Improvement of 

Nonlinear Static Seismic 

Analysis Procedures

FEMA 440 

June 2005

FEMA nehrp

FEMA 440

IMPROVEMENT OF NONLINEAR STATIC

SEISMIC ANALYSIS PROCEDURES

Prepared by:

Applied Technology Council (ATC-55 Project)

201 Redwood Shores Parkway, Suite 240

Redwood City, California 94065

Prepared for:

Department of Homeland Security

Federal Emergency Management Agency

Washington, D.C.

June, 2005
Notice

Any opinions, findings, conclusions, or recommendations expressed in this publication do not necessarily reflect the views of the Applied Technology Council (ATC) 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 440 Improvement of Nonlinear Static Seismic Analysis Procedures iii 

Forward

One of the primary goals of the Department of 

Homeland Security’s Federal Emergency Management Agency (FEMA) and the National Earthquake 

Hazards Reduction Program (NEHRP) is to encourage design and building practices that address the 

earthquake hazard and minimize the resulting damage. This document, Improvement of Nonlinear 

Static Seismic Analysis Procedures (FEMA 440), 

reaffirms FEMA’s ongoing efforts to improve the 

seismic safety of new and existing structures in this 

country. 

The primary goal of this project was the evaluation 

and improvement of the nonlinear static procedures 

(NSPs) contained in the Prestandard and Commentary for Seismic Rehabilitation of Buildings (FEMA 

356) and in the Applied Technology Council ATC40 report, Seismic Evaluation and Retrofit of Concrete Buildings, and the development of guidance on 

when and how each methodology should be used to 

avoid conflicting answers. FEMA initiated this 

project with ATC based on reports of discrepancies 

between the two NSP methodologies. However, in 

the course of this project, several improvements to 

both procedures were also identified and we thought 

it in the best interests of the earthquake engineering 

community to capture those improvements as part of 

this state-of-the-art resource document.

There are some potential differences between this 

document and other FEMA-sponsored products, 

such as the FEMA 356-based Standard for the Seismic Rehabilitation of Existing Buildings currently 

being developed by the American Society of Civil 

Engineers (ASCE-41) and FEMA’s HAZUS standardized loss estimation methodology, which uses 

the procedures of ATC-40 in its fragility functions. 

Some of this document’s recommendations concerning NSPs could bias selection of analysis procedures to linear static procedures (LSPs) unless 

similar modifications are also made to the LSPs. 

These differences are primarily for short-period 

structures, and should not affect the ongoing use of 

those current products. This document is a resource 

guide to capture the current state of the art for improved understanding of NSPs and to generate future improvements to those products, and as such, 

should not take precedence over those products.

Looking ahead, FEMA is already funding ATC to 

perform additional studies of the cyclic and in-cycle 

stiffness and strength degradation nonlinear models 

and their impact on response and response stability. 

Future FEMA-funded ATC studies will focus on the 

differences between linear and nonlinear design for 

short-period buildings and on soil-structure interaction. The results of these studies should be available 

within the next four years, within the time frame for 

submittal to a future update of ASCE 41. 

FEMA is proud to have sponsored the development 

of this resource document through ATC. We are 

particularly grateful for work done by Project Director Craig Comartin, the Project Management Committee, the Project Review Panel, the Project Focus 

Groups and Working Groups, and all of the other 

contributors who made this document possible. 

FEMA also wishes to acknowledge the National 

Science Foundation (NSF) for their funding provided through the Pacific Earthquake Engineering Research Center (PEER) for the investigation of shortperiod building response and soil-structure interaction. We also wish to acknowledge the NSF funding 

of the research of Andrew Guyader on equivalent 

linearization and the NATO science fellowship from 

the Scientific Research and Technical Council of 

Turkey that partially funded research by Sinan 

Akkar. This project is an excellent example of the 

interagency cooperation that is made possible 

through the NEHRP. All of the individuals involved 

in this project are listed at the end of this document, 

and FEMA gratefully appreciates their involvement. 

This product would not have been possible without 

their dedication and professionalism. 

Federal Emergency Management Agency
Forward

iv Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures v 

Preface

Knowledgeable engineers have long recognized that the 

response of buildings to strong ground shaking caused 

by earthquakes results in inelastic behavior. Until 

recently, most structural analysis techniques devised for 

practical application relied on linear procedures to 

predict the seismic behavior of buildings. With the 

publication of the ATC-40 Report, Seismic Evaluation 

and Retrofit of Concrete Buildings, in 1996, the FEMA 

273 Report, Guidelines for the Seismic Rehabilitation of 

Buildings, in 1997, and the FEMA 356 Report, 

Prestandard and Commentary for the Seismic 

Rehabilitation of Buildings (which replaced FEMA 

273), in 2000, nonlinear static analysis procedures 

became available to engineers providing efficient and 

transparent tools for predicting seismic behavior of 

structures.

Both the ATC-40 and FEMA 356 documents present 

similar performance-based engineering methods that 

rely on nonlinear static analysis procedures for 

prediction of structural demands. While procedures in 

both documents involve generation of a “pushover” 

curve to predict the inelastic force-deformation 

behavior of the structure, they differ in the technique 

used to calculate the inelastic displacement demand for 

a given ground motion. The FEMA 356 document uses 

the Coefficient Method, whereby displacement demand 

is calculated by modifying elastic predictions of 

displacement demand. The ATC-40 Report details the 

Capacity-Spectrum Method, whereby modal 

displacement demand is determined from the 

intersection of a capacity curve, derived from the 

pushover curve, with a demand curve that consists of 

the smoothed response spectrum representing the 

design ground motion, modified to account for 

hysteretic damping effects. 

The publication of the above cited documents resulted 

in the widespread use of these two methods, and 

engineers have since reported that the two procedures 

often give different estimates for displacement demand 

for the same building. Hence the Applied Technology 

Council (ATC) proposed to the Federal Emergency 

Management Agency (FEMA) in 2000 that a study be 

conducted to determine the reasons for differing results 

and to develop guidance for practicing engineers on 

improved application of these two methods. FEMA 

agreed to fund the investigation, and in October 2000, 

ATC commenced a project to provide guidance for 

improved applications of these two widely used 

inelastic seismic analysis procedures (ATC-55 Project). 

The ATC-55 Project had two objectives: (1) the 

development of practical recommendations for 

improved prediction of inelastic structural response of 

buildings to earthquakes (i.e., guidance for improved 

application of inelastic analysis procedures) and (2) the 

identification of important issues for future research. 

Intended outcomes of the project included:

1. Improved understanding of the inherent assumptions and theoretical underpinnings of existing and 

proposed updated inelastic analysis procedures.

2. Recognition of the applicability, limitations, and 

reliability of various procedures.

3. Guidelines for practicing engineers to apply the 

procedures to new and existing buildings.

4. Direction for researchers on issues for future 

improvements of inelastic analysis procedures.

The project was conducted in three phases over a 3-year 

time span. Phase 1 consisted of the assembly and 

refinement of important issues relating to the 

improvement of inelastic seismic analysis procedures. 

Activities included (1) the solicitation of input from 

researchers and practicing engineers, and (2) the 

development of study models of typical buildings to 

stimulate discussion, facilitate analytical studies, and 

provide example applications. Phase 2 consisted of 

analytical studies to explore selected key issues, the 

generation of written discussions on important topics, 

and the development of examples of the application of 

inelastic analysis procedures. This phase also included 

assembly of guidelines for the improved practical 

implementation of the procedures. Phase 3 consisted of 

the report development process, under which this 

document was drafted, reviewed, and finalized. 

This report (FEMA 440) is the final and principal 

product of the ATC-55 Project. The document has three 

specific purposes: (1) to provide guidance directly 

applicable to the evaluation and design of actual 

structures by engineering practitioners; (2) to facilitate a 

basic conceptual understanding of underlying principles 

as well as the associated capabilities and limitations of 

the procedures; and (3) to provide additional detailed 

information used in the development of the document 

for future reference and use by researchers and others. 
Preface

vi Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

A wide variety of personnel participated in the project. 

The project was conducted under the direction of ATC 

Senior Consultant Craig Comartin, who served as 

Project Director. Technical and management direction 

were provided by a Project Management Committee 

consisting of Craig Comartin (Chair), Christopher 

Rojahn (Ex-Officio member), Ronald O. Hamburger, 

William T. Holmes, Wilfred D. Iwan, Jack P. Moehle 

and Jonathan Stewart. A Project Review Panel, 

identified by ATC with input from FEMA, provided 

overview and guidance; this Panel consisted of Anthony 

B. Court (ATC Board Representative), Leonard Joseph, 

Daniel Shapiro, Steve Sweeney, Chia-Ming Uang, and 

Michael Valley. 

The Project Management Committee created four Focus 

Groups to assist in developing findings on the following 

specific subtopics: (1) displacement modification; (2) 

equivalent linearization; (3) multi-degree-of-freedom 

effects; and (4) response of short-period buildings, with 

a specific focus on soil-structure interaction. The 

purpose of the Focus Groups was to gather fresh 

perspective from qualified sources that were not 

directly responsible for the project planning or the 

resulting recommendations. Focus Group participants 

reviewed draft materials developed by the project team. 

They then attended a one-day meeting with 

representative members of the Project Management 

Committee and the project team members responsible 

for the subject materials. The meetings allowed for a 

constructive discussion of the subject in general and 

critical feedback – positive and negative – on the draft 

materials. Focus Group members were also afforded an 

opportunity to comment on the final draft of materials 

related to their area of expertise. It is important to note 

that Focus Group members were not asked to endorse 

the project process or the recommendations in 

documents developed as part of the ATC-55 Project. 

These remain the responsibility of ATC and the Project 

Management Committee.

Each Focus Group consisted of three members. John 

Hooper, Gregory A. MacRae, and Stephen A. Mahin 

were members of the Focus Group on Displacement 

Modification. The Focus Group on Equivalent 

Linearization consisted of Terrance Paret, Graham 

Powell, and Andrew S. Whittaker. Anil K. Chopra, Jon 

A. Heintz, and Helmut Krawinkler served on the Focus 

Group on Multi-Degree-of-Freedom Effects, and 

Jacobo Bielak, Gregory L. Fenves, and James Malley 

served on the Focus Group on Soil-structure 

Interaction. 

Detailed work on the project was carried out by several 

Working Groups appointed by the Project Management 

Committee. The Phase 1 Project Working Group 

consisted of Joseph R. Maffei (Group Leader), Mark 

Aschheim, Maureen Coffey, and Mason T. Walters. The 

Phase 2 Project Working Group consisted of Sinan 

Akkar, Mark Aschheim, Andrew Guyader, Mehmet 

Inel, Eduardo Miranda, Junichi Sakai, Jorge RuizGarcia, Tjen Tjhin and Tony Yang. Peter N. Mork 

produced and formatted the electronic files from which 

this report was printed.

The affiliations of the project personnel identified 

above are provided in the list of Project participants. 

The Applied Technology Council gratefully 

acknowledges the cooperation, insight and patience 

provided by the FEMA Project Officer, Michael 

Mahoney, and the FEMA Technical Monitor, Robert D. 

Hanson. ATC also gratefully acknowledges the 

National Science Foundation (NSF)for supplemental 

funding provided through the Pacific Earthquake 

Engineering Research Center to conduct the 

investigation of the response of short-period buildings, 

soil-structure-foundation interaction, and application of 

the proposed methods. NSF also provided funding for 

the research of Andrew Guyader on equivalent 

linearization. A NATO science fellowship from the 

Scientific Research and Technical Council of Turkey 

provided partial support for research by Sinan Akkar.

Christopher Rojahn

ATC Executive Director
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures vii 

Executive Summary

This document records in detail an effort to assess 

current nonlinear static procedures (NSP) for the 

seismic analysis and evaluation of structures. In 

addition, the document presents suggestions that were 

developed to improve these procedures for future 

application by practicing engineers. The elements of 

work included several analytical studies to evaluate 

current procedures and to test potential improvements. 

An extensive review of existing pertinent technical 

literature was compiled. A survey of practicing 

engineers with experience in applying nonlinear static 

procedures was also conducted. Expert practitioners 

and researchers in appropriate fields worked together to 

develop the proposed improvements presented in this 

document. The context for the work was provided by 

two existing documents, the FEMA 356 Prestandard 

and Commentary for the Seismic Rehabilitation of 

Buildings, and the ATC-40 report, Seismic Evaluation 

and Retrofit of Concrete Buildings, each of which 

contain procedures for nonlinear static analysis. These 

procedures were both evaluated and suggestions for 

improvement are made for each. Not all of the portions 

of the two current documents (FEMA 356 and ATC-40) 

were evaluated. Conclusions regarding the relative 

accuracy or technical soundness of these documents 

should not be inferred beyond the specific material and 

discussions contained in this document. 

1. Overview of Inelastic Seismic Analysis 

Procedures

Nonlinear static procedures are one type of inelastic 

analysis that can be used to estimate the response of 

structures to seismic ground shaking. The differences 

between the various approaches relate to the level of 

detail of the structural model and the characterization of 

the seismic ground shaking. Detailed structural models 

can often be simplified into equivalent multi-degree-offreedom (MDOF) models; or, in some cases, singledegree-of-freedom (SDOF) oscillator models, as with 

nonlinear static procedures. The most detailed 

characterizations of seismic ground motion are actual 

ground motion records that comprise accelerations, 

velocities, and displacements expected at the ground 

surface at a specific site. A simplification can be made 

by representing the effects ground motion has in the 

frequency domain with response spectra that plot 

maximum response of an elastic SDOF oscillator as a 

function of period. This is the type of characterization 

normally used for nonlinear static procedures. 

The discussion provided in Chapter 2 includes basic 

descriptions of the two nonlinear static procedures that 

currently are used in practice. FEMA 356 utilizes a 

displacement modification procedure (Coefficient 

Method) in which several empirically derived factors 

are used to modify the response of a single-degree-offreedom model of the structure assuming that it remains 

elastic. The alternative Capacity-Spectrum Method of 

ATC-40 is actually a form of equivalent linearization. 

This technique uses empirically derived relationships 

for the effective period and damping as a function of 

ductility to estimate the response of an equivalent linear 

SDOF oscillator.

2. Evaluation of Current Nonlinear Static 

Procedures

In practice, the current procedures can result in 

estimates of maximum displacement that are 

significantly different from one another. This has 

caused concern on the part of practicing engineers. One 

of the major objectives of the project was to ascertain 

the reason for these differences and to try to correct 

both procedures to produce similar results. Chapter 3 

documents a comprehensive evaluation of both current 

procedures. The basic technique was to develop a series 

of nonlinear single-degree-of-freedom oscillators of 

varying period, strength, and hysteretic behavior. These 

were subjected to ground motion representing different 

site soil conditions. The resulting database of 

approximately 180,000 predictions of maximum 

displacement was used as a benchmark to judge the 

accuracy of the approximate nonlinear static 

procedures. This was accomplished by comparing the 

estimates for each oscillator from both nonlinear static 

procedures to the results of the nonlinear response 

history analyses. Differences in the two estimates were 

compiled and compared in a statistical study. 

3. Strength Degradation

The results of the evaluation of the nonlinear static 

procedures suggest that both procedures would benefit 

from greater clarity with respect to the different types of 

possible degradation in structures subject to seismic 

shaking. This is particularly critical for degradation in 

strength. Chapter 4 presents a discussion of the 

differences between the consequences of strength loss 

within a single cycle of deformation (in-cycle) and that 

which occurs in subsequent cycles (cyclic). In-cycle 

strength degradation, including that associated with P-∆
Executive Summary

viii Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

effects, can lead to dynamic instability. To account for 

this, a limitation on the strength of a structure is 

suggested for use with nonlinear static procedures. The 

limit is a function of the period of the structure and the 

post-elastic stiffness characteristics as modified for incycle strength degradation. If the structure has less 

strength than the limit, nonlinear dynamic analysis is 

recommended.

4. Improved Procedures for 

Displacement Modification 

Based on the evaluation of nonlinear static procedures, 

Chapter 5 proposes modifications to the Coefficient 

Method of FEMA 356. The suggestions relate primarily 

to the coefficients themselves. Improved relationships 

for coefficients C1 and C2 are proposed. It is also 

suggested that the coefficient C3 be replaced with a 

limitation on minimum strength as suggested in the 

previous section.

5. Improved Procedures for Equivalent 

Linearization 

Chapter 6 presents the results of an effort to improve the 

practical application of equivalent linearization 

procedures. The resulting suggestions focus upon 

improved estimates of equivalent period and damping. 

This chapter also includes an optional adjustment to 

generate a modified acceleration-displacement response 

spectrum (MADRS) that does intersect the capacity 

spectrum at the Performance Point. Similar to the 

current ATC-40 procedure, the effective period and 

damping are both dependent on ductility and 

consequently an iterative or graphical technique is 

required to calculate the Performance Point. Several 

options are outlined in Chapter 6. In application, the 

improved procedures are similar to the current ATC-40 

Capacity-Spectrum Method.

6. Evaluation and Comparison of 

Improved Nonlinear Static Procedures

The improved procedures were evaluated in an 

independent study. This study, summarized in 

Chapter 7, utilized nine elastic-perfectly-plastic 

oscillators with three different periods and three 

different strengths. These were subjected to thirteen 

ground motions for class C sites. Estimates of 

maximum displacements were calculated utilizing both 

current procedures and the proposed improved 

procedures. 

This study was not comprehensive enough to make 

broad general conclusions. However, a number of key 

observations can be made:

• The improved procedures do not exhibit large 

differences between displacement modification and 

equivalent linearization approaches.

• The improved procedures also produced more 

accurate estimates of displacements when compared 

to response history analysis (also known as timehistory analysis) results than those produced by the 

current nonlinear procedures.

• Improved procedures also seem to work well, at 

least for the case that was studied, in estimating 

maximum displacement response in conjunction 

with a design spectrum.

• The results of the evaluation of the improved 

nonlinear procedures illustrate the dispersion of 

results from nonlinear response history analysis 

using design level ground motions. 

7. Soil-Structure Interaction Effects

Chapter 8 presents procedures to incorporate soilstructure interaction (SSI) into nonlinear static analyses. 

The objective is to replace the judgmental limits with 

rational technical justifications for reducing seismic 

demand. These SSI techniques address the following 

issues. 

• radiation and material damping in supporting soils; 

• response reduction resulting from structure 

embedment in the ground (i.e., full and partial 

basements); and

• incoherent ground-motion input to buildings with 

relatively large plan dimensions.

The basic principles used for the development of the 

SSI procedures for damping in Chapter 8 have been 

included in the FEMA 368 NEHRP Recommended 

Provisions for Seismic Regulations for New Buildings 

and Other Structures (BSSC, 2000)1 for the linear 

analysis and design of new buildings for a number of 

years. They have been adapted for use with inelastic 

procedures. Both the damping and ground motion 

procedures are applicable to both the displacement 

modification and equivalent linearization forms of 

nonlinear static analysis.

1.Superseded in 2003 with the FEMA 450 Recommended

Provisions for Seismic Regulations for New Buildings

and Other Structures.
Executive Summary

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures ix

8. Multiple-Degree-of Freedom Effects

Chapter 9 reviews the accuracy and practical 

implications of the requirements of ATC-40 and FEMA 

356 related to MDOF effects including:

1. current options for load vectors, and 

2. the conversion of a MDOF pushover curve to an 

equivalent SDOF system.

The results of a comprehensive study of five example 

buildings that examines the differences in response 

predicted using various options compared to a common 

nonlinear dynamic analyses benchmark are also 

summarized. The results are consistent with previous 

research. Practical implications are:

• Nonlinear static procedures generally provide 

reliable estimates of maximum floor and roof 

displacements. 

• Nonlinear static procedures are not particularly 

capable, however, of accurate prediction of 

maximum story drifts, particularly within flexible 

structures.

• Nonlinear static procedures are very poor predictors 

of story forces, including shears and overturning 

moments.

• The use of the first mode load vector is suggested 

due to the relatively good displacement estimates 

made with this assumption. 

• Multi-mode pushover analysis consisting of the use 

of multiple load vectors proportional to the mode 

shapes of the structure and combining them 

statistically shows promise in producing better 

estimates in inter-story drifts over the heights of the 

buildings. 

• The provisions of FEMA 356 as to when higher 

modes are to be considered significant are not 

particularly reliable. 

• Specific limitations as to when nonlinear static 

procedures produce reliable results are elusive. 

• As a result of the study it was observed that, in many 

cases, a single time history response of a multidegree-of-freedom model gave better indications of 

drifts and story forces than any of the approximate 

single-degree-of-freedom estimates.

9. Important Future Developments

The proposed improvements to nonlinear static analysis 

procedures in this document will lead to better results in 

practice. Nonetheless, not all of the shortcomings of 

NSP’s have been addressed. In developing the 

improvements a number of important observations 

about the need for future develop and improvement of 

inelastic seismic analysis procedures have emerged. 

These include the need for additional developmental 

work on:

1. Nonlinear Modeling for Cyclic and In-Cycle Degradation of Strength and Stiffness 

2. Soil and Foundation Structure Interaction

3. Nonlinear Multi-Degree of Freedom Simplified 

Modeling

10. Application Example

Chapter 10 includes an example application of the 

recommended nonlinear static analysis procedures on 

an example building. The application example includes 

a flowchart describing the implementation process, 

along with building plans, calculations, and 

commentary. The example illustrates both the 

displacement modification and the equivalent 

linearization procedures to estimate the maximum 

displacement of a building model. 
Executive Summary

x Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures xi

Table of Contents

Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1. Overview of Inelastic Seismic Analysis Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

2. Evaluation of Current Nonlinear Static Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

3. Strength Degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

4. Improved Procedures for Displacement Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

5. Improved Procedures for Equivalent Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

6. Evaluation and Comparison of Improved Nonlinear Static Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . viii

7. Soil-Structure Interaction Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

8. Multiple-Degree-of Freedom Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

9. Important Future Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

10. Application Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

List of Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.2 Project Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.3 Report Scope, Organization and Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

2. Overview of Inelastic Seismic Analysis Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.1 Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.2 Characterization of Seismic Ground Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

2.3 Options for Inelastic Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5

2.4 Current Nonlinear Static Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8

2.4.1 The Coefficient Method of Displacement Modification from FEMA 356 . . . . . . . . . . . . . . . . 2-9

2.4.2 Capacity-Spectrum Method of Equivalent Linearization in ATC-40 . . . . . . . . . . . . . . . . . . . 2-10

3. Evaluation of Current Nonlinear Static Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.2 Evaluation Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.2.1 Hysteretic Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.2.2 Earthquake Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3

3.2.3 Error Measures and Statistical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4

3.3 Evaluation of Capacity-Spectrum Method of ATC-40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

3.3.1 Summary of the Approximate Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

3.3.2 Iteration Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

3.3.3 Evaluation Using Ground Motion Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8

3.4 Evaluation of Coefficient Method (FEMA 356). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

3.4.1 Summary of the Approximate Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
xii Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

3.4.2 Maximum Displacement Ratio (Coefficient C1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10

3.4.3 Degrading System Response (Coefficient C2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15

3.4.4 P-∆ Effects (Coefficient C3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17

3.5 Nonlinear Elastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19

4. Strength Degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.1 Types of Strength Degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.2 Strength Degradation and SDOF Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.3 Global Force-Deformation Behavior with Strength Degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2

4.4 Limitation on Strength for In-Cycle Strength Degradation Including P-∆ Effects . . . . . . . . . . . . . . . . . 4-3

5. Improved Procedures for Displacement Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.2 Maximum Displacement Ratio (Coefficient C1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.2.1 Simplified Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.2.2 Limits on Maximum Displacements for Short Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2

5.3 Adjustment for Cyclic Degradation (Coefficient C2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3

5.4 Limitation on Strength to Avoid Dynamic Instability for Nonlinear Static Procedures . . . . . . . . . . . . . 5-4

6. Improved Procedures for Equivalent Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.2 Basic Equivalent Linearization Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

6.2.1 Effective Damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

6.2.2 Effective Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

6.2.3 MADRS for Use with Secant Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

6.3 Spectral Reduction for Effective Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

6.4 Solution Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6

6.5 Approximate Solution Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

6.6 Iterative Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

6.7 Limitation on Strength to Avoid Dynamic Instability for Nonlinear Static Procedures . . . . . . . . . . . . 6-10

7. Evaluation and Comparison of Improved Nonlinear Static Procedures . . . . . . . . . . . . . . . 7-1

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2 Summary of Evaluation Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2.1 NEHRP Design Response Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2.2 Ground Motions and Ground-Motion Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2.3 Characteristics of Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

7.2.4 Nonlinear Static Procedure Estimates Using Smoothed or Average Spectra . . . . . . . . . . . . . . 7-3

7.2.5 Response-History Analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

7.3 Results of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

7.4 Summary of Implications of the Results of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10

8. Procedures for Including Soil-Structure Interaction Effects . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.2 Procedures for Kinematic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

8.3 Procedures for Foundation Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4

9. Multiple-Degree-of-Freedom Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures xiii

9.2 Review of Current Simplified Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

9.2.1 Single-Mode Load Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

9.2.2 Multi-Mode Pushover Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2

9.2.3 Summary of Current Provisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2

9.3 Summary of Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

9.3.1 Load Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

9.3.2 Equivalent SDOF Estimates of Global Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-4

9.4 Practical Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6

9.4.1 Single Load Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7

9.4.2 Multi-Mode Pushover Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-10

9.4.3 Roof Displacement Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11

9.4.4 Limitation of Simplified Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11

9.5 Potential Future Improvements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-12

9.5.1 Incremental Response-Spectrum Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-12

9.5.2 Nonlinear Dynamic Procedure Using Scaled Response Histories. . . . . . . . . . . . . . . . . . . . . . 9-12

10. Summary and Application Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1

10.1 Overview of Inelastic Seismic Analysis Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1

10.2 Evaluation of Current Nonlinear Static Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1

10.2.1 Key Observations: ATC-40 Version of Capacity-Spectrum Method . . . . . . . . . . . . . . . . . . . 10-1

10.2.2 Key Observations: FEMA 356 and the Coefficient Method . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2

10.3 Strength Degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3

10.4 Improved Procedures for Displacement Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3

10.4.1 Summary of Findings Pertaining to Coefficient C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3

10.4.2 Summary of Findings Pertaining to Coefficient C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4

10.4.3 Summary of Findings Pertaining to Coefficient C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4

10.5 Improved Procedures for Equivalent Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-5

10.6 Evaluation and Comparison of Improved Nonlinear Static Procedures . . . . . . . . . . . . . . . . . . . . . . . . . 10-5

10.7 Soil-Structure Interaction Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-6

10.8 Multiple-Degree-of Freedom Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-6

10.9 Uncertainty and Reliability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8

10.10 Important Future Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-10

10.10.1 Nonlinear Modeling for Cyclic and In-Cycle Degradation of Strength and Stiffness. . . . . . 10-10

10.10.2 Soil and Foundation Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-11

10.10.3 Nonlinear Multi-Degree of Freedom Simplified Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 10-11

10.11 Application Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-12

10.11.1 Example Building Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-12

10.11.2 Basic Ground Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-12

10.11.3 Kinematic Soil-structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-12

10.11.4 Fixed-Base Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14

10.11.5 Flexible-Base Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14

10.11.6 Foundation Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14

10.11.7 Force-Displacement Relationships (Pushover Curves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-15

10.11.8 Check on Minimum Strength for Strength Degrading Model . . . . . . . . . . . . . . . . . . . . . . . . 10-15

10.11.9 Target Displacement for Displacement Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-15

10.11.10 Calculation of the Performance Point Using Equivalent Linearization . . . . . . . . . . . . . . . . . 10-16

10.11.11 Check on Assumed Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-16

References and Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1
xiv Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Project Participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1

Appendices (on enclosed CD-ROM):

A. Summary of Research on Inelastic Analysis Procedures . . . . . . . . . . . . . . . . . . . . . . . . . A-1

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

A.2 Classification of Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

A.3 Nonlinear Static Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

A.3.1 Overview of Current Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

A.3.2 Fundamental Bases and Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6

A.3.3 Near-Field Effects on SDOF Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

A.3.4 Equivalent SDOF Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

A.3.5 Behavior Mode Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

A.3.6 MDOF and Inelastic Mechanism Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

A.3.7 Pushover Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-11

A.4 Nonlinear Dynamic Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-12

A.4.1 Simplified Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-12

A.4.2 Incremental Dynamic Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-12

A.5 Modeling Limitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-13

A.6 Demand Characterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-14

A.7 Applicability for Performance-Based Earthquake Engineering and Design . . . . . . . . . . . . . . . . . . . . . A-14

A.7.1 Role for Inelastic Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-14

A.7.2 Design Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-15

A.7.3 Quantities to be Determined and Measures of Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . A-16

A.7.4 Statistical Measures and Treatment of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-16

A.8 References and Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-16

B. Summary of Practice using Inelastic Analysis Procedures. . . . . . . . . . . . . . . . . . . . . . . . . B-1

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1

B.2 Typical Buildings and Structural Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1

B.3 Inelastic Analysis Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1

B.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2

B.5 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3

B.6 Use of Limitations on Coefficient C1 in FEMA 356 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4

B.7 Practical Guidance and Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4

C. Supplemental Data on the Evaluation of Current Procedures . . . . . . . . . . . . . . . . . . . . . C-1

C.1 Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1

C.2 Response History Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-6

C.2.1 Effect of Site Class on C1 of SDOF Systems with Elastoplastic Perfectly Plastic (EPP) 

Hysteretic Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C-6

C.2.2 Effect of Site Class on C1 of SDOF Systems with Stiffness Degrading (SD) 

Hysteretic Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C-7

C.2.3 Effect of Site Class on C1 of SDOF Systems with Strength and Stiffness Degrading (SSD) 

Hysteretic Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C-8

C.2.4 Effect of Site Class on C1 of SDOF Systems with Nonlinear Elastic Hysteretic Behavior . . . . C-9

C.2.5 Evaluation of Coefficient C2 for Site Class B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-10

C.2.6 Evaluation of Coefficient C2 for Site Class C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-11

C.2.7 Evaluation of Coefficient C2 for Site Class D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-12
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures xv

C.2.8 Evaluation of Coefficient C2 for Site Class E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-13

C.2.9 Evaluation of Coefficient C2 for Near Fault Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-14

C.2.10 Effect of Site Class on Coefficient C2 (Stiffness Degrading Hysteretic Behavior) . . . . . . . . . C-15

C.2.11 Effect of Site Class on Coefficient C2 (Strength-Stiffness Degrading Hysteretic Behavior) . . C-16

C.2.12 Effect of Site Class on Coefficient C2 (Nonlinear Elastic Hysteretic Behavior) . . . . . . . . . . . C-17

C.2.13 Effect of Hysteretic Behavior on C1 of SDOF Systems (Site Class B). . . . . . . . . . . . . . . . . . . C-18

C.2.14 Effect of Hysteretic Behavior on C1 of SDOF Systems (Site Class C). . . . . . . . . . . . . . . . . . . C-19

C.2.15 Effect of Hysteretic Behavior on C1 of SDOF Systems (Site Class D). . . . . . . . . . . . . . . . . . . C-20

C.2.16 Effect of Hysteretic Behavior on C1 of SDOF Systems (Site Class E). . . . . . . . . . . . . . . . . . . C-21

C.2.17 Effect of Hysteretic Behavior on C1 of SDOF Systems (Near Fault Set) . . . . . . . . . . . . . . . . C-22

C.3 Evaluation of ATC-40 Version of Capacity Spectrum Method: Summary Results . . . . . . . . . . . . . . . . C-23

C.3.1 Comparisons for Site Class B: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-23

C.3.2 Comparisons for Site Class C: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-24

C.3.3 Comparisons for Site Class D: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-25

C.3.4 Comparisons for Site Class E: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-26

C.3.5 Comparisons for Near-Fault Ground Motions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-27

C.4 Evaluation of the Coefficient Method of FEMA 356: Summary Results. . . . . . . . . . . . . . . . . . . . . . . . C-28

C.4.1 FEMA 356 Nonlinear Static Procedure (NSP) C1 Values for Different Ts Values: . . . . . . . . . C-28

C.4.2 FEMA 356 NSP C2 Values for Different Ts Values:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-30

C.4.3 Mean Error of FEMA 356 NSP (Mean of Approximate to Exact Maximum 

Inelastic Displacements):. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-31

C.4.4 Dispersion of the Error in FEMA 356 NSP (Standard Deviation of Approximate to 

Exact Maximum Inelastic Displacements): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-37

D. Supplementary Information and Data on Equivalent Linearization . . . . . . . . . . . . . . . . . D-1

D.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-1

D.2 Capacity-Spectrum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-1

D.2.1 Structural Capacity: Inelastic Pushover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-1

D.2.2 Seismic Demand: Response Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2

D.3 Theoretical Basis for Equivalent Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2

D.4 Starting Point For Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-5

D.5 Alternative Statistical Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-6

D.5.1 Error Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-7

D.5.2 Optimization Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-8

D.6 Effective Linear Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-9

D.7 Performance Point Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-11

D.8 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-11

E. Supplementary Information and Data on Soil-Structure Interaction Effects . . . . . . . . . . . E-1

E.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-1

E.2 Kinematic interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-1

E.2.1 Shallow Foundations at the Ground Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-1

E.2.2 Embedded Shallow Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-3

E.2.3 Application of Transfer Functions to Calculation of Foundation Motions . . . . . . . . . . . . . . . . . E-4

E.2.4 Simplified Procedure for Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-6

E.3 Foundation Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-8

E.3.1 Analysis of Impedance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-9

E.3.2 Analysis of System Damping Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-15

E.3.3 Simplified Procedure for Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-20
xvi Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

E.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-21

F. Supplementary Information and Data on Multi-Degree-of-Freedom Effects . . . . . . . . . . . F-1

F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1

F.1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1

F.1.2 Scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1

F.2 Example Buildings and Demand Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1

F.2.1 Prototype Buildings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-2

F.2.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4

F.2.3 Ground Motions and Demand Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-8

F.2.4 Extensions to Address P-Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-12

F.3 Simplified Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-13

F.3.1 Single Load Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-13

F.3.2 Multiple Mode Pushover Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-17

F.4 Accuracy of Estimates Made Using Simplified Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-19

F.4.1 Error Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-19

F.4.2 Results for Ordinary Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-20

F.4.3 Results for Near Field Motions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-22

F.5 Equivalent SDOF Estimates of Peak Roof Displacement Response . . . . . . . . . . . . . . . . . . . . . . . . . . . F-22

F.5.1 Analysis Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-23

F.5.2 Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-24

F.6 Scaled NDP Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-24

F.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-24

F.6.2 Elaboration of Step 3 and Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-24

F.6.3 Statistical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-26

F.6.4 Observed Coefficients of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-27

F.7 Energy-based Approaches for Pushover Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-28

F.7.1 Peak Displacement Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-28

F.7.2 Multiple Mode Estimates of Response Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-28

F.8 Detailed Figure Sets for the MDOF Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-31

F.8.1 Ground Motion Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-32

F.8.2 Responses to Ordinary (Site Class C) Motions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-47

F.8.3 Errors Associated with Ordinary (Site Class C) Motions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-71

F.8.4 Responses to Near Fault Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-81

F.8.5 Errors Associated with Near Fault Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-113

F.8.6 Observed Coefficients of Variation of the Response Quantities Determined for 

the Ordinary (Site Class C) Motions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-123

F.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-131
xvii Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

List of Figures

Figure 2-1 Schematic depiction of the use of inelastic analysis procedures to estimate forces and 

inelastic deformations for given seismic ground motions and a nonlinear analysis model 

of the building. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

Figure 2-2 Schematic of a detailed 3-dimensional inelastic structural model developed from component 

properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

Figure 2-3 Schematic depictions illustrating how inelastic component strength and stiffness properties 

from test data are used to create idealized force-deformation relationships. . . . . . . . . . . . . . . . . . 2-3

Figure 2-4 Forms of simplified equivalent multiple-degree-of-freedom models. . . . . . . . . . . . . . . . . . . . . . . 2-4

Figure 2-5 Schematics depicting the development of an equivalent SDOF system from a 

pushover/capacity curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4

Figure 2-6 Factors affecting seismic ground motion and various ways to characterize ground motions 

graphically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5

Figure 2-7 Flow chart depicting the nonlinear dynamic analysis process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

Figure 2-8 Incremental dynamic analysis study for thirty ground motion records for a 5-story steelbraced frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

Figure 2-9 Flow chart depicting simplified SDOF nonlinear analysis process. . . . . . . . . . . . . . . . . . . . . . . . . 2-7

Figure 2-10 Flow chart depicting the process followed in nonlinear static procedures. . . . . . . . . . . . . . . . . . . 2-8

Figure 2-11 Matrix depicting possible inelastic seismic analysis procedures for various structural 

models and ground-motion characterizations along with trends of uncertainty in the result. . . . . 2-9

Figure 2-12 Schematic illustrating the process by which the Coefficient Method of displacement 

modification (per FEMA 356) is used to estimate the target displacement for a given 

response spectrum and effective period, Te. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10

Figure 2-13 Graphical representation of the Capacity-Spectrum Method of equivalent linearization, 

as presented in ATC-40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11

Figure 3-1 Basic hysteretic models used in the evaluation of current procedures . . . . . . . . . . . . . . . . . . . . . . 3-2

Figure 3-2 Comparison of experimental results (after Lehman et al., 2000) with the hysteretic 

response computed with the SSD model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3

Figure 3-3 Variation of period shift based on secant stiffness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6

Figure 3-4 Variation of κ-factor with the displacement ductility ratio, μ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6

Figure 3-5 Variation of equivalent (effective) damping ratios with changes in the displacement 

ductility ratio, μ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6

Figure 3-6 Variation of spectral reduction factors SRA for different hysteretic behaviors as a 

function of the displacement ductility ratio, μ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

Figure 3-7 Variation of spectral reduction factors SRV for different hysteretic behaviors as a 

function of the displacement ductility ratio, μ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

Figure 3-8 Mean error associated with the Capacity-Spectrum Method of ATC-40 for hysteretic 

behaviors types A, B, and C for site class C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8

Figure 3-9 Comparison of coefficient C1 in FEMA 356 with and without capping. . . . . . . . . . . . . . . . . . . . 3-10

Figure 3-10 A close up view of the effect of the capping limitation of C1 coefficient. . . . . . . . . . . . . . . . . . . 3-11

Figure 3-11 Variation of mean C1 computed for the elastic perfectly plastic (EPP) model when 

subjected to ground motions recorded on site class C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11

Figure 3-12 Mean coefficient C1 for site classes B, C and D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures xviii

Figure 3-13 Comparison between the mean C1 computed from nonlinear response-history analyses to 

C1 in FEMA 356 (non-capped and capped). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-13

Figure 3-14 Variation of C1 for two individual ground motions recorded on soft soil E. . . . . . . . . . . . . . . . . .3-14

Figure 3-15 Predominant ground motion periods for the soft soil records obtained at Larkspur Ferry 

Terminal and Emeryville during the 1989 Loma Prieta earthquake. . . . . . . . . . . . . . . . . . . . . . . .3-14

Figure 3-16 C1 values of Larkspur Ferry Terminal and Emeryville soft soil records for normalized 

periods of vibration with respect to dominant ground motion periods of each record. . . . . . . . . .3-15

Figure 3-17 The variation of mean C1 values for site class E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-15

Figure 3-18 Mean error statistics of capped and not capped C1 values for the ground motions recorded 

in site classes B and C, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-16

Figure 3-19 A sample variation of C2 values in accordance with FEMA-356 . . . . . . . . . . . . . . . . . . . . . . . . .3-17

Figure 3-20 Mean displacement ratio of SD to EPP models computed with ground motions recorded 

on site class D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-17

Figure 3-21 Mean displacement ratio of SSD to EPP models computed with ground motions recorded 

on site classes B, C, and D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-17

Figure 3-22 The mean error statistics associated with C1 and C2 assuming a Life Safety performance 

level in accordance with FEMA 356 for stiffness degrading (SD) systems. . . . . . . . . . . . . . . . . .3-18

Figure 3-23 The mean error statistics associated with C1 and C2 assuming a Collapse Prevention 

performance level in accordance with FEMA 356 for stiffness and strength (SSD) 

degrading systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-18

Figure 3-24 The variation of C3 from FEMA 356 with respect to R for different negative post-elastic 

stiffness values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-19

Figure 3-25 Bilinear system with in-cycle negative post-elastic stiffness due to P-∆ effects. . . . . . . . . . . . . .3-19

Figure 3-26 Displacement modification factors in SDOF that exhibit in-cycle negative post-yield stiffness. .3-19

Figure 3-27 Ratio of maximum displacement for a nonlinear elastic (NE) oscillator to elastic response 

for site classes B, C, and D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-20

Figure 4-1 Two types of strength degradation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1

Figure 4-2 Example capacity curve for a medium rise concrete structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-2

Figure 4-3 Idealized force-displacement curve for nonlinear static analysis . . . . . . . . . . . . . . . . . . . . . . . . . .4-3

Figure 5-1 Expression for coefficient C1 (Eqn.5-1 with a = 90 for site class C) and current FEMA 356 

expression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-2

Figure 5-2 Comparison of alternative expressions for the coefficient C1 for R = 4 and R = 6 for 

site class C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-3

Figure 5-3 Coefficient C2 from Eq. 4-2 and FEMA 356 for site classes B, C, and D. . . . . . . . . . . . . . . . . . . .5-3

Figure 6-1 Acceleration-displacement response spectrum (ADRS) showing effective period and 

damping parameters of equivalent linear system, along with a capacity curve. . . . . . . . . . . . . . . .6-1

Figure 6-2 Illustration of probability density function of displacement error for a Gaussian distribution. . . . .6-2

Figure 6-3 Types of inelastic behavior considered. BLH=Bilinear Hysteretic STDG=Stiffness 

Degrading, and STRDG=Strength Degrading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-2

Figure 6-4 Modified acceleration-displacement response spectrum (MADRS) for use with secant 

period, Tsec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-5

Figure 6-5 Damping coefficients, B, as a function of damping, βeff, from various resource documents. . . . . .6-6

Figure 6-6 Initial ADRS demand and capacity spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-7

Figure 6-7 Bilinear representation of capacity spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-7

Figure 6-8 Determination of estimated maximum displacement using direct iteration (Procedure A) . . . . . . .6-8
xix Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure 6-9 Determination of estimated maximum displacement using intersection of capacity spectrum 

with MADRS (Procedure B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8

Figure 6-10 Locus of possible performance points using MADRS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

Figure 6-11 Comparison of approximate solution results with results from more detailed procedures. . . . . . . 6-9

Figure 6-12 Tracking iteration for equivalent linearization by comparing assumed displacement to 

calculated displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

Figure 7-1 NEHRP design response spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

Figure 7-2 NEHRP response spectrum and 5%-damped response spectra of scaled motions, used for 

oscillators having T = 0.2 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

Figure 7-3 NEHRP response spectrum and 5%-damped response spectra of scaled motions, used for 

oscillators having T = 0.5 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

Figure 7-4 NEHRP response spectrum and 5%-damped response spectra of scaled motions, used for 

oscillators having T = 1.0 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

Figure 7-5 Bilinear load-displacement relation of oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

Figure 7-6 Linear vibration periods and strength reduction factors for oscillators. . . . . . . . . . . . . . . . . . . . . . 7-4

Figure 7-7 Representative nonlinear response-history analysis result (this example is for oscillator 

period T = 1 s, ground motion DSP090 scaled by factor 1.53, and strength-reduction 

factor R = 4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

Figure 7-8 Comparison of responses for an oscillator with T = 0.2 s calculated using various procedures, 

response spectra scaled to the NEHRP spectrum, and values calculated for the NEHRP 

spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

Figure 7-9 Comparison of responses for an oscillator with T = 0.5 s calculated using various procedures, 

response spectra scaled to the NEHRP spectrum, and values calculated for the NEHRP 

spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

Figure 7-10 Comparison of responses for an oscillator with T = 1.0 s calculated using various procedures, 

response spectra scaled to the NEHRP spectrum, and values calculated for the NEHRP 

spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

Figure 7-11 Comparison of responses of an oscillator with T = 0.2 s calculated using various procedures, 

response spectra scaled to NEHRP spectrum, and values calculated for the average 

spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8

Figure 7-12 Comparison of responses of an oscillator with T = 0.5 s calculated using various procedures,

response spectra scaled to the NEHRP spectrum, and values calculated for the average 

spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9

Figure 7-13 Comparison of responses of an oscillator with T = 1.0 s calculated using various procedures, 

response spectra scaled to the NEHRP spectrum, and values calculated for the average 

spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9

Figure 8-1 Foundation modeling assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2

Figure 8-2 Ratio of response spectra for base slab averaging, RRSbsa, as a function of period, T, and 

effective foundation size, be. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

Figure 8-3 Ratio of response spectra for embedment RRSe, for an embedment, e, of 30 feet as a function 

of period, T, and shear wave velocity, vs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4

Figure 8-4 Example of foundation damping, βf

, as a function of effective period lengthening 

ratio , , for constant embedment, e/rx = 0, and various values of foundation 

stiffness rotational stiffness, h/rθ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6

Figure 8-5 Example of foundation damping, βf

, as a function of effective period lengthening 

ratio, , for constant embedment, e/rx = 0.5, and various values of foundation stiffness 

rotational stiffness, h/rθ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6

 ̃

T T/ eff eff

 ̃

T T/ eff eff
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures xx

Figure 9-1 Example results for displacements predicted by nonlinear static procedures (NSP) compared 

to nonlinear dynamic response-history analyses (NDA). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9-5

Figure 9-2 Dispersion in results for displacement for two levels of global drift. . . . . . . . . . . . . . . . . . . . . . . .9-6

Figure 9-3 Relatively good results for interstory drift predicted using nonlinear static procedures (NSP), 

as compared to nonlinear dynamic response-history analyses (NDA). . . . . . . . . . . . . . . . . . . . . . .9-7

Figure 9-4 Relatively poor results for interstory drift predicted using nonlinear static procedures (NSP) 

compared to nonlinear dynamic response-history analyses (NDA). . . . . . . . . . . . . . . . . . . . . . . . .9-8

Figure 9-5 Story forces and overturning moments in the example three-story frame building when 

different load vectors are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9-9

Figure 9-6 Story forces and overturning moments in eight-story wall and nine-story frame example 

buildings, using various load vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9-9

Figure 10-1 Differences between cyclic and in-cycle strength degradation . . . . . . . . . . . . . . . . . . . . . . . . . . .10-3

Figure 10-2 Acceleration-displacement response spectrum (ADRS) showing effective period and 

damping parameters of equivalent linear system, along with a capacity curve. . . . . . . . . . . . . . .10-5

Figure 10-3 Foundation modeling alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10-7

Figure 10-4 Overturning moments in example 9-story building using various load vectors. . . . . . . . . . . . . . .10-8

Figure 10-5 Error associated with the Coefficient C1 as formulated in FEMA 356 (left) and the potential 

improved formulation (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10-9

Figure 10-6 Dispersion of results for the nonlinear dynamic analysis (NDA) of a SDOF oscillator 

subject to thirteen NEHRP Site Class C ground motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10-10

