differences in peak
displacements using a wide range of periods of vibration, a wide range of
post-elastic stiffnesses, and large numbers of ground motions. Several recent
studies have provided quantitative information on the average effects of
positive post-yield stiffness on response, and on the variability in response
for different ground motion records. They are in agreement that, for
moderate and long-period structures, the presence of a positive post-elastic
stiffness leads to relatively small (less than 5%) reductions in peak
displacement (Ruiz-Garcia and Miranda, 2003; Chopra and Chintanapakdee,
2004). The magnitude of the reduction varies based on the strength of the
system and period of vibration.
System strength is often characterized by a parameter, R, defined as the ratio
between the strength that would be required to keep the system elastic for a
2-4 2: Background Concepts FEMA P440A
given intensity of ground motion, SaT, and the lateral yield strength of the
system, Fy:
aT aT
y y
R S S g
F W F
. . (2-1)
where SaT is expressed as a percentage of gravity. This R factor is related to,
but not the same as, the response-modification coefficient used in code-based
equivalent lateral force design procedures.
For weaker systems (systems with higher values of R), the reduction in
response is greater (more beneficial). For short-period systems, the presence
of a positive post-elastic stiffness can lead to significant reductions in peak
lateral displacements.
Other recent studies have shown that a positive post-elastic stiffness can have
a very large effect in other response parameters. In particular, MacRae and
Kawashima (1997), Kawashima et al., (1998) Pampanin et al. (2002), Ruiz-
Garcia and Miranda, (2006a) have shown that small increments in post-yield
stiffness can lead to substantial reductions in residual drift in structures
across all period ranges.
2.1.3 Stiffness-Degrading Behavior
Many structural components and systems will exhibit some level of stiffness
degradation when subjected to reverse cyclic loading. This is especially true
for reinforced concrete components subjected to several large cyclic load
reversals. Stiffness degradation in reinforced concrete components is usually
the result of cracking, loss of bond, or interaction with high shear or axial
stresses. The level of stiffness degradation depends on the characteristics of
the structure (e.g., material properties, geometry, level of ductile detailing,
connection type), as well as on the loading history (e.g., intensity in each
cycle, number of cycles, sequence of loading cycles).
Figure 2-3 shows three examples of stiffness-degrading models. In the first
model, the loading and unloading stiffness is the same, and the stiffness
degrades as displacement increases. In the second model the loading
stiffness decreases as a function of the peak displacement, but the unloading
stiffness is kept constant and equal to the initial stiffness. In the third model,
both the loading and unloading stiffnesses degrade as a function of peak
displacement, but they are not the same.
In order to evaluate the effects of stiffness degradation, many studies have
compared the peak response of stiffness-degrading systems to that of elastoplastic
and bilinear strength-hardening systems (Clough 1966; Clough and
FEMA P440A 2: Background Concepts 2-5
Johnston 1966; Chopra and Kan, 1973; Powel and Row, 1976; Mahin and
Bertero, 1976; Riddell and Newmark, 1979; Newmark and Riddell, 1980;
Iwan 1980; Otani, 1981; Nassar and Krawinkler 1991; Rahnama and
Krawinkler, 1993; Shi and Foutch, 1997; Foutch and Shi, 1998; Gupta and
Krawinkler, 1998; Gupta and Kunnath, 1998; Medina 2002; Medina and
Krawinkler, 2004; Ruiz-Garcia and Miranda, 2005).
Displacement
Force
K1 K2 K3 K4
K1> K2>K3>K4
Displacement
Force
Displacement
Force
K1 K2 K3 K4
K1> K2>K3>K4
Displacement
Force
Displacement
Force
Displacement
Force
K1 K2 K3
K1= K2=K3
Displacement
Force
Displacement
Force
K1 K2 K3
K1= K2=K3
Displacement
Force
K1 K2 K3
K1> K2>K3
Displacement
Force
Displacement
Force
K1 K2 K3
K1> K2>K3
Displacement
Force
Figure 2-3 Three examples of stiffness-degrading piecewise linear hysteretic models.
These studies have concluded that, in spite of significant reductions in lateral
stiffness and hysteretic energy dissipation capacity (area enclosed within
hysteresis loops), moderate and long-period systems with stiffness-degrading
behavior experience peak displacements that are, on average, similar to those
of structures with elasto-plastic or bilinear strength-hardening hysteretic
behavior. In some cases, peak displacements can even be slightly smaller.
This observation suggests that it is possible to use simpler hysteretic models
that do not incorporate stiffness degradation to estimate lateral displacement
demands for moderate and long-period structures (systems with fundamental
periods longer than 1.0s).
These same studies, however, have concluded that short-period structures
with stiffness degradation experience peak displacements that are, on
average, larger than those experienced by systems with elasto-plastic or
bilinear strength-hardening hysteretic behavior. Differences in peak
displacements between stiffness-degrading and non-degrading systems
increase as the period of vibration decreases and as the lateral strength
decreases.
The above studies examined the effects of stiffness degradation on structures
subjected to ground motions recorded on rock or firm soil sites. Ruiz-Garcia
and Miranda (2006b) examined the effects of stiffness degradation on
structures subjected to ground motions recorded on soft soil sites. This study
concluded that the effects of stiffness degradation are more important for
structures built on soft soil, especially for structures with periods shorter than
the predominant period of the ground motion.
2-6 2: Background Concepts FEMA P440A
2.1.4 Pinching Behavior
Structural components and connections may exhibit a hysteretic phenomenon
called pinching when subjected to reverse cyclic loading (Figure 2-4).
Pinching behavior is characterized by large reductions in stiffness during
reloading after unloading, along with stiffness recovery when displacement is
imposed in the opposite direction.
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
(a) (b)
Figure 2-4 Examples of hysteretic models with: (a) moderate pinching
behavior; and (b) severe pinching behavior.
Pinching behavior is particularly common in reinforced concrete
components, wood components, certain types of masonry components, and
some connections in steel structures. In reinforced concrete, pinching is
typically produced by opening of cracks when displacement is imposed in
one direction. Partial stiffness recovery occurs when cracks are closed
during displacements imposed in the other direction. In wood structures
pinching is primarily caused by opening and closing of gaps in framing
elements due to nail pullout. Pinching also occurs as a result of opening and
closing of flexural cracks in reinforced masonry, opening and closing of gaps
between masonry infill and the surrounding structural frame, and opening
and closing of gaps between plates in steel end-plate connections. The level
of pinching depends on the characteristics of the structure (e.g., material
properties, geometry, level of ductile detailing, and connections), as well as
the loading history (e.g., intensity in each cycle, number of cycles, and
sequence of loading cycles).
Several studies have shown that, for moderate and long-period systems,
pinching alone or in combination with stiffness degradation has only a small
affect on peak displacement demands, as long as the post-yield stiffness
remains positive (Otani, 1981; Nassar and Krawinkler 1991; Rahnama and
Krawinkler, 1993; Shi and Foutch, 1997; Foutch and Shi, 1998; Gupta and
FEMA P440A 2: Background Concepts 2-7
Krawinkler, 1998; Gupta and Kunnath, 1998; Medina 2002; Medina and
Krawinkler, 2004; Ruiz-Garcia and Miranda, 2005).
These and other studies have shown that moderate and long-period systems,
with up to 50% reduction in hysteretic energy dissipation capacity due to
pinching, experience peak displacements that are, on average, similar to
those of structures with elasto-plastic or bilinear strength-hardening
hysteretic behavior. This observation is particularly interesting because it is
contrary to the widespread notion that structures with elasto-plastic or
bilinear behavior exhibit better performance than structures with pinching
behavior because of the presence of additional hysteretic energy dissipation
capacity.
These same studies, however, have also shown that short-period structures
with pinching behavior experience peak displacements that tend to be larger
than those experienced by systems with elasto-plastic or bilinear strengthhardening
hysteretic behavior. Differences in peak displacements increase as
the period of vibration decreases and as the lateral strength decreases.
2.1.5 Cyclic Strength Degradation
Structural components and systems may experience reductions in strength
generically referred to as strength degradation or strength deterioration
(Figure 2-5). One of the most common types of strength degradation is
cyclic strength degradation in which a structural component or system
experiences a reduction in lateral strength as a result of cyclic load reversals.
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
Displacement
Force
(a) (b)
Figure 2-5 Examples of cyclic strength degradation: (a) due to increasing inelastic
displacement; and (b) due to repeated cyclic displacement
In cyclic strength degradation, reductions in lateral strength occur after the
loading has been reversed, or during subsequent loading cycles. Cyclic
reductions in lateral strength are a function of the level of peak displacement
experienced in the system (Park, Reinhorn and Kunnath, 1987; Rahnama and
2-8 2: Background Concepts FEMA P440A
Krawinkler, 1993). This is illustrated in Figure 2-5(a), which shows an
elasto-plastic system experiencing strength degradation in subsequent
loading cycles as the level of inelastic displacement increases. Hysteretic
models that incorporate this type of strength degradation typically specify the
reduction in strength as a function of the ductility ratio, which is taken as the
ratio of peak deformation to yield deformation.
Cyclic strength degradation can also occur in subsequent cycles even if the
level of inelastic displacement is not being increased (Park, Reinhorn and
Kunnath, 1987; Rahnama and Krawinkler, 1993). This is illustrated in
Figure 2-5(b), which shows an elasto-plastic system experiencing cyclic
strength degradation as a result of a single level of inelastic displacement that
is imposed a number of times. The reduction in lateral strength increases as
the number of cycles increases. Hysteretic models that incorporate this type
of strength degradation (Park, Reinhorn and Kunnath, 1987; Rahnama and
Krawinkler, 1993; Mostaghel 1998, 1999; Sivaselvan and Reinhorn 1999,
2000) typically specify the reduction in strength as a function of the total
hysteretic energy demand imposed on the system, taken as the area enclosed
by the hysteresis loops.
Most structural systems exhibit a combination of the types of cyclic strength
degradation shown in Figure 2-5. Several hysteretic models that incorporate
both types of cyclic strength degradation have been developed (Park and
Ang, 1985; Park, Reinhorn and Kunnath, 1987; Rahnama and Krawinkler,
1993; Valles et al., 1996; Shi and Foutch, 1997; Foutch and Shi, 1998; Gupta
and Krawinkler, 1998;Gupta and Kunnath, 1998; Pincheira, Dotiwala, and D’
Souza 1999; Medina 2002; Medina and Krawinkler, 2004; Mostaghel 1998,
19990; Sivaselvan and Reinhorn 1999, 2000; Chenouda, and Ayoub, 2007).
Many of these same investigators have compared the peak response of
systems with cyclic strength degradation to that of elasto-plastic and bilinear
strength-hardening systems. In moderate and long-periods systems, the
effects of cyclic strength degradation have been shown to be very small, and
in many cases can be neglected, even with reductions in strength of 50% or
more. The reason for this can be explained using early observations from
Veletsos and Newmark (1960), which concluded that peak displacement
demands in moderate and long-period systems were not sensitive to changes
in yield strength. This conclusion logically extends to moderate and longperiod
systems that experience cyclic changes (reductions) in lateral strength
during loading.
In short-period structures, however, studies have shown that cyclic strength
degradation can lead to significant increases in peak displacement demands.
FEMA P440A 2: Background Concepts 2-9
This observation can also be explained by results from Veletsos and
Newmark (1960), which concluded that peak displacement demands in shortperiod
systems are very sensitive to changes in yield strength. This
conclusion logically extends to short-period systems that experience cyclic
changes (reductions) in lateral strength during loading.
2.1.6 Combined Stiffness Degradation and Cyclic Strength
Degradation
Several recent studies have examined the effects of stiffness degradation in
combination with cyclic strength degradation (Gupta and Kunnath, 1998;
Song and Pincheira, 2000; Medina 2002; Medina and Krawinkler, 2004;
Ruiz-Garcia and Miranda, 2005; Chenouda, and Ayoub, 2007). Examples of
these behaviors are illustrated in Figure 2-6. Figure 2-6a shows a system
with moderate stiffness and cyclic strength degradation (MSD), and Figure
2-6b shows a system with severe stiffness and cyclic strength degradation
(SSD). In these systems, lateral strength is reduced as a function of both the
peak displacement demand as well as the hysteretic energy demand on the
system.
(a) (b)
Figure 2-6 Hysteretic models combining stiffness degradation and cyclic strength
degradation: (a) moderate stiffness and cyclic strength degradation; and (b)
severe stiffness and cyclic strength degradation (Ruiz-Garcia and Miranda, 2005).
These studies have shown that, for moderate and long-period systems with
combined stiffness and cyclic strength degradation, peak displacements are,
on average, similar to those experienced by elasto-plastic or bilinear strengthhardening
systems. These effects are only observed to be significant for
short-period systems (systems with periods of vibration less than 1.0s).
2.1.7 In-Cycle Strength Degradation
In combination with stiffness degradation, structural components and
systems may experience in-cycle strength degradation (Figure 2-7). In-cycle
(c) SSD
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-15 -12 -9 -6 -3 0 3 6 9 12 15
Displacement Ductility Normalized Force
(b) MSD
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-15 -12 -9 -6 -3 0 3 6 9 12 15
Displacement Ductility
Noormalized Force
2-10 2: Background Concepts FEMA P440A
strength degradation is characterized by a loss of strength within the same
cycle in which yielding occurs. As additional lateral displacement is
imposed, a smaller lateral resistance is developed. This results in a negative
post-yield stiffness within a given cycle.
Figure 2-7 In-cycle strength degradation.
In-cycle strength degradation can occur as a result of geometric
nonlinearities (P-delta effects), material nonlinearities, or a combination of
these. In reinforced concrete components, material nonlinearities that can
lead to in-cycle strength degradation include concrete crushing, shear failure,
buckling or fracture of longitudinal reinforcement, and splice failures. In steel
components, material nonlinearities that can lead to in-cycle strength
degradation include buckling of bracing elements, local buckling in flanges
of columns or beams, and fractures of bolts, welds, or base materials.
2.1.8 Differences Between Cyclic and In-Cycle Strength
Degradation
FEMA 440 identified the distinction between cyclic and in-cycle degradation
to be very important because the consequences of each were observed to be
vastly different. Dynamic response of systems with cyclic strength
degradation is generally stable, while in-cycle strength degradation can lead
to lateral dynamic instability (i.e., collapse) of a structural system.
Figure 2-8 compares the hysteretic behavior of two systems subjected to the
loading protocol shown in Figure 2-9. This loading protocol comprises six
full cycles (twelve half-cycles) with a linearly increasing amplitude of 0.8%
drift in each cycle. The system in Figure 2-8a has cyclic degradation and the
system in Figure 2-8b has in-cycle degradation. When subjected to this
loading protocol, both hysteretic models exhibit similar levels of strength and
stiffness degradation, and similar overall behavior. Their behavior under
different loading protocols, however, can be significantly different.
FEMA P440A 2: Background Concepts 2-11
(a) (b)
Figure 2-8 Hysteretic behavior for models subjected to Loading Protocol 1 with: (a) cyclic
strength degradation; and (b) in-cycle degradation.
Figure 2-9 Loading Protocol 1 used to illustrate the effects of cyclic and incycle
strength degradation.
A second loading protocol, shown in Figure 2-10, is identical to the first
protocol through four half-cycles, but during the fifth half-cycle it continues to
impose additional lateral displacement until a drift ratio of 7.0% is reached.
Figure 2-10 Loading Protocol 2 used to illustrate the effects of cyclic and incycle
strength degradation.
LOADING PROTOCOL 1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Loading Cycle
Drift Ratio
LOADING PROTOCOL 2
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Loading Cycle
Drift Ratio
2-12 2: Background Concepts FEMA P440A
Figure 2-11 compares the hysteretic behavior of both systems subjected to
the second loading protocol. Initially, the responses are similar. During the
fifth half-cycle, however, the responses diverge. The model with cyclic
degradation (Figure 2-11a) is able to sustain lateral strength without loss as
the drift ratio increases. In contrast, the model with in-cycle degradation
(Figure 2-11b) experiences a rapid loss in strength as the drift ratio increases.
While the model with cyclic strength degradation remains stable, the model
with in-cycle strength degradation becomes unstable after losing lateral
resistance.
(a) (b)
Figure 2-11 Hysteretic behavior for models subjected to Loading Protocol 2
with: (a) cyclic strength degradation; and (b) in-cycle
degradation.
Figure 2-12 shows the displacement time histories for these same two
systems when subjected to the north-south component of the Yermo Valley
ground motion of the 1992 Landers Earthquake. The system with cyclic
strength degradation (Figure 2-12a) undergoes a large peak drift ratio during
the record, experiences a residual drift at the end of the record, and yet
remains stable over the duration of the record. In contrast, the system with
in-cycle degradation (Figure 2-12b) undergoes a similar peak drift ratio
during the record, but ratchets further in one direction in subsequent yielding
cycles, and eventually experiences lateral dynamic instability (collapse).
FEMA P440A 2: Background Concepts 2-13
0 5 10 15 20 25 30 35 40 45
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
time (sec)
Interstory Drift
0 5 10 15 20 25 30 35 40 45
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
time (sec)
Interstory Drift
0 5 10 15 20 25 30 35 40 45
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
time (sec)
Interstory Drift
0 5 10 15 20 25 30 35 40 45
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
time (sec)
Interstory Drift
(a) (b)
Figure 2-12 Displacement time histories for models subjected to the 1992 Landers
Earthquake with: (a) cyclic strength degradation; and (b) in-cycle strength
degradation.
2.2 Concepts and Terminology
Historically, the term backbone curve has referred to many different things.
It has been used, for example, to describe limitations on the forcedeformation
behavior of structural components, force-displacement plots
from nonlinear static pushover analyses of structural systems, curves
enveloping the force-displacement response of structural components
undergoing cyclic testing, and curves tracing the force-displacement response
of structural components undergoing monotonic testing.
In the case of component modeling, parameters taken from one definition of
a backbone curve versus another are not interchangeable, and their incorrect
usage can have a significant affect on the predicted nonlinear response. For
this reason, two new terms are introduced to distinguish between different
aspects of hysteretic behavior. These are the force-displacement capacity
boundary, and cyclic envelope.
2.2.1 Force-Displacement Capacity Boundary
Several recent models have been developed to incorporate various types of
degrading phenomena (Kunnath, Reinhorn and Park, 1990; Kunnath, Mander
and Fang, 1997; Mostaghel 1998, 19990; Sivaselvan and Reinhorn 1999,
2000; Ibarra, Medina, Krawinkler, 2005; Chenouda and Ayoub, 2007). A
common feature in all these degrading models is that they start by defining
the maximum strength that a structural member can develop at a given level
of deformation. This results in an effective “boundary” for the strength of a
member in force-deformation space, termed the force-displacement capacity
boundary.
Story Drift Ratio
Story Drift Ratio
2-14 2: Background Concepts FEMA P440A
Figure 2-13 shows examples of two such boundaries commonly used in
structural analysis of degrading components. These curves resemble the
conceptual force-displacement relationship used to express component
modeling and acceptability criteria in ASCE/SEI 41-06 Seismic
Rehabilitation of Existing Buildings (ASCE, 2006b), commonly referred to
as “backbones.”
K1> K2>K3>K4
Force
Displacement
K1> K2>K3>K4
Force
Displacement
K1> K2>K3>K4
Force
Displacement
K1> K2>K3>K4
Force
Displacement
Figure 2-13 Examples of commonly used force-displacement capacity boundaries.
A cyclic load path cannot cross a force-displacement capacity boundary. If a
member is subjected to increasing deformation and the boundary is reached,
then the strength that can be developed in the member is limited and the
response must continue along the boundary. This behavior is in-cycle
strength degradation, and is shown in Figure 2-14. Note that only
displacement excursions intersecting portions of the capacity boundary with
a negative slope will result in in-cycle strength degradation.
Figure 2-14 Interaction between the cyclic load path and the force-displacement
capacity boundary.
FEMA P440A 2: Background Concepts 2-15
In most cases, the force-displacement capacity boundary will not be static.
More advanced models consider that the force-displacement capacity
boundary will degrade (move inward) as a result of cyclic degradation
(Figure 2-15). In some cases, it is also possible for the boundary to move
outward due to cyclic strain hardening, such as in the case of structural steel
elements subjected to large strains, but this behavior is not considered here.
In order to define the cyclic behavior of a component model, one must define
where the force-displacement capacity boundary begins, and how it degrades
under cyclic loading. In the absence of cyclic strain hardening, the initial
force-displacement capacity boundary is simply the monotonic response of a
component. Accordingly, the ideal source for estimating the parameters of
the initial force-displacement capacity boundary comes from monotonic
tests.
Figure 2-15 Degradation of the force-displacement capacity boundary.
Once the initial capacity boundary is defined, then cyclic degradation
parameters must be estimated based on the results of cyclic tests. The use of
several cyclic protocols is desirable to ensure that the calibrated component
model is general enough to represent component response under any type of
loading. This requires the availability of multiple identical specimens that
are tested under several different loading protocols, which is a significant
undertaking, and is rarely done.
When utilizing existing test data to calibrate a component model, it is
uncommon to find sets of test data that include both monotonic and cyclic
tests on identical specimens. It is even more uncommon to find sets of data
that include monotonic tests and cyclic tests using multiple loading protocols
2-16 2: Background Concepts FEMA P440A
on identical specimens. As such, there are only a small number of cases in
which this kind of data exists (Tremblay et al., 1997; El-Bahy, 1999; Ingham
et al., 2001; Uang et al., 2000; Uang et al., 2003).
Most existing data is based only on a single cyclic loading protocol. In such
cases, cyclic degradation can be approximated directly from the test data. In
the absence of monotonic test data, the initial force-displacement capacity
boundary must be extrapolated from the cyclic data (since the monotonic
response is unknown). Considerable judgment must be exercised in
extrapolating an initial force-displacement capacity boundary because there
may be several combinations of initial parameters and cyclic degradation
parameters that result in good agreement with the observed cyclic test data.
Such an approach has been used by Haselton et al. (2007) for reinforced
concrete components and Lignos (2008) for steel components.
2.2.2 Cyclic Envelope
A cyclic envelope is a force-deformation curve that envelopes the hysteretic
behavior of a component or assembly that is subjected to cyclic loading.
Figure 2-16 shows a cyclic envelope, which is defined by connecting the
peak force responses at each displacement level.
Figure 2-16 Example of a cyclic envelope.
Where loading protocols have included multiple cycles at each displacement
increment, a different curve (often referred to as cyclic “backbone”) has been
defined based on the force at either the second or third cycle at each
displacement level. Such a definition was included in FEMA 356
Prestandard and Commentary for the Seismic Rehabilitation of Buildings
FEMA P440A 2: Background Concepts 2-17
(FEMA, 2000). In ASCE/SEI 41-06 (with Supplement No. 1) this has been
changed to be more consistent with the concept of a cyclic envelope, as
described above.
2.2.3 Influence of Loading Protocol on the Cyclic Envelope
The characteristics of a cyclic envelope are strongly influenced by the points
at which unloading occurs in a test, and are therefore strongly influenced by
the loading protocol that was used in the experimental program. Studies by
Takemura and Kawashima (1997) illustrate the influence that the loading
protocol can have on the characteristics of the cyclic envelope. In these
studies, six nominally identical reinforced concrete bridge piers were tested
using six different loading protocols, yielding six significantly different
hysteretic behaviors. The loading protocols and resulting hysteretic plots are
shown in Figure 2-17 through Figure 2-19.
TP 01
-9.0
-6.0
-3.0
0.0
3.0
6.0
9.0
Loading Cycle
Drift % TP 02
-9.0
-6.0
-3.0
0.0
3.0
6.0
9.0
Loading Cycle
Drift %
TP 01
-195
-130
-65
0
65
130
195
-9.0 -6.0 -3.0 0.0 3.0 6.0 9.0
Drift %
Force kN
TP 02
-195
-130
-65
0
65
130
195
-9.0 -6.0 -3.0 0.0 3.0 6.0 9.0
Drift %
Force kN
(a) (b)
Figure 2-17 Loading protocols and resulting hysteretic plots for identical reinforced
concrete bridge pier specimens: (a) Loading Protocol TP01; and (b) Loading
Protocol TP02 (adapted from Takemura and Kawashima, 1997).
2-18 2: Background Concepts FEMA P440A
TP 03
-9.0
-6.0
-3.0
0.0
3.0
6.0
9.0
Loading Cycle
Drift % TP 04
-9.0
-6.0
-3.0
0.0
3.0
6.0
9.0
Loading Cycle
Drift %
TP 03
-195
-130
-65
0
65
130
195
-9.0 -6.0 -3.0 0.0 3.0 6.0 9.0
Drift %
Force kN
TP 04
-195
-130
-65
0
65
130
195
-9.0 -6.0 -3.0 0.0 3.0 6.0 9.0
Drift %
Force kN
(a) (b)
Figure 2-18 Loading protocols and resulting hysteretic plots for identical reinforced concrete
bridge pier specimens: (a) Loading Protocol TP03; and (b) Loading Protocol TP04
(adapted from Takemura and Kawashima, 1997).
TP 05
-9.0
-6.0
-3.0
0.0
3.0
6.0
9.0
Loading Cycle
Drift % TP 06
-9.0
-6.0
-3.0
0.0
3.0
6.0
9.0
Loading Cycle
Drift %
TP 05
-195
-130
-65
0
65
130
195
-9.0 -6.0 -3.0 0.0 3.0 6.0 9.0
Drift %
Force kN
TP 06
-195
-130
-65
0
65
130
195
-9.0 -6.0 -3.0 0.0 3.0 6.0 9.0
Drift %
Force kN
(a) (b)
Figure 2-19 Loading protocols and resulting hysteretic plots for identical reinforced concrete
bridge pier specimens: (a) Loading Protocol TP05; and (b) Loading Protocol TP06
(adapted from Takemura and Kawashima, 1997).
FEMA P440A 2: Background Concepts 2-19
The resulting cyclic envelopes are plotted together in Figure 2-20 for
comparison. Loading protocols with more cycles and increasing amplitudes
in each cycle (e.g., TP 01, TP 02, and TP 03) resulted in smaller cyclic
envelopes. Loading protocols with fewer cycles and decreasing in
amplitudes in each cycle (e.g., TP 04 and TP 06) resulted in larger cyclic
envelopes.
These studies show that if nominally identical specimens are loaded with
different loading protocols, their cyclic envelope will change depending on
the number of cycles used in the loading protocol, the amplitude of each
cycle, and the sequence of the loading cycles.
Figure 2-20 Comparison of cyclic envelopes of reinforced concrete bridge
pier specimens subjected to six different loading protocols
(adapted from Takemura and Kawashima, 1997).
2.2.4 Relationship between Loading Protocol, Cyclic Envelope,
and Force-Displacement Capacity Boundary
For analytical purposes, a series of hysteretic rules can be specified to control
the hysteretic behavior of a component within a force-displacement capacity
boundary. Unless a loading protocol has forced the structural component or
system to reach the force-displacement capacity boundary, the resulting
cyclic envelope will be smaller, and in some cases significantly smaller, than
the actual capacity boundary.
Figure 2-21 shows the cyclic envelope for a structural component subjected
to a single loading protocol. In Figure 2-21a, the cyclic envelope is equal to
the force-displacement capacity boundary. In Figure 2-21b, the forcedisplacement
capacity boundary extends beyond the cyclic envelope (which
2-20 2: Background Concepts FEMA P440A
would be the case if the component actually had more force-displacement
capacity than indicated by a single cyclic envelope).
(a) (b)
Figure 2-21 Examples of a force-displacement capacity boundary that is (a) equal to the cyclic
envelope, and (b) extends beyond the cyclic envelope.
Figure 2-22 shows the hysteretic behavior of the same component subjected
to a different loading protocol. In this protocol the first four cycles are the
same, but in the fifth cycle additional lateral displacement is imposed up to a
peak story drift ratio of 5.5%. In Figure 2-22a, the component reaches the
force-displacement capacity boundary and the response is forced to follow a
downward slope along the boundary (in-cycle strength degradation).
Eventually, zero lateral resistance is reached before the unloading cycle can
begin.
(a) (b)
Figure 2-22 Comparison of hysteretic behavior when the force-displacement capacity boundary is:
(a) equal to the cyclic envelope, and (b) extends beyond the cyclic envelope.
In Figure 2-22b, however, because the force-displacement capacity boundary
extends beyond the cyclic envelope, the component has additional capacity to
resist deformation. As the lateral displacement approaches 5.5%, the
FEMA P440A 2: Background Concepts 2-21
response continues to gain strength until the force-displacement capacity
boundary is reached. The response is then forced to follow along the
boundary (in-cycle strength degradation) until the unloading cycle
commences at peak story drift ratio of 5.5%. In this case the component can
continue to resist 70% of its peak lateral strength at a story drift ratio of
5.5%, rather than degrading to zero lateral resistance before unloading
occurs.
Under lateral displacements that are less than or equal to those used to
generate the cyclic envelope, differences between the cyclic envelope and the
force-displacement capacity boundary are of no consequence. However,
under larger lateral displacements these differences will affect the potential
for in-cycle degradation to occur, which will significantly affect system
behavior and response. Determining the force-displacement capacity
boundary based on the results of a single cyclic loading protocol can result in
overly conservative results due to significant underestimation of the actual
force-displacement capacity and subsequent overestimation of lateral
displacement demands.
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-1
for Focused Analytical Studies
Chapter 3
Development of Single-Degree-of-
Freedom Models for Focused
Analytical Studies
This chapter describes the development of single-degree-of-freedom models,
and explains the analytical procedures used in the conduct of focused
analytical studies.
3.1 Overview of Focused Analytical Studies
3.1.1 Purpose
From past research, it is apparent that in-cycle strength and stiffness
degradation are real phenomena that have been observed and documented to
cause instability in individual components. Little experimental information
exists, however, on whether or not larger assemblies of components of mixed
hysteretic behavior experience similar negative stiffness that could lead to
dynamic instability. In order to further investigate the response of systems
with degrading components, focused analytical studies were conducted. The
purpose of these studies was to test and quantify the effects of different
degrading behaviors on the dynamic stability of structural systems.
3.1.2 Process
Studies consisted of nonlinear dynamic analyses of single-degree-of-freedom
oscillators with varying system characteristics. Characteristics under
investigation included differences in hysteretic behavior, such as cyclic
versus in-cycle degradation, and variations in the features of the forcedisplacement
capacity boundary, such as the point at onset of degradation,
the slope of degradation, the level of residual strength, length of the residual
strength plateau, and ultimate deformation capacity (Figure 3-1).
The process used for developing, analyzing, and comparing structural system
models in the focused analytical studies was as follows:
. A set of single-degree-of-freedom (SDOF) springs were developed
featuring different hysteretic and force-displacement capacity boundary
characteristics. While not an exact representation of the mechanical
3-2 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
behavior of any one specific structural component, springs were intended
to capture the major characteristics of force-displacement capacity
boundaries for systems that would typically be encountered in practice.
Figure 3-1 Features of the force-displacement capacity boundary varied in
focused analytical studies.
. Multiple spring models were used to represent the behavior of more
complex structural systems containing subsystems with different
hysteretic and force-displacement capacity boundary characteristics.
Multi-spring SDOF systems were developed by placing two individual
springs in parallel, linked by a rigid diaphragm.
. Nonlinear response history analyses were performed using the Open
System for Earthquake Engineering Simulation (OpenSEES) software
(Fenves and McKenna, 2004). In OpenSEES, structural system models
were subjected to the Incremental Dynamic Analysis (IDA) procedure
(Vamvatsikos and Cornell, 2002) in which the nonlinear dynamic
response of individual and multiple spring systems were evaluated at
incrementally increasing levels of ground motion intensity.
. Results were compared in two ways: (1) among systems with different
components that were tuned to have the same global yield strength and
the same period of vibration; and (2) among systems composed of the
same two components but having different relative contributions from
each, thus exhibiting different strength and stiffness characteristics.
Comparisons between systems tuned to the same yield strength and
period of vibration were used to observe the influence of different
hysteretic rules and force-displacement capacity boundary
characteristics. Comparisons between systems composed of the same
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-3
for Focused Analytical Studies
components, but with different strength and stiffness characteristics,
were used to observe the relative contribution from each subsystem on
overall system response.
3.1.3 Incremental Dynamic Analysis Procedure
Focused analytical studies were conducted using the Incremental Dynamic
Analysis (IDA) procedure (Vamvatsikos and Cornell, 2002). Incremental
dynamic analysis is a type of response history analysis in which a system is
subjected to ground motion records scaled to increasing levels of intensity
until lateral dynamic instability is observed. In incremental dynamic
analysis, intensity is characterized by a selected intensity measure (IM), and
lateral dynamic instability occurs as a rapid, nearly infinite increase in the
engineering demand parameter (EDP) of interest, given a small increment in
ground motion intensity.
3.1.3.1 Intensity measures
Two intensity measures were used in conducting incremental dynamic
analyses. One was taken as the 5% damped spectral acceleration at the
fundamental period of vibration of the oscillator, Sa(T,5%). This measure is
generally appropriate for single-degree-of-freedom systems. It, however,
does not allow comparison among systems having different periods of
vibration. For this reason, a normalized intensity measure, R =
Sa(T,5%)/Say(T,5%) was also used, where Say(T,5%) is the intensity that
causes first yield to occur in the system. This places first yield at a
normalized intensity of one.
The normalized intensity measure Sa(T,5%)/Say(T,5%) closely resembles the
strength ratio, R, which is a normalized strength that is often used in studies
of SDOF systems (see Chapter 2). Higher values of the normalized intensity
measure Sa(T,5%)/Say(T,5%) represent systems with lower lateral strength.
Note that the R-factor discussed here is not the same as the responsemodification
coefficient used in code-based equivalent lateral force design
procedures. Rather, it is essentially the system ductility reduction factor, Rd,
as defined in the NEHRP Recommended Provisions for Seismic Regulations
for New Buildings and Other Structures, Part 2: Commentary (FEMA,
2004b).
3.1.3.2 Engineering Demand Parameters
The engineering demand parameter of interest was taken as story drift ratio.
This parameter is a normalized measure of lateral displacement that allows
for non-dimensional comparison of results. Lateral dynamic instability
3-4 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
occurs when solutions to the input ground motion fail to converge, implying
infinite lateral displacements.
3.1.3.3 Collapse
Lateral dynamic instability is manifested in structural systems as sidesway
collapse caused by loss of lateral-force-resisting capacity. Sidesway collapse
mechanisms can be explicitly simulated in incremental dynamic analyses,
and comparisons of analytical results are based on this limit state.
It should be noted, however, that behavior of real structures can include loss
of vertical-load-carrying capacity at lateral displacements that are
significantly smaller than those associated with sidesway collapse. Inelastic
deformation of structural components can result in shear and flexural-shear
failures in members, and failures in joints and connections, which can lead to
an inability to support vertical loads (vertical collapse) long before sidesway
collapse can be reached. Differences between sidesway and vertical collapse
behaviors are shown in Figure 3-2.
(a) (b)
Figure 3-2 Different collapse behaviors: (a) vertical collapse due to loss of verticalload-
carrying capacity; and (b) incipient sidesway collapse due to loss of
lateral-force-resisting capacity (reproduced with permission of EERI).
Consideration of vertical collapse modes is beyond the scope of this
investigation, however, collapse simulation and explicit consideration of both
vertical and sidesway collapse modes are described in FEMA P695
Quantification of Building Seismic Performance Factors (FEMA, 2009).
3.1.3.4 Incremental Dynamic Analysis Curves
By plotting discrete intensity measure/engineering demand parameter pairs in
an IM-EDP plane, the results of incremental dynamic analyses can be
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-5
for Focused Analytical Studies
displayed as a suite of IDA curves, one curve corresponding to each ground
motion record. An example of one such suite of curves is shown in Figure
3-3, where IDA curves computed from 30 different ground motions are
shown. Curves in this figure are plotted with the normalized intensity
measure R = Sa/Sa
yield on the vertical axis, and normalized engineering
demand parameter . = ./.yield on the horizontal axis.
The IDA curves in Figure 3-3a have a common characteristic in that they all
terminate with a distinctive horizontal segment, referred to as “flatline.”
Horizontal segments in IDA curves mean that large displacements occur at
small increments in ground motion intensity, which is indicative of lateral
dynamic instability. The intensity (or normalized intensity) at which IDA
curves become horizontal is taken as the sidesway collapse capacity of the
system.
As shown in Figure 3-3a, the sidesway collapse capacity varies significantly
from one ground motion record to another. This variability in response is
known as record-to-record variability. Because of record-to-record
variability, the response due to any one record is highly uncertain. For this
reason, statistical information on response due to a suite of ground motions is
used to quantify the central tendency (median) and variability (dispersion) of
the behavior of a structural system.
Figure 3-3 Examples depicting incremental dynamic analysis results; (a) suite of individual IDA
curves from 30 different ground motion records; and (b) statistically derived quantile
curves given . or R (Vamvatsikos and Cornell 2006)
Figure 3-3b shows quantiles (i.e., 16th, 50th (median) and 84th percentiles) of
collapse capacity derived from the results of the 30 IDA curves shown in
Figure 3-3a. Also shown in Figure 3-3b, are the 16th, 50th (median) and 84th
percentile curves of normalized deformation demands for given normalized
3-6 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
ground motion intensities (. given R), and normalized ground motion
intensities for given lateral deformation demands (R given .). In the figure,
the median curve for . given R is approximately the same as the median
curve for R given .; the 16th percentile curve for . given R is approximately
the same as the 84th percentile curve for R given .; and the 84th percentile
curve for . given R is approximately the same as the 16th percentile curve for
R given ....
