Effects of Strength and 

Stiffness Degradation on 

Seismic Response 

FEMA P440A 

June 2009

FEMA

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-ofFreedom 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 ForceDisplacement 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-ofFreedom 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 postNorthridge 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-Δ. ................................ 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 preNorthridge 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

S Sg R

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), RuizGarcia 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

(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

 (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-ofFreedom 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 

φ1 4

Deg 

φ2

 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 

φ1

4

Deg 

φ2

 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

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, θ, chosen to characterize the force-displacement capacity 

boundary for each of the single-spring system models are listed in Table 3-2. 

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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

θ

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)

θ

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)

θ

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)

θ

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)

θ

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)

θ

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)

θ

7a

1a

7a+1a

Figure 3-41 Combined force-displacement capacity boundary for spring 

7a +1a (normalized by the strength of Spring 1a). 

Each multi-spring combination was subjected to an ATC-24 type loading 

protocol with a degrading force-displacement capacity boundary (cyclic 

degradation). The resulting hysteretic behaviors for the combination of 

Nx2a+1a for (N = 1, 2, 3, 5, and 9) are shown in Figure 3-42 through Figure 

3-44. In addition, individual Spring 2a is shown in Figure 3-44 for 

comparison. In each figure, the initial combined force-displacement capacity 

boundary (before cyclic degradation) is overlaid onto the hysteretic plots. 

 (a) (b) 

Figure 3-42 Initial force-displacement capacity boundary overlaid onto hysteretic behavior for: 

(a) Spring 1x2a+1a; and (b) Spring 2x2a+1a; both with cyclic degradation.
3-32 3: Development of Single-Degree-of-Freedom Models FEMA P440A 

for Focused Analytical Studies 

 (a) (b) 

Figure 3-43 Initial force-displacement capacity boundary overlaid onto hysteretic behavior for: (a) Spring 

1x3a+1a; and (b) Spring 5x2a+1a; both with cyclic degradation. 

 (a) (b) 

Figure 3-44 Initial force-displacement capacity boundary overlaid onto hysteretic behavior for: (a) Spring 

9x2a+1a; and (b) individual Spring 2a; both with cyclic degradation. 

As might be expected, the more the multiplier “N” for Spring 2a increases, 

the more the combined system resembles Spring 2a itself (Figure 3-44), and 

the more the behavior of the combined system would be expected to be 

dominated by the characteristics of the lateral-force-resisting spring 

component. Conversely, for lower multiples of “N”, the characteristics of 

the gravity system are more visible in the combined system properties 

(Figure 3-42), and would be expected to play a more significant role in the 

behavior of the combined system. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-1 

 Focused Analytical Studies 

Chapter 4 

Results from Single-Degree-ofFreedom Focused Analytical 

Studies 

This chapter summarizes the results of focused analytical studies on singlespring and multi-spring systems, compares results to recommendations 

contained in FEMA 440, and explains the development of a new equation 

measuring the potential for lateral dynamic instability. 

4.1 Summary of Analytical Results 

There were 160 single-spring systems (eight basic spring types, “a” and “b” 

versions of each, with and without cyclic degradation, tuned to five different 

periods of vibration). Each system was subjected to incremental dynamic 

analysis using 56 ground motion records scaled to multiple levels of 

increasing intensity. This resulted in over 600,000 nonlinear response 

history analyses on single-spring systems. 

There were 600 multi-spring systems (six lateral-force-resisting springs, “a” 

and “b” versions of each, five relative strength multipliers, five different 

gravity spring combinations, tuned with two different story masses). Each 

system was subjected to incremental dynamic analysis using 56 ground 

motion records scaled to multiple levels of increasing intensity. This resulted 

in over 2,000,000 nonlinear response history analyses on multi-spring 

systems. 

In total, results from over 2.6 million nonlinear response history analyses 

were available for review. Given the large volume of analytical data, 

customized algorithms were developed for post-processing, statistical 

analysis, and visualization of results. Results are summarized in the sections 

that follow. More complete sets of data are presented in the appendices. A 

customized visualization tool that was developed to view results of multispring studies, along with all available data, is included on the CD 

accompanying this report. Use of the visualization tool is described in 

Appendix C and Appendix D. 

4.2 Observations from Single-Spring Studies 

This section summarizes the results from nonlinear dynamic analyses of 

single-spring systems. Results from these studies were used to: 
4-2 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

 identify predominant characteristics of median incremental dynamic 

analysis (IDA) curves for these systems, 

 demonstrate a relationship between IDA curves and features of the forcedisplacement capacity boundaries, and 

 qualitatively determine the effects of different degrading behaviors on 

the dynamic stability of structural systems. 

Only selected results are presented here. Quantile (16th, 50th and 84th

percentile) IDA curves for each of the single-spring systems are provided in 

Appendix B. The horizontal axis for all single-spring IDA results is the 

maximum story drift ratio, max, in radians. 

4.3 Characteristics of Median IDA Curves 

Individual incremental dynamic analysis (IDA) curves for single ground 

motion records are very sensitive to dynamic interaction between the 

properties of the system and the characteristics of the ground motion. 

Quantile (16th, 50th and 84th percentile) IDA curves, however, are much more 

stable and provide better information on the central tendency (median) and 

variability (dispersion) in system response. In general, median IDA curves 

exhibit the following characteristics (Figure 4-1): 

 An initial linear segment corresponding to linear-elastic behavior in 

which in lateral deformation demand is proportional to ground motion 

intensity, regardless of the characteristics of the system or the ground 

motion. This segment extends from the origin to the onset of yielding. 

 A second curvilinear segment corresponding to inelastic behavior in 

which lateral deformation demand is no longer proportional to ground 

motion intensity. As intensity increases, lateral deformation demands 

increase at a faster rate. This segment corresponds to softening of the 

system, or reduction in stiffness (reduction in the slope of the IDA 

curve). In this segment, the system “transitions” from linear behavior to 

eventual dynamic instability. Although a curvilinear segment is always 

present, in some cases the transition can be relatively long and gradual, 

while in other cases it can be very short and abrupt. 

 A final linear segment that is horizontal, or nearly horizontal, in which 

infinitely large lateral deformation demands occur at small increments in 

ground motion intensity. This segment corresponds to the point at which 

a system becomes unstable (lateral dynamic instability). For SDOF 

systems, this point corresponds to the ultimate deformation capacity at 

which the system loses all lateral-force-resisting capacity. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-3 

 Focused Analytical Studies 

Figure 4-1 Characteristic segments of a median IDA curve. 

In some systems, the initial linear segment can be extended beyond yield into 

the inelastic range (Figure 4-2). In this segment lateral deformation demand 

is approximately proportional to ground motion intensity, which is consistent 

with the familiar equal-displacement approximation for estimating inelastic 

displacements. The range of lateral deformation demands over which the 

equal-displacement approximation is applicable depends on the 

characteristics of the force-displacement capacity boundary of the system and 

the period of vibration. 

Figure 4-2 Characteristic segments of a median IDA curve with a pseudolinear segment. 
4-4 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

4.3.1 Dependence on Period of Vibration 

Figure 4-3 shows the force-displacement capacity boundary and resulting 

IDA curves for Spring 3a with different periods of vibration. Each system is 

tuned to a different lateral strength and stiffness so results are compared 

using the normalized intensity measure R = Sa(T,5%)/Say(T,5%). Intensities 

larger than R = 1.0 mean the system is behaving inelastically. 

Figure 4-3 Force-displacement capacity boundary and median IDA curves for Spring 3a with various 

periods of vibration.

In general, moderate and long period systems with zero or positive post-yield 

stiffness in the force-displacement capacity boundary follow the equal 

displacement trend well into the nonlinear range. For systems with periods 

longer than 0.5s, Spring 3a exhibits an extension of the initial linear segment 

well beyond the yield drift of 0.01. In contrast, the short period system 

(T=0.2s) diverges from the initial linear segment just after yielding, even at 

deformations within the strength-hardening segment of the forcedisplacement capacity boundary (drifts between 0.01 and 0.04). 

4.3.2 Dispersion in Response 

Nonlinear response is sensitive to the characteristics of the ground motion 

record, and will vary from one ground motion to the next, even when scaled 

to the same intensity. For a given level of ground motion intensity, the 

lateral deformation demand can be significantly smaller or significantly 

larger than the value shown on median IDA curves. As the level of ground 

motion intensity increases, the dispersion in response tends to increase. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-5 

 Focused Analytical Studies 

Figure 4-4 shows three quantile IDA curves for Spring 3b with period of 

vibration of 2.0s. The 50% (median) IDA curve indicates that, for a given 

level of ground motion intensity (Sa), half of all deformation demands are 

smaller and half are larger than values along this curve. Because the 

distribution of demands is lognormally distributed, the dispersion about the 

median is not symmetric. The upper (16%) curve in the figure indicates that, 

for a given level of ground motion intensity, 16% of all lateral deformation 

demands are to the left of this curve while 84% are to the right. This means 

that lateral deformation demands along this curve have an 84% chance of 

being exceeded. Similarly the lower (84%) curve corresponds to lateral 

deformation demands with a 16% chance of being exceeded. 

Figure 4-4 Force-displacement capacity boundary and 16th, 50th and 84th percentile IDA curves for 

Spring 3b with a period of vibration T=2.0s. 

4.4 Influence of the Force-Displacement Capacity 

Boundary 

Comparisons between force-displacement capacity boundaries and median 

IDA curves show a strong correlation between the shape of the resulting 

curves and key features of the force-displacement capacity boundary, such as 

post-yield behavior and onset of degradation, slope of degradation, ultimate 

deformation capacity, and presence of cyclic degradation. 

Figure 4-5 shows the force-displacement capacity boundary and resulting 

median IDA curve for Spring 3b with a period of 2.0s. With a positive postyield slope, delayed onset of degradation, and robust residual strength 

plateau with an extended maximum deformation capacity, the resulting IDA 

curve includes both linear and pseudo-linear segments and a gradual 

transition to lateral dynamic instability. 
4-6 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

 

Figure 4-5 Force-displacement capacity boundary and median IDA curve for Spring 3b with a 

period of vibration T=2.0s. 

Figure 4-6 shows the force-displacement capacity boundary and resulting 

median IDA curve for Spring 2a with a period of 2.0s. With the onset of 

degradation occurring immediately after yielding, the shape of the resulting 

IDA curve changes. The pseudo-linear segment disappears, but with the 

presence of a residual strength plateau, the transition segment remains 

somewhat gradual until lateral dynamic instability. 

 

Figure 4-6 Force-displacement capacity boundary and median IDA curve for Spring 2a with a 

period of vibration T=2.0s. 

Figure 4-7 shows the force-displacement capacity boundary and resulting 

median IDA curve for Spring 6a with a period of 2.0s. With a broad yielding 

plateau, the pseudo-linear segment extends well into the inelastic range. 

Without a residual strength plateau, however, the system abruptly transitions 

into lateral dynamic instability. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-7 

 Focused Analytical Studies 

Figure 4-7 Force-displacement capacity boundary and median IDA curve for Spring 6a with a period of 

vibration T=2.0s. 

Figure 4-8 shows the force-displacement capacity boundary and resulting 

median IDA curve for Spring 8a with a period of 2.0s. With severe strength 

degradation occurring immediately after yielding, and the absence of a 

residual strength plateau, the system abruptly transitions from linear elastic 

behavior directly into lateral dynamic instability with little or no transition. 

 

Figure 4-8 Force-displacement capacity boundary and median IDA curve for Spring 8a with a period of 

vibration T=2.0s. 

These observed relationships suggest that dynamic response is directly 

influenced by the features of a force-displacement capacity boundary. Figure 

4-9 shows how the characteristic segments of a median IDA curve relate to 

these features. Note that the relationship depicted in this idealized graphical 

representation is dependent upon the period of the system, as described in 

Section 4.3.1. 
4-8 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

 

Figure 4-9 Relationship between IDA curves and the features of a typical 

force-displacement capacity boundary 

For low levels of ground motion intensity, the initial linear segment of the 

IDA curve is controlled by the effective stiffness of the system, Ke . Since 

the response is linear there is no dispersion evident in this segment. As the 

intensity increases the system reaches its yield point, , Fy y  . Systems with a 

non-negative post-elastic stiffness, Ke , will likely exhibit a pseudo-linear 

segment. Beyond yield, dispersion appears in the nonlinear response due to 

ground motion variability, and the 16th and 84th percentile IDA curves begin 

to diverge from the median curve. 

The extent of the pseudo-linear segment depends on the initial post elastic 

stiffness, Ke , and ends prior to reaching the strength hardening limit, 

, Fc c  (also known as the capping point). For systems that exhibit negative 

stiffness, Ke  , immediately after yielding, the pseudo-linear segment may be 

very short or non-existent. Also, for short-period systems, the pseudo-linear 

segment can be very short, even if the system has positive post-yield 

stiffness. 

As the ground motion intensity increases further, deformation demands 

increase at a faster rate, the IDA curve begins to flatten, and the curvilinear 

softening segment emerges. Dispersion between the quantile curves also 

increases. Beyond the strength hardening limit, , Fc c  , degradation occurs, 

and the softening increases at a faster rate. The presence of a residual 

INSTABILITY

LINEAR

PSEUDO

LINEAR

SOFTENING



boundar

16%

84%

50% INSTABILITY

LINEAR



CAPACITY

BOUNDARY

SaT (g)
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-9 

 Focused Analytical Studies 

strength plateau, , Fr r  , can extend the softening segment and delay the 

eventual transition into lateral dynamic instability. The point at which 

instability occurs corresponds to the ultimate deformation capacity, u , at 

which the system loses all lateral force resistance. 

This relationship suggests that it is possible to estimate the nonlinear 

dynamic behavior of a system based on knowledge of the characteristics of 

the force-displacement capacity boundary of the system. The influence that 

important features of the force-displacement capacity boundary have on 

nonlinear response is explained in more detail in the sections that follow. 

4.4.1 Post-Yield Behavior and Onset of Degradation 

The three systems shown in Figure 4-10 have the same elastic stiffness, same 

yield strength, but different post-yield characteristics. The forcedisplacement capacity boundary of Spring 2a experiences strength 

degradation immediately after yielding. In contrast, Spring 3a has a 

moderate yielding plateau before the onset of similar strength degradation, 

while Spring 6a has elastic-perfectly-plastic behavior up to the ultimate 

deformation capacity. 

These three systems have the same elastic behavior, but at drift ratios larger 

than 0.02, their relative potential for in-cycle strength degradation, and their 

resulting collapse behaviors, are all very different. Key parameters related to 

the observed change in response are the post-yield slope and the strength 

hardening limit (capping point). The presence of a non-negative post-yield 

slope and any delay before the onset of degradation reduces potential 

in-cycle strength degradation and improves the collapse capacity of a system. 

Figure 4-10 Effect of post-yield behavior on the collapse capacity of a system (Springs 2a, 3a and 6a with 

T=2.0s). 
4-10 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

4.4.2 Slope of Degradation 

Figure 4-11 shows the force-displacement capacity boundaries of Spring 2a 

and Spring 2b along with the corresponding IDA curves. These two systems 

have the same elastic stiffness, same yield strength, but differ in the negative 

slope of the strength-degrading segment and, therefore, in their potential for 

in-cycle strength degradation. They also have the same ultimate deformation 

capacity, but Spring 2b has a shorter residual strength plateau than Spring 2a 

because of the different slope. 

Figure 4-11 Effect of slope of degradation on the collapse capacity of a system (Springs 2a 

and 2b with T=1.0s). 

The two systems have the same elastic behavior, but their response at drift 

demands larger than 0.01 is very different. Spring 2a, with a steeper 

degrading slope, likely experiences in-cycle strength degradation and reaches 

its collapse capacity relatively early, while Spring 2b, with a more shallow 

degrading slope, reaches a collapse capacity that is approximately 50% 

larger. 

Figure 4-12 shows the force-displacement capacity boundaries of Spring 5a 

and Spring 5b along with the corresponding IDA curves. As in the case of 

Springs 2a and 2b, these two systems differ in the negative slope of the 

strength-degrading segments. They also differ in the presence of a residual 

strength plateau, which exists in Spring 5a, but not in Spring 5b. 

As shown in the figure, the median IDA curves are similar up to 0.005 drift, 

at which both systems reach their peak strength. Beyond this point, the 

curves diverge as a result of the change in negative slope. Spring 5a, with 

steeper degrading slopes, reaches its collapse capacity sooner, while Spring 

5b, with more shallow degrading slopes, reaches a higher collapse capacity. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-11 

 Focused Analytical Studies 

The key parameter related to the observed change in response is the negative 

slope of the strength-degrading segment. In both examples, the change in 

negative slope changed the magnitude of potential in-cycle strength 

degradation, and overshadowed any changes in the residual strength plateau, 

as long as the ultimate deformation capacity remained the same. 

Figure 4-12 Effect of slope of degradation on the collapse capacity of a system (Springs 5a and 5b with 

T=1.0s). 

4.4.3 Ultimate Deformation Capacity 

Figure 4-13 shows the force-displacement capacity boundaries and 

corresponding IDA curves for Springs 1a and 1b. Figure 4-14 shows the 

force-displacement capacity boundaries and corresponding IDA curves for 

Springs 6a and 6b. These spring systems have very different post-yield 

behaviors, one with strength degradation (Springs 1a and 1b) and the other 

with elasto-plastic behavior (Springs 6a and 6b). In both cases, the “b” 

versions of each spring have higher ultimate deformation capacities. 

Figure 4-13 Effect of ultimate deformation capacity on the collapse capacity of a system (Springs 1a and 1b 

with T=1.0s). 
4-12 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

Figure 4-14 Effect of ultimate deformation capacity on the collapse capacity of a system 

(Springs 6a and 6b with T=1.0s). 

In both examples, increasing the ultimate deformation capacity resulted in 

more than a 50% increase in collapse capacity. The key parameter related to 

the observed change in response is the increment in the ultimate deformation 

capacity. Observed changes in collapse capacity resulting from increases in 

the ultimate deformation capacity were insensitive to the other characteristics 

of the post-yield behavior of the springs. 

4.4.4 Degradation of the Force-Displacement Capacity 

Boundary (Cyclic Degradation) 

In general, most components will exhibit some level of cyclic degradation. 

To investigate the effects of cyclic degradation, the “a” and “b” versions of 

each spring (except Spring 6) were analyzed with both a constant forcedisplacement capacity boundary and a degrading force-displacement capacity 

boundary. 

Consistent with observations from past studies, comparison of results 

between springs both with and without cyclic degradation show that the 

effects of cyclic degradation (as measured by gradual movement of the 

capacity boundary) are relatively unimportant in comparison with in-cycle 

degradation (as measured by the extent and steepness of negative slopes in 

the capacity boundary). This trend is illustrated for Spring 3b in Figure 4-15, 

but can be observed in the results for many spring systems. Although the 

system without cyclic degradation has a higher median collapse capacity, the 

difference is not very large. For the single-spring systems studied, the 

difference between median collapse capacity with and without cyclic 

degradation is shown in Appendix B. In general, this difference was 

typically less than 10%. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-13 

 Focused Analytical Studies 

Figure 4-15 Effect of degradation of the force-displacement capacity boundary on the 

collapse capacity of a system (Spring 3b, T=2.0s, with and without cyclic 

degradation). 

This observation has two important exceptions. First, the effect of cyclic 

degradation increases as the level of in-cycle degradation increases. Systems 

such as Spring 2b with a steep negative slope in the capacity boundary, 

indicating a strong potential for severe in-cycle strength degradation, showed 

as much as 30% difference in median collapse capacity between systems 

with and without cyclic degradation (Figure 4-16). Second, the effect of 

cyclic degradation increases as the period of vibration decreases. The short 

period (T=0.5s) versions of each spring showed more influence from cyclic 

degradation than the corresponding longer period (T=1.0s or T=2.0s) 

versions. This can be seen in the plots in Appendix B. 

Figure 4-16 Effect of degradation of the force-displacement capacity boundary on the 

collapse capacity of a system (Spring 2b, T=0.2s, with and without cyclic 

degradation). 
4-14 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

4.5 Observations from Multi-Spring Studies 

This section summarizes the results from nonlinear dynamic analyses of 

multi-spring systems. Results from these studies were used to qualitatively: 

 understand the influence of key features of the combined forcedisplacement capacity boundary on the nonlinear response of multispring systems, 

 determine the effects of lateral strength on the dynamic stability of multispring systems, and 

 determine the effects of secondary systems on the dynamic stability of 

multi-spring systems. 

Only selected results are presented here. Combinations Nx2a+1a and 

Nx3a+1a, for N = 1, 2, 3, 5, or 9, are used to highlight trends observed to be 

generally applicable for the set of multi-spring combinations studied in this 

investigation. Results for each combination, plotted versus normalized and 

non-normalized intensity measures, are provided in Appendix C and 

Appendix D. 

4.5.1 Normalized versus Non-Normalized Results 

Two intensity measures were used in conducting incremental dynamic 

analyses. One was the 5% damped spectral acceleration at the fundamental 

period of vibration of the oscillator, Sa(T,5%). While generally appropriate 

for single-degree-of-freedom systems, this measure does not allow 

comparison between systems having different periods of vibration. For this 

reason, a normalized intensity measure, R = Sa(T,5%)/Say(T,5%) was also 

used, where Say(T,5%) is the intensity that causes first yield to occur in the 

system. 

In order to compare the response of different spring systems, it is necessary 

to plot the IDA curves from several springs in a single figure using a 

common intensity measure. This can be done in two ways. One way is to 

plot them using the normalized intensity measure, R = Sa(T,5%)/Say(T,5%). 

First yield occurs at a normalized intensity of one, and increasing values of 

Sa(T,5%)/Say(T,5%) represent increasing values of ground motion intensity 

with respect to the intensity required to initiate yielding in the system. 

Normalized plots provide a measure of system capacity relative to the yield 

intensity, and are useful for comparing results across different spring types 

when evaluating the influence of the key features of the force-displacement 

capacity boundary on the response of the system. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-15 

 Focused Analytical Studies 

A second way to compare results is to plot them using an absolute (nonnormalized) intensity measure that is somewhere in the middle of the range 

that would be suitable for the systems being plotted (e.g., T=1.0s). When 

evaluating the effects of increasing or decreasing the relative contribution of 

one subsystem with respect to another, use of a single absolute intensity 

measure allows comparison of results based on the relative strengths of 

different systems. 

Since each method has advantages for viewing results and drawing 

comparisons, results for multi-spring systems were plotted using both the 

normalized intensity measure, Sa(T,5%)/Say(T,5%), and non-normalized 

intensity measures, Sa(1s,5%) for stiff systems and Sa(2s,5%) for flexible 

systems. Results for normalized intensity measures, IM = 

Sa(T,5%)/Say(T,5%), are provided in Appendix C, and results for nonnormalized intensity measures, IM = Sa(1s,5%) or Sa(2s,5%), are provided in 

Appendix D. The horizontal axis in all cases is the maximum story drift 

ratio, max, in radians. 

4.5.2 Comparison of Multi-Spring Force-Displacement Capacity 

Boundaries 

Figure 4-17 shows the force-displacement capacity boundaries for multispring systems Nx2a+1a and Nx3a+1a, normalized by the yield strength, Fy, 

of the combined system. Figure 4-18 shows the force-displacement capacity 

boundaries for the same two systems, normalized by the strength of the 

weakest system. Depending on the normalizing parameter used along the 

vertical axis, the resulting curves look very different. 

In Figure 4-17, the use of a normalized base shear, F/Fy or Sa/Say, along the 

vertical axis allows for a better qualitative comparison of the relative shapes 

of the force-displacement capacity boundaries, without the added complexity 

caused by the different yield strengths of the systems. In this figure, it is 

easier to see how increasing the multiplier “N” on the lateral-force-resisting 

spring causes the combined system to more closely resemble the lateral 

spring itself (i.e., as “N” increases from 1 to 9, the combination Nx2a+1a 

begins to look more like Spring 2a). 

Figure 4-17, however, is misleading with regard to the relative strengths of 

the combined systems. In normalizing to the yield strength of the combined 

system, higher values of yield strength will reduce the plotted values by a 

larger ratio, so curves for higher strength systems will plot below curves for 

lower strength systems in F/Fy coordinates. 
4-16 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

0 0.02 0.04 0.06 0.08 0.1

0

0.5

1

1.5



F / Fy

2a+1a

2x2a+1a

3x2a+1a

5x2a+1a

9x2a+1a

 

0 0.02 0.04 0.06 0.08 0.1

0

0.5

1

1.5



F / Fy

3a+1a

2x3a+1a

3x3a+1a

5x3a+1a

9x3a+1a

 (Nx2a+1a) (Nx3a+1a) 

Figure 4-17 Force-displacement capacity boundaries for multi-spring systems Nx2a+1a and 

Nx3a+1a, normalized by the yield strength, Fy, of the combined system. 

 (Nx2a+1a) (Nx3a+1a) 

Figure 4-18 Force-displacement capacity boundaries for multi-spring systems Nx2a+1a and 

Nx3a+1a, normalized by the yield strength of the weakest system 

In Figure 4-18, normalizing to the strength of the weakest system allows for 

a better comparison of the relative strength between the systems. In this 

figure it is easier to see how increasing the multiplier “N” on the lateralforce-resisting spring increases the strength of the combined system. 

4.5.3 Influence of the Combined Force-Displacement Capacity 

Boundary in Multi-Spring Systems 

Regardless of the normalizing parameter, Figure 4-17 and Figure 4-18 show 

how the combined force-displacement capacity boundaries change as the 

relative contributions of the springs vary. Results from single-spring studies 

demonstrated the influence of key features of the force-displacement capacity 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-17 

 Focused Analytical Studies 

boundary on the nonlinear dynamic response of a single-spring system. 

Results from multi-spring studies followed the same relationships. Multispring systems in which the combined force-displacement capacity boundary 

had more favorable features (e.g., delayed onset of degradation, more gradual 

slope of degradation, higher residual strength, and higher ultimate 

deformation capacity) performed better. 

Figure 4-19 shows median IDA curves plotted versus the normalized 

intensity measure R = Sa(T,5%)/Say(T,5%) for multi-spring systems Nx2a+1a 

and Nx3a+1a with a mass of 8.87 tons, representing a series of relatively stiff 

systems. As “N” increases, the yield strength of the combined system 

increases, and each system has a correspondingly shorter period of vibration. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2a+1a (1.53s)

2x2a+1a (1.18s)

3x2a+1a (1.00s)

5x2a+1a (0.80s)

9x2a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3a+1a (1.53s)

2x3a+1a (1.18s)

3x3a+1a (1.00s)

5x3a+1a (0.80s)

9x3a+1a (0.61s)

Figure 4-19 Median IDA curves plotted versus the normalized intensity measure 

Sa(T,5%)/Say(T,5%) for systems Nx2a+1a and Nx3a+1a with a mass of 8.87 tons. 

Figure 4-20 shows median IDA curves for the same two systems with a mass 

of 35.46 tons, representing a series of relatively flexible systems. Because 

each system has a different period of vibration, normalized plots are used to 

qualitatively compare IDA curves between systems. Normalized curves, 

however, can be somewhat misleading with regard to the effect of changing 

“N” in the different spring combinations. The plotting positions in Figure 

4-19, for example, are not an indication of the absolute collapse capacity of 

each system. Rather, they are a measure of collapse capacity relative to the 

intensity required to initiate yielding. Systems with high yield strengths may 

actually collapse at higher absolute intensities than systems with lower yield 

strengths, but because of the normalization to yield intensity, they plot out at 

lower ratios. 
4-18 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

2a+1a (3.05s)

2x2a+1a (2.37s)

3x2a+1a (2.00s)

5x2a+1a (1.60s)

9x2a+1a (1.21s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

3a+1a (3.05s)

2x3a+1a (2.37s)

3x3a+1a (2.00s)

5x3a+1a (1.60s)

9x3a+1a (1.21s)

Figure 4-20 Median IDA curves plotted versus the normalized intensity measure 

Sa(T,5%)/Say(T,5%) for systems Nx2a+1a and Nx3a+1a with a mass of 35.46 tons. 

For system Nx2a+1a, Figure 4-19 and Figure 4-20 show that as “N” 

increases, collapse capacity decreases. The reason for this can be seen in the 

combined force-displacement capacity boundaries for system Nx2a+1a 

shown in Figure 4-17. Because of the characteristics of Spring 2a, 

combinations with higher multiples of “N” have steeper negative slopes. As 

was the case with single-spring systems, steeper slopes in the strengthdegrading segment of the force-displacement capacity boundary result in 

lower collapse capacities. 

For system Nx3a+1a, the results are the same, but less pronounced. Similar 

to system Nx2a+1a, the force-displacement capacity boundaries shown in 

Figure 4-17 for system Nx3a+1a with higher multiples of “N” have steeper 

negative slopes, but the differences are less significant. 

Figure 4-19 and Figure 4-20 also show that, in general, combinations with 

systems that have more favorable characteristics result in higher median 

collapse capacities relative to yield intensity. For example, in Figure 4-19, 

system 9x2a+1a exhibits a median collapse capacity that is approximately 2.3 

times the yield intensity while system 9x3a+1a exhibits a median collapse 

capacity that is approximately 3.5 times the yield intensity. The reason for 

this can be seen by comparing the combined force-displacement capacity 

boundaries for systems Nx2a+1a and Nx3a+1a shown in Figure 4-17. The 

post-yield characteristics of system Nx3a+1a are more favorable in terms of 

the post-yield slope, onset of degradation, and slope of degradation, resulting 

in better performance. 

A more direct illustration of this behavior can be seen by comparing 

combinations using the “a” and “b” versions of primary spring components. 

Figure 4-21 shows the median IDA curves for systems Nx3a+1a and 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-19 

 Focused Analytical Studies 

Nx3b+1a. By definition, the “b” version of each spring was created to have 

more favorable characteristics than the “a” version of the same spring, with 

all other parameters being equal. As shown in the figure, the curves for 

system Nx3b+1a outperform all corresponding combinations of Nx3a+1a in 

terms of collapse capacity relative to yield intensity, for all values of “N” 

from 1 to 9. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3a+1a (1.53s)

2x3a+1a (1.18s)

3x3a+1a (1.00s)

5x3a+1a (0.80s)

9x3a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3b+1a (1.53s)

2x3b+1a (1.18s)

3x3b+1a (1.00s)

5x3b+1a (0.80s)

9x3b+1a (0.61s)

Figure 4-21 Median IDA curves plotted versus the normalized intensity measure Sa(T,5%)/Say(T,5%) 

for systems Nx3a+1a and Nx3b+1a with a mass of 8.87 tons. 

4.5.4 Effects of the Lateral Strength of Multi-Spring Systems 

Plotting of results using absolute (non-normalized) intensity measures allows 

for comparison of results based on the relative strengths of different systems. 

Non-normalized intensity measures of Sa(1s,5%) for stiff systems and 

Sa(2s,5%) for flexible systems were used to identify the effects of the lateral 

strength of the multi-spring system on the lateral dynamic stability of the 

system. 

Figure 4-22 shows median IDA curves for multi-spring systems Nx2a+1a 

and Nx3a+1a tuned with a mass of 8.87 tons. They are plotted versus 

Sa(1s,5%), which is an intensity measure keyed to a period of T=1.0s, located 

in the middle of the range of periods for the relatively stiff set of multi-spring 

systems. Figure 4-23 shows median IDA curves for same set of systems 

Nx2a+1a and Nx3a+1a, tuned with a mass of 35.46 tons. In this figure, the 

curves are plotted versus Sa(2s,5%), which is keyed to a period of T=2.0s, 

located in the middle of the range of periods for the relatively flexible set of 

multi-spring systems. 
4-20 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

Figure 4-22 Median IDA curves plotted versus the common intensity measure Sa(1s,5%) 

for systems Nx2a+1a and Nx3a+1a with a mass of 8.87 tons. 

Figure 4-23 Median IDA curves plotted versus the common intensity measure Sa(2s,5%) 

for systems Nx2a+1a and Nx3a+1a with a mass of 35.46 tons. 

In comparing non-normalized plots of IDA curves for various multi-spring 

combinations, the following observations were made: 

 Increases in the lateral strength of a system change the intensity that 

initiates yielding in the system as well as the intensity at collapse (lateral 

dynamic instability). The incremental change in collapse capacity, 

however, is less than proportional to the increase in yield strength. 

 The effectiveness of increasing the lateral strength of a system is a 

function of the shape of the force-displacement capacity boundary. 

Incremental changes in yield strength are more effective for ductile 

systems than they are for systems with less ductile behavior. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-21 

 Focused Analytical Studies 

 The effectiveness of increasing the lateral strength of a system is also a 

function of the period of system. Incremental changes in yield strength 

are more effective for stiff systems than they are for flexible systems. 

These effects can be observed by comparing the combined forcedisplacement capacity boundaries in Figure 4-18 with the resulting IDA 

curves in Figure 4-22 and Figure 4-23. Figure 4-22 shows that as “N” 

increases, the yield intensity increases significantly, however, increases in 

intensity at lateral dynamic instability are not as significant. For example, 

Figure 4-18 shows that the yield strength of system 9x3a+1a is 

approximately 6.5 times higher than the yield strength of system 3a+1a, but 

Figure 4-22 shows that the collapse capacity is only about two times higher. 

Comparing results between systems Nx2a+1a and Nx3a+1a in Figure 4-22 

shows that increases in collapse capacity that do occur as a result of changes 

in lateral strength are more pronounced for the more ductile Spring 3a than 

they are for the less ductile Spring 2a. For example, the increase in collapse 

capacity for system Nx3a+1a, as “N” increases from 1 to 9, is a factor of 

approximately 2.0. For system Nx2a+1a the corresponding increase in 

collapse capacity is a factor of approximately 1.25. 