Figure 10-7 Application flowchart for nonlinear static seismic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .10-13

Figure D-1 SDOF oscillator model subjected to ground motion, u(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2

Figure D-2 Components of the ADRS format for representing Seismic Demand - PSA versus SD . . . . . . . . D-3

Figure D-3 SDOF oscillator model represented by Equation D-7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-3

Figure D-4 Linear SDOF oscillator model with effective linear parameters as represented by 

Equation D-8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-4

Figure D-5 Bilinear hysteretic system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-4

Figure D-6 Early effort to define optimal equivalent linear parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-5

Figure D-7 Distribution of percent error in Performance Point displacement. Bilinear system with 

alpha=0, T0 = 0.1-2.0 sec (0.1 sec increments), μ=2, 28 far-field earthquakes. . . . . . . . . . . . . . . D-6

Figure D-8 Contour values of εD over the two-dimensional parameter space of Teff and βeff for a 

single combination of inelastic system and ground excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . D-7

Figure D-9 Illustration of assembling εD error distributions at every combination of Teff and βeff over 

an ensemble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-8

Figure D-10 Illustration of probability density functions of displacement error for a Normal distribution. . . . D-8

Figure D-11 Contours of REAR over the Teff, βeff parameter space. The optimum point is marked by 

a square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-9

Figure D-12 Example of optimal effective linear parameters - discrete points and the curve fitted 

to the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-10

Figure D-13 Types of inelastic behavior considered. BLH=Bilinear hysteretic, STDG=Stiffness 

Degrading, and STRDG=Strength Degrading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-10

Figure D-14 Summary of Performance Point errors for bilinear hysteretic (BLH) model . . . . . . . . . . . . . . . D-12

Figure D-15 Summary of Performance Point Errors for Strength Degrading (STDG) model . . . . . . . . . . . . . D-13

Figure E-1 Amplitude of transfer function between free-field motion and foundation input motion 

for vertically incident incoherent waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2
xxi Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure E-2 Relationship between effective incoherence parameter ka and small-strain shear wave 

velocity νs from case histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E-3

Figure E-3 (a) Transfer function amplitudes for embedded cylinders from Day (1978) and Elsabee 

and Morray (1977) along with approximate solution by Elsabee and Morray; 

(b) Transfer function amplitude model by Elsabee and Morray (1977). . . . . . . . . . . . . . . . . . . . .E-4

Figure E-4 Comparison of transfer function amplitude to ratios of response spectra (RRS) at different 

damping ratios.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E-5

Figure E-5 RRS for foundation with be = 330 ft. Simplified model (ka /νs = n1) vs. exact solution 

for ka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E-7

Figure E-6 RRS from simplified model as function of foundation size, be . . . . . . . . . . . . . . . . . . . . . . . . . . .E-7

Figure E-7 (a) RRS for foundation embedded to depth e = 30 ft in different site categories; 

(b) RRS for foundations with variable depths in Site Classes C and D. . . . . . . . . . . . . . . . . . . . . .E-8

Figure E-8 Foundation stiffness and damping factors for elastic and viscoelastic halfspaces (υ = 0.4). . . . .E-10

Figure E-9 Foundation damping factors for halfspace with and without hysteretic damping 

and for soil profiles with indicated shear modulus profiles and no hysteretic damping. . . . . . . .E-11

Figure E-10 Dashpot coefficients for radiation damping vs. normalized frequency for different 

foundation shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E-14

Figure E-11 Oscillator model for analysis of inertial interaction under lateral excitation. . . . . . . . . . . . . . . . .E-16

Figure E-12 Foundation damping for single degree-of-freedom structures on elastic halfspace with 

various aspect ratios (h/rθ) and foundation shapes (rθ/rx), non-embedded foundation 

case (e/rx = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E-18

Figure E-13 Foundation damping for single degree-of-freedom structures on elastic halfspace with 

various aspect ratios (h/rθ) and foundation shapes (rθ/rx), small foundation embedment 

case (e/rx = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E-19

Figure E-14 Foundation damping factor βf

 expressed as a function of period lengthening for different 

building aspect ratios (h/rθ) and embedment ratios (e/rx ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E-20

Figure F-1 Elevation view of the 3-story (regular and weak-story) steel frames used in the study. . . . . . . . . F-2

Figure F-2 Elevation view of the 9-story (regular and weak-story) steel frames used in the study. . . . . . . . . F-3

Figure F-3 Elevation and plan views of the 8-story reinforced concrete shear wall used in the study . . . . . . . F-5

Figure F-4 Drain model of the 3-story (regular and weak story) steel frames. . . . . . . . . . . . . . . . . . . . . . . . . . F-6

Figure F-5 Drain model of the 9-story (regular and weak-story) steel frames. . . . . . . . . . . . . . . . . . . . . . . . . F-7

Figure F-6 Drain model of the 8-story reinforced concrete shear wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-8

Figure F-7 Idealized material stress-strain relationships used in drain model of the 8-story reinforced 

concrete shear wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-8

Figure F-8 Capacity curves for the five model building examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-11

Figure F-9 Shape vectors of the 1st mode shape load pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-14

Figure F-10 Shape vectors of the triangular load pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-15

Figure F-11 Shape vectors of the rectangular load pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-15

Figure F-12 Shape vectors of the code load pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-16

Figure F-13 Shape vectors of the SRSS load pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-17

Figure F-14 First, second, and third mode pushover results for the 3-story regular steel frame. . . . . . . . . . . . F-19

Figure F-15 Example statistical distributions of displacement ratios for the ordinary ground motions. . . . . . F-25

Figure F-16 Example comparisons of energy-based and conventional multiple mode calculations. . . . . . . . . F-29

Figure F-17 Characteristics of the ICC000 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-32

Figure F-18 Characteristics of the LOS000 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-33
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures xxii

Figure F-19 Characteristics of the G02090 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-34

Figure F-20 Characteristics of the TCU122N ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-35

Figure F-21 Characteristics of the G03090 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-36

Figure F-22 Characteristics of the CNP196 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-37

Figure F-23 Characteristics of the CHY101W ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-38

Figure F-24 Characteristics of the ICC090 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-39

Figure F-25 Characteristics of the CNP106 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-40

Figure F-26 Characteristics of the E02140 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-41

Figure F-27 Characteristics of the E11230 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-42

Figure F-28 Characteristics of the ERZMV1 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-43

Figure F-29 Characteristics of the RRSMV1 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-44

Figure F-30 Characteristics of the LUCMV1 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-45

Figure F-31 Characteristics of the SCHMV1 ground motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-46

Figure F-32 Response quantities of the 3-story building for 0.5% drift level . . . . . . . . . . . . . . . . . . . . . . . . . F-47

Figure F-33 Response quantities of the 3-story building for 2% drift level . . . . . . . . . . . . . . . . . . . . . . . . . . F-48

Figure F-34 Response quantities of the 3-story building for 4% drift level . . . . . . . . . . . . . . . . . . . . . . . . . . F-49

Figure F-35 Response quantities of the 3-story weak-story building for 0.5% drift level . . . . . . . . . . . . . . . F-50

Figure F-36 Response quantities of the 3-story weak-story building for 2% drift level . . . . . . . . . . . . . . . . . F-51

Figure F-37 Response quantities of the 3-story weak-story building for 4% drift level . . . . . . . . . . . . . . . . . F-52

Figure F-38 Response quantities of the 8-story building for 0.2% drift level . . . . . . . . . . . . . . . . . . . . . . . . . F-53

Figure F-39 Response quantities of the 8-story building for 1% drift level . . . . . . . . . . . . . . . . . . . . . . . . . . F-55

Figure F-40 Response quantities of the 8-story building for 2% drift level . . . . . . . . . . . . . . . . . . . . . . . . . . F-57

Figure F-41 Response quantities of the 9-story building for 0.5% drift level . . . . . . . . . . . . . . . . . . . . . . . . . F-59

Figure F-42 Response quantities of the 9-story building for 2% drift level . . . . . . . . . . . . . . . . . . . . . . . . . . F-61

Figure F-43 Response quantities of the 9-story building for 4% drift level . . . . . . . . . . . . . . . . . . . . . . . . . . F-63

Figure F-44 Response quantities of the 9-story weak-story building for 0.5% drift level . . . . . . . . . . . . . . . F-65

Figure F-45 Response quantities of the 9-story weak-story building for 2% drift level . . . . . . . . . . . . . . . . . F-67

Figure F-46 Response quantities of the 9-story weak-story building for 4% drift level . . . . . . . . . . . . . . . . . F-69

Figure F-47 Mean and maximum errors for the 3-story building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-71

Figure F-48 Mean and maximum errors for the 3-story weak-story building . . . . . . . . . . . . . . . . . . . . . . . . . F-73

Figure F-49 Mean and maximum errors for the 8-story building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-75

Figure F-50 Mean and maximum errors for the 9-story building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-77

Figure F-51 Mean and maximum errors for the 9-story weak-story building . . . . . . . . . . . . . . . . . . . . . . . . . F-79

Figure F-52 Response quantities of the 3-story building under ERZMV1 ground motion . . . . . . . . . . . . . . . F-81

Figure F-53 Response quantities of the 3-story building under RRSMV1 ground motion . . . . . . . . . . . . . . . F-82

Figure F-54 Response quantities of the 3-story building under LUCMV1 ground motion . . . . . . . . . . . . . . . F-83

Figure F-55 Response quantities of the 3-story building under SCHMV1 ground motion . . . . . . . . . . . . . . . F-84

Figure F-56 Response quantities of the 3-story weak-story building under ERZMV1 ground motion . . . . . F-85

Figure F-57 Response quantities of the 3-story weak-story building under RRSMV1 ground motion . . . . . F-86

Figure F-58 Response quantities of the 3-story weak-story building under LUCMV1 ground motion . . . . . F-87

Figure F-59 Response quantities of the 3-story weak-story building under SCHMV1 ground motion . . . . . F-88

Figure F-60 Response quantities of the 8-story building under ERZMV1 ground motion . . . . . . . . . . . . . . . F-89
xxiii Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-61 Response quantities of the 8-story building under RRSMV1 ground motion . . . . . . . . . . . . . . . F-91

Figure F-62 Response quantities of the 8-story building under LUCMV1 ground motion . . . . . . . . . . . . . . . F-93

Figure F-63 Response quantities of the 8-story building under SCHMV1 ground motion . . . . . . . . . . . . . . . F-95

Figure F-64 Response quantities of the 9-story building under ERZMV1 ground motion . . . . . . . . . . . . . . . F-97

Figure F-65 Response quantities of the 9-story building under RRSMV1 ground motion . . . . . . . . . . . . . . . F-99

Figure F-66 Response quantities of the 9-story building under LUCMV1 ground motion . . . . . . . . . . . . . . F-101

Figure F-67 Response quantities of the 9-story building under SCHMV1 ground motion . . . . . . . . . . . . . . F-103

Figure F-68 Response quantities of the 9-story weak-story building under ERZMV1 ground motion . . . . . F-105

Figure F-69 Response quantities of the 9-story weak-story building under RRSMV1 ground motion . . . . . F-107

Figure F-70 Response quantities of the 9-story weak-story building under LUCMV1 ground motion . . . . . F-109

Figure F-71 Response quantities of the 9-story weak-story building under SCHMV1 ground motion . . . . . F-111

Figure F-72 Mean and maximum errors for the 3-story building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-113

Figure F-73 Mean and maximum errors for the 3-story weak-story building . . . . . . . . . . . . . . . . . . . . . . . . F-115

Figure F-74 Mean and maximum errors for the 8-story building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-117

Figure F-75 Mean and maximum errors for the 9-story building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-119

Figure F-76 Mean and maximum errors for the 9-story weak-story building . . . . . . . . . . . . . . . . . . . . . . . . F-121

Figure F-77 Observed COVs for the 3-story frame building. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-123

Figure F-78 Observed COVs for the 3-story weak story frame building . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-124

Figure F-79 Observed COVs for the 8-story wall building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-125

Figure F-80 Observed COVs for the 9-story frame building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-127

Figure F-81 Observed COVs for the 9-story weak story frame building . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-129
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures xxiv
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures xxv

List of Tables

Table 3-1 Variation of κ-Value in ATC-40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-6

Table 3-2 Minimum Allowable Spectral Reduction Factors for Displacement Ductility 

Ratios Larger than 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3-7

Table 6-1 Coefficients for use in Equations for Effective Damping . . . . . . . . . . . . . . . . . . . . . . . . . . .6-3

Table 6-2 Coefficients for use in Equations for Effective Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6-4

Table 7-1 Ground Motion Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7-2

Table 8-1 Approximate Values of Shear Wave Velocity Reduction Factor, n . . . . . . . . . . . . . . . . . . .8-4

Table 10-1 Coefficients for Use in Equations for Effective Damping . . . . . . . . . . . . . . . . . . . . . . . . . .10-4

Table A-1 Investigator Research Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

Table B-1 Seismic Systems of Example Buildings Submitted by Respondees . . . . . . . . . . . . . . . . . . B-2

Table B-2 Gravity Systems of Example Buildings Submitted by Respondees . . . . . . . . . . . . . . . . . . B-2

Table B-3 Foundation Systems of Example Buildings Submitted by Respondees . . . . . . . . . . . . . . . B-2

Table B-4 Inelastic Analysis Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2

Table B-5 Computer Program Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3

Table C-1 Ground Motions Recorded on Site Class B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1

Table C-2 Ground Motions Recorded on Site Class C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2

Table C-3 Ground Motions Recorded on Site Class D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-3

Table C-4 Ground Motions Recorded on Very Soft Soil Sites Used in This Study . . . . . . . . . . . . . . . C-4

Table C-5 Near-Fault Records with Forward Directivity Used in this Study . . . . . . . . . . . . . . . . . . . . C-5

Table E-1 Approximate values of n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-6

Table F-1 Assumed Loading for the 3- and 9-Story Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .F-2

Table F-2 Properties of the 8-Story Reinforced Concrete Structural Wall . . . . . . . . . . . . . . . . . . . . . .F-4

Table F-3 Assumed Loading for the 8-Story Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .F-5

Table F-4 Periods and Mode Shapes for the Frame and Wall Buildings . . . . . . . . . . . . . . . . . . . . . . . .F-9

Table F-5 Base Shear Coefficient and Drift At Yield for Each Building Model . . . . . . . . . . . . . . . . .F-10

Table F-6 Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .F-10

Table F-7 Scale Factors Applied to Each of the Ordinary Ground Motions for the Dynamic 

Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .F-13

Table F-8 Peak Roof Drift Ratios for the Five Building Models (%) . . . . . . . . . . . . . . . . . . . . . . . . .F-22

Table F-9 Values of c at the 90% Confidence Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .F-26

Table F-10 Approximate Upper Bounds to the COVs over the Height of each Building Model . . . . .F-27

Table F-11 Means of the Ratio of Roof Displacements: SDOF Estimate / Actual MDOF . . . . . . . . . .F-28
xxvi Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 1-1 

1. Introduction

This report documents the results of a project for the 

Department of Homeland Security’s Federal 

Emergency Management Agency (FEMA) by the 

Applied Technology Council (ATC) to evaluate and 

improve the application of simplified inelastic analysis 

procedures for use with performance-based engineering 

methods for seismic design, evaluation, and upgrade of 

buildings. Chapters 1 through 9 summarize the 

developmental efforts and results in concise language to 

facilitate application of the project findings in practice. 

Chapter 10 contains a summary and a practical 

application example using the improved procedures. 

Supporting information describing the project findings 

in detail are provided in the appendices. 

This document has been published in two formats: (1) a 

printed version, which summarizes the developmental 

efforts and project findings and includes the application 

example (Chapters 1 through 10), and (2) a complete 

version of the report on CD-ROM (inside back cover), 

which includes all of the material in the printed version 

plus six appendices containing project results and 

findings. The printed version of the report is relatively 

brief to facilitate use by design professionals.

1.1 Background

During the past decade, significant progress has been 

made in performance-based engineering methods that 

rely on nonlinear static analysis procedures (NSPs). In 

1996, ATC published the ATC-40 report, Seismic 

Evaluation and Retrofit of Concrete Buildings, which 

was developed with funding from the California 

Seismic Safety Commission. In a larger project funded 

by FEMA, ATC (under contract to the Building Seismic 

Safety Council) prepared the FEMA 273 Guidelines for 

the Seismic Rehabilitation of Buildings, and the 

companion FEMA 274 Commentary, which were 

published in 1997 by FEMA. Soon thereafter, the 

American Society of Civil Engineers (ASCE) prepared 

the FEMA 356 report, Prestandard and Commentary 

for the Seismic Rehabilitation of Buildings (the 

successor to FEMA 273/274), which was published by 

FEMA in 2000. All of these documents present similar 

approaches. FEMA 273 and FEMA 356 use a procedure 

known as the Coefficient Method, and ATC-40 details 

the Capacity-Spectrum Method. The two approaches 

are essentially the same when it comes to generating a 

“pushover” curve to represent the inelastic forcedeformation behavior of a building. They differ, 

however, in the technique used to calculate the inelastic 

displacement demand for a given representation of 

ground motion.

The development of this report was instigated by 

several factors. The use of NSPs in engineering practice 

has accelerated since the publication of ATC-40 and 

FEMA 356. Consequently, there is valuable information 

available on the practical application of these inelastic 

analysis procedures. In addition to experience with the 

initial application of these performance-based methods 

by practicing professionals, ongoing research promises 

important modifications, improvements, and 

alternatives to current NSPs. 

There has also been a large national investment in 

performance-based engineering, because of the tangible 

prospect of vastly improving seismic design practices. 

The future effective use of performance-based 

engineering depends on the continued development of 

reliable and credible inelastic analysis procedures. 

The intent of the ATC-55 project has been to gather the 

results of practical experience and relevant research and 

to develop guidance for improving the application of 

nonlinear static analysis procedures to both existing and 

new structures. 

1.2 Project Purpose and Scope

The purpose of the ATC-55 project was to evaluate 

current NSPs, as described in FEMA 356 and ATC-40 

and to develop improvements where feasible. The 

primary objectives were:

• to improve understanding of the inherent 

assumptions and theoretical underpinnings of 

existing and proposed new simplified analysis 

procedures;

• to recognize the applicability, limitations, and 

reliability of various procedures;

• to develop guidelines for practicing engineers on 

how to apply the procedures to new and existing 

buildings; and 

• to provide direction for researchers on issues to 

consider for future improvements of simplified 

inelastic analysis procedures.

Project activities also were guided by the fact that 

engineers and researchers have similar concerns with 
Chapter 1: Introduction

1-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

respect to inelastic analysis procedures. Some of the 

more prominent issues considered are listed below.

• In some cases, different nonlinear static procedures 

produce significantly different results for the same 

building model and ground motion representation. 

• Current procedures for addressing the degradation of 

stiffness and strength in structures are ambiguous 

and unclear.

• The predicted response of short-period structures 

seems to be extreme when compared with observed 

performance.

• Since they are based on single-degree-of-freedom 

(SDOF) approximations, nonlinear static procedures 

may not reliably predict important response 

parameters for some multi-degree-of-freedom 

(MDOF) structures.

1.3 Report Scope, Organization and 

Contents

The document is intended to be useful from the 

practical, educational, and archival standpoints. Its 

fundamental purpose is to provide guidance that can be 

used directly by engineering practitioners. From an 

educational perspective, the report is intended to 

facilitate a basic conceptual understanding of 

underlying principles, as well as the associated 

capabilities and limitations of the procedures, so that 

practicing structural engineers can apply the procedures 

appropriately. Finally, the archival aspect recognizes 

that the development of inelastic procedures will 

continue, and that it is important to record detailed 

information from the project for future reference and 

use.

The scope of the evaluation of inelastic analysis 

procedures and the development of recommendations 

for improvement, as presented in this document, focus 

on nonlinear static procedures (NSPs). In light of the 

concerns identified by practicing engineers and 

researchers, the document specifically addresses the 

following questions: 

• How well do current NSPs predict maximum global 

displacement (elastic plus inelastic)?

• How well do current NSPs predict effects arising 

from the multiple-degree-of-freedom (MDOF) 

response of structures?

• What modifications might be incorporated into 

NSPs to improve accuracy and to reduce uncertainty 

associated with the first two questions?

The initial phase of the project, during early 2001, 

focused on the identification and refinement of 

important issues related to the improvement of inelastic 

seismic analysis procedures. Activities included the 

solicitation of input from researchers (see Appendix A.) 

and practicing engineers (see Appendix B.). This 

information was used to formulate a plan for the 

subsequent phases of the project, comprising the 

evaluation of current procedures and the development 

of proposed improvements. 

Several analytical efforts formed the basis for the 

evaluation of current procedures and the development 

of improvements. The first tested the accuracy of the 

Coefficient Method of FEMA 356 and the CapacitySpectrum Method of ATC-40 in predicting global 

displacement demands, when compared to responsehistory analysis of SDOF oscillators. This effort is 

described in Chapter 3, with detailed results provided in 

Appendix C. 

During evaluations of both the Coefficient Method and 

Capacity-Spectrum Method, it became evident that 

important clarifications regarding strength degradation 

are applicable to both NSP approaches. This issue is 

addressed in Chapter 4. 

Improved procedures for use with the Coefficient 

Method are described in Chapter 5. Improved 

procedures for use with the Capacity-Spectrum Method, 

are described in Chapter 6. Supplementary information 

and data on the equivalent linearization approach are 

provided in Appendix D. 

Chapter 7 describes an independent analysis that was 

implemented to test the accuracy of the procedural 

improvements described in Chapters 5 and 6. 

Comparisons with results using the original procedures 

are provided.

For many years, researchers have observed that the 

predicted inelastic displacement response of oscillators, 

with periods in excess of about 1 second, is often very 

similar to the predicted displacement response of elastic 

oscillators having the same period. This has led to the 

so-called “equal displacement approximation.” 

Researchers have also recognized that the predicted 

inelastic response of oscillators with short periods, less 

than approximately 0.5 seconds, are often significantly 

larger than the predicted response of elastic structures 

of the same period, particularly if the structures are both 

very stiff and very weak. When this principle is applied 

using nonlinear analysis techniques to the performance 
Chapter 1: Introduction

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 1-3

evaluation of small, stiff buildings, such as those that 

comprise much of the building inventory in the United 

States, very poor performance and extreme damage is 

often predicted. This has created a paradox, in that such 

buildings have generally been observed to experience 

limited damage in past earthquakes. Several factors 

contribute to this conflict between predicted and 

observed performance of such structures, including:

• models used to predict performance of such 

structures commonly neglect many elements that 

contribute to their strength;

• fixed base models used to predict structural response 

neglect foundation flexibility, resulting in 

predictions of smaller periods than that of the actual 

structures;

• stiff buildings will experience small displacements 

even at large ductility demand and thus may 

experience only limited damage; and

• in addition to foundation flexibility, other soilstructure interaction effects can significantly reduce 

the response of some stiff structures to ground 

shaking.

In part, these effects can be addressed by more accurate 

analytical models that incorporate all structural and 

nonstructural elements significant to structural response 

as well as the flexibility of foundations. Soil-structure 

interaction effects are of particular importance. 

Chapter 8 describes analysis techniques for SSI effects 

that have been adapted for use with nonlinear static 

procedures and detailed supporting information on soilstructure interaction is provided in Appendix E.

Multi-degree-of-freedom effects are addressed in 

Chapter 9, which summarizes a comprehensive analysis 

of five example buildings to illustrate the application 

and limitations of simplified techniques to account for 

MDOF effects within current NSPs. Details are 

provided in Appendix F.

Finally, Chapter 10 comprises a complete summary of 

the results of the efforts and the suggested 

improvements from a practical perspective. Chapter 10 

concludes with a detailed example application of the 

suggested improved procedures to a building structure.
Chapter 1: Introduction

1-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 2-1 

2. Overview of Inelastic Seismic Analysis 

Procedures

Practicing engineers use inelastic analysis procedures 

for the seismic evaluation and design of upgrades of 

existing buildings and other structures, as well as design 

of new construction. The practical objective of inelastic 

seismic analysis procedures is to predict the expected 

behavior of the structure in future earthquake shaking. 

This has become increasingly important with the 

emergence of performance-based engineering (PBE) as 

a technique for seismic evaluation and design (ATC, 

1996; BSSC, 2000). PBE uses the prediction of 

performance to inform decisions regarding safety and 

risk. For this purpose, PBE characterizes performance 

primarily in terms of expected damage to structural and 

nonstructural components and contents. Since structural 

damage implies inelastic behavior, traditional design 

and analysis procedures that use linear elastic 

techniques can predict performance only implicitly. By 

contrast, the objective of inelastic seismic analysis 

procedures is to directly estimate the magnitude of 

inelastic deformations and distortions.

The generic process of inelastic analysis is similar to 

conventional linear procedures in that the engineer 

develops a model of the building or structure, which is 

then subjected to a representation of the anticipated 

seismic ground motion (see Figure 2-1). The results of 

analysis are predictions of engineering demand 

parameters within the structural model that are 

subsequently used to determine performance based on 

acceptance criteria. The engineering demand 

parameters normally comprise global displacements 

(e.g., roof or other reference point), story drifts, story 

forces, component distortions, and component forces. 

There are several basic inelastic analysis procedures 

that differ primarily on the types of structural models 

used for analysis and the alternatives for characterizing 

seismic ground shaking.

2.1 Structural Modeling

Detailed structural models for inelastic analysis are 

similar to linear elastic finite-element (component) 

models (see Figure 2-2). The primary difference is that 

Figure 2-1 Schematic depiction of the use of inelastic analysis procedures to estimate forces and inelastic 

deformations for given seismic ground motions and a nonlinear analysis model of the building.
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

2-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

the properties of some or all of the components of the 

model include post-elastic strength and deformation 

characteristics in addition to the initial elastic 

properties. These are normally based on approximations 

derived from test results on individual components or 

theoretical analyses (see Figure 2-3). Information of 

this type is tabulated in ATC-40 and FEMA 356. In 

many instances, it is important to include the structural 

and geotechnical components of the foundation in the 

analysis model.

As detailed as these models may be, they inevitably 

introduce approximations and associated uncertainties 

into the analysis process. In most instances with 

inelastic analysis, it is preferable to base the model on 

the best estimate of the expected properties of the 

structure. In this manner, the overall analysis results in 

the estimate of central values (e.g., median or mean) of 

engineering demand parameters with minimum bias. 

Subsequently, the engineer may decide on the 

appropriate interpretation of the results in light of all the 

uncertainties involved and the specific decision in 

question.

In some instances, engineers simplify detailed structural 

models into equivalent multi-degree-of-freedom

models. These can be used to consolidate properties 

into what have been termed “fish bone” models (see 

Figure 2-4a). In some cases, the model can simplified 

further. For example, when rotational coupling among 

various vertical flexural elements is negligible (e.g., 

cantilever shear walls or braced frames) or when story 

shear mechanisms are anticipated (e.g., strong beam/

weak column frames) a “stick” model can be used (see 

Figure 2-4b and c). Often, substructuring techniques are 

helpful in developing simplified models. The purpose 

of the simplified models is to reduce computational and 

data management efforts. More importantly, they can 

also provide an improved visualization tool for the 

engineer. The negative aspect to simplified models is 

that they introduce additional approximations and 

uncertainty into the analysis.

Another important simplification to detailed structural 

models is what have become known as “pushover” or 

“capacity” curves. These curves form the basis of 

nonlinear static procedures discussed below. They are 

generated by subjecting a detailed structural model to 

one or more lateral load patterns (vectors) and then 

increasing the magnitude of the total load to generate a 

nonlinear inelastic force-deformation relationship for 

the structure at a global level (see Figure 2-5). The load 

vector is usually an approximate representation of the 

relative accelerations associated with the first mode of 

vibration for the structure. In the Coefficient Method of 

FEMA 356, the global parameters are normally base 

shear and roof displacement. For the CapacitySpectrum Method of ATC-40, these are transformed to 

spectral acceleration and spectral displacement. 

Nonlinear static procedures use these force-deformation 

relationships to represent the behavior of a simple 

single-degree-of-freedom (SDOF) oscillator. 

2.2 Characterization of Seismic Ground 

Motion

When an earthquake occurs, the amplitude, phasing, 

and frequency content of the shaking depend strongly 

on source characteristics (e.g., magnitude, rupture 

mechanism, fault plane orientation with respect to site). 

In addition, the characteristics of shaking are affected 

by attenuation that occurs as seismic waves propagate 

through rock from the source to the site and by local site 

effects. Site characteristics that may be important 

include potential 3-D basin structure, dynamic 

properties of relatively shallow sediments, and surface 

topography. The source, attenuation, and site effects, 

which are depicted schematically in the left frame of 

Figure 2-6, affect the character of ground shaking as 

expressed by ground motion records (i.e., plots of the 

acceleration, velocity and displacement of a point on 

the ground surface as a function of time (center frame in 

Figure 2-6)). 

Ground motion records can be used to define elastic 

response spectra (right frame in Figure 2-6), which 

comprise a relationship of the maximum response 

Figure 2-2 Schematic of a detailed 3-dimensional 

inelastic structural model developed from 

component properties.

Component strength and 

stiffness properties

δj

δi

θi

θj

Global displacement, d

Component strength and 

stiffness properties

δj

δi

θi

θj

δj

δi

θi

θj

Global displacement, d
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 2-3

Figure 2-3 Schematic depictions illustrating how inelastic component strength and stiffness properties from test data 

are used to create idealized force-deformation relationships.

deformation

force

a) Hysteretic force-deformation behavior from tests 

Force

Deformation

actual hysteretic 

behavior

backbone curve

b) Backbone representation of hysteretic behavior 

Ductile

(deformation controlled)

Backbone

curve

A

B C

D E

Idealized component

behavior

Semi-ductile Brittle

(force contolled)

A

B, C, D

A E

B

C, D

E

Force

Deformation

Ductile

(deformation controlled)

Backbone

curve

A

B C

D E

Idealized component

behavior

Semi-ductile Brittle

(force contolled)

A

B, C, D

A E

B

C, D

E

Force

Deformation

c) Idealized properties for analysis models 
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

2-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

(acceleration, velocity, and displacement) over the 

entire response-history record of a single-degree-offreedom oscillator and the frequency, or more 

commonly the period, of the oscillator, for a specified 

level of damping. Response spectral ordinates are 

commonly used to represent seismic demand for 

structural design. It should be noted that in this 

document, as in conventional structural engineering 

practice, pseudo-acceleration is used in place of actual 

spectral acceleration. The notation Sa actually 

represents the pseudo-acceleration.

The response spectrum for a single ground motion 

record is typically highly variable (jagged), depending 

on the assumed level of damping. For this reason, 

multiple records representative of a single source at a 

specified distance from the site and of a specified 

magnitude are often combined and smoothed, as 

implied in Figure 2-6. The results of this type of seismic 

Figure 2-4 Forms of simplified equivalent multiple-degree-of-freedom models.

Figure 2-5 Schematics depicting the development of an equivalent SDOF system from a pushover/capacity curve.
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 2-5

hazard analysis that provide an estimate of ground 

motion for a specified set of source and path parameters 

is a deterministic spectrum. 

The level of uncertainty in source, path, and site effects 

associated with deterministic spectra is relatively poorly 

defined. These uncertainties are accounted for directly 

in probabilistic seismic hazard analyses that provide 

estimates of ground motion parameters (such as 

response spectral ordinates) with a specified probability 

of being exceeded within a specified time period. The 

analysis includes all earthquakes (magnitudes and 

faults) that potentially could cause significant seismic 

shaking at a given site. When response spectral 

ordinates for a range of periods are evaluated for a 

specified probability of being exceeded, the result is an 

equal-hazard spectrum. 

Modern standards and guidelines (FEMA 356, ATC-40, 

and the NEHRP Recommended Provisions for New 

Buildings), allow the use of approximate design spectra

that represent a simplification of equal-hazard spectra 

on a location-specific basis. Design spectra have 

standardized shapes, and can be evaluated based on 

nationally mapped values of spectral accelerations for 

short and long periods. 

Deterministic spectra, equal-hazard spectra, and design 

spectra commonly exhibit smooth shapes with respect 

to period in contrast with the highly variable (jagged) 

shape of actual ground motion spectral records 

(particularly for low levels of damping). Structural 

response to an actual ground motion record is likely to 

be sensitive to the complex nature of the resulting 

spectrum. This uncertainty is not eliminated by the use 

of smooth spectra. 

2.3 Options for Inelastic Analysis

Various combinations of structural model types and 

characterizations of seismic ground motion define a 

number of options for inelastic analysis. The selection 

of one option over another depends on the purpose of 

the analysis, the anticipated performance objectives, the 

acceptable level of uncertainty, the availability of 

resources, and the sufficiency of data. In some cases, 

applicable codes and standards may dictate the analysis 

procedure. 

The primary decision is whether to choose inelastic 

procedures over more conventional linear elastic 

analysis. In general, linear procedures are applicable 

when the structure is expected to remain nearly elastic 

for the level of ground motion of interest or when the 

design results in nearly uniform distribution of 

nonlinear response throughout the structure. In these 

cases, the level of uncertainty associated with linear 

procedures is relatively low. As the performance 

objective of the structure implies greater inelastic 

demands, the uncertainty with linear procedures 

increases to a point that requires a high level of 

conservatism in demand assumptions and/or 

acceptability criteria to avoid unintended performance. 

Inelastic procedures facilitate a better understanding of 

actual performance. This can lead to a design that 

focuses upon the critical aspects of the building, leading 

to more reliable and efficient solutions.

Nonlinear dynamic analysis using the combination of 

ground motion records with a detailed structural model 

theoretically is capable of producing results with 

relatively low uncertainty (see Figure 2-7). In nonlinear 

dynamic analyses, the detailed structural model 

subjected to a ground-motion record produces estimates 

of component deformations for each degree of freedom 

in the model. Higher-level demands (element 

Figure 2-6 Factors affecting seismic ground motion and various ways to characterize ground motions graphically.
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

2-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

distortions, story drifts, roof displacement) derive 

directly from the basic component actions, as illustrated 

in Figure 2-7. There is still uncertainty with the detailed 

models, associated primarily with the lack of data on 

actual component behavior, particularly at high 

ductilities. In addition, the variability of ground motion 

results in significant dispersion in engineering demand 

parameters. This is illustrated in Figure 2-8, which 

depicts results from a series of nonlinear dynamic 

analyses for increasingly larger intensities of ground 

shaking (Vamvatsikos and Cornell, 2002). At each level 

of intensity, the multiple time histories produce a 

distribution of results in terms of a selected engineering 

demand parameter. Note that the dispersion increases 

with higher shaking intensity and with greater elasticity.

Simplified nonlinear dynamic analysis with equivalent 

multi-degree-of-freedom models also use ground motion 

records to characterize seismic demand. However, these 

techniques produce engineering demand parameters 

above the basic component level only. For example, a 

“stick” model produces story displacements or drifts. 

The engineer can estimate corresponding component 

actions using the assumptions that were originally the 

basis of the simplified model. Thus the uncertainty 

associated with the component actions in the simplified 

model is greater than those associated with the detailed 

model.

Simplified nonlinear dynamic analysis with equivalent 

single-degree-of-freedom (SDOF) models are a further 

simplification using ground motion records to 

characterize seismic shaking (see Figure 2-9). The 

result of the analysis is an estimate of global 

displacement demand. It is important to recognize that 

the resulting lower-level engineering demands (e.g., 

story drifts, component actions) are calculated from the 

global displacement using the force-deformation 

relationship for the oscillator. In contrast to the use of 

the more detailed model (see Figure 2-7), they are 

directly related to the assumptions, and associated 

uncertainties, made to convert the detailed structural 

model to an equivalent SDOF model in the first place. 

This adds further to the overall uncertainty associated 

with the simplified nonlinear dynamic analysis. Note 

that if the SDOF model is subjected to multiple time 

histories a statistical representation of response can be 

generated. 

Nonlinear static procedures (NSPs) convert MDOF 

models to equivalent SDOF structural models and 

represent seismic ground motion with response spectra 

as opposed to ground-motion records (see Figure 2-10). 

They produce estimates of the maximum global 

displacement demand. Story drifts and component 

actions are related subsequently to the global demand 

parameter by the pushover or capacity curve that was 

Figure 2-7 Flow chart depicting the nonlinear dynamic analysis process. Note that component actions are used to 

determine higher-level effects, such as story drifts and roof displacement, ∆.

Detailed model

Story drifts and forces

δij

Global displacement

Ground motion records ∆

Component actions for

each degree of freedom

δj

δi

θi

θj

Detailed model

Story drifts and forces

δij

Global displacement

Ground motion records ∆

Component actions for

each degree of freedom

δj

δi

θi

θj

Time (sec.)

Ground Acceleration (g)
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 2-7

Figure 2-8 Incremental dynamic analysis study for thirty ground motion records for a 5-story steel braced frame 

(Vamvatsikos and Cornell, 2002)

Figure 2-9 Flow chart depicting simplified SDOF nonlinear analysis process. Note that component actions are 

estimated from global displacement demand using the pushover curve.

Component actions

δj

δi

θi

θj

Story drifts and forces

δij

Capacity curve

(pushover)

V

∆

Global displacement

∆

Equivalent

SDOF Oscillator

Capacity curve

(pushover)

V

∆

Ground motion records

Component actions

δj

δi

θi

θj

Component actions

δj

δi

θi

θj

Story drifts and forces

δij

Capacity curve

(pushover)

V

∆

Global displacement

∆

Equivalent

SDOF Oscillator

Capacity curve

(pushover)

V

∆

Ground motion records
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

2-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

used to generate the equivalent SDOF model. This is 

similar to simplified nonlinear dynamic analyses using 

SDOF models. In contrast to the use of simplified 

dynamic analyses using multiple ground motion 

records, the use of nonlinear static procedures implies 

greater uncertainty due to the empirical procedures used 

to estimate the maximum displacement. This is true 

even if spectra representative of the multiple ground 

motion records are used in the nonlinear static analysis. 

Figure 2-11 summarizes the relationship among the 

normal options for inelastic seismic analysis procedures 

with respect to the type of structural model and 

characterization of ground motion. Also noted in the 

figure is the relative uncertainty associated with each 

option. The actual uncertainty inherent in any specific 

analysis depends on a number of considerations. 

Nonlinear dynamic analyses can be less uncertain than 

other techniques if the nonlinear inelastic properties of 

the components in the detailed structural model are 

accurate and reliable. If the component properties are 

poorly characterized, however, the results might not be 

an improvement over other alternatives. Some analysis 

options are better than others, depending on the 

parameter of interest. For example, with simplified 

dynamic analyses, a SDOF oscillator can be subjected 

to a relatively large number of ground motion records to 

provide a good representation of the uncertainty 

associated with global displacement demand due to the 

variability of the ground motion. On the other hand, if 

the engineer is comfortable with the estimate of 

maximum global displacement from a nonlinear static 

procedure, a multi-mode pushover analysis might 

provide improved estimates of inter-story drift that 

would not necessarily be available from the simplified 

SDOF dynamic analyses.

2.4 Current Nonlinear Static Procedures

Nonlinear static procedures are popular with practicing 

engineers, as demonstrated by the voluntary state-ofpractice internet query results in Appendix B. Two 

options are used predominantly. Equivalent 

linearization techniques are based on the assumption 

that the maximum total displacement (elastic plus 

inelastic) of a SDOF oscillator can be estimated by the 

elastic response of an oscillator with a larger period and 

damping than the original. These procedures use 

estimates of ductility to estimate effective period and 

damping. The Coefficient Method is fundamentally a 

displacement modification procedure that is presented 

in FEMA 356. Alternatively, displacement modification 

procedures estimate the total maximum displacement of 

the oscillator by multiplying the elastic response, 

assuming initial linear properties and damping, by one 

or more coefficients. The coefficients are typically 

derived empirically from series of nonlinear responsehistory analyses of oscillators with varying periods and 

strengths. 

A form of equivalent linearization known as the 

Capacity-Spectrum Method is documented in ATC-40. 

Other variations and versions of these two procedures 

Figure 2-10 Flow chart depicting the process followed in nonlinear static procedures. Note that component actions 

are based on global displacement demand and a pushover/capacity curve.

Component actions

δj

δi

θi

θj

Story drifts and forces

δij

Capacity curve

(pushover)

V

∆

Global displacement

∆

Elastic spectrum

Equivalent

SDOF Oscillator

Capacity curve

(pushover)

V

∆

Component actions

δj

δi

θi

θj

Component actions

δj

δi

θi

θj

Story drifts and forces

δij

Capacity curve

(pushover)

V

∆

Global displacement

∆

Elastic spectrum

Equivalent

SDOF Oscillator

Capacity curve

(pushover)

V

∆
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 2-9

have been suggested (see Appendices A and B), but all 

are related fundamentally to either displacement 

modification or equivalent linearization. Both 

approaches use nonlinear static analysis (pushover 

analysis) to estimate the lateral force-deformation 

characteristics of the structure. In both procedures the 

global deformation (elastic and inelastic) demand on the 

structure is computed from the response of an 

equivalent single-degree-of-freedom system having the 

load-deformation properties determined from the 

pushover analysis. They differ, however, in the 

technique used to estimate the maximum deformation 

demand (elastic and inelastic).

2.4.1 The Coefficient Method of Displacement 

Modification from FEMA 356

The Coefficient Method is the primary nonlinear static 

procedure presented in FEMA 356. This approach 

modifies the linear elastic response of the equivalent 

SDOF system by multiplying it by a series of 

coefficients C0 through C3 to generate an estimate of 

the maximum global displacement (elastic and 

inelastic), which is termed the target displacement. The 

process begins with an idealized force-deformation 

curve (i.e., pushover curve) relating base shear to roof 

displacement (see Figure 2-12). An effective period, Te, 

is generated from the initial period, Ti

, by a graphical 

procedure that accounts for some loss of stiffness in the 

transition from elastic to inelastic behavior. The 

effective period represents the linear stiffness of the 

equivalent SDOF system. When plotted on an elastic 

response spectrum representing the seismic ground 

motion as peak acceleration, Sa, versus period, T, the 

effective period identifies a maximum acceleration 

response for the oscillator. The assumed damping, often 

five percent, represents a level that might be expected 

for a typical structure responding in the elastic range. 

Figure 2-11 Matrix depicting possible inelastic seismic analysis procedures for various structural models and groundmotion characterizations along with trends of uncertainty in the result.

S

T

R

U

C

T

U

R

A

L

M

O

D

E

L

GROUND MOTION

Detailed

Equivalent MDOF

Equivalent SDOF

Corresponding response spectra Multiple records

Nonlinear static

procedures

(NSP’s)

Multi-mode pushover

analysis 

(MPA)

Simplified SDOF

dynamic analysis

Simplified MDOF 

dynamic analysis

Dynamic 

analysis

high RELATIVE UNCERTAINTY low

Spectral Acceleration (g)

Period (sec) Time (sec.) Ground Acceleration (g)
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

2-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

The peak elastic spectral displacement is directly 

related to the spectral acceleration by the relationship 

. (2-1)

The coefficient C0 is a shape factor (often taken as the 

first mode participation factor) that simply converts the 

spectral displacement to the displacement at the roof. 

The other coefficients each account for a separate 

inelastic effect.

The coefficient C1 is the ratio of expected displacement 

(elastic plus inelastic) for a bilinear inelastic oscillator 

to the displacement for a linear oscillator. This ratio 

depends on the strength of the oscillator relative to the 

response spectrum and the period of the SDOF system, 

Te. The coefficient C2 accounts for the effect of 

pinching in load-deformation relationships due to 

degradation in stiffness and strength. Finally, the 

coefficient C3 adjusts for second-order geometric 

nonlinearity (P-∆) effects. The coefficients are 

empirical and derived primarily from statistical studies 

of the nonlinear response-history analyses of SDOF 

oscillators and adjusted using engineering judgment. 

The coefficients are described in greater detail in 

Chapter 3.

2.4.2 Capacity-Spectrum Method of Equivalent 

Linearization in ATC-40

The basic assumption in equivalent linearization 

techniques is that the maximum inelastic deformation of 

a nonlinear SDOF system can be approximated from the 

maximum deformation of a linear elastic SDOF system 

that has a period and a damping ratio that are larger than 

the initial values of those for the nonlinear system. In 

the Capacity-Spectrum Method of ATC-40, the process 

begins with the generation of a force-deformation 

relationship for the structure. This process is virtually 

identical to that for the Coefficient Method of FEMA 

356, except that the results are plotted in accelerationFigure 2-12 Schematic illustrating the process by which the Coefficient Method of displacement modification (per 

FEMA 356) is used to estimate the target displacement for a given response spectrum and effective period, 

Te.

Co = converts SDOF spectral 

displacement to MDOF roof 

displacement (elastic)

C1 = expected maximum inelastic 

displacement divided by elastic 

displacement

C2 = effects of pinched hysteretic 

shape, stiffness degradation and 

strength deterioration

C3 = increased displacements due to 

dynamic P-∆ effects

T period, T e

Sa

Response spectrum

roof 

displacement, δ

Ke

Ki

i

e i

e

K T T

K =

base shear, V

δ

π = =

2

0123 2 4

e

t a

T CCCCS g

Pushover curve

Target displacement

Co

C1 = expected maximum inelastic 

displacement divided by elastic 

displacement

C2 = effects of pinched hysteretic 

shape, stiffness degradation and 

strength deterioration

C3 = increased displacements due to 

dynamic P-∆ effects

T period, T e

Sa

Response spectrum

roof 

displacement, δ

Ke

Ki

i

e i

e

K T T

K =

base shear, V

δ

π = =

2

0123 2 4

e

t a

T CCCCS g

Pushover curve

Target displacement

S

T

d S eff = a

2

2 4π
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 2-11

displacement response spectrum (ADRS) format (see 

Figure 2-13). This format is a simple conversion of the 

base-shear-versus-roof-displacement relationship using 

the dynamic properties of the system, and the result is 

termed a capacity curve for the structure. The seismic 

ground motion is also converted to ADRS format. This 

enables the capacity curve to be plotted on the same 

axes as the seismic demand. In this format, period can 

be represented as radial lines emanating from the origin.

The Capacity-Spectrum Method of equivalent 

linearization assumes that the equivalent damping of the 

system is proportional to the area enclosed by the 

capacity curve. The equivalent period, Teq, is assumed 

to be the secant period at which the seismic ground 

motion demand, reduced for the equivalent damping, 

intersects the capacity curve. Since the equivalent 

period and damping are both a function of the 

displacement, the solution to determine the maximum 

inelastic displacement (i.e., performance point) is 

iterative. ATC-40 imposes limits on the equivalent 

damping to account for strength and stiffness 

degradation. These limits are reviewed in greater detail 

in Chapter 3.

Figure 2-13 Graphical representation of the Capacity-Spectrum Method of equivalent linearization, as presented in 

ATC-40.