Computing normalized ground motion intensities for given lateral
deformation demands (i.e., R given .) is an iterative process (Ruiz-Garcia
and Miranda, 2003). Further complicating this process is that, in certain
cases, there can be multiple intensity levels corresponding to a given lateral
deformation demand (Vamvatsikos and Cornell, 2002). For these reasons,
results in this investigation are reported using quantiles of lateral deformation
demand given ground motion intensity (i.e., . given R).
Use of quantiles of deformation given intensity (i.e., . given R) means that
16% of the lateral deformation demands for a given level of ground motion
intensity would be to the left of the 16th percentile IDA curve, and that 84%
would be to the right. Thus, the 16th percentile IDA curve for . given R will
always be above the median curve. Similarly, the 84th percentile IDA curve
for . given R will always be below the median curve.
3.1.4 Ground Motion Records
Analyses were performed using an early version of the ground motion record
set selected for use in the ATC-63 Project, and provided in FEMA P695
Quantification of Building Seismic Performance Factors (FEMA, 2009). In
general this set is intended to include far-field records from all largemagnitude
events in the PEER NGA database (PEER, 2006). To avoid event
bias, no more than two records were taken from any one earthquake.
In total 28 sets of two horizontal components were used (see Table 3-1).
This record set is similar, but not identical, to the set ultimately selected for
use in FEMA P695. All records are from firm soil sites, and none include
any traces of near source directivity. Values of peak ground acceleration
(PGA) and peak ground velocity (PGV) shown in the table correspond to the
largest of the two horizontal components.
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-7
for Focused Analytical Studies
Table 3-1 Earthquake Records Used in Focused Analytical Studies (Both Horizontal Components)
Event1
Station
R 2
Km
Vs30 3
m/s
f1 4
Deg
f2
4
deg
PGA
g
PGV
cm/s
Northridge 1994 (M=6.7)
1. Beverly Hills, Mullholland Dr. 9.4 356 009 279 0.52 57.2
2. Canyon Country, W Lost Canyon 11.4 309 000 270 0.48 44.8
Kern County 1952 (M=7.4)
3. Taft Lincoln School 38.4 385 021 111 0.18 15.6
Borrego Mtn 1968 (M=6.6)
4. El Centro Array #9 45.1 213 180 270 0.13 18.5
Duzce Turkey 1999 (M=7.1)
5. Bolu 12 326 000 090 0.82 59.2
Hector Mine 1999 (M=7.1)
6. Armboy 41.8 271 090 360 0.18 23.2
7. Hector 10.4 685 000 090 0.34 34.1
Imperial Valley 1979 (M=6.5)
8. Delta 22 275 262 352 0.35 28.4
9. El centro Array #11 12.5 196 140 230 0.38 36.7
Kobe, Japan 1995 (M=6.9)
10. Nishi-Akashi 7.1 609 000 090 0.51 36.1
11. Shin-Osaka 19.1 256 000 090 0.24 33.9
Kocaeli, Turkey 1999 (M=7.5)
12. Duzce 13.6 276 180 270 0.36 54.1
13. Arcelik 10.6 523 000 090 0.22 27.4
Landers 1992 (M=7.3)
14. Yermo Fire Station 23.6 354 270 360 0.24 37.7
15. Coolwater 19.7 271 long trans 0.42 32.4
Loma Prieta 1989 (M=6.9)
16. Capitola 8.7 289 000 090 0.53 34.2
17. Gilroy Array #3 12.2 350 000 090 0.56 42.3
Manjil Iran 1990 (M=7.4)
18. Abbar 12.6 724 long trans 0.51 47.3
3-8 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
Table 3-1 Earthquake Records Used in Focused Analytical Studies (Both Horizontal Components)
(continued)
Event1
Station
R 2
Km
Vs30 3
m/s
f1 4
Deg
f2
4
deg
PGA
g
PGV
cm/s
Superstition Hills 1987 (M=6.7)
19. El Centro Imp. Co Cent 18.2 192 000 090 0.36 42.8
20. Poe Road 11.2 208 270 360 0.45 31.7
Cape Mendocino 1992 (M=7.0)
21. Eureka – Myrtle and West 40.2 339 000 090 0.18 24.2
22. Rio Dell Overpass – FF 7.9 312 270 360 0.55 45.4
Chi-Chi, Taiwan 1999 (M=7.6)
23. CHY101 10.0 259 090 000 0.44 90.7
24. TCU045 26.0 705 090 000 0.51 38.8
San Fernando, 1971 (M=6.6)
25. LA Hollywood Sto FF 22.8 316 090 180 0.21 17.8
St Elias, Alaska 1979 (M=7.5)
26. Yakutat 80.0 275 009 279 0.08 34.3
27. Icy Bay 26.5 275 090 180 0.18 26.6
Friuli, Italy 1976 (M=6.5)
28. Tolmezzo 15.0 425 000 270 0.35 25.9
1 Moment magnitude
2 Closest distance to surface projection of fault rupture
3 S-wave speed in upper 30m of soil
4 Component
3.1.5 Analytical Models
The basis of the focused analytical studies is a set of idealized spring models
representative of the hysteretic and force-displacement capacity boundary
characteristics of different structural systems. The springs were modeled
using the Pinching4, ElasticPP and MinMax uniaxial materials in
OpenSEES. The Pinching4 material allows the definition of a complex
multi-linear force-displacement capacity boundary composed of four linear
segments. The ElasticPP material defines a system with an elasto-plastic
force-displacement capacity boundary. The MinMax material allows the
setting of an ultimate drift at which a system loses all its lateral-forceresisting
capacity in both loading directions. The Pinching4 and ElasticPP
materials in combination with MinMax were used to define springs with the
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-9
for Focused Analytical Studies
desired force-displacement capacity boundary characteristics along with
finite ultimate deformation capacities.
Parametric studies were conducted on single-degree-of-freedom (SDOF)
oscillators constructed with these springs and their variants. Generic storymodels
were developed using single-spring systems or multi-spring systems
consisting of two springs in parallel. Story models were intended to
approximate the behavior of single-story systems composed of an individual
subassembly or a mixture of subassemblies having complex hysteretic and
force-displacement capacity boundary characteristics linked by rigid
diaphragms.
3.2 Single-Spring Models
Each single-spring system model is defined by a hysteretic model confined
within a force-displacement boundary (Figure 3-4) developed from
information available in the literature. The single-spring systems are based
on the following set of eight different hysteretic behaviors and forcedisplacement
capacity boundary characteristics:
. Spring 1 – typical gravity frame system (e.g., steel)
. Spring 2 – non-ductile moment frame system (e.g., steel or concrete)
. Spring 3 – ductile moment frame system (e.g., steel or concrete)
. Spring 4 – stiff non-ductile system (e.g., steel concentric braced frame)
. Spring 5 – stiff, highly-pinched non-ductile system (e.g., brittle infill
wall)
. Spring 6 – elastic-perfectly-plastic system (for comparison)
. Spring 7 – limited-ductility moment frame system (e.g., concrete)
. Spring 8 – non-ductile gravity frame system (e.g., concrete)
While intended to be representative of realistic degrading response that has
been observed to occur in some structural components, these idealized
springs are not intended to be a detailed characterization of the mechanical
behavior of any one specific structural component or structural subassembly.
Rather, they are used to capture the main response characteristics of
components or subassemblies that are often present and combined in real
structural systems. The focus was not on investigating the seismic
performance of a particular structural system, but on identifying the effects
of various aspects of degrading behavior on the response of one-story singledegree-
of-freedom system models.
3-10 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
Figure 3-4 Hysteretic model confined by a force-displacement capacity
boundary.
An “a” and “b” version of each spring was developed. The “a” and “b”
versions differ by one or two characteristics of the force-displacement
capacity boundary so that the “b” version always possesses the more
favorable characteristics of the two. Sources of variation included the point
at onset of degradation, the slope of degradation, the level of residual
strength, and length of the residual strength plateau. To investigate period
dependency, systems utilizing the “a” and “b” versions of each individual
spring were tuned to periods of 0.5s, 1.0s, 1.5s, 2.0s, and 2.5s.
All springs were defined to be symmetrical, using the same forcedisplacement
capacity boundary in both the positive and negative loading
directions. All have a finite ultimate deformation capacity at which all
lateral-force-resisting capacity is lost, and all, except for Spring 6 (which is
elastic-perfectly-plastic), include in-cycle strength degradation.
In addition, the “a” and “b” versions of each spring (except for Spring 6)
were analyzed with both a constant force-displacement capacity boundary
and a degrading force-displacement capacity boundary in order to quantify
the effect of cyclic degradation on system response. To do this, springs were
subjected to an ATC-24 type loading protocol (ATC, 1992), consisting of
two cycles at each level of drift starting at 0.5% drift, and increasing in
increments of 1% drift up to a maximum of 8% drift.
Displacement
Force
Displacement
HYSTERETIC
MODEL
CAPACITY
BOUNDARY
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-11
for Focused Analytical Studies
The generic force-displacement capacity boundary used for all springs is
shown in Figure 3-5. The values of normalized base shear, F/Fy, and story
drift ratio, ., chosen to characterize the force-displacement capacity
boundary for each of the single-spring system models are listed in Table 3-2.
.
F / Fy
E
D
F
A G
B
C
Figure 3-5 Generic force-displacement capacity boundary used for all
single-spring system models.
For purposes of comparison, one version of each spring is shown in Figure
3-6. The parameters that define each spring, and the variations in each
spring, are described in more detail in the sections that follow.
-0.05 0 0.05 0.1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
.
F / Fy
Spring 1a (nodeg, Fy=1)
pushover
cyclic
-0.05 0 0.05 0.1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
.
F / Fy
Spring 2a (nodeg, Fy=1)
pushover
cyclic
-0.05 0 0.05 0.1
-1.5
-1
-0.5
0
0.5
1
1.5
.
F / Fy
Spring 3a (nodeg, Fy=1)
pushover
cyclic
-0.05 0 0.05 0.1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
.
F / Fy
Spring 4a (nodeg, Fy=1)
pushover
cyclic
Spring 1a Spring 2a Spring 3a Spring 4a
-0.05 0 0.05 0.1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
.
F / Fy
Spring 5a (nodeg, Fy=1)
pushover
cyclic
-0.05 0 0.05 0.1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
.
F / Fy
Spring 6a (nodeg, Fy=1)
pushover
cyclic
-0.05 0 0.05 0.1
-1.5
-1
-0.5
0
0.5
1
1.5
.
F / Fy
Spring 7a (nodeg, Fy=1)
pushover
cyclic
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
-1.20
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
.
F / Fy
Spring 8a (nodeg, Fy=1)
capacity boundary
cyclic test
Spring 5a Spring 6a Spring 7a Spring 8a
Figure 3-6 Comparison of eight basic single-spring system models.
3-12 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
Table 3-2 Force-Displacement Capacity Boundary Control Points for Single-Spring System
Models.
Prototype Type Quantity Points of the force-deformation capacity boundary
A B C D E F G
Typical gravity frame 1a F/Fy 0 0.25 1 0.55 0.55 0.55 0
.. 0 0.005 0.025 0.04 0.07 0.07 0.07
1b F/Fy 0 0.25 1 0.55 0.55 0.55 0
. 0 0.005 0.025 0.04 0.12 0.12 0.12
Non-ductile moment
frame
2a F/Fy 0 1 0.15 0.15 0.15 0.15 0
. 0 0.01 0.03 0.05 0.06 0.06 0.06
2b F/Fy 0 1 0.15 0.15 0.15 0.15 0
. 0 0.01 0.05 0.055 0.06 0.06 0.06
Ductile moment frame 3a F/Fy 0 1 1.05 0.45 0.45 0.45 0
. 0 0.01 0.04 0.06 0.08 0.08 0.08
3b F/Fy 0 1 1.05 0.8 0.8 0.8 0
. 0 0.01 0.04 0.06 0.08 0.08 0.08
Stiff non-ductile system 4a F/Fy 0 1 0.3 0.3 0.3 0.3 0
. 0 0.004 0.02 0.06 0.08 0.08 0.08
4b F/Fy 0 1 0.5 0.5 0.5 0.5 0
. 0 0.004 0.04 0.06 0.08 0.08 0.08
Stiff, highly pinched nonductile
system
5a F/Fy 0 0.67 1 0.6 0.067 0.067 0
. 0 0.002 0.005 0.028 0.04 0.06 0.06
5b F/Fy 0 0.67 1 0.6 0.067 0.067 0
. 0 0.002 0.005 0.042 0.06 0.06 0.06
Elastic-perfectly-plastic 6a F/Fy 0 1 1 1 1 1 0
. 0 0.01 0.02 0.03 0.07 0.07 0.07
6b F/Fy 0 1 1 1 1 1 0
. 0 0.01 0.02 0.03 0.12 0.12 0.12
Limited-ductile moment
frame
7a F/Fy 0 1 1 0.2 0.2 0.2 0
. 0 0.01 0.02 0.025 0.04 0.04 0.04
7b F/Fy 0 1 1 0.2 0.2 0.2 0
. 0 0.01 0.02 0.04 0.06 0.06 0.06
Non-ductile gravity frame 8a F/Fy 0 1 1 0 0 0 0
. 0 0.025 0.025 0.025 0.025 0.025 0.025
8b F/Fy 0 1 1 0.55 0.55 0.55 0
. 0 0.025 0.025 0.03 0.04 0.04 0.04
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-13
for Focused Analytical Studies
3.2.1 Springs 1a and 1b – Typical Gravity Frame Systems
Springs 1a and 1b are intended to model the behavior of typical gravity frame
systems in buildings. The force-displacement capacity boundary includes a
strength drop immediately after yielding that terminates on a plateau with a
residual strength of 55% of the yield strength (Figure 3-7). The “a” and “b”
versions of this spring differ in the length of the residual strength plateau,
which extends to an ultimate deformation capacity of 7% drift in Spring 1a
and 12% drift in Spring 1b. This represents the maximum ductility that is
achieved by any of the spring subsystems.
The hysteretic behaviors of Spring 1a and Spring 1b, both with and without
cyclic degradation, are shown in Figure 3-8 and Figure 3-9. In each figure,
the initial force-displacement capacity boundary (before cyclic degradation)
is overlaid onto the hysteretic plots.
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
(Spring 1a) (Spring 1b)
Figure 3-7 Force-displacement capacity boundaries for Spring 1a and Spring 1b.
(a) (b)
Figure 3-8 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring 1a:
(a) without cyclic degradation; and (b) with cyclic degradation.
3-14 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
(a) (b)
Figure 3-9 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for
Spring 1b: (a) without cyclic degradation; and (b) with cyclic degradation.
Springs 1a and 1b are consistent with steel gravity frame systems with classic
simple shear-tab connections. Experiments have shown that the gap between
the beam and column flange is a critical parameter in determining forcedisplacement
behavior of these systems. When a joint achieves enough
rotation to result in contact between the beam and column flanges, bolts in
the shear tab will be subjected to bearing strength failure, and the shear
connection fails (Liu and Astaneh, 2003). This limit state marks the end of
the residual strength plateau.
Spring 1a is consistent with a system in which beam/column flange contact
occurs relatively early (7% drift), while Spring 1b is consistent with a system
in which this contact occurs later (12% drift). Results from experimental
tests on steel shear tab connections (Figure 3-10) exhibit a behavior that is
similar to behavior the modeled in Springs 1a and 1b.
Figure 3-10 Hysteretic behavior from experimental tests on beam-to-column
shear tab connections (Liu and Astaneh, 2003).
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-15
for Focused Analytical Studies
3.2.2 Springs 2a and 2b – Non-Ductile Moment Frame Systems
Springs 2a and 2b are intended to model the behavior of non-ductile
moment-resisting frame systems in buildings. They are characterized by a
force-displacement capacity boundary that includes strength degradation
immediately after yielding, a low residual strength plateau at 15% of the
yield strength, and an ultimate deformation capacity of 6% drift (Figure
3-11). The “a” and “b” versions of this spring differ in the negative slope of
the strength-degrading segment, which is negative 43% in Spring 2a and
negative 21% in Spring 2b.
The hysteretic behaviors of Spring 2a and Spring 2b, both with and without
cyclic degradation, are shown in Figure 3-12 and Figure 3-13. In each
figure, the initial force-displacement capacity boundary (before cyclic
degradation) is overlaid onto the hysteretic plots.
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
(Spring 2a) (Spring 2b)
Figure 3-11 Force-displacement capacity boundaries for Spring 2a and Spring 2b.
(a) (b)
Figure 3-12 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring 2a:
(a) without cyclic degradation; and (b) with cyclic degradation.
3-16 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
(a) (b)
Figure 3-13 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring 2b:
(a) without cyclic degradation; and (b) with cyclic degradation.
Systems with this behavior could be constructed in either steel or concrete.
In the case of steel, these springs would be representative of momentresisting
frames with pre-Northridge welded beam-column connections, in
which connection behavior is characterized by fracture and a large reduction
in lateral force resistance. In the case of concrete, these springs would be
representative of older (pre-1975) concrete frames with inadequate joint
reinforcement, minimal concrete confinement and other poor detailing
characteristics that would be prone to shear failure. Results from
experimental tests on pre-Northridge welded steel connections and shearcritical
reinforced concrete columns (Figure 3-14) exhibit a behavior that is
similar to the behavior modeled in Springs 2a and 2b.
(a) (b)
Figure 3-14 Hysteretic behavior from experimental tests on: (a) pre-Northridge welded steel beam-column
connections (Goel and Stojadinovic, 1999); and (b) shear-critical reinforced concrete columns
(Elwood and Moehle, 2003).
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-17
for Focused Analytical Studies
3.2.3 Springs 3a and 3b – Ductile Moment Frame Systems
Springs 3a and 3b are intended to model the behavior of moderately-ductile
moment-resisting frame systems in buildings. They are characterized by a
force-displacement capacity boundary that includes a strength-hardening
segment with a positive slope equal to 2% of the elastic stiffness, a strengthdegrading
segment that begins at 4% drift and ends at 6% drift, and a residual
strength plateau with an ultimate deformation capacity of 8% drift (Figure
3-15). The “a” and “b” versions of this spring differ in the negative slope of
the strength-degrading segment, which is negative 30% in Spring 3a and
negative 13% in Spring 3b, and in the height of the residual strength plateau,
which is 50% of yield in Spring 3a and 80% in Spring 3b.
The hysteretic behaviors of Spring 3a and Spring 3b, both with and without
cyclic degradation, are shown in Figure 3-16 and Figure 3-17. In each
figure, the initial force-displacement capacity boundary (before cyclic
degradation) is overlaid onto the hysteretic plots.
Systems with this type of behavior could include special steel momentresisting
frames with ductile (e.g., post-Northridge) beam-column
connections, or well-detailed reinforced concrete moment-resisting frames.
Results from experimental tests on post-Northridge reduced beam steel
moment connections (Figure 3-18) exhibit a behavior that is similar to the
behavior modeled in Springs 3a and 3b.
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
(Spring 3a) (Spring 3b)
Figure 3-15 Force-displacement capacity boundaries for Spring 3a and Spring 3b.
3-18 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
(a) (b)
Figure 3-16 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring
3a: (a) without cyclic degradation; and (b) with cyclic degradation.
(a) (b)
Figure 3-17 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring
3b: (a) without cyclic degradation and (b) with cyclic degradation.
Figure 3-18 Hysteretic behavior from experimental tests on post-Northridge reduced-beam steel
moment connections (Venti and Engelhardt, 1999).
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-19
for Focused Analytical Studies
3.2.4 Springs 4a and 4b – Stiff, Non-Ductile Systems
Springs 4a and 4b are intended to model the behavior of relatively stiff
lateral-force-resisting systems that are subject to significant in-cycle strength
degradation at small levels of deformation. They are characterized by a
force-displacement capacity boundary that includes a strength-degrading
segment beginning at 0.4% drift and terminating on a residual strength
plateau with an ultimate deformation capacity of 8% drift (Figure 3-19). The
“a” and “b” versions of this spring differ in the negative slope of the
strength-degrading segment, which is negative 18% in Spring 4a and
negative 6% in Spring 4b, and in the height of the residual strength plateau,
which is 30% of yield in Spring 4a and 50% in Spring 4b.
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
(Spring 4a) (Spring 4b)
Figure 3-19 Force-displacement capacity boundaries for Spring 4a and Spring 4b.
The hysteretic behaviors of Spring 4a and Spring 4b, both with and without
cyclic degradation, are shown in Figure 3-20 and Figure 3-21. They
resemble a typical peak-oriented model with severe cyclic degradation of
strength, unloading, and reloading stiffness parameters. In each figure, the
initial force-displacement capacity boundary (before cyclic degradation) is
overlaid onto the hysteretic plots.
Systems with this type of behavior could include steel concentric braced
frames, which experience a sharp drop in strength following buckling of the
braces at small levels of lateral deformation demand. Results from
experimental tests on steel concentric braced frames (Figure 3-22) exhibit a
behavior that is similar to the behavior modeled in Springs 4a and 4b.
3-20 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
(a) (b)
Figure 3-20 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring 4a:
(a) without cyclic degradation; and (b) with cyclic degradation.
(a) (b)
Figure 3-21 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring 4b:
(a) without cyclic degradation; and (b) with cyclic degradation.
Figure 3-22 Hysteretic behavior from experimental tests on steel concentric braced frames (Uriz and
Mahin, 2004).
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-21
for Focused Analytical Studies
3.2.5 Springs 5a and 5b – Stiff, Highly-Pinched Non-Ductile
Systems
Springs 5a and 5b are intended to model the behavior of stiff and highlypinched
non-ductile lateral-force-resisting systems in buildings. They are
characterized by a force-displacement capacity boundary with the highest
initial stiffness of any of the spring subsystems, followed by varying levels of
strength degradation and an ultimate deformation capacity of 6% drift
(Figure 3-23). In both the “a” and “b” versions of this spring, peak strength
occurs at 0.5% drift, and initial cracking occurs at 67% of peak strength at a
drift ratio of 0.2%. The “a” and “b” versions of this spring differ in the
slopes of the two strength-degrading segments, which are 5% and 13% (of
the initial elastic stiffness) in Spring 5a, and 3% and 9% in Spring 5b. They
also differ in the presence of a residual strength plateau, which exists in
Spring 5a, but not in Spring 5b.
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
(Spring 5a) (Spring 5b)
Figure 3-23 Force-displacement capacity boundaries for Spring 5a and Spring 5b.
The hysteretic behaviors of Spring 5a and Spring 5b, both with and without
cyclic degradation, are shown in Figure 3-24 and Figure 3-25. They
resemble a sliding system with cyclic degradation of strength, unloading, and
reloading stiffness parameters. In each figure, the initial force-displacement
capacity boundary (before cyclic degradation) is overlaid onto the hysteretic
plots.
Systems with this type of behavior could include masonry walls and concrete
frames with masonry infill. Results from experimental tests on these systems
(Figure 3-26) exhibit a behavior that is similar to the behavior modeled in
Springs 5a and 5b.
3-22 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
(a) (b)
Figure 3-24 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring 5a:
(a) without cyclic degradation; and (b) with cyclic degradation.
(a) (b)
Figure 3-25 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring 5b:
(a) without cyclic degradation; and (b) with cyclic degradation.
(a) (b)
Figure 3-26 Hysteretic behavior from experimental tests on: (a) reinforced masonry walls (Shing et al.,
1991); and (b) concrete frames with masonry infill (Dolsek and Fajfar, 2005).
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-23
for Focused Analytical Studies
3.2.6 Springs 6a and 6b – Elastic-Perfectly-Plastic Systems
Springs 6a and 6b are intended to model the behavior of idealized elasticperfectly-
plastic systems with full, kinematic hysteresis loops, without any
cyclic or in-cycle degradation of strength or stiffness. The forcedisplacement
capacity boundaries are shown in Figure 3-27. The “a” and “b”
versions of this spring differ in their finite ultimate deformation capacity,
which is 7% drift in Spring 6a and 12% drift in Spring 6b.
Spring 6a and Spring 6b were analyzed with a constant force-displacement
capacity boundary (no cyclic degradation). The resulting hysteretic
behaviors are shown in Figure 3-28, with initial force-displacement capacity
boundaries overlaid onto the hysteretic plots.
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
(Spring 6a) (Spring 6b)
Figure 3-27 Force-displacement capacity boundaries for Spring 6a and Spring 6b.
(a) (b)
Figure 3-28 Force-displacement capacity boundary overlaid onto hysteretic behaviors for:
(a) Spring 6a without cyclic degradation; and (b) Spring 6b without cyclic
degradation.
3-24 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
This is a highly idealized system developed for comparison of results.
Practically speaking, only selected buckling-restrained braces or baseisolated
systems would be capable of emulating this behavior under repeated
cycles of large deformation demand.
3.2.7 Springs 7a and 7b – Limited-Ductility Moment Frame
Systems
Springs 7a and 7b are intended to model the behavior of limited-ductility
moment-resisting frame systems in buildings. They are characterized by a
force-displacement capacity boundary with a short yielding plateau that
maintains strength until a drift of 2%, followed strength degradation that
terminates on a short residual strength plateau set at 20% of the yield strength
(Figure 3-29). The “a” and “b” versions of this spring differ in the negative
slope of the strength-degrading segment, which is negative 160% in Spring
7a and negative 40% in Spring 7b, and in the ultimate deformation capacity,
which is 4% drift in Spring 7a and 6% drift in Spring 7b.
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
(Spring 7a) (Spring 7b)
Figure 3-29 Force-displacement capacity boundaries for Spring 7a and Spring 7b.
The hysteretic behaviors of Spring 7a and Spring 7b, both with and without
cyclic degradation, are shown in Figure 3-30 and Figure 3-31. In each
figure, the initial force-displacement capacity boundary (before cyclic
degradation) is overlaid onto the hysteretic plots.
Systems with this type of behavior could include older reinforced concrete
frames not designed for seismic loads, which can be lightly reinforced, and
may have inadequate joint reinforcement or concrete confinement. Results
from experimental tests on lightly reinforced concrete columns (Figure 3-32)
exhibit a behavior that is similar to the behavior modeled in Springs 7a and
7b.
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-25
for Focused Analytical Studies
(a) (b)
Figure 3-30 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring 7a:
(a) without cyclic degradation; and (b) with cyclic degradation.
(a) (b)
Figure 3-31 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for Spring 7b:
(a) without cyclic degradation; and (b) with cyclic degradation.
Figure 3-32 Hysteretic behavior from experimental tests on lightly reinforced concrete columns (Elwood
and Moehle, 2006; Sezen, 2002).
3-26 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
3.2.8 Springs 8a and 8b – Non-Ductile Gravity Frame
Systems
Springs 8a and 8b are intended to model the behavior of non-ductile gravity
frame systems in buildings. The force-displacement capacity boundary
includes significant strength degradation immediately after yielding, and
limited ultimate deformation capacity (Figure 3-33). The “a” and “b”
versions of this spring differ in the strength that is lost after yield, which is
100% in Spring 8a, and 45% in Spring 8b, and in the ultimate deformation
capacity, which is 2.5% drift in Spring 8a and 4% drift in Spring 8b. They
also differ in the presence of a residual strength plateau, which does not exist
in Spring 8a, but does in Spring 8b.
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
0
0.2
0.4
0.6
0.8
1
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
.
F / Fy
(Spring 8a) (Spring 8b)
Figure 3-33 Force-displacement capacity boundaries for Spring 8a and Spring 8b.
The hysteretic behaviors of Spring 8a and Spring 8b, both with and without
cyclic degradation, are shown in Figure 3-34 and Figure 3-35. In each
figure, the initial force-displacement capacity boundary (before cyclic
degradation) is overlaid onto the hysteretic plots.
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-27
for Focused Analytical Studies
(a) (b)
Figure 3-34 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for
Spring 8a: (a) without cyclic degradation; and (b) with cyclic degradation.
(a) (b)
Figure 3-35 Initial force-displacement capacity boundary overlaid onto hysteretic behaviors for
Spring 8b: (a) without cyclic degradation; and (b) with cyclic degradation.
3.3 Multiple Spring Models
Multiple spring models were used to represent the behavior of more complex
structural systems containing subsystems with different hysteretic and forcedisplacement
capacity boundary characteristics linked by rigid diaphragms.
Multi-spring SDOF systems were developed by placing individual springs in
parallel. Combinations were performed in a manner consistent with
combinations that would be encountered in real structural systems. For each
such combination, variations in the relative contribution of individual springs
to the initial stiffness and maximum lateral strength over a range of periods
were considered.
3-28 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
3.3.1 Multi-Spring Combinations of Single-Spring Systems
Of the numerous combinations possible, only assemblages consisting of two
springs in parallel were considered in this investigation. Furthermore, only
springs including cyclic degradation were considered in multi-spring
combinations. This was done to limit the number of possible permutations
under consideration, but also because, in general, realistic systems
experiencing strong in-cycle degradation will also experience cyclic
degradation.
Two-spring assemblages consisting of a lateral-force-resisting system
(Springs 2, 3, 4, 5, 6, or 7), working in combination with a gravity frame
system (Springs 1a, 1b, 8a, or 8b), were used. For example, a combination of
Spring 2a with Spring 1a would be representative of a non-ductile moment
frame system with a typical gravity frame back-up system in parallel.
In general, it is not realistic to assume that the contribution of each
subsystem to the peak lateral strength of the combined system would be
equal. In most cases, the lateral-force-resisting system in a building would
be expected to be stronger and stiffer than the gravity system. For this
reason, systems were combined using an additional parameter, N, as a
multiplier on the contribution of lateral-force-resisting springs in the
combined system. Multi-spring systems then carry a designation of
“NxJa+1a” or “NxJa+1b” where “N” is the peak strength multiplier (N = 1,
2, 3, 5, or 9), “J” is the lateral-force-resisting spring number (J = 2, 3, 4, 5, 6,
or 7), and 1a or 1b is the gravity system identifier. Using this designation,
“3x2a+1a” would identify a system consisting of a multiple of three nonductile
moment frame springs (Spring 2a) in combination with a single
gravity system spring (Spring 1a).
To investigate potential period-dependency, multi-spring systems were tuned
to center the resulting periods of vibration for each set of “NxJa” lateralforce-
resisting systems approximately around T=1.0s (representing relatively
stiff systems) and T=2.0s (representing relatively flexible systems). This was
accomplished by assuming two different story masses of M=8.87 tons or
M=35.46 tons, respectively.
In summary the following series of multi-spring systems were investigated:
. Series 1: NxJa + 1a (M=8.87 ton; relatively stiff)
. Series 2: NxJb + 1a (M=35.46 ton; relatively flexible)
. Series 3: NxJa + 1b (M=8.87 ton; relatively stiff)
. Series 4: NxJb + 1b (M=35.46 ton; relatively flexible)
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-29
for Focused Analytical Studies
Multi-spring combinations using Spring 8a and Spring 8b were created and
analyzed, however, the resulting behavior was not substantially different
from other systems analyzed. As a result, this data was not investigated in
detail, and information on these combinations has not been provided. As part
of the series of investigations, each “NxJa” lateral-force-resisting system was
analyzed without the 1a or 1b gravity system in order to compare results both
with and without the contribution of the back-up system. A representative
force-displacement capacity boundary from each multi-spring system is
shown in Figure 3-36 through Figure 3-41.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.00 0.02 0.04 0.06 0.08 0.10
F/(Fy of 1a)
.
2a
1a
2a+1a
Figure 3-36 Combined force-displacement capacity boundary for spring
2a +1a (normalized by the strength of Spring 1a).
0
0.5
1
1.5
2
2.5
0.00 0.02 0.04 0.06 0.08 0.10
F/(Fy of 1a)
.
3a
1a
3a+1a
Figure 3-37 Combined force-displacement capacity boundary for spring
3a +1a (normalized by the strength of Spring 1a).
3-30 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.00 0.02 0.04 0.06 0.08 0.10
F/(Fy of 1a)
.
4a
1a
4a+1a
Figure 3-38 Combined force-displacement capacity boundary for spring
4a +1a (normalized by the strength of Spring 1a).
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.00 0.02 0.04 0.06 0.08 0.10
F/(Fy of 1a)
.
5a
1a
5a+1a
Figure 3-39 Combined force-displacement capacity boundary for spring
5a +1a (normalized by the strength of Spring 1a).
0
0.5
1
1.5
2
2.5
0.00 0.02 0.04 0.06 0.08 0.10
F/(Fy of 1a)
.
6a
1a
6a+1a
Figure 3-40 Combined force-displacement capacity boundary for spring
6a +1a (normalized by the strength of Spring 1a).
FEMA P440A 3: Development of Single-Degree-of-Freedom Models 3-31
for Focused Analytical Studies
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.00 0.02 0.04 0.06 0.08 0.10
F/(Fy of 1a)
.
7a
1a
7a+1a
Figure 3-41 Combined force-displacement capacity boundary for spring
7a +1a (normalized by the strength of Spring 1a).
Each multi-spring combination was subjected to an ATC-24 type loading
protocol with a degrading force-displacement capacity boundary (cyclic
degradation). The resulting hysteretic behaviors for the combination of
Nx2a+1a for (N = 1, 2, 3, 5, and 9) are shown in Figure 3-42 through Figure
3-44. In addition, individual Spring 2a is shown in Figure 3-44 for
comparison. In each figure, the initial combined force-displacement capacity
boundary (before cyclic degradation) is overlaid onto the hysteretic plots.
(a) (b)
Figure 3-42 Initial force-displacement capacity boundary overlaid onto hysteretic behavior for:
(a) Spring 1x2a+1a; and (b) Spring 2x2a+1a; both with cyclic degradation.
3-32 3: Development of Single-Degree-of-Freedom Models FEMA P440A
for Focused Analytical Studies
(a) (b)
Figure 3-43 Initial force-displacement capacity boundary overlaid onto hysteretic behavior for: (a) Spring
1x3a+1a; and (b) Spring 5x2a+1a; both with cyclic degradation.
(a) (b)
Figure 3-44 Initial force-displacement capacity boundary overlaid onto hysteretic behavior for: (a) Spring
9x2a+1a; and (b) individual Spring 2a; both with cyclic degradation.
As might be expected, the more the multiplier “N” for Spring 2a increases,
the more the combined system resembles Spring 2a itself (Figure 3-44), and
the more the behavior of the combined system would be expected to be
dominated by the characteristics of the lateral-force-resisting spring
component. Conversely, for lower multiples of “N”, the characteristics of
the gravity system are more visible in the combined system properties
(Figure 3-42), and would be expected to play a more significant role in the
behavior of the combined system.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-1
Focused Analytical Studies
Chapter 4
Results from Single-Degree-of-
Freedom Focused Analytical
Studies
This chapter summarizes the results of focused analytical studies on singlespring
and multi-spring systems, compares results to recommendations
contained in FEMA 440, and explains the development of a new equation
measuring the potential for lateral dynamic instability.
4.1 Summary of Analytical Results
There were 160 single-spring systems (eight basic spring types, “a” and “b”
versions of each, with and without cyclic degradation, tuned to five different
periods of vibration). Each system was subjected to incremental dynamic
analysis using 56 ground motion records scaled to multiple levels of
increasing intensity. This resulted in over 600,000 nonlinear response
history analyses on single-spring systems.
There were 600 multi-spring systems (six lateral-force-resisting springs, “a”
and “b” versions of each, five relative strength multipliers, five different
gravity spring combinations, tuned with two different story masses). Each
system was subjected to incremental dynamic analysis using 56 ground
motion records scaled to multiple levels of increasing intensity. This resulted
in over 2,000,000 nonlinear response history analyses on multi-spring
systems.
In total, results from over 2.6 million nonlinear response history analyses
were available for review. Given the large volume of analytical data,
customized algorithms were developed for post-processing, statistical
analysis, and visualization of results. Results are summarized in the sections
that follow. More complete sets of data are presented in the appendices. A
customized visualization tool that was developed to view results of multispring
studies, along with all available data, is included on the CD
accompanying this report. Use of the visualization tool is described in
Appendix C and Appendix D.
4.2 Observations from Single-Spring Studies
This section summarizes the results from nonlinear dynamic analyses of
single-spring systems. Results from these studies were used to:
4-2 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
. identify predominant characteristics of median incremental dynamic
analysis (IDA) curves for these systems,
. demonstrate a relationship between IDA curves and features of the forcedisplacement
capacity boundaries, and
. qualitatively determine the effects of different degrading behaviors on
the dynamic stability of structural systems.
Only selected results are presented here. Quantile (16th, 50th and 84th
percentile) IDA curves for each of the single-spring systems are provided in
Appendix B. The horizontal axis for all single-spring IDA results is the
maximum story drift ratio, .max, in radians.
4.3 Characteristics of Median IDA Curves
Individual incremental dynamic analysis (IDA) curves for single ground
motion records are very sensitive to dynamic interaction between the
properties of the system and the characteristics of the ground motion.
Quantile (16th, 50th and 84th percentile) IDA curves, however, are much more
stable and provide better information on the central tendency (median) and
variability (dispersion) in system response. In general, median IDA curves
exhibit the following characteristics (Figure 4-1):
. An initial linear segment corresponding to linear-elastic behavior in
which in lateral deformation demand is proportional to ground motion
intensity, regardless of the characteristics of the system or the ground
motion. This segment extends from the origin to the onset of yielding.