Comparing results between Figure 4-22 and Figure 4-23 shows that as the 

period increases, the increment in collapse capacity caused by a change in 

lateral strength decreases. For example, the increase in collapse capacity 

shown in Figure 4-22 for the relatively stiff combinations of system Nx3a+1a 

is a factor of approximately 2.0. The increase in collapse capacity shown in 

Figure 4-23 for the relatively flexible combinations of system Nx3a+1a is a 

factor of approximately 1.3. 

4.5.5 Effects of Secondary System Characteristics 

The contribution of a secondary (“gravity”) system acting in parallel with a 

primary lateral-force-resisting system always results in an improvement in 

post-yield performance, especially close to collapse. This result was 

observed both qualitatively and quantitatively (i.e., both in normalized and 

non-normalized coordinates). 

The improvement is larger when considering secondary systems with larger 

ultimate deformation capacities. Figure 4-24 shows median IDA curves 

plotted versus the normalized intensity measure R = Sa(T,5%)/Say(T,5%) for 

multi-spring systems Nx2a+1a and Nx2a+1b with a mass of 8.87 tons. In the 

figure it can be seen that combinations with Spring 1b (with a larger ultimate 

deformation capacity) perform significantly better than combinations with 
4-22 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

Spring 1a. This result was observed in combinations with all lateral-forceresisting springs. 

Near collapse, secondary systems with larger deformation capacities have an 

even greater influence, even if the lateral strength is small compared to that 

of the primary system. This can be observed by comparing differences 

between systems 9x2a+1a and 9x2a+1b in Figure 4-24. Even though the 

relative contribution of Spring 1 in these combinations is small, the resulting 

collapse capacity is increased significantly. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2a+1a (1.53s)

2x2a+1a (1.18s)

3x2a+1a (1.00s)

5x2a+1a (0.80s)

9x2a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2a+1b (1.53s)

2x2a+1b (1.18s)

3x2a+1b (1.00s)

5x2a+1b (0.80s)

9x2a+1b (0.61s)

Figure 4-24 Median IDA curves plotted versus the normalized intensity measure 

Sa(T,5%)/Say(T,5%) for systems Nx2a+1a and Nx2a+1b with a mass of 8.87 tons. 

The contribution of the secondary system is more noticeable and significant 

in systems where the primary lateral resisting system is less ductile. Figure 

4-25 shows median IDA curves plotted versus the normalized intensity 

measure R = Sa(T,5%)/Say(T,5%) for multi-spring systems Nx2a+1a and 

Nx3a+1a with a mass of 8.87 tons. 

Comparing the systems in Figure 4-25 shows a much wider spread between 

the median IDA curves for system Nx2a+1a than the curves for system 

Nx3a+1a. This means that the behavior of Spring 2a is more heavily 

influenced by the combination with Spring 1a than Spring 3a. The reason for 

this can be explained by the relative contributions of each spring to the 

combined force-displacement capacity boundaries in Figure 4-17. 

Spring 2a, which represents a non-ductile moment frame system, has less 

favorable post-yield behavior in its force-displacement capacity boundary 

than does Spring 3a, which represents a ductile moment frame system. As 

such, Spring 2a is more favorably impacted by the characteristics of Spring 

1a, and combinations with Spring 1a result in greater changes in 

performance. However, as “N” increases from 1 to 9, system Nx2a+1a 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-23 

 Focused Analytical Studies 

becomes more like Spring 2a, and the positive influences of Spring 1a 

diminish. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2a+1a (1.53s)

2x2a+1a (1.18s)

3x2a+1a (1.00s)

5x2a+1a (0.80s)

9x2a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3a+1a (1.53s)

2x3a+1a (1.18s)

3x3a+1a (1.00s)

5x3a+1a (0.80s)

9x3a+1a (0.61s)

Figure 4-25 Median IDA curves plotted versus the normalized intensity measure 

Sa(T,5%)/Say(T,5%) for systems Nx2a+1a and Nx3a+1a with a mass of 8.87 tons. 

4.6 Comparison with FEMA 440 Limitations on 

Strength for Lateral Dynamic Instability 

In FEMA 440, a minimum strength requirement (maximum value of R) was 

developed as an approximate measure of the need to further investigate the 

potential for lateral dynamic instability caused by in-cycle strength 

degradation and P-delta effects. The recommended limitation is shown in 

Equation 4-1, with terms defined in Equation 4-2 and Equation 4-3, and 

illustrated in Figure 4-26: 

 4

Rmax

 





 

t

d e

y

 (4-1) 

where 

 t T  1 0.15 ln (4-2) 

and 

     eP P     2   (4-3) 

for 0 <  < 1.0. 

In-cycle strength degradation caused by P-delta is represented by P . The 

effects from all other sources of cyclic and in-cycle strength and stiffness 

degradation are represented by the term   2  P . At the time, it was 

apparent that modeling rules specified the use of hysteretic envelopes 

idealized from cyclic test results and would, consequently, overestimate 
4-24 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

actual in-cycle losses. For this reason, these effects were reduced by factor 

, which was less than 1.0. 

Figure 4-26 Idealized force-displacement curve for nonlinear static analysis 

(from FEMA 440). 

According to FEMA 440, the idealized force-displacement relationship 

(Figure 4-26) and the factor  were based on judgment, and significant 

variability should be expected in the value predicted using the equation for 

Rmax. As such, Rmax was intended only for identification of cases where 

further investigation using nonlinear response history analysis should be 

performed, and not as an accurate measure of the strength required to avoid 

lateral dynamic instability. 

To further investigate correlation between the FEMA 440 equation for Rmax

and lateral dynamic instability, the results of this equation were compared to 

quantile IDA curves for selected multi-spring systems included in this 

investigation. In making this comparison, parameters in the FEMA 440 

equation for Rmax were estimated from multi-spring force-displacement 

capacity boundaries idealized as shown in Figure 4-27. 

Results from this comparison indicate that values predicted by the FEMA 

440 equation for Rmax are variable, but generally plot between the median and 

84th percentile results for lateral dynamic instability of the systems 

investigated. The trends observed in this comparison indicate that an 

improved equation, in a form similar to Rmax, could be developed as a more 

accurate and reliable (less variable) predictor of lateral dynamic instability 

for use in current nonlinear static analysis procedures. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-25 

 Focused Analytical Studies 

0

1

2

3

4

5

6

7

0.00 0.02 0.04 0.06 0.08

θ

F/h

max 4

t

d e

y

R

 



 



t T  1 0.15 ln 

Approximate degrading stiffness

0

1

2

3

4

5

6

7

0.00 0.02 0.04 0.06 0.08

θ

F/h

max 4

t

d e

y

R

 



 



t T  1 0.15 ln 

Approximate degrading stiffness

Δy Δd

a e

0

1

2

3

4

5

6

7

0.00 0.02 0.04 0.06 0.08

θ

F/h

max 4

t

d e

y

R

 



 



t T  1 0.15 ln 

Approximate degrading stiffness

0

1

2

3

4

5

6

7

0.00 0.02 0.04 0.06 0.08

θ

F/h

max 4

t

d e

y

R

 



 



t T  1 0.15 ln 

Approximate degrading stiffness

Δy Δd

a e

Figure 4-27 Idealization of multi-spring force-displacement capacity 

boundaries to estimate effective negative stiffness for use in the 

FEMA 440 equation for Rmax. 

4.6.1 Improved Equation for Evaluating Lateral Dynamic 

Instability 

An improved estimate for the strength ratio at which lateral dynamic 

instability might occur was developed based on nonlinear regression of the 

extensive volume of data generated during this investigation. In performing 

this regression, results were calibrated to the median response of the SDOF 

spring systems studied in this investigation. 

A median-targeted strength ratio for lateral dynamic instability, Rdi, is 

defined as: 

3

3

a

c e ur r

di e

y cy

T F R b T

 F

     

    

    

 (4-4) 

where Te is the effective fundamental period of vibration of the structure, y, 

c, r, and u are displacements corresponding to the yield strength, Fy, 

capping strength, Fc, residual strength, Fr, and ultimate deformation capacity 

at the end of the residual strength plateau, as shown in Figure 4-28. 

Parameters a and b are functions given by: 

a  dTe )exp(1 (4-5) 

2

1 













 

c

r

F

F b (4-6) 

aeKe
4-26 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

where the parameter d is a constant equal to 4 for systems with stiffness 

degradation, and 5 for systems without stiffness degradation. The parameter 

 is the ratio of the post-capping slope (degrading stiffness) to the initial 

effective slope (elastic stiffness). 

y c r u

Fr

Fy

Fc

F

Ke

Ke

Ke

y c r u

Fr

Fy

Fc

F

Ke

Ke

Ke

Figure 4-28 Simplified force-displacement boundary for estimating the 

median collapse capacity associated with lateral dynamic 

instability. 

u r c y

R



1st term

2nd term

3rd term

u r c y

R



1st term

2nd term

3rd term

Figure 4-29 Relationship between Equation 4-4 and the segments of a 

typical IDA curve. 

The three terms in Equation 4-4 relate to the segments of a typical forcedisplacement capacity boundary (Figure 4-28) and typical IDA curve (Figure 

4-29). The first term provides an estimate of the median ground motion 

intensity corresponding to the end of the pseudo-linear segment of an IDA 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-27 

 Focused Analytical Studies 

curve (i.e., intensity at the onset of degradation). The second term provides 

an estimate of the increment in ground motion intensity required to push the 

structure onto the residual strength plateau. The third term provides an 

estimate of the increment in ground motion intensity required produce lateral 

dynamic instability (collapse). 

As developed, the equation for Rdi is intended to be a more reliable (less 

variable) predictor of median response at lateral dynamic instability. The 

resulting equation was compared to the FEMA 440 equation for Rmax and 

overlaid onto results for selected multi-spring systems. With few exceptions, 

Figure 4-30 through Figure 4-35 show that the equation for Rdi consistently 

predicts median response over a range of system types and periods of 

vibration. 

Figure 4-30 Comparison of Rdi with FEMA 440 Rmax and IDA results for 

system 2x2a+1a with T=1.18s. 
4-28 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

Figure 4-31 Comparison of Rdi with FEMA 440 Rmax and IDA results for 

system 3x3b+1b with T=1.0s. 

Figure 4-32 Comparison of Rdi with FEMA 440 Rmax and IDA results for 

system 9x3b+1b with T=0.61s. 
FEMA P440A 4: Results from Single-Degree-of-Freedom 4-29 

 Focused Analytical Studies 

Figure 4-33 Comparison of Rdi with FEMA 440 Rmax and IDA results for 

system 5x5a+1a with T=1.15s. 

Figure 4-34 Comparison of Rdi with FEMA 440 Rmax and IDA results for 

system 5x5a+1a with T=0.58s. 
4-30 4: Results from Single-Degree-of-Freedom FEMA P440A 

 Focused Analytical Studies 

Figure 4-35 Comparison of Rdi with FEMA 440 Rmax and IDA results for 

system 9x5a+1a with T=0.34s. 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-1 

Chapter 5 

Findings, Conclusions, and 

Recommendations 

This chapter summarizes the findings, conclusions, and recommendations 

resulting from the literature review and focused analytical studies of this 

investigation. Information from other chapters is collected and repeated here 

for ease of reference. In this chapter, findings have been grouped into the 

following categories: 

 Findings related to improved understanding of nonlinear degrading 

response and judgment in implementation of nonlinear analysis results in 

engineering practice. 

 Recommended improvements to current nonlinear 

analysis procedures 

 Suggestions for further study 

From the literature review, it is apparent that in-cycle strength and stiffness 

degradation are real phenomena that have been observed and documented to 

cause instability in individual components. Focused analytical studies have 

shown that larger assemblies of components of mixed hysteretic behavior 

experience similar negative stiffness that can lead to lateral dynamic 

instability. These studies have been able to link nonlinear dynamic response 

to major characteristics of component and system degrading behavior. 

These studies have also confirmed many of the conclusions regarding 

degradation and lateral dynamic instability presented in FEMA 440: (1) incycle strength degradation is a significant contributor to dynamic instability; 

(2) cyclic degradation can increase the potential for dynamic instability, but 

its effects are far less significant in comparison with in-cycle degradation; 

and (3) an equation, such as Rmax, could be used as an indicator of potential 

lateral dynamic instability for use in current nonlinear static analysis 

procedures. 
5-2 5: Findings, Conclusions, and Recommendations FEMA P440A 

5.1 Findings Related to Improved Understanding and 

Judgment 

This section summarizes observations and conclusions related to improved 

understanding of nonlinear degrading response and judgment in 

implementation of nonlinear analysis results in engineering practice. 

Findings, and practical ramifications for engineering practice, are 

summarized in the sections that follow. 

5.1.1 Sidesway Collapse versus Vertical Collapse 

Lateral dynamic instability is manifested in structural systems as sidesway 

collapse caused by loss of lateral-force-resisting capacity. Most sidesway 

collapse mechanisms can be explicitly simulated in nonlinear response 

history analyses. It should be noted, however, that inelastic deformation of 

structural components can result in shear and flexural-shear failures in 

members, and failures in joints and connections, which can lead to an 

inability to support vertical loads (vertical collapse) long before sidesway 

collapse can be reached. 

5.1.1.1 Practical Ramifications 

Behavior of real structures can include loss of vertical-load-carrying capacity 

at lateral displacements that are significantly smaller than those associated 

with sidesway collapse. Use of the findings of this investigation with regard 

to lateral dynamic instability (sidesway collapse) in engineering practice 

should include consideration of possible vertical collapse modes that could 

be present in the structure under consideration. 

5.1.2 Relationship between Loading Protocol, Cyclic Envelope, 

and Force-Displacement Capacity Boundary 

Historically, the term “backbone curve” has referred to many different 

things. For this reason, two new terms have been introduced to distinguish 

between different aspects of hysteretic behavior. These are the forcedisplacement capacity boundary, and cyclic envelope. 

5.1.2.1 Force-Displacement Capacity Boundary 

A force-displacement capacity boundary defines the maximum strength that a 

structural member can develop at a given level of deformation, resulting in 

an effective “boundary” for the strength of a member in force-deformation 

space (Figure 5-1). In many cases, the force-displacement capacity boundary 

corresponds to the monotonic force-deformation curve. 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-3 

A cyclic load path cannot cross a force-displacement capacity boundary. If a 

member is subjected to increasing deformation and the boundary is reached, 

then the strength that can be developed in the member is limited and the 

response must continue along the boundary (in-cycle strength degradation). 

Only displacement excursions intersecting portions of the capacity boundary 

with a negative slope will result in in-cycle strength degradation. 

Figure 5-1 Example of a force-displacement capacity boundary. 

5.1.2.2 Cyclic Envelope 

A cyclic envelope is a force-deformation curve that envelopes the hysteretic 

behavior of a component or assembly that is subjected to cyclic loading 

(Figure 5-2). 

Figure 5-2 Example of a cyclic envelope. 
5-4 5: Findings, Conclusions, and Recommendations FEMA P440A 

The characteristics of the cyclic envelope are strongly influenced by the 

points at which unloading occurs in a test, and are therefore strongly 

influenced by the loading protocol that was used in the experimental 

program. Nominally identical specimens loaded with different loading 

protocols will have different cyclic envelopes depending on the number of 

cycles used in the loading protocol, the amplitude of each cycle, and the 

sequence of the loading cycles, as illustrated in Section 2.2.3. 

Under lateral deformations that are less than or equal to those used to 

generate the cyclic envelope, differences between the cyclic envelope and the 

force-displacement capacity boundary are of no consequence. However, 

under larger lateral displacements these differences will affect the potential 

for in-cycle degradation to occur, and will significantly affect system 

behavior and response (Figure 5-3). 

 (a) (b) 

Figure 5-3 Comparison of hysteretic behavior when the force-displacement capacity 

boundary is: (a) equal to the cyclic envelope, and (b) extends beyond the cyclic 

envelope. 

Constraining nonlinear hysteretic behavior within the limits of a cyclic 

envelope that does not capture the full range of permissible forcedeformation response, as defined by the force-displacement capacity 

boundary, will result in overly pessimistic predictions of the nonlinear 

dynamic response of a system. 

5.1.2.3 Practical Ramifications 

Nonlinear component parameters should be based on the force-displacement 

capacity boundary, which is different from a cyclic envelope. Determining 

the force-displacement capacity boundary from test results using a single 

cyclic loading protocol can result in significant underestimation of the actual 

capacity for force-deformation response and subsequent overestimation of 

nonlinear displacement demands. 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-5 

5.1.3 Characteristics of Median IDA Curves 

Observed relationships between IDA curves and degrading component 

characteristics suggest that dynamic response is directly influenced by the 

features of a force-displacement capacity boundary. This relationship, which 

is dependent upon the period of vibration of the system, is depicted in the 

idealized graphical representation of Figure 5-4. 

Figure 5-4 Relationship between IDA curves and the features of a typical 

force-displacement capacity boundary. 

In general, median IDA curves were observed to exhibit the following 

characteristics: 

 An initial linear segment corresponding to linear-elastic behavior in 

which in lateral deformation demand is proportional to ground motion 

intensity, regardless of the characteristics of the system or the ground 

motion. This segment extends from the origin to the onset of yielding. 

 A second curvilinear segment corresponding to inelastic behavior in 

which lateral deformation demand is no longer proportional to ground 

motion intensity. As intensity increases, lateral deformation demands 

increase at a faster rate. This segment corresponds to softening of the 

system, or reduction in stiffness (reduction in the slope of the IDA 

curve). In this segment, the system “transitions” from linear behavior to 

eventual dynamic instability. Although a curvilinear segment is always 

present, in some cases the transition can be relatively long and gradual, 

while in other cases it can be very short and abrupt. 

INSTABILITY

LINEAR

PSEUDO 

LINEAR

SOFTENING



boundar

16%

84%

50% INSTABILITY

LINEAR



CAPACITY

BOUNDARY

SaT (g)
5-6 5: Findings, Conclusions, and Recommendations FEMA P440A 

 A final linear segment that is horizontal, or nearly horizontal, in which 

infinitely large lateral deformation demands occur at small increments in 

ground motion intensity. This segment corresponds to the point at which 

a system becomes unstable (lateral dynamic instability). For SDOF 

systems, this point corresponds to the ultimate deformation capacity at 

which the system loses all lateral-force-resisting capacity. 

In some systems, the initial linear segment can be extended beyond yield into 

the inelastic range. In this pseudo-linear segment, lateral deformation 

demand is approximately proportional to ground motion intensity, which is 

consistent with the familiar equal-displacement approximation for estimating 

inelastic displacements. The range of lateral deformation demands over 

which the equal-displacement approximation is applicable depends on the 

characteristics of the force-displacement capacity boundary of the system and 

the period of vibration. 

5.1.3.1 Practical Ramifications 

The observed relationships support the conclusion that it is possible to 

estimate nonlinear dynamic response based on knowledge of the 

characteristics of the force-displacement capacity boundary. 

5.1.4 Dependence on Period of Vibration 

In general, moderate and long period systems with zero or positive post-yield 

stiffness in the force-displacement capacity boundary follow the equal 

displacement trend well into the nonlinear range, as shown for Spring 3a in 

Figure 5-5. For systems with periods longer than 0.5s, Spring 3a exhibits an 

extension of the initial linear segment well beyond the yield drift of 0.01. In 

contrast, the short period system (T=0.2s) diverges from the initial linear 

segment just after yielding, even at deformations within the strengthhardening segment of the force-displacement capacity boundary (drifts 

between 0.01 and 0.04 in the figure). 

5.1.4.1 Practical Ramifications 

It is important to consider the dependence on period of vibration in 

conjunction with the effects of other parameters identified in this 

investigation. The generalized effect of any one single parameter can be 

misleading. 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-7 

Figure 5-5 Force-displacement capacity boundary and median IDA curves for Spring 3a with various periods 

of vibration.

5.1.5 Dispersion in Response 

Nonlinear response is sensitive to the characteristics of the ground motion 

record, and will vary from one ground motion to the next, even when scaled 

to the same intensity (Figure 5-6). For a given level of ground motion 

intensity, the lateral deformation demand can be significantly smaller or 

significantly larger than the value shown on median IDA curves, as indicated 

by the 16th and 84th percentile curves in the figure. As the level of ground 

motion intensity increases, the dispersion in response tends to increase. 

Figure 5-6 Force-displacement capacity boundary and 16th, 50th and 84th percentile IDA 

curves for Spring 3b with a period of vibration T=2.0s. 

5.1.5.1 Practical Ramifications 

It is important to recognize the level of uncertainty that is inherent in 

nonlinear analysis, particularly regarding variability in response due to 
5-8 5: Findings, Conclusions, and Recommendations FEMA P440A 

ground motion uncertainty. It may not be sufficient to rely on median (50%) 

estimates of response for certain design or evaluation quantities of interest, 

unless the intensity of the ground motion is associated with an appropriately 

rare probability of exceedance. 

5.1.6 Influence of the Force-Displacement Capacity Boundary 

Key features of a force-displacement capacity boundary that were observed 

to influence the shape of median IDA curves included post-yield behavior 

and onset of degradation, slope of degradation, ultimate deformation 

capacity, and presence of cyclic degradation. Systems in which the forcedisplacement capacity boundary had more favorable post-yield 

characteristics (e.g., delayed onset of degradation, more gradual slope of 

degradation, higher residual strength, and higher ultimate deformation 

capacity) were observed to perform better. 

5.1.6.1 Post-Yield Behavior and Onset of Degradation 

The presence of a non-negative post-yield slope and delay before the onset of 

degradation reduced potential in-cycle strength degradation and significantly 

improved the collapse capacity of a system (Figure 5-7). 

Figure 5-7 Effect of post-yield behavior on the collapse capacity of a system (Springs 2a, 3a 

and 6a with T=2.0s). 

5.1.6.2 Slope of Degradation 

Differences in the negative slope of the strength-degrading segment 

significantly affected the collapse capacity of a system. Systems with more 

shallow degrading slopes reached higher collapse capacities than systems 

with steeper degrading slopes (Figure 5-8). Changes in negative slope 

changed the magnitude of potential in-cycle strength degradation, and 

overshadowed any changes in other parameters (e.g., the residual strength 

plateau), as long as the ultimate deformation capacity remained the same. 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-9 

Figure 5-8 Effect of slope of degradation on the collapse capacity of a system (Springs 2a and 2b 

with T=1.0s). 

5.1.6.3 Ultimate Deformation Capacity 

Increasing the ultimate deformation capacity resulted in significant increases 

in collapse capacity (Figure 5-9). The key parameter related to the observed 

change in response is the increment in the ultimate deformation capacity. 

Observed changes in collapse capacity resulting from increases in the 

ultimate deformation capacity were insensitive to other characteristics of the 

post-yield behavior of the springs. 

Figure 5-9 Effect of ultimate deformation capacity on the collapse capacity of a system (Springs 

1a and 1b with T=1.0s). 

5.1.6.4 Practical Ramifications 

Observed relationships between selected features of the force-displacement 

capacity boundary and the resulting characteristics of median IDA curves 

support the conclusion that the nonlinear dynamic response of a system can 

be correlated to the parameters of the force-displacement capacity boundary 

of that system. Of particular interest is the relationship between global 
5-10 5: Findings, Conclusions, and Recommendations FEMA P440A 

deformation demand and the intensity of the ground motion at lateral 

dynamic instability (collapse). Results indicate that it is possible to use 

nonlinear static procedures to estimate the potential for lateral dynamic 

instability of systems exhibiting in-cycle degradation. 

5.1.7 Cyclic Degradation of the Force-Displacement Capacity 

Boundary 

In general, most components will exhibit some level of cyclic degradation. 

Consistent with observations from past studies, comparison of results 

between springs both with and without cyclic degradation show that the 

effects of cyclic degradation (as measured by gradual movement of the 

capacity boundary) are relatively unimportant in comparison with in-cycle 

degradation (as measured by the extent and steepness of negative slopes in 

the capacity boundary). This trend is illustrated for Spring 3b in Figure 5-10, 

but can be observed in the results for many spring systems in Appendix B. 

Figure 5-10 Effect of degradation of the force-displacement capacity boundary on the collapse 

capacity of a system (Spring 3b, T=2.0s, with and without cyclic degradation). 

5.1.7.1 Practical Ramifications 

In most cases the effects of in-cycle strength degradation dominate the 

nonlinear dynamic behavior of a system. This suggests that in many cases 

the effects of cyclic degradation can be neglected. Instead, the focus should 

be on more accurately characterizing the force-displacement capacity 

boundary, which controls the onset of in-cycle degradation (where it occurs). 

Two situations in which the effects of cyclic degradation were observed to be 

important include: (1) short period systems; and (2) systems with very strong 

in-cycle strength degradation effects (very steep and very large drops in 

lateral strength). In these cases, the effects of cyclic degradation can be 

important and should be considered. 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-11 

5.1.8 Effects of Secondary System Characteristics 

The contribution of a secondary (“gravity”) system acting in parallel with a 

primary lateral-force-resisting system always resulted in an improvement in 

nonlinear response, especially close to collapse. This result was observed 

both qualitatively and quantitatively (i.e., both in normalized and nonnormalized coordinates). 

The improvement was larger when considering secondary systems with 

larger ultimate deformation capacities, even if the lateral strength of the 

secondary system was small in comparison to that of the primary system. 

This result is illustrated in Figure 5-11, and is supported by results described 

in Section 5.1.6.3. In the figure, as the system combination ratio increases, 

the relative combination of the secondary system diminishes, yet the 

resulting collapse capacities for combinations with Spring 1b (larger ultimate 

deformation capacity) are significantly higher than combinations with Spring 

1a (smaller ultimate deformation capacity). 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2a+1a (1.53s)

2x2a+1a (1.18s)

3x2a+1a (1.00s)

5x2a+1a (0.80s)

9x2a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2a+1b (1.53s)

2x2a+1b (1.18s)

3x2a+1b (1.00s)

5x2a+1b (0.80s)

9x2a+1b (0.61s)

Figure 5-11 Median IDA curves plotted versus the normalized intensity measure Sa(T,5%)/Say(T,5%) for 

systems Nx2a+1a and Nx2a+1b with a mass of 8.87 tons. 

5.1.8.1 Practical Ramifications 

Consideration of the contribution of secondary (“gravity”) systems acting in 

parallel with primary lateral resisting systems is important and should be 

included in nonlinear modeling for collapse simulation. For seismic retrofit 

of existing structures, this suggests that adding a relatively weak (but ductile) 

system in parallel with the primary system could substantially increase 

collapse capacity and delay the onset of lateral dynamic instability. The 

introduction of such a secondary system could be significantly less 

complicated and less expensive than direct improvements to the strength, 

stiffness and deformation capacity of the primary system. 
5-12 5: Findings, Conclusions, and Recommendations FEMA P440A 

5.1.9 Effects of Lateral Strength 

Increasing the lateral strength of a system was observed to increase collapse 

capacity, with the following limitations: 

 Increases in the lateral strength of a system changed the intensity that 

initiated yielding in the system and the intensity at collapse (lateral 

dynamic instability). The incremental change in collapse capacity, 

however, was less than proportional to the increase in yield strength 

(Figure 5-12). 

 

Figure 5-12 Force-displacement capacity boundaries and median IDA curves plotted versus the common 

intensity measure Sa(2s,5%) for system Nx3a+1a with a mass of 35.46 tons. 

 The effectiveness of increasing the lateral strength of a system was a 

function of the shape of the force-displacement capacity boundary. 

Incremental changes in yield strength were more effective for ductile 

systems than they were for systems with less ductile behavior. 

 The effectiveness of increasing the lateral strength of a system was also a 

function of the period of system. Incremental changes in yield strength 

were more effective for stiff systems than they were for flexible systems. 

5.1.9.1 Practical Ramifications 

Increasing the lateral strength of a system can improve collapse behavior, but 

will not result in equal increases in collapse capacity. The effectiveness of 

seismic retrofit strategies that involve increasing the lateral strength will 

depend on the characteristics of the force-displacement capacity boundary of 

the existing system as well as the period of vibration. 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-13 

5.2 Recommended Improvements to Current 

Nonlinear Analysis Procedures 

Prevailing practice for performance-based seismic design is based on the 

FEMA 273 NEHRP Guidelines for the Seismic Rehabilitation of Buildings

(FEMA, 1997) and its successor documents, FEMA 356 Prestandard and 

Commentary for the Seismic Rehabilitation of Buildings (FEMA, 2000), and 

ASCE/SEI Standard 41-06 Seismic Rehabilitation of Existing Buildings

(ASCE, 2006b). Recommendations contained in FEMA 440 Improvement of 

Nonlinear Static Seismic Analysis Procedures (FEMA, 2005) were 

incorporated into the developing ASCE/SEI Standard 41-06 in 2005. 

ASCE/SEI Standard 41-06 Supplement No. 1 was published in 2007. 

Together these resource documents form the basis of nonlinear analysis in 

current engineering practice. This section summarizes recommended 

clarifications and improvements to current nonlinear analysis procedures as 

characterized in these documents. 

5.2.1 Current Nonlinear Static Procedures 

The Coefficient Method is one method of estimating maximum inelastic 

displacements of a system. The process begins with the generation of an 

idealized force-deformation curve (i.e., static pushover curve) relating base 

shear to roof displacement. From this curve, an effective period, Te, is 

obtained, and the maximum global displacement (target displacement) for a 

specified level of ground motion intensity is estimated using Equation 5-1: 

2

012 2 4

e

t a

T  CCC S g

  (5-1) 

In this expression the first three terms are coefficients that modify the elastic 

displacement of the system. C0 is the first mode participation factor. This 

coefficient essentially converts from spectral ordinates to roof displacement. 

This C1 coefficient (Equation 5-2) increases elastic displacements in short 

period systems, essentially accounting for exceptions to the equal 

displacement approximation. 

1 2

1 1 eR C

aT

   (5-2) 

where: 

/

a

m

y

S R C

V W

  (5-3) 
5-14 5: Findings, Conclusions, and Recommendations FEMA P440A 

and Cm is the effective mass factor to account for higher mode mass 

participation effects. The C2 coefficient (Equation 5-4) increases elastic 

displacements in short period and weak systems to account for stiffness 

degradation, hysteretic pinching, and cyclic strength degradation. 

2

2

1 1 1

800 e

R C

T

          

 (5-4) 

Importantly, C2 does not account for displacement amplification due to incycle strength degradation, which can result in lateral dynamic instability. 

In-cycle strength degradation is addressed by a minimum strength 

requirement (maximum value of R) used as a trigger for the need to further 

investigate the potential for lateral dynamic instability using nonlinear 

response history analysis. The minimum strength requirement in current 

nonlinear analysis procedures was described in Section 4.6 (and is repeated 

in the equations that follow): 

 4

Rmax

 





 

t

d e

y

 (5-5) 

where 

 1 0.15 ln t T   (5-6) 

and 

     eP P      2   (5-7) 

for 0 <  < 1.0. 

Values of R (Equation 5-3) are compared to Rmax. Systems in which R < Rmax

are deemed to meet the minimum strength requirement to avoid lateral 

dynamic instability, and nonlinear response history analysis is not required. 

5.2.2 Clarification of Terminology and Use of the ForceDisplacement Capacity Boundary for Component 

Modeling 

For nonlinear analysis, ASCE/SEI 41-06 specifies component modeling and 

acceptability criteria based on the conceptual force-displacement relationship 

(“backbone”) depicted in Figure 5-13. Since the term “backbone curve” has 

been used to refer to many different things, its definition related to nonlinear 

component modeling is not clear. In Section 2.8 of the standard, it is 

permitted to derive modeling parameters and acceptance criteria using 

experimentally obtained cyclic response characteristics from subassembly 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-15 

testing. So defined, the standard can be interpreted to condone the use of 

cyclic envelopes from component tests to generate the necessary forcedisplacement relationships. 

Figure 5-13 Conceptual force-displacement relationship (“backbone”) used 

in ASCE/SEI 41-06 (adapted from FEMA 356). 

The use of a cyclic envelope, as opposed to a force-displacement capacity 

boundary, has been shown to result in underestimation of the actual capacity 

for force-deformation response and subsequent overestimation of nonlinear 

deformation demands. In some cases the resulting conservatism can be very 

large. 

For this reason, introduction and use of two new terms are recommended to 

distinguish between different aspects of hysteretic behavior. These are the 

force-displacement capacity boundary, and cyclic envelope, defined in 

Section 5.1.2. Important conceptual differences between the forcedisplacement capacity boundary and a loading protocol-specific cyclic 

envelope should be clarified in future revisions to ASCE/SEI 41, and the use 

of an appropriate force-displacement capacity boundary should be specified 

for characterizing component hysteretic behavior. 

Proper definition of the hysteretic behavior in a component model requires an 

understanding of: (1) the initial force-displacement capacity boundary; and 

(2) how the force-displacement capacity boundary degrades under cyclic 

loading. The ideal method for establishing an initial force-displacement 

capacity boundary is through monotonic testing. Once the initial forcedisplacement capacity boundary is defined, degradation parameters should be 

established based on results from cyclic tests. 

There is no recognized testing protocol that incorporates realistic 

consideration of the force-displacement capacity boundary. The use of 

several cyclic loading protocols is desirable to ensure that the degradation 

parameters are properly identified and the calibrated component model is 

general enough to represent response under any type of loading. 
5-16 5: Findings, Conclusions, and Recommendations FEMA P440A 

In Commentary Section C6.3.1.2.2 of ASCE/SEI 41-06 Supplement No. 1, it 

is suggested that the sudden drop from Point C to Point D (in Figure 5-10) 

can be overly pessimistic, and that a more gradual slope from Point C to 

Point E might be more realistic for concrete components. Some 

experimental results suggest that such an adjustment could be applicable for 

other types of components. If the actual monotonic curve is not available, or 

cannot be estimated, use of a force-displacement capacity boundary with this 

alternate slope can be considered. 

