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.