. A second curvilinear segment corresponding to inelastic behavior in
which lateral deformation demand is no longer proportional to ground
motion intensity. As intensity increases, lateral deformation demands
increase at a faster rate. This segment corresponds to softening of the
system, or reduction in stiffness (reduction in the slope of the IDA
curve). In this segment, the system “transitions” from linear behavior to
eventual dynamic instability. Although a curvilinear segment is always
present, in some cases the transition can be relatively long and gradual,
while in other cases it can be very short and abrupt.
. A final linear segment that is horizontal, or nearly horizontal, in which
infinitely large lateral deformation demands occur at small increments in
ground motion intensity. This segment corresponds to the point at which
a system becomes unstable (lateral dynamic instability). For SDOF
systems, this point corresponds to the ultimate deformation capacity at
which the system loses all lateral-force-resisting capacity.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-3
Focused Analytical Studies
Figure 4-1 Characteristic segments of a median IDA curve.
In some systems, the initial linear segment can be extended beyond yield into
the inelastic range (Figure 4-2). In this segment lateral deformation demand
is approximately proportional to ground motion intensity, which is consistent
with the familiar equal-displacement approximation for estimating inelastic
displacements. The range of lateral deformation demands over which the
equal-displacement approximation is applicable depends on the
characteristics of the force-displacement capacity boundary of the system and
the period of vibration.
Figure 4-2 Characteristic segments of a median IDA curve with a pseudolinear
segment.
4-4 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
4.3.1 Dependence on Period of Vibration
Figure 4-3 shows the force-displacement capacity boundary and resulting
IDA curves for Spring 3a with different periods of vibration. Each system is
tuned to a different lateral strength and stiffness so results are compared
using the normalized intensity measure R = Sa(T,5%)/Say(T,5%). Intensities
larger than R = 1.0 mean the system is behaving inelastically.
Figure 4-3 Force-displacement capacity boundary and median IDA curves for Spring 3a with various
periods of vibration.
In general, moderate and long period systems with zero or positive post-yield
stiffness in the force-displacement capacity boundary follow the equal
displacement trend well into the nonlinear range. For systems with periods
longer than 0.5s, Spring 3a exhibits an extension of the initial linear segment
well beyond the yield drift of 0.01. In contrast, the short period system
(T=0.2s) diverges from the initial linear segment just after yielding, even at
deformations within the strength-hardening segment of the forcedisplacement
capacity boundary (drifts between 0.01 and 0.04).
4.3.2 Dispersion in Response
Nonlinear response is sensitive to the characteristics of the ground motion
record, and will vary from one ground motion to the next, even when scaled
to the same intensity. For a given level of ground motion intensity, the
lateral deformation demand can be significantly smaller or significantly
larger than the value shown on median IDA curves. As the level of ground
motion intensity increases, the dispersion in response tends to increase.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-5
Focused Analytical Studies
Figure 4-4 shows three quantile IDA curves for Spring 3b with period of
vibration of 2.0s. The 50% (median) IDA curve indicates that, for a given
level of ground motion intensity (Sa), half of all deformation demands are
smaller and half are larger than values along this curve. Because the
distribution of demands is lognormally distributed, the dispersion about the
median is not symmetric. The upper (16%) curve in the figure indicates that,
for a given level of ground motion intensity, 16% of all lateral deformation
demands are to the left of this curve while 84% are to the right. This means
that lateral deformation demands along this curve have an 84% chance of
being exceeded. Similarly the lower (84%) curve corresponds to lateral
deformation demands with a 16% chance of being exceeded.
Figure 4-4 Force-displacement capacity boundary and 16th, 50th and 84th percentile IDA curves for
Spring 3b with a period of vibration T=2.0s.
4.4 Influence of the Force-Displacement Capacity
Boundary
Comparisons between force-displacement capacity boundaries and median
IDA curves show a strong correlation between the shape of the resulting
curves and key features of the force-displacement capacity boundary, such as
post-yield behavior and onset of degradation, slope of degradation, ultimate
deformation capacity, and presence of cyclic degradation.
Figure 4-5 shows the force-displacement capacity boundary and resulting
median IDA curve for Spring 3b with a period of 2.0s. With a positive postyield
slope, delayed onset of degradation, and robust residual strength
plateau with an extended maximum deformation capacity, the resulting IDA
curve includes both linear and pseudo-linear segments and a gradual
transition to lateral dynamic instability.
4-6 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
Figure 4-5 Force-displacement capacity boundary and median IDA curve for Spring 3b with a
period of vibration T=2.0s.
Figure 4-6 shows the force-displacement capacity boundary and resulting
median IDA curve for Spring 2a with a period of 2.0s. With the onset of
degradation occurring immediately after yielding, the shape of the resulting
IDA curve changes. The pseudo-linear segment disappears, but with the
presence of a residual strength plateau, the transition segment remains
somewhat gradual until lateral dynamic instability.
Figure 4-6 Force-displacement capacity boundary and median IDA curve for Spring 2a with a
period of vibration T=2.0s.
Figure 4-7 shows the force-displacement capacity boundary and resulting
median IDA curve for Spring 6a with a period of 2.0s. With a broad yielding
plateau, the pseudo-linear segment extends well into the inelastic range.
Without a residual strength plateau, however, the system abruptly transitions
into lateral dynamic instability.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-7
Focused Analytical Studies
Figure 4-7 Force-displacement capacity boundary and median IDA curve for Spring 6a with a period of
vibration T=2.0s.
Figure 4-8 shows the force-displacement capacity boundary and resulting
median IDA curve for Spring 8a with a period of 2.0s. With severe strength
degradation occurring immediately after yielding, and the absence of a
residual strength plateau, the system abruptly transitions from linear elastic
behavior directly into lateral dynamic instability with little or no transition.
Figure 4-8 Force-displacement capacity boundary and median IDA curve for Spring 8a with a period of
vibration T=2.0s.
These observed relationships suggest that dynamic response is directly
influenced by the features of a force-displacement capacity boundary. Figure
4-9 shows how the characteristic segments of a median IDA curve relate to
these features. Note that the relationship depicted in this idealized graphical
representation is dependent upon the period of the system, as described in
Section 4.3.1.
4-8 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
Figure 4-9 Relationship between IDA curves and the features of a typical
force-displacement capacity boundary
For low levels of ground motion intensity, the initial linear segment of the
IDA curve is controlled by the effective stiffness of the system, Ke . Since
the response is linear there is no dispersion evident in this segment. As the
intensity increases the system reaches its yield point, Fy ,.y . Systems with a
non-negative post-elastic stiffness, . Ke , will likely exhibit a pseudo-linear
segment. Beyond yield, dispersion appears in the nonlinear response due to
ground motion variability, and the 16th and 84th percentile IDA curves begin
to diverge from the median curve.
The extent of the pseudo-linear segment depends on the initial post elastic
stiffness, . Ke , and ends prior to reaching the strength hardening limit,
Fc ,.c (also known as the capping point). For systems that exhibit negative
stiffness, . Ke , immediately after yielding, the pseudo-linear segment may be
very short or non-existent. Also, for short-period systems, the pseudo-linear
segment can be very short, even if the system has positive post-yield
stiffness.
As the ground motion intensity increases further, deformation demands
increase at a faster rate, the IDA curve begins to flatten, and the curvilinear
softening segment emerges. Dispersion between the quantile curves also
increases. Beyond the strength hardening limit, Fc ,.c , degradation occurs,
and the softening increases at a faster rate. The presence of a residual
INSTABILITY
LINEAR
PSEUDO
LINEAR
SOFTENING
.
boundar
16%
84%
50% INSTABILITY
LINEAR
.
CAPACITY
BOUNDARY
SaT (g)
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-9
Focused Analytical Studies
strength plateau, Fr ,.r , can extend the softening segment and delay the
eventual transition into lateral dynamic instability. The point at which
instability occurs corresponds to the ultimate deformation capacity, .u , at
which the system loses all lateral force resistance.
This relationship suggests that it is possible to estimate the nonlinear
dynamic behavior of a system based on knowledge of the characteristics of
the force-displacement capacity boundary of the system. The influence that
important features of the force-displacement capacity boundary have on
nonlinear response is explained in more detail in the sections that follow.
4.4.1 Post-Yield Behavior and Onset of Degradation
The three systems shown in Figure 4-10 have the same elastic stiffness, same
yield strength, but different post-yield characteristics. The forcedisplacement
capacity boundary of Spring 2a experiences strength
degradation immediately after yielding. In contrast, Spring 3a has a
moderate yielding plateau before the onset of similar strength degradation,
while Spring 6a has elastic-perfectly-plastic behavior up to the ultimate
deformation capacity.
These three systems have the same elastic behavior, but at drift ratios larger
than 0.02, their relative potential for in-cycle strength degradation, and their
resulting collapse behaviors, are all very different. Key parameters related to
the observed change in response are the post-yield slope and the strength
hardening limit (capping point). The presence of a non-negative post-yield
slope and any delay before the onset of degradation reduces potential
in-cycle strength degradation and improves the collapse capacity of a system.
Figure 4-10 Effect of post-yield behavior on the collapse capacity of a system (Springs 2a, 3a and 6a with
T=2.0s).
4-10 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
4.4.2 Slope of Degradation
Figure 4-11 shows the force-displacement capacity boundaries of Spring 2a
and Spring 2b along with the corresponding IDA curves. These two systems
have the same elastic stiffness, same yield strength, but differ in the negative
slope of the strength-degrading segment and, therefore, in their potential for
in-cycle strength degradation. They also have the same ultimate deformation
capacity, but Spring 2b has a shorter residual strength plateau than Spring 2a
because of the different slope.
Figure 4-11 Effect of slope of degradation on the collapse capacity of a system (Springs 2a
and 2b with T=1.0s).
The two systems have the same elastic behavior, but their response at drift
demands larger than 0.01 is very different. Spring 2a, with a steeper
degrading slope, likely experiences in-cycle strength degradation and reaches
its collapse capacity relatively early, while Spring 2b, with a more shallow
degrading slope, reaches a collapse capacity that is approximately 50%
larger.
Figure 4-12 shows the force-displacement capacity boundaries of Spring 5a
and Spring 5b along with the corresponding IDA curves. As in the case of
Springs 2a and 2b, these two systems differ in the negative slope of the
strength-degrading segments. They also differ in the presence of a residual
strength plateau, which exists in Spring 5a, but not in Spring 5b.
As shown in the figure, the median IDA curves are similar up to 0.005 drift,
at which both systems reach their peak strength. Beyond this point, the
curves diverge as a result of the change in negative slope. Spring 5a, with
steeper degrading slopes, reaches its collapse capacity sooner, while Spring
5b, with more shallow degrading slopes, reaches a higher collapse capacity.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-11
Focused Analytical Studies
The key parameter related to the observed change in response is the negative
slope of the strength-degrading segment. In both examples, the change in
negative slope changed the magnitude of potential in-cycle strength
degradation, and overshadowed any changes in the residual strength plateau,
as long as the ultimate deformation capacity remained the same.
Figure 4-12 Effect of slope of degradation on the collapse capacity of a system (Springs 5a and 5b with
T=1.0s).
4.4.3 Ultimate Deformation Capacity
Figure 4-13 shows the force-displacement capacity boundaries and
corresponding IDA curves for Springs 1a and 1b. Figure 4-14 shows the
force-displacement capacity boundaries and corresponding IDA curves for
Springs 6a and 6b. These spring systems have very different post-yield
behaviors, one with strength degradation (Springs 1a and 1b) and the other
with elasto-plastic behavior (Springs 6a and 6b). In both cases, the “b”
versions of each spring have higher ultimate deformation capacities.
Figure 4-13 Effect of ultimate deformation capacity on the collapse capacity of a system (Springs 1a and 1b
with T=1.0s).
4-12 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
Figure 4-14 Effect of ultimate deformation capacity on the collapse capacity of a system
(Springs 6a and 6b with T=1.0s).
In both examples, increasing the ultimate deformation capacity resulted in
more than a 50% increase in collapse capacity. The key parameter related to
the observed change in response is the increment in the ultimate deformation
capacity. Observed changes in collapse capacity resulting from increases in
the ultimate deformation capacity were insensitive to the other characteristics
of the post-yield behavior of the springs.
4.4.4 Degradation of the Force-Displacement Capacity
Boundary (Cyclic Degradation)
In general, most components will exhibit some level of cyclic degradation.
To investigate the effects of cyclic degradation, the “a” and “b” versions of
each spring (except Spring 6) were analyzed with both a constant forcedisplacement
capacity boundary and a degrading force-displacement capacity
boundary.
Consistent with observations from past studies, comparison of results
between springs both with and without cyclic degradation show that the
effects of cyclic degradation (as measured by gradual movement of the
capacity boundary) are relatively unimportant in comparison with in-cycle
degradation (as measured by the extent and steepness of negative slopes in
the capacity boundary). This trend is illustrated for Spring 3b in Figure 4-15,
but can be observed in the results for many spring systems. Although the
system without cyclic degradation has a higher median collapse capacity, the
difference is not very large. For the single-spring systems studied, the
difference between median collapse capacity with and without cyclic
degradation is shown in Appendix B. In general, this difference was
typically less than 10%.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-13
Focused Analytical Studies
Figure 4-15 Effect of degradation of the force-displacement capacity boundary on the
collapse capacity of a system (Spring 3b, T=2.0s, with and without cyclic
degradation).
This observation has two important exceptions. First, the effect of cyclic
degradation increases as the level of in-cycle degradation increases. Systems
such as Spring 2b with a steep negative slope in the capacity boundary,
indicating a strong potential for severe in-cycle strength degradation, showed
as much as 30% difference in median collapse capacity between systems
with and without cyclic degradation (Figure 4-16). Second, the effect of
cyclic degradation increases as the period of vibration decreases. The short
period (T=0.5s) versions of each spring showed more influence from cyclic
degradation than the corresponding longer period (T=1.0s or T=2.0s)
versions. This can be seen in the plots in Appendix B.
Figure 4-16 Effect of degradation of the force-displacement capacity boundary on the
collapse capacity of a system (Spring 2b, T=0.2s, with and without cyclic
degradation).
4-14 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
4.5 Observations from Multi-Spring Studies
This section summarizes the results from nonlinear dynamic analyses of
multi-spring systems. Results from these studies were used to qualitatively:
. understand the influence of key features of the combined forcedisplacement
capacity boundary on the nonlinear response of multispring
systems,
. determine the effects of lateral strength on the dynamic stability of multispring
systems, and
. determine the effects of secondary systems on the dynamic stability of
multi-spring systems.
Only selected results are presented here. Combinations Nx2a+1a and
Nx3a+1a, for N = 1, 2, 3, 5, or 9, are used to highlight trends observed to be
generally applicable for the set of multi-spring combinations studied in this
investigation. Results for each combination, plotted versus normalized and
non-normalized intensity measures, are provided in Appendix C and
Appendix D.
4.5.1 Normalized versus Non-Normalized Results
Two intensity measures were used in conducting incremental dynamic
analyses. One was the 5% damped spectral acceleration at the fundamental
period of vibration of the oscillator, Sa(T,5%). While generally appropriate
for single-degree-of-freedom systems, this measure does not allow
comparison between systems having different periods of vibration. For this
reason, a normalized intensity measure, R = Sa(T,5%)/Say(T,5%) was also
used, where Say(T,5%) is the intensity that causes first yield to occur in the
system.
In order to compare the response of different spring systems, it is necessary
to plot the IDA curves from several springs in a single figure using a
common intensity measure. This can be done in two ways. One way is to
plot them using the normalized intensity measure, R = Sa(T,5%)/Say(T,5%).
First yield occurs at a normalized intensity of one, and increasing values of
Sa(T,5%)/Say(T,5%) represent increasing values of ground motion intensity
with respect to the intensity required to initiate yielding in the system.
Normalized plots provide a measure of system capacity relative to the yield
intensity, and are useful for comparing results across different spring types
when evaluating the influence of the key features of the force-displacement
capacity boundary on the response of the system.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-15
Focused Analytical Studies
A second way to compare results is to plot them using an absolute (nonnormalized)
intensity measure that is somewhere in the middle of the range
that would be suitable for the systems being plotted (e.g., T=1.0s). When
evaluating the effects of increasing or decreasing the relative contribution of
one subsystem with respect to another, use of a single absolute intensity
measure allows comparison of results based on the relative strengths of
different systems.
Since each method has advantages for viewing results and drawing
comparisons, results for multi-spring systems were plotted using both the
normalized intensity measure, Sa(T,5%)/Say(T,5%), and non-normalized
intensity measures, Sa(1s,5%) for stiff systems and Sa(2s,5%) for flexible
systems. Results for normalized intensity measures, IM =
Sa(T,5%)/Say(T,5%), are provided in Appendix C, and results for nonnormalized
intensity measures, IM = Sa(1s,5%) or Sa(2s,5%), are provided in
Appendix D. The horizontal axis in all cases is the maximum story drift
ratio, .max, in radians.
4.5.2 Comparison of Multi-Spring Force-Displacement Capacity
Boundaries
Figure 4-17 shows the force-displacement capacity boundaries for multispring
systems Nx2a+1a and Nx3a+1a, normalized by the yield strength, Fy,
of the combined system. Figure 4-18 shows the force-displacement capacity
boundaries for the same two systems, normalized by the strength of the
weakest system. Depending on the normalizing parameter used along the
vertical axis, the resulting curves look very different.
In Figure 4-17, the use of a normalized base shear, F/Fy or Sa/Say, along the
vertical axis allows for a better qualitative comparison of the relative shapes
of the force-displacement capacity boundaries, without the added complexity
caused by the different yield strengths of the systems. In this figure, it is
easier to see how increasing the multiplier “N” on the lateral-force-resisting
spring causes the combined system to more closely resemble the lateral
spring itself (i.e., as “N” increases from 1 to 9, the combination Nx2a+1a
begins to look more like Spring 2a).
Figure 4-17, however, is misleading with regard to the relative strengths of
the combined systems. In normalizing to the yield strength of the combined
system, higher values of yield strength will reduce the plotted values by a
larger ratio, so curves for higher strength systems will plot below curves for
lower strength systems in F/Fy coordinates.
4-16 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
0 0.02 0.04 0.06 0.08 0.1
0
0.5
1
1.5
.
F / Fy
2a+1a
2x2a+1a
3x2a+1a
5x2a+1a
9x2a+1a
0 0.02 0.04 0.06 0.08 0.1
0
0.5
1
1.5
.
F / Fy
3a+1a
2x3a+1a
3x3a+1a
5x3a+1a
9x3a+1a
(Nx2a+1a) (Nx3a+1a)
Figure 4-17 Force-displacement capacity boundaries for multi-spring systems Nx2a+1a and
Nx3a+1a, normalized by the yield strength, Fy, of the combined system.
(Nx2a+1a) (Nx3a+1a)
Figure 4-18 Force-displacement capacity boundaries for multi-spring systems Nx2a+1a and
Nx3a+1a, normalized by the yield strength of the weakest system
In Figure 4-18, normalizing to the strength of the weakest system allows for
a better comparison of the relative strength between the systems. In this
figure it is easier to see how increasing the multiplier “N” on the lateralforce-
resisting spring increases the strength of the combined system.
4.5.3 Influence of the Combined Force-Displacement Capacity
Boundary in Multi-Spring Systems
Regardless of the normalizing parameter, Figure 4-17 and Figure 4-18 show
how the combined force-displacement capacity boundaries change as the
relative contributions of the springs vary. Results from single-spring studies
demonstrated the influence of key features of the force-displacement capacity
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-17
Focused Analytical Studies
boundary on the nonlinear dynamic response of a single-spring system.
Results from multi-spring studies followed the same relationships. Multispring
systems in which the combined force-displacement capacity boundary
had more favorable features (e.g., delayed onset of degradation, more gradual
slope of degradation, higher residual strength, and higher ultimate
deformation capacity) performed better.
Figure 4-19 shows median IDA curves plotted versus the normalized
intensity measure R = Sa(T,5%)/Say(T,5%) for multi-spring systems Nx2a+1a
and Nx3a+1a with a mass of 8.87 tons, representing a series of relatively stiff
systems. As “N” increases, the yield strength of the combined system
increases, and each system has a correspondingly shorter period of vibration.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1a (1.53s)
2x2a+1a (1.18s)
3x2a+1a (1.00s)
5x2a+1a (0.80s)
9x2a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3a+1a (1.53s)
2x3a+1a (1.18s)
3x3a+1a (1.00s)
5x3a+1a (0.80s)
9x3a+1a (0.61s)
Figure 4-19 Median IDA curves plotted versus the normalized intensity measure
Sa(T,5%)/Say(T,5%) for systems Nx2a+1a and Nx3a+1a with a mass of 8.87 tons.
Figure 4-20 shows median IDA curves for the same two systems with a mass
of 35.46 tons, representing a series of relatively flexible systems. Because
each system has a different period of vibration, normalized plots are used to
qualitatively compare IDA curves between systems. Normalized curves,
however, can be somewhat misleading with regard to the effect of changing
“N” in the different spring combinations. The plotting positions in Figure
4-19, for example, are not an indication of the absolute collapse capacity of
each system. Rather, they are a measure of collapse capacity relative to the
intensity required to initiate yielding. Systems with high yield strengths may
actually collapse at higher absolute intensities than systems with lower yield
strengths, but because of the normalization to yield intensity, they plot out at
lower ratios.
4-18 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
2a+1a (3.05s)
2x2a+1a (2.37s)
3x2a+1a (2.00s)
5x2a+1a (1.60s)
9x2a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3a+1a (3.05s)
2x3a+1a (2.37s)
3x3a+1a (2.00s)
5x3a+1a (1.60s)
9x3a+1a (1.21s)
Figure 4-20 Median IDA curves plotted versus the normalized intensity measure
Sa(T,5%)/Say(T,5%) for systems Nx2a+1a and Nx3a+1a with a mass of 35.46 tons.
For system Nx2a+1a, Figure 4-19 and Figure 4-20 show that as “N”
increases, collapse capacity decreases. The reason for this can be seen in the
combined force-displacement capacity boundaries for system Nx2a+1a
shown in Figure 4-17. Because of the characteristics of Spring 2a,
combinations with higher multiples of “N” have steeper negative slopes. As
was the case with single-spring systems, steeper slopes in the strengthdegrading
segment of the force-displacement capacity boundary result in
lower collapse capacities.
For system Nx3a+1a, the results are the same, but less pronounced. Similar
to system Nx2a+1a, the force-displacement capacity boundaries shown in
Figure 4-17 for system Nx3a+1a with higher multiples of “N” have steeper
negative slopes, but the differences are less significant.
Figure 4-19 and Figure 4-20 also show that, in general, combinations with
systems that have more favorable characteristics result in higher median
collapse capacities relative to yield intensity. For example, in Figure 4-19,
system 9x2a+1a exhibits a median collapse capacity that is approximately 2.3
times the yield intensity while system 9x3a+1a exhibits a median collapse
capacity that is approximately 3.5 times the yield intensity. The reason for
this can be seen by comparing the combined force-displacement capacity
boundaries for systems Nx2a+1a and Nx3a+1a shown in Figure 4-17. The
post-yield characteristics of system Nx3a+1a are more favorable in terms of
the post-yield slope, onset of degradation, and slope of degradation, resulting
in better performance.
A more direct illustration of this behavior can be seen by comparing
combinations using the “a” and “b” versions of primary spring components.
Figure 4-21 shows the median IDA curves for systems Nx3a+1a and
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-19
Focused Analytical Studies
Nx3b+1a. By definition, the “b” version of each spring was created to have
more favorable characteristics than the “a” version of the same spring, with
all other parameters being equal. As shown in the figure, the curves for
system Nx3b+1a outperform all corresponding combinations of Nx3a+1a in
terms of collapse capacity relative to yield intensity, for all values of “N”
from 1 to 9.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3a+1a (1.53s)
2x3a+1a (1.18s)
3x3a+1a (1.00s)
5x3a+1a (0.80s)
9x3a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3b+1a (1.53s)
2x3b+1a (1.18s)
3x3b+1a (1.00s)
5x3b+1a (0.80s)
9x3b+1a (0.61s)
Figure 4-21 Median IDA curves plotted versus the normalized intensity measure Sa(T,5%)/Say(T,5%)
for systems Nx3a+1a and Nx3b+1a with a mass of 8.87 tons.
4.5.4 Effects of the Lateral Strength of Multi-Spring Systems
Plotting of results using absolute (non-normalized) intensity measures allows
for comparison of results based on the relative strengths of different systems.
Non-normalized intensity measures of Sa(1s,5%) for stiff systems and
Sa(2s,5%) for flexible systems were used to identify the effects of the lateral
strength of the multi-spring system on the lateral dynamic stability of the
system.
Figure 4-22 shows median IDA curves for multi-spring systems Nx2a+1a
and Nx3a+1a tuned with a mass of 8.87 tons. They are plotted versus
Sa(1s,5%), which is an intensity measure keyed to a period of T=1.0s, located
in the middle of the range of periods for the relatively stiff set of multi-spring
systems. Figure 4-23 shows median IDA curves for same set of systems
Nx2a+1a and Nx3a+1a, tuned with a mass of 35.46 tons. In this figure, the
curves are plotted versus Sa(2s,5%), which is keyed to a period of T=2.0s,
located in the middle of the range of periods for the relatively flexible set of
multi-spring systems.
4-20 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
Figure 4-22 Median IDA curves plotted versus the common intensity measure Sa(1s,5%)
for systems Nx2a+1a and Nx3a+1a with a mass of 8.87 tons.
Figure 4-23 Median IDA curves plotted versus the common intensity measure Sa(2s,5%)
for systems Nx2a+1a and Nx3a+1a with a mass of 35.46 tons.
In comparing non-normalized plots of IDA curves for various multi-spring
combinations, the following observations were made:
. Increases in the lateral strength of a system change the intensity that
initiates yielding in the system as well as the intensity at collapse (lateral
dynamic instability). The incremental change in collapse capacity,
however, is less than proportional to the increase in yield strength.
. The effectiveness of increasing the lateral strength of a system is a
function of the shape of the force-displacement capacity boundary.
Incremental changes in yield strength are more effective for ductile
systems than they are for systems with less ductile behavior.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-21
Focused Analytical Studies
. The effectiveness of increasing the lateral strength of a system is also a
function of the period of system. Incremental changes in yield strength
are more effective for stiff systems than they are for flexible systems.
These effects can be observed by comparing the combined forcedisplacement
capacity boundaries in Figure 4-18 with the resulting IDA
curves in Figure 4-22 and Figure 4-23. Figure 4-22 shows that as “N”
increases, the yield intensity increases significantly, however, increases in
intensity at lateral dynamic instability are not as significant. For example,
Figure 4-18 shows that the yield strength of system 9x3a+1a is
approximately 6.5 times higher than the yield strength of system 3a+1a, but
Figure 4-22 shows that the collapse capacity is only about two times higher.
Comparing results between systems Nx2a+1a and Nx3a+1a in Figure 4-22
shows that increases in collapse capacity that do occur as a result of changes
in lateral strength are more pronounced for the more ductile Spring 3a than
they are for the less ductile Spring 2a. For example, the increase in collapse
capacity for system Nx3a+1a, as “N” increases from 1 to 9, is a factor of
approximately 2.0. For system Nx2a+1a the corresponding increase in
collapse capacity is a factor of approximately 1.25.
Comparing results between Figure 4-22 and Figure 4-23 shows that as the
period increases, the increment in collapse capacity caused by a change in
lateral strength decreases. For example, the increase in collapse capacity
shown in Figure 4-22 for the relatively stiff combinations of system Nx3a+1a
is a factor of approximately 2.0. The increase in collapse capacity shown in
Figure 4-23 for the relatively flexible combinations of system Nx3a+1a is a
factor of approximately 1.3.
4.5.5 Effects of Secondary System Characteristics
The contribution of a secondary (“gravity”) system acting in parallel with a
primary lateral-force-resisting system always results in an improvement in
post-yield performance, especially close to collapse. This result was
observed both qualitatively and quantitatively (i.e., both in normalized and
non-normalized coordinates).
The improvement is larger when considering secondary systems with larger
ultimate deformation capacities. Figure 4-24 shows median IDA curves
plotted versus the normalized intensity measure R = Sa(T,5%)/Say(T,5%) for
multi-spring systems Nx2a+1a and Nx2a+1b with a mass of 8.87 tons. In the
figure it can be seen that combinations with Spring 1b (with a larger ultimate
deformation capacity) perform significantly better than combinations with
4-22 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
Spring 1a. This result was observed in combinations with all lateral-forceresisting
springs.
Near collapse, secondary systems with larger deformation capacities have an
even greater influence, even if the lateral strength is small compared to that
of the primary system. This can be observed by comparing differences
between systems 9x2a+1a and 9x2a+1b in Figure 4-24. Even though the
relative contribution of Spring 1 in these combinations is small, the resulting
collapse capacity is increased significantly.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1a (1.53s)
2x2a+1a (1.18s)
3x2a+1a (1.00s)
5x2a+1a (0.80s)
9x2a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1b (1.53s)
2x2a+1b (1.18s)
3x2a+1b (1.00s)
5x2a+1b (0.80s)
9x2a+1b (0.61s)
Figure 4-24 Median IDA curves plotted versus the normalized intensity measure
Sa(T,5%)/Say(T,5%) for systems Nx2a+1a and Nx2a+1b with a mass of 8.87 tons.
The contribution of the secondary system is more noticeable and significant
in systems where the primary lateral resisting system is less ductile. Figure
4-25 shows median IDA curves plotted versus the normalized intensity
measure R = Sa(T,5%)/Say(T,5%) for multi-spring systems Nx2a+1a and
Nx3a+1a with a mass of 8.87 tons.
Comparing the systems in Figure 4-25 shows a much wider spread between
the median IDA curves for system Nx2a+1a than the curves for system
Nx3a+1a. This means that the behavior of Spring 2a is more heavily
influenced by the combination with Spring 1a than Spring 3a. The reason for
this can be explained by the relative contributions of each spring to the
combined force-displacement capacity boundaries in Figure 4-17.
Spring 2a, which represents a non-ductile moment frame system, has less
favorable post-yield behavior in its force-displacement capacity boundary
than does Spring 3a, which represents a ductile moment frame system. As
such, Spring 2a is more favorably impacted by the characteristics of Spring
1a, and combinations with Spring 1a result in greater changes in
performance. However, as “N” increases from 1 to 9, system Nx2a+1a
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-23
Focused Analytical Studies
becomes more like Spring 2a, and the positive influences of Spring 1a
diminish.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1a (1.53s)
2x2a+1a (1.18s)
3x2a+1a (1.00s)
5x2a+1a (0.80s)
9x2a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3a+1a (1.53s)
2x3a+1a (1.18s)
3x3a+1a (1.00s)
5x3a+1a (0.80s)
9x3a+1a (0.61s)
Figure 4-25 Median IDA curves plotted versus the normalized intensity measure
Sa(T,5%)/Say(T,5%) for systems Nx2a+1a and Nx3a+1a with a mass of 8.87 tons.
4.6 Comparison with FEMA 440 Limitations on
Strength for Lateral Dynamic Instability
In FEMA 440, a minimum strength requirement (maximum value of R) was
developed as an approximate measure of the need to further investigate the
potential for lateral dynamic instability caused by in-cycle strength
degradation and P-delta effects. The recommended limitation is shown in
Equation 4-1, with terms defined in Equation 4-2 and Equation 4-3, and
illustrated in Figure 4-26:
max 4 R
. .
.
.
. .
t
d e
y
(4-1)
where
t .1. 0.15 ln T (4-2)
and
. . e P 2 P . . . . . .. .. . . . (4-3)
for 0 < ..< 1.0.
In-cycle strength degradation caused by P-delta is represented by P . .. . The
effects from all other sources of cyclic and in-cycle strength and stiffness
degradation are represented by the term . . 2 P . . .. . . At the time, it was
apparent that modeling rules specified the use of hysteretic envelopes
idealized from cyclic test results and would, consequently, overestimate
4-24 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
actual in-cycle losses. For this reason, these effects were reduced by factor
., which was less than 1.0.
Figure 4-26 Idealized force-displacement curve for nonlinear static analysis
(from FEMA 440).
According to FEMA 440, the idealized force-displacement relationship
(Figure 4-26) and the factor . were based on judgment, and significant
variability should be expected in the value predicted using the equation for
Rmax. As such, Rmax was intended only for identification of cases where
further investigation using nonlinear response history analysis should be
performed, and not as an accurate measure of the strength required to avoid
lateral dynamic instability.
To further investigate correlation between the FEMA 440 equation for Rmax
and lateral dynamic instability, the results of this equation were compared to
quantile IDA curves for selected multi-spring systems included in this
investigation. In making this comparison, parameters in the FEMA 440
equation for Rmax were estimated from multi-spring force-displacement
capacity boundaries idealized as shown in Figure 4-27.
Results from this comparison indicate that values predicted by the FEMA
440 equation for Rmax are variable, but generally plot between the median and
84th percentile results for lateral dynamic instability of the systems
investigated. The trends observed in this comparison indicate that an
improved equation, in a form similar to Rmax, could be developed as a more
accurate and reliable (less variable) predictor of lateral dynamic instability
for use in current nonlinear static analysis procedures.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-25
Focused Analytical Studies
0
1
2
3
4
5
6
7
0.00 0.02 0.04 0.06 0.08
.
F/h
max 4
t
d e
y
R
. . .
. .
.
t . 1. 0.15 lnT
Approximate degrading stiffness
0
1
2
3
4
5
6
7
0.00 0.02 0.04 0.06 0.08
.
F/h
max 4
t
d e
y
R
. . .
. .
.
t . 1. 0.15 lnT
Approximate degrading stiffness
.y .d
a e
0
1
2
3
4
5
6
7
0.00 0.02 0.04 0.06 0.08
.
F/h
max 4
t
d e
y
R
. . .
. .
.
t . 1. 0.15 lnT
Approximate degrading stiffness
0
1
2
3
4
5
6
7
0.00 0.02 0.04 0.06 0.08
.
F/h
max 4
t
d e
y
R
. . .
. .
.
t . 1. 0.15 lnT
Approximate degrading stiffness
.y .d
a e
Figure 4-27 Idealization of multi-spring force-displacement capacity
boundaries to estimate effective negative stiffness for use in the
FEMA 440 equation for Rmax.
4.6.1 Improved Equation for Evaluating Lateral Dynamic
Instability
An improved estimate for the strength ratio at which lateral dynamic
instability might occur was developed based on nonlinear regression of the
extensive volume of data generated during this investigation. In performing
this regression, results were calibrated to the median response of the SDOF
spring systems studied in this investigation.
A median-targeted strength ratio for lateral dynamic instability, Rdi, is
defined as:
3
3
a
c e r u r
di e
y c y
R b T F T
. F
. . . . . . . .
. .. .. . . .. .. . . . . . .
(4-4)
where Te is the effective fundamental period of vibration of the structure, .y
,
.c
, .r
, and .u
are displacements corresponding to the yield strength, Fy,
capping strength, Fc, residual strength, Fr, and ultimate deformation capacity
at the end of the residual strength plateau, as shown in Figure 4-28.
Parameters a and b are functions given by:
1 exp( ) e a . . .dT (4-5)
2
1 . .. .
. ..
.
. .
c
r
F
b F (4-6)
aeKe
4-26 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
where the parameter d is a constant equal to 4 for systems with stiffness
degradation, and 5 for systems without stiffness degradation. The parameter
. is the ratio of the post-capping slope (degrading stiffness) to the initial
effective slope (elastic stiffness).
.u .r .c .y
Fr
Fy
Fc
F
Ke
.Ke
.Ke
.u .r .c .y
Fr
Fy
Fc
F
Ke
.Ke
.Ke
Figure 4-28 Simplified force-displacement boundary for estimating the
median collapse capacity associated with lateral dynamic
instability.
.y .c .r .u
R
.
1st term
2nd term
3rd term
.y .c .r .u
R
.
1st term
2nd term
3rd term
Figure 4-29 Relationship between Equation 4-4 and the segments of a
typical IDA curve.
The three terms in Equation 4-4 relate to the segments of a typical forcedisplacement
capacity boundary (Figure 4-28) and typical IDA curve (Figure
4-29). The first term provides an estimate of the median ground motion
intensity corresponding to the end of the pseudo-linear segment of an IDA
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-27
Focused Analytical Studies
curve (i.e., intensity at the onset of degradation). The second term provides
an estimate of the increment in ground motion intensity required to push the
structure onto the residual strength plateau. The third term provides an
estimate of the increment in ground motion intensity required produce lateral
dynamic instability (collapse).
As developed, the equation for Rdi is intended to be a more reliable (less
variable) predictor of median response at lateral dynamic instability. The
resulting equation was compared to the FEMA 440 equation for Rmax and
overlaid onto results for selected multi-spring systems. With few exceptions,
Figure 4-30 through Figure 4-35 show that the equation for Rdi consistently
predicts median response over a range of system types and periods of
vibration.
Figure 4-30 Comparison of Rdi with FEMA 440 Rmax and IDA results for
system 2x2a+1a with T=1.18s.
4-28 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
Figure 4-31 Comparison of Rdi with FEMA 440 Rmax and IDA results for
system 3x3b+1b with T=1.0s.