5.2.3 Improved Equation for Evaluating Lateral Dynamic 

Instability 

In comparison with results for selected multi-spring systems in this 

investigation, the FEMA 440 equation for Rmax was shown to predict values 

that are variable, but generally fall between the median and 84th percentile 

results for lateral dynamic instability. This result suggests that the current 

equation for Rmax would be conservative if used in conjunction with a 

capacity boundary generated from a pushover analysis. It could be very 

conservative if the pushover curve was based on component modeling 

parameters determined using a cyclic envelope rather than a forcedisplacement capacity boundary. 

The trends observed in this comparison indicate that an improved equation, 

in a form similar to Rmax, could be developed as a more accurate and reliable 

(less variable) predictor of lateral dynamic instability for use in current 

nonlinear static analysis procedures. An improved estimate for the strength 

ratio at which lateral dynamic instability might occur was developed based 

on nonlinear regression of the extensive volume of data generated during this 

investigation. In performing this regression, results were calibrated to the 

median response of the SDOF spring systems studied in this investigation. 

A median-targeted minimum strength requirement (maximum value of R) for 

lateral dynamic instability, Rdi, is proposed in Equation 5-8: 

3

3

a

c e ur r

di e

y cy

T F R b T

 F

     

    

    

 (5-8) 

where Te is the effective fundamental period of vibration of the structure, y, 

c, r, and u are displacements corresponding to the yield strength, Fy, 

capping strength, Fc, residual strength, Fr, and ultimate deformation capacity 

at the end of the residual strength plateau, as shown in Figure 5-14. 

Parameters a and b are functions given by: 

1 exp( ) a dT   e (5-9) 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-17 

2

1 r

c

F b

F

       

 (5-10) 

The parameter d is a constant equal to 4 for systems with stiffness 

degradation, and 5 for systems without stiffness degradation. The parameter 

 is the ratio of the post-capping slope (degrading stiffness) to the initial 

effective slope (elastic stiffness). 

y c r u

Fr

Fy

Fc

F

Ke

Ke

Ke

y c r u

Fr

Fy

Fc

F

Ke

Ke

Ke

Figure 5-14 Simplified force-displacement boundary for estimating the 

median collapse capacity associated with dynamic instability. 

5.2.3.1 Practical Ramifications 

Since Rdi has been calibrated to median response, use of this equation could 

eliminate some of the conservatism built into the current Rmax limitation on 

use of nonlinear static procedures. Calibrated using the extensive volume of 

data generated during this investigation, use of this equation could improve 

the reliability of current nonlinear static procedures with regard to cyclic and 

in-cycle degradation. 

In conjunction with a pushover curve used as a system force-displacement 

capacity boundary, the equation for Rdi could be used to determine if a 

system is susceptible to lateral dynamic instability for a specified level of 

spectral acceleration, SaT. Similar to Rmax, use of Rdi would involve 

comparison with R (Equation 5-3). If R < Rdi the system could be deemed 

satisfactory without additional nonlinear dynamic analysis. This capability 

is, of course, limited to systems for which the assumption of SDOF behavior 

is appropriate (i.e., MDOF effects are not significant). 

Calculated values of Rdi should be viewed carefully with respect to the 

intensity measure (SaT) considered. Collapse limit states (i.e., lateral dynamic 
5-18 5: Findings, Conclusions, and Recommendations FEMA P440A 

instability) should be evaluated for intensities associated with rare ground 

motions (long return periods). Evaluation of collapse limit states at lower 

ground motion intensities leaves open the possibility that collapse could 

occur during events in which those intensities are exceeded. 

In addition, the development of the proposed equation for Rdi targeted 

median response, which was intentionally less conservative than the level at 

which the FEMA 440 equation for Rmax appeared to be predicting. Median 

response implies a fifty percent chance of being above or below the specified 

value. Use of Rdi in engineering practice should consider whether or not a 

median predictor represents an appropriate level of safety against the 

potential for lateral dynamic instability. If needed, a reduction factor could 

be applied to Equation 5-8 to reduce the resulting values of Rdi and ultimately 

achieve a correspondingly higher level of safety. 

5.2.4 Simplified Nonlinear Dynamic Analysis Procedure 

From empirical relationships for characteristic segments of IDA curves for 

many systems, Vamvatsikos and Cornell (2006) suggested that static 

pushover curves could be used to estimate nonlinear dynamic response. The 

open source software tool, Static Pushover 2 Incremental Dynamic Analysis

(SPO2IDA), was created as a product of that research, and can be obtained at 

http://www.ucy.ac.cy/~divamva/software.html. A Microsoft Excel version 

of the SPO2IDA application has also been provided on the CD 

accompanying this report. 

As the name suggests, SPO2IDA transforms static pushover (SPO) curves to 

incremental dynamic analysis (IDA) plots. It utilizes a large database of IDA 

results to fit representative 16th, 50th, and 84th percentile IDA curves to a 

given idealized single-degree-of-freedom (SDOF) oscillator subjected to a 

static pushover analysis. The relationships between force-displacement 

capacity boundaries and IDA curves observed in this investigation are 

consistent with this notion. 

Focused analytical studies comparing force-displacement capacity 

boundaries to incremental dynamic analysis results led to the concept of a 

simplified nonlinear dynamic analysis procedure. In this procedure, a 

nonlinear static analysis is used to generate an idealized force-deformation 

curve (i.e., static pushover curve). The resulting curve is then used as a 

force-displacement capacity boundary to constrain the hysteretic behavior of 

an equivalent SDOF oscillator. This SDOF oscillator is then subjected to 

incremental dynamic analysis (or approximate IDA results are obtained using 

SPO2IDA). 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-19 

The concept of a simplified nonlinear dynamic analysis procedure is 

described in the steps outlined below. 

 Develop an analytical model of the system. 

Models can be developed in accordance with prevailing practice for 

seismic evaluation, design, and rehabilitation of buildings described in 

ASCE/SEI 41-06. Component properties should be based on forcedisplacement capacity boundaries, rather than cyclic envelopes. 

 Perform a nonlinear static pushover analysis. 

Subject the model to a conventional pushover analysis in accordance 

with prevailing practice. Lateral load increments and resulting 

displacements are recorded to generate an idealized force-deformation 

curve. 

 Conduct an incremental dynamic analysis of the system based on an 

equivalent SDOF model. 

The idealized force-deformation curve is, in effect, a system forcedisplacement capacity boundary that can be used to constrain a hysteretic 

model of an equivalent SDOF oscillator. This SDOF oscillator is then 

subjected to incremental dynamic analysis to check for lateral dynamic 

instability and other limit states of interest. Alternatively, an 

approximate incremental dynamic analysis can be accomplished using 

the idealized force-deformation curve and SPO2IDA. 

 Determine probabilities associated with limit states of interest. 

Results from incremental dynamic analysis can be used to obtain 

response statistics associated with limit states of interest in addition to 

lateral dynamic instability. SPO2IDA can also be used to obtain median, 

16th, and 84th percentile IDA curves relating displacements to intensity. 

Using the fragility relationships described in Appendix E in conjunction 

with a site hazard curve, this information can be converted into annual 

probabilities of exceedance for each limit state. Probabilistic information 

in this form can be used to make enhanced decisions based on risk and 

uncertainty, rather than on discrete threshold values of acceptance. 

The procedure is simplified because only a SDOF oscillator is subjected to 

nonlinear dynamic analysis. Further simplification is achieved through the 

use of SPO2IDA, which avoids the computational effort associated with 

incremental dynamic analysis. This simplified procedure has several 

advantages over nonlinear static analysis procedures: (1) lateral dynamic 

instability is investigated explicitly; (2) results include the effects of recordto-record variability in ground motion; (3) response can be characterized 
5-20 5: Findings, Conclusions, and Recommendations FEMA P440A 

probabilistically; and (4) uncertainty can be considered explicitly. Results 

can be investigated for any limit state that can be linked to the demand 

parameter of interest (e.g. roof displacement). 

Use of the procedure is explained in more detail in the example application 

contained in Appendix F. 

5.3 Suggestions for Further Study 

This section summarizes suggestions for further study that will expand the 

application of results to more complex systems, fill in gaps in existing 

knowledge, and enhance future practice. 

5.3.1 Application of Results to Multiple-Degree-of-Freedom 

Systems 

Multi-story buildings are more complex dynamic systems whose seismic 

response is more difficult to estimate than that of SDOF systems. Recent 

studies have suggested that it may be possible to estimate the collapse 

capacity of multiple-degree-of-freedom (MDOF) systems through dynamic 

analysis of equivalent SDOF systems. 

As part of the focused analytical work, preliminary studies of MDOF 

systems were performed as summarized in Appendix G. These studies 

investigated the use of nonlinear static analyses combined with incremental 

dynamic analyses of equivalent SDOF systems to evaluate dynamic 

instability of multi-story buildings ranging in height from 4 to 20 stories. 

Preliminary results indicate that many of the findings for SDOF systems in 

this investigation (e.g., the relationship between force-displacement capacity 

boundary and IDA curves; the equation for Rdi) may be applicable to MDOF 

systems. More detailed study of the application of these results to MDOF 

systems is recommended as a result of this investigation, and additional 

MDOF investigations are planned under a project funded by the National 

Institute of Standards and Technology (NIST). 

5.3.2 Development of Physical Testing Protocols for 

Determination of Force-Displacement Capacity 

Boundaries 

Important conceptual differences exist between force displacement capacity 

boundaries and loading protocol-specific cyclic envelopes. Proper definition 

of hysteretic behavior in a component model requires an understanding of the 

initial force-displacement capacity boundary and how that boundary 

degrades under cyclic loading. The use of several loading protocols is 

desirable, but there is no recognized testing procedure that accomplishes this. 
FEMA P440A 5: Findings, Conclusions, and Recommendations 5-21 

The loading protocol for experimental investigations described in Section 2.8 

of ASCE/SEI 41-06 is not specific enough to produce a true forcedisplacement capacity boundary. For a set of identical specimens, necessary 

testing could conceivably include: (1) monotonic loading to get an initial 

capacity boundary; (2) multiple symmetric cyclic loading cases to calibrate 

cyclic degradation; (3) high frequency or long duration cyclic loading cases 

to check for fracture or fatigue; (4) cyclic loading followed by a monotonic 

push to more clearly observe changes due to cyclic degradation; and (5) 

unsymmetrical cyclic loading. Development of a specification for physical 

testing protocols necessary to generate appropriate force-displacement 

capacity boundaries is recommended. 

5.3.3 Development and Refinement of Tools for Approximate 

Nonlinear Dynamic Analysis 

Nonlinear dynamic analysis has obvious advantages over nonlinear static 

analysis procedures. Disadvantages are related to increased computational 

effort. Studies have shown that the characteristics of nonlinear dynamic 

response can be estimated through simplified approximate relationships 

based on the results of static pushover analyses. 

Software tools such as SPO2IDA have the capability to estimate dynamic 

response without the computational effort associated with incremental 

dynamic analysis. This approximation facilitates the use of dynamic analysis 

results to supplement and inform more simplified analysis procedures (e.g., 

nonlinear static procedures). Development and refinement of similar 

approximate tools for performing nonlinear dynamic analyses is 

recommended. 

5.4 Concluding Remarks 

Using FEMA 440 as a starting point, this investigation has advanced the 

understanding of degradation and lateral dynamic instability by: 

 Investigating and documenting currently available empirical and 

theoretical knowledge on nonlinear cyclic and in-cycle strength and 

stiffness degradation, and their affects on the stability of structural 

systems 

 Supplementing and refining the existing knowledge base with focused 

analytical studies 

This investigation has resulted in an extensive collection of available 

research on component modeling of degradation, and a database of analytical 

results from over 2.6 million nonlinear response history analyses 
5-22 5: Findings, Conclusions, and Recommendations FEMA P440A 

documenting the effects of a variety of parameters on the overall response of 

SDOF systems with degrading components. 

Results have confirmed conclusions regarding degradation and dynamic 

instability presented in FEMA 440, provided updated information on 

modeling to differentiate between cyclic and in-cycle strength and stiffness 

degradation, and linked nonlinear dynamic response to major characteristics 

of component and system degrading behavior. This information has resulted 

in: 

 an improved understanding of nonlinear degrading response and the 

practical ramifications of this information for engineering practice 

 recommendations to better account for nonlinear degrading response in 

the context of current nonlinear analysis procedures 

 suggestions for further study 

Results from this investigation will ultimately improve the modeling of 

structural components considering cyclic and in-cycle degrading behavior, 

improve the characterization of lateral dynamic instability, and reduce the 

conservatism in current analysis procedures making it more cost-effective to 

strengthen existing buildings for improved seismic resistance in the future. 
FEMA P440A A: Detailed Summary of Previous Research A-1 

Appendix A 

Detailed Summary of 

Previous Research 

This appendix contains a detailed summary of the development of hysteretic 

models for nonlinear analysis. It also contains summaries of publications that 

were extensively reviewed for guidance on appropriately targeting and 

scoping focused analytical studies. 

A.1 Summary of the Development of Hysteretic Models 

A.1.1 Non-Deteriorating Models 

Early studies that incorporated nonlinear behavior in seismic response of 

structures assumed the structure to have an elastoplastic hysteretic behavior 

or a bilinear hysteretic behavior (e.g., Berg and Da Deppo, 1960; Penzien 

1960a, 1960b; Iwan 1961). These might be perfectly plastic with no postelastic stiffness or with some strain hardening. More accurate models were 

also developed with smooth rounded transitions from elastic to plastic 

regions. (Ramberg and Osgood, 1943; Pinto and Guiffre, 1970; Menegotto 

and Pinto, 1973). Ramberg-Osgood and Giuffre-Menegotto-Pinto models 

continue to be used today for modeling non-degrading structures such as 

steel moment-frame structures when fracture and buckling do not occur, and 

have recently been used successfully to model the hysteretic behavior of 

buckling-restrained braces (e.g., Lin et al., 2004). 

Other examples of non-degrading smooth hysteretic models commonly used 

are the Bouc-Wen model (Bouc, 1967a, 1967b; Wen, 1976, 1989) and the 

Ozdemir model (1976). Unlike the Ramberg-Osgood and GiuffreMenegotto-Pinto models in which the force-deformation relationship is 

described by an algebraic equation, in the Bouc-Wen and Ozdemir models 

the force-displacement relationship and the force-deformation characteristics 

are described by a differential equation. These models are relatively easy to 

implement and are capable of describing, relatively well, non-degrading 

hysteretic behavior. An extension of smooth models to a three-dimensional 

tensorial idealization of Prager's model was developed by Casciati and 

Faravelli (1985, Casciati, 1989). Although models based on differential 
A-2 A: Detailed Summary of Previous Research FEMA P440A 

equations with smooth loading curves, such as the Bouc-Wen or Ozdemir 

models, are relatively easy to implement, they generally exhibit a local 

violation of Drucker’s stability postulate. In particular, Thyagarajan and 

Iwan (1990) concluded that the Wen-Bouc model tends to exhibit a 

pronounced drift, particularly when post-yield stiffness is small. 

A.1.2 Piecewise Linear Deteriorating Models 

Many structural materials and structural elements will exhibit some level of 

degradation of stiffness or strength or both, or may also exhibit other 

phenomena such as pinching, when subjected to cyclic reversed loading, this 

is especially true for reinforced concrete elements subjected to several large 

cyclic reversals. Deterioration can be the result of, for example, cracking, 

crushing, rebar buckling, crack or gap opening and closing, loss of bond, 

and interaction with high shear or axial stresses. The level of degradation 

depends, on the one hand, on the characteristics of the structural element 

such as properties of the materials, geometry, level of detailing, and type and 

characteristics of the connections, and on the other hand, on the loading 

history (e.g., loading intensity on each cycle, number of cycles, and sequence 

of loading cycles). 

One of the earliest attempts to model deterioration of structural elements 

subjected to cyclic reversals was conducted by Jacobsen (1958) who 

proposed a behaviorist model to study the response of connections to cyclic 

loading. His model consisted of a combination of sliding blocks arranged in 

series which experienced frictional forces of different amplitudes and which 

were joined by Hookean springs with different elastic stiffnesses. Although 

this model allowed the mathematical description of observed static behavior, 

earthquake response of deteriorating structures was not studied until 1962, 

when Hisada proposed a degrading model (Hisada, Nakagawa and Izumi, 

1962) for studying earthquake response of degrading structures. 

Concerned with the stiffness degradation observed in reinforced concrete 

elements, Clough and Johnston (Clough, 1966; Clough and Johnston, 1966) 

developed a degrading model which incorporated stiffness degradation after 

reloading. In this model unloading occurred with a stiffness equal to the 

initial stiffness but reloading was aimed towards the largest excursion in 

previous cycles. They used the model to study the response of SDOF systems 

subjected to four recorded acceleration time histories. In particular, this study 

computed ratios of maximum deformation of elastoplastic systems to 

maximum deformation of stiffness-degrading systems. When evaluating 

these ratios they wrote “these ratios demonstrate conclusively that there is no 
FEMA P440A A: Detailed Summary of Previous Research A-3 

significant difference between the yield amplitudes generated in the two 

materials. The ratios vary between 0.8 and 1.2 except in a few cases.” They 

concluded that “earthquake ductility requirements in the degrading stiffness 

systems are not materially different from those observed in ordinary 

elastoplastic structures, except for structures having a period of vibration 

less than 1⁄2 second.” Based on their study they also concluded that “the 

ductility required in the members of reinforced concrete frame buildings will 

be about the same as is required in equivalent steel frame buildings.” 

An unrealistic feature of the Clough model when experiencing large load 

reversals followed by small load reversals was pointed out by Mahin and 

Bertero (1976) and by Riddell and Newmark (1979) who showed that after a 

small unloading the model would unrealistically reload toward the point of 

maximum deformation. They modified the model to reload along the same 

unloading branch until the reloading branch was reached and then aim 

toward the point of peak deformation. Mahin and Bertero (1976) also made 

the model more versatile by incorporating a positive post-yield stiffness and 

variable unloading stiffness as a function of the peak deformation. The model 

proposed by Mahin and Bertero (1972), which is often referred to as the 

modified-Clough model, has been incorporated in several general nonlinear 

analysis programs and has been used extensively to model the behavior of 

flexurally controlled reinforced concrete elements. 

An early model proposed for nonlinear analysis of reinforced concrete is the 

Takeda model (Takeda, Sozen and Nielsen, 1970). This model incorporated 

some of the features of the Clough model but also added other features such 

as a trilinear loading curve to incorporate pre-cracking and post-cracking 

stiffnesses, a variable unloading stiffness which was a function of the peak 

deformation, and improved hysteretic rules for inner cyclic loops. This model 

has also been incorporated in several general analysis programs and has been 

extensively used in earthquake engineering to study the seismic response of 

reinforced concrete structures. 

A slight modification to the Takeda model was proposed by Otani and Sozen 

(1972) who replaced the trilinear initial loading segments of the Takeda 

model by a bilinear relationship. The resulting model is known as bilinear 

Takeda model. Otani (1981) compared the response of six different hysteretic 

models (Ramberg-Osgood, degrading bilinear, modified-Clough, bilinear 

Takeda, Takeda and degrading trilinear) when subjected to horizontal 

components of the 1940 El Centro and the 1954 Taft records. He concluded 
A-4 A: Detailed Summary of Previous Research FEMA P440A 

that “maximum response amplitudes are not as sensitive to details in the 

differences in hysteretic rules of these models.” 

Other early models developed specifically for reinforced concrete structures 

include the model developed by Nielsen and Imbeault (1970) who proposed a 

degrading bilinear system whose stiffness would change only when a prior 

maximum displacement was exceeded, and the degrading model proposed by 

Anagnostopoulos (1972) which combines Nielsen’s degrading bilinear model 

and the Clough model. Models where reloading is aimed at the point of 

maximum deformation in prior cycles are sometimes also referred to as 

“peak-oriented” models (Rahnama and Krawinkler, 1993; Medina and 

Krawinkler, 2004). The Clough model, the modified-Clough, Takeda, 

bilinear Takeda and Anagnostopoulos models are all peak-oriented models. 

Iwan developed a general class of stiffness-degrading and pinching models 

(Iwan, 1973, 1977, 1978). Similar to prior models developed by him (Iwan, 

1966, 1967) and by Jacobsen (1958), this model consisted of a collection of 

linear elastic and Coulomb slip elements. He then studied the response of a 

wide range of stiffness-degrading and pinching models when subjected to an 

ensemble of 12 accelerograms recorded in various earthquakes (Iwan and 

Gates, 1979a, 1979b). After comparing the response of the various degrading 

and non-degrading systems they noted “despite the quite different load 

deformation characteristic the overall effect for a given ductility is nearly the 

same. This is a rather surprising result which may be useful in design, for it 

implies that it may not be necessary to know the precise details of the loaddeflection behavior of a structure in order to make a reasonably accurate 

estimate of its response.” In another study aimed at estimating inelastic 

spectra from elastic spectra using equivalent linear methods, Iwan (1980) 

concluded that “the differences in hysteretic behavior considered herein 

appear to have only a secondary effect on the accuracy of the results.” 

Chopra and Kan (1973) studied the effects of stiffness degradation on 

ductility requirements of two idealized multistory buildings, one having a 

period of vibration of 0.5 s and the other 2.0 s. They concluded that “stiffness 

degradation has little influence on ductility requirements for flexible 

buildings, but it leads to increased ductility requirements for stiff buildings.” 

Riddell and Newmark (Riddell and Newmark, 1979; Newmark and Riddell, 

1980) studied the influence of hysteretic behavior on inelastic spectral 

ordinates. They considered an elastoplastic system, a bilinear system and a 

stiffness degrading system. They compared average inelastic spectral 
FEMA P440A A: Detailed Summary of Previous Research A-5 

ordinates for the three systems for ground motions scaled to peak ground 

acceleration, peak ground velocity and peak ground displacement in the 

acceleration, velocity and displacement-controlled spectral regions, 

respectively. They arrived at similar conclusions to those of Clough or those 

of Iwan; in particular they concluded that “the ordinates of the average 

spectra do not vary significantly when various nonlinear models are used.” 

They also noted “It is particularly significant that, on the average, the 

stiffness degradation phenomenon is not as critical as one might expect ” and 

concluded that “the use of the elastoplastic idealization provides, in almost 

every case, a conservative estimate of the average response to a number of 

earthquake ground motions.” 

Mahin and Bertero (1981) used the modified Clough model to also study the 

difference in response of elastoplastic systems and stiffness-degrading 

systems. They noted that “ductility demands for a stiffness degrading system 

subjected to a particular ground motion can differ significantly from those 

obtained for elastoplastic systems in some period ranges. However, it 

appears that, on average, the differences are generally small.” Similar 

conclusions were also reached by Powel and Row (1976) who studied the 

influence of analysis assumptions on computed inelastic response of three 

different types of reinforced concrete ten-story buildings. They concluded 

that “degrading stiffness appears to have no substantial influence on 

interstory drift demands.” Nassar and Krawinkler (1991), used the modified 

Clough model to study the difference of strength reduction factors associated 

to increasing levels of ductility demands in bilinear and stiffness degrading 

models. In their report they wrote “... except for very short period systems, 

the stiffness-degrading models allow higher reduction factors than the 

bilinear model, for systems without strain hardening. This difference 

diminishes with strain hardening. This is a very interesting result in that it 

suggests that the stiffness degrading model behaves “better” than the 

bilinear model, i.e., it has a smaller inelastic strength demand for the same 

ductility ratio.” 

Other piece-wise models that incorporate degradation include the Park and 

Ang mechanistic model (Park and Ang, 1985) and the three-parameter model 

(Park, Reinhorn and Kunnath, 1987; Valles et al., 1996). The three-parameter 

model includes strain hardening, variable unloading stiffness, pinching and 

cyclic load degradation (that is, decreasing yielding strength as a function of 

maximum deformation, hysteretic energy demand, or a combination of the 

two). The model was further improved in Kunnath and Reinhorn (1990). 

Rahnama and Krawinkler (1993) developed a general piece-wise linear 
A-6 A: Detailed Summary of Previous Research FEMA P440A 

hysteretic model which was incorporated into a SDOF analysis program 

referred to as SNAP (SDOF Nonlinear Analysis Program). The model has a 

bilinear skeleton relationship and includes variable unloading stiffness, peakoriented stiffness degradation at reloading, pinching, cyclic strength 

deterioration as a function of hysteretic energy demands, and also the 

capability of accelerating the degradation of loading stiffness beyond the 

peak-oriented degradation. 

They used this model to study the influence of hysteretic behavior of SDOF 

systems subjected to 15 recorded ground motions recorded at firm sites 

during California earthquakes to study constant-ductility strength-reduction 

factors. Their results confirmed observations of Nassar and Krawinkler 

(1991) and of previous investigators who noted that the effect of stiffness 

degradation was small, on average, leading to smaller displacement demands 

except for short-period structures where displacement demands in systems 

with stiffness degradation were larger than those in bilinear systems. In their 

study they also noted that cyclic strength deterioration increased 

displacement demands but that the increase was not large unless the strength 

deteriorates to a small value, and noted that further research was needed 

before quantitative conclusions could be drawn. 

Rahnama and Krawinkler also studied the effect of in-cyclic degradation by 

considering a negative post-elastic slope (that is, negative strain hardening) 

in bilinear and degrading models. They concluded that “ratios of reduction 

factors for degrading and bilinear systems become significantly larger than 

1.0 when negative hardening is present, particularly if the periods of 

vibration are short and the ductility demands are high.” 

Gupta and Krawinkler (1998, 1999) investigated the effects of pinching and 

stiffness degradation in SDOF and MDOF structures using the hysteretic 

model previous developed by Rahnama (Rahnama and Krawinkler, 1993) 

which was incorporated in the DRAIN-2DX analysis program (Allahabadi 

and Powell, 1988). They concluded that “for SDOF systems, pinching leads 

to a relatively small amplification of the displacement response for systems 

with medium and long periods, regardless of the yielding strength. For short 

period structures, which are subject to a larger number of cycles, the 

displacement amplification increases significantly.” They also noted that “the 

effect of the pinched force-deformation relationship on the displacement 

ratio is not very sensitive to the severity of the ground motion.” For MDOF 

structures they concluded that “pinching of the force-deformation 

characteristics of inelastic systems has a global (roof) drift similar to that 
FEMA P440A A: Detailed Summary of Previous Research A-7 

observed in SDOF systems.” They also investigated the effect of negative 

post-yield stiffness in SDOF and MDOF systems. They concluded that “for 

SDOF systems a negative post-yield stiffness (which could represent P-

effects) has a large effect on the displacement demand for systems with 

bilinear characteristics. The effect increases rapidly with an increase in the 

negative stiffness ratio , with decrease in the yield strength of the system, 

and a decrease in the period. Dynamic instability, caused by attainment of 

zero lateral resistance, is a distinct possibility and was observed under 

several of the ground motion records.” For systems with negative strain 

hardening they noted that the pinching model exhibits better behavior than 

the bilinear model. 

Recently, Ruiz-Garcia and Miranda (2005) examined the effect of hysteretic 

behavior on maximum deformations of SDOF systems subjected to an 

ensemble of 240 ground motions recorded in California. They considered 

seven different types of hysteretic behavior: elastoplastic, bilinear, modified 

Clough, Takeda, origin-oriented, moderate degrading, and severely 

degrading. The modified Clough, Takeda and origin-oriented models only 

exhibit stiffness degradation while the moderate degrading and severely 

degrading systems exhibit both stiffness and cyclic strength degradation. 

They found that the effect of positive post-yield stiffness was relatively small 

except for systems with very short periods of vibration (T<0.2s). When 

subjected to firm soil records they found that the effects of hysteretic 

behavior were relatively small for structures with periods of vibration larger 

than about 0.7s. 

The same authors used the modified Clough model to examine the effect of 

stiffness degradation on single-degree-of-freedom systems subjected to 

ground motions recorded on soft soil sites (Miranda and Ruiz-Garcia, 2002, 

Ruiz-Garcia and Miranda, 2004, 2006b). They concluded that the effects of 

stiffness degradation were larger for structures on soft soil sites than those 

observed for structures on firm sites. In particular, they concluded that for 

structures with periods of vibration shorter than the predominant period of 

the ground motion, the lateral displacement demands in stiffness degrading 

systems on average are 25% larger than those of non-degrading systems and 

that, in order to control lateral deformations to comparable levels of those in 

non-degrading structures, stiffness-degrading structures in this spectral 

region need to be designed for higher lateral forces. 

A model similar to the one developed by Krawinkler and his coworkers but 

with additional capabilities to model connection fracture was developed and 
A-8 A: Detailed Summary of Previous Research FEMA P440A 

incorporated into DRAIN-2DX by Shi and Foutch (Shi and Foutch, 1997; 

Foutch and Shi, 1998). They studied the influence of hysteretic behavior on 

the seismic response of buildings by considering seven different hysteretic 

behaviors, which included non-degrading and degrading models, and nine 

steel moment-resisting frame models of buildings with three, six, and nine 

stories. They concluded that “Hysteresis type has only a minimum effect on 

ductility demands of structures.” When evaluating ratios of deformations of 

degrading to bilinear behavior they noted that “For the non-pinching 

hysteresis models, the maximum ratios of ductility demand to the bilinear 

hysteresis model range from 1.10 to 1.15 when the period of the structure is 

less than 1.0 second. For pinching hysteresis types the maximum ratios are 

on the order of 1.25 to 1.30.” 

Gupta and Kunnath (1998) arrived at similar conclusions. More recently 

Medina and Krawinkler (2004) studied the effects of hysteretic behavior (i.e., 

bilinear, peak oriented and pinching) in the evaluation of peak deformation 

demands and their distribution over the height for regular frame structures 

over a wide range of stories (from 3 to 18) and fundamental periods (from 

0.3 s. to 3.6 s.). The study did not consider monotonic in-cycle deterioration. 

The ground motions used were those with frequency content characteristic of 

what they referred to as “ordinary ground motions” (that is, no near-fault or 

soft soil effects). They concluded that “the degree of stiffness degradation is 

important for the seismic performance evaluation of regular frames because 

systems with a large degree of stiffness degradation tend to exhibit larger 

peak drift demands and a less uniform distribution of peak drifts over the 

height.” 

Based on the general class of non-degrading and degrading models 

developed by Iwan (1966, 1967, 1973), Mostaghel (1998, 1999) developed a 

general hysteretic model by providing an analytical description (that is, with 

differential equations) of physical models consisting of a series of linear 

springs, dashpots, and sliders. The model includes the effects of pinching, 

stiffness degradation, and load deterioration. He showed that complex multilinear hysteretic behavior can be obtained by solving (2n-1) differential 

equations where n is the number of linear segments in the model. 

A.1.3 Smooth Deteriorating Hysteretic Models 

Degradation and pinching have also been incorporated in smooth hysteretic 

models. Some examples of degrading smooth models are the Baber model 

(Baber and Wen, 1981; Baber and Noori 1985, 1986) which extends the 

Bouc-Wen model to include stiffness degradation and pinching. 
FEMA P440A A: Detailed Summary of Previous Research A-9 

More recently, Sivaselvan and Reinhorn (1999, 2000) developed a versatile 

hysteretic model that is conceptually based on a general class of nondegrading and degrading models developed by Iwan (1966, 1967, 1973) but 

extended the model developed by Mostaghel (1998, 1999) to include smooth 

curvilinear segments. Stiffness degradation is incorporated using a pivot rule 

analogous to the one incorporated in the three-parameter model (Park, 

Reinhorn and Kunnath, 1987). Cyclic strength degradation is modeled by 

reducing the capacity in the backbone curve while pinching is achieved by 

adding an additional slip-lock spring in series with the main smooth 

hysteretic spring, which is similar to the Bouc-Wen model. The hysteretic 

behavior is then described by the solution of four time-independent 

differential equations which are solved using Runge-Kutta’s method. 

Although they provided specific rules for controlling stiffness degradation, 

cyclic degradation, and pinching, they showed that other rules could be 

implemented as well. This hysteretic model has been incorporated in recent 

versions of IDARC (Valles et al., 1996). 

A.1.4 Hysteretic Models for Steel Braces 

Experimental research on the behavior of steel braces has shown that their 

behavior under severe cyclic loading is complicated and not fully understood. 

Cyclic nonlinear behavior of steel brace members is complex as a result of 

various phenomena occurring in the braces and their connections, such as 

yielding in tension, buckling in compression, post-buckling deterioration of 

compressive load capacity, deterioration of axial stiffness with cycling, lowcycle fatigue fractures at plastic hinge regions, Bauschinger effect, and 

buckling and fracture in the gusset plates. As in the models previously 

described, element models for steel braces can be classified as either 

phenomenological models in which the load-deformation behavior of steel 

braces is described through a series of hysteretic rules that try to reproduce 

behavior observed experimentally, or material-based models such as finite 

element models and fiber element models where the steel brace is discretized 

into small elements and the overall behavior of the brace is obtained from 

uniaxial, biaxial or triaxial material behavior of the material. 

A significant amount of both experimental and analytical work on the 

behavior of steel bracing has been conducted at the University of Michigan 

under the direction of Professors Goel and Hanson. One of the first analytical 

models for predicting the force-deformation behavior of axially-loaded 

members with intermediate slenderness ratios was developed by 

Higginbotham and Hanson (1976). Prathuangsit, Goel, and Hanson, (1978) 

proposed a model with rotational end springs to simulate the end restraint 
A-10 A: Detailed Summary of Previous Research FEMA P440A 

resulting from the flexural rigidity of the connections of axially loaded (that 

is, bracing) members. They showed that members with balanced strength 

connections, (that is, which form plastic hinges simultaneously at midspan 

and at the ends), have more efficient compressive load and energy dissipation 

capacities than members of the same length and same cross-sectional 

properties with unbalanced strength connections. They concluded that the 

hysteresis behavior of a balanced strength member can be represented 

adequately by that of a pin-connected member of the same cross section and 

same effective slenderness ratio 

Jain, Goel, and Hanson tested 17 tube specimens and eight angle specimens 

under repeated axial loading (Jain, Goel and Hanson, 1976, 1978a, b). The 

objective of this experimental investigation was to quantify the reduction in 

maximum compressive loads and increase in member length, and to study the 

influence of the buckling mode and the shape of the cross-section on the 

hysteretic behavior and dissipation of energy through the hysteretic cycles. 