Sa

Capacity spectrum

dpi

api

ay

dy

Kinitial

ESo = Maximum strain energy

= Area of hatched triangle

= apidpi / 2

ED = Energy dissipated by damping 

= Area of enclosed by hysteresis loop

= Area of parallelogram

β 0 = Equivalent viscous damping

associated with full

hysteresis loop area

= 1 ED

4π ESo

ESo

ED

Sd

Seismic demand 

for β = 5%

Seismic demand 

for βeq = β 0 + 5%

Teq = Tsec

Sa

Capacity spectrum

dpi

api

ay

dy

Kinitial

ESo = Maximum strain energy

= Area of hatched triangle

= apidpi / 2

= Maximum strain energy

= Area of hatched triangle

= apidpi / 2

ED = Energy dissipated by damping 

= Area of enclosed by hysteresis loop

= Area of parallelogram

β 0 = Equivalent viscous damping

associated with full

hysteresis loop area

= 1 ED

4π ESo

= Equivalent viscous damping

associated with full

hysteresis loop area

= 1 ED

4π ESo

ESo

ED

Sd

Seismic demand 

for β = 5%

Seismic demand 

for βeq = β 0 + 5%

Teq = Tsec
Chapter 2: Overview of Inelastic Seismic Analysis Procedures

2-12 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-1 

3. Evaluation of Current Nonlinear Static Procedures

3.1 Introduction

This chapter summarizes the results of studies to assess 

the ability of current approximate nonlinear static 

procedures to estimate the maximum displacement of 

inelastic structural models. Initial studies evaluated 

both the Coefficient Method of FEMA 356 and the 

Capacity-Spectrum Method of ATC-40.

The use of NSPs (nonlinear static procedures) has 

accelerated in the United States since the publication of 

ATC-40, FEMA 273/274 and FEMA 356 documents. 

As a consequence there is valuable information 

available on the practical application of these inelastic 

analysis procedures (see Appendix B, “Summary of 

Practice using Inelastic Analysis Procedures”). Various 

researchers and practicing engineers have found that, in 

some cases, different inelastic analysis methods give 

substantially different estimates for displacement 

demand for the same ground motion and same SDOF 

oscillator (Aschheim et al., 1998; Chopra and Goel 

1999a,b, 2000; Albanessi et al., 2000; Kunnath and 

Gupta, 2000; Lew and Kunnath, 2000; Yu et al, 2001; 

Zamfirescu and Fajfar, 2001; MacRae and Tagawa, 

2002). The disparities in displacement predictions 

highlight the need for comparison and further study of 

these different approaches (see Appendix A, “Summary 

of Research on Inelastic Analysis Procedures”). 

The objective of this evaluation was to study the 

accuracy of the approximate methods described in 

ATC-40 and FEMA 356 for estimating the maximum 

displacement demand of inelastic single-degree-offreedom (SDOF) systems. This global displacement is a 

spectral displacement, termed the Performance Point in 

ATC-40. It is the roof displacement, termed the Target 

Displacement in FEMA 356. In particular, this study 

was aimed at identifying and quantifying the errors in 

these procedures when applied to SDOF systems. For 

this purpose, approximate total displacements 

computed with ATC-40 and with FEMA 356 were 

compared with the results of nonlinear response-history 

analyses of SDOF oscillators. The nonlinear responsehistory analyses are “exact” for the assumptions made 

for the properties of the oscillator (damping ratio and 

type of hysteretic behavior) and for the particular 

ground motion record. Thus these results are a useful 

benchmark to evaluate the approximate procedures.

Of particular interest is the extent to which the 

approximate methods might tend to overestimate or 

underestimate displacement demands (introduce bias) 

and the spectral regions or strength levels for which 

these biases are likely to occur. Errors were quantified 

through statistical analyses. A large number of SDOF 

systems (with a wide range of periods of vibration, 

lateral strengths, and hysteretic behavior) were 

subjected to a relatively large number of recorded 

earthquake ground motions. Ground motions included 

near-fault and far-fault records representative of site 

conditions ranging from rock to very soft soil. 

However, it is recognized that there may be some 

situations that deviate from those used in this 

investigation. Caution should be used when 

extrapolating the results presented in this evaluation for 

ground motions and site conditions that differ 

substantially.

Section 3.2 describes the period of vibration, damping 

ratio, lateral strength, and hysteretic behavior of the 

SDOF systems that were considered in this 

investigation. This section also describes the types and 

characteristics of the recorded ground motion records 

that were used as well as the error measures computed 

in this study. Section 3.3 describes the evaluation of the 

simplified inelastic analysis procedure in ATC-40 to 

estimate the maximum displacement of inelastic 

systems using equivalent linearization. Section 3.4 

provides a corresponding evaluation of the simplified 

analysis procedure in FEMA 356. In particular, this 

chapter provides an evaluation of coefficients C1, C2

and C3 in this method. Finally, Section 3.5 summarizes 

the dynamic response of nonlinear elastic, or rocking, 

oscillators. A complete compilation of the evaluation 

study data is provided in Appendix C, “Supplemental 

Data on the Evaluation of Current Procedures.”

3.2 Evaluation Procedures

3.2.1 Hysteretic Characteristics 

SDOF systems with initial periods of vibration between 

0.05 s and 3.0 s were used in this investigation. A total 

of 50 periods of vibration were considered (40 periods 

between 0.05 s and 2.0 s, equally spaced at 0.05 s, and 

10 periods between 2.0 s and 3.0 s, equally spaced at 0.1 

s intervals). The initial damping ratio, β, was assumed 

to be equal to 5% for all systems.
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

In this study the lateral strength is normalized by the 

strength ratio R, which is defined as

(3-1)

where m is the mass of the SDOF oscillator, Sa is the 

spectral acceleration ordinate corresponding to the 

initial period of the system, and Fy is the lateral yield 

strength of the system. The numerator in Equation 3-1 

represents the lateral strength required to maintain the 

system elasticity, which sometimes is also referred to as 

the elastic strength demand. Note that this R-factor is 

not the same as the response-modification coefficient 

conventionally used for design purposes. This R-factor 

is the design R-factor divided by the overstrength factor, 

omega sub-zero. This is discussed on page 105 of 

FEMA 450-2, NEHRP Recommended Provisions for 

Seismic Regulations for New Buildings and Other 

Structures, Part 2: Commentary (BSSC, 2003). Nine 

levels of normalized lateral strength were considered, 

corresponding to R = 1, 1.5, 2, 3, 4, 5, 6, 7 and 8. 

Four different hysteretic behaviors were used in this 

study (see Figure 3-1): 

• The elastic perfectly plastic (EPP) model is used as 

a reference model. This model has been used widely 

in previous investigations and therefore it represents 

a benchmark to study the effect of hysteretic 

behavior. Furthermore, recent studies have shown 

that this is a reasonable hysteretic model for steel 

beams that do not experience lateral or local 

buckling or connection failure (Foutch and Shi, 

1998).

• The stiffness-degrading (SD) model corresponds to 

the modified-Clough model, as originally proposed 

by Clough (1966) and as modified by Mahin and Lin 

(1983). This model was originally proposed as 

Figure 3-1 Basic hysteretic models used in the evaluation of current procedures: elastic perfectly plastic (EPP); 

stiffness-degrading (SD); strength and stiffness degrading (SSD), and nonlinear elastic (NE).

Elastoplastic Perfectly Plastic 

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

ecr oF

EPP

Stiffness Degrading

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

ecr oF

SD

Strength and Stiffness Degrading

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

ecr oF

SSD

Nonlinear Elastic 

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

ecr oF

NE

R mS

F

a

y

=
Chapter 3: Evaluation of Current Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-3

representative of well detailed and flexurally 

controlled reinforced concrete structures in which 

the lateral stiffness decreases as the level of lateral 

displacement increases. 

• The strength and stiffness-degrading (SSD) model is 

aimed at approximately reproducing the hysteretic 

behavior of structures in which lateral stiffness and 

lateral strength decrease when subjected to cyclic 

reversals. In this model, the amount of strength and 

stiffness degradation is a function of the maximum 

displacement in previous cycles as well as a function 

of the hysteretic energy dissipated. This model is 

similar to the three-parameter model implemented in 

IDARC (Kunnath et al., 1992). When properly 

calibrated, this model can reproduce the response of 

poorly detailed reinforced concrete structures 

relatively well. An example is shown in Figure 3-2, 

in which the load-deformation relationship of a 

poorly detailed beam-column joint tested at the 

University of Washington (Lehman et al., 2000) is 

compared with the response computed with the SSD 

model. A single set of parameters representing 

severe strength and stiffness degradation was used 

for this model. The type of degradation that is 

captured by this model only includes cyclic 

degradation. Note that the post-elastic stiffness in 

any cycle is always equal to zero or greater. Thus, 

the strength never diminishes in the current cycle of 

deformation. The degradation of strength occurs in 

subsequent cycles (or half-cycles) of deformation. 

Oscillators that have in-cycle negative post-elastic 

stiffnesses and in-cycle degradation of strength can 

be prone to dynamic instability. They are covered in 

Section 3.4.4 and in Chapter 4.

• The nonlinear elastic (NE) model unloads on the 

same branch as the loading curve and therefore 

exhibits no hysteretic energy dissipation. This model 

approximately reproduces the behavior of pure 

rocking structures. Most instances of rocking in real 

structures are a combination of this type of behavior 

with one of the other hysteretic types that include 

hysteretic energy losses.

In summary, the combinations of period of vibration, 

lateral strength, and hysteretic behavior represent a total 

of 1,800 different SDOF systems.

3.2.2 Earthquake Ground Motions

A total of 100 earthquake ground motions recorded on 

different site conditions were used in this study. Ground 

motions were divided into five groups with 20 

accelerograms in each group. The first group consisted 

of earthquake ground motions recorded on stations 

located on rock with average shear wave velocities 

between 760 m/s (2,500 ft/s) and 1,525 m/s (5,000 ft/s). 

Figure 3-2 Comparison of experimental results (after Lehman et al., 2000) with the hysteretic response computed 

with the SSD model.

Specimen 1 

Lehman and Stanton

-80

-60

-40

-20

0

20

40

60

80

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0

Drift [in]

Force [kips]

Experimental

Analytical
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

These are representative of site class B, as defined by 

the NEHRP Recommended Provisions for Seismic 

Regulations for New Buildings and Other Structures, 

Part I, Provisions (BSSC, 2000)1. The second group 

consisted of records obtained on stations on very dense 

soil or soft rock with average shear wave velocities 

between 360 m/s (1,200 ft/s) and 760 m/s, while the 

third group consisted of ground motions recorded on 

stations on stiff soil with average shear wave velocities 

between 180 m/s (600 ft/s) and 360 m/s. These are 

consistent with site class C and D respectively. The 

fourth group corresponds to ground motions recorded 

on very soft soil conditions with shear wave velocities 

smaller than 180 m/s, which can be classified as site 

class E. Finally, the fifth group corresponds to 20 

ground motions influenced by near-field forwarddirectivity effects. Detailed listings of the ground 

motions are presented in Appendix C. 

3.2.3 Error Measures and Statistical Study 

The maximum displacement of each inelastic SDOF 

system was estimated with the simplified inelastic 

procedures in ATC-40 and FEMA 356 when subjected 

to each of the ground motions. The maximum 

displacement of each inelastic SDOF system was then 

computed using nonlinear response-history analyses. 

The maximum displacement is defined as the maximum 

of the absolute value of the displacement response. A 

total of 180,000 nonlinear response-history analyses 

were run as part of this investigation. In this study, the 

results computed with nonlinear response-history 

analyses are the benchmark maximum displacements, 

(∆i

)ex. The maximum displacements estimated with 

simplified inelastic procedures of ATC-40 and FEMA 

356 are the approximate maximum displacements,

(∆i

)app of the inelastic system. It should be noted that 

the nonlinear response-history analyses are “exact” only 

for the SDOF oscillator with the assumed properties and 

for the particular ground motion. The uncertainty of the 

modeling assumptions with respect to the actual 

building is not included in either the nonlinear 

response-history analyses or the approximate analyses. 

The nonlinear response-history results are a convenient 

benchmark.

In order to evaluate the accuracy of these approximate 

procedures, an error measure was defined as the ratio of 

approximate, (∆i

)app, to benchmark, (∆i)ex, maximum 

displacement as follows: 

(3-2)

This error measure was computed for each period of 

vibration T and each level of normalized lateral strength 

R. Values of ET,R larger than one indicate that the 

approximate method overestimates the maximum 

displacement of the SDOF system and values smaller 

than one indicate underestimation. A total of 320,000 

individual errors were computed in this study. 

In order to identify whether the approximate methods, 

on average, tend to overestimate or underestimate 

maximum displacements of inelastic systems, mean 

errors were computed as follows:

(3-3)

where n is the number of records in each group of 

ground motions. Mean errors were computed for each 

hysteretic behavior type, each period of vibration (or for 

each normalized period of vibration as will be explained 

later) and each level of normalized lateral strength. 

Therefore, mean errors computed with Equation 3-3 do 

not allow for underestimations in a spectral region to be 

compensated by overestimations in another spectral 

region. Information on the bias for each period, for each 

type of hysteretic behavior, for each level of normalized 

lateral strength, and for each site class is retained. 

The sample mean error computed with Equation 3-3 is 

an unbiased estimator of the mean error of the 

population. Therefore, it provides an estimate of the 

average error produced by the approximate methods. 

However, it provides no information on the dispersion 

of the error. In order to obtain a measure of the 

dispersion of the errors produced by the approximate 

methods, the standard deviation of the error was 

computed as

(3-4)

The square of the sample standard deviation of the error 

computed with Equation 3-4 is an unbiased estimator of 

the variance of the error in the population. The standard 

deviation of the error was computed for each period, for 1.Superseded in 2003 with the FEMA 450 Recommended

Provisions for Seismic Regulations for New Buildings

and Other Structures.

ET R

i

i

T R

,

,

= ( )

( )

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

∆

∆

app

ex

E

n

TR TR E

i

i

n

, , = ( ) =

∑

1

1

σTR TR i T R

i

n

n , ,, = E E − ⎡( ) − ⎣ ⎤

⎦ =

∑ 1

1

2

1
Chapter 3: Evaluation of Current Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-5

each type of hysteretic behavior, for each level of 

normalized lateral strength, and for each site class. 

3.3 Evaluation of Capacity-Spectrum 

Method of ATC-40

3.3.1 Summary of the Approximate Method

The simplified inelastic analysis procedure in ATC-40, 

a version of the Capacity-Spectrum Method (CSM), is 

based on equivalent linearization. The basic assumption 

in equivalent linear methods is that the maximum 

displacement of a nonlinear SDOF system can be 

estimated from the maximum displacement of a linear 

elastic SDOF system that has a period and a damping 

ratio that are larger than those of the initial values for 

the nonlinear system. The elastic SDOF system that is 

used to estimate the maximum inelastic displacement of 

the nonlinear system is usually referred to as the 

equivalent or substitute system. Similarly, the period of 

vibration and damping ratio of the elastic system are 

commonly referred to as equivalent period and 

equivalent damping ratio, respectively.

The concept of equivalent viscous damping was first 

proposed by Jacobsen (1930) to obtain approximate 

solutions for the steady forced vibration of damped 

SDOF systems with linear force-displacement 

relationships but with damping forces proportional to 

the nth power of the velocity of motion when subjected 

to sinusoidal forces. In this pioneering study, the 

stiffness of the equivalent system was set equal to the 

stiffness of the real system and the equivalent viscous 

damping ratio was based on equating the dissipated 

energy per cycle of the real damping force to that of the 

equivalent damping force. Years later, the same author 

extended the concept of equivalent viscous damping to 

yielding SDOF systems (Jacobsen, 1960). Since then, 

there have been many methods proposed in the 

literature. Review of the earlier equivalent linear 

methods can be found in Jennings (1968), Iwan and 

Gates (1979), Hadjian (1982), Fardis and Panagiatakos 

(1996), while a review of some recent methods can be 

found in Miranda and Ruiz-García (2003). The 

Capacity Spectrum Method as documented in ATC-40 

is based primarily on the work of Freeman et al. (1975).

In equivalent linear methods, the equivalent period is 

computed from the initial period of vibration of the 

nonlinear system and from the maximum displacement 

ductility ratio, μ. Similarly, the equivalent damping 

ratio is computed as a function of damping ratio in the 

nonlinear system and the displacement ductility ratio. 

The main differences among the many equivalent linear 

methods that are available in the literature stem 

primarily from the functions used to compute the 

equivalent period and equivalent damping ratio.

As discussed in Section 2.4.2, the Capacity-Spectrum 

Method according to ATC-40 uses the secant stiffness at 

maximum displacement to compute the effective period 

and relates effective damping to the area under the 

hysteresis curve (see Figure 2-13).These assumptions 

result in an equivalent period, Teq, and equivalent 

damping ratio (referred to as effective viscous damping, 

βeq, in ATC-40) given by

(3-5)

(3-6)

where T0 is the initial period of vibration of the 

nonlinear system, α is the post-yield stiffness ratio and 

κ is an adjustment factor to approximately account for 

changes in hysteretic behavior in reinforced concrete 

structures. ATC-40 proposes three equivalent damping 

levels that change according to the hysteretic behavior 

of the system. Type A hysteretic behavior denotes 

structures with reasonably full hysteretic loops, similar 

to the EPP oscillator in Figure 3-1. The corresponding 

equivalent damping ratios take the maximum values. 

Type C hysteretic behavior represents severely 

degraded hysteretic loops (e.g., SSD), resulting in the 

smallest equivalent damping ratios. Type B hysteretic 

behavior is an intermediate hysteretic behavior between 

types A and C (e.g., SD). The value of κ decreases for 

degrading systems (hysteretic behavior types B and C). 

ATC-40 suggests an initial elastic viscous damping ratio 

(first term on the right hand side of Equation 3-6) of 

0.05 (5%) for reinforced concrete buildings. The terms 

to the right of κ in Equation 3-6 represent the equivalent 

hysteretic viscous damping for an idealized bilinear 

system designated as β0 in ATC-40 documentation. 

Table 3-1 shows the variation of κ with respect to β0 for 

different hysteretic behaviors types.

The equivalent period in Equation 3-5 is based on a 

lateral stiffness of the equivalent system that is equal to 

the secant stiffness at the maximum displacement. It 

only depends on the displacement ductility ratio and the 

T T eq = + − 0 1

μ

αμ α

ββ κ

π

μ α

μ αμ α eq eff == + ( − )( − )

( + − ) 0 05 2 1 1

1 .
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

post-yield stiffness ratio of the inelastic system. 

Figure 3-3 shows the variation of equivalent periods for 

different post-yield stiffness ratios for a wide range of 

displacement ductility ratios. The equivalent period 

becomes longer as the displacement ductility ratio 

increases and as the post-yield stiffness ratio decreases. 

Figures 3-4 and 3-5 present the variation of κ and 

effective damping value, βeq, with changes in the 

ductility ratio, respectively. The calculations were done 

assuming elastic perfectly plastic (EPP) behavior to 

represent a system that has full hysteretic loops (i.e., a 

non-degrading system). It can be seen that for structures 

with type A behavior (systems having full hysteretic 

loops), the κ value is 1.0 for displacement ductility 

ratios less than 1.3. For ductility ratios larger than 1.3, 

κ decreases up to a value of 0.77 at a displacement 

ductility ratio of 3.4 and remains constant at 0.77 for 

larger ductilities. Similarly, for structures with type B 

hysteretic behavior, the value of κ is constant and equal 

to 0.67 for displacement ductility ratios less than 1.6, 

decreases to 0.53 for ductility ratio of 3.4, and remains 

constant for larger ductilities. For structures with type C 

hysteretic behavior, the κ factor is equal to 0.33 

regardless of the level of ductility demand.

The equivalent damping ratio in the equivalent linear 

spectrum method documented in ATC-40 rapidly 

increases once the structures yields and remains 

constant for ductility ratios higher than 3.4. The 

maximum equivalent damping ratios for hysteretic 

behavior types A, B, and C are 0.40, 0.29 and 0.20, 

respectively. According to Equations 3-5 and 3-6, 

structures with hysteretic behaviors type B and C will 

Table 3-1 Variation of κ-Value in ATC-40

Hysteretic 

Behavior β0 κ

Type A ≤ 0.1625 1.0

> 0.1625 1.13 – 0.51 × (π/2) × β0

Type B ≤ 0.25 0.67

> 0.25 0.845 – 0.446 × (π/2) × β0

Type C Any value 0.33

Figure 3-3 Variation of period shift based on secant 

stiffness.

0.0

1.0

2.0

3.0

4.0

0 2 4 6 8 10

Displacement Ductility, μ

Teq / T0

α = 0.00

α = 0.03

α = 0.05

Figure 3-4 Variation of κ-factor with the 

displacement ductility ratio, μ.

Figure 3-5 Variation of equivalent (effective) 

damping ratios with changes in the 

displacement ductility ratio, μ.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10

Displacement ductility, μ

κ

Type A

Type B

Type C

0.0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10

Displacement Ductility, μ

β eff

Type A

Type B

Type C
Chapter 3: Evaluation of Current Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-7

have much larger displacement demands because of the 

reduced hysteretic energy dissipation capacity produced 

by narrower hysteretic loops.

When applied to design spectra, ATC-40 provides 

reduction factors to reduce spectral ordinates in the 

constant-acceleration region and constant-velocity 

region as a function of the effective damping ratio. 

These spectral reduction factors are given by 

(3-7)

(3-8)

where βeff, is the effective or equivalent damping ratio 

computed with Equation 3-6. SRA is the spectralreduction factor to be applied to the constantacceleration region in the linear elastic design spectrum, 

and SRV is the spectral reduction factor to be applied to 

the constant-velocity region (descending branch) in the 

linear elastic design spectrum. These spectral-reduction 

factors are shown in Figures 3-6 and 3-7. It can be seen 

that for displacement ductility demands larger than 3.4, 

the spectral ordinates no longer decrease. Consequently, 

the ATC-40 procedures impose limits on the amount of 

hysteretic damping-related reduction in spectral 

response that can be achieved. Table 3-2 shows these 

limiting values.

3.3.2 Iteration Procedures

Equivalent linearization equations, in general, require 

prior knowledge of the displacement ductility ratio in 

order to compute the equivalent period of vibration and 

equivalent damping ratio, μ, which are then needed to 

estimate the maximum inelastic displacement demand 

on a SDOF system when subjected to a particular 

ground motion. Specifically, in Equations 3-5 and 3-6, 

μ must be known in order to compute βeff and Teq. 

However, when evaluating a structure, the maximum 

displacement ductility ratio is not known. 

Consequently, iteration is required in order to estimate 

the maximum displacement.

ATC-40 describes three iterative procedures to reach a 

solution for the approximation. Procedures A and B are 

described as the most transparent and most convenient 

for programming, as they are based on an analytical 

method. Procedure C is a graphical method that is not 

convenient for spreadsheet programming. ATC-40 

presents Procedure A as the most straightforward and 

Figure 3-6 Variation of spectral reduction factors 

SRA for different hysteretic behaviors as a 

function of the displacement ductility 

ratio, μ.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10

Displacement Ductility, μ

SRA

SRAmin for Type A

SRAmin for Type B

SRAmin for Type C

SRA = 3 21 0 68 100 − ( )

2 12

. . ln

.

βeff

SRV = 2 31 0 41 100 − ( )

1 65

. . ln

.

βeff

Figure 3-7 Variation of spectral reduction factors 

SRV for different hysteretic behaviors as a 

function of the displacement ductility 

ratio, μ.

Table 3-2 Minimum Allowable Spectral Reduction 

Factors for Displacement Ductility 

Ratios Larger than 3.4

Behavior Type SRA SRV

Type A 0.33 0.50

Type B 0.44 0.56

Type C 0.56 0.67

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10

Displacement Ductility, μ

SRV

SRVmin for Type C

SRVmin for Type B

SRVmin for Type A
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

easy in application among the three procedures. In a 

recent study, Chopra and Goel (1999a,b, 2000) 

investigated the iteration methods implemented in ATC40. By using various SDOF examples, they showed that 

Procedure A did not always converge when using actual 

earthquake spectra, as opposed to smooth design 

spectra. They also concluded that the displacement 

computed with Procedure B was unique and the same as 

that determined with Procedure A, provided that the 

latter converged. In a more recent study, Miranda and 

Akkar (2002) provide further discussion of the 

convergence issues in equivalent linearization 

procedures. They also note that equivalent linearization 

procedures can lead to multiple results for some specific 

earthquake ground motions.

An iteration procedure based on secant iteration that is 

guaranteed to converge was used for the evaluation 

study. As noted in the previous section, multiple 

equivalent linearization solutions may exist for actual 

ground motion records that were used for the study, as 

opposed to smoothed spectra normally used by 

engineers. For the purposes of this investigation, the 

first computed displacement encountered within 1% of 

the assumed displacement was taken as the approximate 

inelastic displacement without verifying whether this 

was the only possible solution. 

3.3.3 Evaluation Using Ground Motion Records

In order to evaluate the Capacity-Spectrum Method 

when applied to structures with hysteretic behavior type 

A, approximate results were compared with responsehistory analysis (RHA) benchmark results computed 

with the EPP hysteretic model. Similarly, the 

approximate results computed for behavior type B were 

compared with RHA benchmark results of the stiffness 

degrading (SD) model, and the approximate results 

computed for behavior type C were compared with 

RHA benchmark results of the strength-and-stiffnessdegrading (SSD) model. Mean errors corresponding to 

ground motions recorded in site class C and for 

hysteretic behaviors type A, B, and C are shown in 

Figure 3-8. Based on the complete results presented in 

Appendix C, it was found that the Capacity-Spectrum 

Method implemented in ATC-40 leads to very large 

overestimations of the maximum displacement for 

relatively short-period systems (periods smaller than 

about 0.5 s). Approximate maximum displacements in 

this period range can be, on average, larger than twice 

the RHA benchmark displacements. These large 

overestimations of displacement in the short-period 

range have also been reported previously for other 

equivalent linearization methods that are based on 

secant stiffness (Miranda and Ruiz-García, 2003; Akkar 

and Miranda, 2005).

Figure 3-8 Mean error associated with the CapacitySpectrum Method of ATC-40 for hysteretic 

behaviors types A, B, and C for site class C.

SITE CLASS C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

SITE CLASS C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

SITE CLASS C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING

R = 8.0

R = 1.5

R = 8.0

R = 1.5

R = 8.0

R = 1.5
Chapter 3: Evaluation of Current Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-9

The complete results indicate that, for periods longer 

than about 0.6 s, ATC-40 behavior type A tends to 

underestimate the maximum displacements. Maximum 

displacements computed with the ATC-40 procedure 

are, on average, about 25% to 35% smaller than those 

computed with RHA using elasto-plastic systems. 

Underestimations are slightly smaller for site class B 

and slightly larger for site class D. Mean errors for 

ATC-40 behavior type A are not significantly 

influenced by changes in the normalized lateral strength 

R. 

For systems with ATC-40 hysteretic behavior type B 

and periods longer than about 0.8 s, the CapacitySpectrum Method tends to underestimate displacements 

compared with those of inelastic systems with stiffnessdegrading (SD) models for site class B. 

Underestimations are small and tend to decrease as R

increases. Average underestimations range from 5% to 

25%. For site classes C and D, ATC-40 may 

underestimate or overestimate lateral deformation of 

systems with type B hysteretic behavior depending on 

the normalized lateral strength, R. 

In the case of systems with hysteretic behavior type C, 

the approximate ATC-40 procedure tends to 

overestimate inelastic displacements for practically all 

periods when compared to those computed for inelastic 

systems with strength-and-stiffness-degrading (SSD) 

hysteretic models. Overestimations increase as R

increases. The level of overestimation varies from one 

site class to another. Detailed information on the actual 

errors are contained in Appendix C. 

Dispersion of the error is very large for periods smaller 

than about 0.5 s and is moderate and approximately 

constant for periods longer than 0.5 s. In general, 

dispersion increases as R increases. Mean errors 

computed with ground motions recorded on very soft 

soil sites or with near-fault ground motions are strongly 

influenced by the predominant period of the ground 

motion. Detailed results of dispersion for site classes B, 

C, and D and behavior types A, B, and C are also 

presented in Appendix C. 

3.4 Evaluation of Coefficient Method 

(FEMA 356)

3.4.1 Summary of the Approximate Method

The determination of the target displacement in the 

simplified nonlinear static procedure (NSP) known as 

the displacement Coefficient Method is primarily 

described in the FEMA 356 document (Section 

3.3.3.3.2). According to this document, the target 

displacement, δt

, which corresponds to the 

displacement at roof level, can be estimated as 

(3-9)

where:

C0 = Modification factor to relate spectral displacement of an equivalent SDOF system to 

the roof displacement of the building MDOF 

system. It can be calculated from

• the first modal participation factor,

• the procedure described in Section 

3.3.3.2.3 in FEMA 356, or

• the appropriate value from Table 3.2 in 

FEMA 356.

C1 = Modification factor to relate the expected 

maximum displacements of an inelastic 

SDOF oscillator with EPP hysteretic properties to displacements calculated for the linear 

elastic response.

but not greater than the values given in Section 3.3.1.3.1 (Linear Static Procedure, LSP 

section) nor less than 1. Values of C1 in Section 3.3.1.3.1 are

with linear interpolation used to calculate C1

for the intermediate values of Te.

The limit imposed on C1 by Section 3.3.1.3.1 

is often referred to as “C1 capping.”

C2 = Modification factor to represent the effect of 

pinched hysteretic shape, stiffness degradation, and strength deterioration on the maxiδ

π t a e CCCCS T = g 0123

2

2 4

C

T T

R T

T

R

T T

e s

s

e e s

1

1 0

1 0

1 =

≥

( − )

<

⎧

⎨

⎪

⎪

⎩

. for

. +

for ⎪

⎪

C

T

T T

e

e s 1

15 01

1 = <≥

⎧

⎨

⎩

. . for s

.0 for
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

mum displacement response. Values of C2

for different framing systems and structural 

performance levels (i.e., immediate occupancy, life safety, and collapse prevention) 

are obtained from Table 3.3 of the FEMA 

356 document. Alternatively, C2 can take the 

value of one in nonlinear procedures.

C3 = Modification factor to represent increased 

displacements due to dynamic P-∆ effects. 

For buildings with positive post-yield stiffness, C3 is set equal to 1. For buildings with 

negative post-yield stiffness, values of C3 are 

calculated using the following expression:

 (3-10)

where:

Te = Effective fundamental period of the 

building computed in accordance with section 3.3.3.2.5.

Ts = Characteristic period of the response 

spectrum, defined as the period associated 

with the transition from the constant-acceleration segment of the spectrum to the constant-velocity segment of the spectrum.

R = Ratio of elastic strength demand to calculated strength capacity.

Sa = Response spectrum acceleration, at the effective fundamental period and damping ratio 

of the building.

g = Gravitational acceleration.

3.4.2 Maximum Displacement Ratio 

(Coefficient C1)

Coefficient C1 is the ratio of the maximum 

displacement for inelastic response of a SDOF 

oscillator with non-degrading hysteretic behavior to the 

maximum displacement had the oscillator remained 

elastic. Figure 3-9 shows the variation of C1 for site 

class B using a characteristic period Ts equal to 0.4 s. 

This characteristic period value is computed by 

applying the procedure described in Sections 1.6.1.5 

and 1.6.2.1 of the FEMA 356 document. For the 

evaluation of the FEMA 356 Coefficient Method, this 

study utilized characteristic periods equal to 0.4 s, 

0.55 s, 0.6 s and 1.0 s for site classes B, C, D, and E, 

respectively. These characteristic periods are 

representative of the periods computed according to 

FEMA 356 when using large ground motion intensities 

for which the system is expected to behave nonlinearly. 

Figure 3-9 shows a comparison between the values of 

C1 with the limitation (capping), as defined in FEMA 

356 Section 3.3.3.3.2, and without the limitation. 

C

R

Te

3

3 2

1 0

1

= + ( − ) . α

Figure 3-9 Comparison of coefficient C1 in FEMA 

356 with and without capping.

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

PERIOD [s]

C1

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

Ts = 0.4 s

WITH CAPPING

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

PERIOD [s]

C1

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

Ts = 0.4 s

WITHOUT CAPPING

R=1.5

R=2.0/3.0

R=1.5

R=8.0
Chapter 3: Evaluation of Current Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-11

The most important observation that can be made from 

Figure 3-9 is that with the limitations on C1 imposed by 

FEMA 356 for structures with short periods of vibration 

(often referred to as “capping”), the C1 coefficient 

becomes independent of the lateral strength of the 

structure. This means that changes in R do not produce 

changes in lateral displacement demand. Figure 3-10 

shows a close-up view of the C1 coefficients for site 

class B as a function of period. For R = 1.5 (top graph) 

the equation specified in the NSP will control this 

coefficient for periods between 0.2 and 0.4 s, while for 

R = 2.0 (bottom graph) the NSP equation has only a 

minimal effect for periods between 0.3 and 0.4 s. For 

values of R approximately larger than 2.5, the capping 

equation will always control the value of C1. 

Mean values of the computed ratio of the maximum 

displacement for inelastic response of a SDOF 

oscillator with non-degrading hysteretic behavior, to the 

maximum displacement had the oscillator remained 

elastic when subjected to 20 ground motions recorded 

on site class C, is shown in Figure 3-11. It can be seen 

that this ratio is clearly different in two spectral regions. 

Based on this figure, the following observations can be 

made:

• For periods longer than about 1.0 s, the computed C1

ratio is on average fairly insensitive to the level of 

strength (i.e., the value of C1 does not change much 

with changes in R).

• In the long-period spectral region, the computed C1

ratio is on average independent of the period of 

vibration (i.e., the value of C1 does not change much 

with changes in T).

• The equal-displacement approximation is a 

relatively good approximation of the expected value 

of C1 in the long-period spectral region (i.e., the 

value of C1 is approximately equal to one when T > 

1.3). 

Figure 3-10 A close up view of the effect of the 

capping limitation of C1 coefficient.

FEMA 356

0.8

1.0

1.2

1.4

1.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6

PERIOD [s]

C1

LSP (Section 3.3.1.3.1)

NSP (Section 3.3.3.3.2)

SITE CLASS B (Ts = 0.4 s) R=1.5

FEMA 356

0.8

1.0

1.2

1.4

1.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6

PERIOD [s]

C1

LSP (Section 3.3.1.3.1)

NSP (Section 3.3.3.3.2)

SITE CLASS B (Ts = 0.4 s) R=2.0

LSP NSP

LSP NSP

Figure 3-11 Variation of mean C1 computed for the 

elastic perfectly plastic (EPP) model when 

subjected to ground motions recorded on 

site class C. 

SITE CLASS C

(mean of 20 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

C1,EPP

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

• C1 IS APPROXIMATELY CONSTANT 

WITH CHANGES IN T 

• C1 DOES NOT CHANGE MUCH WITH 

CHANGES IN R 

• C1 IS ON AVERAGE APPROXIMATELY 

EQUAL TO ONE 

TWO BASIC SPECTRAL REGIONS 

• C1 IS ON AVERAGE 

LARGER THAN ONE 

• C1 INCREASES WITH 

DECREASING T 

• C1 INCREASES WITH 

INCREASING R

R = 8.0

R = 1.5
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-12 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

• In the short-period region, inelastic displacements 

are on average larger than elastic displacements (i.e., 

C1 is larger than one).

• In the short-period region, the value of C1 is highly 

dependent (i.e., very sensitive) on the level of lateral 

strength. In general, C1 increases as R increases (i.e., 

as the lateral strength decreases). 

• In the short-period region, the value of C1 is 

sensitive to changes in the period of vibration. In 

general, for a given R, a decrease in period produces 

an increase in C1.

• The transition period dividing the region in which 

the equal-displacement approximation 

underestimates displacement, from the region in 

which this approximation applies (short- versus 

long-period region), increases as the lateral strength 

decreases (as R increases).

Figure 3-12 presents a comparison of mean values of 

coefficient C1 generated from the nonlinear responsehistory analyses for site classes B, C, and D. The 

transition period dividing the region in which the equaldisplacement approximation underestimates 

displacements, from the region in which this 

approximation is valid, increases as the site becomes 

softer. For site classes B and R smaller than 8, this 

period is approximately 1.0 s; for site class C it is 

approximately 1.1 s; and for site class D it is 

approximately 1.4 s.

Figure 3-13 compares mean values of the computed 

ratio of the maximum displacement for inelastic 

response of a SDOF oscillator with elasto-plastic 

hysteretic behavior to the maximum displacement had 

the oscillator remained elastic when subjected to 20 

ground motions recorded on site B to the approximate 

coefficient C1 specified in FEMA 356. 

The FEMA 356 transition period, dividing the region in 

which the equal-displacement approximation 

underestimates displacements, from the region in which 

this approximation is valid, is shorter than that observed 

for the ground motions used in this study. For example, 

for site class B, the transition period in FEMA 356 is 

0.4 s while results from nonlinear response-history 

analyses suggest that this period should be about twice 

as long. The transition periods that can be observed 

from these nonlinear response-history analyses in 

Figure 3-12 (approximately 1.0 s, 1.1 s and 1.4 s for site 

classes B, C and D, respectively) are all significantly 

longer than those specified in FEMA 356 (0.4 s, 0.55 s, 

0.6 s, for site classes B, C, and D, respectively).

While results from nonlinear response-history analyses 

indicate a strong sensitivity of the computed C1 ratio 

with changes in R for short periods, the capping in 

FEMA 356 practically eliminates this sensitivity to 

lateral strength. For example, mean inelastic 

displacement ratios computed from response-history 

analyses for a period of 0.3 s suggest that a change in R

Figure 3-12 Mean coefficient C1 for site classes B, C 

and D.

SITE CLASSES B

(mean of 20 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

C1,EPP

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS C

(mean of 20 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

C1,EPP

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS D

(mean of 20 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

C1,EPP

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

R = 8.0

R = 1.5

R = 8.0

R = 1.5

R = 8.0

R = 1.5
Chapter 3: Evaluation of Current Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-13

from 2 to 8 almost triples the value of C1, while the 

capped coefficient in FEMA 356 leads to the conclusion 

that the displacement of these systems is the same 

regardless of the lateral strength of the structure.

In the absence of the cap on C1, the equation currently 

used in FEMA 356 to estimate this coefficient in section 

3.3.3.3.2 does not capture the effect of changes in 

lateral strength on displacement demands. For example, 

for SDOF systems with periods of 0.3 s, one with R = 2 

and the other with R = 8, the expression in FEMA 356 

would indicate that the displacement demand in the 

weaker system would be only about 15% larger than the 

displacement demand in the stronger system, while 

response-history analyses indicate a much larger 

sensitivity to lateral strength.

Figure 3-14 shows inelastic displacement ratios 

computed for two ground motions recorded in very soft 

soil sites in the San Francisco Bay Area during the 1989 

Loma Prieta earthquake. It can be seen that despite 

being in the same site class, the inelastic displacement 

ratios can be very different. For example, for a structure 

with a 1 s period and R = 6 at the Larkspur site C1 can 

reach 2.8 (displacement for the inelastic oscillator 2.8 

times larger than the maximum elastic), while at the 

Emeryville site it is 0.65 (displacement for the inelastic 

oscillator smaller than the maximum elastic). In order to 

obtain a better characterization of maximum 

displacement ratios, periods of vibration were 

normalized by the predominant period of the ground 

motion, as first proposed by Miranda (1991, 1993). The 

predominant period, Tg, of the ground motion is computed as the period of vibration corresponding to the 

maximum 5% damped relative-velocity spectral 

ordinate. Examples of the computation of Tg for these 

two recording stations are shown in Figure 3-15. The 

resulting inelastic displacement ratios are shown in 

Figure 3-16, where it can be seen that when the periods 

of vibration are normalized, a better characterization of 

displacement demands is obtained. As shown, inelastic 

displacement ratios at soft soil sites are characterized by 

values larger than one for normalized periods smaller 

than about 0.7, values smaller than one for normalized 

periods between 0.7 and 1.5 s, and values 

approximately equal to one for longer normalized 

periods. 

Mean inelastic ratios computed for 20 ground motions 

for site class E are shown in Figure 3-17. The same 

trend observed in individual records is preserved for the 

mean. Additional information on inelastic displacement 

demands of structures on very soft soil can be found in 

Ruiz-García and Miranda (2004).

Figure 3-13 Comparison between the mean C1

computed from nonlinear responsehistory analyses to C1 in FEMA 356 (noncapped and capped).

SITE CLASSES B

(mean of 20 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

C1

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

C1

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

Ts = 0.4s 

WITHOUT CAPPING

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

C1

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

Ts = 0.4s 

WITH CAPPING

R = 6.0

R = 1.5

R = 8.0

R = 1.5

R = 1.5

R=2.0/3.0
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-14 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Inelastic displacement ratios for near-fault ground 

motions influenced by forward directivity effects can be 

computed in an analogous manner by normalizing the 

periods of vibration by the pulse period, which was 

computed using the same procedure as for soft soils 

(refer to Figure 3-15).

The procedure described in Section 3.2 was used to 

calculate mean errors associated with the FEMA 356 

specifications for the coefficient C1 when compared 

with the nonlinear response-history benchmark. 

Figure 3-18 shows mean errors corresponding to 

maximum displacement demands computed using 

FEMA 356 with and without capping when subjected to 

ground motions recorded on site classes B and C. These 

mean errors correspond to displacements computed 

with C2 = C3 = 1, normalized by the benchmark 

displacement demands computed with an EPP 

hysteretic model. It can be seen that, in general, the 

results are very good for periods of vibration larger than 

1.0, where the equal-displacement approximation 

provides acceptable results with only small 

overestimations.

In Figure 3-18, it is evident that for site class B and 

periods between 0.4 s and 1.0 s, the underestimation of 

the transition period leads to underestimation of 

maximum displacement. Underestimation increases as 

R increases. For example, for a period of 0.4 s, 

benchmark displacements are on average 1.8 times 

larger than approximate displacements for R = 8. 

Similar underestimations are produced for site class C.

For periods smaller than 0.4 s in the case of site class B, 

and for periods smaller than 0.55 s in the case of site 

class C, the use of capping on C1 leads to large 

underestimation of displacements when R is larger than 

2. When the capping is removed, in some cases large 

Figure 3-14 Variation of C1 for two individual ground 

motions recorded on soft soil E.

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD 

C1

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD 

C1

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

EMERYVILLE 

LARKSPUR FERRY TERMINAL 

R = 6.0

R = 1.5

R = 6.0

R = 1.5

Figure 3-15 Predominant ground motion periods for 

the soft soil records obtained at Larkspur 

Ferry Terminal and Emeryville during the 

1989 Loma Prieta earthquake.

0

20

40

60

80

100

120

140

160

0.0 1.0 2.0 3.0 4.0

PERIOD

Sv [cm/s]

0

20

40

60

80

100

120

140

160

0.0 1.0 2.0 3.0 4.0

PERIOD

Sv [cm/s]

 Tg = 2.09 s 

LARKSPUR FERRY TERMINAL 

 Tg = 1.50 s 

EMERYVILLE 
Chapter 3: Evaluation of Current Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-15

underestimations of displacements are produced while 

in other cases large overestimations of displacements 

are computed. This suggests that the variation of C1

with changes in period and lateral strength as specified 

in FEMA 356 could be improved.

3.4.3 Degrading System Response (Coefficient 

C2)

The coefficient C2 is a modification factor to represent 

the effect of pinched hysteretic shape, stiffness 

degradation, and strength deterioration on the 

maximum displacement response according to FEMA 

356. Values of C2 for implementation in FEMA 356 

depend on the type of structural framing system and 

structural performance levels being considered (i.e., 

immediate occupancy, life safety, and collapse 

prevention). Values of coefficient C2, computed 

according to Table 3-3 in FEMA 356, are shown in 

Figure 3-19.

Benchmark ratios of the maximum displacement 

demand were calculated by dividing the maximum 

displacement for the stiffness-degrading oscillator (SD) 

model by that for the EPP model when both were 

subject to actual ground motions. This ratio thus 

corresponds with the coefficient C2. Mean ratios were 

calculated for the different site classes. An example for 

ground motions recorded on site class D is shown in 

Figure 3-20. With the exception of periods of vibration 

smaller than about 0.6 s, the maximum displacements of 

SD models are on average slightly smaller (3% to 12%) 

than that of the EPP systems. Although this may seem 

surprising considering the smaller hysteresis loops of 

the SD model, the results shown in this figure are 

consistent with previous investigations (Clough, 1966; 

Clough and Johnston, 1966; Chopra and Kan, 1973; 

Powell and Row, 1976; Riddel and Newmark, 1979; 

Mahin and Bertero, 1981; Gupta and Kunnath, 1998; 

Foutch and Shi, 1998; and Gupta and Krawinkler, 

1998). The coefficient C2 specified in FEMA 356, in 

contrast, increases lateral displacements in this period 

range.

For periods of vibration smaller than about 0.6 s, lateral 

displacement of SD systems are generally larger than 

those of non-degrading EPP systems. Differences 

increase with increasing R. This observation is similar 

to observations of several of the studies mentioned 

previously. Values of C2 in the period range specified in 

FEMA 356 are generally higher than those computed 

for relatively strong SD systems (R < 3) but smaller 

than those computed for relatively weak SD systems. 

Figure 3-16 C1 values of Larkspur Ferry Terminal and 

Emeryville soft soil records for normalized 

periods of vibration with respect to 

dominant ground motion periods of each 

record.

Figure 3-17 The variation of mean C1 values for site 

class E.

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

C1

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

C1

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

LARKSPUR FERRY TERMINAL 

EMERYVILLE 

R = 6.0

R = 1.5

R = 6.0

R = 1.5

SITE CLASS E

(mean of 20 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Tg

C1,EPP

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

R = 8.0

R = 1.5
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-16 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Mean ratios of maximum displacements of strengthand-stiffness degrading (SSD) systems to those of EPP 

systems are shown in Figure 3-21, which shows very 

similar trends. However, in the case of periods shorter 

than 0.8 s, the increase in lateral displacement produced 

by SSD behavior is larger than that produced by 

stiffness degradation only. For periods longer than 0.8 s, 

the maximum displacement of SSD systems is on 

average equal to that of EPP systems. It should be noted 

that displacement ratios shown in Figures 3-20 and 3-21 

only correspond to mean (average) values and that a 

very large uncertainty exists, particularly for periods 

smaller than 0.6 s. 

Figure 3-22 presents mean errors calculated from the 

ratio of the displacements computed with FEMA 356 

(with and without capping of C1) for C2 computed 

assuming a life safety structural performance level to 

the maximum displacements computed with nonlinear 

response-history analyses using the SD model. Results 

Figure 3-18 Mean error statistics of capped and not capped C1 values for the ground motions recorded in site classes 

B and C, respectively.