Figure 4-32 Comparison of Rdi with FEMA 440 Rmax and IDA results for
system 9x3b+1b with T=0.61s.
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-29
Focused Analytical Studies
Figure 4-33 Comparison of Rdi with FEMA 440 Rmax and IDA results for
system 5x5a+1a with T=1.15s.
Figure 4-34 Comparison of Rdi with FEMA 440 Rmax and IDA results for
system 5x5a+1a with T=0.58s.
4-30 4: Results from Single-Degree-of-Freedom FEMA P440A
Focused Analytical Studies
Figure 4-35 Comparison of Rdi with FEMA 440 Rmax and IDA results for
system 9x5a+1a with T=0.34s.
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-1
Chapter 5
Findings, Conclusions, and
Recommendations
This chapter summarizes the findings, conclusions, and recommendations
resulting from the literature review and focused analytical studies of this
investigation. Information from other chapters is collected and repeated here
for ease of reference. In this chapter, findings have been grouped into the
following categories:
. Findings related to improved understanding of nonlinear degrading
response and judgment in implementation of nonlinear analysis results in
engineering practice.
. Recommended improvements to current nonlinear
analysis procedures
. Suggestions for further study
From the literature review, it is apparent that in-cycle strength and stiffness
degradation are real phenomena that have been observed and documented to
cause instability in individual components. Focused analytical studies have
shown that larger assemblies of components of mixed hysteretic behavior
experience similar negative stiffness that can lead to lateral dynamic
instability. These studies have been able to link nonlinear dynamic response
to major characteristics of component and system degrading behavior.
These studies have also confirmed many of the conclusions regarding
degradation and lateral dynamic instability presented in FEMA 440: (1) incycle
strength degradation is a significant contributor to dynamic instability;
(2) cyclic degradation can increase the potential for dynamic instability, but
its effects are far less significant in comparison with in-cycle degradation;
and (3) an equation, such as Rmax, could be used as an indicator of potential
lateral dynamic instability for use in current nonlinear static analysis
procedures.
5-2 5: Findings, Conclusions, and Recommendations FEMA P440A
5.1 Findings Related to Improved Understanding and
Judgment
This section summarizes observations and conclusions related to improved
understanding of nonlinear degrading response and judgment in
implementation of nonlinear analysis results in engineering practice.
Findings, and practical ramifications for engineering practice, are
summarized in the sections that follow.
5.1.1 Sidesway Collapse versus Vertical Collapse
Lateral dynamic instability is manifested in structural systems as sidesway
collapse caused by loss of lateral-force-resisting capacity. Most sidesway
collapse mechanisms can be explicitly simulated in nonlinear response
history analyses. It should be noted, however, that inelastic deformation of
structural components can result in shear and flexural-shear failures in
members, and failures in joints and connections, which can lead to an
inability to support vertical loads (vertical collapse) long before sidesway
collapse can be reached.
5.1.1.1 Practical Ramifications
Behavior of real structures can include loss of vertical-load-carrying capacity
at lateral displacements that are significantly smaller than those associated
with sidesway collapse. Use of the findings of this investigation with regard
to lateral dynamic instability (sidesway collapse) in engineering practice
should include consideration of possible vertical collapse modes that could
be present in the structure under consideration.
5.1.2 Relationship between Loading Protocol, Cyclic Envelope,
and Force-Displacement Capacity Boundary
Historically, the term “backbone curve” has referred to many different
things. For this reason, two new terms have been introduced to distinguish
between different aspects of hysteretic behavior. These are the forcedisplacement
capacity boundary, and cyclic envelope.
5.1.2.1 Force-Displacement Capacity Boundary
A force-displacement capacity boundary defines the maximum strength that a
structural member can develop at a given level of deformation, resulting in
an effective “boundary” for the strength of a member in force-deformation
space (Figure 5-1). In many cases, the force-displacement capacity boundary
corresponds to the monotonic force-deformation curve.
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-3
A cyclic load path cannot cross a force-displacement capacity boundary. If a
member is subjected to increasing deformation and the boundary is reached,
then the strength that can be developed in the member is limited and the
response must continue along the boundary (in-cycle strength degradation).
Only displacement excursions intersecting portions of the capacity boundary
with a negative slope will result in in-cycle strength degradation.
Figure 5-1 Example of a force-displacement capacity boundary.
5.1.2.2 Cyclic Envelope
A cyclic envelope is a force-deformation curve that envelopes the hysteretic
behavior of a component or assembly that is subjected to cyclic loading
(Figure 5-2).
Figure 5-2 Example of a cyclic envelope.
5-4 5: Findings, Conclusions, and Recommendations FEMA P440A
The characteristics of the cyclic envelope are strongly influenced by the
points at which unloading occurs in a test, and are therefore strongly
influenced by the loading protocol that was used in the experimental
program. Nominally identical specimens loaded with different loading
protocols will have different cyclic envelopes depending on the number of
cycles used in the loading protocol, the amplitude of each cycle, and the
sequence of the loading cycles, as illustrated in Section 2.2.3.
Under lateral deformations that are less than or equal to those used to
generate the cyclic envelope, differences between the cyclic envelope and the
force-displacement capacity boundary are of no consequence. However,
under larger lateral displacements these differences will affect the potential
for in-cycle degradation to occur, and will significantly affect system
behavior and response (Figure 5-3).
(a) (b)
Figure 5-3 Comparison of hysteretic behavior when the force-displacement capacity
boundary is: (a) equal to the cyclic envelope, and (b) extends beyond the cyclic
envelope.
Constraining nonlinear hysteretic behavior within the limits of a cyclic
envelope that does not capture the full range of permissible forcedeformation
response, as defined by the force-displacement capacity
boundary, will result in overly pessimistic predictions of the nonlinear
dynamic response of a system.
5.1.2.3 Practical Ramifications
Nonlinear component parameters should be based on the force-displacement
capacity boundary, which is different from a cyclic envelope. Determining
the force-displacement capacity boundary from test results using a single
cyclic loading protocol can result in significant underestimation of the actual
capacity for force-deformation response and subsequent overestimation of
nonlinear displacement demands.
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-5
5.1.3 Characteristics of Median IDA Curves
Observed relationships between IDA curves and degrading component
characteristics suggest that dynamic response is directly influenced by the
features of a force-displacement capacity boundary. This relationship, which
is dependent upon the period of vibration of the system, is depicted in the
idealized graphical representation of Figure 5-4.
Figure 5-4 Relationship between IDA curves and the features of a typical
force-displacement capacity boundary.
In general, median IDA curves were observed to exhibit the following
characteristics:
. An initial linear segment corresponding to linear-elastic behavior in
which in lateral deformation demand is proportional to ground motion
intensity, regardless of the characteristics of the system or the ground
motion. This segment extends from the origin to the onset of yielding.
. A second curvilinear segment corresponding to inelastic behavior in
which lateral deformation demand is no longer proportional to ground
motion intensity. As intensity increases, lateral deformation demands
increase at a faster rate. This segment corresponds to softening of the
system, or reduction in stiffness (reduction in the slope of the IDA
curve). In this segment, the system “transitions” from linear behavior to
eventual dynamic instability. Although a curvilinear segment is always
present, in some cases the transition can be relatively long and gradual,
while in other cases it can be very short and abrupt.
INSTABILITY
LINEAR
PSEUDO
LINEAR
SOFTENING
.
boundar
16%
84%
50% INSTABILITY
LINEAR
.
CAPACITY
BOUNDARY
SaT (g)
5-6 5: Findings, Conclusions, and Recommendations FEMA P440A
. A final linear segment that is horizontal, or nearly horizontal, in which
infinitely large lateral deformation demands occur at small increments in
ground motion intensity. This segment corresponds to the point at which
a system becomes unstable (lateral dynamic instability). For SDOF
systems, this point corresponds to the ultimate deformation capacity at
which the system loses all lateral-force-resisting capacity.
In some systems, the initial linear segment can be extended beyond yield into
the inelastic range. In this pseudo-linear segment, lateral deformation
demand is approximately proportional to ground motion intensity, which is
consistent with the familiar equal-displacement approximation for estimating
inelastic displacements. The range of lateral deformation demands over
which the equal-displacement approximation is applicable depends on the
characteristics of the force-displacement capacity boundary of the system and
the period of vibration.
5.1.3.1 Practical Ramifications
The observed relationships support the conclusion that it is possible to
estimate nonlinear dynamic response based on knowledge of the
characteristics of the force-displacement capacity boundary.
5.1.4 Dependence on Period of Vibration
In general, moderate and long period systems with zero or positive post-yield
stiffness in the force-displacement capacity boundary follow the equal
displacement trend well into the nonlinear range, as shown for Spring 3a in
Figure 5-5. For systems with periods longer than 0.5s, Spring 3a exhibits an
extension of the initial linear segment well beyond the yield drift of 0.01. In
contrast, the short period system (T=0.2s) diverges from the initial linear
segment just after yielding, even at deformations within the strengthhardening
segment of the force-displacement capacity boundary (drifts
between 0.01 and 0.04 in the figure).
5.1.4.1 Practical Ramifications
It is important to consider the dependence on period of vibration in
conjunction with the effects of other parameters identified in this
investigation. The generalized effect of any one single parameter can be
misleading.
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-7
Figure 5-5 Force-displacement capacity boundary and median IDA curves for Spring 3a with various periods
of vibration.
5.1.5 Dispersion in Response
Nonlinear response is sensitive to the characteristics of the ground motion
record, and will vary from one ground motion to the next, even when scaled
to the same intensity (Figure 5-6). For a given level of ground motion
intensity, the lateral deformation demand can be significantly smaller or
significantly larger than the value shown on median IDA curves, as indicated
by the 16th and 84th percentile curves in the figure. As the level of ground
motion intensity increases, the dispersion in response tends to increase.
Figure 5-6 Force-displacement capacity boundary and 16th, 50th and 84th percentile IDA
curves for Spring 3b with a period of vibration T=2.0s.
5.1.5.1 Practical Ramifications
It is important to recognize the level of uncertainty that is inherent in
nonlinear analysis, particularly regarding variability in response due to
5-8 5: Findings, Conclusions, and Recommendations FEMA P440A
ground motion uncertainty. It may not be sufficient to rely on median (50%)
estimates of response for certain design or evaluation quantities of interest,
unless the intensity of the ground motion is associated with an appropriately
rare probability of exceedance.
5.1.6 Influence of the Force-Displacement Capacity Boundary
Key features of a force-displacement capacity boundary that were observed
to influence the shape of median IDA curves included post-yield behavior
and onset of degradation, slope of degradation, ultimate deformation
capacity, and presence of cyclic degradation. Systems in which the forcedisplacement
capacity boundary had more favorable post-yield
characteristics (e.g., delayed onset of degradation, more gradual slope of
degradation, higher residual strength, and higher ultimate deformation
capacity) were observed to perform better.
5.1.6.1 Post-Yield Behavior and Onset of Degradation
The presence of a non-negative post-yield slope and delay before the onset of
degradation reduced potential in-cycle strength degradation and significantly
improved the collapse capacity of a system (Figure 5-7).
Figure 5-7 Effect of post-yield behavior on the collapse capacity of a system (Springs 2a, 3a
and 6a with T=2.0s).
5.1.6.2 Slope of Degradation
Differences in the negative slope of the strength-degrading segment
significantly affected the collapse capacity of a system. Systems with more
shallow degrading slopes reached higher collapse capacities than systems
with steeper degrading slopes (Figure 5-8). Changes in negative slope
changed the magnitude of potential in-cycle strength degradation, and
overshadowed any changes in other parameters (e.g., the residual strength
plateau), as long as the ultimate deformation capacity remained the same.
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-9
Figure 5-8 Effect of slope of degradation on the collapse capacity of a system (Springs 2a and 2b
with T=1.0s).
5.1.6.3 Ultimate Deformation Capacity
Increasing the ultimate deformation capacity resulted in significant increases
in collapse capacity (Figure 5-9). The key parameter related to the observed
change in response is the increment in the ultimate deformation capacity.
Observed changes in collapse capacity resulting from increases in the
ultimate deformation capacity were insensitive to other characteristics of the
post-yield behavior of the springs.
Figure 5-9 Effect of ultimate deformation capacity on the collapse capacity of a system (Springs
1a and 1b with T=1.0s).
5.1.6.4 Practical Ramifications
Observed relationships between selected features of the force-displacement
capacity boundary and the resulting characteristics of median IDA curves
support the conclusion that the nonlinear dynamic response of a system can
be correlated to the parameters of the force-displacement capacity boundary
of that system. Of particular interest is the relationship between global
5-10 5: Findings, Conclusions, and Recommendations FEMA P440A
deformation demand and the intensity of the ground motion at lateral
dynamic instability (collapse). Results indicate that it is possible to use
nonlinear static procedures to estimate the potential for lateral dynamic
instability of systems exhibiting in-cycle degradation.
5.1.7 Cyclic Degradation of the Force-Displacement Capacity
Boundary
In general, most components will exhibit some level of cyclic degradation.
Consistent with observations from past studies, comparison of results
between springs both with and without cyclic degradation show that the
effects of cyclic degradation (as measured by gradual movement of the
capacity boundary) are relatively unimportant in comparison with in-cycle
degradation (as measured by the extent and steepness of negative slopes in
the capacity boundary). This trend is illustrated for Spring 3b in Figure 5-10,
but can be observed in the results for many spring systems in Appendix B.
Figure 5-10 Effect of degradation of the force-displacement capacity boundary on the collapse
capacity of a system (Spring 3b, T=2.0s, with and without cyclic degradation).
5.1.7.1 Practical Ramifications
In most cases the effects of in-cycle strength degradation dominate the
nonlinear dynamic behavior of a system. This suggests that in many cases
the effects of cyclic degradation can be neglected. Instead, the focus should
be on more accurately characterizing the force-displacement capacity
boundary, which controls the onset of in-cycle degradation (where it occurs).
Two situations in which the effects of cyclic degradation were observed to be
important include: (1) short period systems; and (2) systems with very strong
in-cycle strength degradation effects (very steep and very large drops in
lateral strength). In these cases, the effects of cyclic degradation can be
important and should be considered.
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-11
5.1.8 Effects of Secondary System Characteristics
The contribution of a secondary (“gravity”) system acting in parallel with a
primary lateral-force-resisting system always resulted in an improvement in
nonlinear response, especially close to collapse. This result was observed
both qualitatively and quantitatively (i.e., both in normalized and nonnormalized
coordinates).
The improvement was larger when considering secondary systems with
larger ultimate deformation capacities, even if the lateral strength of the
secondary system was small in comparison to that of the primary system.
This result is illustrated in Figure 5-11, and is supported by results described
in Section 5.1.6.3. In the figure, as the system combination ratio increases,
the relative combination of the secondary system diminishes, yet the
resulting collapse capacities for combinations with Spring 1b (larger ultimate
deformation capacity) are significantly higher than combinations with Spring
1a (smaller ultimate deformation capacity).
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1a (1.53s)
2x2a+1a (1.18s)
3x2a+1a (1.00s)
5x2a+1a (0.80s)
9x2a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1b (1.53s)
2x2a+1b (1.18s)
3x2a+1b (1.00s)
5x2a+1b (0.80s)
9x2a+1b (0.61s)
Figure 5-11 Median IDA curves plotted versus the normalized intensity measure Sa(T,5%)/Say(T,5%) for
systems Nx2a+1a and Nx2a+1b with a mass of 8.87 tons.
5.1.8.1 Practical Ramifications
Consideration of the contribution of secondary (“gravity”) systems acting in
parallel with primary lateral resisting systems is important and should be
included in nonlinear modeling for collapse simulation. For seismic retrofit
of existing structures, this suggests that adding a relatively weak (but ductile)
system in parallel with the primary system could substantially increase
collapse capacity and delay the onset of lateral dynamic instability. The
introduction of such a secondary system could be significantly less
complicated and less expensive than direct improvements to the strength,
stiffness and deformation capacity of the primary system.
5-12 5: Findings, Conclusions, and Recommendations FEMA P440A
5.1.9 Effects of Lateral Strength
Increasing the lateral strength of a system was observed to increase collapse
capacity, with the following limitations:
. Increases in the lateral strength of a system changed the intensity that
initiated yielding in the system and the intensity at collapse (lateral
dynamic instability). The incremental change in collapse capacity,
however, was less than proportional to the increase in yield strength
(Figure 5-12).
Figure 5-12 Force-displacement capacity boundaries and median IDA curves plotted versus the common
intensity measure Sa(2s,5%) for system Nx3a+1a with a mass of 35.46 tons.
. The effectiveness of increasing the lateral strength of a system was a
function of the shape of the force-displacement capacity boundary.
Incremental changes in yield strength were more effective for ductile
systems than they were for systems with less ductile behavior.
. The effectiveness of increasing the lateral strength of a system was also a
function of the period of system. Incremental changes in yield strength
were more effective for stiff systems than they were for flexible systems.
5.1.9.1 Practical Ramifications
Increasing the lateral strength of a system can improve collapse behavior, but
will not result in equal increases in collapse capacity. The effectiveness of
seismic retrofit strategies that involve increasing the lateral strength will
depend on the characteristics of the force-displacement capacity boundary of
the existing system as well as the period of vibration.
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-13
5.2 Recommended Improvements to Current
Nonlinear Analysis Procedures
Prevailing practice for performance-based seismic design is based on the
FEMA 273 NEHRP Guidelines for the Seismic Rehabilitation of Buildings
(FEMA, 1997) and its successor documents, FEMA 356 Prestandard and
Commentary for the Seismic Rehabilitation of Buildings (FEMA, 2000), and
ASCE/SEI Standard 41-06 Seismic Rehabilitation of Existing Buildings
(ASCE, 2006b). Recommendations contained in FEMA 440 Improvement of
Nonlinear Static Seismic Analysis Procedures (FEMA, 2005) were
incorporated into the developing ASCE/SEI Standard 41-06 in 2005.
ASCE/SEI Standard 41-06 Supplement No. 1 was published in 2007.
Together these resource documents form the basis of nonlinear analysis in
current engineering practice. This section summarizes recommended
clarifications and improvements to current nonlinear analysis procedures as
characterized in these documents.
5.2.1 Current Nonlinear Static Procedures
The Coefficient Method is one method of estimating maximum inelastic
displacements of a system. The process begins with the generation of an
idealized force-deformation curve (i.e., static pushover curve) relating base
shear to roof displacement. From this curve, an effective period, Te, is
obtained, and the maximum global displacement (target displacement) for a
specified level of ground motion intensity is estimated using Equation 5-1:
2
0 1 2 4 2
e
t a
. C C C S T g
.
. (5-1)
In this expression the first three terms are coefficients that modify the elastic
displacement of the system. C0 is the first mode participation factor. This
coefficient essentially converts from spectral ordinates to roof displacement.
This C1 coefficient (Equation 5-2) increases elastic displacements in short
period systems, essentially accounting for exceptions to the equal
displacement approximation.
1 2
1 1
e
C R
aT
.
. . (5-2)
where:
/
a
m
y
R S C
V W
. . (5-3)
5-14 5: Findings, Conclusions, and Recommendations FEMA P440A
and Cm is the effective mass factor to account for higher mode mass
participation effects. The C2 coefficient (Equation 5-4) increases elastic
displacements in short period and weak systems to account for stiffness
degradation, hysteretic pinching, and cyclic strength degradation.
2
2
1 1 1
800 e
C R
T
. . .
. . .. ..
. .
(5-4)
Importantly, C2 does not account for displacement amplification due to incycle
strength degradation, which can result in lateral dynamic instability.
In-cycle strength degradation is addressed by a minimum strength
requirement (maximum value of R) used as a trigger for the need to further
investigate the potential for lateral dynamic instability using nonlinear
response history analysis. The minimum strength requirement in current
nonlinear analysis procedures was described in Section 4.6 (and is repeated
in the equations that follow):
max 4 R
. .
.
.
. .
t
d e
y
(5-5)
where
t .1. 0.15 ln T (5-6)
and
. . e P 2 P . . . . . .. .. . . . (5-7)
for 0 < ..< 1.0.
Values of R (Equation 5-3) are compared to Rmax. Systems in which R < Rmax
are deemed to meet the minimum strength requirement to avoid lateral
dynamic instability, and nonlinear response history analysis is not required.
5.2.2 Clarification of Terminology and Use of the Force-
Displacement Capacity Boundary for Component
Modeling
For nonlinear analysis, ASCE/SEI 41-06 specifies component modeling and
acceptability criteria based on the conceptual force-displacement relationship
(“backbone”) depicted in Figure 5-13. Since the term “backbone curve” has
been used to refer to many different things, its definition related to nonlinear
component modeling is not clear. In Section 2.8 of the standard, it is
permitted to derive modeling parameters and acceptance criteria using
experimentally obtained cyclic response characteristics from subassembly
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-15
testing. So defined, the standard can be interpreted to condone the use of
cyclic envelopes from component tests to generate the necessary forcedisplacement
relationships.
Figure 5-13 Conceptual force-displacement relationship (“backbone”) used
in ASCE/SEI 41-06 (adapted from FEMA 356).
The use of a cyclic envelope, as opposed to a force-displacement capacity
boundary, has been shown to result in underestimation of the actual capacity
for force-deformation response and subsequent overestimation of nonlinear
deformation demands. In some cases the resulting conservatism can be very
large.
For this reason, introduction and use of two new terms are recommended to
distinguish between different aspects of hysteretic behavior. These are the
force-displacement capacity boundary, and cyclic envelope, defined in
Section 5.1.2. Important conceptual differences between the forcedisplacement
capacity boundary and a loading protocol-specific cyclic
envelope should be clarified in future revisions to ASCE/SEI 41, and the use
of an appropriate force-displacement capacity boundary should be specified
for characterizing component hysteretic behavior.
Proper definition of the hysteretic behavior in a component model requires an
understanding of: (1) the initial force-displacement capacity boundary; and
(2) how the force-displacement capacity boundary degrades under cyclic
loading. The ideal method for establishing an initial force-displacement
capacity boundary is through monotonic testing. Once the initial forcedisplacement
capacity boundary is defined, degradation parameters should be
established based on results from cyclic tests.
There is no recognized testing protocol that incorporates realistic
consideration of the force-displacement capacity boundary. The use of
several cyclic loading protocols is desirable to ensure that the degradation
parameters are properly identified and the calibrated component model is
general enough to represent response under any type of loading.
5-16 5: Findings, Conclusions, and Recommendations FEMA P440A
In Commentary Section C6.3.1.2.2 of ASCE/SEI 41-06 Supplement No. 1, it
is suggested that the sudden drop from Point C to Point D (in Figure 5-10)
can be overly pessimistic, and that a more gradual slope from Point C to
Point E might be more realistic for concrete components. Some
experimental results suggest that such an adjustment could be applicable for
other types of components. If the actual monotonic curve is not available, or
cannot be estimated, use of a force-displacement capacity boundary with this
alternate slope can be considered.
5.2.3 Improved Equation for Evaluating Lateral Dynamic
Instability
In comparison with results for selected multi-spring systems in this
investigation, the FEMA 440 equation for Rmax was shown to predict values
that are variable, but generally fall between the median and 84th percentile
results for lateral dynamic instability. This result suggests that the current
equation for Rmax would be conservative if used in conjunction with a
capacity boundary generated from a pushover analysis. It could be very
conservative if the pushover curve was based on component modeling
parameters determined using a cyclic envelope rather than a forcedisplacement
capacity boundary.
The trends observed in this comparison indicate that an improved equation,
in a form similar to Rmax, could be developed as a more accurate and reliable
(less variable) predictor of lateral dynamic instability for use in current
nonlinear static analysis procedures. An improved estimate for the strength
ratio at which lateral dynamic instability might occur was developed based
on nonlinear regression of the extensive volume of data generated during this
investigation. In performing this regression, results were calibrated to the
median response of the SDOF spring systems studied in this investigation.
A median-targeted minimum strength requirement (maximum value of R) for
lateral dynamic instability, Rdi, is proposed in Equation 5-8:
3
3
a
c e r u r
di e
y c y
R b T F T
. F
. . . . . . . .
. .. .. . . .. .. . . . . . .
(5-8)
where Te is the effective fundamental period of vibration of the structure, .y
,
.c
, .r
, and .u
are displacements corresponding to the yield strength, Fy,
capping strength, Fc, residual strength, Fr, and ultimate deformation capacity
at the end of the residual strength plateau, as shown in Figure 5-14.
Parameters a and b are functions given by:
a .1. exp(.dTe ) (5-9)
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-17
2
1 r
c
b F
F
. .
. .. .
. .
(5-10)
The parameter d is a constant equal to 4 for systems with stiffness
degradation, and 5 for systems without stiffness degradation. The parameter
. is the ratio of the post-capping slope (degrading stiffness) to the initial
effective slope (elastic stiffness).
.y .c .r .u
Fr
Fy
Fc
F
Ke
.Ke
.Ke
.y .c .r .u
Fr
Fy
Fc
F
Ke
.Ke
.Ke
Figure 5-14 Simplified force-displacement boundary for estimating the
median collapse capacity associated with dynamic instability.
5.2.3.1 Practical Ramifications
Since Rdi has been calibrated to median response, use of this equation could
eliminate some of the conservatism built into the current Rmax limitation on
use of nonlinear static procedures. Calibrated using the extensive volume of
data generated during this investigation, use of this equation could improve
the reliability of current nonlinear static procedures with regard to cyclic and
in-cycle degradation.
In conjunction with a pushover curve used as a system force-displacement
capacity boundary, the equation for Rdi could be used to determine if a
system is susceptible to lateral dynamic instability for a specified level of
spectral acceleration, SaT. Similar to Rmax, use of Rdi would involve
comparison with R (Equation 5-3). If R < Rdi the system could be deemed
satisfactory without additional nonlinear dynamic analysis. This capability
is, of course, limited to systems for which the assumption of SDOF behavior
is appropriate (i.e., MDOF effects are not significant).
Calculated values of Rdi should be viewed carefully with respect to the
intensity measure (SaT) considered. Collapse limit states (i.e., lateral dynamic
5-18 5: Findings, Conclusions, and Recommendations FEMA P440A
instability) should be evaluated for intensities associated with rare ground
motions (long return periods). Evaluation of collapse limit states at lower
ground motion intensities leaves open the possibility that collapse could
occur during events in which those intensities are exceeded.
In addition, the development of the proposed equation for Rdi targeted
median response, which was intentionally less conservative than the level at
which the FEMA 440 equation for Rmax appeared to be predicting. Median
response implies a fifty percent chance of being above or below the specified
value. Use of Rdi in engineering practice should consider whether or not a
median predictor represents an appropriate level of safety against the
potential for lateral dynamic instability. If needed, a reduction factor could
be applied to Equation 5-8 to reduce the resulting values of Rdi and ultimately
achieve a correspondingly higher level of safety.
5.2.4 Simplified Nonlinear Dynamic Analysis Procedure
From empirical relationships for characteristic segments of IDA curves for
many systems, Vamvatsikos and Cornell (2006) suggested that static
pushover curves could be used to estimate nonlinear dynamic response. The
open source software tool, Static Pushover 2 Incremental Dynamic Analysis
(SPO2IDA), was created as a product of that research, and can be obtained at
http://www.ucy.ac.cy/~divamva/software.html. A Microsoft Excel version
of the SPO2IDA application has also been provided on the CD
accompanying this report.
As the name suggests, SPO2IDA transforms static pushover (SPO) curves to
incremental dynamic analysis (IDA) plots. It utilizes a large database of IDA
results to fit representative 16th, 50th, and 84th percentile IDA curves to a
given idealized single-degree-of-freedom (SDOF) oscillator subjected to a
static pushover analysis. The relationships between force-displacement
capacity boundaries and IDA curves observed in this investigation are
consistent with this notion.
Focused analytical studies comparing force-displacement capacity
boundaries to incremental dynamic analysis results led to the concept of a
simplified nonlinear dynamic analysis procedure. In this procedure, a
nonlinear static analysis is used to generate an idealized force-deformation
curve (i.e., static pushover curve). The resulting curve is then used as a
force-displacement capacity boundary to constrain the hysteretic behavior of
an equivalent SDOF oscillator. This SDOF oscillator is then subjected to
incremental dynamic analysis (or approximate IDA results are obtained using
SPO2IDA).
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-19
The concept of a simplified nonlinear dynamic analysis procedure is
described in the steps outlined below.
. Develop an analytical model of the system.
Models can be developed in accordance with prevailing practice for
seismic evaluation, design, and rehabilitation of buildings described in
ASCE/SEI 41-06. Component properties should be based on forcedisplacement
capacity boundaries, rather than cyclic envelopes.
. Perform a nonlinear static pushover analysis.
Subject the model to a conventional pushover analysis in accordance
with prevailing practice. Lateral load increments and resulting
displacements are recorded to generate an idealized force-deformation
curve.
. Conduct an incremental dynamic analysis of the system based on an
equivalent SDOF model.
The idealized force-deformation curve is, in effect, a system forcedisplacement
capacity boundary that can be used to constrain a hysteretic
model of an equivalent SDOF oscillator. This SDOF oscillator is then
subjected to incremental dynamic analysis to check for lateral dynamic
instability and other limit states of interest. Alternatively, an
approximate incremental dynamic analysis can be accomplished using
the idealized force-deformation curve and SPO2IDA.
. Determine probabilities associated with limit states of interest.
Results from incremental dynamic analysis can be used to obtain
response statistics associated with limit states of interest in addition to
lateral dynamic instability. SPO2IDA can also be used to obtain median,
16th, and 84th percentile IDA curves relating displacements to intensity.
Using the fragility relationships described in Appendix E in conjunction
with a site hazard curve, this information can be converted into annual
probabilities of exceedance for each limit state. Probabilistic information
in this form can be used to make enhanced decisions based on risk and
uncertainty, rather than on discrete threshold values of acceptance.
The procedure is simplified because only a SDOF oscillator is subjected to
nonlinear dynamic analysis. Further simplification is achieved through the
use of SPO2IDA, which avoids the computational effort associated with
incremental dynamic analysis. This simplified procedure has several
advantages over nonlinear static analysis procedures: (1) lateral dynamic
instability is investigated explicitly; (2) results include the effects of recordto-
record variability in ground motion; (3) response can be characterized
5-20 5: Findings, Conclusions, and Recommendations FEMA P440A
probabilistically; and (4) uncertainty can be considered explicitly. Results
can be investigated for any limit state that can be linked to the demand
parameter of interest (e.g. roof displacement).
Use of the procedure is explained in more detail in the example application
contained in Appendix F.
5.3 Suggestions for Further Study
This section summarizes suggestions for further study that will expand the
application of results to more complex systems, fill in gaps in existing
knowledge, and enhance future practice.
5.3.1 Application of Results to Multiple-Degree-of-Freedom
Systems
Multi-story buildings are more complex dynamic systems whose seismic
response is more difficult to estimate than that of SDOF systems. Recent
studies have suggested that it may be possible to estimate the collapse
capacity of multiple-degree-of-freedom (MDOF) systems through dynamic
analysis of equivalent SDOF systems.
As part of the focused analytical work, preliminary studies of MDOF
systems were performed as summarized in Appendix G. These studies
investigated the use of nonlinear static analyses combined with incremental
dynamic analyses of equivalent SDOF systems to evaluate dynamic
instability of multi-story buildings ranging in height from 4 to 20 stories.
Preliminary results indicate that many of the findings for SDOF systems in
this investigation (e.g., the relationship between force-displacement capacity
boundary and IDA curves; the equation for Rdi) may be applicable to MDOF
systems. More detailed study of the application of these results to MDOF
systems is recommended as a result of this investigation, and additional
MDOF investigations are planned under a project funded by the National
Institute of Standards and Technology (NIST).
5.3.2 Development of Physical Testing Protocols for
Determination of Force-Displacement Capacity
Boundaries
Important conceptual differences exist between force displacement capacity
boundaries and loading protocol-specific cyclic envelopes. Proper definition
of hysteretic behavior in a component model requires an understanding of the
initial force-displacement capacity boundary and how that boundary
degrades under cyclic loading. The use of several loading protocols is
desirable, but there is no recognized testing procedure that accomplishes this.
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-21
The loading protocol for experimental investigations described in Section 2.8
of ASCE/SEI 41-06 is not specific enough to produce a true forcedisplacement
capacity boundary. For a set of identical specimens, necessary
testing could conceivably include: (1) monotonic loading to get an initial
capacity boundary; (2) multiple symmetric cyclic loading cases to calibrate
cyclic degradation; (3) high frequency or long duration cyclic loading cases
to check for fracture or fatigue; (4) cyclic loading followed by a monotonic
push to more clearly observe changes due to cyclic degradation; and (5)
unsymmetrical cyclic loading. Development of a specification for physical
testing protocols necessary to generate appropriate force-displacement
capacity boundaries is recommended.
5.3.3 Development and Refinement of Tools for Approximate
Nonlinear Dynamic Analysis
Nonlinear dynamic analysis has obvious advantages over nonlinear static
analysis procedures. Disadvantages are related to increased computational
effort. Studies have shown that the characteristics of nonlinear dynamic
response can be estimated through simplified approximate relationships
based on the results of static pushover analyses.
Software tools such as SPO2IDA have the capability to estimate dynamic
response without the computational effort associated with incremental
dynamic analysis. This approximation facilitates the use of dynamic analysis
results to supplement and inform more simplified analysis procedures (e.g.,
nonlinear static procedures). Development and refinement of similar
approximate tools for performing nonlinear dynamic analyses is
recommended.
5.4 Concluding Remarks
Using FEMA 440 as a starting point, this investigation has advanced the
understanding of degradation and lateral dynamic instability by:
. Investigating and documenting currently available empirical and
theoretical knowledge on nonlinear cyclic and in-cycle strength and
stiffness degradation, and their affects on the stability of structural
systems
. Supplementing and refining the existing knowledge base with focused
analytical studies
This investigation has resulted in an extensive collection of available
research on component modeling of degradation, and a database of analytical
results from over 2.6 million nonlinear response history analyses
5-22 5: Findings, Conclusions, and Recommendations FEMA P440A
documenting the effects of a variety of parameters on the overall response of
SDOF systems with degrading components.
Results have confirmed conclusions regarding degradation and dynamic
instability presented in FEMA 440, provided updated information on
modeling to differentiate between cyclic and in-cycle strength and stiffness
degradation, and linked nonlinear dynamic response to major characteristics
of component and system degrading behavior. This information has resulted
in:
. an improved understanding of nonlinear degrading response and the
practical ramifications of this information for engineering practice
. recommendations to better account for nonlinear degrading response in
the context of current nonlinear analysis procedures
. suggestions for further study
Results from this investigation will ultimately improve the modeling of
structural components considering cyclic and in-cycle degrading behavior,
improve the characterization of lateral dynamic instability, and reduce the
conservatism in current analysis procedures making it more cost-effective to
strengthen existing buildings for improved seismic resistance in the future.
FEMA P440A A: Detailed Summary of Previous Research A-1
Appendix A
Detailed Summary of
Previous Research
This appendix contains a detailed summary of the development of hysteretic
models for nonlinear analysis. It also contains summaries of publications that
were extensively reviewed for guidance on appropriately targeting and
scoping focused analytical studies.
A.1 Summary of the Development of Hysteretic Models
A.1.1 Non-Deteriorating Models
Early studies that incorporated nonlinear behavior in seismic response of
structures assumed the structure to have an elastoplastic hysteretic behavior
or a bilinear hysteretic behavior (e.g., Berg and Da Deppo, 1960; Penzien
1960a, 1960b; Iwan 1961). These might be perfectly plastic with no postelastic
stiffness or with some strain hardening. More accurate models were
also developed with smooth rounded transitions from elastic to plastic
regions. (Ramberg and Osgood, 1943; Pinto and Guiffre, 1970; Menegotto
and Pinto, 1973). Ramberg-Osgood and Giuffre-Menegotto-Pinto models
continue to be used today for modeling non-degrading structures such as
steel moment-frame structures when fracture and buckling do not occur, and
have recently been used successfully to model the hysteretic behavior of
buckling-restrained braces (e.g., Lin et al., 2004).
Other examples of non-degrading smooth hysteretic models commonly used
are the Bouc-Wen model (Bouc, 1967a, 1967b; Wen, 1976, 1989) and the
Ozdemir model (1976). Unlike the Ramberg-Osgood and Giuffre-
Menegotto-Pinto models in which the force-deformation relationship is
described by an algebraic equation, in the Bouc-Wen and Ozdemir models
the force-displacement relationship and the force-deformation characteristics
are described by a differential equation. These models are relatively easy to
implement and are capable of describing, relatively well, non-degrading
hysteretic behavior. An extension of smooth models to a three-dimensional
tensorial idealization of Prager's model was developed by Casciati and
Faravelli (1985, Casciati, 1989). Although models based on differential
A-2 A: Detailed Summary of Previous Research FEMA P440A
equations with smooth loading curves, such as the Bouc-Wen or Ozdemir
models, are relatively easy to implement, they generally exhibit a local
violation of Drucker’s stability postulate. In particular, Thyagarajan and
Iwan (1990) concluded that the Wen-Bouc model tends to exhibit a
pronounced drift, particularly when post-yield stiffness is small.