They concluded that local buckling and shape of the cross section can have a 

significant influence on the hysteretic behavior of axially loaded steel 

members. Based on their experimental results they developed a hysteresis 

model for steel tubular members that included a reduction in compressive 

strength and an increase in member length with the number of cycles (Jain 

and Goel, 1978, 1980) which was then incorporated in the DRAIN-2D 

analysis program. This model has been extensively used by investigators at 

the University of Michigan and elsewhere to study the seismic response of 

concentrically braced steel frames. 

Astaneh-Asl and Goel investigated the behavior of double-angle bracing 

members subjected to out-of-plane buckling due to severe cyclic load 

reversals (Astaneh-Asl et al. 1982; Astaneh-Asl and Goel 1984). Nine fullsize test specimens were subjected to severe inelastic axial deformations. 

Test specimens were made of back-to-back A36 steel angle sections 

connected to the end gusset plates by fillet welds or high-strength bolts. Five 

of the test specimens were designed according to current design procedures 

and code requirements. These specimens experienced fracture in gusset 

plates and stitches during early cycles of loading. Based on observations and 

analysis of the behavior of the specimens, new design procedures were 

proposed for improved ductility and energy dissipation capacity of doubleangle bracing members. Goel and El-Tayem (1986) investigated the behavior 

of angle cross-bracing. 
FEMA P440A A: Detailed Summary of Previous Research A-11 

Gugerili and Goel (1982) tested nine commercially available wide-flange 

shapes and structural tubes with different slenderness and width-to-thickness 

ratios in order to investigate the effects of cross section and slenderness ratio. 

A general rule was developed for transitioning between compression and 

tension mechanism lines of constant shape and elastic segments. The new 

rule also included the effects of residual elongation. The theoretical model 

was used as a basis for a developing a semi-empirical model for predicting 

experimental hysteresis loops more accurately than previous models. This 

model included the decrease in compressive strength with cumulative plastic 

hinge deformation. 

Tang and Goel (1987) developed a procedure to predict the fracture life of 

bracing members. Their empirically based procedure was refined with an 

energy approach using Jain's hysteresis model and was then also incorporated 

in the the DRAIN-2D program. Based on the experimental results of 

previous researchers Hassan and Goel (1991) formulated a refined and 

practical hysteresis model for bracing the members of concentrically 

(chevron) braced steel structures subjected to severe earthquakes. A more 

recent model was implemented in a Structural Nonlinear Analysis Program 

(SNAP) (Rai et al, 1996), which eliminates the brace once it is estimated that 

it has fractured (while this program has the same name, it is different from 

the program developed by Krawinkler and his coworkers). 

Professors Popov and Mahin at the University of California at Berkeley have 

also conducted a significant amount of experimental and analytical research 

on steel braces. Zayas, Popov, and Mahin, (1979, 1980) conducted a series of 

experimental tests on scaled tubular brace members subjected to severe 

inelastic cyclic loading. The tubular brace specimens considered were onesixth scale models of braces of the type used for offshore platforms. A 

method of predicting the reduction in buckling load was presented. Maison 

and Popov (1980) performed an experimental and analytical investigation of 

the behavior of structural steel frames with K-braces subjected to severe 

cyclic loadings simulating seismic effects. They developed an empirical 

brace model to analyze steel frames with K-braces. 

Black and Popov, E.P. (1980, 1981) conducted an experimental study on 24 

commercially available steel struts, commonly used as bracing members. 

They investigated the effects of loading patterns, end conditions, crosssectional shapes, and slenderness ratios on the hysteresis response of 

members. A large variety of shapes were tested, including wide flanges, 

structural tees, double-angles, a double-channel, and thick-walled and thin-
A-12 A: Detailed Summary of Previous Research FEMA P440A 

walled square and round tubes. Two types of boundary conditions were 

considered, pinned-pinned and fixed-pinned, with effective slenderness ratios 

of 40, 80, and 120. An explanation was provided regarding the fundamental 

mechanisms responsible for the observed degradation in the buckling load 

capacity during inelastic cycling. 

Ikeda and Mahin (1984) developed an analytical model for simulation of the 

inelastic buckling behavior of steel braces. In their model, buckling is 

simulated by the use of predefined straight-line segments and simple rules 

regarding factors such as buckling load deterioration and plastic growth. The 

model, based on the approach of Maison and Popov (1980) overcomes some 

of the limitations of earlier models. They proposed a systematic method for 

selecting input parameters along with several rules governing the values of 

certain parameters. The model was subsequently refined in 1986 by 

combining analytical formulations describing plastic hinge behavior with 

empirical formulas that are based on a study of experimental data. Analytical 

expressions for the axial force versus axial deformation behavior of braces 

were derived as solutions of the basic beam-column equation based on 

specified assumptions. While analytical expressions form the basis of this 

model, several empirical behavioral characteristics were implemented in the 

modeling to achieve better representation of observed cyclic inelastic 

behavior. 

Khatib and Mahin (1987, 1988) conducted an analytical investigation on 

concentrically braced frames. Their study stressed that concentrically braced 

steel frames designed by conventional methods may exhibit several 

undesirable modes of behavior. In particular, they showed that chevronbraced frames have an inelastic cyclic behavior that is often characterized by 

a rapid redistribution of internal forces, a deterioration of strength, a 

tendency to form soft stories, and fracture due to excessive deformation 

demand. They identified parameters having a significant influence on these 

phenomena and provided recommendations for preferable ranges of brace 

slenderness, approaches for designing beams, and a simplified capacity 

design approach for proportioning columns and connections. They also 

proposed several alternative brace configurations with improved behavior 

including zipper bracing systems which incorporate vertical linkage elements 

in a conventional chevron-braced frame to decrease concentrations of 

interstory drift demands. 

More recently Uriz and Mahin (2004, Uriz, 2005) conducted experimental 

testing of a nearly full-size, two-story Special Concentric Braced Frame 
FEMA P440A A: Detailed Summary of Previous Research A-13 

(SCBF) specimen. They also conducted an analytical investigation using the 

same reliability framework used to assess Special Moment Resisting Frame 

(SMRF) structures during the FEMA/SAC Steel Project in order to assess the 

confidence with which SCBFs might achieve the seismic performance 

expected of new SMRF construction. 

Other institutions have also been actively involved in research on the 

behavior of steel braces. For example, Nonaka (1973, 1977) at Kyoto 

University, conducted elastic, perfectly plastic analyses of a bar under 

repeated axial loading. The bar was taken as a one-dimensional continuum 

with both ends simply supported. His analysis considered the plastic 

interaction for the combined action of bending and axial deformation, based 

on a piecewise-linear yield condition. With a number of simplifying 

assumptions, a closed form solution was derived that can describe the 

hysteretic behavior of a bar, such as a structural brace or a truss member, 

under any given history of tension and/or compression or of corresponding 

displacements. Nonaka’s closed form solution was later extended by Shibata 

(1982) for a bar of ideal I-section with bilinear stress-strain relationship. For 

a bar of arbitrary solid cross-section with a piecewise linear stress-strain 

relationship, an incremental load-displacement relationship was also obtained 

in analytical form. 

At the University of Canterbury in New Zealand, Remennikov and Walpole 

(1997a, 1997b) developed an analytical model for the inelastic response 

analysis of braced steel structures. Their model combines the analytical 

formulation of plastic hinge behavior with empirical formulas developed on 

the basis of experimental data. The brace is modeled as a pin-ended member, 

with a plastic hinge located at the midspan and braces with other end 

conditions are handled using the effective length concept. Step-wise 

regression analysis is employed to approximate the plastic conditions for the 

steel UC section. Verification of the brace model is performed on the basis of 

quasi-static analyses of individual struts and a one-bay one-story crossbraced steel frame. 

At the Ecole Polytechnique de Montreal in Canada, Archambault, Tremblay, 

and Filiatrault, (1995, 2003) conducted experimental and analytical studies 

on the seismic performance of concentrically braced steel frames made with 

cold-formed rectangular tubular bracing members. They tested a total of 24 

quasi-static cyclic tests on full scale X bracing and single diagonal bracing 

systems. They developed simplified models to predict the out-of-plane 

deformation of the braces as a function of the ductility level. They then used 
A-14 A: Detailed Summary of Previous Research FEMA P440A 

their models to develop an empirical expression to assess the inelastic 

deformation capacity before fracture of bracing members made of 

rectangular hollow sections. 

More recently, Jin and El-Tawil, (2003) developed a beam-column element 

to model the inelastic cyclic behavior of steel braces. In their model a 

bounding surface plasticity model in stress-resultant space coupled with a 

backward Euler algorithm is used to keep track of the spread of plasticity 

through the cross-section. Deterioration of the cross-section stiffness due to 

local buckling is accounted for through a damage model. 

Further information on the experimental and analytical response of steel 

bracing is available in Tremblay (2002), Jin and El-Tawil, (2003) and Uriz 

(2005), which provide summaries of experimental and analytical work. In 

particular Tremblay (2002) conducted a survey of past experimental studies 

on the inelastic response of diagonal steel bracing members subjected to 

cyclic inelastic loading to collect data for the seismic design of concentrically 

braced steel frames for which a ductile response is required during 

earthquakes. He examined the buckling strength of the bracing members, the 

brace post-buckling compressive resistance at various ductility levels, the 

brace maximum tensile strength including strain hardening effects, and the 

lateral deformations of the braces upon buckling. Additionally he proposed 

equations for each of these parameters and examined the maximum ductility 

that can be achieved by rectangular hollow bracing members. 

Nakashima and Wakabayashi (1992) provide an overview of Japanese 

experimental and analytical research on steel braces and braced frames. 

Current Japanese practice is also briefly summarized. 

A.2 Detailed Summaries of Relevant Publications 

This section presents summaries of publications that were judged to be 

particularly relevant to the subject of nonlinear degrading response, and 

were, therefore, reviewed in detail. Each summary includes the list of 

authors, an abstract, a narrative summary of the work, relevant figures, a 

summary of important findings, and a listing of relevant publications 

included in the list of references. The following publications were selected 

for detailed review: 

 Bernal, D., 1998, “Instability of buildings during seismic response,” 

Engineering Structures, Vol. 20, No. 4-6, pp. 496-502. 
FEMA P440A A: Detailed Summary of Previous Research A-15 

 Pincheira, J.A, Dotiwala, F.S., and D’ Souza J.T., 1999, “Spectral 

displacement demands of stiffness- and strength-degrading systems,” 

Earthquake Spectra, 15(2), 245–272. 

 Song, J.-K., and Pincheira, J.A, 2000, “Seismic analysis of older 

reinforced concrete columns,” Earthquake Spectra, 16(4), 817–851. 

 Miranda, E. and Akkar, S.D., 2003 “Dynamic instability of simple 

structural systems,” Journal of Structural Engineering, ASCE, 129(12), 

pp 1722-1727. 

 Vian, D. and Bruneau, M., 2003, “Tests to structural collapse of singledegree-of-freedom frames subjected to earthquake excitations,” Journal 

of Structural Engineering, ASCE, 129(12), 1676-1685. 

 Kanvinde, A.M., 2003, “Methods to evaluate the dynamic stability of 

structures – shake table tests and nonlinear dynamic analyses,” EERI 

Annual Student Paper Competition, Proceedings of 2003 EERI Annual 

Meeting, Portland, Oregon. 

 Vamvatsikos, D. and Cornell, C.A., 2005, Seismic performance, capacity 

and reliability of structures as seen through incremental dynamic 

analysis, John A. Blume Earthquake Engineering Research Center, 

Report No. 151, Department of Civil and Environmental Engineering, 

Stanford University, Stanford, California. 

 Ibarra, L., Medina, R., and Krawinkler, H., 2005, “Hysteretic models that 

incorporate strength and stiffness deterioration, Earthquake Engineering 

and Structural Dynamics, Vol. 34, no. 12, pp. 1489-1511. 

 Ibarra, L.F., and Krawinkler, H., 2005, Global collapse of frame 

structures under seismic excitations, John A. Blume Earthquake 

Engineering Research Center, Report No. 152, Department of Civil and 

Environmental Engineering, Stanford University, Stanford, California. 

 Kaul, R., 2004, Object-oriented development of strength and stiffness 

degrading models for reinforced concrete structures, Ph.D. Thesis, 

Department of Civil and Environmental Engineering, Stanford 

University, Stanford, California. 

 Elwood, K.J., 2002, Shake table tests and analytical studies on the 

gravity load collapse of reinforced concrete frames, Ph.D. Dissertation, 

University of California, Berkeley, California. 

 Lee, L.H., Han, S.W., and Oh, Y.H., 1999, “Determination of ductility 

factor considering different hysteretic models,” Earthquake Engineering 

and Structural Dynamics, Vol. 28, 957–977. 
A-16 A: Detailed Summary of Previous Research FEMA P440A 

 Foutch, D.A. and Shi, S., 1998, “Effects of hysteresis type on the seismic 

response of buildings,” Proc. 6th U.S. National Conference on 

Earthquake Engineering, Seattle, Washington, Earthquake Engineering 

Research Institute, Oakland, California. 

 Ruiz-Garcia, J. and Miranda, E., 2003, “Inelastic displacement ratio for 

evaluation of existing structures,” Earthquake Engineering and 

Structural Dynamics. 32(8), 1237-1258. 

 Dolsek, M. and Fajfar, P., 2004, “Inelastic spectra for infilled reinforced 

concrete frames,” Earthquake Engineering and Structural Dynamics, 

Vol. 33, 1395–1416. 

A.2.1 Instability of Buildings During Seismic Response 

Authors: 

Bernal, D. (1998) 

Abstract: 

The issue of gravity-induced instability during response to severe seismic 

excitation is examined. While static instability is fully determined by the 

existence of at least one negative eigenvalue in the second-order tangent 

stiffness, this condition is necessary but not sufficient for instability during 

dynamic response. The likelihood of collapse is strongly dependent on the 

shape of the mechanism that controls during the critical displacement cycle 

and this shape can be reasonably identified using a pushover analysis with an 

appropriately selected lateral load distribution. A characterization of the 

instability limit state based on the reduction of a multistory building to an 

equivalent SDOF system is presented 

Summary: 

Dynamic instability is a phenomenon whereby the response changes from 

vibration to drift in a single direction. In this study a structure is defined as 

stable if small increases in the ground motion intensity result in small 

changes in the response. The study shows that the distribution of inelastic 

action along the height of the building plays a critical role in the likelihood of 

instability. The study emphasizes that gravity generally has little effect on the 

dynamic response of structures, except when failure from instability is near. 

In particular, the study shows that the static based approach of accounting for 

second-order effects through amplifications of the first-order solution is not 

appropriate in a dynamic setting. Specifically, design in a region where 
FEMA P440A A: Detailed Summary of Previous Research A-17 

amplifications from P- are significant implies unacceptably low safety 

factors. 

Representative Figures: 

Figure A-1 Effect of mechanism shape on the monotonic work vs. 

amplitude relationship. 

Figure A-2 Illustration of the definition of stability coefficient: (a) general 

load deformation relationship, (b) elasto-plastic system. 

Summary of Findings: 

Dynamic instability takes place when the strength of the structure is below a 

certain threshold and is strongly dependent on the shape of the failure (or 

collapse) mechanism that controls. Safety against dynamic instability cannot 

be guaranteed by placing controls on initial elastic stiffness; a rational check 

of the safety against collapse must contemplate the strength and shape of the 

critical mechanism. In particular, a rational approach is to estimate the 

strength level associated with the instability threshold and to ensure that the 

strength level provided exceeds the required limit by an appropriate safety 

margin. 
A-18 A: Detailed Summary of Previous Research FEMA P440A 

Relevant Publications: 

Bernal, D., 1998, "Instability of buildings during seismic response," 

Engineering Structures, Vol. 20, No. 4-6, pp. 496-502. 

Bernal, D., 1992, "Instability of buildings subjected to earthquakes." Journal 

of Structural Engineering , ASCE, Vol. 118, No. 8, pp. 2239-2260. 

Bernal, D., 1987, “Amplification factors for inelastic dynamic P-Δ effects in 

earthquake analysis,” Earthquake Eng. Struct. Dyn., 15(5), pp. 117-144. 

A.2.2 Seismic Analysis of Older Reinforced Concrete Columns 

Authors: 

Pincheira, J.A., Dotiwala, F.S., and D’Souza J.T. (1999) 

Abstract: 

A nonlinear model and an analytical procedure for calculating the cyclic 

response of nonductile reinforced concrete columns are presented. The main 

characteristics of the model include the ability to represent flexure or shear 

failure under monotonically increasing or reversed cyclic loading. Stiffness 

degradation with cyclic loading can also be represented. The model was 

implemented in a multipurpose analysis program and was used to calculate 

the response of selected columns representative of older construction. A 

comparison of the calculated response with experimental results shows that 

the strength, failure mode and general characteristics of the measured cyclic 

response can be well represented by the model. 

Summary: 

A beam-column element was created in order to simulate the behavior of 

older non-ductile or shear-critical reinforced concrete columns in 2D frames. 

This is a lumped plasticity element using two flexural springs at the beam 

ends and a shear spring at the midpoint (Figure A-3). For the flexural springs 

a Takeda hysteresis law is used together with a quadrilinear backbone curve 

that incorporates a hardening post-yield segment followed by a negative 

post-peak slope that stops at a residual plateau; essentially, only in-cycle 

strength degradation is considered (Figure A-4a). On the other hand, the 

shear spring uses a similar quadrilinear backbone but a pinching hysteresis 

together with cyclic degradation of the post-peak strength (Figure A-4b). 

The element was incorporated in the Drain-2D analysis program, resulting in 

important limitations in its implementation and applicability. The solution 
FEMA P440A A: Detailed Summary of Previous Research A-19 

algorithm of Drain-2D cannot handle negative stiffness, thus necessitating 

the use of numerical techniques to find an approximate solution. Specifically, 

on a negative slope (for any of the three springs), the load steps are 

performed first with an arbitrary positive stiffness, and the load unbalance is 

then subtracted from the resulting increased load to move down to the actual 

negative slope (Figure A-5). The results can be considered reliable only 

under small load-steps and they may indeed lead to gross numerical errors 

and possible numerical instabilities at the MDOF level. Furthermore, the use 

of Drain-2D means that only a load-control pushover is possible, thus 

severely reducing the applicability of this element for anything but timehistory analysis with small time steps. 

Significant effort has gone into the definition of the spring backbones, using 

modified compression field theory for the shear backbone, and considering 

anchorage slip, lap-splice slip, and section degradation for the flexural 

springs. Calibration and testing of the element were performed with regard to 

the experimental results. 

Representative Figures: 

 

Figure A-3 The RC column element formulation. 
A-20 A: Detailed Summary of Previous Research FEMA P440A 

Figure A-4 The hysteretic laws for shear and moment springs. 

Figure A-5 The force (or moment) unbalance is subtracted after an arbitrary 

positive stiffness step towards the “correct” displacement. Very 

small load steps are needed for accuracy, even at the SDOF 

level. 
FEMA P440A A: Detailed Summary of Previous Research A-21 

Figure A-6 Observed versus calculated response for a column specimen SC3 (shear critical). 

Figure A-7 Observed versus calculated response for a column specimen 2CLH18 (fails in shear 

after considerable flexural deformation). 

Summary of Findings: 

The comparison with experimental results showed that good correlation of 

the observed versus the calculated post-peak response was in many cases not 

possible (Figures A-6, A-7). The cyclic degradation parameters were shown 

to have a large influence on the post-peak response while significant 

epistemic uncertainty was identified in the cyclic degradation. 

The column failure mode was captured in every test considered, but the 

estimated failure loads and drifts were generally conservative. Nonetheless, 

the model was able to capture satisfactorily the overall strength degradation, 
A-22 A: Detailed Summary of Previous Research FEMA P440A 

stiffness degradation, and in-cycle and cyclic degradation properties of the 

specimens. 

Relevant Publications: 

Pincheira, J.A, Dotiwala, F.S., and D’Souza J.T., 1999, “Spectral 

displacement demands of stiffness- and strength-degrading systems,” 

Earthquake Spectra, 15(2), 245–272. 

Song J.-K., and Pincheira, J.A., 2000, “Seismic analysis of older reinforced 

concrete columns,” Earthquake Spectra, 16(4), 817–851. 

Dotiwala, F.S., 1996, A nonlinear flexural-shear model for RC columns 

subjected to earthquake loads, MS Thesis, Department of Civil and 

Environmental Engineering, University of Wisconsin-Madison. 

Pincheira, J.A., and Dotiwala, F.S., 1996, “Modeling of nonductile R/C 

columns subjected to earthquake loading,” Proc. 11th World Conf. on 

Earthquake Engineering, Paper No. 316, Acapulco, Mexico. 

A.2.3 Spectral Displacement Demands of Stiffness- and 

Strength-Degrading Systems 

Authors: 

Song, J.-K., and Pincheira, J.A. (2000) 

Abstract: 

The effect of stiffness and strength degradation on the maximum inelastic

displacement of single-degree-of-freedom (SDOF) systems was investigated. 

The SDOF model included strength and stiffness degradation with increasing 

deformation amplitude and upon reversal of loading cycles. Pinching of the 

hysteresis loops was also considered. Spectral displacements were calculated 

for oscillators with a range of degrading characteristics subjected to twelve 

ground motions on rock, firm, and soft soils. The results show that the 

maximum displacements of degrading oscillators are, on average, largerthan 

those of non-degrading systems. The displacement amplification depends 

significantly with the period, strength coefficient, degradation rate, and 

ground motion considered. Nonetheless, the amplification due to the 

degradation characteristics ofthe system is more important in the shortperiod range where average amplification factors of two or three are credible. 

The amplification factors proposed in the FEMA 273 report by the ATC-33 

project provided conservative estimates for oscillators with periods greater 

than 0.3 seconds subjected to motions on rock or firm soil. On soft soils, a 
FEMA P440A A: Detailed Summary of Previous Research A-23 

good correlation was found for periods greater than 1.5 seconds. At shorter 

periods, the ATC 33 factors underestimate the displacement amplification. 

Summary: 

About 7,600 SDOF dynamic analyses were performed for 12 ground motions 

including rock, firm soil, soft soil, and near-field and far-field records. The 

oscillator had a quadrilinear backbone with a hardening, a softening and a 

residual strength segment. Still, only a limited set of backbones were 

considered, all having hardening stiffness 5% of the elastic, residual strength 

10% of yield strength and reaching 1.25 ductility at peak strength. Some 

gentle negative slopes were investigated, namely -1% and -3% of the elastic. 

The oscillator had 5% damping and used a pinching hysteresis with or 

without cyclic strength degradation. During the investigation an ad hoc

collapse limit-state was considered when the post-peak strength reached 10% 

of the yield strength. 

Representative Figures: 

 

Figure A-8 (a) Hysteresis law used for the SDOF system and (b) ratio of degrading to nondegrading displacement amplification factors for the post-peak stiffness equal 

to -1% or -3% of the elastic stiffness. 

Summary of Findings: 

The post-peak stiffness and the unloading stiffness were found to be the most 

important parameters, while the degree of pinching was important except for 

soft-soil records. Cyclic degradation generally increases the dynamic 

response, but mostly for the short periods. Significant differences in the 

SDOF response and its dependence on cyclic degradation were found 

between soft soil and firm soil (or rock) and maybe between near-field versus 

far-field record response, although only one near-source record was used. 
A-24 A: Detailed Summary of Previous Research FEMA P440A 

Finally, for a given strength reduction factor, collapse (as defined in the 

paper) was consistently observed below a certain oscillator period. Some 

duration effects were also reported. 

The study seriously suffers from the lack of records. Only twelve were used 

and they were selected from soft and firm soil sites without differentiating 

between near-field and far-field. Thus, the statistics on the results are not 

reliable, although the general observations provided may prove useful. 

Relevant Publications: 

Song J.-K., and Pincheira, J.A., 2000, “Seismic analysis of older reinforced 

concrete columns,” Earthquake Spectra, 16(4), 817–851. 

Pincheira, J.A, Dotiwala, F.S., and D’Souza, J.T., 1999, “Spectral 

displacement demands of stiffness- and strength-degrading systems,” 

Earthquake Spectra, 15(2), 245–272. 

A.2.4 Dynamic Instability of Simple Structural Systems 

Authors: 

Miranda, E. and Akkar, S.D., (2003) 

Abstract: 

Lateral strengths required to avoid dynamic instability of SDOF systems are 

examined. Oscillators with a bilinear hysteretic behavior with negative postyield stiffness are considered. Mean lateral strengths, normalized by the 

lateral strength required to maintain the system elastic, are computed for 

systems with periods ranging from 0.2 to 3.0 s when subjected to 72 

earthquake ground motions recorded on firm soil. The effect of period of 

vibration and post-yield stiffness is investigated. Results indicate that mean 

normalized lateral strengths required to avoid dynamic instability decrease as 

negative post-yield stiffness increases, and that the reductions are much 

larger for small negative post-yield stiffness than for severe negative postyield stiffness. It is concluded that there is a significant influence of the 

period of vibration for short-period systems and for systems with mildly 

negative post-yield stiffness. Dispersion of normalized lateral strengths 

required to avoid dynamic instability are found to increase as the negative 

post-yield stiffness decreases and as the period of vibration increases. Simple 

equations that capture the effects of period and post-yield stiffness to aid in 

the evaluation of existing structures are obtained through nonlinear 

regression analyses. 
FEMA P440A A: Detailed Summary of Previous Research A-25 

Summary: 

The objective of this study was to assess the minimum lateral strength 

required to avoid dynamic instability in SDOF systems. The minimum lateral 

strength is computed as a function of linear elastic spectral ordinates, that is, 

the lateral strength required to maintain the system elastic. Specific goals of 

the study were: (a) to study the effect of the post-yield negative stiffness on 

the minimum strength required to avoid collapse, (b) to study the effect of 

period of vibration, (c) to compute mean normalized strengths required to 

avoid dynamic instability, and (d) to develop approximate expressions to 

assist practicing engineers in evaluating the minimum lateral strengths 

required in existing structures to avoid dynamic instability. 

The study considered SDOF systems with a bilinear forcedisplacement 

relationship characterized by a linear segment with initial stiffness K

followed by a post-yield linear segment with negative stiffness -K. When 

subjected to earthquake ground motions the likelihood of experiencing 

dynamic instability in a system with a given negative slope increases as the 

lateral strength decreases. Lateral strengths required to avoid dynamic 

instability of bilinear SDOF systems with negative post-yield stiffness were 

investigated. Mean lateral strengths normalized by the lateral strength 

required to maintain the system elastic are computed for systems with 

periods ranging from 0.2 to 3.0 s and post-yield negative stiffness ratios 

ranging from 0.03 to 2.0 when subjected to 72 earthquake ground motions 

recorded on firm soil. 

Representative Figures: 

Figure A-9 Force-displacement characteristics of bilinear systems 

considered 
A-26 A: Detailed Summary of Previous Research FEMA P440A 

Figure A-10 Effect of period of vibration and post-yield stiffness on the mean 

strength ratio at which dynamic instability is produced. 

Summary of Findings: 

Lateral strengths required to avoid dynamic instability of bilinear SDOF 

systems with negative post-yield stiffness were investigated. Mean lateral 

strengths, normalized by the lateral strength required to maintain the system 

elastic, are computed for systems with periods ranging from 0.2 to 3.0 s and 

post-yield negative stiffness ratios ranging from 0.03 to 2.0 when subjected 

to 72 earthquake ground motions recorded on firm soil. The following 

conclusions are drawn from this study. 

The strength ratio at which dynamic instability is produced decreases as the 

post-yield negative stiffness ratio  increases. This means that the lateral 

strength required to avoid collapse increases as the post-yield descending 

branch of the force-deformation relationship is steeper. When  is smaller 

than about 0.2 small increases in  can produce significant increases in 

required lateral strength to avoid instability. Meanwhile, for values of  >1, 

the system must remain practically elastic in order to avoid collapse. 

The collapse strength ratio increases with increasing period, particularly for 

post-yielding negative stiffness ratios smaller than 0.3. Mean collapse 

strength ratios of short period structures are relatively strong, particularly 

when >0.1. Dispersion of collapse strength ratios decreases as  increases 

and as the period of vibration decreases. Coefficients of variation of collapse 

strength ratios are particularly small for >0.5. An approximate equation to 

estimate lateral strengths required to avoid dynamic instability of bilinear 

SDOF system is proposed. 
FEMA P440A A: Detailed Summary of Previous Research A-27 

Relevant Publications: 

Miranda, E. and Akkar, S.D., 2003 "Dynamic instability of simple structural 

systems," Journal of Structural Engineering, ASCE, 129(12) , pp 1722-

1727. 

Vamvatsikos, D., and Cornell, C.A., 2005, Seismic performance, capacity 

and reliability of structures as seen through incremental dynamic 

analysis, John A. Blume Earthquake Engineering Research Center 

Report No. 151, Department of Civil and Environmental Engineering, 

Stanford University, Stanford, California. 

A.2.5 Tests to Structural Collapse of Single-Degree-of-Freedom 

Frames Subjected to Earthquake Excitations 

Authors: 

Vian, D. and Bruneau, M. (2003) 

Abstract: 

This paper presents and analyzes experimental results of tests of 15 fourcolumn frame specimens subjected to progressively increasing uniaxial 

ground shaking until collapse. The specimens were subdivided into groups of 

three different column slenderness ratios: 100, 150, and 200. Within each 

group, the column dimensions and supported mass varied. Ground motion of 

different severity was required to collapse the structures tested. The 

experimental setup is briefly described and results are presented. Test 

structure performance is compared with the proposed limits for minimizing 

P– effects in highway bridge piers. The stability factor is found to have a 

strong relation to the relative structural performance in this regard. 

Performance is also compared with the capacity predicted by currently used 

design equations dealing with axial and moment interactions for strength and 

stability by expressing these capacities in terms of acceleration and 

maximum base shear (represented as a fraction of the system’s weight). The 

experimental results exceeded the maximum spectral accelerations calculated 

when considering second-order effects, but did not when considering only 

member strength. Finally, an example of how to use the experimental data 

for analytical model verification is presented, illustrating the shortcomings 

and inaccuracies of using a particular simplified model with constant 

structural damping. 
A-28 A: Detailed Summary of Previous Research FEMA P440A 

Summary: 

Although the first and foremost objective of this project was to provide welldocumented data (freely available on the web to be used by others) of tests to 

collapse, this paper includes results from a preliminary investigation of 

behavioral trends observed from the shake table results. In particular, peak 

responses are compared with limits proposed by others to minimize P–

effects in bridge piers. Specimen behavior is also investigated with respect to 

axial and moment interaction limits considering strength and stability. 

Finally, to illustrate how the generated experimental data could be used to 

develop or calibrate analytical models of inelastic behavior to collapse, 

experimental results are compared with those obtained using a simple 

analytical model. Progressive bilinear dynamic analyses are performed in 

two different ways and are compared with the shake table test results. 

Representative Figures: 

 

Figure A-11 (a) Schematic of test setup 

 Figure A-12 Simplified bilinear force deformation model 
FEMA P440A A: Detailed Summary of Previous Research A-29 

Figure A-13 Comparison of experimental (left) and analytical (right) results. 

Summary of Findings: 

Specimens showed an approximate bilinear behavior with a tendency to drift 

to one side and ultimately experience collapse. The stability factor, , was 

observed to have the most significant effect on the structure’s propensity to 

collapse. As  increases, the maximum attainable ductility, maximum 

sustainable drift, and maximum spectral acceleration reached before collapse, 

all decrease. When this factor  was larger than 0.1, the ultimate values of 

maximum spectral acceleration, displacement ductility, and drift reached 

before collapse were all grouped below values of 0.75 g, 5, and 20%, 

respectively. 

Relevant Publications: 

Vian, D. and Bruneau, M., 2003, "Tests to structural collapse of single 

degree of freedom frames subjected to earthquake excitations." Journal 

of Structural Engineering, ASCE, 129(12), 1676-1685. 

Bruneau, M. and Vian, D., 2002, “Tests to collapse of simple structures and 

comparison with existing codified procedures,” Proc. 7th U.S. National 

Conference on Earthquake Engineering, Boston, MA. 

Bruneau, M. and Vian, D., 2002, “Experimental investigation of P−Δ effects 

to collapse during earthquakes,” Proc. 12th European Conference on 

Earthquake Engineering, London, UK. 
A-30 A: Detailed Summary of Previous Research FEMA P440A 

Vian, D. and Bruneau, M., 2001, Experimental investigation of P−Δ effects 

to collapse during earthquakes, Report MCEER-01-0001, 

Multidisciplinary Research for Earthquake Engineering Research Center, 

Buffalo, N.Y. 