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

SITE CLASS B

Ts = 0.4 s

(C2 = 1.0)

R=2.0

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS B

Ts = 0.4 s

(C2 = 1.0)

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

(C2 = 1.0)

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

SITE CLASS C

Ts = 0.55 s

(C2 = 1.0)

R=2.0

R = 8.0

R = 1.5

R = 1.5

R = 8.0

R = 1.5

R = 8.0

R = 8.0

R = 1.5

R = 8.0

R = 1.5

R = 1.5

R = 8.0

R = 8.0

R = 1.5

R = 1.5

R = 8.0
Chapter 3: Evaluation of Current Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-17

presented in this figure are for site class B. For periods 

of vibration larger that 1.0 s, the simplified method in 

FEMA 356 overestimates displacements by about 25%. 

For short periods of vibration, maximum displacements 

tend to be overestimated for small values of R and 

underestimated for large values of R. This trend is more 

pronounced when capping is included.

Figure 3-23 presents mean errors calculated from the 

ratio of the displacements computed using C1 and C2 as 

determined from FEMA 356 to maximum 

displacements computed with nonlinear responsehistory analyses for the SSD model. Results in this case 

correspond to site class C. The trends are in general 

similar to those presented in Figure 3-22; however, in 

this case overestimations are larger and 

underestimations are smaller.

3.4.4 P-∆ Effects (Coefficient C3)

The displacement modification factor C3 is intended to 

account for increased displacements due to dynamic P-

∆ effects. Displacement modification factors (C3) 

computed using Equation 3-10 of FEMA 356 are shown 

in Figure 3-24. Displacement amplifications increase as 

the post-yield negative stiffness ratio α decreases 

(becomes more negative), as R increases, and as the 

period of vibration decreases. 

In order to evaluate this coefficient, the model shown in 

Figure 3-25 was considered. Several studies have 

shown that systems with negative post-elastic stiffness 

may exhibit dynamic instability when subjected to 

earthquake ground motions (Jennings and Husid, 1968; 

Husid, 1969; Bernal 1987, 1992; MacRae, 1994; and 

Miranda and Akkar, 2003). An example from Miranda 

and Akkar (2003) is shown in Figure 3-26. In this 

figure, the ratio of maximum displacement of the 

system with negative post-yield stiffness to the 

maximum displacement in an elastic system is plotted 

for two systems with a period of 1.0 s as a function of R

when subjected to a recorded earthquake ground 

motion. The darker line represents a system with 

relatively severe negative post-elastic stiffness, while 

the light line represents a system with more moderate 

negative post-elastic stiffness. It can be seen that in the 

system with moderate negative stiffness (α = –0.06), R

Figure 3-19 A sample variation of C2 values in 

accordance with FEMA-356

Figure 3-20 Mean displacement ratio of SD to EPP 

models computed with ground motions 

recorded on site class D.

SITE CLASS B

0.0

1.0

2.0

3.0

0.0 0.5 1.0 1.5

PERIOD [s]

C2 FRAMING TYPE 1

Collapse Prevention Life Safety

Immediate Occupancy

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,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 6.0 R = 1.5

R = 1.5

R = 1.5

R = 6.0

Figure 3-21 Mean displacement ratio of SSD to EPP 

models computed with ground motions 

recorded on site classes B, C, and D.

SITE CLASSES B,C,D

(mean of 240 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

∆i,SSD / ∆i,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

R = 6.0

R = 1.5
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-18 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

can be increased to approximately 4 without any 

significant increase in lateral displacement. Note that α

is a ratio of the post-elastic stiffness to the elastic 

stiffness. Thus, a negative value of α indicates an 

effective decrease of strength with increasing 

displacement. If the lateral strength is further decreased 

(R is further increased), a large, abrupt increase in 

lateral displacements is produced, and soon after 

dynamic instability occurs. For the system with more 

severe negative stiffness (α = –0.21), R can only be 

increased to about 1.8. From this and other similar data, 

it is clear that systems that may exhibit negative 

stiffness need to have a minimum lateral strength (an R

smaller than a maximum critical value) in order to avoid 

collapse. Comparison of Figures 3-24 and 3-26 

illustrates that this phenomenon is not adequately 

captured by coefficient C3 in FEMA 356.

It should be noted that P-∆ effects are equivalent to a 

type of strength degradation that occurs in a single cycle 

(in-cycle) of vibratory motion. This differs from cyclic 

strength degradation that occurs in subsequent cycles 

modeled with the SSD type oscillator. These two types 

of strength degradation have different implications with 

respect to dynamic behavior. Further discussion of this 

subject is contained in Chapter 4. 

Figure 3-22 The mean error statistics associated with C1 and C2 assuming a Life Safety performance level in 

accordance with FEMA 356 for stiffness degrading (SD) systems.

Figure 3-23 The mean error statistics associated with C1 and C2 assuming a Collapse Prevention performance level in 

accordance with FEMA 356 for stiffness and strength (SSD) degrading systems.

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS B

Ts = 0.40 s

C2 as for Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS B

Ts = 0.40 s

C2 as for Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

R = 8.0

R = 1.5

R = 1.5

R = 8.0

R = 8.0

R = 1.5

R = 1.5

R = 8.0

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

R = 8.0

R = 1.5

R = 1.5

R = 8.0

R = 8.0

R = 1.5

R = 1.5

R = 8.0
Chapter 3: Evaluation of Current Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 3-19

3.5 Nonlinear Elastic Behavior

The results of the response-history analyses for the 

nonlinear elastic (NE) model are illustrated in 

Figure 3-27. Comparison with Figure 3-12 indicates 

that the maximum nonlinear elastic (NE) response is 

generally greater than the EPP. The difference varies 

with both period and strength and can exceed 40% in 

some cases. Neither ATC-40 nor FEMA 356 explicitly 

address nonlinear elastic behavior. In reality, it is not 

found often for typical structural systems. It represents 

a pure rocking response. Virtually all structures exhibit 

some hysteretic damping that tends to reduce response 

from that predicted for pure rocking.

Figure 3-24 The variation of C3 from FEMA 356 with 

respect to R for different negative postelastic stiffness values.

Figure 3-25 Bilinear system with in-cycle negative 

post-elastic stiffness due to P-∆ effects.

T = 1.0s

0

1

2

3

4

5

6

7

8

0123456

R

C3

α = − 0.21

α = − 0.06

F 

Fy

K 

αK 

∆y

∆si

∆

Figure 3-26 Displacement modification factors in 

SDOF that exhibit in-cycle negative postyield stiffness.

T = 1.0s

0

1

2

3

4

5

6

7

8

0123456

R

∆ i/∆ e

α = − 0.21

α = − 0.06

1984 Morgan Hill, California Earthquake

Gilroy #3, Sewage Treatment Plant, Comp. 0°
Chapter 3: Evaluation of Current Nonlinear Static Procedures

3-20 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure 3-27 Ratio of maximum displacement for a 

nonlinear elastic (NE) oscillator to elastic 

response for site classes B, C, and D.

SITE CLASS B

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,NLE

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASSES C

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,NLE

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASSES D

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,NLE

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

R = 6.0

R = 1.5

R = 6.0

R = 1.5

R = 6.0

R = 1.5
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 4-1 

4. Strength Degradation

Loss of lateral strength in structures during an 

earthquake is an issue of concern for engineers. In 

general, the nonlinear hysteretic characteristics of most 

buildings include both stiffness degradation and 

strength degradation to some extent. Strength 

degradation, including P-∆ effects, can lead to an 

apparent negative post-elastic stiffness in a forcedeformation relationship for a structural model using 

nonlinear static procedures. The performance 

implications depend on the type of strength 

degradation. For structures that are affected by 

component strength losses, including P-∆ effects, 

occurring in the same cycle as yielding, the negative 

post-elastic slope can lead to dynamic instability of the 

structural model. For this reason, a suggestion for a 

minimum strength for such structures is presented in 

Section 4.4

4.1 Types of Strength Degradation

Two types of strength degradation during hysteretic 

response are shown in Figure 4-1. Both oscillators 

exhibit inelastic stiffness and strength degradation. The 

oscillator in Figure 4-1a (cyclic strength degradation) 

maintains its strength during a given cycle of 

deformation, but loses strength in the subsequent 

cycles. The effective stiffness also decreases in the 

subsequent cycles. The slope of the post-elastic portion 

of the curve during any single cycle of deformation is 

not negative. Figure 4-1b (in-cycle strength 

degradation) illustrates a different type of strength 

degradation. Note that the degradation occurs during 

the same cycle of deformation in which yielding occurs, 

resulting in a negative post-elastic stiffness. This can be 

due to actual degradation in the properties of the 

component due to damage. It is also the consequence of 

P-∆ effects that increase demand on components and 

effectively reduce strength available to resist inertial 

loads. 

4.2 Strength Degradation and SDOF 

Performance

The strength and stiffness degrading (SSD) oscillators 

used to evaluate current nonlinear static procedures (see 

Section 3.2) were similar to those in Figure 4-1a. The 

results of the evaluation demonstrate that these cyclic 

strength-degrading oscillators often exhibit maximum 

displacements that are comparable with those that do 

not exhibit strength degradation. More importantly, 

responses are dynamically stable in general, even for 

relatively weak systems and large ductility.

The in-cycle strength-degrading counterpart discussed 

in Section 3.4.4, in contrast, can be prone to dynamic 

instability. Velocity pulses often associated with nearFigure 4-1 Two types of strength degradation.

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

Force

Strength loss occurs in subsequent cycles;

not in the same cycle as yield.

Strength loss occurs 

in same cycle as yield.

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

Force

b) In-cycle strength degradation

Strength loss occurs in subsequent cycles;

not in the same cycle as yield.

Strength loss occurs 

in same cycle as yield.

a) Cyclic strength degradation

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

Force

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

Force

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

Force

Strength loss occurs in subsequent cycles;

not in the same cycle as yield.

Strength loss occurs 

in same cycle as yield.

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

Force

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

Force

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400 Displacement

Force

b) In-cycle strength degradation

Strength loss occurs in subsequent cycles;

not in the same cycle as yield.

Strength loss occurs 

in same cycle as yield.

a) Cyclic strength degradation
Chapter 4: Strength Degradation

4-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

field ground motion records can exacerbate the 

problem. These pulses can drive the oscillator far into 

the post-elastic, strength-degrading branch in a single 

cycle of motion. 

4.3 Global Force-Deformation Behavior 

with Strength Degradation

In many structures, strength degradation is complex. A 

pushover curve for an example medium-rise reinforced 

concrete building is shown in Figure 4-2. There is an 

apparent negative post-elastic stiffness. This might be 

due to three effects. First, there could be cyclic (that is, 

from cycle to cycle) strength degradation associated 

with low-cycle fatigue damage of various components 

in the lateral-force-resisting system. Interspersed might 

be in-cycle strength losses due to component damage as 

deformations increase monotonically. Superimposed on 

this is the negative slope associated with P-∆ effects, 

which may or may not be significant. Unfortunately, it 

is not possible to distinguish between cyclic and incycle strength losses solely from information normally 

available from a nonlinear static analysis. The P-∆

effects are always present and contribute to real 

negative post-elastic stiffness. The P-∆ effects are 

simple to separate from the others. Precise separation 

of the remaining constituents of strength degradation 

cannot be inferred directly, since the distribution 

depends on the nature of individual ground motions and 

the sequence of inelastic behavior among the various 

components as a lateral mechanism develops.

For purposes of nonlinear static analysis, the calculated 

relationship between base shear and displacement of a 

control node (e.g. roof) may be replaced with an 

idealized relationship to calculate the effective lateral 

stiffness (Ke), effective yield strength (Vy), and effective 

positive (α1) and/or negative (α2) stiffnesses of the 

building model, as shown in Figure 4-3. The initial 

linear portion of the idealized force-displacement curve 

begins at the origin. A second linear portion ends at a 

point on the calculated force-displacement curve at the 

calculated target displacement, or the point of 

maximum base shear (Vd), whichever is least. The 

intersection of the two idealized segments defines 

effective lateral stiffness (Ke), the effective yield 

strength (Vy), and effective positive post-yield stiffness 

(α1 Ke). The intersection point is determined by 

satisfying two constraints. First, the effective stiffness, 

Ke, must be such that the first segment passes through 

the calculated curve at a point where the base shear is 

60% of the effective yield strength. Second, the areas 

above and below the calculated curve should be 

approximately equal. For models that exhibit negative 

post-elastic stiffness, a third idealized segment can be 

determined by the point of maximum base shear on the 

calculated force-displacement curve and the point at 

which the base shear degrades to 60% of the effective 

yield strength [the same strength that was used to 

establish Ke]. This segment defines the maximum 

negative post-elastic stiffness (α2 Ke). This negative 

slope approximates the effects of cyclic and in-cycle 

degradation of strength. Note that the selection of 60% 

of the yield strength to define this slope is based purely 

on judgement.

As noted, nonlinear static procedures are not capable of 

distinguishing completely between cyclic and in-cycle 

strength losses. However, insight can be gained by 

separating the in-cycle P-∆ effects from α2 (see 

Figure 4-3). An effective post-elastic stiffness can then 

be determined as

Figure 4-2 Example capacity curve for a medium rise concrete structure

Degradation of strength 

including both cyclic and incycle losses

Displacement

Base shear
Chapter 4: Strength Degradation

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 4-3

(4-1)

where . 

Current knowledge of component behavior as well as 

unknown characteristics of the future ground motion 

make it impossible at present to know the correct value 

of λ. For the present, it is recommended that λ be 

assigned a value of 0.2 for sites not subject to near field 

effects and 0.8 for those that are. These values, solely 

based on judgment, are intended to recognize the 

potential for dynamic instability that might arise from 

in-cycle strength losses associated with large impulsive 

near-field motions while, at the same time, avoid 

penalizing structures with predominantly cyclic 

strength loss associated with non-impulsive motions.

4.4 Limitation on Strength for In-Cycle 

Strength Degradation Including P-∆

Effects

When using displacement modification techniques 

similar to the coefficient method of FEMA 356, it is 

recommended that the displacement prediction be 

modified to account for cyclic degradation of stiffness 

and strength. Chapter 5 presents an improved 

procedure for calculating the coefficient C2 for this 

purpose. It is also suggested that the current coefficient 

C3 be eliminated and replaced with a limit on minimum 

strength (maximum value of R) required to avoid 

dynamic instability. The same limitation on Rmax is 

recommended for the equivalent linearization 

alternative in ATC-40 as modified in Chapter 6 of this 

document.

The recommended limitation on the design force 

reduction, Rmax, is as follows (see also Figure 4-3 for 

notation):

(4-2)

where

(4-3)

If this limitation is not satisfied, then a nonlinear 

dynamic analysis using representative ground motion 

records for the site should be implemented to 

investigate the potential for dynamic instability. The 

structural model must appropriately model the strength 

degradation characteristics of the structure and its 

components. 

Equation 4-2 is a simplification of an expression 

derived by Miranda and Akkar (2003), which was 

obtained using single-degree-of-freedom systems. It 

should be noted that significant variability exists in the 

strength required to avoid dynamic instability; hence, 

this equation is aimed only at identifying cases where 

dynamic instability should be further investigated using 

response history analyses and not as an accurate 

measure of the lateral strength required to avoid 

dynamic instability in MDOF structures.

The use of the equivalent linearization techniques (see 

Chapter 6) can provide initial insight into whether the 

α α λα α ( ) e P 2P =+− −∆ −∆

0 1.0 ≤ ≤ λ

max 4

α −

∆

= +

∆

t

d e

y

R

t lnT =1 0.15 +

Figure 4-3 Idealized force-displacement curve for nonlinear static analysis

Displacement

Base shear

Ke

α1Ke

α2Ke

αe e K

α P e −∆K

y ∆d ∆

Vd

Vy

0.6Vy

Actual force-displacement 

curve

Displacement

Base shear

Ke

α1Ke

α2Ke

αe e K

α P e −∆K

y ∆d ∆

Vd

Vy

0.6Vy

Actual force-displacement 

curve
Chapter 4: Strength Degradation

4-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

nonlinear dynamic analysis is worthwhile. In particular, 

solution procedure C produces a locus of potential 

performance points. If this locus tends to be parallel to 

and above the capacity curve, then dynamic instability 

is indicated according to that procedure. However, if the 

locus intersects the capacity curve, instability is not 

indicated; nonlinear dynamic analysis may be fruitful in 

demonstrating this stability.
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 5-1 

5. Improved Procedures for Displacement 

Modification

5.1 Introduction

Based on the evaluation summarized in Chapter 3 and 

available research data, suggested improvements to the 

Coefficient Method of FEMA 356 have been developed 

and are presented in this chapter. Recommendations 

include several improved alternatives for the basic ratio 

of the maximum displacement (elastic plus inelastic) for 

an elastic perfectly plastic SDOF oscillator to the 

maximum displacement for a completely linear elastic 

oscillator that is designated as the coefficient C1 in 

FEMA 356. This chapter also recommends that the 

current limitations (capping) allowed by FEMA 356 to 

the coefficient C1 be abandoned. In addition, a 

distinction is recognized between two different types of 

strength degradation that have different effects on 

system response and performance, as discussed in 

Chapter 4. This distinction leads to recommendations 

for the coefficient C2 to account for cyclic degradation 

in strength and stiffness. It is also suggested that the 

coefficient C3 be eliminated and replaced with a 

limitation on strength in accordance with Section 4.4.

The proposed expressions for coefficients in this section 

are based on empirical data. They have been formulated 

to provide estimates of expected values based on 

available analytical results on the response of SDOF 

oscillators subjected to ground motion records. As 

noted in the subsequent text, there is dispersion, at times 

large, in the data. The user should be cognizant of this 

when applying these procedures in practice. 

5.2 Maximum Displacement Ratio 

(Coefficient C1)

The coefficient C1 in FEMA 356 is used along with 

other coefficients in a nonlinear static procedure known 

as the Coefficient Method. This form of displacement 

modification is described in more detail in Section 3.4.1 

of this document and in Chapter 3 of FEMA 356. As a 

result of the work summarized in Chapter 3 and a 

review of available pertinent research, improvements to 

the coefficient C1 can be made. A relatively simple 

expression is proposed here. As noted in Section 3.4.1, 

FEMA 356 currently allows the coefficient C1 to be 

limited (capped) for relatively short-period structures. It 

is suggested that this limitation not be used. This may 

increase estimates of displacement for some structures. 

However, Chapter 8 presents rational procedures to 

account for some of the characteristics of short-period 

structures that may reduce their response to ground 

motions in lieu of the current limitations on the 

coefficient C1.

5.2.1 Simplified Expression

For most structures the following simplified expression 

may be used for the coefficient C1:

(5-1)

where Te is the effective fundamental period of the 

SDOF model of the structure in seconds and R is the 

strength ratio computed with Equation 3-16 of the 

FEMA 356 document. The constant a is equal to 130, 

90, and 60 for site classes B, C, and D, respectively. For 

periods less than 0.2 s, the value of the coefficient C1

for 0.2 s may be used. For periods greater than 1.0 s, C1

may be assumed to be 1.0. 

This expression provides improved estimation of the 

ratio of peak deformations of inelastic SDOF systems 

with elasto-plastic behavior to peak deformations of 

linear single-degree-of-freedom systems. Equation 5-1 

is plotted in Figure 5-1. This equation estimates mean 

values of this ratio. Considerable dispersion (scatter) 

exists about the mean. For information and discussion 

of the dispersion of C1 see Ruiz-Garcia and Miranda 

(2003). When interpreting results and assessing 

structural performance, engineers should consider the 

implications of these uncertainties. For example, the 

expression can be used with a = 60 for softer sites (class 

E and F) to estimate displacements, but it is less reliable 

due to very high dispersion of results in studies of 

SDOF oscillators for soft sites. Similarly, this equation 

may not provide completely adequate results for ground 

motions strongly influenced by forward directivity 

effects, for the same reason. 

Systems with nonlinear elastic hysteretic behavior (e.g. 

rocking) can have deformation ratios larger than those 

computed with Equation 5-1. Results of the studies for 

nonlinear elastic systems (NE) summarized in 

Section 3.5 indicate that these oscillators can exhibit 

displacements up to 40% larger than their elasto-plastic 

counterparts. However, most systems that exhibit 

rocking also have some hysteretic energy dissipation (as 

C R

aTe

1 2 1 1

= + −
Chapter 5: Improved Procedures for Displacement Modification

5-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

opposed to the “pure” rocking of the NE oscillator) that 

would likely reduce this tendency. Specific 

recommendations cannot be made at this point and 

further study is warranted.

Recently, various studies have proposed simplified 

expressions for C1. Figure 5-2 compares the C1

computed with Equation 5-1 assuming site class C to 

that proposed by other investigators (Aydinoglu and 

Kacmaz, 2002; Ramirez et al., 2002; Ruiz-Garcia and 

Miranda, 2003; Chopra and Chintanapakdee, 2003). 

With exception of the study by Ramirez et al., all 

deformation ratios plotted in Figure 5-2 are for EPP 

hysteretic behavior. Deformation ratios by Ramirez et 

al. shown in Figure 5-2 were computed using constants 

recommended for systems with post-elastic stiffnesses 

of 5% of the elastic. The simplified equation proposed 

here leads to results that are similar to those of previous 

investigations.

5.2.2 Limits on Maximum Displacements for 

Short Periods

FEMA 356 currently contains a limitation (cap) on the 

maximum value of the coefficient C1 as described in 

Section 3.4.1. As noted in Appendix B, the limitation is 

used by many engineers. The evaluation of the 

Coefficient Method in Chapter 3 demonstrates that the 

limitation contributes to inaccuracy in the prediction of 

maximum displacements. The authors of FEMA 356

included the limitations for two related reasons. First, 

there is a belief in the practicing engineering 

community that short, stiff buildings do not respond to 

seismic shaking as adversely as might be predicted 

using simplified analytical models. Indeed, there may 

be logical explanations for this phenomenon, including 

various aspects of soil-structure interaction. These 

factors are often cited qualitatively, along with the 

observed good performance of such buildings in past 

earthquakes, as justification for less onerous demand 

parameters in codes and analytical procedures. 

Traditional design procedures have evolved 

accordingly, giving rise to a second reason. The authors 

of FEMA 356 felt that the required use of the empirical 

equation without relief in the short-period range would 

motivate practitioners to revert to the more traditional, 

and apparently less conservative, linear procedures. 

FEMA 357, Global Topics Report on the Prestandard 

and Commentary for the Seismic Rehabilitation of 

Buildings (ASCE, 2000b), has a discussion of the issue 

and addresses the concern about the limitations 

(capping) on C1 and the potential for underestimating 

the displacement response of weak structures. 

In an effort to deal more logically with the 

characteristics of short-period structures that may 

reduce their response to strong ground motions from 

that predicted by current analysis procedures, this 

document includes the development of rational 

procedures in Chapter 8. It is suggested that these be 

used in lieu of the limitation in FEMA 356 to estimate 

the response of short-period structures.

Figure 5-1 Expression for coefficient C1 (Eqn.5-1 with a = 90 for site class C) and current FEMA 356 expression. 

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Period, T(sec)

C1

Eq.5-1,R=2

Eq.5-1,R=4

Eq.5-1,R=6

FEMA 356, R=2

FEMA 356, R=4

FEMA 356, R=6

FEMA 356, capped

R=2

R=4

R=6

FEMA, R=2

FEMA, R=4

FEMA, R=6

FEMA, 

capped
Chapter 5: Improved Procedures for Displacement Modification

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 5-3

5.3 Adjustment for Cyclic Degradation 

(Coefficient C2) 

As discussed in Chapter 4, two types of degradation of 

stiffness and/or strength can affect response. Also, the 

effects of each type differ from one another. For the 

purposes of displacement modification procedures in 

accordance with FEMA 356, it is suggested that the C2

coefficient represent the effects of stiffness degradation 

only. The effects of strength degradation are addressed 

by the suggested limitation presented in Chapter 4. It is 

recommended that the C2 coefficient be as follows:

(5-2)

For periods less than 0.2 s, the value of the coefficient 

C2 for 0.2 s may be used. For periods greater than 0.7 

sec, C2 may be assumed equal to 1.0. The expression is 

plotted in Figure 5-3. The coefficient C2 need only be 

applied to structures that exhibit significant stiffness 

and/or strength degradation.

The degree by which deformation demands are 

increased by cyclic degradation depends on the 

characteristics of the hysteretic behavior, which are 

very sensitive to the structural material, detailing, and 

ground motion characteristics. Because of the many 

parameters involved, it is difficult to capture the effects 

of all possible types of cyclic degradation with a single 

modifying factor. Equation 5-2 represents a 

simplification and interpretation of many statistical 

results with various kinds of cyclically degrading 

systems. The dispersion of results of SDOF oscillator 

studies used to formulate the C2 factor is larger than that 

of the C1 factor. It is important to consider this large 

dispersion when interpreting the results obtained from 

simplified procedures recommended in this document, 

particularly for structures with periods of vibration 

smaller than 0.5s.

Figure 5-2 Comparison of alternative expressions for 

the coefficient C1 for R = 4 and R = 6 for 

site class C.

R = 4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

PERIOD [s]

C 1

 Equation 5-1 for site class C

 Aydinoglu and Kacmaz

 Chopra and Chintanapakdee

 Ramirez et al.

 Ruiz-Garcia and Miranda

R = 6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

PERIOD [s]

C 1

 Equation 5-1 for site class C

 Aydinoglu and Kacmaz

 Chopra and Chintanapakdee

 Ramirez et al.

 Ruiz-Garcia and Miranda

Figure 5-3 Coefficient C2 from Eq. 4-2 and FEMA 

356 for site classes B, C, and D. 

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0

Period, T (sec)

C 2

 Equation 5-2, R=6

 Equation 5-2, R=4

FEMA 356 Collapse Prevention (CP)

FEMA 356 Life Safety LS)

R=4

R=6

FEMA 356, CP

FEMA 356, LS

FEMA 356, CP

2

2

1 1

1

800

R

C

T

− = + ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
Chapter 5: Improved Procedures for Displacement Modification

5-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

5.4 Limitation on Strength to Avoid 

Dynamic Instability for Nonlinear 

Static Procedures

The studies of the Coefficient Method in Chapter 3 

indicate that global displacement demand is not 

significantly amplified by degrading strength until a 

critical point at which dynamic instability may occur. 

This point is related to the initial strength and period of 

the oscillator as well as the magnitude of the negative 

post-elastic stiffness caused by in-cycle strength 

degradation.

It is suggested that the current coefficient C3 be 

eliminated and replaced with a limit on minimum 

strength (maximum R) required to avoid dynamic 

instability. The proposed limitation is presented in 

Section 4.4.
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 6-1 

6. Improved Procedures for Equivalent Linearization

6.1 Introduction

This chapter presents an improved equivalent 

linearization procedure as a modification to the 

Capacity-Spectrum Method (CSM) of ATC-40. The 

CSM is a form of equivalent linearization briefly 

summarized in Sections 2.4 and 3.3.1. Detailed 

information on equivalent linearization in general and 

the derivation of the improved procedures are included 

in Appendix D. 

When equivalent linearization is used as a part of a 

nonlinear static procedure that models the nonlinear 

response of a building with a SDOF oscillator, the 

objective is to estimate the maximum displacement 

response of the nonlinear system with an “equivalent” 

linear system using an effective period, Teff, and 

effective damping, βeff (see Figure 6-1). The global 

force-deformation relationship shown in Figure 6-1 for 

a SDOF oscillator in acceleration-displacement 

response spectrum (ADRS) format is termed a capacity 

curve. The capacity curve shown in Figure 6-1 is 

developed using the conventional procedures of FEMA 

356 or ATC-40. The effective linear parameters are 

functions of the characteristics of the capacity curve, 

the corresponding initial period and damping, and the 

ductility demand, μ, as specified in the following 

sections.

Recommendations for the improved equivalent 

linearization procedures rely on the previous procedures 

in ATC-40, and much of the process remains the same. 

This chapter focuses on the parts that change. The 

following section presents new expressions to 

determine effective period and effective damping. It 

also includes a technique to modify the resulting 

demand spectrum to coincide with the familiar CSM 

technique of using the intersection of the modified 

demand with the capacity curve to generate a 

performance point for the structural model. The 

reduction in the initial demand spectrum resulting from 

the effective damping may be determined using 

conventional techniques outlined in Section 6.3. The 

previous limits on effective damping of ATC-40 should 

not be applied to these new procedures. However, the 

user must recognize that the results are an estimate of 

median response and imply no factor of safety for 

structures that may exhibit poor performance and/or 

large uncertainty in behavior. The effective parameters 

for equivalent linearization are functions of ductility. 

Since ductility (the ratio of maximum displacement to 

yield displacement) is the object of the analysis, the 

solution must be found using iterative or graphical 

techniques. Three of these are presented in Section 6.4. 

They have been developed to be similar to those of 

ATC-40.

Finally, it should be noted that these procedures may 

not be reliable for extremely high ductilities (e.g., 

greater than 10 to 12).

Figure 6-1 Acceleration-displacement response spectrum (ADRS) showing effective period and damping parameters 

of equivalent linear system, along with a capacity curve.

Sa

Sd dmax

amax capacity spectrum

βeff (CS,β 0,μ )

Teff (CS,T0,μ )

Initial elastic demand with 

damping = β0

dy

ductility μ = dmax/dy

T0

Spectral Acceleration

Spectral Displacement

Sa

Sd dmax

amax capacity spectrum

βeff (CS,β 0,μ )

Teff (CS,T0,μ )

Initial elastic demand with 

damping = β0

dy

ductility μ = dmax/dy

T0

Spectral Acceleration

Spectral Displacement
Chapter 6: Improved Procedures for Equivalent Linearization

6-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

6.2 Basic Equivalent Linearization 

Parameters 

Optimal equivalent linear parameters (i.e., effective 

period, Teff, and effective damping, βeff) are determined 

through a statistical analysis that minimizes, in a 

rigorous manner, the extreme occurrences of the 

difference (i.e., error) between the maximum response 

of an actual inelastic system and its equivalent linear 

counterpart. Conventionally, the measurement of error 

has been the mean of the absolute difference between 

the displacements. Although this seems logical, it might 

not lead to particularly good results from an engineering 

standpoint in which the difference between 

conservative or unconservative estimates is important. 

This is illustrated in Figure 6-2. It is possible to select 

linear parameters for which the mean error is zero, as 

for the broad, flat distribution. However, the narrower 

curve might represent equivalent linear parameters that 

provide better results from an engineering standpoint, 

since the chance of errors outside a –10% to +10% 

range, for example, are much lower, even accounting 

for the –5% mean error. This is owing to the smaller 

standard deviation. See Appendix D for details on the 

optimization process. 

A variety of different inelastic hysteretic systems have 

been studied including bilinear hysteretic (BLH), 

stiffness- degrading (STDG), and strength-degrading 

behavior as shown in Figure 6-3. Note that the bilinear 

model (BLH) is the same as the elastic perfectly plastic 

(EPP) discussed in Chapter 3. Similarly, the stiffness 

degrading model (STDG) is the same as the SD model 

in Chapter 3. In contrast, the strength-degrading model 

(STRDG) differs from the SSD model of Chapter 3. A 

negative value of the post-elastic stiffness ratio, α, is 

indicative of in-cycle degradation (see Chapter 4). Also 

included are parameters that have been optimized for all 

types of behavior.

6.2.1 Effective Damping

Effective viscous damping values, expressed as a 

percentage of critical damping, for all hysteretic model 

types and alpha values have the following form:

For 1.0 < μ < 4.0:

(6-1)

Figure 6-2 Illustration of probability density function 

of displacement error for a Gaussian 

distribution.

βμ μβ eff = − A B ( 1 1 ) + − ( ) +

2 3

0

Figure 6-3 Types of inelastic behavior considered. BLH=Bilinear Hysteretic STDG=Stiffness Degrading, and 

STRDG=Strength Degrading.

BLH (α=0%) STDG (α=0%) STRDG (α=−5%)
Chapter 6: Improved Procedures for Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 6-3

For 4.0 ≤ μ ≤ 6.5:

(6-2)

For μ > 6.5:

 (6-3)

Values of the coefficients in the equations for effective 

damping of the model oscillators are tabulated in 

Table 6-1. Note that these are a function of the 

characteristics of the capacity curve for the oscillator in 

terms of basic hysteretic type and post-elastic stiffness, 

α.

The coefficients in Table 6-1 have been optimized to fit 

the empirical results for idealized model oscillators 

having well defined hysteretic behavior designated 

earlier in this document as Elastic Perfectly Plastic 

(EPP), Stiffness Degrading (SD) and Strength and 

Stiffness Degrading (SSD). Real buildings, comprised 

of a combination of many elements, each of which may 

have somewhat different strength and stiffness 

characteristics, will seldom display hysteretic behaviors 

that match those of the oscillators, exactly. Adaptation 

of these coefficients to building models with a number 

of components may be done with caution. If all 

components exhibit similar behavior (e.g., flexurally 

controlled concrete with stiffness degradation and strain 

hardening), then it is reasonable to infer that hysteretic 

behavior of the overall building will be similar to the 

behavior of the simple idealized oscillators on which 

this table is based. For building models in which 

components exhibit disparate force-deformation 

behavior, it is less clear which coefficients to use. When 

in doubt, the practitioner should use the more generally 

optimized equations presented in the following 

paragraph.

The following approximate equations for the effective 

damping value have been optimized for application to 

any capacity curve, independent of hysteretic model 

type or alpha value used for the study:

For 1.0 < μ < 4.0:

(6-4)

For 4.0 ≤ μ ≤ 6.5:

(6-5)

β μβ eff =+ − C D( 1) + 0

β μ

μ

eff

eff = ( − ) −

( − )]

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎛

⎝

⎜

⎞

⎠

⎟ E +

F

[F

T

T

1 1

1 2

0

2

β0

βeff = − 49 1 11 1 (μ μ ) − − ( ) + β 2 3

0 . .

β μβ eff =+ − 14 0 0 32 1 ( ) + 0 . .

Table 6-1 Coefficients for use in Equations for Effective Damping

Model α (%) A B C D E F

Bilinear hysteretic 0 3.2 –0.66 11 0.12 19 0.73

Bilinear hysteretic 2 3.3 –0.64 9.4 1.1 19 0.42

Bilinear hysteretic 5 4.2 –0.83 10 1.6 22 0.40

Bilinear hysteretic 10 5.1 –1.1 12 1.6 24 0.36

Bilinear hysteretic 20 4.6 –0.99 12 1.1 25 0.37

Stiffness degrading 0 5.1 –1.1 12 1.4 20 0.62

Stiffness degrading 2 5.3 –1.2 11 1.6 20 0.51

Stiffness degrading 5 5.6 –1.3 10 1.8 20 0.38

Stiffness degrading 10 5.3 –1.2 9.2 1.9 21 0.37

Stiffness degrading 20 4.6 –1.0 9.6 1.3 23 0.34

Strength degrading -3a 5.3 –1.2 14 0.69 24 0.90

Strength degrading –5a 5.6 –1.3 14 0.61 22 0.90

a. Negative values of post-elastic stiffness should be limited to αe, as discussed in Section 4.3
Chapter 6: Improved Procedures for Equivalent Linearization

6-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

For μ > 6.5:

(6-6)

6.2.2 Effective Period 

Effective period values for all hysteretic model types 

and alpha values have the following form:

For 1.0 < μ < 4.0:

 (6-7)

For 4.0 ≤ μ ≤ 6.5:

 (6-8)

For μ > 6.5:

(6-9)

Values of the coefficients in the equations for effective 

period of the model oscillators are tabulated in 

Table 6-2. Note that these are a function of the 

characteristics of the capacity spectrum for the 

oscillator in terms of basic hysteretic type and postelastic stiffness, α.

The use of these coefficients in Table 6-2 for actual 

buildings is subject to the same limitations as for 

effective damping, as discussed in Section 6.2.1. When 

in doubt, the practitioner should use the following 

equations for the effective period value that have been 

optimized for application to any capacity spectrum, 

independent of the hysteretic model type or alpha value: 

For 1.0 < μ < 4.0:

(6-10)

For 4.0 ≤ μ ≤ 6.5:

(6-11)

For μ > 6.5:

(6-12)

Note that these expressions apply only for T0 = 0.2 to 

2.0 s. 

Table 6-2 Coefficients for use in Equations for Effective Period

Model α(%) G H I J K L

Bilinear hysteretic 0 0.11 –0.017 0.27 0.090 0.57 0.00

Bilinear hysteretic 2 0.10 –0.014 0.17 0.12 0.67 0.02

Bilinear hysteretic 5 0.11 –0.018 0.09 0.14 0.77 0.05

Bilinear hysteretic 10 0.13 –0.022 0.27 0.10 0.87 0.10

Bilinear hysteretic 20 0.10 –0.015 0.17 0.094 0.98 0.20

Stiffness degrading 0 0.17 –0.032 0.10 0.19 0.85 0.00

Stiffness degrading 2 0.18 –0.034 0.22 0.16 0.88 0.02

Stiffness degrading 5 0.18 –0.037 0.15 0.16 0.92 0.05

Stiffness degrading 10 0.17 –0.034 0.26 0.12 0.97 0.10

Stiffness degrading 20 0.13 –0.027 0.11 0.11 1.0 0.20

Strength degrading –3a 0.18 –0.033 0.17 0.18 0.76 –0.03

Strength degrading –5a 0.20 –0.038 0.25 0.17 0.71 –0.05

a.Negative values of post-elastic stiffness may be limited to αe, as discussed in Section 4.3

β μ

μ

eff = ( − ) −

⎡ ( − ) ⎣ ⎤

⎦

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ 19

0 64 1 1

0 64 1 2

.

.

T

T

eff

0

2

0

⎛

⎝

⎜

⎞

⎠

⎟ + β

TG H T eff = − ( ) + − ( ) + [ ] μ μ 1 11 2 3

0

T IJ T eff =+ − [ ] (μ 1 1 ) + 0

T K

L eff = T −

+ − ( ) − ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

( ) μ

μ

1

1 2

1 1 0

T T eff = − { } 0 20 1 0 038 1 1 ( ) − − ( ) +

2 3

0 . . μ μ

T T eff =+ − ⎡ ( ) + ⎣ ⎤

⎦ 0 28 0 13 1 1 0 . . μ

Teff = −

+ − ( ) − ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

+

⎧

⎨

⎪

⎩⎪

⎫

0 89 1

1 0 05 2 . 1 1 ( )

.

μ

μ

⎬

⎪

⎭⎪

T0
Chapter 6: Improved Procedures for Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 6-5

6.2.3 MADRS for Use with Secant Period 

The conventional Capacity-Spectrum Method (ATC-40) 

uses the secant period as the effective linear period in 

determining the maximum displacement (performance 

point). This assumption results in the maximum 

displacement occurring at the intersection of the 

capacity curve for the structure and a demand curve for 

the effective damping in ADRS format. This feature is 

useful for two reasons. First, it provides the engineer 

with a visualization tool by facilitating a direct 

graphical comparison of capacity and demand. Second, 

there are very effective solution strategies for 

equivalent linearization that rely on a modified ADRS 

demand curve (MADRS) that intersects the capacity 

curve at the maximum displacement. 

The use of the effective period and damping equations 

in Sections 6.2.1 and 6.2.2 generate a maximum 

displacement that coincides with the intersection of the 

radial effective period line and the ADRS demand for 

the effective damping (see Figure 6-4). The effective 

period of the improved procedure, Teff, is generally 

shorter than the secant period, Tsec, defined by the point 

on the capacity curve corresponding to the maximum 

displacement, dmax. The effective acceleration, aeff, is 

not meaningful since the actual maximum acceleration, 

amax, must lie on the capacity curve and coincide with 

the maximum displacement, dmax. Multiplying the 

ordinates of the ADRS demand corresponding to the 

effective damping, βeff, by the modification factor

(6-13)

results in the modified ADRS demand curve (MADRS) 

that may now intersect the capacity curve at the 

performance point. Since the acceleration values are 

directly related to the corresponding periods, the 

modification factor can be calculated as: 

, (6-14)

using the equations in Section 6.2.2 for the effective 

period and recognizing from Equation 3-5 that

(6-15)

where α is the post-elastic stiffness from Equation 6-18.

6.3 Spectral Reduction for Effective 

Damping

Equivalent linearization procedures applied in practice 

normally require the use of spectral reduction factors to 

adjust an initial response spectrum to the appropriate 

level of effective damping, βeff. They are also a 

practical way to adjust for foundation damping as 

presented in Chapter 8. In the case of foundation 

damping, the initial damping value, β0, for a flexiblebase structural model is modified from the fixed-base 

linear value, βi

 (e.g., 5%). These factors are a function 

of the effective damping and are termed damping 

coefficients, B(βeff). They are used to adjust spectral 

acceleration ordinates as follows: 

(6-16)

There are a number of options in current procedures for 

determining B(βeff). Some of these are plotted in 

Figure 6-5. Also shown in the figure is the following 

expression: 

(6-17)

This simple expression is very close to equations 

specified in both the NEHRP Recommended Provisions 

for Seismic Regulations for New Buildings and Other 

Structures and the ATC-40 document. It is suggested 

that Equation 6-17 replace the current specifications. 

Figure 6-4 Modified acceleration-displacement 

response spectrum (MADRS) for use with 

secant period, Tsec.

Sa

Sd

MADRS (β eff ,M)

Tsec

dmax

amax

capacity curve

ADRS (β eff)

Teff

aeff

ADRS (β0)

Spectral Acceleration

Spectral Displacement

Sa

Sd

MADRS (β eff ,M)

Tsec

dmax

amax

capacity curve

ADRS (β eff)

Teff

aeff

ADRS (β0)

Spectral Acceleration

Spectral Displacement

M a

a = max

eff

M T

T

T

T

T

T = ⎛

⎝

⎜ ⎞

⎠

⎟ = ⎛

⎝

⎜ ⎞

⎠

⎟ ⎛

⎝

⎜ ⎞

⎠

⎟ eff

sec

eff

0

0

sec

2 2 2

( ) 2

0

sec

T 1 1

T

α μ

μ

⎛ ⎞ + −

⎜ ⎟ =

⎝ ⎠

S

S

B a

a ( ) = ( )

( ) β β

0

eff

B = − ( )

4

5 6. ln βeff in %
Chapter 6: Improved Procedures for Equivalent Linearization

6-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Note that if the ATC-40 equations are used, then the 

limits on the reduction should not be applied.

6.4 Solution Procedures

Since the effective period, Teff, and effective damping, 

βeff, are both functions of ductility demand, the 

calculation of a maximum displacement using 

equivalent linearization is not direct and requires an 

iterative or graphical solution procedure. This is the 

same as the previous situation with the CapacitySpectrum Method of ATC-40. This section presents 

three alternate procedures. Other procedures are 

possible.

All of the solution procedures presented here require 

initial steps listed below. 

1. Select a spectral representation of the ground 

motion of interest with an initial damping, βi (normally 5%). This may be a design spectrum from 

ATC-40 or FEMA 356, a site-specific deterministic 

spectrum, or an equal hazard probabilistic spectrum. 

2. Modify the selected spectrum, as appropriate, for 

soil-structure interaction (SSI) in accordance with 

the procedures in Chapter 9. This involves both 

potential reduction in spectral ordinates for kinematic interaction and a modification in the system 

damping from the initial value, βi to β0, to account 

for foundation damping. If foundation damping is 

ignored, β0 is equal to βi

.

3. Convert the selected spectrum, modified for SSI 

when appropriate, to an acceleration-displacement 

response spectrum format in accordance with the 

guidance in ATC-40. This spectrum is the initial 

ADRS demand (see Figure 6-6).

4. Generate a capacity curve for the structure to be 

analyzed. This is a fundamental relationship for a 

SDOF model of the structure between spectral 

acceleration and spectral displacement (see 

Figure 6-6). Detailed guidance is available in 

ATC-40 and FEMA 356. Note that the FEMA 356 

procedures result in a relationship between base 

shear and roof displacement. This requires conversion to ADRS format for equivalent linearization 

procedures (see ATC-40). 

Figure 6-5 Damping coefficients, B, as a function of damping, βeff, from various resource documents.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0 10 20 30 40 50

2000 NEHRP Table 13A.3.1

FEMA 356 Table 1.6

ATC 40 Eqn 8-10 (no limits)

FEMA 440 Eqn 6-17

t nei ciff e

oc gni p

m

a

D

Damping in percent β eff

B(β eff)
Chapter 6: Improved Procedures for Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 6-7

5. Select an initial performance point (maximum 

acceleration, api, and displacement, dpi). This may 

be based on an equal-displacement approximation 

as shown in Figure 6-6 or any other point based on 

engineering judgment.

6. Develop a bilinear representation of the capacity 

spectrum in accordance with the procedures in 

ATC-40. This defines the initial period, T0, yield 

displacement, dy, and yield acceleration, ay. (see 

Figure 6-7). Note that these parameters may vary 

for differing assumptions api and dpi

7. For the bilinear representation developed in Step 6, 

calculate the values of post-elastic stiffness, α, and 

ductility, μ, as follows:

(6-18)

(6-19)

8. Using the calculated values for post-elastic stiffness, α, and ductility, μ, from Step 7, calculate the 

corresponding effective damping, βeff, (see 

Section 6.2.1). Similarly calculate the corresponding effective period, Teff, (see Section 6.2.2).

After this step in the procedures, a number of options 

are available for identifying a single solution. Three 

possible procedures are described below. 

Procedure A (Direct Iteration). In this procedure, the 

iteration is done to converge directly on a performance 

point. The ADRS demand spectra generated for the 

various values of effective damping are not modified to 

intersect the capacity spectrum, as outlined in 

Section 6.2.3.

A9. Using the effective damping determined from 

Step 8, adjust the initial ADRS to βeff (see 

Section 6.3). 

A10. Determine the estimated maximum displacement, di

, using the intersection of the radial 

effective period, Teff, with the ADRS for βeff. 

The estimated maximum acceleration, ai

, is that 

corresponding to di on the capacity curve (see 

Figure 6-8). 

A11. Compare the estimated maximum displacement,

di

, with the initial (or previous) assumption. If it 

is within acceptable tolerance, the performance 

point corresponds to ai

 and di. If it is not within 

acceptable tolerance, then repeat the process 

from Step 5 using ai

 and di, or some other 

selected assumption (see Section 6.6), as a starting point.

Procedure B (Intersection with MADRS). In this 

procedure, the performance point is defined as the 

intersection of the capacity spectrum with the modified 

ADRS (MADRS). The MADRS demand spectrum is 

generated by modifying the ADRS for the various 

values of effective damping, as outlined in 

Section 6.2.3.

Figure 6-6 Initial ADRS demand and capacity 

spectrum. 

Figure 6-7 Bilinear representation of capacity 

spectrum. 

noit ar el eccAl art cepS

Spectral Displacement

capacity curve for 

structure

initial ADRS demand 

with damping β o

Sa

Sd

dpi

api

equal displacement approximation 

(arbitrary initial assumption)

noit ar el eccAl art cepS

Spectral Displacement

capacity curve for 

structure

initial ADRS demand 

with damping β o

Sa

Sd

dpi

api

equal displacement approximation 

(arbitrary initial assumption)

noit ar el eccAl art cepS

Spectral Displacement

capacity curve for 

structure

Sa

Sd

dpi

api

dy

ay

T0

bilinear representation 

of capacity curve

noit ar el eccAl art cepS

Spectral Displacement

capacity curve for 

structure

Sa

Sd

dpi

api

dy

ay

T0

bilinear representation 

of capacity curve

α =

−

−

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

a a

d d

a

d

pi y

pi y

y

y

μ = d pi

dy
Chapter 6: Improved Procedures for Equivalent Linearization

6-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

B9. Using the effective damping determined from 

Step 8, adjust the initial ADRS to βeff (see 

Section 6.3). 