A.1.2 Piecewise Linear Deteriorating Models
Many structural materials and structural elements will exhibit some level of
degradation of stiffness or strength or both, or may also exhibit other
phenomena such as pinching, when subjected to cyclic reversed loading, this
is especially true for reinforced concrete elements subjected to several large
cyclic reversals. Deterioration can be the result of, for example, cracking,
crushing, rebar buckling, crack or gap opening and closing, loss of bond,
and interaction with high shear or axial stresses. The level of degradation
depends, on the one hand, on the characteristics of the structural element
such as properties of the materials, geometry, level of detailing, and type and
characteristics of the connections, and on the other hand, on the loading
history (e.g., loading intensity on each cycle, number of cycles, and sequence
of loading cycles).
One of the earliest attempts to model deterioration of structural elements
subjected to cyclic reversals was conducted by Jacobsen (1958) who
proposed a behaviorist model to study the response of connections to cyclic
loading. His model consisted of a combination of sliding blocks arranged in
series which experienced frictional forces of different amplitudes and which
were joined by Hookean springs with different elastic stiffnesses. Although
this model allowed the mathematical description of observed static behavior,
earthquake response of deteriorating structures was not studied until 1962,
when Hisada proposed a degrading model (Hisada, Nakagawa and Izumi,
1962) for studying earthquake response of degrading structures.
Concerned with the stiffness degradation observed in reinforced concrete
elements, Clough and Johnston (Clough, 1966; Clough and Johnston, 1966)
developed a degrading model which incorporated stiffness degradation after
reloading. In this model unloading occurred with a stiffness equal to the
initial stiffness but reloading was aimed towards the largest excursion in
previous cycles. They used the model to study the response of SDOF systems
subjected to four recorded acceleration time histories. In particular, this study
computed ratios of maximum deformation of elastoplastic systems to
maximum deformation of stiffness-degrading systems. When evaluating
these ratios they wrote “these ratios demonstrate conclusively that there is no
FEMA P440A A: Detailed Summary of Previous Research A-3
significant difference between the yield amplitudes generated in the two
materials. The ratios vary between 0.8 and 1.2 except in a few cases.” They
concluded that “earthquake ductility requirements in the degrading stiffness
systems are not materially different from those observed in ordinary
elastoplastic structures, except for structures having a period of vibration
less than ½ second.” Based on their study they also concluded that “the
ductility required in the members of reinforced concrete frame buildings will
be about the same as is required in equivalent steel frame buildings.”
An unrealistic feature of the Clough model when experiencing large load
reversals followed by small load reversals was pointed out by Mahin and
Bertero (1976) and by Riddell and Newmark (1979) who showed that after a
small unloading the model would unrealistically reload toward the point of
maximum deformation. They modified the model to reload along the same
unloading branch until the reloading branch was reached and then aim
toward the point of peak deformation. Mahin and Bertero (1976) also made
the model more versatile by incorporating a positive post-yield stiffness and
variable unloading stiffness as a function of the peak deformation. The model
proposed by Mahin and Bertero (1972), which is often referred to as the
modified-Clough model, has been incorporated in several general nonlinear
analysis programs and has been used extensively to model the behavior of
flexurally controlled reinforced concrete elements.
An early model proposed for nonlinear analysis of reinforced concrete is the
Takeda model (Takeda, Sozen and Nielsen, 1970). This model incorporated
some of the features of the Clough model but also added other features such
as a trilinear loading curve to incorporate pre-cracking and post-cracking
stiffnesses, a variable unloading stiffness which was a function of the peak
deformation, and improved hysteretic rules for inner cyclic loops. This model
has also been incorporated in several general analysis programs and has been
extensively used in earthquake engineering to study the seismic response of
reinforced concrete structures.
A slight modification to the Takeda model was proposed by Otani and Sozen
(1972) who replaced the trilinear initial loading segments of the Takeda
model by a bilinear relationship. The resulting model is known as bilinear
Takeda model. Otani (1981) compared the response of six different hysteretic
models (Ramberg-Osgood, degrading bilinear, modified-Clough, bilinear
Takeda, Takeda and degrading trilinear) when subjected to horizontal
components of the 1940 El Centro and the 1954 Taft records. He concluded
A-4 A: Detailed Summary of Previous Research FEMA P440A
that “maximum response amplitudes are not as sensitive to details in the
differences in hysteretic rules of these models.”
Other early models developed specifically for reinforced concrete structures
include the model developed by Nielsen and Imbeault (1970) who proposed a
degrading bilinear system whose stiffness would change only when a prior
maximum displacement was exceeded, and the degrading model proposed by
Anagnostopoulos (1972) which combines Nielsen’s degrading bilinear model
and the Clough model. Models where reloading is aimed at the point of
maximum deformation in prior cycles are sometimes also referred to as
“peak-oriented” models (Rahnama and Krawinkler, 1993; Medina and
Krawinkler, 2004). The Clough model, the modified-Clough, Takeda,
bilinear Takeda and Anagnostopoulos models are all peak-oriented models.
Iwan developed a general class of stiffness-degrading and pinching models
(Iwan, 1973, 1977, 1978). Similar to prior models developed by him (Iwan,
1966, 1967) and by Jacobsen (1958), this model consisted of a collection of
linear elastic and Coulomb slip elements. He then studied the response of a
wide range of stiffness-degrading and pinching models when subjected to an
ensemble of 12 accelerograms recorded in various earthquakes (Iwan and
Gates, 1979a, 1979b). After comparing the response of the various degrading
and non-degrading systems they noted “despite the quite different load
deformation characteristic the overall effect for a given ductility is nearly the
same. This is a rather surprising result which may be useful in design, for it
implies that it may not be necessary to know the precise details of the loaddeflection
behavior of a structure in order to make a reasonably accurate
estimate of its response.” In another study aimed at estimating inelastic
spectra from elastic spectra using equivalent linear methods, Iwan (1980)
concluded that “the differences in hysteretic behavior considered herein
appear to have only a secondary effect on the accuracy of the results.”
Chopra and Kan (1973) studied the effects of stiffness degradation on
ductility requirements of two idealized multistory buildings, one having a
period of vibration of 0.5 s and the other 2.0 s. They concluded that “stiffness
degradation has little influence on ductility requirements for flexible
buildings, but it leads to increased ductility requirements for stiff buildings.”
Riddell and Newmark (Riddell and Newmark, 1979; Newmark and Riddell,
1980) studied the influence of hysteretic behavior on inelastic spectral
ordinates. They considered an elastoplastic system, a bilinear system and a
stiffness degrading system. They compared average inelastic spectral
FEMA P440A A: Detailed Summary of Previous Research A-5
ordinates for the three systems for ground motions scaled to peak ground
acceleration, peak ground velocity and peak ground displacement in the
acceleration, velocity and displacement-controlled spectral regions,
respectively. They arrived at similar conclusions to those of Clough or those
of Iwan; in particular they concluded that “the ordinates of the average
spectra do not vary significantly when various nonlinear models are used.”
They also noted “It is particularly significant that, on the average, the
stiffness degradation phenomenon is not as critical as one might expect ” and
concluded that “the use of the elastoplastic idealization provides, in almost
every case, a conservative estimate of the average response to a number of
earthquake ground motions.”
Mahin and Bertero (1981) used the modified Clough model to also study the
difference in response of elastoplastic systems and stiffness-degrading
systems. They noted that “ductility demands for a stiffness degrading system
subjected to a particular ground motion can differ significantly from those
obtained for elastoplastic systems in some period ranges. However, it
appears that, on average, the differences are generally small.” Similar
conclusions were also reached by Powel and Row (1976) who studied the
influence of analysis assumptions on computed inelastic response of three
different types of reinforced concrete ten-story buildings. They concluded
that “degrading stiffness appears to have no substantial influence on
interstory drift demands.” Nassar and Krawinkler (1991), used the modified
Clough model to study the difference of strength reduction factors associated
to increasing levels of ductility demands in bilinear and stiffness degrading
models. In their report they wrote “... except for very short period systems,
the stiffness-degrading models allow higher reduction factors than the
bilinear model, for systems without strain hardening. This difference
diminishes with strain hardening. This is a very interesting result in that it
suggests that the stiffness degrading model behaves “better” than the
bilinear model, i.e., it has a smaller inelastic strength demand for the same
ductility ratio.”
Other piece-wise models that incorporate degradation include the Park and
Ang mechanistic model (Park and Ang, 1985) and the three-parameter model
(Park, Reinhorn and Kunnath, 1987; Valles et al., 1996). The three-parameter
model includes strain hardening, variable unloading stiffness, pinching and
cyclic load degradation (that is, decreasing yielding strength as a function of
maximum deformation, hysteretic energy demand, or a combination of the
two). The model was further improved in Kunnath and Reinhorn (1990).
Rahnama and Krawinkler (1993) developed a general piece-wise linear
A-6 A: Detailed Summary of Previous Research FEMA P440A
hysteretic model which was incorporated into a SDOF analysis program
referred to as SNAP (SDOF Nonlinear Analysis Program). The model has a
bilinear skeleton relationship and includes variable unloading stiffness, peakoriented
stiffness degradation at reloading, pinching, cyclic strength
deterioration as a function of hysteretic energy demands, and also the
capability of accelerating the degradation of loading stiffness beyond the
peak-oriented degradation.
They used this model to study the influence of hysteretic behavior of SDOF
systems subjected to 15 recorded ground motions recorded at firm sites
during California earthquakes to study constant-ductility strength-reduction
factors. Their results confirmed observations of Nassar and Krawinkler
(1991) and of previous investigators who noted that the effect of stiffness
degradation was small, on average, leading to smaller displacement demands
except for short-period structures where displacement demands in systems
with stiffness degradation were larger than those in bilinear systems. In their
study they also noted that cyclic strength deterioration increased
displacement demands but that the increase was not large unless the strength
deteriorates to a small value, and noted that further research was needed
before quantitative conclusions could be drawn.
Rahnama and Krawinkler also studied the effect of in-cyclic degradation by
considering a negative post-elastic slope (that is, negative strain hardening)
in bilinear and degrading models. They concluded that “ratios of reduction
factors for degrading and bilinear systems become significantly larger than
1.0 when negative hardening is present, particularly if the periods of
vibration are short and the ductility demands are high.”
Gupta and Krawinkler (1998, 1999) investigated the effects of pinching and
stiffness degradation in SDOF and MDOF structures using the hysteretic
model previous developed by Rahnama (Rahnama and Krawinkler, 1993)
which was incorporated in the DRAIN-2DX analysis program (Allahabadi
and Powell, 1988). They concluded that “for SDOF systems, pinching leads
to a relatively small amplification of the displacement response for systems
with medium and long periods, regardless of the yielding strength. For short
period structures, which are subject to a larger number of cycles, the
displacement amplification increases significantly.” They also noted that “the
effect of the pinched force-deformation relationship on the displacement
ratio is not very sensitive to the severity of the ground motion.” For MDOF
structures they concluded that “pinching of the force-deformation
characteristics of inelastic systems has a global (roof) drift similar to that
FEMA P440A A: Detailed Summary of Previous Research A-7
observed in SDOF systems.” They also investigated the effect of negative
post-yield stiffness in SDOF and MDOF systems. They concluded that “for
SDOF systems a negative post-yield stiffness (which could represent P-.
effects) has a large effect on the displacement demand for systems with
bilinear characteristics. The effect increases rapidly with an increase in the
negative stiffness ratio ., with decrease in the yield strength of the system,
and a decrease in the period. Dynamic instability, caused by attainment of
zero lateral resistance, is a distinct possibility and was observed under
several of the ground motion records.” For systems with negative strain
hardening they noted that the pinching model exhibits better behavior than
the bilinear model.
Recently, Ruiz-Garcia and Miranda (2005) examined the effect of hysteretic
behavior on maximum deformations of SDOF systems subjected to an
ensemble of 240 ground motions recorded in California. They considered
seven different types of hysteretic behavior: elastoplastic, bilinear, modified
Clough, Takeda, origin-oriented, moderate degrading, and severely
degrading. The modified Clough, Takeda and origin-oriented models only
exhibit stiffness degradation while the moderate degrading and severely
degrading systems exhibit both stiffness and cyclic strength degradation.
They found that the effect of positive post-yield stiffness was relatively small
except for systems with very short periods of vibration (T<0.2s). When
subjected to firm soil records they found that the effects of hysteretic
behavior were relatively small for structures with periods of vibration larger
than about 0.7s.
The same authors used the modified Clough model to examine the effect of
stiffness degradation on single-degree-of-freedom systems subjected to
ground motions recorded on soft soil sites (Miranda and Ruiz-Garcia, 2002,
Ruiz-Garcia and Miranda, 2004, 2006b). They concluded that the effects of
stiffness degradation were larger for structures on soft soil sites than those
observed for structures on firm sites. In particular, they concluded that for
structures with periods of vibration shorter than the predominant period of
the ground motion, the lateral displacement demands in stiffness degrading
systems on average are 25% larger than those of non-degrading systems and
that, in order to control lateral deformations to comparable levels of those in
non-degrading structures, stiffness-degrading structures in this spectral
region need to be designed for higher lateral forces.
A model similar to the one developed by Krawinkler and his coworkers but
with additional capabilities to model connection fracture was developed and
A-8 A: Detailed Summary of Previous Research FEMA P440A
incorporated into DRAIN-2DX by Shi and Foutch (Shi and Foutch, 1997;
Foutch and Shi, 1998). They studied the influence of hysteretic behavior on
the seismic response of buildings by considering seven different hysteretic
behaviors, which included non-degrading and degrading models, and nine
steel moment-resisting frame models of buildings with three, six, and nine
stories. They concluded that “Hysteresis type has only a minimum effect on
ductility demands of structures.” When evaluating ratios of deformations of
degrading to bilinear behavior they noted that “For the non-pinching
hysteresis models, the maximum ratios of ductility demand to the bilinear
hysteresis model range from 1.10 to 1.15 when the period of the structure is
less than 1.0 second. For pinching hysteresis types the maximum ratios are
on the order of 1.25 to 1.30.”
Gupta and Kunnath (1998) arrived at similar conclusions. More recently
Medina and Krawinkler (2004) studied the effects of hysteretic behavior (i.e.,
bilinear, peak oriented and pinching) in the evaluation of peak deformation
demands and their distribution over the height for regular frame structures
over a wide range of stories (from 3 to 18) and fundamental periods (from
0.3 s. to 3.6 s.). The study did not consider monotonic in-cycle deterioration.
The ground motions used were those with frequency content characteristic of
what they referred to as “ordinary ground motions” (that is, no near-fault or
soft soil effects). They concluded that “the degree of stiffness degradation is
important for the seismic performance evaluation of regular frames because
systems with a large degree of stiffness degradation tend to exhibit larger
peak drift demands and a less uniform distribution of peak drifts over the
height.”
Based on the general class of non-degrading and degrading models
developed by Iwan (1966, 1967, 1973), Mostaghel (1998, 1999) developed a
general hysteretic model by providing an analytical description (that is, with
differential equations) of physical models consisting of a series of linear
springs, dashpots, and sliders. The model includes the effects of pinching,
stiffness degradation, and load deterioration. He showed that complex multilinear
hysteretic behavior can be obtained by solving (2n-1) differential
equations where n is the number of linear segments in the model.
A.1.3 Smooth Deteriorating Hysteretic Models
Degradation and pinching have also been incorporated in smooth hysteretic
models. Some examples of degrading smooth models are the Baber model
(Baber and Wen, 1981; Baber and Noori 1985, 1986) which extends the
Bouc-Wen model to include stiffness degradation and pinching.
FEMA P440A A: Detailed Summary of Previous Research A-9
More recently, Sivaselvan and Reinhorn (1999, 2000) developed a versatile
hysteretic model that is conceptually based on a general class of nondegrading
and degrading models developed by Iwan (1966, 1967, 1973) but
extended the model developed by Mostaghel (1998, 1999) to include smooth
curvilinear segments. Stiffness degradation is incorporated using a pivot rule
analogous to the one incorporated in the three-parameter model (Park,
Reinhorn and Kunnath, 1987). Cyclic strength degradation is modeled by
reducing the capacity in the backbone curve while pinching is achieved by
adding an additional slip-lock spring in series with the main smooth
hysteretic spring, which is similar to the Bouc-Wen model. The hysteretic
behavior is then described by the solution of four time-independent
differential equations which are solved using Runge-Kutta’s method.
Although they provided specific rules for controlling stiffness degradation,
cyclic degradation, and pinching, they showed that other rules could be
implemented as well. This hysteretic model has been incorporated in recent
versions of IDARC (Valles et al., 1996).
A.1.4 Hysteretic Models for Steel Braces
Experimental research on the behavior of steel braces has shown that their
behavior under severe cyclic loading is complicated and not fully understood.
Cyclic nonlinear behavior of steel brace members is complex as a result of
various phenomena occurring in the braces and their connections, such as
yielding in tension, buckling in compression, post-buckling deterioration of
compressive load capacity, deterioration of axial stiffness with cycling, lowcycle
fatigue fractures at plastic hinge regions, Bauschinger effect, and
buckling and fracture in the gusset plates. As in the models previously
described, element models for steel braces can be classified as either
phenomenological models in which the load-deformation behavior of steel
braces is described through a series of hysteretic rules that try to reproduce
behavior observed experimentally, or material-based models such as finite
element models and fiber element models where the steel brace is discretized
into small elements and the overall behavior of the brace is obtained from
uniaxial, biaxial or triaxial material behavior of the material.
A significant amount of both experimental and analytical work on the
behavior of steel bracing has been conducted at the University of Michigan
under the direction of Professors Goel and Hanson. One of the first analytical
models for predicting the force-deformation behavior of axially-loaded
members with intermediate slenderness ratios was developed by
Higginbotham and Hanson (1976). Prathuangsit, Goel, and Hanson, (1978)
proposed a model with rotational end springs to simulate the end restraint
A-10 A: Detailed Summary of Previous Research FEMA P440A
resulting from the flexural rigidity of the connections of axially loaded (that
is, bracing) members. They showed that members with balanced strength
connections, (that is, which form plastic hinges simultaneously at midspan
and at the ends), have more efficient compressive load and energy dissipation
capacities than members of the same length and same cross-sectional
properties with unbalanced strength connections. They concluded that the
hysteresis behavior of a balanced strength member can be represented
adequately by that of a pin-connected member of the same cross section and
same effective slenderness ratio
Jain, Goel, and Hanson tested 17 tube specimens and eight angle specimens
under repeated axial loading (Jain, Goel and Hanson, 1976, 1978a, b). The
objective of this experimental investigation was to quantify the reduction in
maximum compressive loads and increase in member length, and to study the
influence of the buckling mode and the shape of the cross-section on the
hysteretic behavior and dissipation of energy through the hysteretic cycles.
They concluded that local buckling and shape of the cross section can have a
significant influence on the hysteretic behavior of axially loaded steel
members. Based on their experimental results they developed a hysteresis
model for steel tubular members that included a reduction in compressive
strength and an increase in member length with the number of cycles (Jain
and Goel, 1978, 1980) which was then incorporated in the DRAIN-2D
analysis program. This model has been extensively used by investigators at
the University of Michigan and elsewhere to study the seismic response of
concentrically braced steel frames.
Astaneh-Asl and Goel investigated the behavior of double-angle bracing
members subjected to out-of-plane buckling due to severe cyclic load
reversals (Astaneh-Asl et al. 1982; Astaneh-Asl and Goel 1984). Nine fullsize
test specimens were subjected to severe inelastic axial deformations.
Test specimens were made of back-to-back A36 steel angle sections
connected to the end gusset plates by fillet welds or high-strength bolts. Five
of the test specimens were designed according to current design procedures
and code requirements. These specimens experienced fracture in gusset
plates and stitches during early cycles of loading. Based on observations and
analysis of the behavior of the specimens, new design procedures were
proposed for improved ductility and energy dissipation capacity of doubleangle
bracing members. Goel and El-Tayem (1986) investigated the behavior
of angle cross-bracing.
FEMA P440A A: Detailed Summary of Previous Research A-11
Gugerili and Goel (1982) tested nine commercially available wide-flange
shapes and structural tubes with different slenderness and width-to-thickness
ratios in order to investigate the effects of cross section and slenderness ratio.
A general rule was developed for transitioning between compression and
tension mechanism lines of constant shape and elastic segments. The new
rule also included the effects of residual elongation. The theoretical model
was used as a basis for a developing a semi-empirical model for predicting
experimental hysteresis loops more accurately than previous models. This
model included the decrease in compressive strength with cumulative plastic
hinge deformation.
Tang and Goel (1987) developed a procedure to predict the fracture life of
bracing members. Their empirically based procedure was refined with an
energy approach using Jain's hysteresis model and was then also incorporated
in the the DRAIN-2D program. Based on the experimental results of
previous researchers Hassan and Goel (1991) formulated a refined and
practical hysteresis model for bracing the members of concentrically
(chevron) braced steel structures subjected to severe earthquakes. A more
recent model was implemented in a Structural Nonlinear Analysis Program
(SNAP) (Rai et al, 1996), which eliminates the brace once it is estimated that
it has fractured (while this program has the same name, it is different from
the program developed by Krawinkler and his coworkers).
Professors Popov and Mahin at the University of California at Berkeley have
also conducted a significant amount of experimental and analytical research
on steel braces. Zayas, Popov, and Mahin, (1979, 1980) conducted a series of
experimental tests on scaled tubular brace members subjected to severe
inelastic cyclic loading. The tubular brace specimens considered were onesixth
scale models of braces of the type used for offshore platforms. A
method of predicting the reduction in buckling load was presented. Maison
and Popov (1980) performed an experimental and analytical investigation of
the behavior of structural steel frames with K-braces subjected to severe
cyclic loadings simulating seismic effects. They developed an empirical
brace model to analyze steel frames with K-braces.
Black and Popov, E.P. (1980, 1981) conducted an experimental study on 24
commercially available steel struts, commonly used as bracing members.
They investigated the effects of loading patterns, end conditions, crosssectional
shapes, and slenderness ratios on the hysteresis response of
members. A large variety of shapes were tested, including wide flanges,
structural tees, double-angles, a double-channel, and thick-walled and thin-
A-12 A: Detailed Summary of Previous Research FEMA P440A
walled square and round tubes. Two types of boundary conditions were
considered, pinned-pinned and fixed-pinned, with effective slenderness ratios
of 40, 80, and 120. An explanation was provided regarding the fundamental
mechanisms responsible for the observed degradation in the buckling load
capacity during inelastic cycling.
Ikeda and Mahin (1984) developed an analytical model for simulation of the
inelastic buckling behavior of steel braces. In their model, buckling is
simulated by the use of predefined straight-line segments and simple rules
regarding factors such as buckling load deterioration and plastic growth. The
model, based on the approach of Maison and Popov (1980) overcomes some
of the limitations of earlier models. They proposed a systematic method for
selecting input parameters along with several rules governing the values of
certain parameters. The model was subsequently refined in 1986 by
combining analytical formulations describing plastic hinge behavior with
empirical formulas that are based on a study of experimental data. Analytical
expressions for the axial force versus axial deformation behavior of braces
were derived as solutions of the basic beam-column equation based on
specified assumptions. While analytical expressions form the basis of this
model, several empirical behavioral characteristics were implemented in the
modeling to achieve better representation of observed cyclic inelastic
behavior.
Khatib and Mahin (1987, 1988) conducted an analytical investigation on
concentrically braced frames. Their study stressed that concentrically braced
steel frames designed by conventional methods may exhibit several
undesirable modes of behavior. In particular, they showed that chevronbraced
frames have an inelastic cyclic behavior that is often characterized by
a rapid redistribution of internal forces, a deterioration of strength, a
tendency to form soft stories, and fracture due to excessive deformation
demand. They identified parameters having a significant influence on these
phenomena and provided recommendations for preferable ranges of brace
slenderness, approaches for designing beams, and a simplified capacity
design approach for proportioning columns and connections. They also
proposed several alternative brace configurations with improved behavior
including zipper bracing systems which incorporate vertical linkage elements
in a conventional chevron-braced frame to decrease concentrations of
interstory drift demands.
More recently Uriz and Mahin (2004, Uriz, 2005) conducted experimental
testing of a nearly full-size, two-story Special Concentric Braced Frame
FEMA P440A A: Detailed Summary of Previous Research A-13
(SCBF) specimen. They also conducted an analytical investigation using the
same reliability framework used to assess Special Moment Resisting Frame
(SMRF) structures during the FEMA/SAC Steel Project in order to assess the
confidence with which SCBFs might achieve the seismic performance
expected of new SMRF construction.
Other institutions have also been actively involved in research on the
behavior of steel braces. For example, Nonaka (1973, 1977) at Kyoto
University, conducted elastic, perfectly plastic analyses of a bar under
repeated axial loading. The bar was taken as a one-dimensional continuum
with both ends simply supported. His analysis considered the plastic
interaction for the combined action of bending and axial deformation, based
on a piecewise-linear yield condition. With a number of simplifying
assumptions, a closed form solution was derived that can describe the
hysteretic behavior of a bar, such as a structural brace or a truss member,
under any given history of tension and/or compression or of corresponding
displacements. Nonaka’s closed form solution was later extended by Shibata
(1982) for a bar of ideal I-section with bilinear stress-strain relationship. For
a bar of arbitrary solid cross-section with a piecewise linear stress-strain
relationship, an incremental load-displacement relationship was also obtained
in analytical form.
At the University of Canterbury in New Zealand, Remennikov and Walpole
(1997a, 1997b) developed an analytical model for the inelastic response
analysis of braced steel structures. Their model combines the analytical
formulation of plastic hinge behavior with empirical formulas developed on
the basis of experimental data. The brace is modeled as a pin-ended member,
with a plastic hinge located at the midspan and braces with other end
conditions are handled using the effective length concept. Step-wise
regression analysis is employed to approximate the plastic conditions for the
steel UC section. Verification of the brace model is performed on the basis of
quasi-static analyses of individual struts and a one-bay one-story crossbraced
steel frame.
At the Ecole Polytechnique de Montreal in Canada, Archambault, Tremblay,
and Filiatrault, (1995, 2003) conducted experimental and analytical studies
on the seismic performance of concentrically braced steel frames made with
cold-formed rectangular tubular bracing members. They tested a total of 24
quasi-static cyclic tests on full scale X bracing and single diagonal bracing
systems. They developed simplified models to predict the out-of-plane
deformation of the braces as a function of the ductility level. They then used
A-14 A: Detailed Summary of Previous Research FEMA P440A
their models to develop an empirical expression to assess the inelastic
deformation capacity before fracture of bracing members made of
rectangular hollow sections.
More recently, Jin and El-Tawil, (2003) developed a beam-column element
to model the inelastic cyclic behavior of steel braces. In their model a
bounding surface plasticity model in stress-resultant space coupled with a
backward Euler algorithm is used to keep track of the spread of plasticity
through the cross-section. Deterioration of the cross-section stiffness due to
local buckling is accounted for through a damage model.
Further information on the experimental and analytical response of steel
bracing is available in Tremblay (2002), Jin and El-Tawil, (2003) and Uriz
(2005), which provide summaries of experimental and analytical work. In
particular Tremblay (2002) conducted a survey of past experimental studies
on the inelastic response of diagonal steel bracing members subjected to
cyclic inelastic loading to collect data for the seismic design of concentrically
braced steel frames for which a ductile response is required during
earthquakes. He examined the buckling strength of the bracing members, the
brace post-buckling compressive resistance at various ductility levels, the
brace maximum tensile strength including strain hardening effects, and the
lateral deformations of the braces upon buckling. Additionally he proposed
equations for each of these parameters and examined the maximum ductility
that can be achieved by rectangular hollow bracing members.
Nakashima and Wakabayashi (1992) provide an overview of Japanese
experimental and analytical research on steel braces and braced frames.
Current Japanese practice is also briefly summarized.
A.2 Detailed Summaries of Relevant Publications
This section presents summaries of publications that were judged to be
particularly relevant to the subject of nonlinear degrading response, and
were, therefore, reviewed in detail. Each summary includes the list of
authors, an abstract, a narrative summary of the work, relevant figures, a
summary of important findings, and a listing of relevant publications
included in the list of references. The following publications were selected
for detailed review:
. Bernal, D., 1998, “Instability of buildings during seismic response,”
Engineering Structures, Vol. 20, No. 4-6, pp. 496-502.
FEMA P440A A: Detailed Summary of Previous Research A-15
. Pincheira, J.A, Dotiwala, F.S., and D’ Souza J.T., 1999, “Spectral
displacement demands of stiffness- and strength-degrading systems,”
Earthquake Spectra, 15(2), 245–272.
. Song, J.-K., and Pincheira, J.A, 2000, “Seismic analysis of older
reinforced concrete columns,” Earthquake Spectra, 16(4), 817–851.
. Miranda, E. and Akkar, S.D., 2003 “Dynamic instability of simple
structural systems,” Journal of Structural Engineering, ASCE, 129(12),
pp 1722-1727.
. Vian, D. and Bruneau, M., 2003, “Tests to structural collapse of singledegree-
of-freedom frames subjected to earthquake excitations,” Journal
of Structural Engineering, ASCE, 129(12), 1676-1685.
. Kanvinde, A.M., 2003, “Methods to evaluate the dynamic stability of
structures – shake table tests and nonlinear dynamic analyses,” EERI
Annual Student Paper Competition, Proceedings of 2003 EERI Annual
Meeting, Portland, Oregon.
. Vamvatsikos, D. and Cornell, C.A., 2005, Seismic performance, capacity
and reliability of structures as seen through incremental dynamic
analysis, John A. Blume Earthquake Engineering Research Center,
Report No. 151, Department of Civil and Environmental Engineering,
Stanford University, Stanford, California.
. Ibarra, L., Medina, R., and Krawinkler, H., 2005, “Hysteretic models that
incorporate strength and stiffness deterioration, Earthquake Engineering
and Structural Dynamics, Vol. 34, no. 12, pp. 1489-1511.
. Ibarra, L.F., and Krawinkler, H., 2005, Global collapse of frame
structures under seismic excitations, John A. Blume Earthquake
Engineering Research Center, Report No. 152, Department of Civil and
Environmental Engineering, Stanford University, Stanford, California.
. Kaul, R., 2004, Object-oriented development of strength and stiffness
degrading models for reinforced concrete structures, Ph.D. Thesis,
Department of Civil and Environmental Engineering, Stanford
University, Stanford, California.
. Elwood, K.J., 2002, Shake table tests and analytical studies on the
gravity load collapse of reinforced concrete frames, Ph.D. Dissertation,
University of California, Berkeley, California.
. Lee, L.H., Han, S.W., and Oh, Y.H., 1999, “Determination of ductility
factor considering different hysteretic models,” Earthquake Engineering
and Structural Dynamics, Vol. 28, 957–977.
A-16 A: Detailed Summary of Previous Research FEMA P440A
. Foutch, D.A. and Shi, S., 1998, “Effects of hysteresis type on the seismic
response of buildings,” Proc. 6th U.S. National Conference on
Earthquake Engineering, Seattle, Washington, Earthquake Engineering
Research Institute, Oakland, California.
. Ruiz-Garcia, J. and Miranda, E., 2003, “Inelastic displacement ratio for
evaluation of existing structures,” Earthquake Engineering and
Structural Dynamics. 32(8), 1237-1258.
. Dolsek, M. and Fajfar, P., 2004, “Inelastic spectra for infilled reinforced
concrete frames,” Earthquake Engineering and Structural Dynamics,
Vol. 33, 1395–1416.
A.2.1 Instability of Buildings During Seismic Response
Authors:
Bernal, D. (1998)
Abstract:
The issue of gravity-induced instability during response to severe seismic
excitation is examined. While static instability is fully determined by the
existence of at least one negative eigenvalue in the second-order tangent
stiffness, this condition is necessary but not sufficient for instability during
dynamic response. The likelihood of collapse is strongly dependent on the
shape of the mechanism that controls during the critical displacement cycle
and this shape can be reasonably identified using a pushover analysis with an
appropriately selected lateral load distribution. A characterization of the
instability limit state based on the reduction of a multistory building to an
equivalent SDOF system is presented
Summary:
Dynamic instability is a phenomenon whereby the response changes from
vibration to drift in a single direction. In this study a structure is defined as
stable if small increases in the ground motion intensity result in small
changes in the response. The study shows that the distribution of inelastic
action along the height of the building plays a critical role in the likelihood of
instability. The study emphasizes that gravity generally has little effect on the
dynamic response of structures, except when failure from instability is near.
In particular, the study shows that the static based approach of accounting for
second-order effects through amplifications of the first-order solution is not
appropriate in a dynamic setting. Specifically, design in a region where
FEMA P440A A: Detailed Summary of Previous Research A-17
amplifications from P-. are significant implies unacceptably low safety
factors.
Representative Figures:
Figure A-1 Effect of mechanism shape on the monotonic work vs.
amplitude relationship.
Figure A-2 Illustration of the definition of stability coefficient: (a) general
load deformation relationship, (b) elasto-plastic system.
Summary of Findings:
Dynamic instability takes place when the strength of the structure is below a
certain threshold and is strongly dependent on the shape of the failure (or
collapse) mechanism that controls. Safety against dynamic instability cannot
be guaranteed by placing controls on initial elastic stiffness; a rational check
of the safety against collapse must contemplate the strength and shape of the
critical mechanism. In particular, a rational approach is to estimate the
strength level associated with the instability threshold and to ensure that the
strength level provided exceeds the required limit by an appropriate safety
margin.
A-18 A: Detailed Summary of Previous Research FEMA P440A
Relevant Publications:
Bernal, D., 1998, "Instability of buildings during seismic response,"
Engineering Structures, Vol. 20, No. 4-6, pp. 496-502.
Bernal, D., 1992, "Instability of buildings subjected to earthquakes." Journal
of Structural Engineering , ASCE, Vol. 118, No. 8, pp. 2239-2260.
Bernal, D., 1987, “Amplification factors for inelastic dynamic P-. effects in
earthquake analysis,” Earthquake Eng. Struct. Dyn., 15(5), pp. 117-144.
A.2.2 Seismic Analysis of Older Reinforced Concrete Columns
Authors:
Pincheira, J.A., Dotiwala, F.S., and D’Souza J.T. (1999)
Abstract:
A nonlinear model and an analytical procedure for calculating the cyclic
response of nonductile reinforced concrete columns are presented. The main
characteristics of the model include the ability to represent flexure or shear
failure under monotonically increasing or reversed cyclic loading. Stiffness
degradation with cyclic loading can also be represented. The model was
implemented in a multipurpose analysis program and was used to calculate
the response of selected columns representative of older construction. A
comparison of the calculated response with experimental results shows that
the strength, failure mode and general characteristics of the measured cyclic
response can be well represented by the model.
Summary:
A beam-column element was created in order to simulate the behavior of
older non-ductile or shear-critical reinforced concrete columns in 2D frames.
This is a lumped plasticity element using two flexural springs at the beam
ends and a shear spring at the midpoint (Figure A-3). For the flexural springs
a Takeda hysteresis law is used together with a quadrilinear backbone curve
that incorporates a hardening post-yield segment followed by a negative
post-peak slope that stops at a residual plateau; essentially, only in-cycle
strength degradation is considered (Figure A-4a). On the other hand, the
shear spring uses a similar quadrilinear backbone but a pinching hysteresis
together with cyclic degradation of the post-peak strength (Figure A-4b).
The element was incorporated in the Drain-2D analysis program, resulting in
important limitations in its implementation and applicability. The solution
FEMA P440A A: Detailed Summary of Previous Research A-19
algorithm of Drain-2D cannot handle negative stiffness, thus necessitating
the use of numerical techniques to find an approximate solution. Specifically,
on a negative slope (for any of the three springs), the load steps are
performed first with an arbitrary positive stiffness, and the load unbalance is
then subtracted from the resulting increased load to move down to the actual
negative slope (Figure A-5). The results can be considered reliable only
under small load-steps and they may indeed lead to gross numerical errors
and possible numerical instabilities at the MDOF level. Furthermore, the use
of Drain-2D means that only a load-control pushover is possible, thus
severely reducing the applicability of this element for anything but timehistory
analysis with small time steps.
Significant effort has gone into the definition of the spring backbones, using
modified compression field theory for the shear backbone, and considering
anchorage slip, lap-splice slip, and section degradation for the flexural
springs. Calibration and testing of the element were performed with regard to
the experimental results.
Representative Figures:
Figure A-3 The RC column element formulation.
A-20 A: Detailed Summary of Previous Research FEMA P440A
Figure A-4 The hysteretic laws for shear and moment springs.
Figure A-5 The force (or moment) unbalance is subtracted after an arbitrary
positive stiffness step towards the “correct” displacement. Very
small load steps are needed for accuracy, even at the SDOF
level.
FEMA P440A A: Detailed Summary of Previous Research A-21
Figure A-6 Observed versus calculated response for a column specimen SC3 (shear critical).
Figure A-7 Observed versus calculated response for a column specimen 2CLH18 (fails in shear
after considerable flexural deformation).
Summary of Findings:
The comparison with experimental results showed that good correlation of
the observed versus the calculated post-peak response was in many cases not
possible (Figures A-6, A-7). The cyclic degradation parameters were shown
to have a large influence on the post-peak response while significant
epistemic uncertainty was identified in the cyclic degradation.
The column failure mode was captured in every test considered, but the
estimated failure loads and drifts were generally conservative. Nonetheless,
the model was able to capture satisfactorily the overall strength degradation,
A-22 A: Detailed Summary of Previous Research FEMA P440A
stiffness degradation, and in-cycle and cyclic degradation properties of the
specimens.