A.2.6 Methods to Evaluate the Dynamic Stability of Structures – 

Shake Table Tests and Nonlinear Dynamic Analyses 

Author: 

Kanvinde, A.M. (2003) 

Abstract: 

This paper aims to understand the phenomenon of dynamic instability in 

structures better, and to suggest and evaluate methods to predict collapse 

limit states of structures during earthquakes, based on findings of recent 

shake table tests and nonlinear dynamic analyses conducted at Stanford 

University. Simple models that collapsed due to a story mechanism were 

used as test specimens. Data from nineteen experiments suggest that current 

methods of nonlinear dynamic analysis (using the OpenSees program in this 

case) are accurate and reliable for predicting collapse and tracing the path of 

the structure down to the ground during collapse. Moreover, it is found from 

the experiments that for non-degrading structures, an estimate of collapse 

drift based on a static pushover analysis can be successfully applied to 

predict the dynamic collapse or instability due to P-Δ effects. The rationale 

for this is that the structure has an elongated period at the point of global 

instability, virtually insulating it from the ground motion and justifying the 

use of a static-analysis-based drift. Finally, the paper directs the readers to a 

valuable database of test data from collapse tests of a “clean” structure, 

which can be used for further verification studies. 

Summary: 

This was a brief assessment of the collapse performance of two 2D singlestory single-bay frames using results from static pushover and incremental 

dynamic analyses (IDA) and correlating them with experimental tests. The 

frames tested and simulated had a rigid beam that forced the creation of 

plastic hinges in the two columns. By using easily replaceable steel plates for 

the columns it was possible to repeat the shake table tests at various 

intensities and in effect experimentally reproduce IDA-like results. A total of 

19 uniaxial shake table tests was performed using two ground motion records 

and two different structures (that is, two different column types). 
FEMA P440A A: Detailed Summary of Previous Research A-31 

Structure A was ductile and, having a ratio of yield base shear to weight of 

1.03, was practically impossible to collapse. Structure B had weaker 

columns, produced by drilling holes at the bottom and top of the steel plates, 

and was thus prone to a story-mechanism collapse due to significant P-Δ

effects (Figure A-14a). The two structures were also simulated in OpenSEES 

using a Giufre-Menegotto-Pinto hysteresis model for the column hinges 

(Figure A-14b) without any cyclic deterioration, and an exact corrotational 

formulation for geometric nonlinearities. 

Representative Figures: 

 

Figure A-14 (a) Static pushover curves for the two frames and (b) modeling 

of the column plastic hinges in OpenSEES. 

Summary of Findings: 

The evidence presented shows that nonlinear dynamic analysis is a reliable 

tool to predict the actual behavior of the two structures. The usefulness of 

static pushover was also proven, at least when cyclic deterioration is not an 

issue, as the collapse drift calculated statically was accurately matched by 

both incremental dynamic analysis and the shake table experiments. 

Compared to the response at lower intensities, larger scatter was observed 
A-32 A: Detailed Summary of Previous Research FEMA P440A 

close to collapse both in the experiments and in the dynamic analyses, even 

when using the same earthquake record. This suggests an increased 

sensitivity of the actual results to the uncertainties in the initial condition of 

the structure, and an increased difficulty in predicting the collapse drift or 

intensity level even for such simple specimens. 

Relevant Publications: 

Kanvinde, A.M., 2003, “Methods to evaluate the dynamic stability of 

structures – shake table tests and nonlinear dynamic analyses,” EERI 

Annual Student Paper Competition, Proceedings of 2003 EERI Meeting, 

Portland, OR. 

Vian, D. and Bruneau, M., 2001, Experimental investigation of P-Δ effects to 

collapse during earthquakes, Report No. MCEER-01-0001, 

Multidisciplinary Research for Earthquake Engineering Research Center, 

Buffalo, N.Y. 

A.2.7 Seismic Performance, Capacity and Reliability of 

Structures as Seen Through Incremental Dynamic Analysis 

Authors: 

Vamvatsikos, D. and Cornell, C.A. (2005) 

Abstract: 

Incremental Dynamic Analysis (IDA) is an emerging structural analysis 

method that offers thorough seismic demand and limit-state capacity 

prediction capability by using a series of nonlinear dynamic analyses under a 

suite of multiply scaled ground motion records. Realization of its 

opportunities is enhanced by several innovations, such as choosing suitable 

ground motion intensity measures and representative structural demand 

measures. In addition, proper interpolation and summarization techniques for 

multiple records need to be employed, providing the means for estimating the 

probability distribution of the structural demand given the seismic intensity. 

Limit-states, such as the dynamic global system instability, can be naturally 

defined in the context of IDA. The associated capacities are calculated so that 

when properly combined with probabilistic seismic hazard analysis, they 

allow the estimation of the mean annual frequencies of limit-state 

exceedance. 

IDA is resource-intensive. Thus the use of simpler approaches becomes 

attractive. The IDA can be related to the computationally faster Static 
FEMA P440A A: Detailed Summary of Previous Research A-33 

Pushover (SPO), enabling a fast and accurate approximation to be established 

for SDOF systems. By investigating oscillators with quadrilinear backbones 

and summarizing the results into a few empirical equations, a new software 

tool, SPO2IDA, is produced here that allows direct estimation of the 

summarized IDA results. Interesting observations are made regarding the 

influence of the period and the backbone shape on the seismic performance 

of oscillators. Taking advantage of SPO2IDA, existing methodologies for 

predicting the seismic performance of first-mode-dominated, MDOF systems 

can be upgraded to provide accurate estimation well beyond the peak of the 

SPO. 

The IDA results may display a large record-to-record variability. By 

incorporating elastic spectrum information, efficient intensity measures can 

be created that reduce such dispersions, resulting in significant computational 

savings. By employing either a single optimal spectral value, a vector of two 

or a scalar combination of several spectral values, significant efficiency is 

achieved. As the structure becomes damaged, the evolution of such optimally 

selected spectral values is observed, providing intuition about the role of 

spectral shape in the seismic performance of structures. 

Summary: 

The research presented is entirely based on the concept of incremental 

dynamic analysis (IDA). The methodology is established and is extensively 

used to derive (among others) the collapse capacity of MDOF frames. Of 

particular importance is the exploration of the connection between the 

fractile IDA curves and the pushover. The authors propose the use of a 5% 

damped SDOF oscillator with a complex quadrilinear backbone (including a 

hardening, a softening and a residual plateau segment) with moderately 

pinching hysteresis to capture the pushover curve shape of actual MDOF 

frames. No cyclic degradation was considered but in the process millions of 

nonlinear dynamic SDOF analyses are performed for 30 records and a wide 

variety of oscillator backbones and periods. The results are fitted and 

incorporated into a complex R-μ-T relationship, realized in the form of the 

SPO2IDA Excel tool. 

The proposed tool is applied to the MDOF prediction problem using the 

worst-case pushover concept. This is defined as the pushover that leads to the 

earliest post-peak collapse, stipulating that it will also help find the collapse 

mechanism that a dynamic analysis would predict. By applying the 

SPO2IDA tool on the worst-case pushover the complete IDA curves of 
A-34 A: Detailed Summary of Previous Research FEMA P440A 

MDOF frames are generated for a 5-story, a 9-story and a 20-story steel 

frame. 

Representative Figures: 

0 1

0

1

ductility,  =  /  yield

strength reduction factor, R = F / F yield

c f

ac ah

r

non−negative

(hardening) negative

residual plateau

elastic fracture 

Figure A-15 The backbone of the studied oscillator. 

Figure A-16 The interface of the SPO2IDA tool for moderate periods. 
FEMA P440A A: Detailed Summary of Previous Research A-35 

0 1 2 3 4 5 6 7 8

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

 ac= −25%

ductility,  =  /  yield

strength reduction factor, R = Sa / Syield

a

 20%

 ac ac

= −200% = −50%

 ac

= −25% (50% IDA)

 ac

= −50% (50% IDA)

 ac

= −200% (50% IDA)

0 1 2 3 4 5 6 7 8

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

 ah= 10%

ductility,  =  /  yield

strength reduction factor, R = Sa / Syield

a

−200%

 ah= 25%

 ah

= 50%

 ah

= 75%

 ah

= 10% (50% IDA)

 ah

= 25%, 50%, 75% (50% IDAs)

 

Figure A-17 Influence of (a) the post-peak and (b) post-yield stiffness on the median dynamic response of the 

oscillator. When the negative segment is the same then the hardening slope has a negligible effect. 

 

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

0.05

0.1

0.15

0.2

0.25

0.3

roof drift ratio,  roof

base shear / mass (g)

first−mode

two modes SRSS

two modes SRSS, then uniform

two modes SRSS, then inverse

0 0.05 0.1 0.15

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

peak roof drift ratio, roof

"first−mode" spectral acceleration Sa(T1,5%) (g)

IDA

SPO2IDA

50% IDA 

84% IDA 

16% IDA 

Figure A-18 (a) Influence of the load pattern on the pushover curve shape and (b) the predicted versus actual 

dynamic response for various intensity levels using SPO2IDA and the worst-case pushover for a 

9-story steel moment frame. 

Summary of Findings: 

Regarding SDOF oscillators, it was found that the shape of the backbone 

curve has a very complex effect on the dynamic response. For example, the 

negative slope, the hardening deformation and the residual strength level are 

the three parameters that dominate (e.g. Figure A-17a). On the other hand, 

the hardening slope is not as important, while the residual plateau is 

significant only when long or high enough. Surprisingly, the peak strength of 

the oscillator is found to be relatively unimportant when the subsequent 

negative-slope segment is fixed. Thus some very different backbones exist 

that have almost the same performance (Figure A-17b). 
A-36 A: Detailed Summary of Previous Research FEMA P440A 

For MDOF application, it was found that the worst-case pushover is not 

always easy to estimate. For a 9-story steel frame building, in order to find 

the dominant collapse mechanism it was necessary to change the load pattern 

after the peak of the pushover and actually try several combinations before 

getting an acceptable shape (Figure A-18a). Under this condition, the use of 

SPO2IDA was found to provide accurate results for first-mode-dominated 

frames. A test conducted on a 20-story frame, where higher modes are a 

significant issue, showed that it was impossible to get good agreement in the 

early inelastic range. Curiously, when close to collapse, this approach still 

managed to provide an accurate answer, leading to the observation that an 

SDOF can predict reliably the collapse capacity of complex buildings. 

Relevant Publications: 

Vamvatsikos, D., and Cornell, C.A., 2005, Seismic performance, capacity 

and reliability of structures as seen through incremental dynamic 

analysis, John A. Blume Earthquake Engineering Research Center, 

Report No. 151, Department of Civil and Environmental Engineering, 

Stanford University, Stanford, California. 

Ibarra, L.F., and Krawinkler, H., 2005, Global collapse of frame structures 

under seismic excitations, John A. Blume Earthquake Engineering 

Research Center, Report No. 152, Department of Civil and 

Environmental Engineering, Stanford University, Stanford, California. 

A.2.8 Hysteretic Models that Incorporate Strength and Stiffness 

Deterioration 

Authors: 

Ibarra, L., Medina, R.A., and Krawinkler, H., (2005) 

Abstract: 

This paper presents the description, calibration and application of relatively 

simple hysteretic models that include strength and stiffness deterioration 

properties, features that are critical for demand predictions as a structural 

system approaches collapse. Three of the basic hysteretic models used in 

seismic demand evaluation are modified to include deterioration properties: 

bilinear, peak-oriented, and pinching models. The modified models include 

most of the sources of deterioration, namely, various modes of cyclic 

deterioration and softening of the post-yielding stiffness, and they also 

account for a residual strength after deterioration. The models incorporate an 

energy-based deterioration parameter that controls four cyclic deterioration 
FEMA P440A A: Detailed Summary of Previous Research A-37 

modes: basic strength, post-capping strength, unloading stiffness, and 

accelerated reloading stiffness deterioration modes. Calibration of the 

hysteretic models on steel, plywood, and reinforced-concrete components 

demonstrates that the proposed models are capable of simulating the main 

characteristics that influence deterioration. An application of a peak-oriented 

deterioration model in the seismic evaluation of SDOF systems is illustrated. 

The advantages of using deteriorating hysteretic models for obtaining the 

response of highly inelastic systems are discussed. 

Summary: 

This study presents an improved piece-wise linear hysteretic model that is 

capable of considering stiffness degradation, pinching cyclic strength 

degradation as well as in-cycle strength degradation. The paper has a 

threefold objective: (a) to describe the properties of proposed hysteretic 

models that incorporate both monotonic and cyclic deterioration; (b) to 

illustrate the calibration of these hysteretic models on component tests of 

steel, plywood, and reinforced-concrete specimens; and (c) to exemplify the 

utilization of the hysteretic models in the seismic response evaluation of 

SDOF systems. In this study the term deteriorating hysteretic models refers 

to models that include strength deterioration of the backbone curve or cyclic 

deterioration or both. 

As shown in Figure A-19, the model considers a backbone curve consisting 

of four linear segments: an elastic segment until a yield displacement, postyield strain-hardening segment until the ‘capping’ displacement is reached, a 

post-capping segment with negative stiffness (that is, in-cycle degradation), 

and a final residual horizontal segment. 

Representative Figures: 

Figure A-19 The backbone of the proposed hysteretic model. 
A-38 A: Detailed Summary of Previous Research FEMA P440A 

Figure A-20 Basic rules for peak-oriented hysteretic model. 

Figure A-21 Pinching hysteretic model: (a) basic model rules; and (b) modification if reloading deformation is 

to the right of break point. 

(a) 

Figure A-22 Examples of comparisons between experimental and analytical results for (a) non-ductile 

reinforced concrete column; and (b) plywood shear wall. 

Summary of Findings: 

The hysteretic models include a post-capping softening branch, residual 

strength, and cyclic deterioration. Cyclic deterioration permits deterioration 

to be traced as a function of past loading history, the rate of deterioration 

(which depends on the hysteretic energy dissipated in past cycles), and on a 
FEMA P440A A: Detailed Summary of Previous Research A-39 

reference energy dissipation capacity. Four modes of cyclic deterioration can 

be simulated: basic strength, postcapping strength, unloading stiffness, and 

accelerated reloading stiffness deterioration. Based on calibrations performed 

with experimental data from component tests of steel, wood, and reinforcedconcrete specimens, they concluded that it appears that, for a given 

component, the backbone characteristics and a single parameter that controls 

all four modes of cyclic deterioration are adequate to represent component 

behavior regardless of the loading history. 

Results from the seismic evaluation of various SDOF systems demonstrate 

that strength deterioration becomes a dominant factor when the response of a 

structure approaches the limit state of collapse. At early stages of inelastic 

behavior, both deteriorating and nondeteriorating systems exhibit similar 

responses. The differences become important when the post-capping stiffness 

is attained in the response. 

Relevant Publications: 

Ibarra, L., Medina, R., Krawinkler, H., 2005, “Hysteretic models that 

incorporate strength and stiffness deterioration, Earthquake Engineering 

and Structural Dynamics, Vol. 34, no. 12, pp. 1489-1511. 

Ibarra, L.F., and Krawinkler, H., 2005, Global collapse of frame structures 

under seismic excitations, John A. Blume Earthquake Engineering 

Research Center, Report No. 152, Department of Civil and 

Environmental Engineering, Stanford University, Stanford, California. 

Ibarra, L., Medina, R., Krawinkler, H., 2002, “Collapse assessment of 

deteriorating SDOF systems,” Proc. 12th European Conference on 

Earthquake Engineering, London, UK, Paper 665, Elsevier Science Ltd. 

A.2.9 Global Collapse of Frame Structures Under Seismic 

Excitations 

Authors: 

Ibarra, L.F. and Krawinkler, H. (2005) 

Abstract: 

Global collapse in earthquake engineering refers to the inability of a 

structural system to sustain gravity loads in the presence of seismic effects. 

This research proposes a methodology for evaluating global incremental 

(sidesway) collapse based on a relative intensity measure instead of an 

Engineering Demand Parameter (EDP). The relative intensity is the ratio of 
A-40 A: Detailed Summary of Previous Research FEMA P440A 

ground motion intensity to a structure strength parameter, which is increased 

until the response of the system becomes unstable, which means that the 

relative intensity - EDP curve becomes flat (that is, with zero slope). The 

largest relative intensity is referred to as “collapse capacity.” 

In order to implement the methodology, deteriorating hysteretic models are 

developed to represent the monotonic and cyclic behavior of structural 

components. Parameter studies that utilize these deteriorating models are 

performed to obtain collapse capacities and quantify the effects of system 

parameters that most influence the collapse for SDOF and MDOF structural 

systems. The range of collapse capacity due to record-to-record variability 

and uncertainty in the system parameters is evaluated. The latter source of 

dispersion is quantified by means of the first order second moment (FOSM) 

method. The studies reveal that softening of the post-yield stiffness in the 

backbone curve (postcapping stiffness) and the displacement at which this 

softening commences (defined by the ductility capacity) are the two system 

parameters that most influence the collapse capacity of a system. Cyclic 

deterioration appears to be an important but not the dominant issue for 

collapse evaluation. P-Δ effects greatly accelerate collapse of deteriorating 

systems and may be the primary source of collapse for flexible, but very 

ductile, structural systems. 

The dissertation presents applications of the proposed collapse methodology 

to the development of collapse fragility curves and the evaluation of the 

mean annual frequency of collapse. 

An important contribution is the development of a transparent methodology 

for the evaluation of incremental collapse, in which the assessment of 

collapse is closely related with the physical phenomena that lead to this limit 

state. The methodology addresses the fact that collapse is caused by 

deterioration in complex assemblies of structural components that should be 

modeled explicitly. 

Summary: 

The authors used an oscillator with a quadrilinear backbone curve with 

hardening, softening and residual segment to conduct an extensive 

parametric study. Pinching, peak-oriented and (bilinear-like) kinematic 

hysteresis rules were considered, while the cyclic degradation of the 

backbone stiffness and strength and of the unloading/reloading stiffness were 

also included. The effects of P-Δ were added separately, as a rotation of the 

backbone around the center of the axes. The investigation used 40 “ordinary” 
FEMA P440A A: Detailed Summary of Previous Research A-41 

ground motion records and it was focused on determining the influence of all 

the parameters on the collapse capacity, which was considered to occur in an 

IDA (incremental dynamic analysis) fashion, when numerical instability 

occurred or when the IDA curve becomes horizontal. 

Additionally a number of 2D single-bay frames with 3, 6, 9, 12, 15 and 18 

stories was considered; they were designed according to a strong column, 

weak beam, concept, with the beam hinges having a hysteretic model of the 

same type as the one used for the SDOF studies. By maintaining a uniform 

hysteretic model for all beam hinges and globally varying its parameters, 

another parametric study was performed, focused now on the effect of the 

hysteretic parameters on the MDOF response. 

Representative Figures: 

Figure A-23 (a) Backbone curve used for the investigations and (b) post-peak 

stiffness cyclic deterioration considered. 

Figure A-24 (a) Effect of the post-peak stiffness to the median collapse capacity spectra for a peak-oriented 

model and (b) the ratio of collapse capacities for different hysteretic models. 
A-42 A: Detailed Summary of Previous Research FEMA P440A 

Figure A-25 Effect of (a) post-yield slope and (b) reloading stiffness cyclic deterioration on the collapse capacity. 

Figure A-26 (a) Effect of the beam-hinge hysteretic model on the median MDOF collapse capacity and (b) the 

generation of an equivalent SDOF system by using an auxiliary backbone curve to incorporate 

P-Δ. 

Summary of Findings: 

For the SDOF oscillator it was found that there is a complex interplay 

between the parameters that defines the dynamic response. Their relative 

values are significant. The cyclic deterioration was found to be unimportant 

in the pre-peak region and only mildly important post-peak. Furthermore, its 

influence does not depend on the type of ground motion, i.e. whether it is 

near or far field or long in duration. The peak-strength ductility and the postpeak slope are the most dominant parameters. Regarding the hysteresis laws, 

kinematic hysteresis produces lower capacities than the pinching or the peakoriented, which are roughly similar. The residual strength becomes important 

only when it is large enough. Finally, the application of damping in a massproportional or stiffness-proportional formulation becomes an important 

issue after yielding, as the tangent stiffness is constantly changing. 
FEMA P440A A: Detailed Summary of Previous Research A-43 

The study of the MDOF frames concluded that these higher-mode-influenced 

frames fail mostly due to a lower-story mechanism. There is a large 

dependence of the collapse capacity on the first-mode period. In general the 

effects of the hinge hysteresis and backbone parameters were similar to the 

SDOF results. It was also observed that the inelastic instability coefficient 

(i.e. the difference between the post-peak slope with and without P-Δ) was 

often much larger than the elastic stability coefficient. Thus, surrogate 

SDOFs need a separate inclusion of the P-Δ effects in the pre- and post-peak 

regions. Such equivalent SDOFs were shown to have good accuracy in 

predicting the collapse capacity. 

Relevant Publications: 

Ibarra, L.F., and Krawinkler, H., 2005, Global collapse of frame structures 

under seismic excitations, John A. Blume Earthquake Engineering 

Research Center, Report No. 152, Department of Civil and 

Environmental Engineering, Stanford University, Stanford, California. 

Rahnama, M. and Krawinkler, H., 1993, Effect of soft soils and hysteresis 

models on seismic design spectra, John A. Blume Earthquake 

Engineering Research Center, Report No. 108, Department of Civil 

Engineering, Stanford University, Stanford, California. 

Medina, R., 200), Seismic demands for nondeteriorating frame structures 

and their dependence on ground motions, Ph.D.. dissertation submitted 

to the Department of Civil and Environmental Engineering, Stanford 

University, Stanford, California. 

Ibarra, L., Medina, R., and Krawinkler, H., 2002, “Collapse assessment of 

deteriorating SDOF systems,” Proc. 12th European Conference on 

Earthquake Engineering, London, UK, Paper 665, Elsevier Science Ltd. 

A.2.10 Object-Oriented Development of Strength and Stiffness 

Degrading Models for Reinforced Concrete Structures 

Author: 

Kaul, R. (2004) 

Abstract: 

The aim of this research is to develop structural simulation models that can 

capture the strength and stiffness degradation of reinforced concrete frames 

up to collapse under earthquake-induced motions. The key modeling aspects 

of the element formulations include: (1) rigorous modeling of large 

deformation response, (2) flexural yielding and inelastic interaction between 
A-44 A: Detailed Summary of Previous Research FEMA P440A 

axial force and moment (3) degradation of the element stiffness under cyclic 

loading and (4) axial force-moment-shear interaction for shear-critical 

reinforced concrete columns. Beam-column models are developed and 

implemented in an object-oriented analysis framework called OpenSees 

(Open System for Earthquake Engineering Simulation). The large 

deformation element formulations employ an updated Lagrangian approach. 

Inelastic models are based on stress-resultant plasticity to simulate inelastic 

hardening and softening response under combined axial loads and bending. A 

two-surface evolution model is proposed for combined nonuniform 

expansion or contraction and kinematic motion of the yield surface. The 

yield surface can be used to simulate inelastic section response at integration 

points along a beam-column element (distributed plasticity) or inelastic 

hinging at the ends of a beam-column element (concentrated plasticity). In 

the concentrated plasticity approach, the element between the hinges is quasielastic, in which hysteretic models are developed to model the cyclic 

degradation. This concentrated plasticity model is extended to simulate 

shear-critical column behavior, including shear strength degradation and 

failure, interaction between axial and shear forces, and pinched cyclic 

response. Implementation of the models in OpenSees is planned and 

structured using object-oriented programming concepts. Individual 

components of the inelastic modeling problem are identified and the 

interactions between the governing classes are established. The models are 

implemented in a hierarchal structure, which provides a modular and 

extensible software design. The accuracy and the capabilities of the proposed 

models are verified by comparing the analytical results with the experimental 

data. The models developed as part of this research provide ideal tools for 

conducting extensive application studies. An extensible framework is 

provided to facilitate tool development for nonlinear or inelastic analysis. 

Summary: 

The objective was the creation, and incorporation into OpenSEES, of a 

beam-column element with concentrated plasticity, that is appropriate for 

multiaxial loading of older, shear-critical RC columns. The element has been 

based on a yield-surface formulation and the focus was on modeling the 

multiaxial response of a complete RC section. The model incorporates incycle strength degradation, allowing for a quadrilinear backbone with a 

negative stiffness segment (Figure A-27). Inelastic hardening and softening 

were formulated according to a combined kinematic and isotropic hardening 

rule with either peak-oriented (for moment) or pinching (for shear) hysteretic 

rules (Figure A-28). There is little provision for cyclic degradation; the 

formulation is entirely based on the peak plastic strains and rotations, and the 
FEMA P440A A: Detailed Summary of Previous Research A-45 

direction of evolution, so depending on the details there may be no cyclic 

degradation. 

The significant advantage of the models is their apparent extensibility and the 

possibilities for easy modification and incorporation into a variety of 

elements, an inherent feature of the object-oriented programming upon which 

OpenSEES has been built. 

The model behavior has been calibrated and tested against a variety of RC 

beam-column experiments, including both shear and moment-critical 

columns, as well as a set of theoretical solutions for large deformation 

response (Figure A-29). 

Representative Figures: 

Figure A-27 Idealization of the (a) flexure spring and (b) shear spring backbones. 

 

Figure A-28 (a) Full and (b) half cycle pinching hysteresis for the shear spring. 
A-46 A: Detailed Summary of Previous Research FEMA P440A 

Figure A-29 Comparison of calculated versus experimental results for (a) a moment-critical column and (b) a 

shear-critical column. 

Summary of Findings: 

A reliable and extensible concentrated-plasticity beam-column element was 

created for RC members. Extensive testing and calibration has shown good 

agreement for a variety of experimental results, including shear-critical 

columns, large deformations, and planar moment and axial-force interaction. 

The only serious limitation is the limited formulation of cyclic degradation, 

an issue that can be potentially solved with the incorporation of damage 

models. The extension to 3D beam-column elements is somewhat hampered. 

Appropriate yield surface and evolution rules have not been incorporated, 

although the hysteretic material models presented are directly usable. The 

absence of the bond-slip effect and longitudinal reinforcement development 

in the element springs, are issues that still remain to be addressed. 

Relevant Publications: 

Kaul, R., 2004, Object oriented development of strength and stiffness 

degrading models for reinforced concrete structures, Ph.D. Thesis, 

Department of Civil and Environmental Engineering, Stanford 

University, Stanford, California. 

McKenna, F.T. (1997), Object-oriented finite element programming: 

framework for analysis, algorithms, parallel computing, Ph.D. 

Dissertation, University of California, Berkeley, California. 

Mehanny, S.S.F., and Deierlein, G.G., 2001, “Seismic collapse assessment of 

composite RCS moment frames,” Journal of Structural Engineering, 

ASCE, Vol. 127(9). 
FEMA P440A A: Detailed Summary of Previous Research A-47 

El-Tawil, S., 1996, Inelastic dynamic analysis of mixed steel-concrete space 

frames, Ph.D. Dissertation, Cornell University, Ithaca, NY. 

Elwood, K.J., 2002, Shake table tests and analytical studies on the gravity 

load collapse of reinforced concrete frames, Ph.D. Dissertation, 

University of California, Berkeley, California. 

A.2.11 Shake Table Tests and Analytical Studies on the Gravity 

Load Collapse of Reinforced Concrete Frames 

Author: 

Elwood, K.J. (2002). 

Abstract: 

An empirical model, based on the evaluation of results from an experimental 

database, is developed to estimate the drift at shear failure of existing 

reinforced concrete building columns. A shear-friction model is also 

developed to represent the general observation from experimental tests that 

the drift at axial failure of a shear-damaged column is directly proportional to 

the amount of transverse reinforcement and is inversely proportional to the 

magnitude of the axial load. The two drift-capacity models are incorporated 

in a nonlinear uniaxial constitutive model implemented in a structural 

analysis platform to allow for the evaluation of the influence of shear and 

axial load column failures on the response of a building. Shake table tests 

were designed to observe the process of dynamic shear and axial load 

failures in reinforced concrete columns when an alternative load path is 

provided for load redistribution. The results from these tests provide data on 

the dynamic shear strength and the hysteretic behavior of columns failing in 

shear, the loss of axial load capacity after shear failure, the redistribution of 

loads in a frame after shear and axial failures of a single column, and the 

influence of axial load on each of the above-mentioned variables. An 

analytical model of the shake table specimens, incorporating the proposed 

drift-capacity models to capture the observed shear and axial load failures, 

provides a good estimate of the measured response of the specimens. 

Summary: 

The objective was the creation and incorporation into OpenSEES of a beamcolumn element with concentrated plasticity that is appropriate for multiaxial 

loading of older, shear-critical RC columns. The element has been based on a 

yield-surface formulation and the focus was on modeling the multiaxial 

response of a complete RC section. The model incorporates in-cycle strength 
A-48 A: Detailed Summary of Previous Research FEMA P440A 

degradation, allowing for a quadrilinear backbone with a negative stiffness 

segment (Figure A-30). Inelastic hardening and softening were formulated 

according to a combined kinematic and isotropic hardening rule with either 

peak-oriented (for moment) or pinching (for shear) hysteretic rules (Figure 

A-31). There is little provision for cyclic degradation; the formulation is 

entirely based on the peak plastic strains and rotations, and the direction of 

evolution, so depending on the details there may be no cyclic degradation. 

The significant advantage of the models is their apparent extensibility and the 

possibilities for easy modification and incorporation into a variety of 

elements, an inherent feature of the object-oriented programming upon which 

OpenSEES has been built. 

Representative Figures: 

 

Figure A-30 Use of Sezen model to estimate (a) shear capacity and (b) displacement ductility capacity. 

 

Figure A-31 Comparison of the Sezen shear strength model and the proposed drift capacity model. 
FEMA P440A A: Detailed Summary of Previous Research A-49 

Figure A-32 Redefinition of backbone in Elwood’s model after shear failure 

is detected. 

Figure A-33 Comparison of calculated versus experimental results for two shear-critical columns. 

Summary of Findings: 

Given the lack of agreement between existing models for the drift at shear 

failure and results from an experimental database of shear-critical building 

columns, two empirical models were developed to provide a more reliable 

estimate of the drift at shear failure for existing reinforced concrete columns: 

Based on shear-friction concepts and the results from 12 columns tested to 

axial failure, a model was also developed to estimate the drift at axial failure 

for a shear-damaged column: 
A-50 A: Detailed Summary of Previous Research FEMA P440A 

The capacity models for the drift at shear and axial load failure were used to 

initiate the strength degradation of a uniaxial material model implemented in 

the OpenSees analytical platform (OpenSees, 2002). When attached in series 

with a beam-column element, the material model can be used to model either 

shear or axial failure, or both if two materials are used in series. Based on 

experimental evidence suggesting that an increase in lateral shear 

deformations may lead to an increase in axial deformations and a loss of 

axial load, shear-to-axial coupling was incorporated in the material model to 

approximate the response of a column after the onset of axial failure. 

Relevant Publications: 

Elwood, K.J., 2002, Shake table tests and analytical studies on the gravity 

load collapse of reinforced concrete frames, Ph.D. Dissertation, 

University of California, Berkeley. 

Elwood, K.J., and Moehle, J.P., 2003, Shake table tests and analytical 

studies on the gravity load collapse of reinforced concrete frames. 

Pacific Earthquake Engineering Research Center, PEER Report 2003/01, 

University of California, Berkeley, Calif. 

Elwood, K.J., 2004, “Modeling failures in existing reinforced concrete 

columns,” Can. J. Civ. Eng., 31: 846–859 (2004) 

Elwood, K.J., and Moehle, J.P., 2005, “Drift capacity of reinforced concrete 

columns with light transverse reinforcement.” Earthquake Spectra, 

Volume 21, No. 1, pp. 71–89, 

A.2.12 Determination of Ductility Factor Considering Different 

Hysteretic Models 

Authors: 

Lee, L.H., Han, S.W., and Oh, Y.H. (2003) 

Abstract: 

In current seismic design procedures, base shear is calculated by the elastic 

strength demand divided by the strength reduction factor. This factor is well 

known as the response modification factor, R, which accounts for ductility, 

overstrength, redundancy, and damping of a structural system. In this study, 

the R factor accounting for ductility is called the ductility factor, Rμ. The Rμ
FEMA P440A A: Detailed Summary of Previous Research A-51 

factor is defined as the ratio of elastic strength demand imposed on the SDOF 

system to inelastic strength demand for a given ductility ratio. The Rμ factor 

allows a system to behave inelastically within the target ductility ratio during 

the design level earthquake ground motion. The objective of this study is to 

determine the ductility factor considering different hysteretic models. It 

usually requires large computational efforts to determine the Rμ factor. In 

order to reduce the computational efforts, the Rμ factor is prepared as a 

functional form in this study. For this purpose, statistical studies are carried 

out using forty different earthquake ground motions recorded at a stiff soil 

site. The Rμ factor is assumed to be a function of the characteristic parameters 

of each hysteretic model, target ductility ratio and structural period. The 

effects of each hysteretic model on the Rμ factor are also discussed. 

Summary: 

The focus of the research was the creation of an R-μ-T relationship that 

would include the effect of several backbone and hysteretic characteristics. 

The authors’ approach was to consider such effects as completely 

independent from each other by considering the following models: (a) 

kinematic hysteresis with a bilinear backbone having positive post-yield 

stiffness; (b) bilinear backbone and kinematic hysteresis with cyclic 

degradation of the reloading/unloading stiffness; (c) bilinear backbone with 

peak-oriented (Clough-like) hysteresis with cyclic strength degradation; and 

(d) bilinear backbone with pinching hysteresis (Figure A-34). Details of the 

models are provided by Kunnath et al., (1990). 

Using the elastic-perfectly-plastic model with kinematic hardening as a basis, 

the researchers investigated the influence of each of the four parameters and 

provided correction factors, in a “coefficient-like” method. These can be 

applied to an R-μ-T relationship based on the elastic-plastic system to 

account for the effect of each of the parameters. 
A-52 A: Detailed Summary of Previous Research FEMA P440A 

Representative Figures: 

 

Figure A-34 The parameters investigated: (a) backbone hardening ratio; 

(b) unloading/reloading cyclic stiffness degradation; (c) strength degradation; 

and (d) degree of pinching. 

Figure A-35 The effect of (a) cyclic strength degradation and (b) degree of pinching on the 

mean R-factor for a given ductility. 