B10. Multiply the acceleration ordinates only (i.e., not 

the displacement ordinates) of the ADRS for βeff

by the modification factor, M, determined using 

the calculated effective period, Teff, in accordance with Section 6.2.3 to generate the modified acceleration-displacement response 

spectrum (MADRS).

B11. Determine the estimate of the maximum acceleration, ai

, and displacement, di, as the intersection 

of the MADRS with the capacity curve (see 

Figure 6-9).

B12. Compare the estimated maximum displacement,

di

, with the initial (or previous) assumption, dpi. 

If it is within acceptable tolerance, the performance point corresponds to ai

 and di. If it is not 

within acceptable tolerance, then repeat the process from Step 5 using ai

 and di, or some other 

selected assumption (see Section 6.6), as a starting point.

Procedure C (MADRS Locus of Possible 

Performance Points). This approach uses the modified 

acceleration-response spectrum for multiple assumed 

solutions (api, dpi) and the corresponding ductilities to 

generate a locus of possible performance points. The 

actual performance point is located at the intersection of 

this locus and the capacity spectrum.

C9. Using the effective damping determined from 

Step 8, adjust the initial ADRS to βeff (see 

Section 6.3).

C10. Multiply the acceleration ordinates of the ADRS 

for βeff by the modification factor, M, determined 

using the calculated effective period, Teff, in 

accordance with Section 6.2.3 to generate the 

modified acceleration-displacement response 

spectrum (MADRS).

C11. A possible performance point is generated by the 

intersection of the radial secant period, Tsec, with 

the MADRS (see Figure 6-10).

C12. Increase or decrease the assumed performance 

point and repeat the process to generate a series 

of possible performance points.

C13. The actual performance point is defined by the 

intersection of the locus of points from Step 12 

and the capacity spectrum.

Note that Procedure C is conducive to an automated 

process wherein the initial solution is assumed to 

correspond to a ductility of 1.0 and subsequent trials are 

set as incrementally greater ductilities (e.g., 2, 3, 4, 

5,....).

Figure 6-8 Determination of estimated maximum 

displacement using direct iteration 

(Procedure A)

noit ar el eccAl art cepS

Spectral Displacement

capacity curve(CS) 

for structure

Sa

Sd

dpi

api

di

T0

initial ADRS, β 0

ADRS, β eff (μ, CS)

Teff (μ, CS)

ai

noit ar el eccAl art cepS

Spectral Displacement

capacity curve(CS) 

for structure

Sa

Sd

dpi

api

di

T0

initial ADRS, β 0

ADRS, β eff (μ, CS)

Teff (μ, CS)

ai

Figure 6-9 Determination of estimated maximum 

displacement using intersection of 

capacity spectrum with MADRS 

(Procedure B)

noit ar el eccAl art cepS

Spectral Displacement

capacity curve (CS) 

for structure

Sa

Sd

dpi

api

di

T0

initial ADRS, β 0

MADRS, β eff (μ, CS, M)

ADRS, βeff (μ, CS)

ai

noit ar el eccAl art cepS

Spectral Displacement

capacity curve (CS) 

for structure

Sa

Sd

dpi

api

di

T0

initial ADRS, β 0

MADRS, β eff (μ, CS, M)

ADRS, βeff (μ, CS)

ai
Chapter 6: Improved Procedures for Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 6-9

6.5 Approximate Solution Procedure

The following procedure is a simplified MADRS 

approach based on approximations to the equations in 

Section 6.2. It uses a MADRS solution procedure 

similar to that of Section 6.4. The approximations are 

based upon an EPP single-degree-of freedom system. 

The results of the approximate procedure are compared 

to the more rigorous procedures for various types of 

hysteretic behavior in Figure 6-11. The first seven steps 

in the procedure are the same as Steps 1 through 7 in the 

beginning of Section 6.4. The next steps in the 

approximate procedure are given below. 

D8. Using the calculated values for ductility, μ, from 

Step 7, calculate the corresponding spectral 

response-reduction factors as

for 1 ≤ μ ≤ 4 (6-20)

for μ > 4 (6-21)

D9. Using the spectral response-reduction factors 

from Step 8, multiply both the spectral accelerations and corresponding spectral displacements 

by the response-reduction factor to generate a 

reduced ADRS corresponding to the assumed 

ductility, μ.

D10. Multiply the spectral acceleration ordinates (not 

the spectral displacement ordinates) of the 

reduced ADRS by a simplified modification factor

(6-22)

to generate the approximate modified 

acceleration-displacement response spectrum 

(MADRS). It should be noted that for ductilities 

greater than 1.6 the bounding limit of 0.64 

controls this step. 

Figure 6-10 Locus of possible performance points 

using MADRS.

Spectral Acceleration

Spectral Displacement

Sa

Sd

dmax

Tsec (μ=1)

Initial ADRS, β 0 (μ=1)

MADRS (μ=2)

Tsec (μ=2)

Tsec (μ=3)

Tsec (μ=4)

Tsec (μ=5)

Tsec (μ=6)

Tsec (μ=7)

(μ=3)

(μ=4)

(μ=5)

(μ=6)

a

Locus of possible 

Performance Points

(μ=7)

Capacity curve

Spectral Acceleration

Spectral Displacement

Sa

Sd

dmax

Tsec (μ=1)

Initial ADRS, β 0 (μ=1)

MADRS (μ=2)

Tsec (μ=2)

Tsec (μ=3)

Tsec (μ=4)

Tsec (μ=5)

Tsec (μ=6)

Tsec (μ=7)

(μ=3)

(μ=4)

(μ=5)

(μ=6)

a

Locus of possible 

Performance Points

(μ=7)

Capacity curve

6

μ + 5

⎡

⎣

⎢ ⎤

⎦

⎥

75

μ +110

⎡

⎣

⎢ ⎤

⎦

⎥

M = ≥

1 0 64

μ .

Figure 6-11 Comparison of approximate solution results with results from more detailed procedures.

Loci of Performance Points for T = 0.2, 0.5, and 1.0 sec

0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14

SD (in)

SA (in/s/s)

Eqns 6-1 to 6-3 and 6-7 to 6-9 for alpha=0%

Eqns 6-1 to 6-3 and 6-7 to 6-9 for alpha=5%

Eqns 6-1 to 6-3 and 6-7 to 6-9 for alpha=10%

Eqns 6-1 to 6-3 and 6-7 to 6-9 for alpha=20%

Approx MADRS Approach (Eqns 6-20 to 6-22)

Loci of Performance Points for T = 0.2, 0.5, and 1.0 sec

0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14

SD (in)

SA (in/s/s)

Eqns 6-1 to 6-3 and 6-7 to 6-9 for alpha=0%

Eqns 6-1 to 6-3 and 6-7 to 6-9 for alpha=5%

Eqns 6-1 to 6-3 and 6-7 to 6-9 for alpha=10%

Eqns 6-1 to 6-3 and 6-7 to 6-9 for alpha=20%

Approx MADRS Approach (Eqns 6-20 to 6-22)
Chapter 6: Improved Procedures for Equivalent Linearization

6-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

D11. A possible performance point is generated by the 

intersection of the radial effective period, Teff, 

with the MADRS (see Figure 6-9).

D12. Increase or decrease the assumed performance 

point and repeat the process to generate a series 

of possible performance points.

D13. The actual performance point is defined by the 

intersection of the locus of points from Step 12 

and the capacity curve. For this approximate procedure, the calculated target displacement must 

be equal to or greater than the elastic target displacement. 

6.6 Iterative Strategy

Subsequent assumptions for the performance point can 

be calculated by averaging the previous value of the 

initial assumption dpi and the calculated result di

, then 

choosing the corresponding acceleration value from the 

capacity curve. However, this is not required and some 

educated guessing and judgment can improve solution 

time. For example, the initial assumption, dpi, and the 

resulting estimated maximum displacement, di

, can be 

plotted as a point, as shown in Figure 6-12. Note that 

the actual performance point will fall along the line 

where the two values are equal. By tracking the 

subsequent trial point with this type of plot, it is easy to 

see solution trends. An example with three iterations is 

shown in Figure 6-12. After the second trial, it is 

apparent that the performance point is larger than the 

estimate, as the track of the trial points has not crossed 

the line of equal displacement. So the third trial 

assumes a relatively large displacement. The results of 

the third trial indicate a solution somewhere between 

the assumptions of Trial 2 and Trial 3.

6.7 Limitation on Strength to Avoid 

Dynamic Instability for Nonlinear 

Static Procedures

The evaluation of current procedures summarized in 

Chapter 3 revealed that oscillators that exhibit in-cycle 

strength degradation can be prone to dynamic instability 

during strong shaking. The subject is covered in detail 

in Chapter 4. When using equivalent linearization 

procedures, the strength of the structural model should 

be checked in accordance with Section 4.4. 

Figure 6-12 Tracking iteration for equivalent 

linearization by comparing assumed 

displacement to calculated displacement.

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10

Assumed d pi

Calculated 

di

Trial 1

Trial 2

Trial 3

Actual Performance Point will fall 

on line where d i

 = dpi
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 7-1 

7. Evaluation and Comparison of Improved 

Nonlinear Static Procedures

7.1 Introduction

Previous chapters have introduced improvements to 

nonlinear static procedures that are useful for estimating 

the peak displacement amplitude for a SDOF oscillator 

subjected to earthquake ground motion. This chapter 

compares results of those methods with results obtained 

using nonlinear response-history analyses for ground 

motion records selected and scaled to be representative 

of a specific hazard level and site conditions. The 

ground motion selection and scaling procedures are 

similar to those specified in the 2000 NEHRP 

Recommended Provisions for Seismic Regulations for 

New Buildings and Other Structures (BSSC, 2000),1

and therefore provide an example of the types of results 

one might obtain in a practice-related application. 

Several nonlinear oscillators were selected, having 

different vibration periods and strengths. The oscillators 

were assumed to be sited on ground classified as 

NEHRP Site Class C, with ground motions generated 

by a fault capable of a strike-slip earthquake of 

magnitude Ms = 7. Smooth design response spectra 

were established using the 2000 NEHRP Recommended 

Provisions for New Buildings, scaled for the designbasis earthquake. Furthermore, ground motion records 

from representative sites and earthquakes were selected 

and scaled. Displacement amplitudes of the oscillators 

were calculated by both the nonlinear static procedures 

and nonlinear dynamic response-history analysis, for 

comparison purposes.

The scope of the study reported in this chapter is limited 

by the periods, strengths, and hysteretic behavior of the 

SDOF oscillators, as well as the number and nature of 

the ground motions used. The results do not represent a 

large statistical sample and broad general conclusions 

should not be drawn solely from these data. 

Nonetheless, they are illustrative of the types of errors 

and variations among procedures that should be 

anticipated when using these simplified analysis 

techniques.

7.2 Summary of Evaluation Procedures

7.2.1 NEHRP Design Response Spectrum

Procedures similar to the 2000 NEHRP Recommended 

Provisions for New Buildings were used to define 

design response spectra. Values for short- and 1-second 

period spectral accelerations at the Maximum 

Considered Earthquake (MCE) level were read from the 

pertinent maps for 5% damping and site class C, 

resulting in values SS = 1.5 g and S1 = 0.6 g. Following 

the 2000 NEHRP procedures, the short- and longperiod values were modified for site class C to 

SMS = FaSS and SM1 = Fv S1, where Fa = 1.0 and 

Fv = 1.3. Design-basis ordinates then were obtained as 

SDS = (2/3)SMS and SD1 = (2/3)SM1. These values were 

used with the spectral shape defined in the NEHRP 

Recommended Provisions for New Buildings

(Figure 7-1) to derive the NEHRP design response 

spectrum. Note that the acceleration values in 

Figure 7-1 and in the rest of the document are actually 

pseudo-acceleration values. 

7.2.2 Ground Motions and Ground-Motion 

Scaling

Ground motions were intended to be representative of 

design-level motions for a facility located 

approximately 10 km from a fault rupturing with strikeslip mechanism at magnitude Ms 7. The soil at the site 

corresponds to NEHRP Site Class C. Ground motions 

were selected from the Pacific Earthquake Engineering 

Research (PEER) Center strong ground motion 

1.Superseded in 2003 with the FEMA 450 Recommended

Provisions for Seismic Regulations for New Buildings

and Other Structures.

Figure 7-1 NEHRP design response spectrum.
Chapter 7: Evaluation and Comparison of Improved Nonlinear Static Procedures

7-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

database (http://peer.berkeley.edu), and were scaled to 

be representative of design-level motions at the site.

The NEHRP Recommended Provisions for New 

Buildings prescribe a scaling procedure to be used when 

ground motion records are used directly for timedomain dynamic analysis. According to this procedure, 

ground motions should be selected that are from similar 

site conditions, rupture mechanism and magnitude, and 

epicentral distance. For the present study, the selected 

records were for sites classified as NEHRP Site Class 

C, having strike-slip mechanism, magnitude Ms ranging 

from 6.3 to 7.5, and closest distance to fault rupture 

ranging from 5 to 25 km. 

The SDOF oscillators were to be analyzed as planar 

structures subjected to a single horizontal component of 

ground motion. Therefore, records were scaled 

individually rather than scaling them as pairs as is 

recommended by the NEHRP Recommended Provisions 

for New Buildings for three-dimensional structures. The 

Provisions stipulate that the ground motions be scaled 

such that the average of the ordinates of the fivepercent-damped linear response spectra does not fall 

below the design spectrum for the period range 0.2Ti

 to 

1.5Ti

, where Ti is the fundamental period of vibration of 

the structure modeled as a linear system. The period 

0.2Ti is selected as the lower bound to ensure that 

important higher modes of vibration are adequately 

excited. This lower bound is not relevant for the present 

study because the structure is an oscillator with a single 

vibration mode. Rather, for the present study, it is more 

important that the average approximate the design 

spectrum in the period range just below Ti

 to values 

higher than Ti

, such that as the oscillator yields, it will, 

on average, experience ground motion intensities close 

to that represented by the design spectrum. Also, 

because this is a study of the procedures, rather than a 

building design, it is preferable to scale the motions so 

that the average of the spectral ordinates follows the 

design spectrum closely, rather than conservatively 

scaling the motions to be above the design spectrum as 

might be done for design purposes.

Sixteen ground motion records were selected for 

consideration. Each was examined to be certain it did 

not contain obvious near-fault directivity effects. Each 

motion was scaled so that the five-percent-damped 

spectral ordinate at the period of the oscillator matched 

that of the NEHRP response spectrum at the same 

period. Ground motions were eliminated selectively to 

avoid motions with unacceptably large scaling factors 

and motions whose response spectra did not appear 

consistent with the NEHRP response spectrum. The 

process of elimination continued until there were ten 

records available for each oscillator. Note that the 

oscillators had three different vibration periods (0.2, 

0.5, and 1.0 s). Within the criteria stated above, it was 

not feasible to use the same ten motions for each 

oscillator. In total, 13 ground motions were used for the 

study. The ground motion records are identified in 

Table 7-1. The response spectra of the scaled ground 

Table 7-1 Ground Motion Records

Earthquake Magnitude Record PGA (g)

PGV 

(cm/s)

PGD 

(cm)

Distance closest to fault 

rupture (km)

Imperial Valley 1979/

10/15 23:16

M 6.5, 

Ml

 6.6, 

Ms 6.9 

IMPVALL/H-PTS315 0.204 16.1 9.94 14.2

IMPVALL/H-CPE147 0.169 11.6 4.25 26.5

IMPVALL/H-CPE237 0.157 18.6 7.95 26.5

Landers 1992/06/28 

11:58

M 7.3

Ms 7.4

LANDERS/CLW-LN 0.283 25.6 13.74 21.2

LANDERS/CLW-TR 0.417 42.3 13.76 21.2

LANDERS/MVH000 0.188 16.6 9.45 19.3

LANDERS/MVH090 0.14 20.2 6.33 19.3

LANDERS/DSP000 0.171 20.2 13.87 23.2

LANDERS/DSP090 0.154 20.9 7.78 23.2

LANDERS/JOS000 0.274 27.5 9.82 11.6

LANDERS/JOS090 0.284 43.2 14.51 11.6

LANDERS/NPS000 0.136 11 4.97 24.2

PGA: peak ground acceleration; PGV: peak ground velocity; PGD: peak ground displacement
Chapter 7: Evaluation and Comparison of Improved Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 7-3

motions used for oscillators having periods 0.2, 0.5, and 

1.0 s are shown in Figures 7-2, 7-3, and 7-4, 

respectively.

7.2.3 Characteristics of Oscillators

Nine SDOF oscillators were used for this study. The 

oscillators had bilinear load-displacement relationships 

with post-elastic stiffness equal to five percent of the 

initial elastic stiffness. Loading and unloading 

characteristics are shown in Figure 7-5 without strength 

or stiffness degradation. Initial damping was five 

percent of critical damping. The oscillators had three 

different yield strengths and three different periods. For 

each period, the spectral acceleration was read from the 

NEHRP response spectrum. The yield strengths were 

then defined as the elastic base shear demand (product 

of the mass and spectral acceleration) divided by a 

strength reduction factor R. R values of 2, 4, and 8 were 

considered. Figure 7-6 summarizes the elastic vibration 

periods and R values selected. 

7.2.4 Nonlinear Static Procedure Estimates 

Using Smoothed or Average Spectra

The improved nonlinear static procedures of Chapters 5 

and 6 were applied to the NEHRP response spectra, as 

well as to the average of the 5%-damped response 

spectra. The former represents more closely how the 

procedures would be used with the NEHRP response 

spectra, whereas the latter represents more closely how 

the procedures might be used when a site-specific 

response spectrum is defined by the average of the 

response spectra for a series of design ground motions 

selected for a site. 

For application of the displacement modification 

method of Chapter 5, the displacement amplitude was 

defined as 

,

Figure 7-2 NEHRP response spectrum and 5%-

damped response spectra of scaled 

motions, used for oscillators having 

T = 0.2 s.

0.05 0.1 0.15 0.2 0.25 0.3 0

2

4

6

8

10

12

14

16

Period (sec)

ces/ m( aS 2)

For T = 0.2 sec ( range 0.04 ~ 0.30 sec)

DBE

average

H-PTS315 , (SF = 2.06)

H-CPE147 , (SF = 2.58)

H-CPE237 , (SF = 1.97)

CLW-LN , (SF = 1.03)

MVH000 , (SF = 2.84)

MVH090 , (SF = 2.64)

DSP000 , (SF = 1.86)

JOS090 , (SF = 2.02)

NPS000 , (SF = 2.94)

NPS090 , (SF = 2.91)

Figure 7-3 NEHRP response spectrum and 5%-

damped response spectra of scaled 

motions, used for oscillators having 

T = 0.5 s.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0

2

4

6

8

10

12

14

16

18

Period (sec)

ces/ m( aS 2)

For T = 0.5 sec ( range 0.10 ~ 0.75 sec)

DBE

average

H-PTS225 , (SF = 3.82)

H-PTS315 , (SF = 2.96)

H-CPE147 , (SF = 2.14)

CLW-LN , (SF = 1.34)

CLW-TR , (SF = 0.90)

MVH000 , (SF = 2.33)

MVH090 , (SF = 2.21)

DSP000 , (SF = 3.24)

DSP090 , (SF = 3.37)

JOS000 , (SF = 1.46)

CC S CC S T

d a i

12 12

2

2 = ⎛

⎝

⎞

π ⎠
Chapter 7: Evaluation and Comparison of Improved Nonlinear Static Procedures

7-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

in which Sa = pseudo-spectral acceleration ordinate at 

the period of the oscillator Ti

. The coefficient C1 was 

defined as 

.

Coefficient C2 was taken equal to 1.0, as it was assumed 

that there was no stiffness or strength degradation.

For application of the equivalent linearization 

procedure of Chapter 6, response spectra were 

converted to the spectral acceleration-spectral 

displacement format. In studies using the average 

response spectra, the spectral ordinates were calculated 

for each ground motion for each of several different 

damping ratios. The results for a given damping ratio 

were averaged for the different ground motions to 

obtain the average response spectrum for that damping 

ratio. In studies using the NEHRP smooth design 

response spectra, spectral ordinates for damping 

exceeding 5% of critical damping were calculated using 

the spectral reduction factors of ATC-40; however, the 

limits on the reductions (ATC-40 Tables 8-1 and 8-2) 

were not imposed. Damping factors and effective 

periods were calculated using the equations and 

tabulated quantities in FEMA 440 Chapter 6, specific to 

the bilinear oscillator behavior with 5% post-elastic 

stiffness, rather than the more generally applicable 

equations. Iteration Procedure A was used with the 

average spectra, while iteration Procedure B was used 

for the smooth spectra, in general accordance with 

Section 6.4. Convergence was assumed when the 

Figure 7-4 NEHRP response spectrum and 5%-

damped response spectra of scaled 

motions, used for oscillators having 

T = 1.0 s.

Figure 7-5 Bilinear load-displacement relation of 

oscillators.

0.2 0.4 0.6 0.8 1 1.2 1.4 0

2

4

6

8

10

12

14

16

18

Period (sec)

ces/ m( aS 2)

For T = 1.0 sec ( range 0.20 ~ 1.50 sec)

DBE

average

H-CPE237 , (SF = 1.77)

CLW-TR , (SF = 1.43)

MVH000 , (SF = 2.51)

MVH090 , (SF = 2.20)

DSP000 , (SF = 2.26)

DSP090 , (SF = 1.53)

JOS000 , (SF = 1.29)

JOS090 , (SF = 0.98)

NPS000 , (SF = 3.35)

NPS090 , (SF = 2.37)

Displacement

Shear

K0

0.05 K0

Figure 7-6 Linear vibration periods and strength 

reduction factors for oscillators.

Initial Period, sec

0.2 0.5 1.0

2

8

4

Strength Reduction Factor

C R

Ti

1 2 1 1

90

= + −
Chapter 7: Evaluation and Comparison of Improved Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 7-5

calculated displacement was not more than 5% different 

from the assumed displacement. Also, solutions were 

generated using the approximate MADRS approach of 

Section 6.5.

Results also are presented using the Coefficient Method 

of FEMA 356 and the Capacity-Spectrum Method of 

ATC-40. For the Coefficient Method, the coefficients 

for the nonlinear static procedure were used with a cap 

on C1 equal to 1.5, as permitted, and all other 

coefficients set equal to 1.0. For the Capacity-Spectrum 

Method, the procedures of ATC-40 were followed 

explicitly, using the response spectra in the same 

manner as for the improved procedure. 

7.2.5 Response-History Analyses

Inelastic responses of the single-degree-of-freedom 

oscillators, with different periods and strengthreduction factors, were calculated for each of the 

ground motion histories. Figure 7-7 presents a 

representative result.

7.3 Results of the Study

Figure 7-8 presents results of the study using ground 

motions scaled to match the NEHRP design response 

spectrum, with the nonlinear static results calculated for 

the NEHRP design response spectrum. Data are 

presented in three sequential graphs, one each for 

oscillator of the initial periods: 0.2 s 0.5 s, and 1.0 s. 

Figure 7-7 Representative nonlinear response-history analysis result (this example is for oscillator period T = 1 s, 

ground motion DSP090 scaled by factor 1.53, and strength-reduction factor R = 4).

0 5 10 15 20 25 30 35 40 45 50 -3

-2

-1

0

1

2

Ag (m/sec2)

time (sec)

Ground motion: DSP090 (SF = 1.53)

0 5 10 15 20 25 30 35 40 45 50 -0.1

-0.05

0

0.05

0.1

0.15

time (sec)

Urelative (m)

T = 1 sec, R = 4, ζ = 5%, Umax = 1.239e-001 m

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 -1500

-1000

-500

0

500

1000

1500

Urelative (m)

Pk (N)

Ke = 3.95e+004 N/m, Dy = 3.23e-002 m, α = 0.050
Chapter 7: Evaluation and Comparison of Improved Nonlinear Static Procedures

7-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Each graph plots maximum relative displacement 

amplitude as a function of the strength-reduction factor 

R. The legend to the right of each graph identifies the 

data in the graph, as follows:

1. NDA mean is the mean of the maximum displacement response amplitudes calculated using nonlinear dynamic analysis (time-domain) for the ten 

ground motions. Each graph also includes a representation of the NDA results for each strength 

value, consisting of the mean plus and minus one 

standard deviation. 

2. FEMA 440 EL is the result obtained by the 

improved equivalent linearization method 

(Section 6.4)

3. FEMA 440 DM is the result obtained by the 

improved displacement modification method of 

Chapter 5.

4. Approximate EL is the result obtained by the 

approach given in Section 6.5. 

5. ATC-40 is the result obtained by the CapacitySpectrum Method of ATC-40.

6. FEMA 356 is the result obtained by the displacement modification method of FEMA 356.

7. μ = 10 plots the displacement corresponding to displacement ductility of 10.

In Figures 7-8, 7-9, and 7-10, the results of primary 

interest are those for which the actual displacement is 

less than approximately10 times the yield displacement. 

Displacements near or beyond this level are unrealistic 

for most actual structures, because their vertical- and 

lateral-force-resisting systems are unlikely to be able to 

sustain such large deformations without failure. The 

coefficients of the FEMA 440 EL method were 

optimized for solutions with displacement ductility less 

than this limit.

The results obtained using nonlinear dynamic analysis 

(NDA) indicate that for short-period oscillators, the 

maximum displacement response amplitude increases 

with decreasing strength (increasing R), while for 

longer-period oscillators the peak displacement 

response is less sensitive to strength. NDA results 

reflect wider dispersion for shorter-period oscillators 

and for lower strength values. This observation is partly 

because the response spectra (Figures 7-2, 7-3, and 7-4) 

show increasing dispersion as the period elongates (as 

occurs for structures with lower strengths). Previous 

studies, including those summarized in Chapter 3, also 

have shown that dispersion of response generally 

Figure 7-8 Comparison of responses for an oscillator with T = 0.2 s calculated using various procedures, response 

spectra scaled to the NEHRP spectrum, and values calculated for the NEHRP spectrum

T = 0.2s

NEHRP Response Spectrum

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0123456789

Strength Reduction Factor, R

) m( t ne mecal psi D

mu mi xa M

NDA

FEMA 440 EL

FEMA 440 DM

Approx. EL

ATC 40

FEMA 356

μ=10

std R=2

std R=8

std R=4

Indicates mean of NDA plus and

minus one standard deviation
Chapter 7: Evaluation and Comparison of Improved Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 7-7

Figure 7-9 Comparison of responses for an oscillator with T = 0.5 s calculated using various procedures, response 

spectra scaled to the NEHRP spectrum, and values calculated for the NEHRP spectrum.

Figure 7-10 Comparison of responses for an oscillator with T = 1.0 s calculated using various procedures, response 

spectra scaled to the NEHRP spectrum, and values calculated for the NEHRP spectrum.

T = 0.5s

NEHRP Response Spectrum

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0123456789

Strength Reduction Factor, R

) m( t ne mecal psi D

mu mi xa M

NDA

FEMA 440 EL

FEMA 440 DM

Approx. EL

ATC 40

FEMA 356

μ=10

std R=2

std R=4

std R=8

Indicates mean of NDA plus and

minus one standard deviation

T = 1.0s

NEHRP Response Spectrum

0.00

0.05

0.10

0.15

0.20

0.25

0123456789

Strength Reduction Factor, R

) m( t ne mecal psi D

mu mi xa M

NDA

FEMA 440 EL

FEMA 440 DM

Approx. EL

ATC 40

FEMA 356

μ=10

std R=2

std R=4

std R=8

Indicates mean of NDA plus and

minus one standard deviation
Chapter 7: Evaluation and Comparison of Improved Nonlinear Static Procedures

7-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

increases for shorter periods and higher R values, 

regardless of the tendency of the response spectra.

The proposed improved procedures generally follow the 

observed mean trends for the NDA results, provided 

that the displacement ductilities remain within 

reasonable bounds. Unreasonable ductility values are 

the cause of overestimates of displacement in some 

instances, using the FEMA 440 EL and the approximate 

EL procedures (e.g., Figure 7-8 with T = 0.2 s and

R = 8, Figure 7-10 with T = 1.0 s and R= 8). This 

tendency is not apparent when the average spectrum is 

used, as noted below. 

As expected, the FEMA 356 procedure does not predict 

the increase in displacement response with increasing R

for short-period oscillators. The ATC-40 procedure 

tends to underestimate the displacement response for 

small R and overestimate the response for large R. 

These results are again consistent with the previous 

studies (Chapter 3).

Figures 7-11, 7-12, and 7-13 present data similar to 

those of Figures 7-8, 7-9, and 7-10. The ground motions 

are identical, having been scaled to match the NEHRP

smooth design response spectrum, and oscillator 

strengths also are identical. However, the nonlinear 

static procedures all are applied using the average of the 

response spectra for the scaled ground motions. For the 

displacement modification methods, the ordinate of the 

5% damped response spectrum at period T of the 

oscillator is unchanged from the previous analyses, so 

the results shown in Figures 7-11, 7-12, and 7-13 for 

those methods are the same as those shown in 

Figures 7-8, 7-9, and 7-10. For the equivalent 

linearization methods, the analysis required the 

calculation of the average of the linear response spectra 

for each scaled ground motion record for each of 

several different damping values. Results for these 

methods therefore differ from those presented in 

Figures 7-8, 7-9, and 7-10. Data are presented in three 

sequential graphs, separated by the oscillator initial 

periods of 0.2, 0.5, and 1.0 s. Each graph plots 

maximum relative displacement amplitude as a function 

of strength-reduction factor, R. The legend to the right 

of each graph identifies the data in the graph, defined as 

described above. 

Results for the improved equivalent linearization 

methods using the average spectrum (Figure 7-8) are 

somewhat improved over those using the NEHRP 

spectrum (Figures 7-8, 7-9, and 7-10), especially for 

larger ductilities. This improvement might be expected 

for two reasons. First, the equivalent linearization 

methods were derived using response spectra calculated 

for individual motions for various specific values of 

Figure 7-11 Comparison of responses of an oscillator with T = 0.2 s calculated using various procedures, response 

spectra scaled to NEHRP spectrum, and values calculated for the average spectrum.

T = 0.2s

Average Response Spectrum

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0123456789

Strength Reduction Factor, R

) m( t ne mecal psi D

mu mi xa M

NDA mean

FEMA 440 EL

FEMA 440 DM

Approx. EL

ATC 40

FEMA 356

μ=10

Indicates mean of NDA plus and 

minus one standard deviation
Chapter 7: Evaluation and Comparison of Improved Nonlinear Static Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 7-9

Figure 7-12 Comparison of responses of an oscillator with T = 0.5 s calculated using various procedures, response 

spectra scaled to the NEHRP spectrum, and values calculated for the average spectrum.

\

Figure 7-13 Comparison of responses of an oscillator with T = 1.0 s calculated using various procedures, response 

spectra scaled to the NEHRP spectrum, and values calculated for the average spectrum

T = 0.5s

Average Response Spectrum

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0123456789

Strength Reduction Factor, R

) m( t ne mecal psi D

mu mi xa M

NDA mean

FEMA 440 EL

FEMA 440 DM

Approx. EL

ATC 40

FEMA 356

μ=10

std R=2

std R=4

std R=8

Indicates mean of NDA plus and 

minus one standard deviation

T = 1.0s

Average Response Spectrum

0.00

0.05

0.10

0.15

0.20

0.25

0123456789

Strength Reduction Factor, R

) m( t ne mecal psi D

mu mi xa M

NDA mean

FEMA 440 EL

FEMA 440 DM

Approx. EL

ATC 40

FEMA 356

μ=10

std R=2

std R=4

std R=8

Indicates mean of NDA plus and

minus one standard deviation
Chapter 7: Evaluation and Comparison of Improved Nonlinear Static Procedures

7-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

damping. When used with the NEHRP design spectrum, 

it was necessary to estimate the effect of damping on 

spectral ordinates using approximate spectral reduction 

factors. Additionally, the effective period relationships 

were optimized from actual spectra as opposed to an 

assumed shape (e.g., NEHRP spectrum). 

7.4 Summary of Implications of the Results 

of the Study

As noted elsewhere in this document, the dispersion of 

maximum displacement responses for nonlinear 

oscillators subjected to earthquake ground motions is 

relatively large, such that a relatively large number of 

analyses with different oscillators and ground motions 

may be required to reach statistically meaningful 

conclusions regarding response statistics. The results 

reported in this chapter based on a relatively small 

number of ground motions and oscillators are 

insufficient to serve as the basis for broad conclusions 

for all cases. Nonetheless, some general observations 

can be made from the results.

Engineers using the Capacity-Spectrum Method of 

ATC-40 and the Coefficient Method of FEMA 356 have 

observed that sometimes the two methods give widely 

different displacement estimates. This observation is 

evident from the results reported in Section 7.3. In some 

cases, the results of the methods differ by as much as a 

factor of two (Figures 7-8 through 7-13). One of the 

objectives of the effort to develop improved nonlinear 

static procedures, reported here, was to reduce the 

discrepancy in the results obtained by the two methods. 

As shown in Figures 7-8 through 7-13, this objective 

has been met for the ground motions and oscillators that 

were studied.

Another objective in developing the improved 

procedures in the frequency domain was to improve the 

accuracy of the methods relative to results for 

maximum global displacements obtained using 

nonlinear dynamic analysis. For this particular sample 

of ground motions and oscillators, the improved 

nonlinear static procedures provide generally better 

estimates of the mean maximum displacement response 

than do the current procedures. For displacement 

ductility less than about 10, which is deemed an 

excessive value for most structures to which these 

procedures would be applied, the improved nonlinear 

static procedures produced results within about one 

standard deviation of mean responses obtained by 

nonlinear dynamic analysis.

Another objective was to investigate whether the 

improved simplified static procedures could be applied 

to design spectra commonly used in practice, with 

sufficient accuracy. As shown in Figures 7-8, 7-9, and 

7-10, for the ground motions, scaling procedure, and 

oscillators considered, the improved simplified static 

procedures effectively estimated the mean of maximum 

displacement response in conjunction with smooth 

design spectra. Again, the procedures probably should 

not be used for excessive displacement ductility values.

Finally, the results reported in this chapter illustrate the 

dispersion typical of nonlinear dynamic analysis using 

design-level ground motions. Actual response for a real 

design-level event may differ significantly from the 

estimate given by the simplified procedures using a 

NEHRP-like design spectrum. The same is true even if 

the spectrum is derived from specific ground motions 

records and even if the simplified procedures are 

capable of reasonably matching the median response. 

When interpreting results and assessing structural 

performance, engineers must consider the implications 

of these uncertainties.
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 8-1 

8. Procedures for Including Soil-Structure Interaction 

Effects

8.1 Introduction

This chapter presents simplified procedures for 

including the effects of interaction between a structure 

and the supporting soils in a structural model for 

nonlinear static analysis procedures. There are three 

primary categories of soil-structure interaction (SSI) 

effects. These include: 

• introduction of flexibility to the soil-foundation 

system (flexible foundation effects),

• filtering of the ground motions transmitted to the 

structure (kinematic effects), and 

• dissipation of energy from the soil-structure system 

through radiation and hysteretic soil damping 

(foundation damping effects). 

Current analysis procedures in FEMA 356 and ATC-40 

partially address the flexible foundation effect through 

guidance on including the stiffness and strength of the 

geotechnical (soil) components of the foundation in the 

structural analysis model. However, these procedures 

do not address the reduction of the shaking demand on 

the structure relative to the free-field motion caused by 

kinematic interaction or the foundation damping effect. 

Guidance on including these effects in NSPs is provided 

in this section. A simple example illustrates the 

application of these procedures. Appendix E provides 

detailed information on these soil-structure interaction 

effects.

Figure 8-1a illustrates the assumption that the structural 

model is mounted on a rigid base that is excited by the 

free-field motion. The free-field motion is the 

theoretical movement of a single point on the surface of 

the ground, assuming that there is no structure near it. 

The fixed-base modeling assumption is inappropriate 

for many structures though. Structural systems that 

incorporate stiff vertical elements for lateral resistance 

(e.g., shear walls, braced frames) can be particularly 

sensitive to even small base rotations and translations 

that are neglected with a fixed base assumption. 

Relatively flexible vertical elements (e.g., moment 

frames) are often not significantly affected by SSI.

Figure 8-1b illustrates the incorporation of foundation 

flexibility into the structural model directly. ATC-40

and FEMA 356 include provisions for estimating the 

flexibility and strength of the foundation (i.e., the 

properties of the springs indicated in Figure 8-1b) in a 

structural model for inelastic analysis. Those provisions 

normally use the free-field motion as the seismic 

demand with 5% damping as the conventional initial 

value. This approach is capable of modeling both the 

structural and geotechnical (soil) components of the 

foundation. The result is that the response of the overall 

structural system includes deformations (elastic and 

inelastic) in the structural and geotechnical parts of the 

foundation system. These deformations are sometimes 

referred to as an inertial SSI effect. These 

improvements in modeling can lead to significant 

departures from fixed-base results and more accurate 

representation of probable structural response. 

Compared with the fixed-base modeling approach, the 

predicted period of the structure lengthens, the 

distribution of forces among various elements changes, 

the sequence of inelasticity and the modes of inelastic 

behavior can change, and foundation mechanisms (e.g., 

rocking, soil bearing failure, and pier/pile slip) can be 

directly evaluated and considered. All of these effects 

result in more realistic evaluation of the probable 

structural behavior and performance. 

Figure 8-1c illustrates the filtering effects that soilstructure interaction can have on the character and 

intensity of ground motion experienced by the structural 

model. Kinematic interaction results from the presence 

of relatively stiff foundation elements on or in soil that 

cause foundation motions to deviate from free-field 

motions. Two effects are commonly identified: base-slab 

averaging and embedment effects. The base-slab 

averaging effect can be visualized by recognizing that 

the instantaneous motion that would have occurred in 

the absence of the structure within and below its 

footprint is not the same at every point. Placement of a 

structure and foundation across these spatially variable 

motions produces an averaging effect in which the 

overall motion is less than the localized maxima that 

would have occurred in the free field. The embedment 

effect is associated with the reduction of ground motion 

that tends to occur with depth in a soil deposit. Both 

base-slab averaging and embedment affect the character 

of the foundation-level motion (sometimes called the 

foundation input motion, or FIM) in a manner that is 

independent of the superstructure (i.e., the portion of the 

structure above the foundation), with one exception. The 

effects are strongly period-dependent, being maximized 

at small periods. The effects can be visualized as a filter 
Chapter 8: Procedures for Including Soil-Structure Interaction Effects

8-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

applied to the high-frequency (short-period) components 

of the free-field ground motion. The impact of those 

effects on superstructure response will tend to be 

greatest for short-period buildings. A simplified 

procedure to apply these principles for reduction of the 

spectral amplitudes of the free-field motion to generate 

the FIM spectrum is presented in Section 8.2. The 

foundation input motion can be applied to a fixed-base 

model or, as depicted in Figure 8-1c, can be combined 

with a flexible-base model. 

Figure 8-1d illustrates foundation damping effects that 

are another result of inertial soil-structure interaction in 

addition to foundation flexibility. Foundation damping 

results from the relative movements of the foundation 

and the supporting soil. It is associated with radiation of 

energy away from the foundation and hysteretic 

damping within the soil. The result is an effective 

decrease in the spectral ordinates of ground motion 

experienced by the structure. Although seldom used in 

practice the NEHRP Recommended Provisions for 

Seismic Regulations for New Buildings and Other 

Structures (BSSC, 2000),1 as well as the ASCE-7 

Standard for Minimum Design Loads for Buildings and 

Other Structures (ASCE, 2002) include procedures to 

account for this effect when using linear analysis 

procedures. Section 8.3 incorporates similar, although 

updated, procedures for use with NSPs. In the 

procedure, the foundation damping is linked to the ratio 

of the fundamental period of the system on the flexiblefoundation to that of a fixed-base model. Other factors 

affecting foundation damping are the foundation size 

and embedment. The foundation damping is combined 

with the conventional initial structural damping to 

generate a revised damping ratio for the entire system, 

including the structure, foundation, and soil. This 

system damping ratio then modifies the foundation 

Figure 8-1 Foundation modeling assumptions.

ug= free field motion (FFM) with 

conventional damping

Infinitely rigid foundation and soil

ug= free field motion (FFM) with 

conventional damping

structural components of 

foundation geotechnical components 

of foundation

a) Rigid base model b) Flexible base model

ug= foundation input motion (FIM) 

with conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

ug= foundation input motion (FIM) 

with system damping including 

Adjust for foundation foundation damping

damping

foundation input motion (FIM) with 

conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

c) Kinematic interaction d) Foundation damping

ug= free field motion (FFM) with 

conventional damping

Infinitely rigid foundation and soil

ug= free field motion (FFM) with 

conventional damping

structural components of 

foundation geotechnical components 

of foundation

a) Rigid base model b) Flexible base model

ug= foundation input motion (FIM) 

with conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

ug= foundation input motion (FIM) 

with conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

ug= foundation input motion (FIM) 

with system damping including 

Adjust for foundation foundation damping

damping

foundation input motion (FIM) with 

conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

c) Kinematic interaction d) Foundation damping

1.Superseded in 2003 with the FEMA 450 Recommended

Provisions for Seismic Regulations for New Buildings

and Other Structures.
Chapter 8: Procedures for Including Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 8-3

input motion imparted to the system model as seismic 

shaking demand.

8.2 Procedures for Kinematic Effects

The ground motions imposed at the foundation of a 

structure can differ from those in the free field due to 

averaging of variable ground motions across the 

foundation slab, wave scattering, and embedment 

effects. These effects are referred to here as kinematic 

interaction effects, and they tend to be important for 

buildings with relatively short fundamental periods 

(i.e., periods < ∼ 0.5 s), large plan dimensions, or 

basements embedded 10 feet or more in soil materials. 

This section presents procedures to account for 

kinematic effects on building structures. 

A ratio of response spectra (RRS) factor can be used to 

represent kinematic interaction effects. An RRS is 

simply the ratio of the response spectral ordinates 

imposed on the foundation (i.e., the foundation input 

motion, FIM) to the free-field spectral ordinates. Two 

phenomena should be considered in evaluating RRS: 

base slab averaging and foundation embedment, both of 

which are introduced in the preceding section. Baseslab averaging occurs to some extent in virtually all 

buildings. The slab-averaging effect occurs at the 

foundation level for mats or spread footings 

interconnected by either grade beams or reinforced 

concrete slabs. Even if a laterally stiff foundation 

system is not present, averaging can occur at the first 

elevated level of buildings with rigid diaphragms. The 

only case in which base-slab averaging effects should 

be neglected is in buildings without a laterally 

connected foundation system and with flexible floor 

and roof diaphragms. Foundation embedment effects 

should be considered for buildings with basements. 

Such effects should not be considered for buildings 

without basements, even if the footings are embedded. 

Embedment effects tend to be significant when the 

depth of basements is greater than about 10 feet. The 

following simplified procedure (adapted from Kim and 

Stewart (2003) and other sources) is recommended for 

analysis of these two kinematic interaction effects as a 

function of period, T, of the structural model:

1. Evaluate the effective foundation size , 

where a and b are the full footprint dimensions (in 

feet) of the building foundation in plan view. 

2. Evaluate the RRS from base-slab averaging 

(RRSbsa) as a function of period (see Figure 8-2). 

An approximation to the curves in Figure 8-2 is 

given by the following:

≥ the value for 

T = 0.2 s (8-1)

3. If the structure has a basement embedded a depth e

from the ground surface, evaluate an additional 

RRS from embedment (RRSe) as a function of 

period (see Figure 8-3). The curves in Figure 8-3 

are described by the following:

 the larger of 0.453 or the 

RRSe value for T = 0.2 s. (8-2)

where

e = foundation embedment (in feet) 

vs = shear wave velocity for site soil conditions, 

taken as average value of velocity to a depth 

of be below foundation (ft/s)

n = shear wave velocity reduction factor for the 

expected PGA as estimated from Table 8-1.

4. Evaluate the product of RRSbsa times RRSe to 

obtain the total RRS for each period of interest. The 

spectral ordinate of the foundation input motion at 

each period is the product of the free-field spectrum 

and the total RRS.

b ab e =

Figure 8-2 Ratio of response spectra for base slab 

averaging, RRSbsa, as a function of period, 

T, and effective foundation size, be.

0 0.2 0.4 0.6 0.8 1 1.2

Period (s)

0.4

0.5

0.6

0.7

0.8

0.9

1

S

R

R dl eif- e

erf/ noit adnuo

F

S

R

R( gni gar e

v

a bal s esab m

orf asb )

Simplified Model

be = 65 ft

be = 130 ft

be = 200 ft

be = 330 ft

RRS b

T bsa e = − ⎛

⎝

⎜ ⎞

⎠ 1 ⎟ 1

14 100

1 2

,

.

RRS e

Tnv e

s

= ⎛

⎝

⎜ ⎞

⎠ cos ⎟ ≥ 2π
Chapter 8: Procedures for Including Soil-Structure Interaction Effects

8-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

5. Repeat steps 2 through 4 for other periods if desired 

to generate a complete spectrum for the foundation 

input motion (FIM).

If desired, more detailed procedures can also be used, 

which are described in Appendix E. 

Limitations associated with application of this approach 

include the following, each of which is explained in 

Appendix E:

• Kinematic interaction effects should be neglected for 

soft clay sites such as Site Class E. 

• Embedment effects can be neglected for foundations 

embedded in firm rock (Site Classes A and B). 

• The base-slab averaging model:

a. underestimates reductions in ground motions 

for foundation materials that consist of firm 

rock (Site Classes A and B).

b. has not been rigorously studied for structures 

without large in-plane stiffness (continuous mat 

foundation or footings interconnected with a 

reinforced slab and/or grade beams); however, 

it is considered reasonable to extend its application to all structures except those without both 

an interconnected foundation and rigid floor 

and roof diaphragms.

c. has not been rigorously studied for structures 

with plan dimensions greater than 200 ft.; however, it is considered reasonable to extend the 

application to these conditions, provided that 

the foundation elements are laterally connected.

d. has not been rigorously studied for structures 

with pile-supported foundations; however it is 

considered reasonable to extend application to 

pile-supported structures in which the cap and 

soil are in contact or in which the caps are laterally connected to one another by a slab or grade 

beams.