Relevant Publications:
Pincheira, J.A, Dotiwala, F.S., and D’Souza J.T., 1999, “Spectral
displacement demands of stiffness- and strength-degrading systems,”
Earthquake Spectra, 15(2), 245–272.
Song J.-K., and Pincheira, J.A., 2000, “Seismic analysis of older reinforced
concrete columns,” Earthquake Spectra, 16(4), 817–851.
Dotiwala, F.S., 1996, A nonlinear flexural-shear model for RC columns
subjected to earthquake loads, MS Thesis, Department of Civil and
Environmental Engineering, University of Wisconsin-Madison.
Pincheira, J.A., and Dotiwala, F.S., 1996, “Modeling of nonductile R/C
columns subjected to earthquake loading,” Proc. 11th World Conf. on
Earthquake Engineering, Paper No. 316, Acapulco, Mexico.
A.2.3 Spectral Displacement Demands of Stiffness- and
Strength-Degrading Systems
Authors:
Song, J.-K., and Pincheira, J.A. (2000)
Abstract:
The effect of stiffness and strength degradation on the maximum inelastic
displacement of single-degree-of-freedom (SDOF) systems was investigated.
The SDOF model included strength and stiffness degradation with increasing
deformation amplitude and upon reversal of loading cycles. Pinching of the
hysteresis loops was also considered. Spectral displacements were calculated
for oscillators with a range of degrading characteristics subjected to twelve
ground motions on rock, firm, and soft soils. The results show that the
maximum displacements of degrading oscillators are, on average, larger than
those of non-degrading systems. The displacement amplification depends
significantly with the period, strength coefficient, degradation rate, and
ground motion considered. Nonetheless, the amplification due to the
degradation characteristics of the system is more important in the shortperiod
range where average amplification factors of two or three are credible.
The amplification factors proposed in the FEMA 273 report by the ATC-33
project provided conservative estimates for oscillators with periods greater
than 0.3 seconds subjected to motions on rock or firm soil. On soft soils, a
FEMA P440A A: Detailed Summary of Previous Research A-23
good correlation was found for periods greater than 1.5 seconds. At shorter
periods, the ATC 33 factors underestimate the displacement amplification.
Summary:
About 7,600 SDOF dynamic analyses were performed for 12 ground motions
including rock, firm soil, soft soil, and near-field and far-field records. The
oscillator had a quadrilinear backbone with a hardening, a softening and a
residual strength segment. Still, only a limited set of backbones were
considered, all having hardening stiffness 5% of the elastic, residual strength
10% of yield strength and reaching 1.25 ductility at peak strength. Some
gentle negative slopes were investigated, namely -1% and -3% of the elastic.
The oscillator had 5% damping and used a pinching hysteresis with or
without cyclic strength degradation. During the investigation an ad hoc
collapse limit-state was considered when the post-peak strength reached 10%
of the yield strength.
Representative Figures:
Figure A-8 (a) Hysteresis law used for the SDOF system and (b) ratio of degrading to nondegrading
displacement amplification factors for the post-peak stiffness equal
to -1% or -3% of the elastic stiffness.
Summary of Findings:
The post-peak stiffness and the unloading stiffness were found to be the most
important parameters, while the degree of pinching was important except for
soft-soil records. Cyclic degradation generally increases the dynamic
response, but mostly for the short periods. Significant differences in the
SDOF response and its dependence on cyclic degradation were found
between soft soil and firm soil (or rock) and maybe between near-field versus
far-field record response, although only one near-source record was used.
A-24 A: Detailed Summary of Previous Research FEMA P440A
Finally, for a given strength reduction factor, collapse (as defined in the
paper) was consistently observed below a certain oscillator period. Some
duration effects were also reported.
The study seriously suffers from the lack of records. Only twelve were used
and they were selected from soft and firm soil sites without differentiating
between near-field and far-field. Thus, the statistics on the results are not
reliable, although the general observations provided may prove useful.
Relevant Publications:
Song J.-K., and Pincheira, J.A., 2000, “Seismic analysis of older reinforced
concrete columns,” Earthquake Spectra, 16(4), 817–851.
Pincheira, J.A, Dotiwala, F.S., and D’Souza, J.T., 1999, “Spectral
displacement demands of stiffness- and strength-degrading systems,”
Earthquake Spectra, 15(2), 245–272.
A.2.4 Dynamic Instability of Simple Structural Systems
Authors:
Miranda, E. and Akkar, S.D., (2003)
Abstract:
Lateral strengths required to avoid dynamic instability of SDOF systems are
examined. Oscillators with a bilinear hysteretic behavior with negative postyield
stiffness are considered. Mean lateral strengths, normalized by the
lateral strength required to maintain the system elastic, are computed for
systems with periods ranging from 0.2 to 3.0 s when subjected to 72
earthquake ground motions recorded on firm soil. The effect of period of
vibration and post-yield stiffness is investigated. Results indicate that mean
normalized lateral strengths required to avoid dynamic instability decrease as
negative post-yield stiffness increases, and that the reductions are much
larger for small negative post-yield stiffness than for severe negative postyield
stiffness. It is concluded that there is a significant influence of the
period of vibration for short-period systems and for systems with mildly
negative post-yield stiffness. Dispersion of normalized lateral strengths
required to avoid dynamic instability are found to increase as the negative
post-yield stiffness decreases and as the period of vibration increases. Simple
equations that capture the effects of period and post-yield stiffness to aid in
the evaluation of existing structures are obtained through nonlinear
regression analyses.
FEMA P440A A: Detailed Summary of Previous Research A-25
Summary:
The objective of this study was to assess the minimum lateral strength
required to avoid dynamic instability in SDOF systems. The minimum lateral
strength is computed as a function of linear elastic spectral ordinates, that is,
the lateral strength required to maintain the system elastic. Specific goals of
the study were: (a) to study the effect of the post-yield negative stiffness on
the minimum strength required to avoid collapse, (b) to study the effect of
period of vibration, (c) to compute mean normalized strengths required to
avoid dynamic instability, and (d) to develop approximate expressions to
assist practicing engineers in evaluating the minimum lateral strengths
required in existing structures to avoid dynamic instability.
The study considered SDOF systems with a bilinear forcedisplacement
relationship characterized by a linear segment with initial stiffness K
followed by a post-yield linear segment with negative stiffness -.K. When
subjected to earthquake ground motions the likelihood of experiencing
dynamic instability in a system with a given negative slope increases as the
lateral strength decreases. Lateral strengths required to avoid dynamic
instability of bilinear SDOF systems with negative post-yield stiffness were
investigated. Mean lateral strengths normalized by the lateral strength
required to maintain the system elastic are computed for systems with
periods ranging from 0.2 to 3.0 s and post-yield negative stiffness ratios
ranging from 0.03 to 2.0 when subjected to 72 earthquake ground motions
recorded on firm soil.
Representative Figures:
Figure A-9 Force-displacement characteristics of bilinear systems
considered
A-26 A: Detailed Summary of Previous Research FEMA P440A
Figure A-10 Effect of period of vibration and post-yield stiffness on the mean
strength ratio at which dynamic instability is produced.
Summary of Findings:
Lateral strengths required to avoid dynamic instability of bilinear SDOF
systems with negative post-yield stiffness were investigated. Mean lateral
strengths, normalized by the lateral strength required to maintain the system
elastic, are computed for systems with periods ranging from 0.2 to 3.0 s and
post-yield negative stiffness ratios ranging from 0.03 to 2.0 when subjected
to 72 earthquake ground motions recorded on firm soil. The following
conclusions are drawn from this study.
The strength ratio at which dynamic instability is produced decreases as the
post-yield negative stiffness ratio . increases. This means that the lateral
strength required to avoid collapse increases as the post-yield descending
branch of the force-deformation relationship is steeper. When . is smaller
than about 0.2 small increases in . can produce significant increases in
required lateral strength to avoid instability. Meanwhile, for values of . >1,
the system must remain practically elastic in order to avoid collapse.
The collapse strength ratio increases with increasing period, particularly for
post-yielding negative stiffness ratios smaller than 0.3. Mean collapse
strength ratios of short period structures are relatively strong, particularly
when ..>0.1. Dispersion of collapse strength ratios decreases as . increases
and as the period of vibration decreases. Coefficients of variation of collapse
strength ratios are particularly small for ..>0.5. An approximate equation to
estimate lateral strengths required to avoid dynamic instability of bilinear
SDOF system is proposed.
FEMA P440A A: Detailed Summary of Previous Research A-27
Relevant Publications:
Miranda, E. and Akkar, S.D., 2003 "Dynamic instability of simple structural
systems," Journal of Structural Engineering, ASCE, 129(12) , pp 1722-
1727.
Vamvatsikos, D., and Cornell, C.A., 2005, Seismic performance, capacity
and reliability of structures as seen through incremental dynamic
analysis, John A. Blume Earthquake Engineering Research Center
Report No. 151, Department of Civil and Environmental Engineering,
Stanford University, Stanford, California.
A.2.5 Tests to Structural Collapse of Single-Degree-of-Freedom
Frames Subjected to Earthquake Excitations
Authors:
Vian, D. and Bruneau, M. (2003)
Abstract:
This paper presents and analyzes experimental results of tests of 15 fourcolumn
frame specimens subjected to progressively increasing uniaxial
ground shaking until collapse. The specimens were subdivided into groups of
three different column slenderness ratios: 100, 150, and 200. Within each
group, the column dimensions and supported mass varied. Ground motion of
different severity was required to collapse the structures tested. The
experimental setup is briefly described and results are presented. Test
structure performance is compared with the proposed limits for minimizing
P–. effects in highway bridge piers. The stability factor is found to have a
strong relation to the relative structural performance in this regard.
Performance is also compared with the capacity predicted by currently used
design equations dealing with axial and moment interactions for strength and
stability by expressing these capacities in terms of acceleration and
maximum base shear (represented as a fraction of the system’s weight). The
experimental results exceeded the maximum spectral accelerations calculated
when considering second-order effects, but did not when considering only
member strength. Finally, an example of how to use the experimental data
for analytical model verification is presented, illustrating the shortcomings
and inaccuracies of using a particular simplified model with constant
structural damping.
A-28 A: Detailed Summary of Previous Research FEMA P440A
Summary:
Although the first and foremost objective of this project was to provide welldocumented
data (freely available on the web to be used by others) of tests to
collapse, this paper includes results from a preliminary investigation of
behavioral trends observed from the shake table results. In particular, peak
responses are compared with limits proposed by others to minimize P–.
effects in bridge piers. Specimen behavior is also investigated with respect to
axial and moment interaction limits considering strength and stability.
Finally, to illustrate how the generated experimental data could be used to
develop or calibrate analytical models of inelastic behavior to collapse,
experimental results are compared with those obtained using a simple
analytical model. Progressive bilinear dynamic analyses are performed in
two different ways and are compared with the shake table test results.
Representative Figures:
Figure A-11 (a) Schematic of test setup
Figure A-12 Simplified bilinear force deformation model
FEMA P440A A: Detailed Summary of Previous Research A-29
Figure A-13 Comparison of experimental (left) and analytical (right) results.
Summary of Findings:
Specimens showed an approximate bilinear behavior with a tendency to drift
to one side and ultimately experience collapse. The stability factor, ., was
observed to have the most significant effect on the structure’s propensity to
collapse. As . increases, the maximum attainable ductility, maximum
sustainable drift, and maximum spectral acceleration reached before collapse,
all decrease. When this factor . was larger than 0.1, the ultimate values of
maximum spectral acceleration, displacement ductility, and drift reached
before collapse were all grouped below values of 0.75 g, 5, and 20%,
respectively.
Relevant Publications:
Vian, D. and Bruneau, M., 2003, "Tests to structural collapse of single
degree of freedom frames subjected to earthquake excitations." Journal
of Structural Engineering, ASCE, 129(12), 1676-1685.
Bruneau, M. and Vian, D., 2002, “Tests to collapse of simple structures and
comparison with existing codified procedures,” Proc. 7th U.S. National
Conference on Earthquake Engineering, Boston, MA.
Bruneau, M. and Vian, D., 2002, “Experimental investigation of P-. effects
to collapse during earthquakes,” Proc. 12th European Conference on
Earthquake Engineering, London, UK.
A-30 A: Detailed Summary of Previous Research FEMA P440A
Vian, D. and Bruneau, M., 2001, Experimental investigation of P-. effects
to collapse during earthquakes, Report MCEER-01-0001,
Multidisciplinary Research for Earthquake Engineering Research Center,
Buffalo, N.Y.
A.2.6 Methods to Evaluate the Dynamic Stability of Structures –
Shake Table Tests and Nonlinear Dynamic Analyses
Author:
Kanvinde, A.M. (2003)
Abstract:
This paper aims to understand the phenomenon of dynamic instability in
structures better, and to suggest and evaluate methods to predict collapse
limit states of structures during earthquakes, based on findings of recent
shake table tests and nonlinear dynamic analyses conducted at Stanford
University. Simple models that collapsed due to a story mechanism were
used as test specimens. Data from nineteen experiments suggest that current
methods of nonlinear dynamic analysis (using the OpenSees program in this
case) are accurate and reliable for predicting collapse and tracing the path of
the structure down to the ground during collapse. Moreover, it is found from
the experiments that for non-degrading structures, an estimate of collapse
drift based on a static pushover analysis can be successfully applied to
predict the dynamic collapse or instability due to P-. effects. The rationale
for this is that the structure has an elongated period at the point of global
instability, virtually insulating it from the ground motion and justifying the
use of a static-analysis-based drift. Finally, the paper directs the readers to a
valuable database of test data from collapse tests of a “clean” structure,
which can be used for further verification studies.
Summary:
This was a brief assessment of the collapse performance of two 2D singlestory
single-bay frames using results from static pushover and incremental
dynamic analyses (IDA) and correlating them with experimental tests. The
frames tested and simulated had a rigid beam that forced the creation of
plastic hinges in the two columns. By using easily replaceable steel plates for
the columns it was possible to repeat the shake table tests at various
intensities and in effect experimentally reproduce IDA-like results. A total of
19 uniaxial shake table tests was performed using two ground motion records
and two different structures (that is, two different column types).
FEMA P440A A: Detailed Summary of Previous Research A-31
Structure A was ductile and, having a ratio of yield base shear to weight of
1.03, was practically impossible to collapse. Structure B had weaker
columns, produced by drilling holes at the bottom and top of the steel plates,
and was thus prone to a story-mechanism collapse due to significant P-.
effects (Figure A-14a). The two structures were also simulated in OpenSEES
using a Giufre-Menegotto-Pinto hysteresis model for the column hinges
(Figure A-14b) without any cyclic deterioration, and an exact corrotational
formulation for geometric nonlinearities.
Representative Figures:
Figure A-14 (a) Static pushover curves for the two frames and (b) modeling
of the column plastic hinges in OpenSEES.
Summary of Findings:
The evidence presented shows that nonlinear dynamic analysis is a reliable
tool to predict the actual behavior of the two structures. The usefulness of
static pushover was also proven, at least when cyclic deterioration is not an
issue, as the collapse drift calculated statically was accurately matched by
both incremental dynamic analysis and the shake table experiments.
Compared to the response at lower intensities, larger scatter was observed
A-32 A: Detailed Summary of Previous Research FEMA P440A
close to collapse both in the experiments and in the dynamic analyses, even
when using the same earthquake record. This suggests an increased
sensitivity of the actual results to the uncertainties in the initial condition of
the structure, and an increased difficulty in predicting the collapse drift or
intensity level even for such simple specimens.
Relevant Publications:
Kanvinde, A.M., 2003, “Methods to evaluate the dynamic stability of
structures – shake table tests and nonlinear dynamic analyses,” EERI
Annual Student Paper Competition, Proceedings of 2003 EERI Meeting,
Portland, OR.
Vian, D. and Bruneau, M., 2001, Experimental investigation of P-. effects to
collapse during earthquakes, Report No. MCEER-01-0001,
Multidisciplinary Research for Earthquake Engineering Research Center,
Buffalo, N.Y.
A.2.7 Seismic Performance, Capacity and Reliability of
Structures as Seen Through Incremental Dynamic Analysis
Authors:
Vamvatsikos, D. and Cornell, C.A. (2005)
Abstract:
Incremental Dynamic Analysis (IDA) is an emerging structural analysis
method that offers thorough seismic demand and limit-state capacity
prediction capability by using a series of nonlinear dynamic analyses under a
suite of multiply scaled ground motion records. Realization of its
opportunities is enhanced by several innovations, such as choosing suitable
ground motion intensity measures and representative structural demand
measures. In addition, proper interpolation and summarization techniques for
multiple records need to be employed, providing the means for estimating the
probability distribution of the structural demand given the seismic intensity.
Limit-states, such as the dynamic global system instability, can be naturally
defined in the context of IDA. The associated capacities are calculated so that
when properly combined with probabilistic seismic hazard analysis, they
allow the estimation of the mean annual frequencies of limit-state
exceedance.
IDA is resource-intensive. Thus the use of simpler approaches becomes
attractive. The IDA can be related to the computationally faster Static
FEMA P440A A: Detailed Summary of Previous Research A-33
Pushover (SPO), enabling a fast and accurate approximation to be established
for SDOF systems. By investigating oscillators with quadrilinear backbones
and summarizing the results into a few empirical equations, a new software
tool, SPO2IDA, is produced here that allows direct estimation of the
summarized IDA results. Interesting observations are made regarding the
influence of the period and the backbone shape on the seismic performance
of oscillators. Taking advantage of SPO2IDA, existing methodologies for
predicting the seismic performance of first-mode-dominated, MDOF systems
can be upgraded to provide accurate estimation well beyond the peak of the
SPO.
The IDA results may display a large record-to-record variability. By
incorporating elastic spectrum information, efficient intensity measures can
be created that reduce such dispersions, resulting in significant computational
savings. By employing either a single optimal spectral value, a vector of two
or a scalar combination of several spectral values, significant efficiency is
achieved. As the structure becomes damaged, the evolution of such optimally
selected spectral values is observed, providing intuition about the role of
spectral shape in the seismic performance of structures.
Summary:
The research presented is entirely based on the concept of incremental
dynamic analysis (IDA). The methodology is established and is extensively
used to derive (among others) the collapse capacity of MDOF frames. Of
particular importance is the exploration of the connection between the
fractile IDA curves and the pushover. The authors propose the use of a 5%
damped SDOF oscillator with a complex quadrilinear backbone (including a
hardening, a softening and a residual plateau segment) with moderately
pinching hysteresis to capture the pushover curve shape of actual MDOF
frames. No cyclic degradation was considered but in the process millions of
nonlinear dynamic SDOF analyses are performed for 30 records and a wide
variety of oscillator backbones and periods. The results are fitted and
incorporated into a complex R-µ-T relationship, realized in the form of the
SPO2IDA Excel tool.
The proposed tool is applied to the MDOF prediction problem using the
worst-case pushover concept. This is defined as the pushover that leads to the
earliest post-peak collapse, stipulating that it will also help find the collapse
mechanism that a dynamic analysis would predict. By applying the
SPO2IDA tool on the worst-case pushover the complete IDA curves of
A-34 A: Detailed Summary of Previous Research FEMA P440A
MDOF frames are generated for a 5-story, a 9-story and a 20-story steel
frame.
Representative Figures:
0 1
0
1
ductility, . = . / . yield
strength reduction factor, R = F / F yield
.c .f
ac ah
r
non-negative
(hardening) negative
residual plateau
elastic fracture
Figure A-15 The backbone of the studied oscillator.
Figure A-16 The interface of the SPO2IDA tool for moderate periods.
FEMA P440A A: Detailed Summary of Previous Research A-35
0 1 2 3 4 5 6 7 8
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
ac= -25%
ductility, . = . / . yield
strength reduction factor, R = S
a
/ S
a
yield
20%
ac a = -50% c= -200%
ac= -25% (50% IDA)
ac= -50% (50% IDA)
ac= -200% (50% IDA)
0 1 2 3 4 5 6 7 8
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
ah= 10%
ductility, . = . / . yield
strength reduction factor, R = S
a
/ S
a
yield
-200%
ah= 25%
ah= 50%
ah= 75%
ah= 10% (50% IDA)
ah= 25%, 50%, 75% (50% IDAs)
Figure A-17 Influence of (a) the post-peak and (b) post-yield stiffness on the median dynamic response of the
oscillator. When the negative segment is the same then the hardening slope has a negligible effect.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
0.05
0.1
0.15
0.2
0.25
0.3
roof drift ratio, . roof
base shear / mass (g)
first-mode
two modes SRSS
two modes SRSS, then uniform
two modes SRSS, then inverse
0 0.05 0.1 0.15
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
peak roof drift ratio, . roof
"first-mode" spectral acceleration S
a
(T
1
,5%) (g)
IDA
SPO2IDA
50% IDA
84% IDA
16% IDA
Figure A-18 (a) Influence of the load pattern on the pushover curve shape and (b) the predicted versus actual
dynamic response for various intensity levels using SPO2IDA and the worst-case pushover for a
9-story steel moment frame.
Summary of Findings:
Regarding SDOF oscillators, it was found that the shape of the backbone
curve has a very complex effect on the dynamic response. For example, the
negative slope, the hardening deformation and the residual strength level are
the three parameters that dominate (e.g. Figure A-17a). On the other hand,
the hardening slope is not as important, while the residual plateau is
significant only when long or high enough. Surprisingly, the peak strength of
the oscillator is found to be relatively unimportant when the subsequent
negative-slope segment is fixed. Thus some very different backbones exist
that have almost the same performance (Figure A-17b).
A-36 A: Detailed Summary of Previous Research FEMA P440A
For MDOF application, it was found that the worst-case pushover is not
always easy to estimate. For a 9-story steel frame building, in order to find
the dominant collapse mechanism it was necessary to change the load pattern
after the peak of the pushover and actually try several combinations before
getting an acceptable shape (Figure A-18a). Under this condition, the use of
SPO2IDA was found to provide accurate results for first-mode-dominated
frames. A test conducted on a 20-story frame, where higher modes are a
significant issue, showed that it was impossible to get good agreement in the
early inelastic range. Curiously, when close to collapse, this approach still
managed to provide an accurate answer, leading to the observation that an
SDOF can predict reliably the collapse capacity of complex buildings.
Relevant Publications:
Vamvatsikos, D., and Cornell, C.A., 2005, Seismic performance, capacity
and reliability of structures as seen through incremental dynamic
analysis, John A. Blume Earthquake Engineering Research Center,
Report No. 151, Department of Civil and Environmental Engineering,
Stanford University, Stanford, California.
Ibarra, L.F., and Krawinkler, H., 2005, Global collapse of frame structures
under seismic excitations, John A. Blume Earthquake Engineering
Research Center, Report No. 152, Department of Civil and
Environmental Engineering, Stanford University, Stanford, California.
A.2.8 Hysteretic Models that Incorporate Strength and Stiffness
Deterioration
Authors:
Ibarra, L., Medina, R.A., and Krawinkler, H., (2005)
Abstract:
This paper presents the description, calibration and application of relatively
simple hysteretic models that include strength and stiffness deterioration
properties, features that are critical for demand predictions as a structural
system approaches collapse. Three of the basic hysteretic models used in
seismic demand evaluation are modified to include deterioration properties:
bilinear, peak-oriented, and pinching models. The modified models include
most of the sources of deterioration, namely, various modes of cyclic
deterioration and softening of the post-yielding stiffness, and they also
account for a residual strength after deterioration. The models incorporate an
energy-based deterioration parameter that controls four cyclic deterioration
FEMA P440A A: Detailed Summary of Previous Research A-37
modes: basic strength, post-capping strength, unloading stiffness, and
accelerated reloading stiffness deterioration modes. Calibration of the
hysteretic models on steel, plywood, and reinforced-concrete components
demonstrates that the proposed models are capable of simulating the main
characteristics that influence deterioration. An application of a peak-oriented
deterioration model in the seismic evaluation of SDOF systems is illustrated.
The advantages of using deteriorating hysteretic models for obtaining the
response of highly inelastic systems are discussed.
Summary:
This study presents an improved piece-wise linear hysteretic model that is
capable of considering stiffness degradation, pinching cyclic strength
degradation as well as in-cycle strength degradation. The paper has a
threefold objective: (a) to describe the properties of proposed hysteretic
models that incorporate both monotonic and cyclic deterioration; (b) to
illustrate the calibration of these hysteretic models on component tests of
steel, plywood, and reinforced-concrete specimens; and (c) to exemplify the
utilization of the hysteretic models in the seismic response evaluation of
SDOF systems. In this study the term deteriorating hysteretic models refers
to models that include strength deterioration of the backbone curve or cyclic
deterioration or both.
As shown in Figure A-19, the model considers a backbone curve consisting
of four linear segments: an elastic segment until a yield displacement, postyield
strain-hardening segment until the ‘capping’ displacement is reached, a
post-capping segment with negative stiffness (that is, in-cycle degradation),
and a final residual horizontal segment.
Representative Figures:
Figure A-19 The backbone of the proposed hysteretic model.
A-38 A: Detailed Summary of Previous Research FEMA P440A
Figure A-20 Basic rules for peak-oriented hysteretic model.
Figure A-21 Pinching hysteretic model: (a) basic model rules; and (b) modification if reloading deformation is
to the right of break point.
(a)
Figure A-22 Examples of comparisons between experimental and analytical results for (a) non-ductile
reinforced concrete column; and (b) plywood shear wall.
Summary of Findings:
The hysteretic models include a post-capping softening branch, residual
strength, and cyclic deterioration. Cyclic deterioration permits deterioration
to be traced as a function of past loading history, the rate of deterioration
(which depends on the hysteretic energy dissipated in past cycles), and on a
FEMA P440A A: Detailed Summary of Previous Research A-39
reference energy dissipation capacity. Four modes of cyclic deterioration can
be simulated: basic strength, postcapping strength, unloading stiffness, and
accelerated reloading stiffness deterioration. Based on calibrations performed
with experimental data from component tests of steel, wood, and reinforcedconcrete
specimens, they concluded that it appears that, for a given
component, the backbone characteristics and a single parameter that controls
all four modes of cyclic deterioration are adequate to represent component
behavior regardless of the loading history.
Results from the seismic evaluation of various SDOF systems demonstrate
that strength deterioration becomes a dominant factor when the response of a
structure approaches the limit state of collapse. At early stages of inelastic
behavior, both deteriorating and nondeteriorating systems exhibit similar
responses. The differences become important when the post-capping stiffness
is attained in the response.
Relevant Publications:
Ibarra, L., Medina, R., Krawinkler, H., 2005, “Hysteretic models that
incorporate strength and stiffness deterioration, Earthquake Engineering
and Structural Dynamics, Vol. 34, no. 12, pp. 1489-1511.
Ibarra, L.F., and Krawinkler, H., 2005, Global collapse of frame structures
under seismic excitations, John A. Blume Earthquake Engineering
Research Center, Report No. 152, Department of Civil and
Environmental Engineering, Stanford University, Stanford, California.
Ibarra, L., Medina, R., Krawinkler, H., 2002, “Collapse assessment of
deteriorating SDOF systems,” Proc. 12th European Conference on
Earthquake Engineering, London, UK, Paper 665, Elsevier Science Ltd.
A.2.9 Global Collapse of Frame Structures Under Seismic
Excitations
Authors:
Ibarra, L.F. and Krawinkler, H. (2005)
Abstract:
Global collapse in earthquake engineering refers to the inability of a
structural system to sustain gravity loads in the presence of seismic effects.
This research proposes a methodology for evaluating global incremental
(sidesway) collapse based on a relative intensity measure instead of an
Engineering Demand Parameter (EDP). The relative intensity is the ratio of
A-40 A: Detailed Summary of Previous Research FEMA P440A
ground motion intensity to a structure strength parameter, which is increased
until the response of the system becomes unstable, which means that the
relative intensity - EDP curve becomes flat (that is, with zero slope). The
largest relative intensity is referred to as “collapse capacity.”
In order to implement the methodology, deteriorating hysteretic models are
developed to represent the monotonic and cyclic behavior of structural
components. Parameter studies that utilize these deteriorating models are
performed to obtain collapse capacities and quantify the effects of system
parameters that most influence the collapse for SDOF and MDOF structural
systems. The range of collapse capacity due to record-to-record variability
and uncertainty in the system parameters is evaluated. The latter source of
dispersion is quantified by means of the first order second moment (FOSM)
method. The studies reveal that softening of the post-yield stiffness in the
backbone curve (postcapping stiffness) and the displacement at which this
softening commences (defined by the ductility capacity) are the two system
parameters that most influence the collapse capacity of a system. Cyclic
deterioration appears to be an important but not the dominant issue for
collapse evaluation. P-. effects greatly accelerate collapse of deteriorating
systems and may be the primary source of collapse for flexible, but very
ductile, structural systems.
The dissertation presents applications of the proposed collapse methodology
to the development of collapse fragility curves and the evaluation of the
mean annual frequency of collapse.
An important contribution is the development of a transparent methodology
for the evaluation of incremental collapse, in which the assessment of
collapse is closely related with the physical phenomena that lead to this limit
state. The methodology addresses the fact that collapse is caused by
deterioration in complex assemblies of structural components that should be
modeled explicitly.
Summary:
The authors used an oscillator with a quadrilinear backbone curve with
hardening, softening and residual segment to conduct an extensive
parametric study. Pinching, peak-oriented and (bilinear-like) kinematic
hysteresis rules were considered, while the cyclic degradation of the
backbone stiffness and strength and of the unloading/reloading stiffness were
also included. The effects of P-. were added separately, as a rotation of the
backbone around the center of the axes. The investigation used 40 “ordinary”
FEMA P440A A: Detailed Summary of Previous Research A-41
ground motion records and it was focused on determining the influence of all
the parameters on the collapse capacity, which was considered to occur in an
IDA (incremental dynamic analysis) fashion, when numerical instability
occurred or when the IDA curve becomes horizontal.
Additionally a number of 2D single-bay frames with 3, 6, 9, 12, 15 and 18
stories was considered; they were designed according to a strong column,
weak beam, concept, with the beam hinges having a hysteretic model of the
same type as the one used for the SDOF studies. By maintaining a uniform
hysteretic model for all beam hinges and globally varying its parameters,
another parametric study was performed, focused now on the effect of the
hysteretic parameters on the MDOF response.
Representative Figures:
Figure A-23 (a) Backbone curve used for the investigations and (b) post-peak
stiffness cyclic deterioration considered.
Figure A-24 (a) Effect of the post-peak stiffness to the median collapse capacity spectra for a peak-oriented
model and (b) the ratio of collapse capacities for different hysteretic models.
A-42 A: Detailed Summary of Previous Research FEMA P440A
Figure A-25 Effect of (a) post-yield slope and (b) reloading stiffness cyclic deterioration on the collapse capacity.
Figure A-26 (a) Effect of the beam-hinge hysteretic model on the median MDOF collapse capacity and (b) the
generation of an equivalent SDOF system by using an auxiliary backbone curve to incorporate
P-..
Summary of Findings:
For the SDOF oscillator it was found that there is a complex interplay
between the parameters that defines the dynamic response. Their relative
values are significant. The cyclic deterioration was found to be unimportant
in the pre-peak region and only mildly important post-peak. Furthermore, its
influence does not depend on the type of ground motion, i.e. whether it is
near or far field or long in duration. The peak-strength ductility and the postpeak
slope are the most dominant parameters. Regarding the hysteresis laws,
kinematic hysteresis produces lower capacities than the pinching or the peakoriented,
which are roughly similar. The residual strength becomes important
only when it is large enough. Finally, the application of damping in a massproportional
or stiffness-proportional formulation becomes an important
issue after yielding, as the tangent stiffness is constantly changing.
FEMA P440A A: Detailed Summary of Previous Research A-43
The study of the MDOF frames concluded that these higher-mode-influenced
frames fail mostly due to a lower-story mechanism. There is a large
dependence of the collapse capacity on the first-mode period. In general the
effects of the hinge hysteresis and backbone parameters were similar to the
SDOF results. It was also observed that the inelastic instability coefficient
(i.e. the difference between the post-peak slope with and without P-.) was
often much larger than the elastic stability coefficient. Thus, surrogate
SDOFs need a separate inclusion of the P-. effects in the pre- and post-peak
regions. Such equivalent SDOFs were shown to have good accuracy in
predicting the collapse capacity.
Relevant Publications:
Ibarra, L.F., and Krawinkler, H., 2005, Global collapse of frame structures
under seismic excitations, John A. Blume Earthquake Engineering
Research Center, Report No. 152, Department of Civil and
Environmental Engineering, Stanford University, Stanford, California.
Rahnama, M. and Krawinkler, H., 1993, Effect of soft soils and hysteresis
models on seismic design spectra, John A. Blume Earthquake
Engineering Research Center, Report No. 108, Department of Civil
Engineering, Stanford University, Stanford, California.
Medina, R., 200), Seismic demands for nondeteriorating frame structures
and their dependence on ground motions, Ph.D.. dissertation submitted
to the Department of Civil and Environmental Engineering, Stanford
University, Stanford, California.
Ibarra, L., Medina, R., and Krawinkler, H., 2002, “Collapse assessment of
deteriorating SDOF systems,” Proc. 12th European Conference on
Earthquake Engineering, London, UK, Paper 665, Elsevier Science Ltd.
A.2.10 Object-Oriented Development of Strength and Stiffness
Degrading Models for Reinforced Concrete Structures
Author:
Kaul, R. (2004)
Abstract:
The aim of this research is to develop structural simulation models that can
capture the strength and stiffness degradation of reinforced concrete frames
up to collapse under earthquake-induced motions. The key modeling aspects
of the element formulations include: (1) rigorous modeling of large
deformation response, (2) flexural yielding and inelastic interaction between
A-44 A: Detailed Summary of Previous Research FEMA P440A
axial force and moment (3) degradation of the element stiffness under cyclic
loading and (4) axial force-moment-shear interaction for shear-critical
reinforced concrete columns. Beam-column models are developed and
implemented in an object-oriented analysis framework called OpenSees
(Open System for Earthquake Engineering Simulation). The large
deformation element formulations employ an updated Lagrangian approach.
Inelastic models are based on stress-resultant plasticity to simulate inelastic
hardening and softening response under combined axial loads and bending. A
two-surface evolution model is proposed for combined nonuniform
expansion or contraction and kinematic motion of the yield surface. The
yield surface can be used to simulate inelastic section response at integration
points along a beam-column element (distributed plasticity) or inelastic
hinging at the ends of a beam-column element (concentrated plasticity). In
the concentrated plasticity approach, the element between the hinges is quasielastic,
in which hysteretic models are developed to model the cyclic
degradation. This concentrated plasticity model is extended to simulate
shear-critical column behavior, including shear strength degradation and
failure, interaction between axial and shear forces, and pinched cyclic
response. Implementation of the models in OpenSees is planned and
structured using object-oriented programming concepts. Individual
components of the inelastic modeling problem are identified and the
interactions between the governing classes are established. The models are
implemented in a hierarchal structure, which provides a modular and
extensible software design. The accuracy and the capabilities of the proposed
models are verified by comparing the analytical results with the experimental
data. The models developed as part of this research provide ideal tools for
conducting extensive application studies. An extensible framework is
provided to facilitate tool development for nonlinear or inelastic analysis.
Summary:
The objective was the creation, and incorporation into OpenSEES, of a
beam-column element with concentrated plasticity, that is appropriate for
multiaxial loading of older, shear-critical RC columns. The element has been
based on a yield-surface formulation and the focus was on modeling the
multiaxial response of a complete RC section. The model incorporates incycle
strength degradation, allowing for a quadrilinear backbone with a
negative stiffness segment (Figure A-27). Inelastic hardening and softening
were formulated according to a combined kinematic and isotropic hardening
rule with either peak-oriented (for moment) or pinching (for shear) hysteretic
rules (Figure A-28). There is little provision for cyclic degradation; the
formulation is entirely based on the peak plastic strains and rotations, and the
FEMA P440A A: Detailed Summary of Previous Research A-45
direction of evolution, so depending on the details there may be no cyclic
degradation.
The significant advantage of the models is their apparent extensibility and the
possibilities for easy modification and incorporation into a variety of
elements, an inherent feature of the object-oriented programming upon which
OpenSEES has been built.
The model behavior has been calibrated and tested against a variety of RC
beam-column experiments, including both shear and moment-critical
columns, as well as a set of theoretical solutions for large deformation
response (Figure A-29).
Representative Figures:
Figure A-27 Idealization of the (a) flexure spring and (b) shear spring backbones.
Figure A-28 (a) Full and (b) half cycle pinching hysteresis for the shear spring.
A-46 A: Detailed Summary of Previous Research FEMA P440A
Figure A-29 Comparison of calculated versus experimental results for (a) a moment-critical column and (b) a
shear-critical column.
Summary of Findings:
A reliable and extensible concentrated-plasticity beam-column element was
created for RC members. Extensive testing and calibration has shown good
agreement for a variety of experimental results, including shear-critical
columns, large deformations, and planar moment and axial-force interaction.
The only serious limitation is the limited formulation of cyclic degradation,
an issue that can be potentially solved with the incorporation of damage
models. The extension to 3D beam-column elements is somewhat hampered.
Appropriate yield surface and evolution rules have not been incorporated,
although the hysteretic material models presented are directly usable. The
absence of the bond-slip effect and longitudinal reinforcement development
in the element springs, are issues that still remain to be addressed.