Summary of Findings: 

The results presented show the effect of the cyclic degradation of strength or 

of reloading and unloading stiffness, and the degree of pinching on the mean 

R-factor observed for a given ductility for SDOF systems (Figure A-35). In 

general the effects are relatively small. Unfortunately the influence of such 

parameters is evaluated separately for each parameter, only for a relatively 

limited range of values, and always in relation to the pure elasto-plastic 

system. Still, the authors do perform verifications for systems having a 

combination of all such characteristics, thus providing evidence that the 
FEMA P440A A: Detailed Summary of Previous Research A-53 

proposed formulas can approximate more complex systems. Perhaps the 

greatest limitation of this research is that it does not apply to systems with incycle strength degradation. Only positive post-yield stiffnesses are 

considered. Therefore, the influence of several investigated parameters is 

small, and the results cannot be applied when negative backbone slopes are 

present. 

Relevant Publications: 

Lee, L.H, Han, S.W. and Oh, Y.H., 1999, “Determination of ductility factor 

considering different hysteretic models,” Earthquake Engineering and 

Structural Dynamics, Vol. 28, 957–977. 

Kunnath, S.K., Reinhorn, A.M. and Park, Y.J., 1990, “Analytical modeling 

of inelastic seismic response of RC structures,” Journal of Structural 

Engineering, ASCE, 116, 996–1017. 

A.2.13 Effects of Hysteresis Type on the Seismic Response of 

Buildings 

Authors: 

Foutch, D.A. and Shi, S. (1998) 

Abstract: 

Current design procedures account for inelastic behavior in a crude manner 

using the R factor. Although different R values are used for different building 

types, the determination of a specific R value was not done in a very 

consistent or scientific manner. The hysteresis behavior of members can be 

different depending on the material and member type. Buildings with 

members that dissipate energy through full hysteresis loops (for example, 

steel moment frames with compact members and no joint fracture) will 

respond differently from buildings with members that demonstrate strengthdegrading hysteresis behavior by having either non-compact steel members, 

concentric braces, or members with fractured joints. This paper will present 

results of a study that has closely examined these effects using both SDOF 

and MDOF systems. A procedure for developing reliability-based design 

methods which incorporates these effects will also be presented. 

Summary: 

A total of nine moment-resisting frames with three different configurations 

(3-story, 6-story and 9-story) were used in this study to examine the effect of 

the beam-hinge model on the seismic behavior of MDOF structures. Using a 
A-54 A: Detailed Summary of Previous Research FEMA P440A 

suite of 12 ground motion records, all structures (numbering 3x9x8) were 

analyzed for several R-factor levels (or approximately an equivalent number 

of earthquake intensity levels) and the results were summarized and 

compared with the buildings having a basic bilinear hinge with kinematic 

hardening (Figure A-36). 

Eight different hinge models were considered: (1) kinematic hardening with 

bilinear backbone (positive post-yield stiffness), (2) same as 1 but with cyclic 

strength degradation, (3) same as 1 but having peak-oriented (Clough-like) 

hysteresis, (4) same as 2 but with peak-oriented hysteresis, (5) same as 1 but 

with pinching hysteresis, (6) same as 2 but with pinching hysteresis, (7) 

fracturing connection model with pinching hysteresis and asymmetric 

backbones including a negative slope and a residual plateau at one direction, 

and (8) a purely elastic bilinear backbone that dissipates no energy (Figure 

A-37). 

Representative Figures: 

Figure A-36 Ratio of maximum displacement for all buildings and hinge 

types versus the hinge type 1 (kinematic hardening, no 

degradation). 
FEMA P440A A: Detailed Summary of Previous Research A-55 

Figure A-37 The hysteresis types considered for the beam-hinges. 
A-56 A: Detailed Summary of Previous Research FEMA P440A 

Summary of Findings: 

The results show that the investigated plastic hinge models have a relatively 

similar effect on the seismic response of the structures, regardless of the 

structural period. Even hinge models with no energy dissipation do not 

produce excessive demands. At most, for an R = 8 reduction factor, the 

differences that appear are in the order of 30-40% and they only appear for a 

limited range of periods and models. Nevertheless, only a single backbone 

with in-cycle strength degradation has been considered, and even then the 

negative slope exists only in one direction of loading. Therefore the 

conclusions may be of limited use in such cases. This is an interesting 

exercise and one of the very few investigations that has produced data on the 

actual impact of local, hinge-level models on the global MDOF response. 

Relevant Publications: 

Foutch, D.A. and Shi, S., 1998, “Effects of hysteresis type on the seismic 

response of buildings,” Proc. 6th U.S. National Conference on 

Earthquake Engineering, EERI, Seattle, WA. 

Shi, S. and Foutch, D.A., 1997, Evaluation of connection fracture and 

hysteresis type on the seismic response of steel buildings, Report No. 

617, Structural Research Series, Civil Engineering Studies, University of 

Illinois at Urbana-Champaign, Urbana, IL. 

A.2.14 Performance-Based Assessment of Existing Structures 

Accounting For Residual Displacements 

Authors: 

Ruiz-Garcia, J. and Miranda, E. (2005) 

Abstract: 

The first part of this investigation describes comprehensive statistical studies 

to quantify residual and maximum displacement demands of inelastic SDOF 

systems, considering a relatively large earthquake ground motion database, 

and considering a large number of structural parameters. The second part of 

this study focuses on the evaluation of permanent (residual) and maximum 

(transient) drift demands of multi-story framed building models under 

different levels of ground motion intensity. Both parts include the 

formulation and implementation of simplified probabilistic approaches to 

estimate maximum and residual displacement demands accounting for the 

uncertainty in the structural response and the ground motion hazard. The 

study provides information towards incorporating explicitly the evaluation of 
FEMA P440A A: Detailed Summary of Previous Research A-57 

residual displacement demands for assessing the seismic performance of 

existing structures, or for the preliminary design phase of new structures, 

where structural damage control is achieved through control of lateral 

deformation demands. 

Summary: 

This study examined the effect of hysteretic behavior of maximum 

deformations of SDOF systems subjected to a large ensemble of 240 ground 

motions recorded on firm sites in California. They considered seven different 

types of hysteretic behavior: elastoplastic, bilinear, modified Clough, 

Takeda, origin-oriented, moderate degrading and severely degrading models. 

The modified Clough, the Takeda and origin-oriented models only exhibit 

stiffness degradation while the moderate degrading and severely degrading 

systems exhibit both stiffness and cyclic strength degradation. This study 

computed mean ratios of maximum deformation of degrading hysteretic 

models to non-degrading ones. They also studied the effect of hysteretic 

behavior for systems subjected to ground motions recorded in very soft soil 

sites and near-fault ground motions influence by forward directivity. 

Representative Figures: 

(b) TK

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-15 -12 -9 -6 -3 0 3 6 9 12 15

Displacement Ductility

Normalized Force

(a) MC

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-15 -10 -5 0 5 10 15

Displacement Ductility 

Normalized Force

(c) O-O 

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-15 -10 -5 0 5 10 15

Displacement Ductility 

Normalized Force

Figure A-38 Hysteretic models used in this investigation that only have stiffness degradation. (a) ModifiedClough (MC); (b) Takeda model (TK); and (c) Origin-Oriented model (O-O). 
A-58 A: Detailed Summary of Previous Research FEMA P440A 

(b) MSD 

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-15 -12 -9 -6 -3 0 3 6 9 12 15

Displacement Ductility 

Noormalized Force 

(c) SSD

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-15 -12 -9 -6 -3 0 3 6 9 12 15

Displacement Ductility 

Normalized Force

Figure A-39 Hysteretic models used in this investigation with stiffness and cyclic strength degradation. 

(b) Moderate Degrading (MSD); and (c) Severely Degrading (SSD). 

(a)  = 3%

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

i,=3%/i,=0

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

(b)  = 5%

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

i,=5/i,=0

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

Figure A-40 Mean ratios of maximum deformation of bilinear to elastoplastic systems: (a) = 3%; 

and (b) = 5%. 

(b) SSD-2 model

SITE CLASS D

(mean of 80 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

i,SSD / i,EP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

(a) SSD-1 model

SITE CLASS D

(mean of 80 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

i,SSD / i,EP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

Figure A-41 Mean ratio of inelastic displacement demands in structural degrading and bilinear 

systems: (a) SSD-1 model; and (b) SSD-2 model. 
FEMA P440A A: Detailed Summary of Previous Research A-59 

(b) HC = 2.5 (Takeda model)

SITE CLASS D 

(mean of 80 ground motions)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

i,SD/i,EP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

(c) HC = 0.1 (origin-oriented model)

SITE CLASS D

(mean of 80 ground motions)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

i,SD/i,EP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

(a) HC (modified-Clough model)

SITE CLASS D

(mean of 80 ground motions)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

i,SD / i,EP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

Figure A-42 Influence of hysteretic behavior on maximum deformation for three types of stiffness-degrading 

systems: (a) Modified-Clough model; (b) Takeda model; and (c) Origin-oriented model. 

Summary of Findings: 

The effect of positive post-yield stiffness was a function of period of 

vibration and level of lateral strength relative to the strength required to 

maintain the system elastic. Positive post-elastic stiffness tended to reduce 

maximum deformation demands but for realistic values of post-yield 

stiffness, with exception of systems with very short periods, reductions were 

small (smaller than 10%). 

Maximum deformation demands of short-period degrading structures are, on 

average, larger than those of non-degrading systems. In general, the 

increment in displacement produced by degradation effects increases as the 

strength ratio increases (i.e., as the system becomes weaker relative to the 

lateral strength required to maintain the system elastic). For structures with 

periods longer than about 0.7 s, maximum deformation of degrading systems 

are on average either similar or slightly smaller than those of non-degrading 

systems. 
A-60 A: Detailed Summary of Previous Research FEMA P440A 

The effects of stiffness degradation were larger for structures on soft soil 

sites than those observed for structures on firm sites. For structures with 

periods of vibration shorter than the predominant period of the ground 

motion, the lateral displacement demands in stiffness-degrading systems are 

on average 25% larger than those of non-degrading systems and in order to 

control lateral deformations to levels comparable to those in non-degrading 

structures, stiffness-degrading structures in this spectral region need to be 

designed for higher lateral forces. 

Maximum inelastic displacement demands of stiffness-degrading systems are 

not significantly affected by the unloading stiffness provided that the 

reduction in unloading stiffness is small or moderate. However, for systems 

that unload toward the origin (that is, origin-oriented systems), or near the 

origin, maximum inelastic displacements are on average larger than 

maximum deformation demands of elastoplastic or bilinear systems and 

therefore the equal displacement rule should not be used for these systems. 

Hysteretic behaviors, in particular post-yield stiffness and unloading 

stiffness, have a large influence on residual displacement demands. 

Relevant Publications: 

Ruiz-Garcia, J. and Miranda, E., 2003, “Inelastic displacement ratio for 

evaluation of existing structures,” Earthquake Engineering and 

Structural Dynamics. 32(8), 1237-1258. 

Ruiz-Garcia, J. and Miranda, E., 2004, “Inelastic displacement ratios for 

structures built on soft soil sites”, Journal of Structural Engineering, 

130(12), December 2004, pp. 2051-2061 

Ruiz-Garcia, J. and Miranda, E., 2005, Performance-based assessment of 

existing structures accounting for residual displacements, John A. Blume 

Earthquake Engineering Center, Report No. 153, Department of Civil 

and Environmental Engineering, Stanford University, Stanford, 

California, 444 p. 

Ruiz-Garcia, J. and Miranda, E., 2006a, “Residual displacement ratios for the 

evaluation of existing structures,” Earthquake Engineering and 

Structural Dynamics, Vol. 35, pp. 315-336. 

Ruiz-Garcia, J. and Miranda, E., 2006b, “Inelastic displacement ratios for 

evaluation of structures built on soft soil sites,” Earthquake Engineering 

and Structural Dynamics, in press. 
FEMA P440A A: Detailed Summary of Previous Research A-61 

A.2.15 Inelastic Spectra for Infilled Reinforced Concrete Frames 

Authors: 

Dolsek, M. and Fajfar, P. (2004) 

Abstract: 

In two companion papers a simplified nonlinear analysis procedure for 

infilled reinforced concrete frames is introduced. In this paper a simple 

relation between strength reduction factor, ductility and period (R–μ–T

relation) is presented. It is intended to be used for the determination of 

inelastic displacement ratios and of inelastic spectra in conjunction with 

idealized elastic spectra. The R–μ–T relation was developed from results of 

an extensive parametric study employing a SDOF mathematical model 

composed of structural elements representing the frame and infill. The 

structural parameters used in the proposed R–μ–T relation, in addition to the 

parameters used in a usual (e.g. elasto-plastic) system, are ductility at the 

beginning of strength degradation, and the reduction of strength after the 

failure of the infills. Formulae depend also on the corner periods of the 

elastic spectrum. The proposed equations were validated by comparing 

results in terms of the reduction factors, inelastic displacement ratios, and 

inelastic spectra in the acceleration–displacement format, with those obtained 

by non-linear dynamic analyses for three sets of recorded and semi-artificial 

ground motions. A new approach was used for generating semi-artificial 

ground motions compatible with the target spectrum. This approach 

preserves the basic characteristics of individual ground motions, whereas the 

mean spectrum of the complete ground motion set fits the target spectrum 

excellently. In the parametric study, the R–μ–T relation was determined by 

assuming a constant reduction factor, while the corresponding ductility was 

calculated for different ground motions. The mean values proved to be 

noticeably different from the mean values as determined when based on a 

constant ductility approach, while the median values determined by the 

different procedures were between the two means. The approach employed in 

the study yields an R–μ–T relation which is conservative both for design and 

performance assessment (compared with a relation based on median values). 

Summary: 

An R-μ-T relationship was developed that is suitable for use with infilled 

frames having a quadrilinear elastic-positive-negative-residual backbone. 

The system used for the analysis contained separate springs, Takeda for the 

frame and shear-slip for the infill (Figure A-43). These were suitably 
A-62 A: Detailed Summary of Previous Research FEMA P440A 

calibrated to generate a backbone similar to the ones observed in pushovers 

of infilled frames (Figure A-44). Thus, only in-cycle strength degradation 

was considered, while any cyclic degradation issues were not investigated. 

The system was analyzed using three suites of ground motion records which 

were spectrum-matched to a target design spectrum. A parametric study of 

the quadrilinear system was then conducted by varying the period and the 

backbone parameters within prescribed values. 

Representative Figures: 

 

Figure A-43 The SDOF system: (a) force-displacement envelope; and (b) mathematic 

model. 

 

Figure A-44 The hysteretic behavior of the equivalent SDOF system. 
FEMA P440A A: Detailed Summary of Previous Research A-63 

Figure A-45 The influence of negative slope and residual plateau on the mean ductility for given R-factor. 

Summary of Findings: 

The negative slope was found to have a very small effect on the seismic 

response of the system when only a short drop in strength to the residual 

plateau is allowed (Figure A-45a). Actually, the level of the plateau 

combined with a worsening negative slope were shown to be quite important, 

lower plateaus increase the ductility demands considerably (Figure A-45b). 

Approximate values were proposed for the dispersion around the mean 

provided by the relationship. 

Relevant Publications: 

Dolsek, M. and Fajfar, P., 2004, “Inelastic spectra for infilled reinforced 

concrete frames,” Earthquake Engineering and Structural Dynamics, 

Vol. 33, 1395–1416. 

Dolsek, M. and Fajfar, P., 2000, “Simplified nonlinear seismic analysis of 

infilled reinforced concrete frames,” Earthquake Engineering and 

Structural Dynamics, Vol. 34, 49–66. 

Dolsek, M., 2002, Seismic response of infilled reinforced concrete frames, 

Ph.D. thesis, University of Ljubljana, Faculty of Civil and Geodetic 

Engineering, Ljubljana, Slovenia [in Slovenian]. 

ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-1 

Appendix B 

Quantile IDA Curves for 

Single-Spring Systems 

This appendix presents quantile (16th, 50th and 84th percentile) incremental 

dynamic analysis (IDA) curves from focused analytical studies on individual 

spring single-degree-of-freedom (SDOF) systems. These systems consist of 

spring types 1 through 8, with characteristics described in Chapter 3. This 

collection of curves is intended to present the range of results for 

representative short (T=0.5s), moderate (T=1.0s), and long (T=2.0s) period 

systems, both with and without cyclic degradation. In the figures, the 

vertical axis is the intensity measure Sa(T,5%), which is not normalized, and 

the horizontal axis is the maximum story drift ratio, max, in radians. IDA 

curves with cyclic degradation (black lines) are shown along with IDA 

curves without cyclic degradation (grey lines). Differences between the 

black and grey lines in the plots indicate the effect of cyclic degradation 

given the characteristics of the particular spring and period of vibration. 
B-2 B: Quantile IDA Curves for Single-Spring Systems ATC-62 

 Figure B-1 Quantile IDA curves plotted versus Sa(T,5%) for Spring 1a and Spring 1b with a period of T = 0.5s 

 Figure B-2 Quantile IDA curves plotted versus Sa(T,5%) for Spring 1a and Spring 1b with a period of T = 1.0s.

 Figure B-3 Quantile IDA curves plotted versus Sa(T,5%) for Spring 1a and Spring 1b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-3 

 Figure B-4 Quantile IDA curves plotted versus Sa(T,5%) for Spring 2a and Spring 2b with a period of T = 0.5s.

 Figure B-5 Quantile IDA curves plotted versus Sa(T,5%) for Spring 2a and Spring 2b with a period of T = 1.0s.

 Figure B-6 Quantile IDA curves plotted versus Sa(T,5%) for Spring 2a and Spring 2b with a period of T = 2.0s.
B-4 B: Quantile IDA Curves for Single-Spring Systems ATC-62 

 Figure B-7 Quantile IDA curves plotted versus Sa(T,5%) for Spring 3a and Spring 3b with a period of T = 0.5s.

 Figure B-8 Quantile IDA curves plotted versus Sa(T,5%) for Spring 3a and Spring 3b with a period of T = 1.0s.

 Figure B-9 Quantile IDA curves plotted versus Sa(T,5%) for Spring 3a and Spring 3b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-5 

 Figure B-10 Quantile IDA curves plotted versus Sa(T,5%) for Spring 4a and Spring 4b with a period of T = 0.5s.

 Figure B-11 Quantile IDA curves plotted versus Sa(T,5%) for Spring 4a and Spring 4b with a period of T = 1.0s.

 Figure B-12 Quantile IDA curves plotted versus Sa(T,5%) for Spring 4a and Spring 4b with a period of T = 2.0s.
B-6 B: Quantile IDA Curves for Single-Spring Systems ATC-62 

 Figure B-13 Quantile IDA curves plotted versus Sa(T,5%) for Spring 5a and Spring 5b with a period of T = 0.5s.

 Figure B-14 Quantile IDA curves plotted versus Sa(T,5%) for Spring 5a and Spring 5b with a period of T = 1.0s.

 Figure B-15 Quantile IDA curves plotted versus Sa(T,5%) for Spring 5a and Spring 5b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-7 

 Figure B-16 Quantile IDA curves plotted versus Sa(T,5%) for Spring 6a and Spring 6b with a period of T = 0.5s.

 Figure B-17 Quantile IDA curves plotted versus Sa(T,5%) for Spring 6a and Spring 6b with a period of T = 1.0s.

 Figure B-18 Quantile IDA curves plotted versus Sa(T,5%) for Spring 6a and Spring 6b with a period of T = 2.0s.
B-8 B: Quantile IDA Curves for Single-Spring Systems ATC-62 

 Figure B-19 Quantile IDA curves plotted versus Sa(T,5%) for Spring 7a and Spring 7b with a period of T = 0.5s.

 Figure B-20 Quantile IDA curves plotted versus Sa(T,5%) for Spring 7a and Spring 7b with a period of T = 1.0s.

 Figure B-21 Quantile IDA curves plotted versus Sa(T,5%) for Spring 7a and Spring 7b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-9 

 Figure B-22 Quantile IDA curves plotted versus Sa(T,5%) for Spring 8a and Spring 8b with a period of T = 0.5s.

 Figure B-23 Quantile IDA curves plotted versus Sa(T,5%) for Spring 8a and Spring 8b with a period of T = 1.0s.

 Figure B-24 Quantile IDA curves plotted versus Sa(T,5%) for Spring 8a and Spring 8b with a period of T = 2.0s.

FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-1 

versus Normalized Intensity Measures 

Appendix C 

Median IDA Curves for 

Multi-Spring Systems versus 

Normalized Intensity Measures 

This appendix contains normalized plots of median incremental dynamic 

analysis (IDA) curves from focused analytical studies on multi-spring singledegree-of-freedom (SDOF) systems. All systems are composed of two 

springs representing a primary lateral-force-resisting system and a secondary 

gravity system with the characteristics described in Chapter 3. Multi-spring 

systems carry a designation of “NxJa+1a” or “NxJa+1b” where “N” is the 

peak strength multiplier (N = 1, 2, 3, 5, or 9), “J” is the lateral-force-resisting 

spring number (J = 2, 3, 4, 5, 6, or 7), and 1a or 1b is the gravity system 

identifier. In all figures, the vertical axis is the normalized intensity measure 

R = Sa(T1,5%)/Say(T1,5%), and the horizontal axis is the maximum story drift 

ratio, max, in radians. The period of vibration for each system is indicated in 

parentheses. 

C.1 Visualization Tool 

Given the large volume of analytical data, customized algorithms were 

developed for post-processing, statistical analysis, and visualization of 

results. The accompanying CD includes an electronic visualization tool that 

was developed to view results of multi-spring studies. The tool is a 

Microsoft Excel based application with a user-interface that accesses a 

database of all available multi-spring data. By selecting a desired spring 

combination (“NxJa+1a” or “NxJa+1b”), stiffness level (stiff or flexible), and 

intensity measure (normalized or non-normalized), users can view the 

resulting quantile (median, 16th, and 84th percentile) IDA curves for the 

combination of interest. 
C-2 C: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

2

2.5

3

3.5

4

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2a (1.87s)

2x2a (1.32s)

3x2a (1.08s)

5x2a (0.84s)

9x2a (0.62s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

2

2.5

3

3.5

4

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2b (1.87s)

2x2b (1.32s)

3x2b (1.08s)

5x2b (0.84s)

9x2b (0.62s)

 

Figure C-1 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx2a and Nx2b with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2a+1a (1.53s)

2x2a+1a (1.18s)

3x2a+1a (1.00s)

5x2a+1a (0.80s)

9x2a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2b+1a (1.53s)

2x2b+1a (1.18s)

3x2b+1a (1.00s)

5x2b+1a (0.80s)

9x2b+1a (0.61s)

Figure C-2 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx2a+1a and Nx2b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2a+1b (1.53s)

2x2a+1b (1.18s)

3x2a+1b (1.00s)

5x2a+1b (0.80s)

9x2a+1b (0.61s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

2b+1b (1.53s)

2x2b+1b (1.18s)

3x2b+1b (1.00s)

5x2b+1b (0.80s)

9x2b+1b (0.61s)

Figure C-3 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx2a+1b and Nx2b+1b with mass M=8.87ton. 
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-3 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3a (1.87s)

2x3a (1.32s)

3x3a (1.08s)

5x3a (0.84s)

9x3a (0.62s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3b (1.87s)

2x3b (1.32s)

3x3b (1.08s)

5x3b (0.84s)

9x3b (0.62s)

 

Figure C-4 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx3a and Nx3b with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3a+1a (1.53s)

2x3a+1a (1.18s)

3x3a+1a (1.00s)

5x3a+1a (0.80s)

9x3a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3b+1a (1.53s)

2x3b+1a (1.18s)

3x3b+1a (1.00s)

5x3b+1a (0.80s)

9x3b+1a (0.61s)

Figure C-5 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx3a+1a and Nx3b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3a+1b (1.53s)

2x3a+1b (1.18s)

3x3a+1b (1.00s)

5x3a+1b (0.80s)

9x3a+1b (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

3b+1b (1.53s)

2x3b+1b (1.18s)

3x3b+1b (1.00s)

5x3b+1b (0.80s)

9x3b+1b (0.61s)

Figure C-6 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx3a+1b and Nx3b+1b with mass M=8.87ton. 
C-4 C: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

4a (1.18s)

2x4a (0.84s)

3x4a (0.68s)

5x4a (0.53s)

9x4a (0.39s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

4b (1.18s)

2x4b (0.84s)

3x4b (0.68s)

5x4b (0.53s)

9x4b (0.39s)

 

Figure C-7 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx4a and Nx4b with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

4a+1a (1.08s)

2x4a+1a (0.80s)

3x4a+1a (0.66s)

5x4a+1a (0.52s)

9x4a+1a (0.39s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

4b+1a (1.08s)

2x4b+1a (0.80s)

3x4b+1a (0.66s)

5x4b+1a (0.52s)

9x4b+1a (0.39s)

Figure C-8 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx4a+1a and Nx4b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

4a+1b (1.08s)

2x4a+1b (0.80s)

3x4a+1b (0.66s)

5x4a+1b (0.52s)

9x4a+1b (0.39s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

4b+1b (1.08s)

2x4b+1b (0.80s)

3x4b+1b (0.66s)

5x4b+1b (0.52s)

9x4b+1b (0.39s)

Figure C-9 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx4a+1b and Nx4b+1b with mass M=8.87ton. 
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-5 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

5a (1.02s)

2x5a (0.72s)

3x5a (0.59s)

5x5a (0.46s)

9x5a (0.34s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

5b (1.02s)

2x5b (0.72s)

3x5b (0.59s)

5x5b (0.46s)

9x5b (0.34s)

 

Figure C-10 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx5a and Nx5b with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

5a+1a (0.95s)

2x5a+1a (0.70s)

3x5a+1a (0.58s)

5x5a+1a (0.45s)

9x5a+1a (0.34s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

5b+1a (0.95s)

2x5b+1a (0.70s)

3x5b+1a (0.58s)

5x5b+1a (0.45s)

9x5b+1a (0.34s)

Figure C-11 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx5a+1a and Nx5b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

5a+1b (0.95s)

2x5a+1b (0.70s)

3x5a+1b (0.58s)

5x5a+1b (0.45s)

9x5a+1b (0.34s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

5b+1b (0.95s)

2x5b+1b (0.70s)

3x5b+1b (0.58s)

5x5b+1b (0.45s)

9x5b+1b (0.34s)

Figure C-12 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx5a+1b and Nx5b+1b with mass M=8.87ton. 
C-6 C: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

6a (1.87s)

2x6a (1.32s)

3x6a (1.08s)

5x6a (0.84s)

9x6a (0.62s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

6b (1.87s)

2x6b (1.32s)

3x6b (1.08s)

5x6b (0.84s)

9x6b (0.62s)

 

Figure C-13 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx6a and Nx6b with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

6a+1a (1.53s)

2x6a+1a (1.18s)

3x6a+1a (1.00s)

5x6a+1a (0.80s)

9x6a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

6b+1a (1.53s)

2x6b+1a (1.18s)

3x6b+1a (1.00s)

5x6b+1a (0.80s)

9x6b+1a (0.61s)

Figure C-14 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx6a+1a and Nx6b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

6a+1b (1.53s)

2x6a+1b (1.18s)

3x6a+1b (1.00s)

5x6a+1b (0.80s)

9x6a+1b (0.61s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

6b+1b (1.53s)

2x6b+1b (1.18s)

3x6b+1b (1.00s)

5x6b+1b (0.80s)

9x6b+1b (0.61s)

Figure C-15 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx6a+1b and Nx6b+1b with mass M=8.87ton. 
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-7 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

2

2.5

3

3.5

4

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

7a (1.87s)

2x7a (1.32s)

3x7a (1.08s)

5x7a (0.84s)

9x7a (0.62s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

2

2.5

3

3.5

4

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

7b (1.87s)

2x7b (1.32s)

3x7b (1.08s)

5x7b (0.84s)

9x7b (0.62s)

 

Figure C-16 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx7a and Nx7b with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

7a+1a (1.53s)

2x7a+1a (1.18s)

3x7a+1a (1.00s)

5x7a+1a (0.80s)

9x7a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

7b+1a (1.53s)

2x7b+1a (1.18s)

3x7b+1a (1.00s)

5x7b+1a (0.80s)

9x7b+1a (0.61s)

Figure C-17 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx7a+1a and Nx7b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

7a+1b (1.53s)

2x7a+1b (1.18s)

3x7a+1b (1.00s)

5x7a+1b (0.80s)

9x7a+1b (0.61s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=8.87

7b+1b (1.53s)

2x7b+1b (1.18s)

3x7b+1b (1.00s)

5x7b+1b (0.80s)

9x7b+1b (0.61s)

Figure C-18 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx7a+1b and Nx7b+1b with mass M=8.87ton. 
C-8 C: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

2

2.5

3

3.5

4

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

2a (3.74s)

2x2a (2.65s)

3x2a (2.16s)

5x2a (1.67s)

9x2a (1.25s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

2

2.5

3

3.5

4

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

2b (3.74s)

2x2b (2.65s)

3x2b (2.16s)

5x2b (1.67s)

9x2b (1.25s)

 

Figure C-19. Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx2a and Nx2b with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

2a+1a (3.05s)

2x2a+1a (2.37s)

3x2a+1a (2.00s)

5x2a+1a (1.60s)

9x2a+1a (1.21s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

2b+1a (3.05s)

2x2b+1a (2.37s)

3x2b+1a (2.00s)

5x2b+1a (1.60s)

9x2b+1a (1.21s)

Figure C-20 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx2a+1a and Nx2b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

2a+1b (3.05s)

2x2a+1b (2.37s)

3x2a+1b (2.00s)

5x2a+1b (1.60s)

9x2a+1b (1.21s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

2b+1b (3.05s)

2x2b+1b (2.37s)

3x2b+1b (2.00s)

5x2b+1b (1.60s)

9x2b+1b (1.21s)

Figure C-21 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx2a+1b and Nx2b+1b with mass M=35.46ton. 
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-9 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

3a (3.74s)

2x3a (2.65s)

3x3a (2.16s)

5x3a (1.67s)

9x3a (1.25s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

3b (3.74s)

2x3b (2.65s)

3x3b (2.16s)

5x3b (1.67s)

9x3b (1.25s)

 

Figure C-22 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx3a and Nx3b with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

3a+1a (3.05s)

2x3a+1a (2.37s)

3x3a+1a (2.00s)

5x3a+1a (1.60s)

9x3a+1a (1.21s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

3b+1a (3.05s)

2x3b+1a (2.37s)

3x3b+1a (2.00s)

5x3b+1a (1.60s)

9x3b+1a (1.21s)

Figure C-23 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx3a+1a and Nx3b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

3a+1b (3.05s)

2x3a+1b (2.37s)

3x3a+1b (2.00s)

5x3a+1b (1.60s)

9x3a+1b (1.21s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

6

7

8

9

10

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

3b+1b (3.05s)

2x3b+1b (2.37s)

3x3b+1b (2.00s)

5x3b+1b (1.60s)

9x3b+1b (1.21s)

Figure C-24 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx3a+1b and Nx3b+1b with mass M=35.46ton. 
C-10 C: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

14

16

18

20

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

4a (2.37s)

2x4a (1.67s)

3x4a (1.37s)

5x4a (1.06s)

9x4a (0.79s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

14

16

18

20

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

4b (2.37s)

2x4b (1.67s)

3x4b (1.37s)

5x4b (1.06s)

9x4b (0.79s)

 

Figure C-25 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx4a and Nx4b with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

20

25

30

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

4a+1a (2.16s)

2x4a+1a (1.60s)

3x4a+1a (1.32s)

5x4a+1a (1.04s)

9x4a+1a (0.78s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

20

25

30

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

4b+1a (2.16s)

2x4b+1a (1.60s)

3x4b+1a (1.32s)

5x4b+1a (1.04s)

9x4b+1a (0.78s)

Figure C-26 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx4a+1a and Nx4b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

20

25

30

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

4a+1b (2.16s)

2x4a+1b (1.60s)

3x4a+1b (1.32s)

5x4a+1b (1.04s)

9x4a+1b (0.78s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

20

25

30

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

4b+1b (2.16s)

2x4b+1b (1.60s)

3x4b+1b (1.32s)

5x4b+1b (1.04s)

9x4b+1b (0.78s)

Figure C-27 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx4a+1b and Nx4b+1b with mass M=35.46ton. 
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-11 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

5a (2.04s)

2x5a (1.45s)

3x5a (1.18s)

5x5a (0.91s)

9x5a (0.68s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

5b (2.04s)

2x5b (1.45s)

3x5b (1.18s)

5x5b (0.91s)

9x5b (0.68s)

 

Figure C-28 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx5a and Nx5b with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

20

25

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

5a+1a (1.91s)

2x5a+1a (1.39s)

3x5a+1a (1.15s)

5x5a+1a (0.90s)

9x5a+1a (0.68s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

20

25

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

5b+1a (1.91s)

2x5b+1a (1.39s)

3x5b+1a (1.15s)

5x5b+1a (0.90s)

9x5b+1a (0.68s)

Figure C-29 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx5a+1a and Nx5b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

20

25

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

5a+1b (1.91s)

2x5a+1b (1.39s)

3x5a+1b (1.15s)

5x5a+1b (0.90s)

9x5a+1b (0.68s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

20

25

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

5b+1b (1.91s)

2x5b+1b (1.39s)

3x5b+1b (1.15s)

5x5b+1b (0.90s)

9x5b+1b (0.68s)

Figure C-30 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx5a+1b and Nx5b+1b with mass M=35.46ton. 
C-12 C: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

6a (3.74s)

2x6a (2.65s)

3x6a (2.16s)

5x6a (1.67s)

9x6a (1.25s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

6b (3.74s)

2x6b (2.65s)

3x6b (2.16s)

5x6b (1.67s)

9x6b (1.25s)

 

Figure C-31 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx6a and Nx6b with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

6a+1a (3.05s)

2x6a+1a (2.37s)

3x6a+1a (2.00s)

5x6a+1a (1.60s)

9x6a+1a (1.21s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

6b+1a (3.05s)

2x6b+1a (2.37s)

3x6b+1a (2.00s)

5x6b+1a (1.60s)

9x6b+1a (1.21s)

Figure C-32 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx6a+1a and Nx6b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

6a+1b (3.05s)

2x6a+1b (2.37s)

3x6a+1b (2.00s)

5x6a+1b (1.60s)

9x6a+1b (1.21s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

6b+1b (3.05s)

2x6b+1b (2.37s)

3x6b+1b (2.00s)

5x6b+1b (1.60s)

9x6b+1b (1.21s)

Figure C-33 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx6a+1b and Nx6b+1b with mass M=35.46ton. 
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-13 

versus Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

7a (3.74s)

2x7a (2.65s)

3x7a (2.16s)

5x7a (1.67s)

9x7a (1.25s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

7b (3.74s)

2x7b (2.65s)

3x7b (2.16s)

5x7b (1.67s)

9x7b (1.25s)

 

Figure C-34 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx7a and Nx7b with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

7a+1a (3.05s)

2x7a+1a (2.37s)

3x7a+1a (2.00s)

5x7a+1a (1.60s)

9x7a+1a (1.21s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

7b+1a (3.05s)

2x7b+1a (2.37s)

3x7b+1a (2.00s)

5x7b+1a (1.60s)

9x7b+1a (1.21s)

Figure C-35 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx7a+1a and Nx7b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

7a+1b (3.05s)

2x7a+1b (2.37s)

3x7a+1b (2.00s)

5x7a+1b (1.60s)

9x7a+1b (1.21s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

2

4

6

8

10

12

Sa(T1,5%) / Sa,y(T1,5%)

max

M=35.5

7b+1b (3.05s)

2x7b+1b (2.37s)

3x7b+1b (2.00s)

5x7b+1b (1.60s)

9x7b+1b (1.21s)

Figure C-36 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for 

systems Nx7a+1b and Nx7b+1b with mass M=35.46ton. 