8.3 Procedures for Foundation Damping

Damping related to foundation-soil interaction can 

significantly supplement damping that occurs in a 

structure due to inelastic action of structural 

components. The damping from foundation-soil 

interaction is associated with hysteretic behavior of soil 

(not to be confused with hysteretic action in structural 

components) as well as radiation of energy into the soil 

from the foundation (i.e., radiation damping). These 

foundation damping effects tend to be important for 

stiff structural systems (e.g., shear walls, braced 

frames), particularly when the foundation soil is 

relatively soft (i.e., Site Classes D-E). 

The effects of foundation damping are represented by a 

modified system-damping ratio. The initial damping 

ratio for the structure neglecting foundation damping is 

referred to as βi

, and is generally taken as 5%. The 

damping attributed to foundation-soil interaction alone 

(i.e., the foundation damping) is referred to as βf

. 

Finally, the damping ratio of the complete structural 

system, accounting for foundation-soil interaction, as 

well as structural damping, is referred to as β0. The 

change in damping ratio from βi

 to β0 modifies the 

elastic response spectrum. The spectral ordinates are 

reduced if β0 > βi

. 

A number of factors influence the foundation damping 

factor βf

 (see Appendix E). Subject to the limitations 

noted below, the following simplified procedure can be 

used to estimate βf

 and the subsequent spectral ordinate 

change due to the modified damping ratio of the 

complete structural system, β0. 

Figure 8-3 Ratio of response spectra for embedment 

RRSe, for an embedment, e, of 30 feet as 

a function of period, T, and shear wave 

velocity, vs.

Table 8-1 Approximate Values of Shear Wave 

Velocity Reduction Factor, n

Peak Ground Acceleration (PGA)

0.10g 0.15g 0.20g 0.30g

n 0.90 0.80 0.70 0.65

0 0.4 0.8 1.2 1.6 2

Period (s)

0

0.2

0.4

0.6

0.8

1

1.2

S

R

R dl ei F- eer F/ noit adnuoF

e = 30 ft

v s = 2500 ft/s

v s = 1200 ft/s

v s = 600 ft/s
Chapter 8: Procedures for Including Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 8-5

1. Evaluate the linear periods for the structural model 

assuming a fixed base, T, and a flexible base, 

using appropriate foundation modeling assumptions. Guidelines for the evaluation of soil spring 

stiffnesses are provided in FEMA 356 and ATC-40. 

In those calculations, the strain-degraded shear 

modulus should be used to represent the soil stiffness. 

2. Calculate the effective structural stiffness of the 

SDOF oscillator for fixed base conditions as

(8-3)

where M*

 is the effective mass for the first mode 

calculated as the total mass times the effective mass 

coefficient (see ATC-40 Eqn. 8-21).

3. Determine the equivalent foundation radius for 

translation as

(8-4)

where Af

 is the area of the foundation footprint if 

the foundation components are inter-connected 

laterally.

4. Calculate the translational stiffness of the foundation, Kx. This can be evaluated using the procedures 

in FEMA 356 (Chapter 4) or ATC-40 (Chapter 10). 

For many applications, the translational stiffness 

can be estimated as 

(8-5)

where G = effective, strain-degraded soil shear 

modulus (see FEMA 356, Table 4.7) and υ = soil 

Poisson’s ratio (∼0.3 for sand, ∼0.45 for clay). 

5. Calculate the equivalent foundation radius for rotation, rθ, by first evaluating the effective rotational 

stiffness of the foundation, Kθ, as

(8-6)

Where h*

 is the effective structure height taken as 

the full height of the building for one-story 

structures, and as the vertical distance from the 

foundation to the centroid of the first mode shape 

for multi-story structures. In the latter case, h* can 

often be well-approximated as 70% of the total 

structure height. The quantity Kx is often much 

larger than K*

fixed, in which case an accurate 

evaluation of Kx is unnecessary and the ratio, 

K*

fixed/Kx, can be approximated as zero. 

The equivalent foundation radius for rotation is 

then calculated as

(8-7)

The soil shear modulus, G, and soil Poisson’s ratio, 

υ, should be consistent with those used in the 

evaluation of foundation spring stiffness.

6. Determine the basement embedment, e, if applicable. 

7. Estimate the effective period-lengthening ratio, 

, using the site-specific structural model 

developed for nonlinear pushover analyses. This 

period-lengthening ratio is calculated for the structure in its degraded state (i.e., accounting for structural ductility and soil ductility). An expression for 

the ratio is

(8-8)

where the term μ is the expected ductility demand 

for the system (i.e., including structure and soil 

effects). Thus, the ductility must be estimated prior 

to the actual solution and subsequently verified. 

8. Evaluate the initial fixed-base damping ratio for the 

structure (βi

), which is often taken as 5%.

9. Determine foundation damping due to radiation 

damping, βf

, based on , e/rx, and h/rθ, 

using the plots in Figures 8-4 and 8-5. An approximation to those curves is given by the following:

 (8-9)

where βf

 is in percent and

T ̃

K M

T fixed

* = ⎛

⎝

⎜ ⎞

⎠

⎟ * 2 2

π

r

A

x

f = π

K Gr x x = −

8

2 υ

K

K h

T

T

K

Kx

θ = ( )

⎛

⎝

⎜

⎞

⎠

⎟ − −

fixed

fixed

* *

*

2

2

1 

r

K

G θ

υ θ = ⎛ ( − )

⎝

⎜

⎞

⎠

⎟

3 1

8

1

3

 ̃

T T/ eff eff

T 

T

T

T

eff

eff

= + ⎛

⎝

⎜

⎞

⎠

⎟ −

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

1 1 1

2

μ ⎪

0 5.

 ̃

T T/ eff eff

β f a

T

T

a

T

T = − ⎛

⎝

⎜

⎞

⎠

⎟ + − ⎛

⎝

⎜

⎞

⎠ 1 2 ⎟

2

1 1  ̃  ̃

eff

eff

eff

eff
Chapter 8: Procedures for Including Soil-Structure Interaction Effects

8-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

(8-9a)

(8-9b)

(8-9c)

The above equations are most applicable for 

 < 1.5, and generally provide conservative 

(low) damping estimates for higher . 

10. Evaluate the flexible-base damping ratio (β0) from 

βi, βf, and as follows:

(8-10)

11. Evaluate the effect on spectral ordinates of the 

change in damping ratio from βi

 to β0 (in accordance with Section 6.3); then modify the spectrum 

of the foundation input motion (recall that foundation input motion is equal to the free-field motion if 

kinematic effects are neglected).

From this point, the maximum expected displacement 

of the nonlinear SDOF oscillator model can be 

estimated using the displacement modification 

procedures of FEMA 356 and Chapter 5 or the 

equivalent linearization procedures of ATC-40 and 

Chapter 6. The ductility demand should be checked 

against the value assumed in Step 7 above. 

The damping ratios determined in accordance with this 

section represent radiation damping effects only. 

Hysteretic soil damping effects are neglected, since 

ductility in soil springs is included as part of structural 

pushover analysis.

Limitations on the damping analysis described above 

include the following:

• The procedure above should not be used when shear 

walls or braced frames are spaced sufficiently 

closely that waves emanating from distinct 

foundation elements will destructively interfere with 

each other across the period range of interest. This 

can effectively decrease the energy dissipated in the 

soil material, and the above formulation could 

overestimate the related damping. Unfortunately, 

this effect has not been investigated sufficiently to 

justify definitive limits. In the absence of such 

limits, a reasonable approximation might be to 

neglect the effect of softly-coupled foundation 

Figure 8-4 Example of foundation damping, βf, as a 

function of effective period lengthening 

ratio, , for constant 

embedment, e/rx = 0, and various values 

of foundation stiffness rotational stiffness, 

h/rθ.

1 1.5 2

0

10

20

30

o

F nu ad it on D p m

a i , gn β ) %(

(radiation damping only)

1.0

2.0

1 1.5 2

0

10

20

30

o

F nu ad it on D p m

a i , gn f ) %(

(radiation damping only)

1.0

2.0

e / rx = 0

h / rθ = 0.5

Period Lengthening, Teff/Teff

~

 ̃

T T/ eff eff

a c hr 1 e = − exp . . / (47 16 θ )

a c hr 2 e = ⎡25 16 ( ) − ⎣ ⎤

⎦ ln / θ

c er e x = 15 1 . / ( ) +

 ̃

T T/ eff eff  ̃

T T/ eff eff

 ̃

T T/ eff eff

β β β

0 3 = +

( ) f

i

T T  ̃ / eff eff

Figure 8-5 Example of foundation damping, βf, as a 

function of effective period lengthening 

ratio, , for constant 

embedment, e/rx = 0.5, and various 

values of foundation stiffness rotational 

stiffness, h/rθ.

1 1.5 2

0

10

20

30

, gni p m

a

Dnoit adnuoF ) %(

(radiation damping only)

1.0

2.0

1 1.5 2

0

10

20

30

, gni p m

a

Dnoit adnuo

F βf ) %(

e / rx = 0.5

(radiation damping only)

1.0

2.0

Period Lengthening, Teff/Teff

~

h / rθ = 0.5

 ̃

T T/ eff eff
Chapter 8: Procedures for Including Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 8-7

components spaced at a distance less than the larger 

dimension of either component in the corresponding 

direction. Further discussion is presented in 

Appendix E, Section E.3.1.5. 

• The analysis can be conservative (underpredicting 

the damping) when foundation aspect ratios exceed 

about 2:1. Further discussion is presented in 

Appendix E, Section E.3.1.4. 

• The analysis is conservative when foundations are 

deeply embedded, e/rx > 0.5. Further discussion is 

presented in Appendix E, Section E.3.1.3.

• The analysis is unconservative (overpredicting the 

damping) if vsT/rx > 2π (where vs = average shear 

wave velocity to a depth of about rx) and the 

foundation soils have significant increases of shear 

stiffness with depth. Further discussion is presented 

in Appendix E, Section E.3.1.2. 

• The analysis is unconservative if the foundation soil 

profile consists of a soil layer overlying a very stiff 

material (i.e., there is a pronounced impedance 

contrast within the soil profile), and if the system 

period is greater than the first-mode period of the 

layer. Further discussion is presented in Appendix E, 

Section E.3.1.2. 
Chapter 8: Procedures for Including Soil-Structure Interaction Effects

8-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 9-1 

9. Multiple-Degree-of-Freedom Effects

9.1 Introduction

One of the primary assumptions of nonlinear static 

analysis procedures is that the behavior of a structure 

with multiple degrees of freedom (MDOF) subject to 

seismic ground motion can be estimated from the 

response of an oscillator with a single degree of 

freedom (SDOF). In order to generate the SDOF model, 

the engineer generates a global force-deformation 

relationship for the structure by subjecting a MDOF 

model to a predetermined lateral load vector. This 

relationship is then converted to an equivalent SDOF 

representation to estimate the maximum global 

displacement of the model using displacementmodification or equivalent linearization techniques. The 

global displacement is typically monitored at the roof 

level or center of mass. The magnitude of localized 

demand in the MDOF model (e.g., story drifts and 

forces or component deformations) are directly related 

to the global displacement. The actual maximum global 

displacement for the MDOF system can differ from the 

equivalent SDOF approximation. The distribution of 

localized demand depends on the assumptions 

associated with the load vector used to generate the 

equivalent SDOF model. The distribution of forces on 

the structure changes continuously during an 

earthquake. In the elastic range, this is attributable to 

the fact that the response comprises contributions from 

multiple modes of vibration. The actual distribution is 

difficult to assess since the dynamic characteristics of 

the ground motion itself are a major influence. 

Inelasticity further complicates the situation. The 

combined deviations of the actual distribution of forces 

and deformations from those associated with the 

equivalent SDOF system and the assumed load vector 

are termed MDOF effects. They can result in maximum 

inelastic response in components or elements that differ 

from the SDOF model predictions in nonlinear static 

analysis. 

This section reviews the accuracy and practical 

implications of the requirements of ATC-40 and FEMA 

356 related to MDOF effects including:

1. current options for load vectors, and 

2. the conversion of a MDOF pushover curve to an 

equivalent SDOF system.

The results of a comprehensive study of five example 

buildings that examines the differences in response 

predicted using various options compared to a common 

nonlinear dynamic analyses benchmark are also 

summarized. Finally, this chapter provides 

recommendations for practical applications and 

identifies promising developments for the future.

9.2 Review of Current Simplified 

Procedures

There are a number of options for the form of the load 

vector used to generate the SDOF model of a structure. 

Some are based on a single vector and one uses several 

vectors applied to comprise a multi-mode pushover 

approach. In all the options, lateral forces are applied 

incrementally to a nonlinear structural model to 

generate a “pushover” or capacity curve representing 

the relationship between the applied lateral force and 

the global displacement at the roof or some other 

control point. The applied lateral force at any level in 

the structure is proportional to the mass at that level and 

an acceleration determined from a specific shape vector 

assumption. The various options are summarized below, 

as are the specifications of ATC-40 and FEMA 356 

related to MDOF effects.

9.2.1 Single-Mode Load Vectors

Concentrated Load. The simplest assumption for a 

load vector is a single concentrated load located 

normally at the top of the structure. 

Uniform. A uniform load vector assumes that the 

acceleration in the MDOF model is constant over its 

height. This alternative is sometimes termed 

“rectangular.”

Triangular. A triangular-shaped vector assumes that the 

acceleration increases linearly from zero at the base to a 

maximum at the top of the MDOF model.

Code Distribution. The “code” load pattern appears in 

many documents. The acceleration pattern varies from 

the triangular shape for periods less than 0.5 s to a 

parabolic shape for periods greater than 2.5 s, as a 

means to account for higher-mode effects. 

First Mode. The first-mode technique applies 

accelerations proportional to the shape of the first mode 

of the elastic MDOF model. 

Adaptive. The adaptive procedure uses the first mode 

and recognizes that softening of the capacity curve 
Chapter 9: Multiple-Degree-of-Freedom Effects

9-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

reflects a reduction in stiffness, which, in turn, causes a 

change in the mode shape. Thus, lateral forces are 

applied in proportion to the amplitude of an evolving 

first-mode shape and the mass at each level within the 

MDOF model.

SRSS. The square-root-of-the-sum-of-the-squares 

(SRSS) technique is based on elastic modal responses. 

The response in each mode has a lateral force pattern, 

which can be summed to obtain story shears associated 

with each mode. An SRSS combination of the modal 

story shears results in a particular shear profile, referred 

to as the SRSS story shears. The lateral forces required 

to generate the SRSS story shear profile are applied to 

the MDOF model in this pushover technique. The 

elastic spectral amplitudes and modal properties are 

used even when nonlinear response is anticipated. A 

sufficient number of modes to represent at least 90% of 

the mass is generally included.

9.2.2 Multi-Mode Pushover Procedures

Multi-mode pushover analysis procedures consider 

response in several modes. Approaches have been 

described by various investigators such as Sasaki et al. 

(1998), Reinhorn (1997), Chopra and Goel (2002), and 

Jan et al. (2004). Chopra and Goel (2001b) describe an 

approach in which pushover analyses are conducted 

independently in each mode, using lateral-force profiles 

that represent the response in each of the first several 

modes. Response values are determined at the target 

displacement associated with each modal pushover 

analysis. Response quantities obtained from each modal 

pushover normally are combined using the SRSS 

method. Although response in each mode may 

potentially be nonlinear, the mode shapes and lateralforce profiles are assumed to be invariant in this 

analysis procedure. Target displacement values may be 

computed by applying displacement modification or 

equivalent linearization procedures to an elastic 

spectrum for an equivalent SDOF system representative 

of each mode to be considered. Chopra and Goel 

(2001d) and Yu et al. (2001) illustrate the method using 

SRSS combinations of floor displacement, interstory 

drift, and component deformation (plastic hinge 

rotations). 

9.2.3 Summary of Current Provisions

FEMA 356. FEMA 356 (Section 3.3.3.2.3) requires 

that two separate nonlinear static analyses be done, each 

using different load vectors. For each response quantity 

of interest, the larger value of the two analyses is 

compared to the applicable acceptability criteria.

One load vector is selected from the following list. 

• Code distribution—Restricted to the cases in which 

more than 75% of mass participates in first mode, 

and the second vector must be the uniform 

distribution.

• First mode—Restricted to the cases in which more 

than 75% of mass participates in first mode. 

• SRSS of modal story loads – This option must be 

used if Te > 1 s.

A second load vector is selected from the following 

options. 

• Uniform distribution or 

• Adaptive load distribution. 

In FEMA 356 (Section 2.4.2.1), the use of NSPs must 

be supplemented with a linear dynamic analysis if any 

SRSS story shear from a response-spectrum analysis 

including modes representing 90% of the mass exceeds 

130% of the corresponding story shear from a firstmode response-spectrum analysis.

The yield displacement, ∆y, of the equivalent SDOF 

system is effectively determined as 

(9-1)

where ∆y,roof = the roof displacement at yield, and 

Γ1 = the first-mode participation factor. 

In the FEMA 356 approximation, it can be shown that 

the yield strength coefficient of the equivalent SDOF 

system is approximated as

(9-2)

where Sa = the pseudo-acceleration associated with 

yield of the ESDOF (Equivalent SDOF) system, g = the 

acceleration of gravity, Vmdof = the yield strength of the 

MDOF system, W = the weight of the MDOF system. 

This simplification relies on the approximation Γ1 ≈ 1/

α1, where α1 is the modal mass coefficient.

ATC-40. The primary recommendation in ATC-40 

(Section 8.2.1) for load vectors is to use the first mode. 

However, the guidelines recognize a hierarchy of other 

options, arranged here in order of preference. 

∆

∆

Γ y

y = ,roof

1

C S

g

V

W y a mdof = = Γ1
Chapter 9: Multiple-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 9-3

1. Concentrated load

2. Code distribution

3. First mode

4. Adaptive

5. Multi-mode pushover

The guidelines also note that pushover analyses using 

the first-mode shape are generally valid for structures 

with fundamental periods up to about one second. They 

suggest that the engineer might want to consider multimode pushover for structures with longer periods.

In the ATC-40 method, the yield displacement of the 

equivalent SDOF is the same as that of FEMA 356; 

however the yield strength coefficient of the equivalent 

SDOF system is given by

(9-3)

where Sa = the pseudo-acceleration associated with 

yield of the ESDOF system, g = the acceleration of 

gravity, Vy, mdof = the yield strength of the MDOF 

system, W = the weight of the MDOF system, and 

α1 = the modal mass coefficient.

9.3 Summary of Illustrative Examples 

In order to compare and illustrate the effects of the 

various options with NSPs related to the effects of 

higher modes, five example buildings were analyzed. 

Detailed information and results of the analyses are 

contained in Appendix F. The basic description of the 

example buildings and the other features of the analysis 

are listed below. 

Example Buildings

• Three-story, steel frame (SAC LA Pre-Northridge 

M1 Model)

• Three-story, weak-story steel frame (lowest story at 

50% of strength)

• Eight-story, shear wall (Escondido Village)

• Nine-story, steel frame (SAC LA Pre-Northridge M1 

Model)

• Nine-story, weak-story steel frame (lowest story at 

60% of strength)

Ground Motions

• Eleven site Class C motions, 8-20 km from the fault 

rupture, five events 

• Four near-field motions: Erzincan, Northridge 

(Rinaldi Receiving Station & Sylmar County 

Hospital), and Landers earthquakes

Global Drift Levels

Ordinary motions (scaled to result in specified global 

drift)

• 0.5, 2, 4% drift, as a percentage of building height, 

for frames 

• 0.2, 1, 2% drift, as a percentage of building height, 

for wall

Near-field (unscaled)

• 1.8 to 5.0% for 3-story frames, 1.7-2.1% for 9-story 

frames

• 0.6 – 2.1% drift, as a percentage of building height, 

for wall

Load Vectors

• Triangular

• Uniform

• Code

• First mode

• Adaptive

• SRSS

• Multi-mode pushover

Response Parameters

• Floor and roof displacement

• Interstory drift

• Story shear

• Overturning moment

Error Measurements

• Mean over all floors

• Maximum over all floors

9.3.1 Load Vectors

For analyses using the ordinary ground motions, each 

motion was scaled to result in the pre-determined levels 

of total drift at the roof for each example building in the 

C S

g

V W

y a mdof = = α1
Chapter 9: Multiple-Degree-of-Freedom Effects

9-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

nonlinear response-history analysis of the MDOF 

models. The resulting response parameters served as the 

basis for comparison with nonlinear static analyses 

using the various options for load vectors. Observations 

from the comparisons are summarized as follows:

• Anomalous capacity curves resulted because the 

roof displacements reversed in two of the highermode pushover analyses. Consequently, the Modal 

Pushover Analysis procedure described by Chopra 

and Goel (2001b) could not be applied without 

modification to the examples. In order to represent 

higher-mode contributions, a multiple mode 

calculation procedure was introduced in the ATC-55 

project. In this procedure, response quantities for the 

2nd and 3rd mode were determined under the 

assumption that the response in these modes is 

elastic. A conventional inelastic pushover analysis 

was used for response in the first mode. Floor 

displacement, interstory drift, story shear, and 

overturning moments were determined as an SRSS 

combination of the modal responses in the first three 

modes. Motivated by review of early results of these 

analyses, Chopra et al. (2004) have investigated this 

approach, described as a “modified MPA,” in 

comparison with the original MPA procedure.

• All the simplified procedures evaluated resulted in 

good estimates of peak displacements over the 

height of the five example buildings (Figure 9-1) 

when compared with nonlinear dynamic responsehistory analysis results. Estimates made using the 

first-mode, triangular, and adaptive load vectors 

were best. A multiple mode procedure may be 

warranted for structures in which displacement 

response is suspected to be predominantly in a 

higher mode. 

• The dispersion in the displaced shapes of the weakstory buildings was pronounced at the moderate drift 

levels. This is likely due to the fact that weak-story 

mechanisms did not always develop at these levels 

of roof drift. This is illustrated by comparing the 

dispersion in floor displacements of the nine-story, 

weak story frame building at 2% roof drift 

(Figure 9-2a) that is actually greater than that for the 

same building at 4% drift (Figure 9-2b).

• Good estimates of interstory drift were obtained 

over the height of the three-story frames and eightstory wall using the first-mode, triangular, code, 

adaptive, and SRSS load vectors, as well as with the 

modified MPA procedure (Figure 9-3). 

• Interstory drifts estimates over the height of the 

nine-story buildings were poor for the single-mode 

load vectors (see Figure 9-4). The results using the 

modified MPA procedure were consistently better 

than those obtained with the single load vectors, 

although the interstory drift values were still 

underestimated at some locations in the nine-story 

frames. Similar results are reported by Goel and 

Chopra (2004). 

• The maximum interstory drift over the height of 

each building model, determined using the singlemode load vectors (excluding the uniform load 

vector), was a reasonable estimate of the maximum 

interstory drift occurring at that particular location in 

the nonlinear dynamic analyses. This drift was also a 

reasonable estimate of the maximum interstory drift 

that developed over the height of each building 

model in the nonlinear dynamic analyses 

(Figures 9-3 and 9-4), although these estimates 

depended to some extent on the load vector selected. 

Also, drifts at other locations predicted with the load 

vectors often did not correspond to those from the 

nonlinear dynamic analyses.

• Estimates of story shear and overturning moment for 

the three-story frames (Figure 9-5) were not as 

accurate as the displacement and interstory drift 

estimates (Figure 9-3a). These quantities typically 

were underestimated using the single load vectors 

and overestimated using the modified MPA 

procedure. The tendency for the modified MPA 

procedure to overestimate forces and moments is not 

surprising, as SRSS combinations of these quantities 

can exceed limits associated with the development 

of an inelastic mechanism and depend on the 

number of modes included in the combination. 

• Estimates were inconsistent and often poor for story 

shears and overturning moment for the eight-story 

wall and nine-story frames (Figure 9-6). Although 

the overall pattern of overturning moments was 

often correct, errors in the estimates of overturning 

moment were often substantial, particularly for the 

upper floors. Similar results are reported by 

Krawinkler and Seneviratna (1998) and Gupta and 

Krawinkler (2003).

• The accuracy of the simplified procedures was 

similar for the set of Site Class C motions and for the 

set of near-field motions that was considered.

9.3.2 Equivalent SDOF Estimates of Global 

Displacement

For each example building, the force-displacement 

relationship generated with the first-mode vector was 

converted to an equivalent SDOF system using the 
Chapter 9: Multiple-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 9-5

Figure 9-1 Example results for displacements predicted by nonlinear static procedures (NSP) compared to nonlinear 

dynamic response-history analyses (NDA). 

(b) Eight-story wall building at 2% drift (c) Nine-story frame at 4% drift 

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

(a) Three-story frame building at 4% drift 

NDA NSP Load Vectors
Chapter 9: Multiple-Degree-of-Freedom Effects

9-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

procedures of both FEMA 356 and ATC-40. These 

models were then subjected to scaled ground-motion 

records. A displacement ratio was defined as the ratio of 

the estimated roof displacement and the peak roof 

displacement obtained in the nonlinear response-history 

analysis. Results are reported in detail in Appendix F 

and summarized below. 

• In cases in which the post-yield stiffness of the 

capacity curve is positive (with or without P-∆

effects present), mean displacement ratios obtained 

using the ATC-40 formulation were between 

approximately 0.95 and 1.25 for the five buildings. 

Within this range, mean displacement ratios tended 

to increase with increasing roof drift. 

• Similar mean displacement ratios were obtained 

with the FEMA 356 formulation, although 

dispersions were larger for this formulation. 

Accuracy was similar for the near-field motions.

• In cases in which the post-yield stiffness of the 

capacity curve is negative (due to P-∆ effects), 

equivalent SDOF systems can have excessive 

displacement response, leading to overestimates of 

the peak roof displacement. For such cases, 

nonlinear dynamic analysis of the MDOF structure 

may be more accurate.

9.4 Practical Implications

NSPs can provide reliable estimates of maximum 

displacement. They are also capable of providing 

reasonable estimates of the largest interstory drift that 

may occur at any location over the height, but are 

Figure 9-2 Dispersion in results for displacement for two levels of global drift.

(a) Nine-story weak story frame at 2% drift (b) Nine-story weak story frame at 4% drift 

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

NDA NSP Load Vectors
Chapter 9: Multiple-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 9-7

limited in the capability to predict drift accurately over 

the full height of relatively tall, flexible MDOF 

structures. In contrast, interstory drift over the height of 

the three-story frames and eight-story shear wall 

example buildings were estimated well. Nonlinear static 

procedures that combine contributions from 

independent modal analyses appear to be poor 

predictors of story shear and overturning moment. 

These observations are consistent with the results of a 

number of other research efforts (Seneviratna and 

Krawinkler, 1994; Krawinkler and Seneviratna, 1998; 

Kunnath and Gupta, 2000; Lew and Kunnath, 2000; Yu 

et al., 2001; Chopra and Goel, 2001b; Gupta and 

Krawinkler, 2003; Goel and Chopra, 2004; and Jan et 

al., 2004). This situation raises a number of questions 

with regard to the practical application of NSPs in cases 

in which MDOF effects are important. First, is there any 

preference for any one load vector over the others? 

Second, when should results of NSPs not be relied upon 

for MDOF effects? Finally, what should be done now 

and in the future?

9.4.1 Single Load Vectors

The first-mode load vector is recommended because of 

the low error obtained for displacement estimates made 

with this assumption and to maintain consistency with 

the derivations of equivalent SDOF systems. The code 

distribution and the triangular vectors may be used as 

alternatives, typically with little increase in error. 

Mean and maximum errors were sometimes smaller and 

sometimes larger using the adaptive load vector. The 

adaptive method requires more computational effort 

and fails for systems exhibiting a negative tangent 

stiffness. 

Figure 9-3 Relatively good results for interstory drift predicted using nonlinear static procedures (NSP), as compared 

to nonlinear dynamic response-history analyses (NDA). 

(a) Interstory drifts for three-story story weak 

 frame building at 4% drift 

(b) Interstory drifts for eight-story wall 

 building at 2% drift 

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

NDA NSP Load Vectors
Chapter 9: Multiple-Degree-of-Freedom Effects

9-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure 9-4 Relatively poor results for interstory drift predicted using nonlinear static procedures (NSP) compared to 

nonlinear dynamic response-history analyses (NDA). 

0 0.2 0.4 0.6 0.8 1

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Story

Interstory Drift (%)

0.5% Drift

0 0.2 0.4 0.6 0.8 1

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Story

Interstory Drift (%)

0.5% Drift

(a) Nine story regular building at 0.5% roof drift

012345

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Story

Interstory Drift (%)

2% Drift

012345

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Story

Interstory Drift (%)

2% Drift

012345

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Story

Interstory Drift (%)

2% Drift

(b) Nine story regular building at 2% roof drift

0 5 10 15 20

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Story

Interstory Drift (%)

4% Drift

0 5 10 15 20

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Story

Interstory Drift (%)

4% Drift

0 5 10 15 20

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Story

Interstory Drift (%)

0 5 10 15 20

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Story

Interstory Drift (%)

4% Drift

(c) Nine story regular building at 4% roof drift

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

NDA NSP Load Vectors
Chapter 9: Multiple-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 9-9

Figure 9-5 Story forces and overturning moments in the example three-story frame building when different load 

vectors are used. 

Figure 9-6 Story forces and overturning moments in eight-story wall and nine-story frame example buildings, using 

various load vectors. 

(a) Story shears for three-story frame building 

 at 4% drift

(b) Overturning moments for three-story frame

 building at 4% drift

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

NDA NSP Load Vectors

(a) Story shears for the eight-story wall 

 building at 1% drift

0 50000 100000 150000 200000 250000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Overturning Moment (kips-ft)

0 50000 100000 150000 200000 250000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Overturning Moment (kips-ft)

0 50000 100000 150000 200000 250000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Overturning Moment (kips-ft)

(b) Overturning moments for the nine-story 

 frame at 4% drift

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

NDA NSP Load Vectors
Chapter 9: Multiple-Degree-of-Freedom Effects

9-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

The SRSS load vector led to small improvements in 

story shear and overturning moment for the example 

frames, had minor and mixed effects for interstory drift, 

and sometimes was worse for estimates of 

displacement, when compared to the first-mode load 

vector. It requires greater computational effort for 

inconsistent improvements.

The uniform load vector led to notably worse errors for 

all four response quantities in the example buildings, 

relative to the first-mode load vector. Thus, it is not 

recommended as a stand-alone option. Although the use 

of the uniform load vector in conjunction with another 

vector as a bounding function (e.g., in the case of a 

shear wall building to ensure flexurally controlled 

behavior) is appealing, peak response quantities often 

exceeded the estimates made with the uniform vector.

The use of multiple load vectors in FEMA 356 implies 

unwarranted accuracy and does not provide reliable 

results. A single first-mode vector is sufficient for 

displacement estimates and for the estimate of response 

quantities that are not significantly affected by higher 

modes.

9.4.2 Multi-Mode Pushover Analysis

It is apparent and logical that the use of multiple mode 

pushover techniques (MPA) should produce generally 

better estimates of interstory drift than single load 

vectors. Although higher modes typically contribute 

little to displacement, multiple mode pushover analyses 

may be useful for identifying cases in which 

displacement responses are dominated by a higher 

mode. 

The application of the multi-mode pushover analysis 

(MPA) procedure in the ATC-55 project was 

encumbered by the reversals observed in two of the 

higher-mode pushover curves. Seeking a single 

approach capable of representing higher-mode 

contributions, a modified MPA procedure was 

introduced in these studies. Although often improved 

over the single-mode vectors, estimates of interstory 

drift over the full height of buildings made with the 

modified MPA procedure may not be consistently 

reliable. However, it is important to note that 

researchers are devoting significant effort to the further 

development of MPA procedures. Some of these are 

briefly described below. 

• Chopra and Goel (2001b) found the original MPA 

provided good estimates of floor displacement and 

story drift, but did not estimate plastic hinge 

rotations with acceptable accuracy for a nine-story 

steel moment-frame building.

• Chintanapakdee and Chopra (2003) applied the 

MPA procedure to estimate interstory drift for socalled “generic” frames having 3, 6, 9, 12, 15, and 

18 stories. They found that the accuracy of interstory 

drift estimates depend on the story level and degree 

of inelasticity. Accuracy was best for shorter 

buildings and for the lower and middle stories of 

taller buildings. For the upper stories of tall frames, 

the MPA procedure was not able to provide a 

reasonable estimate of interstory drift for many 

ground motions. The procedure was not used to 

determine shear, bending moment, axial force, or 

component deformation. 

• Yu et al. (2002) applied the original MPA and two 

modified versions of MPA to estimate the interstory 

drift and plastic hinge rotation for an instrumented 

13-story steel frame building. When target 

displacements were estimated by applying the 

displacement Coefficient Method to the median 

elastic response spectrum, the MPA method tended 

to underestimate story drifts in the upper stories and 

to overestimate drifts in the lower stories; beam and 

column plastic hinge rotations were often 

overestimated, while panel zone deformations were 

estimated reasonably well. 

• Chopra et al. (2004) compared interstory drift 

estimates obtained using the original and modified 

MPA methods for a set of “generic” frames and SAC 

frames and found the modified MPA method is an 

attractive alternative to the original MPA, because it 

leads to a larger estimate of seismic demand, thereby 

improving the accuracy of the MPA results in some 

cases and increasing their conservatism in others.

• Goel and Chopra (2004) describe an “improved” 

version of the MPA, which considers P-∆ effects in 

all modes considered and which adds a specialized 

step for estimating plastic hinge rotation on the basis 

of the estimated interstory drift and an assumed 

inelastic mechanism. The “improved” MPA 

procedure, although better than single-mode 

estimates, is found to lack accuracy for estimating 

plastic hinge rotation, overestimating the rotation in 

the lower stories and underestimating it in the upper 

stories of the 9- and 20-story moment-resisting 

frames that were studied. 

• Jan et al. (2004) propose an alternative technique in 

which potentially inelastic contributions from the 

first two modal pushover analyses are added 

together. Estimates of displacement, interstory drift, 
Chapter 9: Multiple-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 9-11

and plastic hinge rotation were compared with those 

made using a triangular load profile and the original 

MPA procedure for a set of 2-, 5-, 10-, 20-, and 30-

story moment-resisting frames. The triangular load 

profile and the original MPA produced very good 

estimates of interstory drift for the 2- and 5-story 

frames. The proposed technique provided better 

estimates of interstory drift for the 20- and 30-story 

frames, and it was the only technique of those 

considered it that could provide reasonable estimates 

of the location and severity of plastic hinge rotations 

in these frames.

• Hernández-Montes et al. (2004) developed an 

energy-based pushover technique that overcomes the 

problems observed with reversals of the highermode pushover curves that were observed in the 

application of the original MPA procedure in the 

ATC-55 studies. 

The MPA procedures seem to produce results that are 

somewhat more reliable than those obtained from single 

load vectors. However, it is readily apparent from the 

literature that the adequacy of these results depends 

upon the parameter of interest (e.g., drift, plastic hinge 

rotation, force), the characteristics of the structure, and 

the details of the specific procedure. It is also possible 

that future development of the basic MPA procedure 

may improve predictions further. If these improvements 

can be realized with transparent and computationally 

efficient procedures, then they may very well be 

worthwhile. On the other hand, MPA procedures are 

fundamentally limited, as are NSPs more generally. 

From a broader perspective, it is important to develop 

practical versions of nonlinear dynamic responsehistory analyses of detailed and, perhaps, simplified 

MDOF models.

Until other practical nonlinear alternatives are 

available, the recommendation is that experienced 

practitioners, who interpret results with an appropriate 

degree of caution, can utilize MPA procedures for 

comparison with, and possible improvement over, the 

static load vector procedures. 

9.4.3 Roof Displacement Estimation

The results for the estimate of maximum global 

displacement of the example building models are 

consistent with the results of other studies (e.g., 

Miranda, 1991; Collins et al., 1995; Seneviratna and 

Krawinkler, 1997; Cuesta and Aschheim, 2001; Chopra 

et al., 2003). The ATC-40 formulation for the yield 

strength coefficient of an equivalent SDOF 

(Equation 9-3) is recommended, because it resulted in 

smaller dispersions, accurately reflected the frequency 

content of the excitation for elastic response, and 

maintains consistency with derivations of “equivalent” 

SDOF systems. Where the hazard is described by 

smoothed elastic design spectra, displacement estimates 

should make use of the improved procedures that are 

described in Chapters 5 and 6.

9.4.4 Limitation of Simplified Procedures

Nonlinear static pushover procedures appear to be 

reliable for the design and evaluation of low-rise 

buildings. However, MDOF effects associated with the 

presence of significant higher-mode response in 

relatively tall frame buildings, can cause interstory drift, 

story shear, overturning moment, and other response 

quantities to deviate significantly from estimates made 

on the basis of single-mode pushover analyses. Multimode pushover procedures appear capable of more 

reliable estimates than do single-mode procedures; 

however, they cannot be deemed completely reliable 

based on currently available data. The dividing line 

between buildings for which reliable results can be 

obtained using NSPs and those for which the results 

cannot be relied upon is nebulous. The sufficiency of 

nonlinear static procedures and the need for nonlinear 

dynamic analysis depend on a number of related 

considerations. 

• Response quantity of interest. As illustrated in the 

examples, current simplified procedures are often 

adequate for estimating displacements. They seem to 

produce reasonable estimates of interstory drift for 

low-rise frame buildings and wall buildings. 

However, for virtually all cases, the simplified 

procedures produce unreliable estimates of story 

shear and overturning moments. If required for 

evaluation or design, accurate estimates of these 

parameters require more detailed analyses.

• Degree of inelasticity. The example buildings 

indicate that the importance of MDOF effects 

increases with the amount of inelasticity in the 

structure. NSPs may be adequate for situations in 

which the performance goals for a structure are such 

that only slight or moderate levels of inelasticity are 

expected.

• Periods of vibration of the fundamental and higher 

modes relative to the spectral demands at these 

periods. Higher-mode contributions become more 

significant for structures with fundamental periods 

that fall in the constant-velocity portion of the 

response spectrum. It appears that accurate estimates 

of the distribution of interstory drift over the height 

of moment-resisting frames cannot be obtained with 
Chapter 9: Multiple-Degree-of-Freedom Effects

9-12 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

NSPs alone when the fundamental period of the 

structure exceeds approximately twice the 

characteristic site period, Ts. A significantly lower 

limit applies to the determination of story forces in 

both wall and frame structures, however.

• Structural system type. Shear walls and frames have 

different higher-mode periods relative to their 

fundamental modal periods. These systems have 

characteristically different percentages of mass 

participating in the first and higher modes and 

develop characteristically different types of 

mechanisms. As noted previously, NSPs do not 

predict story forces reliably, and more sophisticated 

analytical techniques may be required for systems 

sensitive to these parameters.

• Post-elastic strength. Both the studies on the 

response of SDOF oscillators (Chapter 3) and the 

SDOF examples (Appendix F) demonstrate that 

systems with a critical level of negative post-elastic 

strength degradation are prone to dynamic 

instability. This has been documented in other recent 

research as well. As discussed in Chapter 4, the 

critical post-elastic stiffness should be based on P-∆

effects and other types of in-cycle degradation. 

Systems with strength values less than those 

specified in Chapter 4 require nonlinear responsehistory analysis.

• Inelastic mechanism. Forces associated with 

response in other modes may influence the 

development of an inelastic mechanism, and thus, 

pushover analyses may not always identify the 

governing mechanism (Krawinkler and Seneviratna, 

1998).

• Multi-mode pushover analysis procedures. SRSS 

combinations of force quantities can exaggerate the 

effects of gravity loads and can exceed the limits 

associated with the development of an inelastic 

mechanism. Typically, algebraic signs of the modes 

can be expected to influence the intensity of 

component demands. The use of uniform hazard 

spectra presents inconsistencies, because different 

portions of the spectrum may be driven by vastly 

different events, rather than representing a single 

event.

• FEMA 356 provisions. This document requires 

supplementary linear dynamic analysis if highermode effects are significant. Higher modes are 

considered significant if the SRSS of story shears 

from modes that incorporate at least 90% of the 

mass exceeds 130% of story shear from a first-mode 

response-spectrum analysis. It is important to note 

that all the example buildings, with the minor 

exception of the upper floor of the 9-story frame, 

would have qualified for the nonlinear static 

procedure alone without the linear dynamic 

procedure (LDP) for a NEHRP design spectrum in 

an area of high seismicity and Site Class C site 

conditions. The potential for the NSP to significantly 

underestimate response quantities for structures that 

satisfy this limitation indicates that the current 

limitation is not adequate. 

9.5 Potential Future Improvements

Based on the studies conducted in conjunction with this 

document and results from current research, it is 

apparent that there is a need for improved inelastic 

analysis techniques that can be used to reliably address 

MDOF effects. As noted previously, research on multimode pushover analysis procedures is ongoing. There 

are two examples of potential improvements that have 

not been discussed earlier and that warrant mention 

here.

9.5.1 Incremental Response-Spectrum Analysis

Aydinoglu (2003) describes a multi-modal incremental 

response-spectrum analysis method, in which 

contributions of multiple modes are considered in an 

incremental pushover analysis. The incremental nature 

of the analysis allows the effects of softening due to 

inelasticity in one mode to be reflected in the properties 

of the other modes. An example was used to illustrate 

application of the method to a generic frame model of 

the nine-story SAC building (neglecting gravity loads 

and P-∆ effects), comparing estimates based on four 

modes with those determined by nonlinear dynamic 

analysis. Very good agreement is shown for floor 

displacement, interstory drift, story shear, floor 

overturning moment, and beam plastic hinge rotation. 

Further study is required to establish the generality of 

the findings and potential limitations of the approach.

9.5.2 Nonlinear Dynamic Procedure Using 

Scaled Response Histories

The MDOF example studies summarized in Section 9.3 

revealed that the estimates of response quantities 

obtained by nonlinear static pushover analyses often 

were less accurate than the results obtained by any 

single nonlinear dynamic analysis when comparing 

both to the mean results for all ground motions. This 

observation suggests the possibility of an analytical 

procedure in which response quantities are determined 

by nonlinear dynamic analysis using ground motion 

records that are scaled so that the peak roof 
Chapter 9: Multiple-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 9-13

displacement matches a predetermined target 

displacement. In effect, the seismic hazard would be 

characterized by the maximum inelastic displacement at 

the roof level. This displacement could be estimated for 

a structure using nonlinear static procedures in 

conjunction with the NEHRP maps, for example. Thus, 

nonlinear response-history analyses would be used to 

investigate MDOF effects through nonlinear dynamic 

analyses using a relatively small number of ground 

motion records scaled to give the same roof 

displacement. Such a procedure could avoid both the 

necessity of generating a series of spectrum-compatible 

records and the difficulty of combining the results of the 

analyses for practical use. This potential method, 

termed the “Scaled NDP” method, is summarized here, 

with supporting information provided in Appendix F 

The basic suggested procedure is outlined below. 

Step 1. Given a spectrum representative of the site 

hazard of interest, estimate the peak displacement of the 

roof (or more generally, a “control point”) using the 

displacement modification or equivalent linearization 

procedures, described in Chapters 5 and 6, respectively. 

Step 2. Select n ground motion records that reflect the 

characteristics of the hazard (e.g., magnitude, distance, 

and site class) and for each record, conduct a nonlinear 

dynamic analysis, with the record scaled iteratively 

until the peak displacement of the control point is equal 

to the estimate determined in Step 1. Extract peak 

values of the response quantities of interest from the 

results of each analysis and compute the sample mean, 

, of each peak quantity of interest. At least three 

analyses (n ≥ 3) are suggested.

Step 3. Although the sample mean is the best estimate 

of the true mean, sampling error may be present. 

Furthermore, estimates of some response quantities 

may be desired at the mean plus κ standard deviation 

level. Thus, the sample mean could be multiplied by a 

coefficient that depends on the coefficient of variation 

of the sample, in order to estimate a response quantity at 

the mean plus κ standard deviation level with a desired 

level of confidence. A derivation of such a coefficient is 

provided in Appendix F

Discussion

The proposed analysis method retains the benefits of the 

pushover method, in that the engineer can use the 

pushover to quickly identify the likely nonlinear 

mechanism of the system and the expected peak 

displacement response. The method makes use of 

currently available spectral descriptions of seismic 

hazard as well as the improvements described in 

Chapters 5 and 6 of this document. The dynamic 

analyses indicate the variability in response quantities 

associated with the randomness in the higher mode 

amplitudes and timing relative to the first mode. In 

effect, the static load vector of traditional pushover 

analysis, used to determine the peak of global 

displacement demand, is augmented by a dynamic load, 

represented by the scaled ground motion record. 

Within the limitations of the nonlinear model, each 

analysis faithfully represents the influence of higher 

modes on response quantities such as interstory drift, 

story shear, and overturning moment, and does so in a 

manner that accounts for capacity limits on force and 

moment quantities. Any single dynamic analysis is a 

valid representation of actual response of the model, 

and each analysis helps to establish the central tendency 

and range of peak response quantities.

Refinements and improvements may potentially be 

made in the areas of (1) characterizing and selecting 

site-specific ground motions, (2) determining the 

confidence levels and numbers of standard deviations 

above the mean that should be used in the estimation of 

various response quantities, and (3) improvement of the 

precision of the NSP estimates of peak roof 

displacement. The conservatism of current pushover 

techniques, in their tendency to overestimate the peak 

roof displacements of structures responding 

inelastically, may provide a desirable level of 

conservatism to the method at this stage in its 

development. 

xn
Chapter 9: Multiple-Degree-of-Freedom Effects

9-14 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-1 

10.Summary and Application Example

This document records in detail an effort to assess 

current nonlinear static procedures (NSPs) given in 

FEMA 356 and ATC-40 for the seismic analysis and 

evaluation of structures. In addition, the document 

presents approaches that were developed to improve 

these procedures for future application by practicing 

engineers. Not all of the portions of the two documents 

were evaluated. Conclusions regarding the relative 

accuracy or technical soundness of these documents 

should not be inferred beyond the specific material and 

discussions contained in this document. 

The purpose of this summary is to present a practical 

overview of the results and to illustrate the application 

of NSPs, that include the proposed improvements for an 

example building. Sections 10.1 through 10.8 contain 

key results of analytical studies conducted as part of this 

project and resulting suggestions for specific changes to 

existing procedures for nonlinear static analysis 

procedures. Section 10.9 discusses some important 

aspects of uncertainty and reliability of nonlinear static 

procedures and the suggestions for improvement. 

Section 10.10 summarizes some key observations with 

respect to shortcomings of inelastic seismic analysis 

procedures that were not fully resolved in this project. 

These are areas in which significant improvement might 

be made in the future. Section 10.11 is the application 

example. 

10.1 Overview of Inelastic Seismic Analysis 

Procedures

Nonlinear static procedures are one type of inelastic 

analysis that can be used to estimate the response of 

structures to seismic ground shaking. The differences 

among the various approaches to inelastic analysis in 

general relate to the level of detail of the structural 

model and the characterization of the seismic ground 

shaking. Detailed structural models can often be 

simplified into equivalent multi-degree-of-freedom 

models; or, in some cases, single-degree-of-freedom 

oscillator models, as with nonlinear static procedures. 