Relevant Publications:
Kaul, R., 2004, Object oriented development of strength and stiffness
degrading models for reinforced concrete structures, Ph.D. Thesis,
Department of Civil and Environmental Engineering, Stanford
University, Stanford, California.
McKenna, F.T. (1997), Object-oriented finite element programming:
framework for analysis, algorithms, parallel computing, Ph.D.
Dissertation, University of California, Berkeley, California.
Mehanny, S.S.F., and Deierlein, G.G., 2001, “Seismic collapse assessment of
composite RCS moment frames,” Journal of Structural Engineering,
ASCE, Vol. 127(9).
FEMA P440A A: Detailed Summary of Previous Research A-47
El-Tawil, S., 1996, Inelastic dynamic analysis of mixed steel-concrete space
frames, Ph.D. Dissertation, Cornell University, Ithaca, NY.
Elwood, K.J., 2002, Shake table tests and analytical studies on the gravity
load collapse of reinforced concrete frames, Ph.D. Dissertation,
University of California, Berkeley, California.
A.2.11 Shake Table Tests and Analytical Studies on the Gravity
Load Collapse of Reinforced Concrete Frames
Author:
Elwood, K.J. (2002).
Abstract:
An empirical model, based on the evaluation of results from an experimental
database, is developed to estimate the drift at shear failure of existing
reinforced concrete building columns. A shear-friction model is also
developed to represent the general observation from experimental tests that
the drift at axial failure of a shear-damaged column is directly proportional to
the amount of transverse reinforcement and is inversely proportional to the
magnitude of the axial load. The two drift-capacity models are incorporated
in a nonlinear uniaxial constitutive model implemented in a structural
analysis platform to allow for the evaluation of the influence of shear and
axial load column failures on the response of a building. Shake table tests
were designed to observe the process of dynamic shear and axial load
failures in reinforced concrete columns when an alternative load path is
provided for load redistribution. The results from these tests provide data on
the dynamic shear strength and the hysteretic behavior of columns failing in
shear, the loss of axial load capacity after shear failure, the redistribution of
loads in a frame after shear and axial failures of a single column, and the
influence of axial load on each of the above-mentioned variables. An
analytical model of the shake table specimens, incorporating the proposed
drift-capacity models to capture the observed shear and axial load failures,
provides a good estimate of the measured response of the specimens.
Summary:
The objective was the creation and incorporation into OpenSEES of a beamcolumn
element with concentrated plasticity that is appropriate for multiaxial
loading of older, shear-critical RC columns. The element has been based on a
yield-surface formulation and the focus was on modeling the multiaxial
response of a complete RC section. The model incorporates in-cycle strength
A-48 A: Detailed Summary of Previous Research FEMA P440A
degradation, allowing for a quadrilinear backbone with a negative stiffness
segment (Figure A-30). Inelastic hardening and softening were formulated
according to a combined kinematic and isotropic hardening rule with either
peak-oriented (for moment) or pinching (for shear) hysteretic rules (Figure
A-31). There is little provision for cyclic degradation; the formulation is
entirely based on the peak plastic strains and rotations, and the direction of
evolution, so depending on the details there may be no cyclic degradation.
The significant advantage of the models is their apparent extensibility and the
possibilities for easy modification and incorporation into a variety of
elements, an inherent feature of the object-oriented programming upon which
OpenSEES has been built.
Representative Figures:
Figure A-30 Use of Sezen model to estimate (a) shear capacity and (b) displacement ductility capacity.
Figure A-31 Comparison of the Sezen shear strength model and the proposed drift capacity model.
FEMA P440A A: Detailed Summary of Previous Research A-49
Figure A-32 Redefinition of backbone in Elwood’s model after shear failure
is detected.
Figure A-33 Comparison of calculated versus experimental results for two shear-critical columns.
Summary of Findings:
Given the lack of agreement between existing models for the drift at shear
failure and results from an experimental database of shear-critical building
columns, two empirical models were developed to provide a more reliable
estimate of the drift at shear failure for existing reinforced concrete columns:
Based on shear-friction concepts and the results from 12 columns tested to
axial failure, a model was also developed to estimate the drift at axial failure
for a shear-damaged column:
A-50 A: Detailed Summary of Previous Research FEMA P440A
The capacity models for the drift at shear and axial load failure were used to
initiate the strength degradation of a uniaxial material model implemented in
the OpenSees analytical platform (OpenSees, 2002). When attached in series
with a beam-column element, the material model can be used to model either
shear or axial failure, or both if two materials are used in series. Based on
experimental evidence suggesting that an increase in lateral shear
deformations may lead to an increase in axial deformations and a loss of
axial load, shear-to-axial coupling was incorporated in the material model to
approximate the response of a column after the onset of axial failure.
Relevant Publications:
Elwood, K.J., 2002, Shake table tests and analytical studies on the gravity
load collapse of reinforced concrete frames, Ph.D. Dissertation,
University of California, Berkeley.
Elwood, K.J., and Moehle, J.P., 2003, Shake table tests and analytical
studies on the gravity load collapse of reinforced concrete frames.
Pacific Earthquake Engineering Research Center, PEER Report 2003/01,
University of California, Berkeley, Calif.
Elwood, K.J., 2004, “Modeling failures in existing reinforced concrete
columns,” Can. J. Civ. Eng., 31: 846–859 (2004)
Elwood, K.J., and Moehle, J.P., 2005, “Drift capacity of reinforced concrete
columns with light transverse reinforcement.” Earthquake Spectra,
Volume 21, No. 1, pp. 71–89,
A.2.12 Determination of Ductility Factor Considering Different
Hysteretic Models
Authors:
Lee, L.H., Han, S.W., and Oh, Y.H. (2003)
Abstract:
In current seismic design procedures, base shear is calculated by the elastic
strength demand divided by the strength reduction factor. This factor is well
known as the response modification factor, R, which accounts for ductility,
overstrength, redundancy, and damping of a structural system. In this study,
the R factor accounting for ductility is called the ductility factor, Rµ. The Rµ
FEMA P440A A: Detailed Summary of Previous Research A-51
factor is defined as the ratio of elastic strength demand imposed on the SDOF
system to inelastic strength demand for a given ductility ratio. The Rµ factor
allows a system to behave inelastically within the target ductility ratio during
the design level earthquake ground motion. The objective of this study is to
determine the ductility factor considering different hysteretic models. It
usually requires large computational efforts to determine the Rµ factor. In
order to reduce the computational efforts, the Rµ factor is prepared as a
functional form in this study. For this purpose, statistical studies are carried
out using forty different earthquake ground motions recorded at a stiff soil
site. The Rµ factor is assumed to be a function of the characteristic parameters
of each hysteretic model, target ductility ratio and structural period. The
effects of each hysteretic model on the Rµ factor are also discussed.
Summary:
The focus of the research was the creation of an R-µ-T relationship that
would include the effect of several backbone and hysteretic characteristics.
The authors’ approach was to consider such effects as completely
independent from each other by considering the following models: (a)
kinematic hysteresis with a bilinear backbone having positive post-yield
stiffness; (b) bilinear backbone and kinematic hysteresis with cyclic
degradation of the reloading/unloading stiffness; (c) bilinear backbone with
peak-oriented (Clough-like) hysteresis with cyclic strength degradation; and
(d) bilinear backbone with pinching hysteresis (Figure A-34). Details of the
models are provided by Kunnath et al., (1990).
Using the elastic-perfectly-plastic model with kinematic hardening as a basis,
the researchers investigated the influence of each of the four parameters and
provided correction factors, in a “coefficient-like” method. These can be
applied to an R-µ-T relationship based on the elastic-plastic system to
account for the effect of each of the parameters.
A-52 A: Detailed Summary of Previous Research FEMA P440A
Representative Figures:
Figure A-34 The parameters investigated: (a) backbone hardening ratio;
(b) unloading/reloading cyclic stiffness degradation; (c) strength degradation;
and (d) degree of pinching.
Figure A-35 The effect of (a) cyclic strength degradation and (b) degree of pinching on the
mean R-factor for a given ductility.
Summary of Findings:
The results presented show the effect of the cyclic degradation of strength or
of reloading and unloading stiffness, and the degree of pinching on the mean
R-factor observed for a given ductility for SDOF systems (Figure A-35). In
general the effects are relatively small. Unfortunately the influence of such
parameters is evaluated separately for each parameter, only for a relatively
limited range of values, and always in relation to the pure elasto-plastic
system. Still, the authors do perform verifications for systems having a
combination of all such characteristics, thus providing evidence that the
FEMA P440A A: Detailed Summary of Previous Research A-53
proposed formulas can approximate more complex systems. Perhaps the
greatest limitation of this research is that it does not apply to systems with incycle
strength degradation. Only positive post-yield stiffnesses are
considered. Therefore, the influence of several investigated parameters is
small, and the results cannot be applied when negative backbone slopes are
present.
Relevant Publications:
Lee, L.H, Han, S.W. and Oh, Y.H., 1999, “Determination of ductility factor
considering different hysteretic models,” Earthquake Engineering and
Structural Dynamics, Vol. 28, 957–977.
Kunnath, S.K., Reinhorn, A.M. and Park, Y.J., 1990, “Analytical modeling
of inelastic seismic response of RC structures,” Journal of Structural
Engineering, ASCE, 116, 996–1017.
A.2.13 Effects of Hysteresis Type on the Seismic Response of
Buildings
Authors:
Foutch, D.A. and Shi, S. (1998)
Abstract:
Current design procedures account for inelastic behavior in a crude manner
using the R factor. Although different R values are used for different building
types, the determination of a specific R value was not done in a very
consistent or scientific manner. The hysteresis behavior of members can be
different depending on the material and member type. Buildings with
members that dissipate energy through full hysteresis loops (for example,
steel moment frames with compact members and no joint fracture) will
respond differently from buildings with members that demonstrate strengthdegrading
hysteresis behavior by having either non-compact steel members,
concentric braces, or members with fractured joints. This paper will present
results of a study that has closely examined these effects using both SDOF
and MDOF systems. A procedure for developing reliability-based design
methods which incorporates these effects will also be presented.
Summary:
A total of nine moment-resisting frames with three different configurations
(3-story, 6-story and 9-story) were used in this study to examine the effect of
the beam-hinge model on the seismic behavior of MDOF structures. Using a
A-54 A: Detailed Summary of Previous Research FEMA P440A
suite of 12 ground motion records, all structures (numbering 3x9x8) were
analyzed for several R-factor levels (or approximately an equivalent number
of earthquake intensity levels) and the results were summarized and
compared with the buildings having a basic bilinear hinge with kinematic
hardening (Figure A-36).
Eight different hinge models were considered: (1) kinematic hardening with
bilinear backbone (positive post-yield stiffness), (2) same as 1 but with cyclic
strength degradation, (3) same as 1 but having peak-oriented (Clough-like)
hysteresis, (4) same as 2 but with peak-oriented hysteresis, (5) same as 1 but
with pinching hysteresis, (6) same as 2 but with pinching hysteresis, (7)
fracturing connection model with pinching hysteresis and asymmetric
backbones including a negative slope and a residual plateau at one direction,
and (8) a purely elastic bilinear backbone that dissipates no energy (Figure
A-37).
Representative Figures:
Figure A-36 Ratio of maximum displacement for all buildings and hinge
types versus the hinge type 1 (kinematic hardening, no
degradation).
FEMA P440A A: Detailed Summary of Previous Research A-55
Figure A-37 The hysteresis types considered for the beam-hinges.
A-56 A: Detailed Summary of Previous Research FEMA P440A
Summary of Findings:
The results show that the investigated plastic hinge models have a relatively
similar effect on the seismic response of the structures, regardless of the
structural period. Even hinge models with no energy dissipation do not
produce excessive demands. At most, for an R = 8 reduction factor, the
differences that appear are in the order of 30-40% and they only appear for a
limited range of periods and models. Nevertheless, only a single backbone
with in-cycle strength degradation has been considered, and even then the
negative slope exists only in one direction of loading. Therefore the
conclusions may be of limited use in such cases. This is an interesting
exercise and one of the very few investigations that has produced data on the
actual impact of local, hinge-level models on the global MDOF response.
Relevant Publications:
Foutch, D.A. and Shi, S., 1998, “Effects of hysteresis type on the seismic
response of buildings,” Proc. 6th U.S. National Conference on
Earthquake Engineering, EERI, Seattle, WA.
Shi, S. and Foutch, D.A., 1997, Evaluation of connection fracture and
hysteresis type on the seismic response of steel buildings, Report No.
617, Structural Research Series, Civil Engineering Studies, University of
Illinois at Urbana-Champaign, Urbana, IL.
A.2.14 Performance-Based Assessment of Existing Structures
Accounting For Residual Displacements
Authors:
Ruiz-Garcia, J. and Miranda, E. (2005)
Abstract:
The first part of this investigation describes comprehensive statistical studies
to quantify residual and maximum displacement demands of inelastic SDOF
systems, considering a relatively large earthquake ground motion database,
and considering a large number of structural parameters. The second part of
this study focuses on the evaluation of permanent (residual) and maximum
(transient) drift demands of multi-story framed building models under
different levels of ground motion intensity. Both parts include the
formulation and implementation of simplified probabilistic approaches to
estimate maximum and residual displacement demands accounting for the
uncertainty in the structural response and the ground motion hazard. The
study provides information towards incorporating explicitly the evaluation of
FEMA P440A A: Detailed Summary of Previous Research A-57
residual displacement demands for assessing the seismic performance of
existing structures, or for the preliminary design phase of new structures,
where structural damage control is achieved through control of lateral
deformation demands.
Summary:
This study examined the effect of hysteretic behavior of maximum
deformations of SDOF systems subjected to a large ensemble of 240 ground
motions recorded on firm sites in California. They considered seven different
types of hysteretic behavior: elastoplastic, bilinear, modified Clough,
Takeda, origin-oriented, moderate degrading and severely degrading models.
The modified Clough, the Takeda and origin-oriented models only exhibit
stiffness degradation while the moderate degrading and severely degrading
systems exhibit both stiffness and cyclic strength degradation. This study
computed mean ratios of maximum deformation of degrading hysteretic
models to non-degrading ones. They also studied the effect of hysteretic
behavior for systems subjected to ground motions recorded in very soft soil
sites and near-fault ground motions influence by forward directivity.
Representative Figures:
(b) TK
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-15 -12 -9 -6 -3 0 3 6 9 12 15
Displacement Ductility
Normalized Force
(a) MC
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-15 -10 -5 0 5 10 15
Displacement Ductility
Normalized Force
(c) O-O
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-15 -10 -5 0 5 10 15
Displacement Ductility
NormalizedForce
Figure A-38 Hysteretic models used in this investigation that only have stiffness degradation. (a) Modified-
Clough (MC); (b) Takeda model (TK); and (c) Origin-Oriented model (O-O).
A-58 A: Detailed Summary of Previous Research FEMA P440A
(b) MSD
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-15 -12 -9 -6 -3 0 3 6 9 12 15
Displacement Ductility
Noormalized Force
(c) SSD
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-15 -12 -9 -6 -3 0 3 6 9 12 15
Displacement Ductility
Normalized Force
Figure A-39 Hysteretic models used in this investigation with stiffness and cyclic strength degradation.
(b) Moderate Degrading (MSD); and (c) Severely Degrading (SSD).
(a) . = 3%
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
.i,.=3%/.i,.=0
R = 6.0
R = 5.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
(b) . = 5%
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
.i,.=5/.i,.=0
R = 6.0
R = 5.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
Figure A-40 Mean ratios of maximum deformation of bilinear to elastoplastic systems: (a).. = 3%;
and (b).. = 5%.
(b) SSD-2 model
SITE CLASS D
(mean of 80 ground motions)
0.0
0.5
1.0
1.5
2.0
2.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
.i,SSD / .i,EP
R = 6.0
R = 5.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
(a) SSD-1 model
SITE CLASS D
(mean of 80 ground motions)
0.0
0.5
1.0
1.5
2.0
2.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
.i,SSD / .i,EP
R = 6.0
R = 5.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
Figure A-41 Mean ratio of inelastic displacement demands in structural degrading and bilinear
systems: (a) SSD-1 model; and (b) SSD-2 model.
FEMA P440A A: Detailed Summary of Previous Research A-59
(b) HC = 2.5 (Takeda model)
SITE CLASS D
(mean of 80 ground motions)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
.i,SD/.i,EP
R = 6.0
R = 5.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
(c) HC = 0.1 (origin-oriented model)
SITE CLASS D
(mean of 80 ground motions)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
.i,SD/.i,EP
R = 6.0
R = 5.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
(a) HC (modified-Clough model)
SITE CLASS D
(mean of 80 ground motions)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
.i,SD / .i,EP
R = 6.0
R = 5.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
Figure A-42 Influence of hysteretic behavior on maximum deformation for three types of stiffness-degrading
systems: (a) Modified-Clough model; (b) Takeda model; and (c) Origin-oriented model.
Summary of Findings:
The effect of positive post-yield stiffness was a function of period of
vibration and level of lateral strength relative to the strength required to
maintain the system elastic. Positive post-elastic stiffness tended to reduce
maximum deformation demands but for realistic values of post-yield
stiffness, with exception of systems with very short periods, reductions were
small (smaller than 10%).
Maximum deformation demands of short-period degrading structures are, on
average, larger than those of non-degrading systems. In general, the
increment in displacement produced by degradation effects increases as the
strength ratio increases (i.e., as the system becomes weaker relative to the
lateral strength required to maintain the system elastic). For structures with
periods longer than about 0.7 s, maximum deformation of degrading systems
are on average either similar or slightly smaller than those of non-degrading
systems.
A-60 A: Detailed Summary of Previous Research FEMA P440A
The effects of stiffness degradation were larger for structures on soft soil
sites than those observed for structures on firm sites. For structures with
periods of vibration shorter than the predominant period of the ground
motion, the lateral displacement demands in stiffness-degrading systems are
on average 25% larger than those of non-degrading systems and in order to
control lateral deformations to levels comparable to those in non-degrading
structures, stiffness-degrading structures in this spectral region need to be
designed for higher lateral forces.
Maximum inelastic displacement demands of stiffness-degrading systems are
not significantly affected by the unloading stiffness provided that the
reduction in unloading stiffness is small or moderate. However, for systems
that unload toward the origin (that is, origin-oriented systems), or near the
origin, maximum inelastic displacements are on average larger than
maximum deformation demands of elastoplastic or bilinear systems and
therefore the equal displacement rule should not be used for these systems.
Hysteretic behaviors, in particular post-yield stiffness and unloading
stiffness, have a large influence on residual displacement demands.
Relevant Publications:
Ruiz-Garcia, J. and Miranda, E., 2003, “Inelastic displacement ratio for
evaluation of existing structures,” Earthquake Engineering and
Structural Dynamics. 32(8), 1237-1258.
Ruiz-Garcia, J. and Miranda, E., 2004, “Inelastic displacement ratios for
structures built on soft soil sites”, Journal of Structural Engineering,
130(12), December 2004, pp. 2051-2061
Ruiz-Garcia, J. and Miranda, E., 2005, Performance-based assessment of
existing structures accounting for residual displacements, John A. Blume
Earthquake Engineering Center, Report No. 153, Department of Civil
and Environmental Engineering, Stanford University, Stanford,
California, 444 p.
Ruiz-Garcia, J. and Miranda, E., 2006a, “Residual displacement ratios for the
evaluation of existing structures,” Earthquake Engineering and
Structural Dynamics, Vol. 35, pp. 315-336.
Ruiz-Garcia, J. and Miranda, E., 2006b, “Inelastic displacement ratios for
evaluation of structures built on soft soil sites,” Earthquake Engineering
and Structural Dynamics, in press.
FEMA P440A A: Detailed Summary of Previous Research A-61
A.2.15 Inelastic Spectra for Infilled Reinforced Concrete Frames
Authors:
Dolsek, M. and Fajfar, P. (2004)
Abstract:
In two companion papers a simplified nonlinear analysis procedure for
infilled reinforced concrete frames is introduced. In this paper a simple
relation between strength reduction factor, ductility and period (R–µ–T
relation) is presented. It is intended to be used for the determination of
inelastic displacement ratios and of inelastic spectra in conjunction with
idealized elastic spectra. The R–µ–T relation was developed from results of
an extensive parametric study employing a SDOF mathematical model
composed of structural elements representing the frame and infill. The
structural parameters used in the proposed R–µ–T relation, in addition to the
parameters used in a usual (e.g. elasto-plastic) system, are ductility at the
beginning of strength degradation, and the reduction of strength after the
failure of the infills. Formulae depend also on the corner periods of the
elastic spectrum. The proposed equations were validated by comparing
results in terms of the reduction factors, inelastic displacement ratios, and
inelastic spectra in the acceleration–displacement format, with those obtained
by non-linear dynamic analyses for three sets of recorded and semi-artificial
ground motions. A new approach was used for generating semi-artificial
ground motions compatible with the target spectrum. This approach
preserves the basic characteristics of individual ground motions, whereas the
mean spectrum of the complete ground motion set fits the target spectrum
excellently. In the parametric study, the R–µ–T relation was determined by
assuming a constant reduction factor, while the corresponding ductility was
calculated for different ground motions. The mean values proved to be
noticeably different from the mean values as determined when based on a
constant ductility approach, while the median values determined by the
different procedures were between the two means. The approach employed in
the study yields an R–µ–T relation which is conservative both for design and
performance assessment (compared with a relation based on median values).
Summary:
An R-µ-T relationship was developed that is suitable for use with infilled
frames having a quadrilinear elastic-positive-negative-residual backbone.
The system used for the analysis contained separate springs, Takeda for the
frame and shear-slip for the infill (Figure A-43). These were suitably
A-62 A: Detailed Summary of Previous Research FEMA P440A
calibrated to generate a backbone similar to the ones observed in pushovers
of infilled frames (Figure A-44). Thus, only in-cycle strength degradation
was considered, while any cyclic degradation issues were not investigated.
The system was analyzed using three suites of ground motion records which
were spectrum-matched to a target design spectrum. A parametric study of
the quadrilinear system was then conducted by varying the period and the
backbone parameters within prescribed values.
Representative Figures:
Figure A-43 The SDOF system: (a) force-displacement envelope; and (b) mathematic
model.
Figure A-44 The hysteretic behavior of the equivalent SDOF system.
FEMA P440A A: Detailed Summary of Previous Research A-63
Figure A-45 The influence of negative slope and residual plateau on the mean ductility for given R-factor.
Summary of Findings:
The negative slope was found to have a very small effect on the seismic
response of the system when only a short drop in strength to the residual
plateau is allowed (Figure A-45a). Actually, the level of the plateau
combined with a worsening negative slope were shown to be quite important,
lower plateaus increase the ductility demands considerably (Figure A-45b).
Approximate values were proposed for the dispersion around the mean
provided by the relationship.
Relevant Publications:
Dolsek, M. and Fajfar, P., 2004, “Inelastic spectra for infilled reinforced
concrete frames,” Earthquake Engineering and Structural Dynamics,
Vol. 33, 1395–1416.
Dolsek, M. and Fajfar, P., 2000, “Simplified nonlinear seismic analysis of
infilled reinforced concrete frames,” Earthquake Engineering and
Structural Dynamics, Vol. 34, 49–66.
Dolsek, M., 2002, Seismic response of infilled reinforced concrete frames,
Ph.D. thesis, University of Ljubljana, Faculty of Civil and Geodetic
Engineering, Ljubljana, Slovenia [in Slovenian].
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-1
Appendix B
Quantile IDA Curves for
Single-Spring Systems
This appendix presents quantile (16th, 50th and 84th percentile) incremental
dynamic analysis (IDA) curves from focused analytical studies on individual
spring single-degree-of-freedom (SDOF) systems. These systems consist of
spring types 1 through 8, with characteristics described in Chapter 3. This
collection of curves is intended to present the range of results for
representative short (T=0.5s), moderate (T=1.0s), and long (T=2.0s) period
systems, both with and without cyclic degradation. In the figures, the
vertical axis is the intensity measure Sa(T,5%), which is not normalized, and
the horizontal axis is the maximum story drift ratio, .max, in radians. IDA
curves with cyclic degradation (black lines) are shown along with IDA
curves without cyclic degradation (grey lines). Differences between the
black and grey lines in the plots indicate the effect of cyclic degradation
given the characteristics of the particular spring and period of vibration.
B-2 B: Quantile IDA Curves for Single-Spring Systems ATC-62
Figure B-1 Quantile IDA curves plotted versus Sa(T,5%) for Spring 1a and Spring 1b with a period of T = 0.5s
Figure B-2 Quantile IDA curves plotted versus Sa(T,5%) for Spring 1a and Spring 1b with a period of T = 1.0s.
Figure B-3 Quantile IDA curves plotted versus Sa(T,5%) for Spring 1a and Spring 1b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-3
Figure B-4 Quantile IDA curves plotted versus Sa(T,5%) for Spring 2a and Spring 2b with a period of T = 0.5s.
Figure B-5 Quantile IDA curves plotted versus Sa(T,5%) for Spring 2a and Spring 2b with a period of T = 1.0s.
Figure B-6 Quantile IDA curves plotted versus Sa(T,5%) for Spring 2a and Spring 2b with a period of T = 2.0s.
B-4 B: Quantile IDA Curves for Single-Spring Systems ATC-62
Figure B-7 Quantile IDA curves plotted versus Sa(T,5%) for Spring 3a and Spring 3b with a period of T = 0.5s.
Figure B-8 Quantile IDA curves plotted versus Sa(T,5%) for Spring 3a and Spring 3b with a period of T = 1.0s.
Figure B-9 Quantile IDA curves plotted versus Sa(T,5%) for Spring 3a and Spring 3b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-5
Figure B-10 Quantile IDA curves plotted versus Sa(T,5%) for Spring 4a and Spring 4b with a period of T = 0.5s.
Figure B-11 Quantile IDA curves plotted versus Sa(T,5%) for Spring 4a and Spring 4b with a period of T = 1.0s.
Figure B-12 Quantile IDA curves plotted versus Sa(T,5%) for Spring 4a and Spring 4b with a period of T = 2.0s.
B-6 B: Quantile IDA Curves for Single-Spring Systems ATC-62
Figure B-13 Quantile IDA curves plotted versus Sa(T,5%) for Spring 5a and Spring 5b with a period of T = 0.5s.
Figure B-14 Quantile IDA curves plotted versus Sa(T,5%) for Spring 5a and Spring 5b with a period of T = 1.0s.
Figure B-15 Quantile IDA curves plotted versus Sa(T,5%) for Spring 5a and Spring 5b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-7
Figure B-16 Quantile IDA curves plotted versus Sa(T,5%) for Spring 6a and Spring 6b with a period of T = 0.5s.
Figure B-17 Quantile IDA curves plotted versus Sa(T,5%) for Spring 6a and Spring 6b with a period of T = 1.0s.
Figure B-18 Quantile IDA curves plotted versus Sa(T,5%) for Spring 6a and Spring 6b with a period of T = 2.0s.
B-8 B: Quantile IDA Curves for Single-Spring Systems ATC-62
Figure B-19 Quantile IDA curves plotted versus Sa(T,5%) for Spring 7a and Spring 7b with a period of T = 0.5s.
Figure B-20 Quantile IDA curves plotted versus Sa(T,5%) for Spring 7a and Spring 7b with a period of T = 1.0s.
Figure B-21 Quantile IDA curves plotted versus Sa(T,5%) for Spring 7a and Spring 7b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-9
Figure B-22 Quantile IDA curves plotted versus Sa(T,5%) for Spring 8a and Spring 8b with a period of T = 0.5s.
Figure B-23 Quantile IDA curves plotted versus Sa(T,5%) for Spring 8a and Spring 8b with a period of T = 1.0s.
Figure B-24 Quantile IDA curves plotted versus Sa(T,5%) for Spring 8a and Spring 8b with a period of T = 2.0s.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-1
versus Normalized Intensity Measures
Appendix C
Median IDA Curves for
Multi-Spring Systems versus
Normalized Intensity Measures
This appendix contains normalized plots of median incremental dynamic
analysis (IDA) curves from focused analytical studies on multi-spring singledegree-
of-freedom (SDOF) systems. All systems are composed of two
springs representing a primary lateral-force-resisting system and a secondary
gravity system with the characteristics described in Chapter 3. Multi-spring
systems carry a designation of “NxJa+1a” or “NxJa+1b” where “N” is the
peak strength multiplier (N = 1, 2, 3, 5, or 9), “J” is the lateral-force-resisting
spring number (J = 2, 3, 4, 5, 6, or 7), and 1a or 1b is the gravity system
identifier. In all figures, the vertical axis is the normalized intensity measure
R = Sa(T1,5%)/Say(T1,5%), and the horizontal axis is the maximum story drift
ratio, .max, in radians. The period of vibration for each system is indicated in
parentheses.
C.1 Visualization Tool
Given the large volume of analytical data, customized algorithms were
developed for post-processing, statistical analysis, and visualization of
results. The accompanying CD includes an electronic visualization tool that
was developed to view results of multi-spring studies. The tool is a
Microsoft Excel based application with a user-interface that accesses a
database of all available multi-spring data. By selecting a desired spring
combination (“NxJa+1a” or “NxJa+1b”), stiffness level (stiff or flexible), and
intensity measure (normalized or non-normalized), users can view the
resulting quantile (median, 16th, and 84th percentile) IDA curves for the
combination of interest.
C-2 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a (1.87s)
2x2a (1.32s)
3x2a (1.08s)
5x2a (0.84s)
9x2a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2b (1.87s)
2x2b (1.32s)
3x2b (1.08s)
5x2b (0.84s)
9x2b (0.62s)
Figure C-1 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a and Nx2b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1a (1.53s)
2x2a+1a (1.18s)
3x2a+1a (1.00s)
5x2a+1a (0.80s)
9x2a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2b+1a (1.53s)
2x2b+1a (1.18s)
3x2b+1a (1.00s)
5x2b+1a (0.80s)
9x2b+1a (0.61s)
Figure C-2 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a+1a and Nx2b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1b (1.53s)
2x2a+1b (1.18s)
3x2a+1b (1.00s)
5x2a+1b (0.80s)
9x2a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10 S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2b+1b (1.53s)
2x2b+1b (1.18s)
3x2b+1b (1.00s)
5x2b+1b (0.80s)
9x2b+1b (0.61s)
Figure C-3 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a+1b and Nx2b+1b with mass M=8.87ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-3
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3a (1.87s)
2x3a (1.32s)
3x3a (1.08s)
5x3a (0.84s)
9x3a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3b (1.87s)
2x3b (1.32s)
3x3b (1.08s)
5x3b (0.84s)
9x3b (0.62s)
Figure C-4 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a and Nx3b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3a+1a (1.53s)
2x3a+1a (1.18s)
3x3a+1a (1.00s)
5x3a+1a (0.80s)
9x3a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3b+1a (1.53s)
2x3b+1a (1.18s)
3x3b+1a (1.00s)
5x3b+1a (0.80s)
9x3b+1a (0.61s)
Figure C-5 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a+1a and Nx3b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3a+1b (1.53s)
2x3a+1b (1.18s)
3x3a+1b (1.00s)
5x3a+1b (0.80s)
9x3a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3b+1b (1.53s)
2x3b+1b (1.18s)
3x3b+1b (1.00s)
5x3b+1b (0.80s)
9x3b+1b (0.61s)
Figure C-6 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a+1b and Nx3b+1b with mass M=8.87ton.
C-4 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4a (1.18s)
2x4a (0.84s)
3x4a (0.68s)
5x4a (0.53s)
9x4a (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4b (1.18s)
2x4b (0.84s)
3x4b (0.68s)
5x4b (0.53s)
9x4b (0.39s)
Figure C-7 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a and Nx4b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4a+1a (1.08s)
2x4a+1a (0.80s)
3x4a+1a (0.66s)
5x4a+1a (0.52s)
9x4a+1a (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4b+1a (1.08s)
2x4b+1a (0.80s)
3x4b+1a (0.66s)
5x4b+1a (0.52s)
9x4b+1a (0.39s)
Figure C-8 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a+1a and Nx4b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4a+1b (1.08s)
2x4a+1b (0.80s)
3x4a+1b (0.66s)
5x4a+1b (0.52s)
9x4a+1b (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4b+1b (1.08s)
2x4b+1b (0.80s)
3x4b+1b (0.66s)
5x4b+1b (0.52s)
9x4b+1b (0.39s)
Figure C-9 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a+1b and Nx4b+1b with mass M=8.87ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-5
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5a (1.02s)
2x5a (0.72s)
3x5a (0.59s)
5x5a (0.46s)
9x5a (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5b (1.02s)
2x5b (0.72s)
3x5b (0.59s)
5x5b (0.46s)
9x5b (0.34s)
Figure C-10 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a and Nx5b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5a+1a (0.95s)
2x5a+1a (0.70s)
3x5a+1a (0.58s)
5x5a+1a (0.45s)
9x5a+1a (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5b+1a (0.95s)
2x5b+1a (0.70s)
3x5b+1a (0.58s)
5x5b+1a (0.45s)
9x5b+1a (0.34s)
Figure C-11 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a+1a and Nx5b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5a+1b (0.95s)
2x5a+1b (0.70s)
3x5a+1b (0.58s)
5x5a+1b (0.45s)
9x5a+1b (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5b+1b (0.95s)
2x5b+1b (0.70s)
3x5b+1b (0.58s)
5x5b+1b (0.45s)
9x5b+1b (0.34s)
Figure C-12 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a+1b and Nx5b+1b with mass M=8.87ton.
C-6 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6a (1.87s)
2x6a (1.32s)
3x6a (1.08s)
5x6a (0.84s)
9x6a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6b (1.87s)
2x6b (1.32s)
3x6b (1.08s)
5x6b (0.84s)
9x6b (0.62s)
Figure C-13 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a and Nx6b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6a+1a (1.53s)
2x6a+1a (1.18s)
3x6a+1a (1.00s)
5x6a+1a (0.80s)
9x6a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6b+1a (1.53s)
2x6b+1a (1.18s)
3x6b+1a (1.00s)
5x6b+1a (0.80s)
9x6b+1a (0.61s)
Figure C-14 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a+1a and Nx6b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6a+1b (1.53s)
2x6a+1b (1.18s)
3x6a+1b (1.00s)
5x6a+1b (0.80s)
9x6a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6b+1b (1.53s)
2x6b+1b (1.18s)
3x6b+1b (1.00s)
5x6b+1b (0.80s)
9x6b+1b (0.61s)
Figure C-15 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a+1b and Nx6b+1b with mass M=8.87ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-7
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7a (1.87s)
2x7a (1.32s)
3x7a (1.08s)
5x7a (0.84s)
9x7a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7b (1.87s)
2x7b (1.32s)
3x7b (1.08s)
5x7b (0.84s)
9x7b (0.62s)
Figure C-16 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a and Nx7b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7a+1a (1.53s)
2x7a+1a (1.18s)
3x7a+1a (1.00s)
5x7a+1a (0.80s)
9x7a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7b+1a (1.53s)
2x7b+1a (1.18s)
3x7b+1a (1.00s)
5x7b+1a (0.80s)
9x7b+1a (0.61s)
Figure C-17 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a+1a and Nx7b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7a+1b (1.53s)
2x7a+1b (1.18s)
3x7a+1b (1.00s)
5x7a+1b (0.80s)
9x7a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7b+1b (1.53s)
2x7b+1b (1.18s)
3x7b+1b (1.00s)
5x7b+1b (0.80s)
9x7b+1b (0.61s)
Figure C-18 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a+1b and Nx7b+1b with mass M=8.87ton.
C-8 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2a (3.74s)
2x2a (2.65s)
3x2a (2.16s)
5x2a (1.67s)
9x2a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2b (3.74s)
2x2b (2.65s)
3x2b (2.16s)
5x2b (1.67s)
9x2b (1.25s)
Figure C-19. Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a and Nx2b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2a+1a (3.05s)
2x2a+1a (2.37s)
3x2a+1a (2.00s)
5x2a+1a (1.60s)
9x2a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2b+1a (3.05s)
2x2b+1a (2.37s)
3x2b+1a (2.00s)
5x2b+1a (1.60s)
9x2b+1a (1.21s)
Figure C-20 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a+1a and Nx2b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2a+1b (3.05s)
2x2a+1b (2.37s)
3x2a+1b (2.00s)
5x2a+1b (1.60s)
9x2a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2b+1b (3.05s)
2x2b+1b (2.37s)
3x2b+1b (2.00s)
5x2b+1b (1.60s)
9x2b+1b (1.21s)
Figure C-21 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a+1b and Nx2b+1b with mass M=35.46ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-9
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3a (3.74s)
2x3a (2.65s)
3x3a (2.16s)
5x3a (1.67s)
9x3a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3b (3.74s)
2x3b (2.65s)
3x3b (2.16s)
5x3b (1.67s)
9x3b (1.25s)
Figure C-22 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a and Nx3b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3a+1a (3.05s)
2x3a+1a (2.37s)
3x3a+1a (2.00s)
5x3a+1a (1.60s)
9x3a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3b+1a (3.05s)
2x3b+1a (2.37s)
3x3b+1a (2.00s)
5x3b+1a (1.60s)
9x3b+1a (1.21s)
Figure C-23 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a+1a and Nx3b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3a+1b (3.05s)
2x3a+1b (2.37s)
3x3a+1b (2.00s)
5x3a+1b (1.60s)
9x3a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3b+1b (3.05s)
2x3b+1b (2.37s)
3x3b+1b (2.00s)
5x3b+1b (1.60s)
9x3b+1b (1.21s)
Figure C-24 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a+1b and Nx3b+1b with mass M=35.46ton.