 

FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-1 

versus Non-Normalized Intensity Measures 

Appendix D 

Median IDA Curves for 

Multi-Spring Systems versus 

Non-Normalized Intensity 

Measures 

This appendix contains non-normalized plots of median incremental dynamic 

analysis (IDA) curves from focused analytical studies on multi-spring singledegree-of-freedom (SDOF) systems. All systems are composed of two 

springs representing a primary lateral-force-resisting system and a secondary 

gravity system with the characteristics described in Chapter 3. Multi-spring 

systems carry a designation of “NxJa+1a” or “NxJa+1b” where “N” is the 

peak strength multiplier (N = 1, 2, 3, 5, or 9), “J” is the lateral-force-resisting 

spring number (J = 2, 3, 4, 5, 6, or 7), and 1a or 1b is the gravity system 

identifier. In the figures, the vertical axis is one of two ground motion 

intensities IM = Sa(1s,5%) or Sa(2s,5%), which are not normalized, and the 

horizontal axis is the maximum story drift ratio, max, in radians. The period 

of vibration for each system is indicated in parentheses. 

D.1 Visualization Tool 

Given the large volume of analytical data, customized algorithms were 

developed for post-processing, statistical analysis, and visualization of 

results. The accompanying CD includes an electronic visualization tool that 

was developed to view results of multi-spring studies. The tool is a 

Microsoft Excel based application with a user-interface that accesses a 

database of all available multi-spring data. By selecting a desired spring 

combination (“NxJa+1a” or “NxJa+1b”), stiffness level (stiff or flexible), and 

intensity measure (normalized or non-normalized), users can view the 

resulting quantile (median, 16th, and 84th percentile) IDA curves for the 

combination of interest. 
D-2 D: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(1s,5%) (g)

max

M=8.87

2a (1.87s)

2x2a (1.32s)

3x2a (1.08s)

5x2a (0.84s)

9x2a (0.62s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(1s,5%) (g)

max

M=8.87

2b (1.87s)

2x2b (1.32s)

3x2b (1.08s)

5x2b (0.84s)

9x2b (0.62s)

 

Figure D-1 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx2a and Nx2b 

with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

2a+1a (1.53s)

2x2a+1a (1.18s)

3x2a+1a (1.00s)

5x2a+1a (0.80s)

9x2a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

2b+1a (1.53s)

2x2b+1a (1.18s)

3x2b+1a (1.00s)

5x2b+1a (0.80s)

9x2b+1a (0.61s)

Figure D-2 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx2a+1a and 

Nx2b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

2a+1b (1.53s)

2x2a+1b (1.18s)

3x2a+1b (1.00s)

5x2a+1b (0.80s)

9x2a+1b (0.61s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

2b+1b (1.53s)

2x2b+1b (1.18s)

3x2b+1b (1.00s)

5x2b+1b (0.80s)

9x2b+1b (0.61s)

Figure D-3 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx2a+1b and 

Nx2b+1b with mass M=8.87ton. 
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-3 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

3a (1.87s)

2x3a (1.32s)

3x3a (1.08s)

5x3a (0.84s)

9x3a (0.62s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

3b (1.87s)

2x3b (1.32s)

3x3b (1.08s)

5x3b (0.84s)

9x3b (0.62s)

 

Figure D-4 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx3a and Nx3b 

with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

3a+1a (1.53s)

2x3a+1a (1.18s)

3x3a+1a (1.00s)

5x3a+1a (0.80s)

9x3a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

3b+1a (1.53s)

2x3b+1a (1.18s)

3x3b+1a (1.00s)

5x3b+1a (0.80s)

9x3b+1a (0.61s)

Figure D-5 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx3a+1a and 

Nx3b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

3a+1b (1.53s)

2x3a+1b (1.18s)

3x3a+1b (1.00s)

5x3a+1b (0.80s)

9x3a+1b (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

3b+1b (1.53s)

2x3b+1b (1.18s)

3x3b+1b (1.00s)

5x3b+1b (0.80s)

9x3b+1b (0.61s)

Figure D-6 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx3a+1b and 

Nx3b+1b with mass M=8.87ton. 
D-4 D: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

4a (1.18s)

2x4a (0.84s)

3x4a (0.68s)

5x4a (0.53s)

9x4a (0.39s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

4b (1.18s)

2x4b (0.84s)

3x4b (0.68s)

5x4b (0.53s)

9x4b (0.39s)

 

Figure D-7 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx4a and Nx4b 

with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

Sa(1s,5%) (g)

max

M=8.87

4a+1a (1.08s)

2x4a+1a (0.80s)

3x4a+1a (0.66s)

5x4a+1a (0.52s)

9x4a+1a (0.39s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

Sa(1s,5%) (g)

max

M=8.87

4b+1a (1.08s)

2x4b+1a (0.80s)

3x4b+1a (0.66s)

5x4b+1a (0.52s)

9x4b+1a (0.39s)

Figure D-8 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx4a+1a and 

Nx4b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

Sa(1s,5%) (g)

max

M=8.87

4a+1b (1.08s)

2x4a+1b (0.80s)

3x4a+1b (0.66s)

5x4a+1b (0.52s)

9x4a+1b (0.39s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

Sa(1s,5%) (g)

max

M=8.87

4b+1b (1.08s)

2x4b+1b (0.80s)

3x4b+1b (0.66s)

5x4b+1b (0.52s)

9x4b+1b (0.39s)

Figure D-9 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx4a+1b and 

Nx4b+1b with mass M=8.87ton. 
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-5 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

5a (1.02s)

2x5a (0.72s)

3x5a (0.59s)

5x5a (0.46s)

9x5a (0.34s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

5b (1.02s)

2x5b (0.72s)

3x5b (0.59s)

5x5b (0.46s)

9x5b (0.34s)

 

Figure D-10 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx5a and Nx5b 

with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

Sa(1s,5%) (g)

max

M=8.87

5a+1a (0.95s)

2x5a+1a (0.70s)

3x5a+1a (0.58s)

5x5a+1a (0.45s)

9x5a+1a (0.34s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

Sa(1s,5%) (g)

max

M=8.87

5b+1a (0.95s)

2x5b+1a (0.70s)

3x5b+1a (0.58s)

5x5b+1a (0.45s)

9x5b+1a (0.34s)

Figure D-11 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx5a+1a and 

Nx5b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

Sa(1s,5%) (g)

max

M=8.87

5a+1b (0.95s)

2x5a+1b (0.70s)

3x5a+1b (0.58s)

5x5a+1b (0.45s)

9x5a+1b (0.34s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

Sa(1s,5%) (g)

max

M=8.87

5b+1b (0.95s)

2x5b+1b (0.70s)

3x5b+1b (0.58s)

5x5b+1b (0.45s)

9x5b+1b (0.34s)

Figure D-12 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx5a+1b and 

Nx5b+1b with mass M=8.87ton. 
D-6 D: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sa(1s,5%) (g)

max

M=8.87

6a (1.87s)

2x6a (1.32s)

3x6a (1.08s)

5x6a (0.84s)

9x6a (0.62s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sa(1s,5%) (g)

max

M=8.87

6b (1.87s)

2x6b (1.32s)

3x6b (1.08s)

5x6b (0.84s)

9x6b (0.62s)

 

Figure D-13 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx6a and Nx6b 

with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sa(1s,5%) (g)

max

M=8.87

6a+1a (1.53s)

2x6a+1a (1.18s)

3x6a+1a (1.00s)

5x6a+1a (0.80s)

9x6a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sa(1s,5%) (g)

max

M=8.87

6b+1a (1.53s)

2x6b+1a (1.18s)

3x6b+1a (1.00s)

5x6b+1a (0.80s)

9x6b+1a (0.61s)

Figure D-14 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx6a+1a and 

Nx6b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sa(1s,5%) (g)

max

M=8.87

6a+1b (1.53s)

2x6a+1b (1.18s)

3x6a+1b (1.00s)

5x6a+1b (0.80s)

9x6a+1b (0.61s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sa(1s,5%) (g)

max

M=8.87

6b+1b (1.53s)

2x6b+1b (1.18s)

3x6b+1b (1.00s)

5x6b+1b (0.80s)

9x6b+1b (0.61s)

Figure D-15 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx6a+1b and 

Nx6b+1b with mass M=8.87ton. 
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-7 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Sa(1s,5%) (g)

max

M=8.87

7a (1.87s)

2x7a (1.32s)

3x7a (1.08s)

5x7a (0.84s)

9x7a (0.62s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Sa(1s,5%) (g)

max

M=8.87

7b (1.87s)

2x7b (1.32s)

3x7b (1.08s)

5x7b (0.84s)

9x7b (0.62s)

 

Figure D-16 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx7a and Nx7b 

with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

7a+1a (1.53s)

2x7a+1a (1.18s)

3x7a+1a (1.00s)

5x7a+1a (0.80s)

9x7a+1a (0.61s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

7b+1a (1.53s)

2x7b+1a (1.18s)

3x7b+1a (1.00s)

5x7b+1a (0.80s)

9x7b+1a (0.61s)

Figure D-17 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx7a+1a and 

Nx7b+1a with mass M=8.87ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

7a+1b (1.53s)

2x7a+1b (1.18s)

3x7a+1b (1.00s)

5x7a+1b (0.80s)

9x7a+1b (0.61s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sa(1s,5%) (g)

max

M=8.87

7b+1b (1.53s)

2x7b+1b (1.18s)

3x7b+1b (1.00s)

5x7b+1b (0.80s)

9x7b+1b (0.61s)

Figure D-18 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx7a+1b and 

Nx7b+1b with mass M=8.87ton. 
D-8 D: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Sa(2s,5%) (g)

max

M=35.5

2a (3.74s)

2x2a (2.65s)

3x2a (2.16s)

5x2a (1.67s)

9x2a (1.25s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Sa(2s,5%) (g)

max

M=35.5

2b (3.74s)

2x2b (2.65s)

3x2b (2.16s)

5x2b (1.67s)

9x2b (1.25s)

 

Figure D-19 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx2a and Nx2b 

with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

2a+1a (3.05s)

2x2a+1a (2.37s)

3x2a+1a (2.00s)

5x2a+1a (1.60s)

9x2a+1a (1.21s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

2b+1a (3.05s)

2x2b+1a (2.37s)

3x2b+1a (2.00s)

5x2b+1a (1.60s)

9x2b+1a (1.21s)

Figure D-20 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx2a+1a and 

Nx2b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

2a+1b (3.05s)

2x2a+1b (2.37s)

3x2a+1b (2.00s)

5x2a+1b (1.60s)

9x2a+1b (1.21s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

2b+1b (3.05s)

2x2b+1b (2.37s)

3x2b+1b (2.00s)

5x2b+1b (1.60s)

9x2b+1b (1.21s)

Figure D-21 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx2a+1b and 

Nx2b+1b with mass M=35.46ton. 
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-9 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

Sa(2s,5%) (g)

max

M=35.5

3a (3.74s)

2x3a (2.65s)

3x3a (2.16s)

5x3a (1.67s)

9x3a (1.25s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

Sa(2s,5%) (g)

max

M=35.5

3b (3.74s)

2x3b (2.65s)

3x3b (2.16s)

5x3b (1.67s)

9x3b (1.25s)

 

Figure D-22 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx3a and Nx3b 

with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

3a+1a (3.05s)

2x3a+1a (2.37s)

3x3a+1a (2.00s)

5x3a+1a (1.60s)

9x3a+1a (1.21s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

3b+1a (3.05s)

2x3b+1a (2.37s)

3x3b+1a (2.00s)

5x3b+1a (1.60s)

9x3b+1a (1.21s)

Figure D-23 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx3a+1a and 

Nx3b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

3a+1b (3.05s)

2x3a+1b (2.37s)

3x3a+1b (2.00s)

5x3a+1b (1.60s)

9x3a+1b (1.21s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

3b+1b (3.05s)

2x3b+1b (2.37s)

3x3b+1b (2.00s)

5x3b+1b (1.60s)

9x3b+1b (1.21s)

Figure D-24 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx3a+1b and 

Nx3b+1b with mass M=35.46ton. 
D-10 D: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

4a (2.37s)

2x4a (1.67s)

3x4a (1.37s)

5x4a (1.06s)

9x4a (0.79s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

4b (2.37s)

2x4b (1.67s)

3x4b (1.37s)

5x4b (1.06s)

9x4b (0.79s)

 

Figure D-25 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx4a and Nx4b 

with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

4a+1a (2.16s)

2x4a+1a (1.60s)

3x4a+1a (1.32s)

5x4a+1a (1.04s)

9x4a+1a (0.78s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

4b+1a (2.16s)

2x4b+1a (1.60s)

3x4b+1a (1.32s)

5x4b+1a (1.04s)

9x4b+1a (0.78s)

Figure D-26 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for (systems Nx4a+1a and 

Nx4b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

4a+1b (2.16s)

2x4a+1b (1.60s)

3x4a+1b (1.32s)

5x4a+1b (1.04s)

9x4a+1b (0.78s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

4b+1b (2.16s)

2x4b+1b (1.60s)

3x4b+1b (1.32s)

5x4b+1b (1.04s)

9x4b+1b (0.78s)

Figure D-27 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx4a+1b and 

Nx4b+1b with mass M=35.46ton. 
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-11 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

Sa(2s,5%) (g)

max

M=35.5

5a (2.04s)

2x5a (1.45s)

3x5a (1.18s)

5x5a (0.91s)

9x5a (0.68s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

Sa(2s,5%) (g)

max

M=35.5

5b (2.04s)

2x5b (1.45s)

3x5b (1.18s)

5x5b (0.91s)

9x5b (0.68s)

 

Figure D-28 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx5a and Nx5b 

with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

5a+1a (1.91s)

2x5a+1a (1.39s)

3x5a+1a (1.15s)

5x5a+1a (0.90s)

9x5a+1a (0.68s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

5b+1a (1.91s)

2x5b+1a (1.39s)

3x5b+1a (1.15s)

5x5b+1a (0.90s)

9x5b+1a (0.68s)

Figure D-29 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx5a+1a and 

Nx5b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

5a+1b (1.91s)

2x5a+1b (1.39s)

3x5a+1b (1.15s)

5x5a+1b (0.90s)

9x5a+1b (0.68s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

5b+1b (1.91s)

2x5b+1b (1.39s)

3x5b+1b (1.15s)

5x5b+1b (0.90s)

9x5b+1b (0.68s)

Figure D-30 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx5a+1b and 

Nx5b+1b with mass M=35.46ton. 
D-12 D: Median IDA Curves for Multi-Spring Systems FEMA P440A 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

6a (3.74s)

2x6a (2.65s)

3x6a (2.16s)

5x6a (1.67s)

9x6a (1.25s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

6b (3.74s)

2x6b (2.65s)

3x6b (2.16s)

5x6b (1.67s)

9x6b (1.25s)

 

Figure D-31 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx6a and Nx6b 

with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

6a+1a (3.05s)

2x6a+1a (2.37s)

3x6a+1a (2.00s)

5x6a+1a (1.60s)

9x6a+1a (1.21s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

6b+1a (3.05s)

2x6b+1a (2.37s)

3x6b+1a (2.00s)

5x6b+1a (1.60s)

9x6b+1a (1.21s)

Figure D-32 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx6a+1a and 

Nx6b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

6a+1b (3.05s)

2x6a+1b (2.37s)

3x6a+1b (2.00s)

5x6a+1b (1.60s)

9x6a+1b (1.21s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sa(2s,5%) (g)

max

M=35.5

6b+1b (3.05s)

2x6b+1b (2.37s)

3x6b+1b (2.00s)

5x6b+1b (1.60s)

9x6b+1b (1.21s)

Figure D-33 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx6a+1b and 

Nx6b+1b with mass M=35.46ton. 
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-13 

versus Non-Normalized Intensity Measures 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Sa(2s,5%) (g)

max

M=35.5

7a (3.74s)

2x7a (2.65s)

3x7a (2.16s)

5x7a (1.67s)

9x7a (1.25s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Sa(2s,5%) (g)

max

M=35.5

7b (3.74s)

2x7b (2.65s)

3x7b (2.16s)

5x7b (1.67s)

9x7b (1.25s)

 

Figure D-34 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx7a and Nx7b 

with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

7a+1a (3.05s)

2x7a+1a (2.37s)

3x7a+1a (2.00s)

5x7a+1a (1.60s)

9x7a+1a (1.21s)

 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

7b+1a (3.05s)

2x7b+1a (2.37s)

3x7b+1a (2.00s)

5x7b+1a (1.60s)

9x7b+1a (1.21s)

Figure D-35 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx7a+1a and 

Nx7b+1a with mass M=35.46ton. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

7a+1b (3.05s)

2x7a+1b (2.37s)

3x7a+1b (2.00s)

5x7a+1b (1.60s)

9x7a+1b (1.21s)

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sa(2s,5%) (g)

max

M=35.5

7b+1b (3.05s)

2x7b+1b (2.37s)

3x7b+1b (2.00s)

5x7b+1b (1.60s)

9x7b+1b (1.21s)

Figure D-36 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx7a+1b and 

Nx7b+1b with mass M=35.46ton. 

FEMA P440A E: Uncertainty, Fragility, and Probability E-1 

Appendix E 

Uncertainty, Fragility, and 

Probability 

The concepts presented in this report are compatible with current 

probabilistic trends in performance-based seismic design. A probabilistic 

context allows explicit consideration of the variability and uncertainty 

associated with each of the contributing parameters. An important concept in 

probabilistic procedures is the development and use of fragility curves. 

Use of fragility curves, and explicit consideration of uncertainty in 

performance assessment, is described in ATC-58 Guidelines for Seismic 

Performance Assessment of Buildings (ATC, 2007). Fragility curves are also 

used to determine the margin of safety against collapse in FEMA P695 

Quantification of Building Seismic Performance Factors (FEMA, 2009). 

This appendix explains the conversion of incremental dynamic analysis 

(IDA) results into fragilities, and presents equations that could be used to 

calculate annual probabilities for collapse, or any other limit state of interest. 

E.1 Conversion of IDA Results to Fragilities 

Incremental dynamic analysis results can be readily converted to fragilities. 

Figure E-1 shows an example of IDA results for a single structure subjected 

to a suite of ground motions of varying intensities. 

Figure E-1 IDA results for a single structure subjected to a suite of ground 

motions of varying intensities. 
E-2 E: Uncertainty, Fragility, and Probability FEMA P440A 

In this illustration, sidesway collapse is the governing mechanism, and 

collapse prediction is based on dynamic instability or excessive lateral 

displacements. Using collapse data obtained from IDA results, a collapse 

fragility can be defined through a cumulative distribution function (CDF), 

which relates the ground motion intensity to the probability of collapse 

(Ibarra et al., 2002). Studies have shown that this cumulative distribution 

function can be assumed to be lognormally distributed. Figure E-2 shows an 

example of a cumulative distribution plot obtained by fitting a lognormal 

distribution to the collapse data from Figure E-1. 

Figure E-2 Cumulative distribution plot obtained by fitting a lognormal 

distribution to collapse data from IDA results. 

Lognormal distributions are defined by a median value and a dispersion 

parameter. The median collapse capacity, Sa50% , indicates a ground motion 

intensity that has a 50% chance of producing collapse in the system. It also 

indicates the point at which half of the ground motions will produce collapse 

at higher intensities, and half will produce collapse at lower intensities. For 

each mode of collapse, the record-to-record dispersion can be estimated as: 

84% 16% ( )( )

2

a a

RTR

ln S ln S    (E-1) 

Figure E-3 provides conceptual collapse fragility curves showing the 

probability of collapse due to loss of vertical-load-carrying capacity (LVCC) 

or lateral dynamic instability (LDI). These are events are mutually exclusive, 

meaning that either one or the other can occur, but both events cannot occur 

at the same time. 
FEMA P440A E: Uncertainty, Fragility, and Probability E-3 

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

S a (T 1 )

P( C | Sa )

 combined

 vertical collapse

 lateral collapse

Figure E-3 Conceptual collapse fragility curves for sidesway (lateral) 

collapse, vertical collapse, and a combination of both. 

The total probability of collapse due to either mode can then be represented 

(Cornell et al., 2005) as: 

 P(C | sa) = P(CLDI | sa) + P(CLVCC | NCLDI ,sa)·P(NCLDI | sa (E-2) 

where: 

P(CLDI | sa) is the probability of collapse due to lateral dynamic instability at 

a ground motion intensity level sa , 

P(CLVCC | NCLDI ,sa) is the probability of collapse due to loss of vertical-loadcarrying capacity, given that collapse due to lateral dynamic instability has 

not occurred at intensity sa , and 

P(NCLDI |sa) is the probability of no collapse due to lateral dynamic 

instability at a ground motion intensity level sa. 

Since P(NCLDI |sa) is equal to 1– P(CLDI | sa), then Equation E-2 can also be 

written as: 

P(C | sa) = P(CLVCC | NCLDI , sa)+P(CLDI | sa) - P(CLVCC | NCLDI sa)·P(CLDI | sa) (E-3) 

E.2 Calculation of Annualized Probability 

The results of an incremental dynamic analysis expressed as a cumulative 

distribution function can be used in combination with a seismic hazard curve 

to generate mean annual frequencies (MAF) for collapse (or for other limit 

states of interest). This process is the integration of the limit state CDF (e.g., 
E-4 E: Uncertainty, Fragility, and Probability FEMA P440A 

fragility representing the probability of collapse as a function of spectral 

acceleration) with respect to the probability of occurrence of the intensity 

measure (e.g., hazard curve representing the annual probability of exceeding 

a full range of spectral accelerations). The mean annual frequency of 

collapse, col , or other limit state of interest, can be approximated (Cornell, 

2002) as: 

  1 2 2 exp 2 a col S C RTR    k        (E-4) 

where   S C a

  is the mean annual probability of the median spectral 

acceleration associated with collapse. The parameter k is the slope of the 

hazard curve, and can be calculated as: 

(10/50)

(2/50)

(2/50) (2/50)

(10/50) (10/50)

ln

1.65

ln ln

aT

aT

S

S

aT aT

aT aT

H

H

k

S S

S S

   

         

  

 (E-5) 
FEMA P440A F: Example Application F-1 

Appendix F 

Example Application 

This appendix presents an example application of a simplified nonlinear 

dynamic analysis procedure. The concept originated during the conduct of 

focused analytical studies comparing force-displacement capacity boundaries 

to incremental dynamic analysis results. In this procedure, a nonlinear static 

analysis is used to generate an idealized force-deformation curve (i.e., static 

pushover curve). The resulting curve is then used as a force-displacement 

capacity boundary to constrain the hysteretic behavior of an equivalent 

single-degree-of-freedom (SDOF) oscillator. This SDOF oscillator is then 

subjected to incremental dynamic analysis. 

The steps for conducting a simplified nonlinear dynamic analysis are 

outlined in the following section, and illustrated using an example building. 

Alternative retrofit strategies are evaluated using the same procedure. Use of 

the procedure to develop probabilistic estimates of performance for use in 

making design decisions is also illustrated. 

F.1 Simplified Nonlinear Dynamic Analysis Procedure 

The concept of a simplified nonlinear dynamic analysis procedure includes 

the following steps: 

 Develop an analytical model of the system. 

Models can be developed in accordance with prevailing practice for 

seismic evaluation, design, and rehabilitation of buildings described in 

ASCE/SEI Standard 41-06 Seismic Rehabilitation of Existing Buildings

(ASCE, 2006b). Component properties should be based on forcedisplacement capacity boundaries, rather than cyclic envelopes. 

 Perform a nonlinear static pushover analysis. 

Subject the model to a conventional pushover analysis in accordance 

with prevailing practice. Lateral load increments and resulting 

displacements are recorded to generate an idealized force-deformation 

curve. 
F-2 F: Example Application FEMA P440A 

 Conduct an incremental dynamic analysis of the system based on an 

equivalent SDOF model. 

The idealized force-deformation curve is, in effect, a system forcedisplacement capacity boundary that can be used to constrain a hysteretic 

model of an equivalent SDOF oscillator. This SDOF oscillator is then 

subjected to incremental dynamic analysis to check for lateral dynamic 

instability and other limit states of interest. Alternatively, approximate 

incremental dynamic analysis can be accomplished using the idealized 

force-deformation curve and the Static Pushover 2 Incremental Dynamic 

Analysis open source software tool, SPO2IDA (Vamvatsikos and 

Cornell, 2006). 

 Determine probabilities associated with limit states of interest. 

Results from incremental dynamic analysis can be used to obtain 

response statistics associated with limit states of interest in addition to 

lateral dynamic instability. SPO2IDA can also be used to obtain median, 

16th, and 84th percentile IDA curves relating displacements to intensity. 

Using the fragility relationships described in Appendix E in conjunction 

with a site hazard curve, this information can be converted into annual 

probabilities of exceedance for each limit state. Probabilistic information 

in this form can be used to make enhanced decisions based on risk and 

uncertainty, rather than on discrete threshold values of acceptance. 

F.2 Example Building 

The example building is a five-story reinforced concrete frame residential 

structure with interior unreinforced masonry infill partitions in the upper 

stories, and a soft/weak first-story. An exterior elevation of the building is 

shown in Figure F-1 and first floor plan is shown in Figure F-2. Reinforced 

concrete columns in each orthogonal direction provide lateral resistance to 

seismic forces. As indicated in Figure F-2, the first story includes a mixture 

of components with column-like proportions and components that are more 

like slender shear walls. 

This building is a prototypical example of a soft/weak story structure. 

Concentration of inelastic deformations in the first story presents an obvious 

potential story collapse mechanism. 

F.3 Structural Analysis Model 

To investigate the potential for collapse in this structure, it is reasonable to 

assume that the response can be represented by a SDOF model. The first 

story column components are classified for modeling purposes in accordance 
FEMA P440A F: Example Application F-3 

with ASCE/SEI 41-06. Most of the columns in this example are classified as 

shear-controlled or flexure-shear controlled. Wall-like column components 

are shear-controlled along the strong axis of the member. 

Figure F-1 Example building exterior elevation. 

Figure F-2 Example building first floor plan. 

Modeling parameters for the column components can be characterized by the 

conceptual force-displacement relationship (“backbone”) specified in 

ASCE/SEI 41-06. The modeling parameters selected for the components in 

this example are taken from Chapter 6 of ASCE/SEI 41-06, and depicted in 

Figure F-3. In both cases, the residual strength, c, is taken as zero. 
F-4 F: Example Application FEMA P440A 

The column components are assembled into a model of the structural system 

as shown in Figure F-4. Inelastic response is assumed to occur 

predominantly in the first story. First-story columns are taken as fixed at the 

base on a rigid foundation. The stiffness of the column components are 

based on elastic properties in flexure and shear. The model includes soil 

flexibility, allowing for rigid body rotation due to the response of the 

structure above. Soil stiffness parameters are taken from Chapter 4 of 

ASCE/SEI 41-06, assuming a relatively soft soil site (site Class E). 

Vn

total  columns y  b  0.01

a  0

c  0 A

B,C

D,E

Vn

total  columns y  b  0.01

a  0

c  0 A

B,C

D,E

(a) 

Vn

walls

h



e  0.01

d  0.0075

A c  0

B,C

D,E

Vn

walls

h



e  0.01

d  0.0075

A c  0

B,C

D,E

(b) 

Figure F-3 Force-displacement modeling parameters for: (a) column 

components; and (b) wall-like column components. 

When developing a SDOF representation of a system, it is important to 

account for foundation rotation in assessing column distortions. The 

resulting SDOF model represents the relationship between the total first floor 

drift, including contributions from the foundation, sys fdn cols      , and the 

applied inertial loads, V . 
FEMA P440A F: Example Application F-5 

fdn 

col  Column 

distortion

Foundation 

rotation

1st story mechanism-all 

inelasticity in columns

Rigid foundation on elastic supports

Effective 

height of 

inertial 

forces, h* hcm

h1st

W W

sys fdn col st  h1    

cm col st fdn cm h h 1     

V V

eff K

fdn 

col  Column 

distortion

Foundation 

rotation

1st story mechanism-all 

inelasticity in columns

Rigid foundation on elastic supports

Effective 

height of 

inertial 

forces, h* hcm

h1st

W W

sys fdn col st  h1    

cm col st fdn cm h h 1     

V V

eff K

 

 (a.) (b) 

Figure F-4 Structural analysis model showing: (a) assumptions; and (b) 

distortions. 

F.4 Nonlinear Static Pushover Analysis 

The analytical model is subjected to a conventional pushover analysis. 

Results are shown in Figure F-5. 

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

0.00% 1.00% 2.00% 3.00% 4.00%

System rotation

Base shear (%g)

V

sys 

LVCC 

LDI 

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

0.00% 1.00% 2.00% 3.00% 4.00%

System rotation

Base shear (%g)

V

sys 

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

0.00% 1.00% 2.00% 3.00% 4.00%

System rotation

Base shear (%g)

V

sys 

LVCC 

LDI 

Figure F-5 Pushover curve from nonlinear static analysis. 
F-6 F: Example Application FEMA P440A 

F.5 Evaluation of Limit States of Interest 

Collapse in real structures can be caused by sidesway collapse (lateral 

dynamic instability) or by loss of vertical-load-carrying capacity. In this 

example, the following two limit states are defined (both are shown in Figure 

F-5): 

LVCC  the total system rotation at which loss of vertical-load-carrying 

capacity occurs (i.e., when first story columns fail due to shear 

distortion). In this example, the critical column distortion for loss of 

vertical-load-carrying capacity is taken as 1% inelastic rotation, 

which occurs when the total system rotation sys  = 1.2%. 

LDI  the total system rotation at which lateral dynamic instability occurs 

(i.e., when first story columns lose all lateral-force-resisting 

capacity). In this example, this is taken to occur when the total 

system rotation sys  = 4.0%. 

The target displacement for a given intensity is estimated using the 

Coefficient Method: 

2

012 2 4

e

t a

T  CCC S g

 

Uniform hazard spectra for the example site are shown in Figure F-6 and 

Figure F-7. 

Figure F-6 Uniform hazard spectrum for intensity corresponding to 10% 

chance of exceedance in 50 years (from USGS). 
FEMA P440A F: Example Application F-7 

Figure F-7 Uniform hazard spectrum for intensities corresponding to 2% 

chance of exceedance in 50 years (from USGS). 

For an intensity corresponding to a 10% chance of exceedance in 50 years 

(i.e., 475-year return period), and a period T = 0.3s: 

S g a 1.36

The strength of the model is: 

0.45 Fy  g

which results in: 

3.0 aT

y

S g R

F   for the 10%/50 year hazard level. 

The coefficients are: 

C0  first mode participation factor = 1.0, 

1 2

1 1 1.3 eR C

aT

   , 

where a = 50, and 

2

2

1 1 1 1.04

800 e

R C

T

          

. 

This results in a target displacement of: 
F-8 F: Example Application FEMA P440A 

1.6 t   inches, or 1.6/100 1.6  sys   %. 

This is greater than the acceptable limit for loss of vertical-load-carrying 

capacity ( LVCC  ) taken as sys  = 1.2%. 

To check for lateral dynamic instability, the proposed equation for Rdi is: 

3

3

a

c e ur r

di e

y cy

T F R b T

 F

     

    

    

 

where Te is the effective fundamental period of vibration of the structure, y, 

c, r, and u are displacements corresponding to the yield strength, Fy, 

capping strength, Fc, residual strength, Fr, and ultimate deformation capacity 

at the end of the residual strength plateau. Determination of these parameters 

requires a multi-linear idealization of the pushover curve, as shown in Figure 

F-8. 