The most detailed characterizations of seismic ground 

motion are actual ground motion records that comprise 

accelerations, velocities, and displacements expected at 

the ground surface at a specific site. A simplification 

can be made by representing the effects ground motion 

has in the frequency domain with response spectra that 

plot maximum response of an elastic SDOF oscillator as 

a function of period. This is the type of characterization 

normally used for nonlinear static procedures. 

The discussion in Chapter 2 includes basic descriptions 

of the two nonlinear static procedures that are currently 

used in practice. FEMA 356 uses a displacement 

modification procedure (Coefficient Method) in which 

several empirically derived factors are used to modify 

the response of a single-degree-of-freedom model of the 

structure, assuming that it remains elastic. The 

alternative Capacity-Spectrum Method of ATC-40 is 

actually a form of equivalent linearization. This 

technique uses empirically derived relationships for the 

effective period and damping as a function of ductility 

to estimate the response of an equivalent linear SDOF 

oscillator.

10.2 Evaluation of Current Nonlinear Static 

Procedures

In practice, the current procedures can result in 

estimates of maximum displacement that are 

significantly different from one another. This has 

caused concern on the part of practicing engineers. One 

of the major objectives of the project was to ascertain 

the reasons for these differences and to try to correct 

both procedures to produce similar results. Chapter 3 

documents a comprehensive evaluation of both 

procedures. The basic technique was to develop a 

series of nonlinear single-degree-of-freedom oscillators 

of varying period, strength, and hysteretic behavior. 

These were subjected to ground motion representing 

different site soil conditions. The resulting database of 

approximately 180,000 predictions of maximum 

displacement was used as a benchmark to judge the 

accuracy of the approximate NSPs. This was 

accomplished by comparing the estimates for each 

oscillator from both NSPs to the results of the nonlinear 

response-history analyses. Differences in the two 

estimates were compiled and compared in a statistical 

study. 

10.2.1 Key Observations: ATC-40 Version of 

Capacity-Spectrum Method

Longer-period response. The ATC-40 procedures for 

structures with hysteretic behavior type A tended to 

underestimate the maximum displacement response for 

inelastic systems. The underestimation averages 25% 

to 35% for systems with periods longer than about 0.7 s.

For structures with hysteretic behavior type B, the 

ATC-40 procedures led to small underestimations or 

small overestimations of lateral displacement of 

systems with periods longer than about 0.6 s. Whether 
Chapter 10: Summary and Application Example

10-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

ATC-40 underestimates or overestimates depends on 

the level of lateral strength and on the site class.

For structures with hysteretic behavior type C, the 

ATC-40 procedures led to overestimations of the 

maximum displacement for all periods. The 

overestimation increases as R increases. Average 

overestimations for periods greater than 0.5 s range 

from approximately 5% for systems with R = 1.5 to 

about 35% for systems with R = 8.

Shorter-period response. The ATC-40 procedures can 

lead to significant overestimations of the maximum 

displacements of inelastic oscillators for periods shorter 

than those noted above. The overestimations increase 

with decreasing strength. Estimated displacements in 

the short-period range can be, on average, up to two 

times larger than the benchmark displacements from 

response-history analyses.

Degrading stiffness and strength. ATC-40 assumes 

that the inelastic deformation demands in structures 

with behavior type B will be larger than those in 

structures with behavior type A, while results of 

nonlinear response-history analyses show that the 

deformations are actually approximately the same or 

slightly larger for the elastic-perfectly-plastic (EPP) 

model as compared to the stiffness-degrading (SD) 

model. The current provisions of ATC-40 do not 

address the potential dynamic instability that can arise 

in systems with in-cycle strength degradation and/or Pdelta effects.

Limitations on damping and spectral reduction factors. ATC-40 specifies limits on effective damping that 

result in the imposition of minimum spectral-reduction 

factors based on the anticipated performance of 

building types. These limitations were based on 

engineering judgment that has not been borne out in the 

analytical studies reported here. While the intention of 

these limitations may have been to provide some 

conservatism for degrading structures, the resulting 

estimates of displacement exceed expected mean values 

when compared with actual behavior for many cases. 

10.2.2 Key Observations: FEMA 356 and the 

Coefficient Method

Transition period for the equal-displacement 

approximation. Nonlinear dynamic analyses 

demonstrate that the total displacement experienced by 

long-period structures that undergo inelastic response 

tends to be about the same as structures of the same 

period, responding in an elastic manner, leading to the 

so-called equal displacement approximation. Shortperiod structures do not behave in this manner. FEMA356 defines a characteristic site period to differentiate 

between these two types of behavior. Characteristic site 

periods of FEMA 356 are based on the period at the 

intersection of the constant-acceleration spectral region 

and the constant-velocity spectral region. These 

characteristic periods are shorter than the transition 

periods observed from nonlinear response-history 

analyses. This can result in underestimation of inelastic 

deformations for periods between the characteristic site 

period and periods that are approximately 1.5 times the 

characteristic site period.

Ratio of inelastic to elastic deformation, coefficient 

C1. The use of the equal displacement approximation to 

compute the coefficient C1 for systems with periods 

longer than the characteristic periods leads to relatively 

good approximations of maximum inelastic 

deformations for systems with EPP behavior for periods 

longer than about 1 s. Only small overestimations in the 

order of 5% or 10% are produced with this 

approximation. Note that for very soft soil sites and 

near-fault records, this is only true for systems with 

periods of vibration that are approximately 1.5 times 

longer than the predominant period and the pulse 

periods, respectively.

For systems with R larger than about 2.5, the limiting 

values (capping) of C1 imposed by Section 3.3.1.3.1. of 

the LSP of FEMA 356 will control the estimate of 

maximum inelastic deformation. This can lead to 

theoretically large underestimates of displacements in 

short-period structures, particularly on soft sites.

If capping is not used, and if the transition period is 

lengthened, the FEMA 356 equation to calculate C1

does not adequately capture the changes in inelastic 

deformation demands that are produced with changes in 

R for short-period structures. The magnification of 

inelastic displacement demands with decreasing lateral 

strength for short-period structures was found to be 

larger than that implied by FEMA 356.

Degradation of stiffness and strength (Coefficients 

C2 and C3). There is not a clear division of the intent of 

coefficients C2 and C3. This problem was documented 

in FEMA 357. In particular, C2 is supposed to account 

for changes in lateral displacement produced by 

departures of the hysteretic behavior from an EPP 

hysteretic model (such as pinching, stiffness 

degradation and strength degradation.). P-∆ effects are 

accounted for by C3 in the current provisions of FEMA 

356. FEMA 356 does not distinguish between cyclic 

strength degradation and in-cycle strength degradation. 
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-3

In-cycle degradation produces effects similar to P-∆, 

which can lead to dynamic instability in weak 

structures.

The C2 coefficient of FEMA 356 implies that inelastic 

displacement should increase for stiffness degrading 

systems as compared with EPP systems. With the 

exception of periods of vibration smaller than about 0.7 

s, the maximum displacement of stiffness-degrading 

systems is actually very similar to or slightly smaller 

(5% to 10%) than the maximum displacement of EPP 

systems.

FEMA 356 introduced an alternative recommendation 

for C2 that was not in FEMA 273, as follows: 

“Alternatively, use of C2 = 1 shall be permitted for 

nonlinear procedures”. The ambiguity of conflicting 

recommendations is confusing to users of FEMA 356.

Coefficient C3 does not adequately address the 

possibility of dynamic instability.

10.3 Strength Degradation

The results of the evaluation of the NSPs suggest that 

both procedures would benefit from greater clarity with 

respect to the different types of possible degradation in 

structures subject to seismic shaking. This is 

particularly critical for degradation in strength. 

Chapter 4 discusses the differences between the 

consequences of strength loss within a single cycle of 

deformation (in-cycle) and that which occurs in 

subsequent cycles (cyclic). This important distinction 

illustrated in Figure 10-1. In-cycle strength degradation, 

including that associated with P-∆ effects, can lead to 

dynamic instability. To account for this, a lower limit 

on the strength of structures that exhibit strengthdegrading behavior is suggested for use with nonlinear 

static procedures. The limit is a function of the period 

of the structure and the post-elastic stiffness 

characteristics, as modified for in-cycle strength 

degradation. If the structure has less strength than the 

limit, nonlinear dynamic analysis is recommended.

10.4 Improved Procedures for 

Displacement Modification 

Based on the evaluation of NSPs, Chapter 5 proposes 

modifications to the Coefficient Method of FEMA 356. 

The suggestions relate primarily to the coefficients 

themselves. These are tabulated along with the current 

specifications in Table 10-1. The changes are briefly 

summarized as follows:

10.4.1 Summary of Findings Pertaining to 

Coefficient C1

This coefficient represents the approximate ratio of the 

maximum displacement of an EPP SDOF oscillator 

model to that of a fully elastic counterpart. The 

proposed modification is based on the results of the 

simplified dynamic analyses conducted as a part of the 

Figure 10-1 Differences between cyclic and in-cycle strength degradation

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100Displacement 0 100 200 300 400

Force

Cyclic strength degradation In-cyclic strength degradation 

Strength loss occurs in 

subsequent cycles;

not in the same cycle as yield.

Strength loss occurs 

in same cycle as yield.

Strength and stiffness degrading model

-600

-400

-200

0

200

400

600

-400 -300 -200 -100 0 100 200 300 400

Displacement

Force

Cyclic strength degradation InStrength loss occurs in 

subsequent cycles;

not in the same cycle as yield.
Chapter 10: Summary and Application Example

10-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

evaluation database. The proposed relationship is a 

function of strength (R), period (T), and site class. 

The current provisions of FEMA 356 allow the C1

coefficient to be limited for short-period structures. 

Although this limitation was intended to recognize that 

short-period buildings do not respond as often predicted 

by analysis, the basis of the limitation is subjective. For 

this reason, the use of the “cap” on C1 is not 

recommended. However, the effects of soil-structure 

interaction can have a mitigating effect on maximum 

inelastic displacements of short-period structures. 

Some rational procedures for including the SSI effect in 

nonlinear static analyses are presented in Chapter 8.

10.4.2 Summary of Findings Pertaining to 

Coefficient C2

This coefficient accounts for the change in maximum 

inelastic displacement for systems that exhibit cyclic 

degradation of stiffness and strength. The proposed 

modification is based on the results of the simplified 

dynamic analyses conducted as a part of the evaluation 

database. In many cases, the data suggest that cyclic 

degradation does not increase maximum displacements. 

However, there are exceptions, especially for shortperiod, low-strength structures. 

10.4.3 Summary of Findings Pertaining to 

Coefficient C3

This coefficient was intended to account for P-∆

effects. Review of related research and results of 

detailed analyses indicate that maximum inelastic 

Table 10-1 Coefficients for Use in Equations for Effective Damping

Coefficient Current Specification Modification Purpose of Coefficient

C1 1.0 for Te ≥ Ts

[1.0 +(R-1)Ts/Te]/R for Te<Ts

where a = 

130 for site class B

90 for site class C

60 for site class D

Convert max. elastic 

displacement to estimate for inelastic system

C1

(with short T “cap”)

1.5 for Te < 0.1s

interpolating to

1.0 for Te ≥ Ts

Not recommended

C2

(degrading systems) Immediate Occupancy

1.0

Life Safety

1.3 for T ≤ 0.1

interpolating to

1.1 for T ≥ Ts

Collapse Prevention

1.5 for T ≤ 0.1

interpolating to

1.2 for T ≥ Ts

recommended only for 

structures with significant 

stiffness and/or strength 

degradation

Hysteretic pinching

Cyclic degradation

C2

(non-degrading 

systems)

1.0 1.0

C3 Eliminate in favor of 

strength limit

P-∆

In-cycle degradation

− = + 1 21

1

R

C

aT

− = + ⎡ ⎤ ⎢ ⎥ ⎣ ⎦

2

2

1 1

1

800

R

C

T

α − +

3/2 ( 1) 1.0

e

R

T
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-5

displacements tend to increase abruptly, leading to 

dynamic instability and implied collapse for relatively 

weak structures. The point at which this transition 

occurs is related to the strength, period, and post-elastic 

stiffness of the structure. Although the current 

expression includes these variables, it does not predict 

the instability. The recommendation is for a limit on 

minimum strength (maximum R) for structural models 

that exhibit strength degradation. This limit eliminates 

the need for the C3 coefficient.

10.5 Improved Procedures for Equivalent 

Linearization 

Many engineers favor working with the CapacitySpectrum Method, a form of equivalent linearization. 

This is likely due, at least in part, to the intuitive nature 

of the procedure that graphically relates “capacity” to 

“demand.” Chapter 6 presents the results of an effort to 

improve the practical application of equivalent 

linearization procedures. The resulting suggestions 

focus on improved estimates of equivalent period and 

damping. These differ from the assumptions in ATC40. Generally, the optimal effective period is less than 

the secant period (see Figure 10-2). The optimal 

effective damping is also less than that specified in 

ATC-40. Note also in Figure 10-2 that the optimal 

effective period does not intersect the capacity spectrum 

for the structure at the maximum inelastic displacement 

or Performance Point. In order to preserve this useful 

visualization feature, Chapter 6 also includes an 

optional adjustment to generate a modified 

acceleration-displacement response spectrum 

(MADRS) that does intersect the capacity spectrum at 

the Performance Point. Similar to the current ATC-40 

procedure, the effective period and damping are both 

dependent on ductility and consequently, an iterative or 

graphical technique is required to calculate the 

Performance Point. Several options are outlined in 

Chapter 6. 

10.6 Evaluation and Comparison of 

Improved Nonlinear Static Procedures

The improved procedures were evaluated in an 

independent study. This study utilized nine EPP 

oscillators with three different periods and three 

different strength values. These were subjected to 

thirteen ground motions for class C sites. The motions 

were scaled in accordance with the NEHRP 

Recommended Provisions for Seismic Regulations for 

New Buildings and Other Structures to match a NEHRP

design-response spectrum. Estimates of maximum 

displacements were calculated utilizing both current 

procedures and the proposed improved procedures. 

This was done using both the NEHRP design spectrum 

and the average spectrum for the scaled ground 

motions. This study was not comprehensive enough to 

make broad general conclusions. However, a number 

of key observations were made:

• The improved procedures do not exhibit large 

differences between displacement modification and 

equivalent linearization approaches. This differs 

from previous experience with the ATC-40 

Capacity-Spectrum Method and the FEMA 356 

Coefficient Method. 

• The improved procedures also produced more 

accurate estimates of displacements when compared 

to response-history analysis results than those 

produced by the current nonlinear procedures. For 

displacement ductility of less than ten, the new 

procedures produced estimates that were within one 

standard deviation of the response-history results.

• Improved procedures also seem to work well, at 

least for the case that was studied, in estimating 

maximum displacement response in conjunction 

with a design spectrum. Further investigations are 

warranted to assess the uncertainty involved with 

this approach on a more general basis.

• The results of the evaluation of the improved 

nonlinear procedures illustrate the dispersion of 

results from nonlinear response-history analysis 

using design level ground motions. This dispersion 

is the result of the many uncertainties involved in 

inelastic seismic analysis. It is important for 

practitioners to keep this in mind when interpreting 

the results of inelastic analyses.

Figure 10-2 Acceleration-displacement response 

spectrum (ADRS) showing effective period 

and damping parameters of equivalent linear 

system, along with a capacity curve.

Sa

Sd

MADRS (β eff ,M)

Tsec

dmax

amax

capacity spectrum

ADRS (β eff)

Teff

aeff

ADRS (β0)

Spectral Acceleration

Spectral Displacement

Sa

Sd

MADRS (β eff ,M)

Tsec

dmax

amax

capacity spectrum

ADRS (β eff)

Teff

aeff

ADRS (β0)

Spectral Acceleration

Spectral Displacement
Chapter 10: Summary and Application Example

10-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

10.7 Soil-Structure Interaction Effects

There is a perception among many in the practicing 

engineering community that short, stiff buildings do not 

respond to seismic shaking as adversely as might be 

predicted analytically. There are several reasons why 

short-period structures may not respond as conventional 

analysis procedures predict. Among these are:

• radiation and material damping in supporting soils; 

• structures with basements that experience reduced 

levels of shaking;

• incoherent input to buildings with relatively large 

plan dimensions; and

• inaccuracies in modeling, including lumping of 

masses, neglecting foundation flexibility and some 

elements that contribute to strength.

These factors are often cited qualitatively, along with 

the observed good performance of such buildings in 

past earthquakes, as justification for less onerous 

seismic demand parameters in codes and analytical 

procedures. Traditional design procedures have 

evolved accordingly. Consequently, FEMA 356

currently contains limitations (caps) on the maximum 

value of the coefficient C1 for short-period buildings. 

Many practicing engineers routinely use the limitations 

on C1. Capping leads to prediction of maximum 

inelastic displacements that are less than predicted by 

the current empirical relationship by a margin that 

varies widely depending on period, strength, and site 

conditions. For periods of interest for most buildings (> 

0.2 sec. or so), the margin ranges from relatively small 

(< 20%) for firm (Class B) sites to rather large (> 

200%) for soft (Class E) sites. 

Chapter 8 presents procedures to incorporate soilstructure interaction (SSI) into nonlinear static analyses. 

The objective is to replace the subjective limits with 

rational technical justifications for reducing seismic 

demand. These SSI techniques address the first three 

items listed above. The distribution of mass is not 

addressed in this document; however, it is worthy of 

future investigation to further improve inelastic analysis 

procedures.

FEMA 356 and ATC-40 contain similar procedures for 

incorporating the strength and stiffness of foundations 

into structural models for inelastic analyses. These 

procedures result in changes in response compared to 

fixed base assumptions that can be very significant for 

some structures. These changes include:

• lengthening of period of the system;

• distribution of forces and displacements among 

elements;

• sequence of inelastic behavior; and

• potential foundation modes of inelastic behavior 

(e.g. rocking, soil crushing, pier/pile slip).

Relatively stiff foundation elements on, or in, soil tend 

to average overall shaking effects to an intensity that is 

lower than localized maximums. These kinematic 

effects depend on the plan dimensions of the structure, 

its embedment into the soil, and its period. They can be 

visualized as a low-pass frequency (high-pass period) 

filter on the free-field ground motion. (see Figure 10-3). 

For nonlinear static procedures this leads to a reduced 

spectrum representing a foundation input motion. That 

is, this effect tends to minimize the amplitude of high 

frequency motion experienced by the structure.

Relative movements in the soil beneath structures 

dissipate energy through both radiation damping and 

hysteretic damping. Hysteretic damping is implied in 

the nonlinear force-deformation properties of the 

geotechnical components of foundation models, when 

these elements are modeled. Radiation damping can be 

incorporated into inelastic analysis procedures by 

estimating foundation damping and combining it with 

the conventional assumption for the structure to 

generate an initial system damping ratio for the system. 

For NSPs, the result is a further modification in initial 

spectral ordinates, depending primarily upon the 

foundation area and effective moment of inertia. 

The basic principles used for the development of the 

SSI procedures in Chapter 8 have been included in the 

NEHRP Recommended Provisions for New Buildings 

(BSSC, 2000)1 for the linear analysis and design of new 

buildings for a number of years. They have been 

adapted for use with inelastic procedures. They are 

applicable to both the displacement modification and 

equivalent linearization forms of nonlinear static 

analysis.

10.8 Multiple-Degree-of Freedom Effects

Whether evaluating performance or designing a 

structure, the engineer makes decisions primarily based 

on component forces and deformations. These are 

typically compared to some type of acceptability 

criteria. The intensity of component deformations and 

1.Superseded in 2003 with the FEMA 450 Recommended

Provisions for Seismic Regulations for New Buildings

and Other Structures.
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-7

forces are directly related to a global displacement 

parameter (i.e., roof displacement or first-mode spectral 

displacement) in NSPs. The approximate relative 

distribution of elastic and inelastic forces and 

deformations for the multiple-degree-of-freedom 

(MDOF) structure are controlled by the characteristics 

of the single-degree-of-freedom (SDOF) model 

pushover curve assumed in the analysis. The variations 

of these parameters in a true multi-degree-of-freedom 

system from those of the SDOF approximation are 

known as MDOF effects. The adequacy of simplified 

procedures to address MDOF effects has been 

questioned by a number of researchers.

Chapter 9 summarizes the options for different load 

vectors used to generate SDOF pushover curves for 

structures. In order to investigate and illustrate these 

various options for evaluating MDOF effects, a 

comprehensive study of five buildings compared 

approximate estimates from NSPs for several 

parameters to those obtained from nonlinear MDOF 

response-history analyses. The results are consistent 

with previous research. Practical implications for 

structures with significant MDOF effects are:

• NSPs generally provide reliable estimates of 

maximum floor and roof displacements. They also 

are capable of providing reasonable estimates of the 

largest inter-story drifts that may occur at any 

location over the height. 

• NSPs are not particularly capable, however, of 

accurately predicting maximum drifts at each story, 

particularly within tall flexible structures.

• NSPs are very poor predictors of story forces, 

including shear forces and overturning moments in 

taller structures.

• The use of the first-mode load vector is suggested 

due to the relatively good displacement estimates 

made with this assumption. Other single-load 

vectors were less consistent in producing reliable 

results. The use of two single-load vectors to try to 

envelope response parameters is not particularly 

useful.

• Multi-mode pushover analysis consisting of the use 

of multiple load vectors proportional to the mode 

shapes of the structure that are statistically combined 

shows promise in producing better estimates in interFigure 10-3 Foundation modeling alternatives

ug= free field motion (FFM) with 

conventional damping

Infinitely rigid foundation and soil

a) Rigid base model

ug= free field motion (FFM) with 

conventional damping

structural components of 

foundation geotechnical components 

of foundation

b) Flexible base model

ug= foundation input motion (FIM) 

with conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

c) Kinematic interaction

ug= foundation input motion (FIM) 

with system damping including 

Adjust for foundation foundation damping

damping

foundation input motion (FIM) with 

conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

d) Foundation damping

ug= free field motion (FFM) with 

conventional damping

Infinitely rigid foundation and soil

a) Rigid base model

ug= free field motion (FFM) with 

conventional damping

Infinitely rigid foundation and soil

a) Rigid base model

ug= free field motion (FFM) with 

conventional damping

structural components of 

foundation geotechnical components 

of foundation

b) Flexible base model

ug= free field motion (FFM) with 

conventional damping

structural components of 

foundation geotechnical components 

of foundation

b) Flexible base model

ug= foundation input motion (FIM) 

with conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

c) Kinematic interaction

ug= foundation input motion (FIM) 

with conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

ug= foundation input motion (FIM) 

with conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

c) Kinematic interaction

ug= foundation input motion (FIM) 

with system damping including 

Adjust for foundation foundation damping

damping

foundation input motion (FIM) with 

conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

d) Foundation damping

ug= foundation input motion (FIM) 

with system damping including 

Adjust for foundation foundation damping

damping

foundation input motion (FIM) with 

conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

d) Foundation damping

ug= foundation input motion (FIM) 

with system damping including 

Adjust for foundation foundation damping

damping

foundation input motion (FIM) with 

conventional damping

Kinematic interaction

(high T-pass filter)

free field motion (FFM) with 

conventional damping

d) Foundation damping
Chapter 10: Summary and Application Example

10-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

story drifts over the heights of the buildings. 

Current results documented in the literature 

conclude that the adequacy of results from multimode pushover analyses depends on the parameter 

of interest. It seems that future developments may 

further improve multi-mode pushover analysis.

• The provisions of FEMA 356 as to when higher 

modes are to be considered significant are not 

particularly reliable. All of the example buildings in 

Chapter 9 would have satisfied the criteria (i.e., 

higher modes would not have been significant). 

This is in spite of the fact that all of the buildings, in 

one way or another, showed sensitivity to highermode effects.

• Specific limitations as to when NSPs produce 

reliable results for MDOF structures are elusive. 

Chapter 9 provides a discussion of important 

considerations, but at this time considerable 

judgment on the part of the practicing engineer is 

required.

• As a result of the study, it was observed that, in 

many cases, a single time history response of a 

multi-degree-of-freedom model gave better 

indications of drifts and story forces than any of the 

approximate single-degree-of-freedom estimates 

(see Figure 10-4). This suggests that a future 

procedure might be developed that utilizes a small 

number of response histories to estimate variation 

and MDOF response parameters.

10.9 Uncertainty and Reliability

NSPs are an important part of performance-based 

engineering. Performance-based engineering departs 

from traditional practices in a number of ways. One of 

the more important departures relates to the treatment of 

uncertainty and reliability. Uncertainty arises from the 

seismic ground motion, the structural model, and the 

analysis technique utilized. Traditional prescriptive 

analysis and design procedures (e.g., working stress 

design, load and resistance factor methods) incorporate 

margins of safety in the calculation of demand and 

capacity. These procedures treat uncertainty implicitly 

and they are appropriately conservative with respect to 

the actual potential consequences. In contrast, 

performance-based procedures can be used to predict 

Figure 10-4 Overturning moments in example 9-story building using various load vectors.

Overturning Moments— Weak-story 9-story frame

Weak—2 % Weak—4 %

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

2% Drift

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

Weak—2 % Weak—4 %

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

2% Drift

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

Overturning Moments— Weak-story 9-story frame

Weak—2 % Weak—4 %

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

2% Drift

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

Weak—2 % Weak—4 %

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

2% Drift

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

Weak—2 % Weak—4 %

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

2% Drift

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

2% Drift

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

4% Drift

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD

Weak—2 % Weak—4 %

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

2% Drift

0 50000 100000 150000 200000

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

Floor

Overturning Moment (kips-ft)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular

Inverted Triangular

First Mode

Min Max

Mean

SD SD
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-9

the expected consequences of future ground motions. 

When used in this manner, the results of the analysis are 

central (i.e., mean or median) values. This means that 

they represent the most likely, or “expected,” response. 

However, this also means that the actual response has 

roughly a 50% chance of being greater and a 50% 

chance of being less than the predicted response.

The improvements to existing procedures proposed in 

the document have been developed to optimize 

predictions of expected values. An example is 

illustrated in Figure 10-5 showing the error associated 

with the current FEMA 356 value for coefficient C1 and 

a potential improved formulation. The error is 

determined by dividing the approximate prediction of 

displacement by the expected value from the response 

history analyses. The expected value in this case is the 

mean of results for twenty different ground motion 

records for each period (T) and for each strength (R). 

The closer the error is to 1.0, the better the approximate 

result. Thus the potential improvement clearly provides 

more accurate results than the current procedure. 

However, the dispersion of the results for the twenty 

ground motions is not apparent in this illustration. 

The independent evaluation of the proposed 

improvements is summarized in Chapter 7. In this 

study, a series of bilinear EPP oscillators with postelastic stiffness equal to 5% of the initial stiffness were 

subject to thirteen ground motion records. The results 

of the nonlinear dynamic (response-history) analyses 

for an oscillator with a period of 0.5 s are shown in 

Figure 10-6 for several different strengths. Note the 

dispersion of the results on either side of the mean 

(expected) value. Note also that the dispersion 

increases with lower strength (higher R), as is typical in 

most cases.

In general, it is important to recognize the empirical 

nature of the improved expressions for the proposed 

modifications in this document. They are formulated 

by attempting to match actual analysis data. They may 

appear complex, but they do not imply accuracy beyond 

that associated with the statistical variation in the 

underlying data. Scrutiny of the detailed characteristics 

of the data indicates significant uncertainties in 

expected values. The degree of uncertainty increases 

for:

a. shorter period;

b. lower strength (higher R);

c. degrading hysteretic behavior; and

d. near-source ground motion.

When applying these procedures, it is important to 

estimate basic parameters as carefully as possible. For 

example, using a conservative (low) estimate of the 

strength of a structure may lead to a conservative (high) 

estimate of displacement. It is suggested that realistic 

estimates for all parameters be used to generate 

expected values as a result of the analysis. Then 

engineering judgment may be applied to inject the 

appropriate degree of conservatism, considering the 

Figure 10-5 Error associated with the Coefficient C1 as formulated in FEMA 356 (left) and the potential improved 

formulation (right).

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

ALTERNATIVE I

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[( ∆ i

)app/(∆ i)ex]

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS C

Tg = 0.85 s

Current Potential improvement

E[(∆i)

ap

p

/(∆i)

ex]

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

ALTERNATIVE I

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[( ∆ i

)app/(∆ i)ex]

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS C

Current Potential improvement

E[( ∆ i)app/(∆ i)ex E[( ∆ ] i)app/(∆ i)ex]
Chapter 10: Summary and Application Example

10-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

particular circumstances. With this in mind, it should 

be noted that traditional design equations, including 

some of those in FEMA 356 and ATC-40, are 

conservative and may underestimate strength capacities 

and deformation acceptability for some structures and 

components. More accurate supplemental information 

is available from other sources (FEMA 306/307/308,

FEMA 355C).

10.10 Important Future Developments

The proposed improvements to nonlinear static analysis 

procedures in this document will lead to better results in 

practice. Nonetheless, not all of the shortcomings of 

NSPs have been addressed. In developing the 

improvements, a number of important observations 

about the need for future improvement of inelastic 

seismic analysis procedures have emerged. These are 

summarized in the following sections.

10.10.1 Nonlinear Modeling for Cyclic and InCycle Degradation of Strength and 

Stiffness 

FEMA 440 makes a distinction between two types of 

degradation of stiffness and strength of inelastic single 

degree-of-freedom oscillators (see Figure 10-1). This 

distinction had not previously been addressed explicitly 

by guidelines for nonlinear static procedures. 

Independent studies demonstrate that if strength 

degradation occurs cyclically, then dynamic response of 

SDOF systems is stable. In contrast, in-cycle loss of 

strength can lead to dynamic instability. It is vitally 

important to be able to differentiate between these two 

types of structural degradation. Current nonlinear static 

pushover procedures cannot fully distinguish between 

cyclic and in-cycle strength degradation. FEMA 440 

includes interim recommendations based solely on 

judgment for this purpose.

Important questions include:

• What current data exist on force-deformation 

behavior and strength degradation of components 

subjected to large ductility demands in a single cycle 

of loading?

Figure 10-6 Dispersion of results for the nonlinear dynamic analysis (NDA) of a SDOF oscillator subject to thirteen 

NEHRP Site Class C ground motions

T = 0.5s

Average Response Spectrum

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0123456789

Strength Reduction Factor, R

Maximum Displacement (m)

NDA mean

std R=2

std R=4

std R=8

Indicates mean of NDA plus and 

minus one standard deviation
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-11

• How does in-cycle strength loss in components 

affect the global dynamic stability of structural 

models?

• Can this effect be adequately incorporated into 

NSPs?

• What practical guidance can be provided for the 

incorporation of in-cycle degradation into nonlinear 

response-history analysis procedures?

• How can these effects be incorporated into 

simplified models? 

10.10.2 Soil and Foundation Structure Interaction

While some advances are made in FEMA 440, there is 

not completely adequate guidance for addressing the 

effects of the interaction between structures and 

supporting foundations and soils. This is particularly 

important for short-period or large-footprint structures, 

where current models may over-predict the input 

ground motion. Furthermore, additional guidance on 

force-deformation relationships and damping 

characteristics of foundations is needed. Finally, there 

is an important need for adequate guidance on the effect 

of foundation rocking on structural response.

FEMA 440 supplements existing NSPs with 

preliminary recommendations for the inclusion of soilstructure-interaction effects (see Figure 10-3). These 

recommendations augment the existing guidelines in 

FEMA 356 and ATC-40 for soil-foundation stiffness 

and strength with approximate procedures to account 

for kinematic SSI and soil damping. The provisions for 

soil load-deformation behavior provide a framework 

primarily with some default values for typical materials. 

The documents recommend site-specific studies if 

performance is significantly affected by soil properties.

Important issues include:

• Is the adaptation of linear SSI procedures for 

nonlinear analysis presented in FEMA 440 adequate 

as is, or are further adjustments warranted?

• What information is available on soil loaddeformation characteristics that might be adopted 

for general practical application?

• What analytical procedures are available to 

geotechnical engineers to estimate critical soil 

properties for inelastic seismic analysis?

• What are the effects of foundation rocking on 

inelastic seismic response and how can these effects 

be incorporated into practical analysis procedures?

• What are the effects of foundation sliding on 

inelastic seismic response and how can these effects 

be incorporated into practical analysis procedures?

10.10.3 Nonlinear Multi-Degree of Freedom 

Simplified Modeling 

Current nonlinear static procedures are based on singledegree-of-freedom models, which, while simple to 

understand, are very limited in their ability to address 

complex structures and multiple-degree-of-freedom 

effects from input seismic ground motions. As noted, 

FEMA 440 recognizes that current NSPs are limited in 

the ability to reliably predict the effects of inelastic 

behavior of MDOF systems. Specifically, predictions 

of maximum story drifts, story forces, and inelastic 

component demands (i.e., plastic hinge rotations) are 

not reliable using a single-load vector. FEMA 440 also 

notes that current procedures for using multiple-load 

vectors representative of the fundamental mode and one 

or more higher modes (multi-mode pushover analysis) 

can improve results somewhat, particularly for 

prediction of maximum story drifts. Ongoing research 

suggests that multi-mode pushover procedures might 

be modified to provide better estimates of other demand 

parameters as well. These improvements come at the 

expense of greater computational effort and less 

transparency, however. These barriers have been cited 

as obstacles to the practical application of nonlinear 

analysis techniques (i.e., using response-history 

analysis). This raises the question: why not devote the 

effort to simplified nonlinear response-history analysis?

One of the interesting observations about MDOF effects 

during the preparation of the FEMA 440 report was 

that, in spite of significant dispersion among records, 

any single nonlinear response-history analysis result 

often produced better estimates of maximum 

engineering demand parameters than any of the 

approximate analyses (see Figure 10-4). This 

observation suggests that there may be an analysis 

procedure that characterizes global engineering demand 

as the maximum displacement response of a structural 

model subject to shaking hazard represented by 

currently available regional maps (i.e., by the maps 

currently prepared by the U.S. Geological Survey for 

the National Earthquake Hazards Reduction Program). 

Maximum displacements might be estimated using 

nonlinear static procedures. Story-level and 

component-level engineering demand could then be 

estimated using a simplified MDOF response-history 

analysis for a small number of ground motion records, 

scaled to result in the previously estimated global 
Chapter 10: Summary and Application Example

10-12 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

displacement demand. This approach could greatly 

simplify nonlinear response-history analysis.

Nonlinear response-history analysis might be facilitated 

further by the use of simplified structural models. 

Detailed structural models often can require hundreds 

of degrees of freedom, making the process prone to 

error and complicating the interpretation of results. As 

noted in FEMA 440, many practitioners have used 

innovative sub-structuring techniques to generate 

“stick” or “fishbone” models that greatly simplify data 

management, computational effort, and visualization of 

results.

In summary, this issue presents the following critical 

questions:

• What are the limits (e.g. periods, separation of 

modes, mass participation) for building models 

when MDOF effects must be considered significant?

• Can multi-mode pushover procedures provide 

adequate results for systems with significant MDOF 

effects?

• Can maximum engineering demand below the 

global level (i.e., story and component levels) be 

predicted using a limited number of nonlinear 

response-history analyses?

• How should ground motion records be scaled to 

produce global maximum displacement demands 

that are representative of a specific shaking hazard?

• How can MDOF structural models be simplified 

while still providing reliable results for practical 

application?

• How can strength and stiffness degradation (see 

Section 10.10.1) be adequately represented in 

MDOF structural models?

• How can improved methods for modeling 

foundations and soil structure interaction (see 

Section 10.10.2) be incorporated into MDOF 

structural models?

• What is the effect of concentrating masses at story 

levels on inelastic response, particularly for relative 

short structures?

10.11 Application Example

This section contains structural analysis calculations, 

and related commentary, utilizing nonlinear static 

procedures for the analysis of an example building. The 

steps in this process are presented in the flowchart in 

Figure 10-7. On the flowchart, tags have been used to 

identify pertinent sections of FEMA 440 (this 

document), and also FEMA 356 and ATC-40. The 

calculations also include similar tags for ease of 

reference to these three documents. The example 

illustrates use of both the displacement modification 

and the equivalent linearization procedures to estimate 

the maximum displacement of a building model. 

10.11.1 Example Building Description

In order to illustrate the application of NSPs, including 

the suggested improvements in this document, an 

example building has been developed. It is depicted 

and described on calculation Sheet 1. This type of 

construction is typical for relatively small commercial 

office and/or retail uses. It is assumed to be located in 

an area of relatively high seismicity. This example is 

very simple from an analysis perspective since all of the 

walls are assumed to be identical and the floor and roof 

diaphragms are assumed to be rigid. The building is 

also completely regular and symmetrical. Although 

some actual buildings might be this simple, it is not 

always the case and the user should not infer that all 

structures may be reduced to this level of simplicity.

10.11.2 Basic Ground Motion

The basic ground motion spectrum for the example is 

illustrated on calculation Sheet 2. The selection of the 

basic ground motion does not differ from current 

procedures of FEMA 356, ATC-40 and the 2000 

NEHRP Recommended Provisions for New Buildings. 

The same assumptions used for the evaluation study 

summarized in Chapter 7 are used for the example. 

Values for short- and 1-second period spectral 

accelerations at the Maximum Considered Earthquake 

(MCE) level were assumed for 5% damping and site 

class C, resulting in values SS = 1.5g and S1 = 0.6g. 

Following the procedures in the 2000 NEHRP 

Recommended Provisions for New Buildings, the shortand long-period values were modified for site class C to 

SXS = FaSS and SX1 = FvS1, where Fa = 1.0 and Fv = 1.3. 

Design-basis ordinates then were obtained as 

SDS = 2/3SXS and SD1 = 2/3SX1. These values were 

used with the spectral shape defined in the NEHRP 

Recommended Provisions for New Buildings.

10.11.3 Kinematic Soil-structure Interaction

The next step, also illustrated on Sheet 2, is to modify 

the initial spectrum to account for kinematic soilstructure interaction in accordance with Chapter 8. 

Note that the kinematic effect associated with the baseslab averaging is considered, but not the effect related to 

embedment. This is due to the fact that the building, 
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-13

Figure 10-7 Application flowchart for nonlinear static seismic analysis

Select procedure to 

determine maximum 

displacement

Displacement modification

FEMA 356

Equivalent linearization

ATC 40 

Select ground motion

spectrum

Develop fixed base 

structural model

Model foundation stiffness and 

strength and modify structural 

model for flexible base

Modify spectrum for kinematic

soil-structure interaction

Modify spectrum for 

foundation damping

Select static load vector

Check for minimum strength

to avoid dynamic instability

Generate global forcedeformation curve for 

equivalent SDOF model

Sect. 9.4.1

FEMA 440

Sect. 3.3.3.2.3

FEMA 356

Sect. 8.2.1

ATC 40

Sect. 3.3.3

FEMA 356

Sect. 8.2.1

ATC 40

Chap. 4

FEMA 356

Chap. 10

ATC 40

Chap.3,5,6,7,8

FEMA 356

Chap. 8,9

ATC 40

Sect. 4.5

FEMA 440

Sect. 1.6

FEMA 356

Sect. 4.4

ATC 40

Sect. 8.3

FEMA 440

Sect. 8.2

FEMA 440

Check ductility, μ, 

Sect. 8.3 with assumed value FEMA 440

Sect. 5.2

FEMA 440

Sect. 3.3.3.2.3

FEMA 356

Sect. 3.3.3.3.2

FEMA 356

Sect. 5.3

FEMA 440

Sect. 3.3.3.2.3

FEMA 356

Eqn. 3-15

FEMA 356

Sect. 6.2.1

FEMA 440

Sect. 8.2.2.1

ATC 40

Sect. 6.2.2

FEMA 440

Sect. 8.2.2.1

ATC 40

Sect. 6.4

FEMA 440

Sect. 8.2.2.1

ATC 40

Determine equation for 

effective damping

Determine equation for 

effective period

Select solution procedure and 

calculate maximum displacement

Convert from spectral to roof 

displacement, Coeff. C0

Convert from elastic to inelastic 

displacement, Coeff. C1

Adjust for cyclic strength 

degradation, Coeff. C2

Calculate maximum 

displacement

Select procedure to 

determine maximum 

displacement

Displacement modification

FEMA 356

Equivalent linearization

ATC 40 

Select ground motion

spectrum

Develop fixed base 

structural model

Model foundation stiffness and 

strength and modify structural 

model for flexible base

Modify spectrum for kinematic

soil-structure interaction

Modify spectrum for 

foundation damping

Select static load vector

Check for minimum strength

to avoid dynamic instability

Generate global forcedeformation curve for 

equivalent SDOF model

Sect. 9.4.1

FEMA 440

Sect. 3.3.3.2.3

FEMA 356

Sect. 8.2.1

ATC 40

Sect. 3.3.3

FEMA 356

Sect. 8.2.1

ATC 40

Chap. 4

FEMA 356

Chap. 10

ATC 40

Chap.3,5,6,7,8

FEMA 356

Chap. 8,9

ATC 40

Sect. 4.5

FEMA 440

Sect. 1.6

FEMA 356

Sect. 4.4

ATC 40

Sect. 8.3

FEMA 440

Sect. 8.2

FEMA 440

Check ductility, μ, 

Sect. 8.3 with assumed value FEMA 440

Sect. 5.2

FEMA 440

Sect. 3.3.3.2.3

FEMA 356

Sect. 3.3.3.3.2

FEMA 356

Sect. 5.3

FEMA 440

Sect. 3.3.3.2.3

FEMA 356

Eqn. 3-15

FEMA 356

Sect. 6.2.1

FEMA 440

Sect. 8.2.2.1

ATC 40

Sect. 6.2.2

FEMA 440

Sect. 8.2.2.1

ATC 40

Sect. 6.4

FEMA 440

Sect. 8.2.2.1

ATC 40

Determine equation for 

effective damping

Determine equation for 

effective period

Select solution procedure and 

calculate maximum displacement

Convert from spectral to roof 

displacement, Coeff. C0

Convert from elastic to inelastic 

displacement, Coeff. C1

Adjust for cyclic strength 

degradation, Coeff. C2

Calculate maximum 

displacement
Chapter 10: Summary and Application Example

10-14 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

although supported three feet below grade, does not 

have a basement. The result of this step is a reduced 

spectrum representing the foundation input motion.

10.11.4 Fixed-Base Model

The basic procedures to develop a structural model and 

determine a lateral force and deformation relationship 

(pushover curve) for a structure remain essentially the 

same as in ATC-40 and FEMA 356. For the example 

building, a fixed-based model is relatively simple, as 

shown on Sheet 3. The fixed-based model is necessary, 

even if the intention is to include a flexible foundation, 

due to the fact that foundation damping procedures rely 

on an estimate of the change in period from a fixed base 

to a flexible model. The total masses for the building 

are calculated assuming that the roof weight is 

approximately 140 psf and that the floors are 

approximately 160 psf. These estimates are intended to 

include not only the weight of the structural 

components of the buildings, but also other dead loads 

and actual live loads. In an actual application, the 

weights would be determined in a more detailed takeoff. The effective rigidities of the walls are calculated 

in accordance with the requirements of FEMA 356 and 

ATC-40 for walls that are cracked. The requirements in 

this case are identical in both documents. 

The linear dynamic properties for the model shown on 

Sheet 3 are calculated utilizing the basic equations in 

ATC-40, Section 8.5. In many practical applications, 

these calculations are done using computer software. 

The determination of the linear dynamic properties can 

be simplified by using Rayleigh’s method to reduce the 

number of degrees of freedom in the model to the lateral 

displacement at the roof and that at the floor level. The 

process is initiated by estimating a mode shape 

assuming a uniform acceleration acting on the story 

masses. Then the deflections at the roof and the floor 

are calculated. In this case, moment-area calculations 

were used for this purpose. These displacements are 

then normalized to a unit displacement at the roof to 

determine an initial estimate of the fundamental mode 

shape. Then a first-mode participation factor is 

calculated. Utilizing the participation factor, modal 

story forces for unit spectral acceleration can be 

calculated. This essentially revises the loads that were 

initially assumed using uniform acceleration. 

Application of these revised forces to the model results 

in a change in the displacement at the roof and the floor. 

These are once again normalized to the roof 

displacement to generate a revised mode shape. The 

process continues until the calculated mode shape is 

essentially equal to that which was assumed. Then the 

period may be calculated as shown at the end of the 

calculations on Sheet 3.

10.11.5 Flexible-Base Model

The process continues by repeating the calculations 

with the assumption of a flexible base, as shown on 

calculation Sheet 4. The stiffness of the foundation in 

this case is assumed to be controlled by the soil 

properties (i.e., foundation structure assumed to be rigid 

compared to the supporting soil). The initial shear 

modulus of the soil material is calculated based on the 

shear wave velocity for the material. For a Class C site, 

this ranges from 1200 to 2500 feet per second. The 

effective shear modulus is calculated by reducing the 

initial value, depending on the severity of shaking at the 

site. In this case a ratio was determined in accordance 

with the recommendations of FEMA 356. 

Both FEMA 356 and ATC-40 contain equations for 

calculating rotational and translational stiffness of 

foundations assuming a rigid plate acting on a 

homogeneous elastic half space representing the soil. 

The equations in FEMA 356 differ from those in ATC40 in their formulation. The equations shown Sheet 4 

are from FEMA 356. Essentially the same values can 

be determined by using the equations in ATC-40. Note 

that both the rotational stiffness and the translational 

stiffness are increased due to the embedment of the 

foundation. It should be noted that the translational 

stiffness in this case is calculated only for the six shear 

walls acting in each direction. In reality, the stiffness is 

probably higher, due to the effect of the foundations 

beneath the walls acting in the orthogonal direction, as 

well as the slab on grade that ties all the footings 

together.

Once the foundation stiffness values are calculated, 

Rayleigh’s method can be used once again to reduce the 

degrees of translational freedom to two. The 

calculations to determine dynamic properties for the 

flexible based model as shown on Sheet 4 are then 

analogous to those for a fixed base.

10.11.6 Foundation Damping

The next step is to modify the ground motion spectrum 

further for the effects of foundation damping. The 

calculations to determine foundation damping are 

illustrated on calculation Sheet 5. This process begins 

with an estimate of the effective stiffness of the fixedbase model. Note that the mass must be modified by 

the effective mass coefficient. The equation for this 

may be found in ATC-40. The equivalent foundation 
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-15

radius for translation is calculated for the entire 

footprint of the building. Using this radius, the 

translational stiffness of the foundation can be 

estimated using FEMA 440. Note that this estimate 

corresponds well with that calculated using the actual 

soil properties on Sheet 4. 

The effective height of the building is required to 

estimate the rotational stiffness of the foundation. This 

parameter is essentially the centroid of the first-mode 

shape measured from the base. As noted in Chapter 8, 

70% of the total height of the building is often a good 

approximation. In this example the dynamic properties 

are used to calculate the actual value. Rotational 

stiffness of the foundation can then be estimated. Note 

again that the estimate on Sheet 5 compares reasonably 

well with that calculated for the flexible-base model 

using the soil properties of the foundation directly. This 

leads to an equivalent foundation radius for rotation, 

which can be visualized as a radius of gyration 

representing the effective moment of inertia of the 

foundation.