C-10 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
14
16
18
20
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4a (2.37s)
2x4a (1.67s)
3x4a (1.37s)
5x4a (1.06s)
9x4a (0.79s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
14
16
18
20
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4b (2.37s)
2x4b (1.67s)
3x4b (1.37s)
5x4b (1.06s)
9x4b (0.79s)
Figure C-25 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a and Nx4b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4a+1a (2.16s)
2x4a+1a (1.60s)
3x4a+1a (1.32s)
5x4a+1a (1.04s)
9x4a+1a (0.78s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4b+1a (2.16s)
2x4b+1a (1.60s)
3x4b+1a (1.32s)
5x4b+1a (1.04s)
9x4b+1a (0.78s)
Figure C-26 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a+1a and Nx4b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4a+1b (2.16s)
2x4a+1b (1.60s)
3x4a+1b (1.32s)
5x4a+1b (1.04s)
9x4a+1b (0.78s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4b+1b (2.16s)
2x4b+1b (1.60s)
3x4b+1b (1.32s)
5x4b+1b (1.04s)
9x4b+1b (0.78s)
Figure C-27 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a+1b and Nx4b+1b with mass M=35.46ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-11
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5a (2.04s)
2x5a (1.45s)
3x5a (1.18s)
5x5a (0.91s)
9x5a (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5b (2.04s)
2x5b (1.45s)
3x5b (1.18s)
5x5b (0.91s)
9x5b (0.68s)
Figure C-28 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a and Nx5b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5a+1a (1.91s)
2x5a+1a (1.39s)
3x5a+1a (1.15s)
5x5a+1a (0.90s)
9x5a+1a (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5b+1a (1.91s)
2x5b+1a (1.39s)
3x5b+1a (1.15s)
5x5b+1a (0.90s)
9x5b+1a (0.68s)
Figure C-29 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a+1a and Nx5b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5a+1b (1.91s)
2x5a+1b (1.39s)
3x5a+1b (1.15s)
5x5a+1b (0.90s)
9x5a+1b (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5b+1b (1.91s)
2x5b+1b (1.39s)
3x5b+1b (1.15s)
5x5b+1b (0.90s)
9x5b+1b (0.68s)
Figure C-30 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a+1b and Nx5b+1b with mass M=35.46ton.
C-12 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6a (3.74s)
2x6a (2.65s)
3x6a (2.16s)
5x6a (1.67s)
9x6a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6b (3.74s)
2x6b (2.65s)
3x6b (2.16s)
5x6b (1.67s)
9x6b (1.25s)
Figure C-31 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a and Nx6b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6a+1a (3.05s)
2x6a+1a (2.37s)
3x6a+1a (2.00s)
5x6a+1a (1.60s)
9x6a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6b+1a (3.05s)
2x6b+1a (2.37s)
3x6b+1a (2.00s)
5x6b+1a (1.60s)
9x6b+1a (1.21s)
Figure C-32 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a+1a and Nx6b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6a+1b (3.05s)
2x6a+1b (2.37s)
3x6a+1b (2.00s)
5x6a+1b (1.60s)
9x6a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6b+1b (3.05s)
2x6b+1b (2.37s)
3x6b+1b (2.00s)
5x6b+1b (1.60s)
9x6b+1b (1.21s)
Figure C-33 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a+1b and Nx6b+1b with mass M=35.46ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-13
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7a (3.74s)
2x7a (2.65s)
3x7a (2.16s)
5x7a (1.67s)
9x7a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7b (3.74s)
2x7b (2.65s)
3x7b (2.16s)
5x7b (1.67s)
9x7b (1.25s)
Figure C-34 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a and Nx7b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7a+1a (3.05s)
2x7a+1a (2.37s)
3x7a+1a (2.00s)
5x7a+1a (1.60s)
9x7a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7b+1a (3.05s)
2x7b+1a (2.37s)
3x7b+1a (2.00s)
5x7b+1a (1.60s)
9x7b+1a (1.21s)
Figure C-35 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a+1a and Nx7b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7a+1b (3.05s)
2x7a+1b (2.37s)
3x7a+1b (2.00s)
5x7a+1b (1.60s)
9x7a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7b+1b (3.05s)
2x7b+1b (2.37s)
3x7b+1b (2.00s)
5x7b+1b (1.60s)
9x7b+1b (1.21s)
Figure C-36 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a+1b and Nx7b+1b with mass M=35.46ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-1
versus Non-Normalized Intensity Measures
Appendix D
Median IDA Curves for
Multi-Spring Systems versus
Non-Normalized Intensity
Measures
This appendix contains non-normalized plots of median incremental dynamic
analysis (IDA) curves from focused analytical studies on multi-spring singledegree-
of-freedom (SDOF) systems. All systems are composed of two
springs representing a primary lateral-force-resisting system and a secondary
gravity system with the characteristics described in Chapter 3. Multi-spring
systems carry a designation of “NxJa+1a” or “NxJa+1b” where “N” is the
peak strength multiplier (N = 1, 2, 3, 5, or 9), “J” is the lateral-force-resisting
spring number (J = 2, 3, 4, 5, 6, or 7), and 1a or 1b is the gravity system
identifier. In the figures, the vertical axis is one of two ground motion
intensities IM = Sa(1s,5%) or Sa(2s,5%), which are not normalized, and the
horizontal axis is the maximum story drift ratio, .max, in radians. The period
of vibration for each system is indicated in parentheses.
D.1 Visualization Tool
Given the large volume of analytical data, customized algorithms were
developed for post-processing, statistical analysis, and visualization of
results. The accompanying CD includes an electronic visualization tool that
was developed to view results of multi-spring studies. The tool is a
Microsoft Excel based application with a user-interface that accesses a
database of all available multi-spring data. By selecting a desired spring
combination (“NxJa+1a” or “NxJa+1b”), stiffness level (stiff or flexible), and
intensity measure (normalized or non-normalized), users can view the
resulting quantile (median, 16th, and 84th percentile) IDA curves for the
combination of interest.
D-2 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(1s,5%) (g)
.
max
M=8.87
2a (1.87s)
2x2a (1.32s)
3x2a (1.08s)
5x2a (0.84s)
9x2a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(1s,5%) (g)
.
max
M=8.87
2b (1.87s)
2x2b (1.32s)
3x2b (1.08s)
5x2b (0.84s)
9x2b (0.62s)
Figure D-1 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx2a and Nx2b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
2a+1a (1.53s)
2x2a+1a (1.18s)
3x2a+1a (1.00s)
5x2a+1a (0.80s)
9x2a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
2b+1a (1.53s)
2x2b+1a (1.18s)
3x2b+1a (1.00s)
5x2b+1a (0.80s)
9x2b+1a (0.61s)
Figure D-2 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx2a+1a and
Nx2b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
2a+1b (1.53s)
2x2a+1b (1.18s)
3x2a+1b (1.00s)
5x2a+1b (0.80s)
9x2a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 S
a
(1s,5%) (g)
.
max
M=8.87
2b+1b (1.53s)
2x2b+1b (1.18s)
3x2b+1b (1.00s)
5x2b+1b (0.80s)
9x2b+1b (0.61s)
Figure D-3 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx2a+1b and
Nx2b+1b with mass M=8.87ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-3
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3a (1.87s)
2x3a (1.32s)
3x3a (1.08s)
5x3a (0.84s)
9x3a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3b (1.87s)
2x3b (1.32s)
3x3b (1.08s)
5x3b (0.84s)
9x3b (0.62s)
Figure D-4 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx3a and Nx3b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3a+1a (1.53s)
2x3a+1a (1.18s)
3x3a+1a (1.00s)
5x3a+1a (0.80s)
9x3a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3b+1a (1.53s)
2x3b+1a (1.18s)
3x3b+1a (1.00s)
5x3b+1a (0.80s)
9x3b+1a (0.61s)
Figure D-5 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx3a+1a and
Nx3b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3a+1b (1.53s)
2x3a+1b (1.18s)
3x3a+1b (1.00s)
5x3a+1b (0.80s)
9x3a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3b+1b (1.53s)
2x3b+1b (1.18s)
3x3b+1b (1.00s)
5x3b+1b (0.80s)
9x3b+1b (0.61s)
Figure D-6 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx3a+1b and
Nx3b+1b with mass M=8.87ton.
D-4 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
4a (1.18s)
2x4a (0.84s)
3x4a (0.68s)
5x4a (0.53s)
9x4a (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
4b (1.18s)
2x4b (0.84s)
3x4b (0.68s)
5x4b (0.53s)
9x4b (0.39s)
Figure D-7 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx4a and Nx4b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
4a+1a (1.08s)
2x4a+1a (0.80s)
3x4a+1a (0.66s)
5x4a+1a (0.52s)
9x4a+1a (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
4b+1a (1.08s)
2x4b+1a (0.80s)
3x4b+1a (0.66s)
5x4b+1a (0.52s)
9x4b+1a (0.39s)
Figure D-8 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx4a+1a and
Nx4b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
4a+1b (1.08s)
2x4a+1b (0.80s)
3x4a+1b (0.66s)
5x4a+1b (0.52s)
9x4a+1b (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
4b+1b (1.08s)
2x4b+1b (0.80s)
3x4b+1b (0.66s)
5x4b+1b (0.52s)
9x4b+1b (0.39s)
Figure D-9 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx4a+1b and
Nx4b+1b with mass M=8.87ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-5
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
5a (1.02s)
2x5a (0.72s)
3x5a (0.59s)
5x5a (0.46s)
9x5a (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
5b (1.02s)
2x5b (0.72s)
3x5b (0.59s)
5x5b (0.46s)
9x5b (0.34s)
Figure D-10 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx5a and Nx5b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
5a+1a (0.95s)
2x5a+1a (0.70s)
3x5a+1a (0.58s)
5x5a+1a (0.45s)
9x5a+1a (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
5b+1a (0.95s)
2x5b+1a (0.70s)
3x5b+1a (0.58s)
5x5b+1a (0.45s)
9x5b+1a (0.34s)
Figure D-11 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx5a+1a and
Nx5b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
5a+1b (0.95s)
2x5a+1b (0.70s)
3x5a+1b (0.58s)
5x5a+1b (0.45s)
9x5a+1b (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
5b+1b (0.95s)
2x5b+1b (0.70s)
3x5b+1b (0.58s)
5x5b+1b (0.45s)
9x5b+1b (0.34s)
Figure D-12 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx5a+1b and
Nx5b+1b with mass M=8.87ton.
D-6 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6a (1.87s)
2x6a (1.32s)
3x6a (1.08s)
5x6a (0.84s)
9x6a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6b (1.87s)
2x6b (1.32s)
3x6b (1.08s)
5x6b (0.84s)
9x6b (0.62s)
Figure D-13 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx6a and Nx6b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6a+1a (1.53s)
2x6a+1a (1.18s)
3x6a+1a (1.00s)
5x6a+1a (0.80s)
9x6a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6b+1a (1.53s)
2x6b+1a (1.18s)
3x6b+1a (1.00s)
5x6b+1a (0.80s)
9x6b+1a (0.61s)
Figure D-14 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx6a+1a and
Nx6b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6a+1b (1.53s)
2x6a+1b (1.18s)
3x6a+1b (1.00s)
5x6a+1b (0.80s)
9x6a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6b+1b (1.53s)
2x6b+1b (1.18s)
3x6b+1b (1.00s)
5x6b+1b (0.80s)
9x6b+1b (0.61s)
Figure D-15 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx6a+1b and
Nx6b+1b with mass M=8.87ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-7
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
S
a
(1s,5%) (g)
.
max
M=8.87
7a (1.87s)
2x7a (1.32s)
3x7a (1.08s)
5x7a (0.84s)
9x7a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
S
a
(1s,5%) (g)
.
max
M=8.87
7b (1.87s)
2x7b (1.32s)
3x7b (1.08s)
5x7b (0.84s)
9x7b (0.62s)
Figure D-16 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx7a and Nx7b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
7a+1a (1.53s)
2x7a+1a (1.18s)
3x7a+1a (1.00s)
5x7a+1a (0.80s)
9x7a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
7b+1a (1.53s)
2x7b+1a (1.18s)
3x7b+1a (1.00s)
5x7b+1a (0.80s)
9x7b+1a (0.61s)
Figure D-17 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx7a+1a and
Nx7b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
7a+1b (1.53s)
2x7a+1b (1.18s)
3x7a+1b (1.00s)
5x7a+1b (0.80s)
9x7a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
7b+1b (1.53s)
2x7b+1b (1.18s)
3x7b+1b (1.00s)
5x7b+1b (0.80s)
9x7b+1b (0.61s)
Figure D-18 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx7a+1b and
Nx7b+1b with mass M=8.87ton.
D-8 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
S
a
(2s,5%) (g)
.
max
M=35.5
2a (3.74s)
2x2a (2.65s)
3x2a (2.16s)
5x2a (1.67s)
9x2a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
S
a
(2s,5%) (g)
.
max
M=35.5
2b (3.74s)
2x2b (2.65s)
3x2b (2.16s)
5x2b (1.67s)
9x2b (1.25s)
Figure D-19 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx2a and Nx2b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
2a+1a (3.05s)
2x2a+1a (2.37s)
3x2a+1a (2.00s)
5x2a+1a (1.60s)
9x2a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
2b+1a (3.05s)
2x2b+1a (2.37s)
3x2b+1a (2.00s)
5x2b+1a (1.60s)
9x2b+1a (1.21s)
Figure D-20 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx2a+1a and
Nx2b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
2a+1b (3.05s)
2x2a+1b (2.37s)
3x2a+1b (2.00s)
5x2a+1b (1.60s)
9x2a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
2b+1b (3.05s)
2x2b+1b (2.37s)
3x2b+1b (2.00s)
5x2b+1b (1.60s)
9x2b+1b (1.21s)
Figure D-21 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx2a+1b and
Nx2b+1b with mass M=35.46ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-9
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
S
a
(2s,5%) (g)
.
max
M=35.5
3a (3.74s)
2x3a (2.65s)
3x3a (2.16s)
5x3a (1.67s)
9x3a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
S
a
(2s,5%) (g)
.
max
M=35.5
3b (3.74s)
2x3b (2.65s)
3x3b (2.16s)
5x3b (1.67s)
9x3b (1.25s)
Figure D-22 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx3a and Nx3b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
3a+1a (3.05s)
2x3a+1a (2.37s)
3x3a+1a (2.00s)
5x3a+1a (1.60s)
9x3a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
3b+1a (3.05s)
2x3b+1a (2.37s)
3x3b+1a (2.00s)
5x3b+1a (1.60s)
9x3b+1a (1.21s)
Figure D-23 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx3a+1a and
Nx3b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
3a+1b (3.05s)
2x3a+1b (2.37s)
3x3a+1b (2.00s)
5x3a+1b (1.60s)
9x3a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
3b+1b (3.05s)
2x3b+1b (2.37s)
3x3b+1b (2.00s)
5x3b+1b (1.60s)
9x3b+1b (1.21s)
Figure D-24 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx3a+1b and
Nx3b+1b with mass M=35.46ton.
D-10 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
4a (2.37s)
2x4a (1.67s)
3x4a (1.37s)
5x4a (1.06s)
9x4a (0.79s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
4b (2.37s)
2x4b (1.67s)
3x4b (1.37s)
5x4b (1.06s)
9x4b (0.79s)
Figure D-25 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx4a and Nx4b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
4a+1a (2.16s)
2x4a+1a (1.60s)
3x4a+1a (1.32s)
5x4a+1a (1.04s)
9x4a+1a (0.78s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
4b+1a (2.16s)
2x4b+1a (1.60s)
3x4b+1a (1.32s)
5x4b+1a (1.04s)
9x4b+1a (0.78s)
Figure D-26 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for (systems Nx4a+1a and
Nx4b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
4a+1b (2.16s)
2x4a+1b (1.60s)
3x4a+1b (1.32s)
5x4a+1b (1.04s)
9x4a+1b (0.78s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
4b+1b (2.16s)
2x4b+1b (1.60s)
3x4b+1b (1.32s)
5x4b+1b (1.04s)
9x4b+1b (0.78s)
Figure D-27 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx4a+1b and
Nx4b+1b with mass M=35.46ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-11
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
S
a
(2s,5%) (g)
.
max
M=35.5
5a (2.04s)
2x5a (1.45s)
3x5a (1.18s)
5x5a (0.91s)
9x5a (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
S
a
(2s,5%) (g)
.
max
M=35.5
5b (2.04s)
2x5b (1.45s)
3x5b (1.18s)
5x5b (0.91s)
9x5b (0.68s)
Figure D-28 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx5a and Nx5b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
5a+1a (1.91s)
2x5a+1a (1.39s)
3x5a+1a (1.15s)
5x5a+1a (0.90s)
9x5a+1a (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
5b+1a (1.91s)
2x5b+1a (1.39s)
3x5b+1a (1.15s)
5x5b+1a (0.90s)
9x5b+1a (0.68s)
Figure D-29 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx5a+1a and
Nx5b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
5a+1b (1.91s)
2x5a+1b (1.39s)
3x5a+1b (1.15s)
5x5a+1b (0.90s)
9x5a+1b (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
5b+1b (1.91s)
2x5b+1b (1.39s)
3x5b+1b (1.15s)
5x5b+1b (0.90s)
9x5b+1b (0.68s)
Figure D-30 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx5a+1b and
Nx5b+1b with mass M=35.46ton.
D-12 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6a (3.74s)
2x6a (2.65s)
3x6a (2.16s)
5x6a (1.67s)
9x6a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6b (3.74s)
2x6b (2.65s)
3x6b (2.16s)
5x6b (1.67s)
9x6b (1.25s)
Figure D-31 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx6a and Nx6b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6a+1a (3.05s)
2x6a+1a (2.37s)
3x6a+1a (2.00s)
5x6a+1a (1.60s)
9x6a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6b+1a (3.05s)
2x6b+1a (2.37s)
3x6b+1a (2.00s)
5x6b+1a (1.60s)
9x6b+1a (1.21s)
Figure D-32 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx6a+1a and
Nx6b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6a+1b (3.05s)
2x6a+1b (2.37s)
3x6a+1b (2.00s)
5x6a+1b (1.60s)
9x6a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6b+1b (3.05s)
2x6b+1b (2.37s)
3x6b+1b (2.00s)
5x6b+1b (1.60s)
9x6b+1b (1.21s)
Figure D-33 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx6a+1b and
Nx6b+1b with mass M=35.46ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-13
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
S
a
(2s,5%) (g)
.
max
M=35.5
7a (3.74s)
2x7a (2.65s)
3x7a (2.16s)
5x7a (1.67s)
9x7a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
S
a
(2s,5%) (g)
.
max
M=35.5
7b (3.74s)
2x7b (2.65s)
3x7b (2.16s)
5x7b (1.67s)
9x7b (1.25s)
Figure D-34 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx7a and Nx7b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
7a+1a (3.05s)
2x7a+1a (2.37s)
3x7a+1a (2.00s)
5x7a+1a (1.60s)
9x7a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
7b+1a (3.05s)
2x7b+1a (2.37s)
3x7b+1a (2.00s)
5x7b+1a (1.60s)
9x7b+1a (1.21s)
Figure D-35 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx7a+1a and
Nx7b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
7a+1b (3.05s)
2x7a+1b (2.37s)
3x7a+1b (2.00s)
5x7a+1b (1.60s)
9x7a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
7b+1b (3.05s)
2x7b+1b (2.37s)
3x7b+1b (2.00s)
5x7b+1b (1.60s)
9x7b+1b (1.21s)
Figure D-36 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx7a+1b and
Nx7b+1b with mass M=35.46ton.
FEMA P440A E: Uncertainty, Fragility, and Probability E-1
Appendix E
Uncertainty, Fragility, and
Probability
The concepts presented in this report are compatible with current
probabilistic trends in performance-based seismic design. A probabilistic
context allows explicit consideration of the variability and uncertainty
associated with each of the contributing parameters. An important concept in
probabilistic procedures is the development and use of fragility curves.
Use of fragility curves, and explicit consideration of uncertainty in
performance assessment, is described in ATC-58 Guidelines for Seismic
Performance Assessment of Buildings (ATC, 2007). Fragility curves are also
used to determine the margin of safety against collapse in FEMA P695
Quantification of Building Seismic Performance Factors (FEMA, 2009).
This appendix explains the conversion of incremental dynamic analysis
(IDA) results into fragilities, and presents equations that could be used to
calculate annual probabilities for collapse, or any other limit state of interest.
E.1 Conversion of IDA Results to Fragilities
Incremental dynamic analysis results can be readily converted to fragilities.
Figure E-1 shows an example of IDA results for a single structure subjected
to a suite of ground motions of varying intensities.
Figure E-1 IDA results for a single structure subjected to a suite of ground
motions of varying intensities.
E-2 E: Uncertainty, Fragility, and Probability FEMA P440A
In this illustration, sidesway collapse is the governing mechanism, and
collapse prediction is based on dynamic instability or excessive lateral
displacements. Using collapse data obtained from IDA results, a collapse
fragility can be defined through a cumulative distribution function (CDF),
which relates the ground motion intensity to the probability of collapse
(Ibarra et al., 2002). Studies have shown that this cumulative distribution
function can be assumed to be lognormally distributed. Figure E-2 shows an
example of a cumulative distribution plot obtained by fitting a lognormal
distribution to the collapse data from Figure E-1.
Figure E-2 Cumulative distribution plot obtained by fitting a lognormal
distribution to collapse data from IDA results.
Lognormal distributions are defined by a median value and a dispersion
parameter. The median collapse capacity, Sa50% , indicates a ground motion
intensity that has a 50% chance of producing collapse in the system. It also
indicates the point at which half of the ground motions will produce collapse
at higher intensities, and half will produce collapse at lower intensities. For
each mode of collapse, the record-to-record dispersion can be estimated as:
84% 16% ( ) ( )
2
a a
RTR
. ln S ln S
.
. (E-1)
Figure E-3 provides conceptual collapse fragility curves showing the
probability of collapse due to loss of vertical-load-carrying capacity (LVCC)
or lateral dynamic instability (LDI). These are events are mutually exclusive,
meaning that either one or the other can occur, but both events cannot occur
at the same time.
FEMA P440A E: Uncertainty, Fragility, and Probability E-3
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
S a (T 1 )
P( C | Sa )
combined
vertical collapse
lateral collapse
Figure E-3 Conceptual collapse fragility curves for sidesway (lateral)
collapse, vertical collapse, and a combination of both.
The total probability of collapse due to either mode can then be represented
(Cornell et al., 2005) as:
P(C | sa) = P(CLDI | sa) + P(CLVCC | NCLDI ,sa).P(NCLDI | sa (E-2)
where:
P(CLDI | sa) is the probability of collapse due to lateral dynamic instability at
a ground motion intensity level sa ,
P(CLVCC | NCLDI ,sa) is the probability of collapse due to loss of vertical-loadcarrying
capacity, given that collapse due to lateral dynamic instability has
not occurred at intensity sa , and
P(NCLDI |sa) is the probability of no collapse due to lateral dynamic
instability at a ground motion intensity level sa.
Since P(NCLDI |sa) is equal to 1– P(CLDI | sa), then Equation E-2 can also be
written as:
P(C | sa) = P(CLVCC | NCLDI , sa)+P(CLDI | sa) - P(CLVCC | NCLDI sa).P(CLDI | sa) (E-3)
E.2 Calculation of Annualized Probability
The results of an incremental dynamic analysis expressed as a cumulative
distribution function can be used in combination with a seismic hazard curve
to generate mean annual frequencies (MAF) for collapse (or for other limit
states of interest). This process is the integration of the limit state CDF (e.g.,
E-4 E: Uncertainty, Fragility, and Probability FEMA P440A
fragility representing the probability of collapse as a function of spectral
acceleration) with respect to the probability of occurrence of the intensity
measure (e.g., hazard curve representing the annual probability of exceeding
a full range of spectral accelerations). The mean annual frequency of
collapse, col . , or other limit state of interest, can be approximated (Cornell,
2002) as:
. .exp 1 2 2
col Sa C 2 RTR . .. . .. k . ..
. .
(E-4)
where . . Sa C . . is the mean annual probability of the median spectral
acceleration associated with collapse. The parameter k is the slope of the
hazard curve, and can be calculated as:
(10/ 50)
(2/50)
(2/50) (2/50)
(10/ 50) (10 / 50)
ln
1.65
ln ln
aT
aT
S
S
aT aT
aT aT
H
H
k
S S
S S
. .
.. ..
. . . .
. . . .
.. .. .. ..
. . . .
(E-5)
FEMA P440A F: Example Application F-1
Appendix F
Example Application
This appendix presents an example application of a simplified nonlinear
dynamic analysis procedure. The concept originated during the conduct of
focused analytical studies comparing force-displacement capacity boundaries
to incremental dynamic analysis results. In this procedure, a nonlinear static
analysis is used to generate an idealized force-deformation curve (i.e., static
pushover curve). The resulting curve is then used as a force-displacement
capacity boundary to constrain the hysteretic behavior of an equivalent
single-degree-of-freedom (SDOF) oscillator. This SDOF oscillator is then
subjected to incremental dynamic analysis.
The steps for conducting a simplified nonlinear dynamic analysis are
outlined in the following section, and illustrated using an example building.
Alternative retrofit strategies are evaluated using the same procedure. Use of
the procedure to develop probabilistic estimates of performance for use in
making design decisions is also illustrated.
F.1 Simplified Nonlinear Dynamic Analysis Procedure
The concept of a simplified nonlinear dynamic analysis procedure includes
the following steps:
. Develop an analytical model of the system.
Models can be developed in accordance with prevailing practice for
seismic evaluation, design, and rehabilitation of buildings described in
ASCE/SEI Standard 41-06 Seismic Rehabilitation of Existing Buildings
(ASCE, 2006b). Component properties should be based on forcedisplacement
capacity boundaries, rather than cyclic envelopes.
. Perform a nonlinear static pushover analysis.
Subject the model to a conventional pushover analysis in accordance
with prevailing practice. Lateral load increments and resulting
displacements are recorded to generate an idealized force-deformation
curve.
F-2 F: Example Application FEMA P440A
. Conduct an incremental dynamic analysis of the system based on an
equivalent SDOF model.
The idealized force-deformation curve is, in effect, a system forcedisplacement
capacity boundary that can be used to constrain a hysteretic
model of an equivalent SDOF oscillator. This SDOF oscillator is then
subjected to incremental dynamic analysis to check for lateral dynamic
instability and other limit states of interest. Alternatively, approximate
incremental dynamic analysis can be accomplished using the idealized
force-deformation curve and the Static Pushover 2 Incremental Dynamic
Analysis open source software tool, SPO2IDA (Vamvatsikos and
Cornell, 2006).
. Determine probabilities associated with limit states of interest.
Results from incremental dynamic analysis can be used to obtain
response statistics associated with limit states of interest in addition to
lateral dynamic instability. SPO2IDA can also be used to obtain median,
16th, and 84th percentile IDA curves relating displacements to intensity.
Using the fragility relationships described in Appendix E in conjunction
with a site hazard curve, this information can be converted into annual
probabilities of exceedance for each limit state. Probabilistic information
in this form can be used to make enhanced decisions based on risk and
uncertainty, rather than on discrete threshold values of acceptance.
F.2 Example Building
The example building is a five-story reinforced concrete frame residential
structure with interior unreinforced masonry infill partitions in the upper
stories, and a soft/weak first-story. An exterior elevation of the building is
shown in Figure F-1 and first floor plan is shown in Figure F-2. Reinforced
concrete columns in each orthogonal direction provide lateral resistance to
seismic forces. As indicated in Figure F-2, the first story includes a mixture
of components with column-like proportions and components that are more
like slender shear walls.
This building is a prototypical example of a soft/weak story structure.
Concentration of inelastic deformations in the first story presents an obvious
potential story collapse mechanism.
F.3 Structural Analysis Model
To investigate the potential for collapse in this structure, it is reasonable to
assume that the response can be represented by a SDOF model. The first
story column components are classified for modeling purposes in accordance
FEMA P440A F: Example Application F-3
with ASCE/SEI 41-06. Most of the columns in this example are classified as
shear-controlled or flexure-shear controlled. Wall-like column components
are shear-controlled along the strong axis of the member.
Figure F-1 Example building exterior elevation.
Figure F-2 Example building first floor plan.
Modeling parameters for the column components can be characterized by the
conceptual force-displacement relationship (“backbone”) specified in
ASCE/SEI 41-06. The modeling parameters selected for the components in
this example are taken from Chapter 6 of ASCE/SEI 41-06, and depicted in
Figure F-3. In both cases, the residual strength, c, is taken as zero.
F-4 F: Example Application FEMA P440A
The column components are assembled into a model of the structural system
as shown in Figure F-4. Inelastic response is assumed to occur
predominantly in the first story. First-story columns are taken as fixed at the
base on a rigid foundation. The stiffness of the column components are
based on elastic properties in flexure and shear. The model includes soil
flexibility, allowing for rigid body rotation due to the response of the
structure above. Soil stiffness parameters are taken from Chapter 4 of
ASCE/SEI 41-06, assuming a relatively soft soil site (site Class E).
n V
.total columns y .
b . 0.01
a . 0
A c . 0
B,C
D,E
n V
total . columns y .
b . 0.01
a . 0
A c . 0
B,C
D,E
(a)
n V
walls
h
.
e . 0.01
d . 0.0075
A c . 0
B,C
D,E
n V
walls
h
.
e . 0.01
d . 0.0075
A c . 0
B,C
D,E
(b)
Figure F-3 Force-displacement modeling parameters for: (a) column
components; and (b) wall-like column components.
When developing a SDOF representation of a system, it is important to
account for foundation rotation in assessing column distortions. The
resulting SDOF model represents the relationship between the total first floor
drift, including contributions from the foundation, sys fdn cols . .. .. , and the
applied inertial loads, V .
FEMA P440A F: Example Application F-5
fdn .
col . Column
distortion
Foundation
rotation
1st story mechanism-all
inelasticity in columns
Rigid foundation on elastic supports
Effective
height of
inertial
forces, h* hcm
h1st
W W
sys . fdn col . 1st . . . .. h
cm col 1st fdn cm . .. h .. h
V V
eff K.
fdn .
col . Column
distortion
Foundation
rotation
1st story mechanism-all
inelasticity in columns
Rigid foundation on elastic supports
Effective
height of
inertial
forces, h* hcm
h1st
W W
. . sys fdn col 1st . . . .. h
cm col 1st fdn cm . .. h .. h
V V
eff K. eff K.
(a.) (b)
Figure F-4 Structural analysis model showing: (a) assumptions; and (b)
distortions.
F.4 Nonlinear Static Pushover Analysis
The analytical model is subjected to a conventional pushover analysis.
Results are shown in Figure F-5.
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
V
sys .
LVCC .
LDI .
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
V
sys . 0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
V
sys .
LVCC .
LDI .
Figure F-5 Pushover curve from nonlinear static analysis.
F-6 F: Example Application FEMA P440A
F.5 Evaluation of Limit States of Interest
Collapse in real structures can be caused by sidesway collapse (lateral
dynamic instability) or by loss of vertical-load-carrying capacity. In this
example, the following two limit states are defined (both are shown in Figure
F-5):
LVCC . the total system rotation at which loss of vertical-load-carrying
capacity occurs (i.e., when first story columns fail due to shear
distortion). In this example, the critical column distortion for loss of
vertical-load-carrying capacity is taken as 1% inelastic rotation,
which occurs when the total system rotation sys . = 1.2%.
LDI . the total system rotation at which lateral dynamic instability occurs
(i.e., when first story columns lose all lateral-force-resisting
capacity). In this example, this is taken to occur when the total
system rotation sys . = 4.0%.
The target displacement for a given intensity is estimated using the
Coefficient Method:
2
0 1 2 4 2
e
t a
. C C C S T g
.
.
Uniform hazard spectra for the example site are shown in Figure F-6 and
Figure F-7.
Figure F-6 Uniform hazard spectrum for intensity corresponding to 10%
chance of exceedance in 50 years (from USGS).
FEMA P440A F: Example Application F-7
Figure F-7 Uniform hazard spectrum for intensities corresponding to 2%
chance of exceedance in 50 years (from USGS).
For an intensity corresponding to a 10% chance of exceedance in 50 years
(i.e., 475-year return period), and a period T = 0.3s:
Sa .1.36g
The strength of the model is:
Fy . 0.45g
which results in:
aT 3.0
y
R S g
F
. . for the 10%/50 year hazard level.
The coefficients are:
C0 . first mode participation factor = 1.0,
1 2
1 1 1.3
e
C R
aT
.
. . . ,
where a = 50, and
2
2
1 1 1 1.04
800 e
C R
T
. . .
. . .. .. .
. .
.
This results in a target displacement of:
F-8 F: Example Application FEMA P440A
1.6 t .
. inches, or .sys .1.6/100 .1.6%.
This is greater than the acceptable limit for loss of vertical-load-carrying
capacity ( LVCC . ) taken as sys . = 1.2%.
To check for lateral dynamic instability, the proposed equation for Rdi is:
3
3
a
c e r u r
di e
y c y
R b T F T
. F
. . . . . . . .
. .. .. . . .. .. . . . . . .
where Te is the effective fundamental period of vibration of the structure, .y
,
.c
, .r
, and .u
are displacements corresponding to the yield strength, Fy,
capping strength, Fc, residual strength, Fr, and ultimate deformation capacity
at the end of the residual strength plateau. Determination of these parameters
requires a multi-linear idealization of the pushover curve, as shown in Figure
F-8.
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
Idealization A
Idealization B
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
Idealization A
Idealization B
Figure F-8 Pushover curve from nonlinear static analysis and two idealized
system force-displacement capacity boundaries.
Parameters a and b are functions given by:
1 exp( ) e a . . .dT
2
1 . ..
.
. ..
.
. .
c
r
F
b F
FEMA P440A F: Example Application F-9
and parameter d is a constant equal to 4 for the example building (assuming
the presence of stiffness degradation).
Using the above expressions along with parameters from Idealization ‘A’ in
Figure F-8, results in:
Rdi . 2.6
which is less that the calculated value of R = 3.0 for intensities corresponding
to the 10%/50 year hazard level. The parameter R is a ratio equal to the
strength necessary to keep a system elastic for a given intensity, divided by
the yield strength of the system. Higher values of R imply lower values of
system yield strength. Values of R that exceed Rdi mean that the structure
does not meet the minimum strength necessary to avoid lateral dynamic
instability at this hazard level.
At higher intensities (e.g., 2%/50 year hazard level) the calculated value of R
would be even higher (R = 5.3 >> Rdi = 2.6), illustrating how the comparison
between system strength and the limit on lateral dynamic instability would
change for a different hazard level.
In summary, the example structure does not meet acceptability criteria for
loss of vertical-load-carrying capacity and lateral dynamic instability at the
10%/50 year hazard level. Thus, a nonlinear response-history analysis must
be performed.
F.6 Incremental Dynamic Analysis
The resulting force-displacement relationship from the pushover curve can be
used to generate a force-displacement capacity boundary for the system. An
incremental dynamic analysis (IDA) can then be applied to a SDOF oscillator
constrained by the resulting force-displacement capacity boundary.
Performing an incremental dynamic analysis will allow determination of the
ground motion intensity at which various limit state deformations occur.
For the example building, an approximate incremental dynamic analysis is
performed using the open source software tool, SPO2IDA. Use of SPO2IDA
along with Idealization ‘A’ in Figure F-8 results in the median, 16th, and 84th
percentile IDA curves shown in Figure F-9. The figure also includes the
estimate of Rdi for lateral dynamic instability.
F-10 F: Example Application FEMA P440A
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.0% 1.0% 2.0% 3.0% 4.0% 5.0%
System rotation
Spectral acceleration
IDA-50% IDA-84% IDA-16% Capacity Boundary
Loss of vertical
carrying capacity
(LVCC)
Lateral dynamic
instability (LDI)
Rdi eqn.
Figure F-9 Results of approximate incremental dynamic analysis using
SPO2IDA.
F.7 Determination of Probabilities Associated with
Limit States of Interest
From Figure F-9, median values of intensity causing loss of vertical-loadcarrying
capacity (LVCC) or lateral dynamic instability (LDI) in the example
building can be obtained. Using the expressions in Appendix E, the
dispersion and mean annual frequencies (MAF) associated with these limit
states can be determined. The resulting data is presented Table F-1.
Table F-1 Mean Annual Frequencies for Collapse Limit States
Limit state/collapse mode
Sa50
MAF Sa50 .. MAF
collapse
Loss of vertical load
carrying capability 0.92 0.0050 0.20 0.0060
Lateral dynamic instability 1.26 0.0025 0.32 0.0040
LVCC or LDI 0.88 0.0060 0.17 0.0069
For the example building, fragilities associated with loss of vertical-loadcarrying
capacity (LVCC) or lateral dynamic instability (LDI) are derived
from the median values of spectral acceleration and dispersions in Table F-1,
as illustrated in Figure F-10.
FEMA P440A F: Example Application F-11
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00 1.00 2.00 3.00
Spectral accel., S aT
Probability of collapse, P(C|Sa