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

0.00% 1.00% 2.00% 3.00% 4.00%

System rotation

Base shear (%g)

Idealization A

Idealization B

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

0.00% 1.00% 2.00% 3.00% 4.00%

System rotation

Base shear (%g)

Idealization A

Idealization B

Figure F-8 Pushover curve from nonlinear static analysis and two idealized 

system force-displacement capacity boundaries. 

Parameters a and b are functions given by: 

a  dTe )exp(1

2

1 













 

c

r

F

F b
FEMA P440A F: Example Application F-9 

and parameter d is a constant equal to 4 for the example building (assuming 

the presence of stiffness degradation). 

Using the above expressions along with parameters from Idealization ‘A’ in 

Figure F-8, results in: 

Rdi  2.6

which is less that the calculated value of R = 3.0 for intensities corresponding 

to the 10%/50 year hazard level. The parameter R is a ratio equal to the 

strength necessary to keep a system elastic for a given intensity, divided by 

the yield strength of the system. Higher values of R imply lower values of 

system yield strength. Values of R that exceed Rdi mean that the structure 

does not meet the minimum strength necessary to avoid lateral dynamic 

instability at this hazard level. 

At higher intensities (e.g., 2%/50 year hazard level) the calculated value of R 

would be even higher (R = 5.3 >> Rdi = 2.6), illustrating how the comparison 

between system strength and the limit on lateral dynamic instability would 

change for a different hazard level. 

In summary, the example structure does not meet acceptability criteria for 

loss of vertical-load-carrying capacity and lateral dynamic instability at the 

10%/50 year hazard level. Thus, a nonlinear response-history analysis must 

be performed. 

F.6 Incremental Dynamic Analysis 

The resulting force-displacement relationship from the pushover curve can be 

used to generate a force-displacement capacity boundary for the system. An 

incremental dynamic analysis (IDA) can then be applied to a SDOF oscillator 

constrained by the resulting force-displacement capacity boundary. 

Performing an incremental dynamic analysis will allow determination of the 

ground motion intensity at which various limit state deformations occur. 

For the example building, an approximate incremental dynamic analysis is 

performed using the open source software tool, SPO2IDA. Use of SPO2IDA 

along with Idealization ‘A’ in Figure F-8 results in the median, 16th, and 84th

percentile IDA curves shown in Figure F-9. The figure also includes the 

estimate of Rdi for lateral dynamic instability. 
F-10 F: Example Application FEMA P440A 

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0% 1.0% 2.0% 3.0% 4.0% 5.0%

System rotation

Spectral acceleration

IDA-50% IDA-84% IDA-16% Capacity Boundary

Loss of vertical 

carrying capacity 

(LVCC)

Lateral dynamic 

instability (LDI)

 Rdi eqn.

Figure F-9 Results of approximate incremental dynamic analysis using 

SPO2IDA. 

F.7 Determination of Probabilities Associated with 

Limit States of Interest 

From Figure F-9, median values of intensity causing loss of vertical-loadcarrying capacity (LVCC) or lateral dynamic instability (LDI) in the example 

building can be obtained. Using the expressions in Appendix E, the 

dispersion and mean annual frequencies (MAF) associated with these limit 

states can be determined. The resulting data is presented Table F-1. 

Table F-1 Mean Annual Frequencies for Collapse Limit States 

Limit state/collapse mode Sa50 MAF Sa50  MAF 

collapse 

Loss of vertical load 

carrying capability 0.92 0.0050 0.20 0.0060 

Lateral dynamic instability 1.26 0.0025 0.32 0.0040 

LVCC or LDI 0.88 0.0060 0.17 0.0069 

For the example building, fragilities associated with loss of vertical-loadcarrying capacity (LVCC) or lateral dynamic instability (LDI) are derived 

from the median values of spectral acceleration and dispersions in Table F-1, 

as illustrated in Figure F-10. 
FEMA P440A F: Example Application F-11 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.00 1.00 2.00 3.00

Spectral accel., S aT

Probability of collapse, P(C|Sa<or =SaT)

LVCC

LDI

Total

Figure F-10 Example building collapse fragilities for loss of vertical-loadcarrying capacity (LVCC), lateral dynamic instability (LDI), and a 

combination of both. 

The probabilities of experiencing the limit states of interest for the example 

building are derived in combination with a site-specific seismic hazard curve. 

For a representative soft site, the USGS hazard curve for a period of 0.3 

seconds is shown in Figure F-11. 

Spectra acceleration for T=0.3 sec

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0.000 0.020 0.040 0.060 0.080 0.100

Mean annual frequency of exceedance

Spectral acceleration, SaT

Figure F-11 Hazard curve for representative soft site (from USGS). 

Spectral acceleration for T=0.3 sec
F-12 F: Example Application FEMA P440A 

Using the expressions in Appendix E, the mean annual frequency (MAF) of 

collapse of the example building can be calculated. The MAF of collapse, 

shown in Table F-1, is approaching 1% annually, which can be considered 

high. 

F.8 Retrofit strategies 

As indicated by results of focused analytical studies presented in Chapter 4, 

the probability of collapse, as measured by the potential for lateral dynamic 

instability, can be reduced by adjusting the force-deformation characteristics 

of the system. Retrofit strategies to reduce the annual probability of collapse 

could include: (1) the addition of secondary lateral system; or (2) 

improvement of primary system strength and ductility. 

F.8.1 Addition of a Secondary Lateral System 

Lateral dynamic instability can be improved through the addition of a 

flexible and ductile secondary lateral system. In order to be effective, the 

secondary system does not need to provide much additional lateral strength 

to the overall system. 

Figure F-12 shows a revised pushover curve for the example building after 

retrofit using a ductile moment frame system with a lateral strength 

approximately equal to 10% of the building weight. Calculations similar to 

the above show that this strategy would reduce the mean annual frequency of 

collapse due to lateral dynamic instability to about 0.2%. 

0%

10%

20%

30%

40%

50%

60%

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

System rotation

Base shear (%g)

Retrofit with secondary 

lateral strength

Existing 

building

0%

10%

20%

30%

40%

50%

60%

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

System rotation

Base shear (%g)

Retrofit with secondary 

lateral strength

Existing 

building

Figure F-12 Revised pushover curve for the example building after retrofit 

with a secondary lateral system. 
FEMA P440A F: Example Application F-13 

F.8.2 Improvement of Primary System Strength and Ductility 

Lateral dynamic instability can also be improved by providing additional 

deformation capacity in the system. This can be accomplished by wrapping 

the concrete columns with fiber reinforced polymers (FRP) to increase 

strength and ductility of the primary seismic-force-resisting system. 

Figure F-13 shows a revised pushover curve for the example building after 

retrofitting the existing columns with FRP. Calculations similar to the above 

show that this strategy would reduce the mean annual frequency of collapse 

due to lateral dynamic instability to about 0.03%, which is an order of 

magnitude improvement over the existing structure. 

0%

20%

40%

60%

80%

100%

120%

140%

0.00% 1.00% 2.00% 3.00% 4.00%

System rotation

Base shear (%g)

Retrofit with improved 

strength and ductility 

of columns

Existing 

building

0%

20%

40%

60%

80%

100%

120%

140%

0.00% 1.00% 2.00% 3.00% 4.00%

System rotation

Base shear (%g)

Retrofit with improved 

strength and ductility 

of columns

Existing 

building

Figure F-13 Revised pushover curve for the example building after retrofit 

for improved strength and ductility of columns. 

FEMA P440A G: Preliminary Multiple-Degree-of-Freedom System Studies G-1 

Appendix G 

Preliminary Multiple-Degree-ofFreedom System Studies 

In focused analytical studies on single-degree-of-freedom (SDOF) systems, it 

was observed that nonlinear response of a system depends on the 

characteristics of the force-displacement capacity boundary. It was 

demonstrated that lateral dynamic instability of SDOF systems could be 

evaluated through the use of approximate equations or simplified nonlinear 

dynamic analyses based on the characteristics of the system forcedisplacement capacity boundary. 

Multiple-degree-of-freedom (MDOF) systems are more complex, and their 

dynamic 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 MDOF systems by using static pushover analyses and performing 

dynamic analysis on equivalent SDOF systems (Bernal, 1998; Vamvatsikos, 

2002; Vamvatsikos and Cornell, 2005a, 2005b). In particular, Vamvatsikos 

and Cornell (2005b) suggested that the seismic response of MDOF systems 

could be estimated through the use of incremental dynamic analyses on a 

reference SDOF system whose properties are determined through a nonlinear 

static (pushover) analysis. 

This appendix presents the results of preliminary studies of multiple-degreeof-freedom (MDOF) systems. It explores the application of nonlinear static 

analyses combined with dynamic analyses of SDOF systems to evaluate the 

lateral dynamic instability of MDOF systems. On a preliminary basis, it tests 

how approximate measures of lateral dynamic instability developed for 

SDOF systems might work on more complex MDOF systems. These 

approximate measures include the proposed equation for Rdi (Equation 5-8) 

and the open source software tool Static Pushover 2 Incremental Dynamic 

Analysis, SPO2IDA (Vamvatsikos and Cornell, 2006). 

A total of six buildings ranging in height from 4 to 20 stories are used in this 

investigation. This set includes two steel moment-resisting frame structures 

and four reinforced concrete moment-resisting frame structures. Four were 

previously studied by Haselton (2006), and two were previously studied by 
G-2 G: Preliminary Multiple-Degree-of-Freedom System Studies FEMA P440A 

Vamvatsikos and Cornell (2005b). Results are described in the sections that 

follow. 

G.1 Four-Story Code-Compliant Reinforced Concrete 

Building 

The subject building is a four-story reinforced concrete special perimeter 

moment frame designed in accordance with modern building code provisions 

(ICC 2003, ASCE 2002, ACI 2002). The building has a story height of 15 ft 

in the first story, and 13 ft in the remaining stories. The design base shear 

coefficient was 0.092. The building was modeled in OpenSEES and analyzed 

using incremental dynamic analysis using 80 recorded time histories which 

were scaled at twenty-two different ground motion intensities. The pushover 

analysis was conducted using a lateral force distribution in accordance with 

ASCE/SEI 7-05 Minimum Design Loads for Buildings and Other Structures 

(ASCE, 2006). Ground motions were scaled to increasing values of the 

pseudo-acceleration spectral ordinate at the fundamental period of vibration 

of the building (T1=1.12s). For a more detailed description of the building 

and its modeling, the reader is referred to Haselton (2006). 

Figure G-1 shows the results from the nonlinear static (pushover) analysis. 

The figure on the left shows the force-deformation curve while the figure on 

the right shows the distribution of story drift ratios at a roof drift ratio of 6%. 

It can be seen that story drifts primarily concentrate in the lower two stories. 

The force-deformation pushover curve is characterized by a gradual loss in 

lateral strength for roof drift ratios between 1% and 3.5%, followed by a 

more pronounced loss in lateral strength for roof drift ratios greater than 

3.5%. 

Figure G-1 (a) Monotonic pushover force-deformation curve and (b) story drifts at a roof drift 

ratio of the 0.06 in a four-story concrete frame building (Haselton 2006). 
FEMA P440A G: Preliminary Multiple-Degree-of-Freedom System Studies G-3 

Figure G-2 shows one of three simplified tri-linear force-displacement 

capacity boundaries selected to estimate the seismic response of the structure. 

Alternates are shown in Figure G-4 and Figure G-5. Although a sloping 

intermediate segment might have been somewhat more appropriate for this 

structure, a horizontal intermediate segment was selected in order to evaluate 

the proposed equation for Rdi and results using SPO2IDA. 

Figure G-2 Tri-linear capacity boundary selected for approximate analysis. 

Figure G-3 shows the median seismic behavior computed from incremental 

dynamic analyses conducted by Haselton (2006). These results are indicated 

as MDOF IDA in the figure. Also shown are results computed using the 

proposed equation for Rdi and approximate results from SPO2IDA. In the 

figure, Rdi and SPO2IDA both provide a good approximation of the collapse 

capacity of the building. 

Figure G-3 Comparison of median collapse capacity for a four-story codecompliant concrete frame building computed using incremental 

dynamic analysis and approximate procedures. 

0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Maximum Story Drift Ratio

Sa (T=1.12)/g

 MDOF IDA

 SPO2IDA

Rdi
G-4 G: Preliminary Multiple-Degree-of-Freedom System Studies FEMA P440A 

To explore sensitivity to the idealization of the force-displacement capacity 

boundary, two alternate idealizations, along with corresponding results, are 

shown in Figure G-4 and Figure G-5. Although median collapse capacities 

change with the selection of the force-displacement capacity boundary, the 

observed changes are relatively small. 

 

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. 

 

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. 

The median results shown above represent a measure of the central tendency 

of the response of the system; however, considerable dispersion exists around 

the median. To illustrate record-to-record variability, Figure G-6 shows 

0.0

0.5

1.0

1.5

2.0

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12

Maximum Story Drift Ratio

Sa (T=1.12)/g

 MDOF IDA

 SPO2IDA

Rdi

0.0

0.5

1.0

1.5

2.0

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12

Maximum Story Drift Ratio

Sa (T=1.12)/g

 MDOF IDA

 SPO2IDA

Rdi
FEMA P440A G: Preliminary Multiple-Degree-of-Freedom System Studies G-5 

incremental dynamic analysis results for all 80 ground motions. It can be 

seen that there are ground motions that produce the collapse of the structure 

at intensities equal to one third of the median intensity. Similarly, there are 

ground motions that require an intensity that is twice as large as the median 

intensity in order to produce the collapse of the structure. 

Also shown in Figure G-6 are the 16th and 84th percentiles of the results. 

Approximately 70% of the ground motions fall between these two dashed 

lines. When estimating the collapse probability of a structure, it is important 

to consider this variability. For more information, the reader is referred to 

Haselton (2006). 

0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Max IDR

Sa (T=1.12s) / g

84%

16%

50%

Figure G-6 Incremental dynamic analysis results for a four-story code-compliant 

concrete frame building subjected to 80 ground motions (adapted from 

Haselton, 2006). 

G.2 Eight-Story Code-Compliant Reinforced Concrete 

Building 

The subject building is an eight-story reinforced concrete special perimeter 

moment frame designed in accordance with modern building code provisions 

(ICC 2003, ASCE 2002, ACI 2002). The building has a story height of 15 ft 

in the first story, and 13 ft in the remaining stories. The design base shear 

coefficient was 0.05. The building was modeled in OpenSEES and analyzed 

using incremental dynamic analysis with the same 80 recorded ground 

motions that were used to analyze the four-story building. The pushover 

analysis was performed using a lateral force distribution in accordance with 

Maximum Story Drift Ratio
G-6 G: Preliminary Multiple-Degree-of-Freedom System Studies FEMA P440A 

ASCE/SEI 7-05. The fundamental period of vibration of the building is 

T1=1.71s. For a more detailed description of the building and its modeling, 

the reader is referred to Haselton (2006). 

Figure G-7 shows the results from the nonlinear static (pushover) analysis of 

the building. The figure on the left shows the force-deformation curve while 

the figure on the right shows the distribution of story drift ratios at a roof 

drift ratio of 2.6%. It can be seen that story drifts primarily concentrate in the 

lower four stories. The force-deformation pushover curve is characterized by 

a hardening segment for roof drift ratios between 0.3% and 0.8%, followed 

by softening segment for roof drift ratios greater than 0.8%. 

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 

(Haselton 2006). 

Figure G-8 shows the simplified tri-linear force-displacement capacity 

boundary selected to evaluate the proposed equation for Rdi and results using 

SPO2IDA. 

Figure G-8 Tri-linear capacity boundary selected for approximate analyses using SPO2IDA. 
FEMA P440A G: Preliminary Multiple-Degree-of-Freedom System Studies G-7 

Figure G-9 shows the median seismic behavior computed from incremental 

dynamic analyses conducted by Haselton (2006). These results are indicated 

as MDOF IDA in the figure. Also shown are results computed using the 

proposed equation for Rdi and approximate results from SPO2IDA. In the 

figure, Rdi provides a good estimate of the median collapse capacity, while 

SPO2IDA overestimates the collapse capacity somewhat. Figure G-10 

shows incremental dynamic analysis results for all ground motion records. 

Figure G-9 Comparison of median collapse capacity for an eight-story 

code-compliant concrete frame building computed using 

incremental dynamic analysis and approximate procedures. 

0.0

0.5

1.0

1.5

2.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Max IDR

Sa (T=1.71s) / g

84%

16%

50%

Figure G-10 Incremental dynamic analysis results for an eight-story code-compliant 

concrete frame building subjected to 80 ground motions (adapted from 

Haselton, 2006). 

0.0

0.4

0.8

1.2

1.6

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Maximum Story Drift Ratio

Sa(T=1.71)/g

 MDOF IDA

 SPO2IDA

Rdi

Maximum Story Drift Ratio
G-8 G: Preliminary Multiple-Degree-of-Freedom System Studies FEMA P440A 

G.3 Twelve-Story Code-Compliant Reinforced 

Concrete Building 

The subject building is a twelve-story reinforced concrete special perimeter 

moment frame designed in accordance with modern building code provisions 

(ICC 2003, ASCE 2002, ACI 2002). Similarly to the two previous buildings, 

the story height is 15 ft in the first story and 13 ft in the remaining stories. 

The design base shear coefficient was 0.044. The building was modeled in 

OpenSEES and analyzed using an incremental dynamic analysis using the 

same 80 recorded ground motions that were used to analyze the four-story 

building. The pushover analysis was again done using a lateral force 

distribution in accordance with ASCE/SEI 7-05. The fundamental period of 

vibration of the building is T1=2.01s. For a more detailed description of the 

building and its modeling, the reader is referred to Haselton (2006). 

Figure G-11 shows the results from the nonlinear static (pushover) analysis 

of the building. The figure on the left shows the force-deformation curve 

while the figure on the right shows the distribution of story drift ratios at a 

roof drift ratio of 2.7%. It can be seen that story drifts decrease 

approximately linearly with increasing height with the largest story drifts 

occurring in the two lower stories. The force-deformation pushover curve is 

characterized by a hardening segment for roof drift ratios between 0.3% and 

0.8%, followed by softening segment for roof drift ratios greater than 0.8%. 

 

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 

(Haselton 2006). 

Figure G-12 shows the simplified tri-linear force-displacement capacity 

boundary selected to evaluate the proposed equation for Rdi and results using 

SPO2IDA. It is assumed that at a roof drift ratio of 2.6% the structure 

reaches its maximum deformation capacity and a total loss in strength occurs. 
FEMA P440A G: Preliminary Multiple-Degree-of-Freedom System Studies G-9 

Figure G-13 compares the median seismic behavior computed from 

incremental dynamic analyses conducted by Haselton (2006), indicated in the 

figure as MDOF IDA, with results computed using the proposed equation for 

Rdi and approximate results from SPO2IDA. In the figure, both approximate 

methods somewhat overestimate the collapse capacity of the structure. 

Figure G-14 shows incremental dynamic analysis results for all ground 

motion records. 

Figure G-12 Tri-linear capacity boundary selected for approximate analyses 

using SPO2IDA. 

Figure G-13 Comparison of median collapse capacity for a twelve-story 

code-compliant concrete frame building computed using 

incremental dynamic analysis and approximate procedures. 

0.0

0.4

0.8

1.2

1.6

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Maximum Story Drift Ratio

Sa(T=2.01)/g

 MDOF IDA

 SPO2IDA

Rdi
G-10 G: Preliminary Multiple-Degree-of-Freedom System Studies FEMA P440A 

0.0

0.5

1.0

1.5

2.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Max IDR

Sa (T=2.01s) / g

84%

16%

50%

Figure G-14 Incremental dynamic analysis results for a twelve-story codecompliant concrete frame building subjected to 80 ground 

motions (adapted from Haselton, 2006). 

G.4 Twenty-Story Code-Compliant Reinforced 

Concrete Building 

The subject building is a twenty-story reinforced concrete special perimeter 

moment frame designed in accordance with modern building code provisions 

(ICC 2003, ASCE 2002, ACI 2002). The story height is 15 ft in the first story 

and 13 ft in the remaining stories. The design base shear coefficient was 

0.044. The building was modeled in OpenSEES and analyzed using an 

incremental dynamic analysis using the same 80 recorded ground motions 

that were used to analyze the four-story building. The pushover analysis was 

again done using a lateral force distribution in accordance with ASCE/SEI 7-

05. The fundamental period of vibration of the building is T1=2.63s. For a 

more detailed description of the building and its modeling, the reader is 

referred to Haselton (2006). 

Figure G-15 shows the results from the nonlinear static (pushover) analysis 

of the building. The figure on the left shows the force-deformation curve 

while the figure on the right shows the distribution of story drift ratios at a 

roof drift ratio of 1.8%. It can be seen that story drifts decrease 

approximately linearly with increasing height, with the largest story drifts 

occurring in the lower two stories. The force-deformation pushover curve is 

characterized by a slight softening segment for roof drift ratios between 0.3% 

Maximum Story Drift Ratio
FEMA P440A G: Preliminary Multiple-Degree-of-Freedom System Studies G-11 

and 0.9%, followed by steeper softening segment for roof drift ratios greater 

than 0.9%. 

 

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 twenty-story concrete frame building 

(Haselton 2006). 

Figure G-16 shows the simplified tri-linear force-displacement capacity 

boundary selected to evaluate the proposed equation for Rdi and results using 

SPO2IDA. It is assumed that at a roof drift ratio of 1.85% the structure 

reaches its maximum deformation capacity and a total loss in strength occurs. 

Figure G-16 Tri-linear capacity boundary selected for approximate analyses 

using SPO2IDA. 

Figure G-17 compares the median seismic behavior computed from 

incremental dynamic analyses conducted by Haselton (2006), indicated in the 

figure as MDOF IDA, with results computed using the proposed equation for 

Rdi and approximate results from SPO2IDA. In the figure, proposed equation 

for Rdi provides a good estimate of the median collapse capacity, while 

SPO2IDA somewhat overestimates the collapse capacity. Figure G-18 

shows incremental dynamic analysis results for all ground motion records. 
G-12 G: Preliminary Multiple-Degree-of-Freedom System Studies FEMA P440A 

Figure G-17 Comparison of median collapse capacity for a twenty-story 

code-compliant concrete frame building computed using 

incremental dynamic analysis and approximate procedures. 

0.0

0.5

1.0

1.5

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Max IDR

Sa (T=2.63s) / g

84%

16%

50%

Figure G-18 Incremental dynamic analysis results for a twenty-story codecompliant concrete frame building subjected to 80 ground 

motions (adapted from Haselton, 2006). 

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Maximum Story Drift Ratio

Sa(T=2.63)/g

 MDOF IDA

 SPO2IDA

Rdi

Maximum Story Drift Ratio 
FEMA P440A G: Preliminary Multiple-Degree-of-Freedom System Studies G-13 

G.5 Nine-Story Pre-Northridge Steel Moment-Resisting 

Frame Building 

The subject building is a nine-story steel moment-resisting frame designed 

for the FEMA-funded SAC project in accordance with pre-Northridge code 

requirements for Los Angeles (ICBO, 1994). The building has a single-story 

basement that is 12 ft in height. The first story height is 18 ft and the 

remaining stories are 13 ft uniformly. The building is symmetric in plan with 

six bays of 30 ft in each direction. There is a perimeter moment-resisting 

frame designed for lateral-force-resistance, while internal gravity columns 

carry most of the vertical load. The building was modeled in OpenSEES and 

analyzed using incremental dynamic analysis with 30 “ordinary” ground 

motions. The pushover analysis was done using a triangular lateral force 

distribution. The fundamental period of vibration of the building is T1=2.3s. 

For a more detailed description of the building and its modeling, the reader is 

referred to Gupta and Krawinkler (1999). 

The results from a nonlinear static (pushover) analysis of the building are 

shown in Figure G-19. The force-deformation pushover curve is 

characterized by a hardening segment for roof drift ratios between 1% and 

2.5%, followed by a softening segment that terminates when the building 

reaches zero strength at 5% roof drift. The simplified tri-linear forcedisplacement capacity boundary, also shown in Figure G-19, was selected to 

evaluate the proposed equation for Rdi and results using SPO2IDA. In both 

cases the hardening segment has 13% of the elastic stiffness while the 

negative stiffness is -74% of elastic. 

Figure G-20 shows the median seismic behavior computed from incremental 

dynamic analyses conducted by Vamvatsikos and Fragiadakis (2006). These 

results are indicated as MDOF IDA in the figure. Also shown are results 

computed using the proposed equation for Rdi and approximate results from 

SPO2IDA. In the figure, both Rdi and SPO2IDA provide a good 

approximation of the collapse capacity of the building. 
G-14 G: Preliminary Multiple-Degree-of-Freedom System Studies FEMA P440A 

0

2000

4000

6000

8000

10000

12000

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Roof Drift Ratio

Base Shear (kN) 

pushover

approx

Figure G-19 Monotonic pushover force-deformation curve, and tri-linear 

approximation, for a nine-story pre-Northridge steel moment 

frame building (adapted from Gupta and Krawinkler, 1999). 

Figure G-20 Comparison of median collapse capacity for a nine-story preNorthridge steel moment frame building computed using 

incremental dynamic analysis and approximate procedures. 

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Maximum Story Drift Ratio

Sa(T=2.28s)/g

 MDOF IDA

 SDO2IDA

Rdi
FEMA P440A G: Preliminary Multiple-Degree-of-Freedom System Studies G-15 

G.6 Twenty-Story Pre-Northridge Steel MomentResisting Frame Building 

The subject building is a twenty-story steel moment resisting frame designed 

for the FEMA-funded SAC project in accordance with pre-Northridge code 

requirements for Los Angeles (ICBO, 1994). The building has a basement 

consisting of two stories that are 12 ft in height. The first story height is 18 ft 

and the remaining stories are 13 ft uniformly. The building is slightly 

asymmetric in plan, with five bays of 20 ft in one direction and six bays of 20 

ft in the other direction. There is a perimeter moment-resisting frame 

designed for lateral-force-resistance. Four internal gravity columns carry the 

vertical loads. The building was modeled in Drain-2DX and analyzed using 

incremental dynamic analysis with 30 “ordinary” ground motions. The 

pushover analysis was done using a parabolic (k = 2) lateral force 

distribution. The fundamental period of vibration of the building is T1=4.0s. 

For a more detailed description of the building and its modeling, the reader is 

referred to Gupta and Krawinkler (1999). 

The results from the nonlinear static (pushover) analysis of the building are 

shown in Figure G-21. The force-deformation pushover curve is 

characterized by a short hardening segment (5% stiffness ratio) from 0.7% to 

1.2% roof drift ratio that then turns negative (-24% stiffness ratio) and 

terminates when the building reaches zero strength at 4% roof drift. The 

simplified tri-linear force-displacement capacity boundary, also shown in 

Figure G-21, was selected to evaluate the proposed equation for Rdi and 

results using SPO2IDA. 

Figure G-22 shows the median seismic behavior computed from incremental 

dynamic analyses conducted by Vamvatsikos and Cornell (2006). These 

results are indicated as MDOF IDA in the figure. Also shown are results 

computed using the proposed equation for Rdi and approximate results from 

SPO2IDA. In the figure, Rdi overestimates the collapse capacity of the 

building by about 25%, while SPO2IDA provides a good approximation. 
G-16 G: Preliminary Multiple-Degree-of-Freedom System Studies FEMA P440A 

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0.00 0.01 0.02 0.03 0.04 0.05

Roof Drift Ratio

Base Shear (kN)

 pushover

 approx

Figure G-21 Monotonic pushover force-deformation curve, and tri-linear 

approximation, for a twenty-story pre-Northridge steel moment 

frame building (adapted from Gupta and Krawinkler, 1999). 

Figure G-22 Comparison of median collapse capacity for a twenty-story preNorthridge steel moment frame building computed using 

incremental dynamic analysis and approximate procedures. 

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.15 0.20

Maximum Story Drift Ratio

Sa(T=4.0s)/g

 MDOF IDA

 SDO2IDA

Rdi
FEMA P440A G: Preliminary Multiple-Degree-of-Freedom System Studies G-17 

G.7 Summary and Recommendations 

The studies documented above indicate that the application of procedures 

developed for SDOF systems to several representative MDOF moment frame 

systems produces reasonable approximations of the median intensity causing 

lateral dynamic instability. This was true in the case of both the proposed 

equation for Rdi and simplified nonlinear dynamic analysis using SPO2IDA. 

These results lead to a recommendation for more thorough investigation of 

MDOF systems to modify, or further refine, the procedures presented here. 

FEMA P440A H: References and Bibliography H-1 

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FEMA P440A I: Project Participants I-1 

Project Participants 

ATC Management and Oversight 

Christopher Rojahn 

Project Executive Director 

Applied Technology Council 

201 Redwood Shores Parkway, Suite 240 

Redwood City, California 94065 

Jon A. Heintz 

Project Quality Control Monitor

Applied Technology Council 

201 Redwood Shores Parkway, Suite 240 

Redwood City, California 94065 

William T. Holmes 

Project Technical Monitor

Rutherford & Chekene 

55 Second Street, Suite 600 

San Francisco, California 94105 

Federal Emergency Management Agency 

Michael Mahoney 

Project Officer

Federal Emergency Management Agency 

500 C Street, SW 

Washington, DC 20472 

Mai (Mike) Tong 

Project Monitor

Federal Emergency Management Agency 

500 C Street, SW 

Washington, DC 20472 

Robert D. Hanson 

FEMA Technical Monitor

Federal Emergency Management Agency 

2926 Saklan Indian Drive 

Walnut Creek, California 94595 

Project Management Committee 

Craig D. Comartin 

Project Technical Director 

Comartin Engineers 

7683 Andrea Avenue 

Stockton, California 95207 

Eduardo Miranda 

 Senior Advisor on Strength Degradation 

Stanford University 

Civil & Environmental Engineering 

Terman Room 293 

Stanford, California 94305 

Michael Valley 

 Senior Advisor for Structural Engineering 

Magnusson Klemencic Associates 

1301 Fifth Avenue, Suite 3200 

Seattle, Washington 98101 
I-2 I: Project Participants FEMA P440A 

Working Group 

Dimitrios Vamvatsikos 

University of Cyprus 

75 Kallipoleos Street 

P.O. Box 20537 

Nicosia, 1678, Cyprus 

Project Review Panel 

Kenneth Elwood 

University of British Columbia 

Dept. of Civil Engineering 

6250 Applied Science Lane, Room 2010 

Vancouver, British Columbia V6T 1Z4 Canada 

Subhash C. Goel 

University of Michigan 

Dept. of Civil and Envir. Engineering 

2350 Hayward, 2340 G.G. Brown Building 

Ann Arbor, Michigan 48109-2125 

Farzad Naeim 

John A. Martin & Associates, Inc. 

1212 S. Flower Street, 4th Floor 

Los Angeles, California 90015 

Workshop Participants 

Mark Aschheim 

Santa Clara University 

500 El Camino Real 

Dept. of Civil Engineering 

Santa Clara, California 95053 

Michael Cochran 

Weidlinger Associates 

4551 Glencoe Avenue, Suite 350 

Marina del Rey, California 90292-7927 

Craig D. Comartin 

Comartin Engineers 

7683 Andrea Avenue 

Stockton, California 95207 

Anthony Court 

A. B. Court & Associates 

4340 Hawk Street 

San Diego, California 92103 

Kenneth Elwood 

University of British Columbia 

Department of Civil Engineering 

6250 Applied Science Lane, Room 2010 

Vancouver, British Columbia V6T 1Z4 Canada 

Subhash C. Goel 

University of Michigan 

Department of Civil and Envir. Engineering 

2350 Hayward, 2340 G.G. Brown Building 

Ann Arbor, Michigan 48109 

Robert D. Hanson 

Federal Emergency Management Agency 

2926 Saklan Indian Drive 

Walnut Creek, California 94595 

Curt Haselton 

California State University, Chico 

Department of Civil Engineering 

Langdon 209F 

Chico, California 95929 

Jon A. Heintz 

Applied Technology Council 

201 Redwood Shores Pkwy., Suite 240 

Redwood City, California 94065 
FEMA P440A I: Project Participants I-3 

YeongAe Heo 

University of California at Davis 

Dept. of Civil Engineering 

Davis, California 95616 

William T. Holmes 

Rutherford & Chekene 

55 Second Street, Suite 600 

San Francisco, California 94105 

Sashi Kunnath 

University of California, Davis 

Dept. of Civil & Env. Engineering 

One Shields Ave., 2001 Engr III 

Davis, California 95616 

Joseph Maffei 

Rutherford & Chekene 

55 Second Street, Suite 600 

San Francisco, California 94105 

Stephen Mahin 

University of California, Berkeley 

777 Davis Hall, Dept. of Civil Engineering 

Berkeley, California 94720 

Michael Mahoney 

Federal Emergency Management Agency 

500 C Street, SW 

Washington, DC 20472 

Michael Mehrain 

URS Corporation 

915 Wilshire Blvd., Suite 700 

Los Angeles, California 90017 

Eduardo Miranda 

Stanford University 

Civil & Environmental Engineering 

Terman Room 293 

Stanford, California 94305 

Mark Moore 

ZFA Consulting 

55 Second Street, Suite 600 

San Francisco, California 94105 

Charles Roeder 

University of Washington 

Department of Civil Engineering 

233-B More Hall Box 2700 

Seattle, Washington 98195 

Mark Sinclair 

Degenkolb Engineers 

225 Bush Street, Suite 1000 

San Francisco, California 94104 

Peter Somers 

Magnusson Klemencic Associates 

1301 Fifth Avenue, Suite 3200 

Seattle, Washington 98101 

Mai (Mike) Tong 

Federal Emergency Management Agency 

500 C Street, SW 

Washington, DC 20472 

Luis Toranzo 

KPFF Consulting Engineers 

6080 Center Drive, Suite 300 

Los Angeles, California 90045 

Michael Willford 

ARUP 

901 Market Street, Suite 260 

San Francisco, California 94103