The actual amount of foundation damping depends on 

the relative amount of inelasticity in the foundation 

compared with that in the structure. The procedures in 

Chapter 8 essentially assume that the inelasticity is 

concentrated in the structure, which leads to a 

conservative estimate of foundation damping. The 

calculation requires an estimate of the system ductility 

demand. An initial assumption of 3.0 is made for the 

example. Combining this with an initial damping of 5% 

leads to an effective damping for the flexible-base 

model. Combining the foundation damping with the 

initial assumed damping value (5%) leads to an estimate 

of the total flexible-base system damping. The 

foundation input motion calculated, including the 

effects of kinematic interaction, is based on the initial 

assumption of 5% damping. The foundation input 

motion is then modified to reflect the flexible based 

damping as shown on calculation Sheet 6.

10.11.7 Force-Displacement Relationships 

(Pushover Curves)

The next step in the process is the selection of a lateral 

load vector. FEMA 356 and ATC-40 both require and/

or suggest a number of options for this selection. Based 

on the recommendation in Chapter 8, a vector 

proportional to the first-mode shape is sufficient and 

preferable to the others; thus the first-mode shape for 

the flexible-base model is used to generate the basic 

load-deformation characterizations for the model (see 

calculation Sheet 7). Two different possibilities are 

considered. The first case involves an arbitrary 

assumption that the strength is approximately 0.4 W, 

resulting in an R-factor of 1.52. If the governing 

inelastic mechanism were foundation rocking or some 

other ductile mechanism, the pushover curve might be 

as shown on Sheet 7 for the positive post-elastic 

stiffness model. Note that a positive post-elastic 

stiffness of 5% reflects some strain hardening and 

participation of the slab and columns. If the mechanism 

included modes of behavior that imply the loss of 

strength, the post-elastic portion of the curve would 

have a negative slope for such a degrading system. For 

the second model, this is assumed to be -25% of the 

initial oscillator stiffness for the strength-degrading 

model, as also illustrated in the pushover diagrams on 

Sheet 7. Each of these cases is examined further.

10.11.8 Check on Minimum Strength for Strength 

Degrading Model

The model with degrading strength must be checked to 

determine if there is a potential for dynamic instability, 

as shown on calculation Sheet 8. The maximum 

negative post elastic stiffness evident from the pushover 

curve could be due to cyclic and/or in-cycle loss of 

strength (including P-∆ effects). As noted in Chapter 4, 

there is currently no practical means of separating these 

effects. The suggestion in this document is to assume 

that the effective post-elastic stiffness, for sites located 

in the near field, is equal to that attributable to P-∆

effects plus 80% of the balance evident from the 

pushover curve. For non-near-field sites the percentage 

drops to 20%. This is strictly a subjective provision and 

further research is needed on this issue. For the 

strength-degrading model in the example, dynamic 

analysis would not be required for the building in either 

case. However, as noted on Sheet 8, the assumed 

design level ground motions equal 2/3 of MCE ground 

motions. Larger motions would imply lower relative 

strength (higher R) for the model. In fact, the MCE 

motion likely would result in an R greater than the 

maximum allowable and the potential for dynamic 

instability. This is discussed further below in 

conjunction with equivalent linearization procedures.

10.11.9 Target Displacement for Displacement 

Modification

The target displacement for the positive post-elastic 

stiffness model is calculated using the displacement 

modification as shown on calculation Sheet 9. The 

procedure is the Coefficient Method of FEMA 356

modified with the suggested changes for the 

coefficients C1 and C2. The coefficient C2 is included 

in the calculation since a concrete structure is likely to 
Chapter 10: Summary and Application Example

10-16 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

have stiffness degradation and pinching hysteretic 

behavior. Note that the solution for maximum 

displacement for the strength-degrading model (nearand non-near field) would be the same as the case on 

Sheet 9, since the displacement-modification procedure 

does not directly consider negative post-elastic stiffness 

in the calculation of the coefficients.

10.11.10 Calculation of the Performance Point 

Using Equivalent Linearization

The solution for the maximum displacement of the 

positive post-elastic stiffness model using equivalent 

linearization procedures is shown on calculation Sheets 

10 and 11. The effective damping and period 

calculations for a stiffness-degrading oscillator with 

positive post-elastic stiffness of 5% are shown on Sheet 

10. The selected solution procedure is the construction 

of the locus of performance points, as shown on Sheet 

11. A check using the general equations for effective 

damping (Equations 6-4, 6-5, and 6-6) and effective 

period (Equations 6-10, 6-11, and 6-12) produced 

essentially the same performance point. 

10.11.11 Check on Assumed Ductility

The solutions for the positive post elastic-stiffness 

model are essentially equivalent for displacement 

modification and equivalent linearization. The 

resulting ductility demand is approximately 1.8, as 

opposed to the assumed value of 3.0. This would result 

in an increase from 6.9% to 7.9% in initial flexible-base 

damping for the model. This reduces the maximum 

displacement slightly, but not significantly, in this case. 
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-17
Chapter 10: Summary and Application Example

10-18 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-19
Chapter 10: Summary and Application Example

10-20 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-21
Chapter 10: Summary and Application Example

10-22 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-23
Chapter 10: Summary and Application Example

10-24 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-25
Chapter 10: Summary and Application Example

10-26 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
Chapter 10: Summary and Application Example

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 10-27
Chapter 10: Summary and Application Example

10-28 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 11-1 

References and Bibliography

This section contains references, additional bibliography, and acronyms. Following the references and bibliography section is a list of FEMA reports cited in this 

document.

References and Additional Bibliography

Akkar S.D. and Miranda, E., 2005, “Statistical evaluation of approximate methods for estimating maximum deformation demands on existing structures,” 

Journal of Structural Engineering, American Society of Civil Engineers, Vol. 131, No. 1.

Akkar, S., and Gulkan, P., 2000, “Comparative performance evaluation of displacement based design 

procedures for near field earthquakes,” Proceedings 

of the 12th World Conference on Earthquake Engineering, New Zealand Society for Earthquake Engineering, Upper Hutt, New Zealand.

Albanesi, T., Nuti, C. and Vanzi, I., 2000, “A simplified 

procedure to assess the seismic response of nonlinear structures,” Earthquake Spectra, Vol. 16, No. 4, 

Earthquake Engineering Research Institute, Oakland, California, pp. 715-734.

ASCE, 2000a, Prestandard and Commentary for the 

Seismic Rehabilitation of Buildings, FEMA 356 

Report, prepared by the American Society of Civil 

Engineers, published by the Federal Emergency 

Management Agency, Washington, D.C.

ASCE, 2000b, Global Topics Report on the Prestandard 

and Commentary for the Seismic Rehabilitation of 

Buildings, FEMA 357 Report, prepared by the 

American Society of Civil Engineers, published by 

the Federal Emergency Management Agency, 

Washington, D.C.

ASCE, 2002, Minimum Design Loads for Buildings and 

Other Structures, ASCE 7-02 Report, American 

Society of Civil Engineers, Reston, Virginia.

ATC, 1996, Seismic Evaluation and Retrofit of Concrete 

Buildings, ATC-40 Report, Volumes 1 and 2, 

Applied Technology Council, Redwood City, California.

ATC/BSSC, 1997, NEHRP Guidelines for the Seismic 

Rehabilitation of Buildings, FEMA 273 Report 

(Guidelines) and FEMA 274 Report (Commentary), 

prepared by the Applied Technology Council for 

the Building Seismic Safety Council, published by 

the Federal Emergency Management Agency, 

Washington, D.C.

Aschheim, M.A., Maffei, J., and Black, E.F., 1998, 

“Nonlinear static procedures and earthquake displacement demands,” 6th U.S. National Conference 

on Earthquake Engineering, Seattle, Washington, 

Earthquake Engineering Research Institute, Oakland, California.

Aschheim, M., and Black, E.F., 2000, “Yield point 

spectra for seismic design and rehabilitation,” 

Earthquake Spectra, Vol. 16, No. 2, Earthquake 

Engineering Research Institute, Oakland, California, pp. 317-335.

Aschheim, M.A., Maffei, J., and Black, E.F., 1998, 

“Nonlinear static procedures and earthquake displacement demands,” Proceedings, 6th U.S. 

National Conference on Earthquake Engineering, 

Seattle, Washington, CD-ROM, Paper 167, Earthquake Engineering Research Institute, Oakland, 

California.

Aydinoğlu, N.M., 2003, “An incremental response 

spectrum analysis procedure based on inelastic 

spectral displacements for multi-mode seismic performance evaluation,” Bulletin of Earthquake Engineering, Vol. 1, Kluwer Academic Publishers, 

pp.3-36.

Aydinoglu, N.M., and Kacmaz, U., 2002, Strength 

Based Displacement Amplification Spectra For 

Inelastic Seismic Performance Evaluation, Department of Earthquake Engineering Report, No. 2002/

2, Kandilli Observatory and Earthquake Research 

Institute, Bogazici University, Istanbul, Turkey, 32 

pages.

Baez, J.I., and Miranda, E., 2000, “Amplification factors to estimate inelastic displacement demands for 

the design of structures in the near field,” Proceedings of the 12th World Conference on Earthquake 

Engineering, New Zealand Society for Earthquake 

Engineering, Upper Hutt, New Zealand.

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Bernal, D., 1987, “Amplification factors for inelastic 

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Vol. 15, pp. 635-651. 

Bernal, D., 1992, “Instability of buildings subjected to 

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Vol. 118, No. 8, American Society of Civil Engineers, pp. 2239-2260.
References and Bibliography

11-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Black, E.F., and Aschheim, M., 2000, Seismic Design 

and Evaluation of Multistory Buildings using Yield 

Point Spectra, CD Release 00-04, Mid-America 

Earthquake Center, University of Illinois, Urbana. 

Blume, J.A., Newmark, N.M., and Corning, L.H., 1961, 

Design of Multistory Reinforced Concrete Buildings 

for Earthquake Motions, Portland Cement Association, Skokie, Illinois.

Bracci, J.B., Kunnath, S.K., and Reinhorn, A.M., 1997, 

“Seismic performance and retrofit evaluation of 

reinforced concrete structures,” Journal of Structural Engineering, Vol. 123, American Society of 

Civil Engineers, pp. 3-10.

Browning, J., 2000, “Implications of a proportioning 

method for earthquake-resistant RC frames subjected to strong ground motion,” Proceedings of the 

12th World Conference on Earthquake Engineering, New Zealand Society for Earthquake Engineering, Upper Hutt, New Zealand.

Browning, J., Li, R., Lynn, A., and Moehle, J.P., 2000, 

“Performance assessment for a reinforced concrete 

frame building,” Earthquake Spectra, Vol. 16, No. 

3, Earthquake Engineering Research Institute, Oakland, California, pp. 541-555.

BSSC, 2000, NEHRP Recommended Provisions for 

Seismic Regulations for New Buildings and Other 

Structures, Part 1, Provisions, and Part 2, Commentary, FEMA 368 and 369, prepared by the Building 

Seismic Safety Council, published by the Federal 

Emergency Management Agency, Washington, 

D.C.

BSSC, 2003, NEHRP Recommended Provisions for 

Seismic Regulations for New Buildings and Other 

Structures, Part 1, Provisions, and Part 2, Commentary, FEMA 450-1 and 450-2, prepared by the 

Building Seismic Safety Council, published by the 

Federal Emergency Management Agency, Washington, D.C.

Chintanapakdee, C., and Chopra, A.K., 2003, “Evaluation of modal pushover analysis using generic 

frames,” Earthquake Engineering and Strucural 

Dynamics, Vol. 32, Wiley Publishers, Hoboken, 

New Jersey, pp. 417–442.

Chopra, A.K., and Chintanapakdee, C., 2004, “Inelastic 

deformation ratios for design and evaluation of 

structures: single-degree-of-freedom bilinear systems,” Journal of Structural Engineering., Vol. 130, 

No. 9, American Society of Civil Engineers, 

Reston, Virginia, pp. 1309-1319.

Chopra, A.K., and Goel, R.K., 1999a, CapacityDemand Diagram Methods for Estimating Seismic 

Deformation of Inelastic Structure: SDF Systems, 

Report No. PEER-1999/02, Pacific Earthquake 

Engineering Center, University of California, Berkeley, California.

Chopra, A.K., and Goel, R.K., 1999b, “Capacitydemand diagram methods based on inelastic design 

spectrum,” Earthquake Spectra, Vol. 15, No. 4, 

Earthquake Engineering Research Institute, Oakland, California, pp. 637-656.

Chopra, A.K. and Goel, R.K., 2000, “Evaluation of 

NSP to estimate seismic deformation: SDF systems,” Journal of Structural Engineering, Vol. 126, 

No. 4, American Society of Civil Engineers, 

pp. 482-490.

Chopra, A.K., and Goel, R.K., 2001a, “Direct displacement based design: Use of inelastic vs. elastic 

design spectra,” Earthquake Spectra, Vol. 17, 

No. 1, Earthquake Engineering Research Institute, 

Oakland, California, pp. 47-64.

Chopra, A.K., and Goel, R.K., 2001b, A Modal Pushover Analysis Procedure to Estimate Seismic 

Demands for Buildings: Theory and Preliminary 

Evaluation, Report No. PEER 2001/03, Pacific 

Earthquake Engineering Research Center, University of California, Berkeley.

Chopra, A.K., and Goel, R.K., 2001c, “Modal pushover 

analysis of SAC buildings,” Third U.S.-Japan 

Workshop on Performance-Based Earthquake 

Engineering Methodology for Reinforced Concrete 

Building Structures, August 16-18, Seattle, Washington.

Chopra, A.K., and Goel, R.K., 2001d, A Modal Pushover Analysis Procedure to Estimate Seismic 

Demands for Buildings: Theory and Preliminary 

Evaluation, PEER-2001/03, Berkeley: Pacific 

Earthquake Engineering Research Center, University of California, 87 pages.

Chopra, A.K., and Goel, R.K., 2002, “A modal pushover analysis procedure for estimating seismic 

demands for buildings,” Earthquake Engineering 

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Chopra, A.K. and Kan, C., 1973, “Effect of stiffness 

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buildings,” Earthquake Engineering and Structural 

Dynamics, Vol. 2, pp. 35-45. 

Chopra, A.K., Goel, R.K., and Chintanapakdee, C., 

2003, “Statistics of single-degree-of-freedom estimate of displacement for pushover analysis of 
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buildings,” Journal of Structural Engineering, Vol. 

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Chopra A.K., Goel, R.K., and Chintanapakdee, C., 

2004, “Evaluation of a modified MPA procedure 

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Iwan, W.D., and Chen, X.D., 1995, “Important nearfield ground motion data from the Landers earthquake,” Proceedings, 10th European Conference on 

Earthquake Engineering, Vol. 1, A. A. Balkema, 

Rotterdam, pp. 229-234.

Iwan, W.D. and N.C. Gates, 1978, “The effective period 

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Earthquake Engineering and Structural Dynamics, 

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Iwan, W.D., and Gates, N.C., 1979, “Estimating earthquake response of simple hysteretic structures,” 

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Iwan, W.D. and Guyader, A.C., 2001, “A study of the 

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Iwan, W.D., Huang, C-T., and Guyader, A.C., 1999, 

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Iwan, W.D., Huang, C-T., and Guyader, A.C., 2000, 

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Krawinkler, H., and Alavi, B., 1998, “Development of 

improved design procedures for near fault ground 

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Krawinkler, H., and Seneviratna, G.D.P.K., 1998, “Pros 

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No. 4-6, pp. 452-464.

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Kunnath, S.K., Nghiem, Q., John Jr., A., and El-Tawil, 

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report on research conducted under PGE/PEER 

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MacRae, G. and Tagawa, H., 2002, Methods to Estimate 

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Miranda, E., 1993, “Evaluation of site-dependent 

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Miranda, E., 2000, “Inelastic displacement ratios for 

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Miranda, E., 2001, “Estimation of inelastic deformation 

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Miranda, E. and Akkar, S.D., 2002, “Evaluation of iterative schemes in equivalent linear methods,” submitted for review and possible publication to 

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Miranda, E., and Bertero, V., 1994, “Evaluation of 

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Naeim, F., Skliros, K, and Reinhorn, A.M., 2000, 

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Porter, K.A., Beck, J.L., and Shaidhutdinov, R.V., 2002,

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Powell, G.H., 1993, Drain-2DX Element Description 

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FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 11-7

Prakash, V., Powell, G.H., and Campbell, S., 1993, 

Drain-2DX Base Program Description and User 

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Qi, X., and Moehle, J.P., 1991, Displacement Design 

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Ramirez, O. M., Constantinou, M. C., Whittaker, A. S., 

Kircher, C. A., Johnson, M. W., and Chrysostomou, 

C. Z., 2003, “Validation of the 2000 NEHRP Provisions' equivalent lateral force and modal analysis 

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Earthquake Spectra, Vol. 19, No. 4, Earthquake 

Engineering Research Institute, Oakland, California, pp. 981-999. 

Reinhorn, A., 1997, “Inelastic techniques in seismic 

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Ruiz-Garcia, J., and Miranda, E., 2004, “Inelastic displacement ratios for design of structures on soft 

soils sites,” Journal of Structural Engineering, Vol. 

130, No. 12, American Society of Civil Engineers, 

Reston, Virginia, pp. 2051-2061.

SAC, 2000, State of the Art Report on Systems Performance of Steel Moment Frames Subject to Earthquake Ground Shaking, FEMA 355C Report, 

prepared by the SAC Joint Venture (a partnership of 

the Structural Engineers Association of California, 

the Applied Technology Council, and California 

Universities for Research in Earthquake Engineering) for the Federal Emergency Management 

Agency, Washington, D.C.

Sasaki, K.K., Freeman, S.A., and Paret, T.F., 1998, 

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Seneviratna, G.D.P.K. and Krawinkler, H., 1994, 

“Strength and displacement demands for seismic 

design of structural walls,” Proceedings of the 5th

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Seneviratna, G.D.P.K., and Krawinkler, H., 1997, Evaluation of Inelastic MDOF Effects for Seismic 

Design, BLUME-120 Report, John A. Blume 

Earthquake Engineering Center, Stanford, California, 177 pages.

Song, J.K., and Pincheira, J.A., 2000, “Spectral displacement demands of stiffness and strength 

degrading systems,” Earthquake Spectra, Vol. 16 

No. 4, Earthquake Engineering Research Institute, 

Oakland, California, pp. 817-851.

Tagawa, H., and MacRae, G., 2000, “Capacity spectra 

method for estimating SDOF oscillator demands,” 

Proceedings of the 2001 Structures Congress & 

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Tsopelas, P., Constantinou, M.C., Kircher, C.A., and 

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Earthquake Engineering Research, State University 

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Uetani, K., and Tagawa, H., 1998, “Criteria for suppression of deformation concentration of building 

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Sons, Ltd.

Villaverde, R., 1996, “Simplified response spectrum 

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neering Mechanics, Vol. 122, American Society of 

Civil Engineers, pp. 282-285.

Wen, Y.K., and Wu, C.L., 2001, “Uniform hazard 

ground motions for mid-America cities,” Earthquake Spectra, Vol. 17, No. 2, Earthquake Engineering Research Institute, Oakland, California, 

pp. 359-384.

Wu, J., and Hanson, R.D., 1989, “Study of inelastic 

spectra with high damping,” Journal of Structural 

Engineering, American Society of Civil Engineers, 

Vol. 115, No. 6, pp. 1412-1431.

Yu, K., Heintz, J., and Poland, C., 2001, “Assessment of 

nonlinear static analysis procedures for seismic 

evaluation of building structures,” Proceedings 

U.S.-Japan Joint Workshop and Third Grantees 

Meeting, U.S.-Japan Cooperative Research on 

Urban Earthquake Disaster Mitigation, 15-16 

August 2001, University of Washington, Seattle, 

Washington, pp. 431-450.

Zamfirescu, D. and Fajfar, P., 2001, “Comparison of 

simplified procedures for nonlinear seismic analysis of structures,” Proceedings, 3rd U.S. Japan 

Workshop on Performance-Based Earthquake 

Engineering Methodology for Reinforced Concrete 

Buildings, Seattle, Washington.

FEMA Reports

FEMA reports are listed by report number.

FEMA 306, 1998, Evaluation of Earthquake Damaged 

Concrete and Masonry Wall Buildings, Basic Procedures Manual, prepared by the Applied Technology Council for the Partnership for Response and 

Recovery, published by the Federal Emergency 

Management Agency, Washington D.C.

FEMA 307, 1998, Evaluation of Earthquake Damaged 

Concrete and Masonry Wall Buildings, Technical 

Resources, prepared by the Applied Technology 

Council for the Partnership for Response and 

Recovery, published by the Federal Emergency 

Management Agency, Washington D.C.

FEMA 308, 1998, Repair of Earthquake Damaged 

Concrete and Masonry Wall Buildings, prepared by 

the Applied Technology Council for the Partnership 

for Response and Recovery, published by the Federal Emergency Management Agency, Washington 

D.C.

FEMA 355C, 2000, State of the Art Report on Systems 

Performance of Steel Moment Frames Subject to 

Earthquake Ground Shaking, prepared by the SAC 

Joint Venture (a partnership of the Structural Engineers Association of California, the Applied Technology Council, and California Universities for 

Research in Earthquake Engineering) for the Federal Emergency Management Agency, Washington, 

D.C.

FEMA 356, 2000, Prestandard and Commentary for the 

Seismic Rehabilitation of Buildings, prepared by the 

American Society of Civil Engineers for the Federal Emergency Management Agency, Washington, 

D.C.

FEMA 357, 2000, Global Topics Report on the Prestandard and Commentary for the Seismic Rehabilitation of Buildings, prepared by the American Society 

of Civil Engineers for the Federal Emergency Management Agency, Washington, D.C.

FEMA 368, 2000, NEHRP Recommended Provisions 

for Seismic Regulations for New Buildings and 

Other Structures, Part 1 - Provisions, prepared by 

the Building Seismic Safety Council for the Federal 

Emergency Management Agency, Washington, 

D.C.

FEMA 369, 2000, NEHRP Recommended Provisions 

for Seismic Regulations for New Buildings and 

Other Structures, Part 2 - Commentary, prepared by 

the Building Seismic Safety Council for the Federal 

Emergency Management Agency, Washington, 

D.C.

FEMA 450-1, 2003, NEHRP Recommended Provisions 

for Seismic Regulations for New Buildings and 

Other Structures, Part 1 - Provisions, prepared by 

the Building Seismic Safety Council for the Federal 

Emergency Management Agency, Washington, 

D.C.

FEMA 450-2, 2003, NEHRP Recommended Provisions 

for Seismic Regulations for New Buildings and 

Other Structures, Part 2 - Commentary, prepared by 

the Building Seismic Safety Council for the Federal 

Emergency Management Agency, Washington, 

D.C.
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 12-1 

Project Participants

Project Management

Christopher Rojahn (Principal Investigator)

Applied Technology Council

201 Redwood Shores Parkway, Suite 240

Redwood City, California 94065-1175

Craig D. Comartin (Project Director)

CDComartin, Inc.

7683 Andrea Avenue

Stockton, California 95207-1705

Bernadette Mosby (Project Coordinator)

Applied Technology Council

201 Redwood Shores Parkway, Suite 240

Redwood City, California 94065-1175

FEMA Oversight

Michael Mahoney (Project Officer)

Federal Emergency Management Agency

500 “C” Street, SW, Room 416

Washington, DC 20472

Robert D. Hanson (Technical Monitor)

2926 Saklan Indian Drive

Walnut Creek, California 94595

Project Management Committee

Craig D. Comartin (Chair)

Christopher Rojahn (Ex-officio)

Ronald O. Hamburger

Simpson Gumpertz & Heger

The Landmark @ One Market, Suite 600

San Francisco, California 94105

William T. Holmes

Rutherford & Chekene

427 Thirteenth Street

Oakland, California 94612

Wilfred D. Iwan

California Institute of Technology

Earthquake Engineering Research Lab.

MC 104-44

Pasadena, California 91125

Jack P. Moehle

(University of California at Berkeley)

3444 Echo Springs Road

Lafayette, California 94549

Jonathan P. Stewart

University of California, Los Angeles

Dept. of Civil & Environmental Engineering

5731 Boelter Hall

Los Angeles, California 90095

Project Review Panel

Anthony B. Court*

Curry Price Court

444 Camino Del Rio South, #201

San Diego, California 92108

Leonard Joseph

Thornton-Tomasetti Group

2415 Campus Drive, Suite 110

Irvine, California 92612

*ATC Board Representative

Daniel Shapiro

SOHA Engineers

48 Colin P. Kelley Street

San Francisco, California 94107

Steve Sweeney

US Army Engineer Research and Development Center

Construction Engineering Research Laboratory

2902 Farber Drive

Champaign, Illinois 61821
Project Participants

12-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Chia-Ming Uang

University of California, San Diego

Structural Engineering Department

409 University Center

La Jolla, California 92093-0085

Michael Valley

Magnusson Klemencic Associates

1301 Fifth Avenue, Suite 3200

Seattle, Washington 98101

Phase 1 Project Working Group

Joseph R. Maffei (Group Leader)

Consulting Structural Engineer

Rutherford & Chekene Consulting Engrs.

427 Thirteenth Street

Oakland, California 94612

Mark Aschheim

Santa Clara University

Department of Civil Engineering

500 El Camino Real

Santa Clara, California 95053

Maureen Coffey

Rutherford & Chekene

427 Thirteenth Street

Oakland, California 94612

Mason T. Walters

Forell/Elsesser, Inc.

160 Pine Street, Suite 600

San Francisco, California 94111

Phase 2 Project Working Group

Sinan Akkar

Department of Civil Engineering

Middle East Technical University

06531 Ankara, Turkey

Mark Aschheim

Santa Clara University

Department of Civil Engineering

500 El Camino Real

Santa Clara, California 95053

Andrew Guyader

California Institute of Technology

Earthquake Engineering Research Lab.

MC104-44

Pasadena, California 91125

Mehmet Inel

Pamukkale University

Denizli, Turkey

Eduardo Miranda

Stanford University

Civil & Environmental Engineering

Terman Room 293

Stanford, California 94305-3707

Jorge Ruiz-Garcia

Stanford University

52D Escondido Village

Dudley Lane

Stanford, California 94306

Junichi Sakai

Pacific Earthquake Engineering Research Center

University of California at Berkeley

1301 South 46th Street

Richmond, California 94804-4698

Tjen Tjhin

Department of Civil and Environmental Engineering

University of Illinois

Urbana, Illinois 61820

Tony Yang

Pacific Earthquake Engineering Research Center

University of California at Berkeley

1301 South 46th Street

Richmond, California 94804-4698
Project Participants

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures 12-3

Phase 2 Focus Groups

Focus Group on Displacement Modification 

John Hooper

Magnusson Klemencic Associates

1301 Fifth Avenue, Suite 3200

Seattle, Washington 98101

Gregory A. MacRae

Assistant Professor

University of Washington

Department of Civil Engineering

Seattle, Washington 98195-2700

Stephen A. Mahin

University of California at Berkeley

Dept. of Civil Engineering

777 Davis Hall,

Berkeley, California 94720

Focus Group on Equivalent Linearization 

Terrence Paret

Wiss, Janey, Elstner Associates, Inc.

2200 Powell Street, Suite 925

Emeryville, California 94608-1836

Graham Powell

Graham H. Powell, Inc.

140 Birchbark Place

Danville, California 94506

Andrew S. Whittaker

University at Buffalo, SUNY

Department of Civil, Structural, and

Environmental Engineering

212 Ketter Hall

Buffalo, New York 14260-4300

Focus Group on Multi-Degree-of-Freedom (MDOF) Effects 

Anil K. Chopra

University of California at Berkeley

Department of Civil Engineering

Berkeley, California 94720

Jon A. Heintz

Degenkolb Engineers

225 Bush Street, Suite 1000

San Francisco, California 94104

Helmut Krawinkler

Stanford University

Civil Engineering Department

Stanford, California 94305-4020

Focus Group on Soil-Structure Interaction

Jacobo Bielak

Carnegie Mellon University

Dept. of Civil Engineering

Pittsburgh, Pennsylvania 15213-3890

Gregory L. Fenves

University of California at Berkeley

Dept. of Civil Engineering

729 Davis Hall, #1710

Berkeley, California 94820-1710

James O. Malley

Degenkolb Engineers

225 Bush Street, Suite 1000

San Francisco, California 94104
Applied Technology Council

The Applied Technology Council (ATC) is a nonprofit, tax-exempt corporation established in 1971 through the efforts of the Structural Engineers Association of California. ATC’s mission is to develop state-of-the-art, userfriendly engineering resources and applications for use in mitigating the effects 

of natural and other hazards on the built environment. ATC also identifies and 

encourages needed research and develops consensus opinions on structural engineering issues in a non-proprietary format. ATC thereby fulfills a unique role 

in funded information transfer.

ATC is guided by a Board of Directors consisting of representatives appointed 

by the American Society of Civil Engineers, the National Council of Structural 

Engineers Associations, the Structural Engineers Association of California, the 

Western Council of Structural Engineers Associations, and four at-large representatives concerned with the practice of structural engineering. Each director 

serves a three-year term.

Project management and administration are carried out by a full-time Executive Director and support staff. Project work is conducted by a wide range of 

highly qualified consulting professionals, thus incorporating the experience of 

many individuals from academia, research, and professional practice who 

would not be available from any single organization. Funding for ATC projects 

is obtained from government agencies and from the private sector in the form 

of tax-deductible contributions.

2005-2006 Board of Directors

Lawrence G. Griffis, President Eve Hinman

Christopher P. Jones, Vice President David A. Hutchinson

Pat Buscovich, Secretary/Treasurer Mark H. Larsen

David C. Breiholz Ugo Morelli

Gregory G. Deierlein H. John Price

Ramon Gilsanz Charles Thornton

James R. Harris
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-1 

A. Summary of Research on Inelastic Analysis 

Procedures

A.1 Introduction

Inelastic analysis of structures is a field with a number 

of active researchers. In order to gauge the current 

situation with respect to information and data that might 

be used to evaluate and improve procedures, ATC 

solicited information from investigators on relevant 

published research as well as on research in progress 

and unpublished insights and perspectives. The 

purpose was to obtain insight for the direction of the 

project. This section summarizes the information 

obtained from these investigators including limited 

review of literature relevant to the objectives of the 

project. Announcements in professional newsletters 

informed the engineering community of the project and 

directed interested persons to the ATC web site. There 

they found a description of the project and a form for 

initial researcher input. Project team members met with 

investigators in face-to-face meetings from March 

through June 2001. This effort was supplemented by 

interviews and e-mail discussions. In total, the project 

team members contacted over fifty researchers in the 

United States, Europe, and Japan. Over thirty in 

structural engineering, risk and reliability; geotechnical 

engineering, and engineering seismology provided 

input, for which ATC, FEMA and PEER are most 

thankful. Table A-1 contains a partial list of the 

researchers contacted, along with an indication of the 

format in which information was collected.

Table A-1 Investigator Research Data 

Researchers Contacted

Researcher 

Input Form Discussion Notes

Meeting 

Notes

Telephone 

Notes E-mail Other

Risk and Reliability

Allin Cornell 9 9

Y.K. Wen 9

Structural Engineering

Mark Aschheim 9

John Bonacci 9

Joe Bracci 9

Anil Chopra 9 9

Michael Constantinou 9

Greg Deierlein 9

Amr Elnashai 9

Peter Fajfar 9

Greg Fenves 9

Andre Filiatrault 9

Doug Foutch 9

Sig Freeman 9

Rakesh Goel 9
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

The summary that follows is a composite of information 

submitted by or obtained from interviews of 

researchers, and the results of literature review by 

project team members. Last name references are used 

to attribute thoughts and opinions of individual 

researchers contacted. In a few instances, the work of 

other researchers not contacted is mentioned without a 

formal citation. Information from published and 

pending articles and reports is cited by lead author and 

date as listed in the References section. 

A.2 Classification of Analysis Methods

The research summary addresses various methods of 

inelastic analysis, but focuses upon the nonlinear static 

procedures known as the Capacity-Spectrum Method, 

as described in ATC-40 (ATC, 1996), and the 

Displacement Coefficient Method, as described in 

FEMA 273 (ATC/BSSC, 1997) and FEMA 356 (ASCE, 

2000). Inelastic analysis methods differ based in the 

approximations used to model the structural system 

(e.g. “equivalent” single-degree-of-freedom (SDOF) 

Researchers Contacted

Researcher 

Input Form Discussion Notes

Meeting 

Notes

Telephone 

Notes E-mail Other

Ron Hamburger 9

Bill Iwan 9

Mervyn Kowalsky 9

Helmut Krawinkler 9

Sashi Kunnath 9

Andres Lepage 9

Greg MacRae 9

Joe Maffei 9

Eduardo Miranda 9

Jose Pincheira 9

Graham Powell 9 9

Nigel Priestley 9

Andrei Reinhorn 9

Mete Sozen

John Stanton 9

Masaomi Teshigawara 9

Mike Valley 9

Ed Wilson 9

Geotechnical Engineering and Site Effects

Youssef Hashash 9

Engineering Seismology and Ground Motions

Bruce Bolt 9

Mehmet Celebi 9

Table A-1 Investigator Research Data (continued)
Appendix A: Summary of Research on Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-3

considering one or more modes, stick models, 

“fishbone” models, 2D planar models, 3D models) and 

the form of the representation of demand (e.g. elastic 

spectra, approximate inelastic spectra, one or more 

elastic or nonlinear time histories, or energy-based 

methods). Both the Capacity-Spectrum Method and 

Displacement Coefficient Method use “equivalent” 

SDOF systems to estimate the response of an inelastic 

system from the response of an elastic system to an 

elastic response spectrum. 

MacRae and Maffei observe that methods for 

estimating the response of inelastic systems based on 

elastic response spectra may be classified further based 

on the assumption used for the elastic period in the 

estimate of peak inelastic displacement response. 

Lateral force reduction factors, R-μ-T relations, and the 

Displacement Coefficient Method are similar in their 

adoption of the initial elastic period in this regard. In 

contrast, one of the fundamental features of equivalent 

linearization techniques is the use of a period longer 

than the initial to reflect inelastic effects. The CapacitySpectrum Method, for example, uses the secant period 

corresponding to the maximum inelastic displacement. 

Notably, there is a simple algebraic relationship relating 

the initial and periods if they are both known. 

Consequently, the graphic representation of a method 

can be de-coupled from the underlying relationships 

that are used to estimate displacements. For example 

the use of inelastic spectra in a Capacity-Spectrum 

Method format allows secant stiffnesses to be used with 

R-μ-T relations based on the initial, elastic stiffness for 

estimating displacements. A variant of the CapacitySpectrum Method known as Yield Point Spectra 

(Aschheim) allows initial stiffnesses to be used with the 

same R-μ-T relations. In both cases, the underlying 

displacement estimate is based on the initial stiffness, 

because the initial stiffness is used in the definition of 

the R-μ-T relationships. The original CapacitySpectrum Method bases the displacement estimate on 

the elastic response of an oscillator having the secant 

stiffness of the nonlinear system.

A.3 Nonlinear Static Procedures

A.3.1 Overview of Current Procedures

A.3.1.1 Capacity-Spectrum Method

Description. The peak displacement of a nonlinear 

system is estimated as the intersection of the capacity 

curve and an elastic response spectrum that is reduced 

to account for energy dissipated by the yielding 

structure. The underlying basis of the CapacitySpectrum Method (CSM) is the concept of an 

“equivalent linear” system, wherein a linear system 

having reduced stiffness (Keff = Ksecant), and increased 

damping proportional to hysteretic energy, is used to 

estimate the response of the nonlinear system. The 

CSM is documented thoroughly in ATC-40. It is also 

described in further detail in Section 7 of this document.

Advantages 

1. The intersection of “capacity” and “demand” 

curves implies a sense of dynamic equilibrium.

2. The influence of strength and stiffness on peak displacement is represented by the graphic nature of 

the procedure.

3. As currently presented in ATC-40, the procedure 

equates viscous damping to hysteretic damping providing an appealing link to the actual characteristics 

of the structure.

4. The interpretation of the graphic solution can provide insight for an effective retrofit strategy.

Drawbacks

1. There is a controversy over the accuracy of displacement estimates.

2. The iterative procedure is time-consuming and may 

sometimes lead to no solution or multiple solutions 

(Chopra).

3. The equating of hysteretic energy dissipation to viscous damping energy dissipation provides a somewhat specious sense that the procedure is 

“theoretically” based on fundamental physical 

properties.

Accuracy. Some investigators find the CSM 

overestimates displacement response while others find 

the CSM underestimates displacement response. 

Albanesi et al. (2000) find significant disparities 

between estimates of response made with the CSM, 

equal energy, and equal displacement assumptions 

when compared to the results from nonlinear response 

history analyses. Chopra and Goel (1999a, 1999b) 

report that the CSM procedure significantly 

underestimates displacement response for a wide range 

of periods and ductility values, relative to the Newmark 

Hall and other R-μ-T relations. Tsopelas et al. (1997) 

finds that the CSM either accurately estimates or 

overestimates the mean displacements obtained from 

nonlinear reponse history analysis. MacRae observes 

that the CSM both overestimates the effective damping 

for a given ductility and reduces the 5% damped 
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

spectrum too much for a given level of damping. Iwan 

reports that the CSM is accurate in a mean sense for 

some cases, but the scatter in displacement estimates is 

large because the combination of effective stiffness and 

damping used in the CSM is not optimal. Iwan et al. 

(2000) report that the use of equivalent viscous 

damping to account for inelastic behavior in the CSM 

yields satisfactory results for the limited period ranges 

where a resonance type of response occurs; but the 

CSM is not generally valid for near-field ground 

motions. Akkar and Gulkan (2000) also report that the 

CSM underestimates response to near field earthquakes. 

Freeman notes the intersection point is least ambiguous 

when the capacity and demand curve form a sharp 

intersection; whereas when the curves approach each 

other gradually, the expected displacement may be less 

certain. 

The range of findings on the accuracy of the CSM 

reflect the various strategies and assumptions used in 

the evaluation studies. The assessment of the accuracy 

of the method is likely to vary with the ground motions 

used to evaluate it, with clear differences emerging, for 

example, for near-field motions relative to far-field 

motions. As a part of the current work the CapacitySpectrum Method as presented in ATC-40 was 

evaluated for a wide range of parametric values (see 

Chapter 3)

Theory . The apparent theoretical basis of the 

Capacity-Spectrum Method of ATC-40 is a strength and 

a weakness. In reality the underpinnings of the current 

procedure are obscure. As discussed in Section A.3.2.1, 

the CSM is a form of the empirically based equivalent 

linearization. Also the need for empirically-determined 

coefficients (e.g. to account for structure framing type 

[Valley]) adds an element of empiricism to the method. 

The use of spectral reduction factors to be applied to a 

designated spectral shape makes it unclear if the method 

is even applicable to site specific spectra that depart 

from the designated spectral shape (Iwan).

Enhancements. Improvements and modifications to 

the basic CSM have been suggested by some 

investigators. For example, Albanesi et al. (2000) 

suggest the use of a variable damping response 

spectrum, in which the damping level increases as the 

ductility of the system increases. Potential 

enhancements to the method involve using re-calibrated 

ductility/damping spectral reduction factor relationships 

or, more directly, ductility-related spectral reduction 

factor relationships in:

1. The so-called Direct Capacity Spectra Method 

(MacRae and Tagawa, 2001).

2. The use of inelastic spectra based on R-μ-T relationships.

3. Using a graphic representation in which an intersection of a “demand curve” and a “capacity curve” 

defines the maximum displacement similar to the 

Capacity-Spectrum Method (Chopra and Fajfar).

4. The use of inelastic spectra based on R-μ-T relationships, plotted with yield displacement on the 

abscissa, in a format known as Yield Point Spectra 

(Aschheim). 

Further discussion may be found in Sections A.3.2.1 

and A.3.2.2. Finally, Direct Displacement Based Design 

is a design version of the CSM (Kowalsky) and is 

discussed in Section A.3.1.4

A.3.1.2 Displacement Coefficient Method

Description. The peak displacement of a nonlinear 

system is estimated as the peak displacement of an 

elastic system (Keff = Kinitial) multiplied by a series of 

coefficients. Of primary interest here are the 

coefficients C1, the ratio of the peak displacement of the 

inelastic system and the peak displacement of the elastic 

system having the same period of vibration; C2, which 

accounts for the effect of pinching in the loaddeformation relation; and C3, which accounts for 

second-order (P-Delta) effects. FEMA 356 is the 

primary source of documentation for the Displacement 

Coefficient Method (DCM). It is described further in 

Chapter 5 of this document. The coefficients are 

empirical and derived primarily from statistical studies 

of the nonlinear response history analyses of SDOF 

oscillators. In particular, C1 is conventionally 

characterized in terms of strength, ductility, and period 

(R-μ-T relationships). The coefficient C1 usually is 

greater than 1.0 for relatively short periods and 

converges to approximately 1.0 as period increases 

(equal displacement approximation). In the DCM, 

initial stiffness has a predominant effect on peak 

displacement response, while strength has little effect.

Advantages. The principal advantages of the method 

are that it is direct and simple to apply. It is also based 

on empirical parameters (R-μ-T relationships) that have 

been studied and generally accepted in the technical 

community for some time, leading to relatively little 

controversy.
Appendix A: Summary of Research on Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-5

Drawbacks. The Displacement Coefficient Method has 

received little scrutiny in recent literature, at least 

relative to the Capacity-Spectrum Method. Thus, 

potential drawbacks of the method may not be as 

apparent. The product formulation for representing the 

effects of strength, pinching, and P-Delta effects may be 

questioned. Krawinkler stated that P-Delta effects in 

multistory structures can not be accounted for 

accurately using simplified procedures, and realistic 

spectra should be used for soft soil sites, rather than 

using a coefficient approach.

Accuracy. Many compromises were required to 

transform research results into the FEMA 273/356 

nonlinear static procedure (Krawinkler). The C1 factor 

as defined in FEMA 356 is smaller than research 

indicates, as noted in FEMA 274. Miranda (2001) 

points out that the C1 term should be derived from 

oscillator response values and not from the R-μ-T 

relations that are based on these responses, to avoid 

statistical bias in the results. MacRae and Tagawa 

(2001) note that the coefficient C2 should approach 

unity as the strength of the pinched system approaches 

the strength required for elastic response.

Song and Pincheira (2000) find that the FEMA 273 

recommendations provide conservative estimates of the 

displacement amplification factors for degrading 

oscillators with periods greater than 0.3 sec on firm 

soils, and are unconservative at shorter periods. 

Lew and Kunnath (2000) compare demands computed 

using the LSP, LDP, NSP, and NDP of FEMA 273 with 

the acceptance criteria of the document for an 

instrumented 7-story reinforced concrete frame building 

(the Holiday Inn, Van Nuys, California) subjected to 

ground motions having a 10% probability of 

exceedance in 50 years, as developed for the FEMAfunded SAC1 project. A triangular load pattern was 

used in the pushover analysis, and member plastic 

rotations were calculated from chord rotations as 

suggested in FEMA 273. The beam plastic rotation 

demands determined in this way were similar to the 

mean beam plastic rotations determined by nonlinear 

dynamic analysis, with pushover analysis 

underestimating the plastic rotation demands in the 

columns relative to those determined by nonlinear 

dynamic analysis, particularly in the upper stories.

As a part of the current work for the ATC-55 project, 

the Displacement Coefficient Method as presented in 

FEMA 356 was evaluated for a wide range of 

parametric values (see Appendix B).

Enhancements. The Displacement Coefficient Method 

could be improved by deriving C1 factors directly from 

nonlinear response data. The expression for C1 could 

be made a function of the degree of degradation of the 

oscillator load-deformation response and the degree of 

P-∆ effects present. MacRae and Tagawa (2001) 

suggest improved C1 factors to account for near-field 

effects. See the discussion in Section A.3.3 below.

A.3.1.3 Drift-Based Approaches

Simple methods to consider drift in the design of 

multistory buildings have been put forward by Sozen 

and his coworkers. Lepage (1998) describes a method 

for estimating peak drift, in which an effective period 

and an elastic displacement spectrum determined for 

2% damping are used with a linear model of the 

structure. Browning (2000) uses the technique to 

determine a target period to limit the expected roof drift 

and interstory drifts during a design level event.

A.3.1.4 Direct Displacement-Based Design

Kowalksy considers Direct Displacement Based Design 

to be a design-oriented implementation of CSM. Rather 

than estimating peak displacements, a limit on 

displacement is used to determine the required 

properties of the system. Lepage noted that this method 

has been criticized for overestimating the effective 

damping present, but at the same time, Fenwick finds 

that uncracked properties make individual components 

stiffer than is considered in the method. As a result, the 

higher damping tends to compensate for the use of a 

more flexible building, resulting in the apparently 

reasonable displacement estimates. Chopra and Goel 

(2001,a,b,c) report that the use of linear elastic spectra 

with increased damping, as recommended by Priestley, 

does not work well in comparison with an inelastic 

design spectrum derived using the Newmark-Hall 

relations. 

Filiatrault and Folz (2001) have adapted this procedure 

to wood frame construction. Because wood softens 

gradually, a sharp yield point does not exist. This makes 

the use of R-μ-T relationships very difficult, while 

approaches that use the CSM format can handle 

softening more easily. 1. SAC: a joint venture partnership of the Structural 

Engineers Association of California, the Applied Technology Council, and California Universities for Research in 

Earthquake Engineering 
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

A.3.2 Fundamental Bases and Relationships

The Capacity-Spectrum Method and Displacement 

Coefficient Method rely on different underlying 

relationships to estimate the response of nonlinear 

systems based on an elastic response spectrum. The 

Capacity-Spectrum Method relies on the concept of 

equivalent linearization while the Displacement 

Coefficient Method uses R-μ-T relationships (where R 

= the strength required for elastic response divided by 

the effective yield strength of the system and μ= the 

displacement ductility response of the system). These 

fundamental relations are reviewed in the following 

sections. As presented and utilized currently, the 

graphical characteristics of the two procedures are also 

different. However, these differences are not 

fundamental and results from either approach may be 

readily transformed into various graphical 

representations. 

A.3.2.1 Equivalent Linearization

The basis of the Capacity-Spectrum Method is the 

premise that the peak response of an inelastic system 

can be estimated as the peak response of a linear elastic 

system having reduced stiffness and increased damping. 

Different approaches have been used to determine the 

properties of the “equivalent” linear system. In some 

cases, relationships between the energy dissipated by 

material nonlinearity and the energy dissipated by 

viscous damping are used, while in others, empirical 

calibrations are used to identify the viscous damping 

(and, in some cases, stiffness) that result in the best 

estimates of peak displacement response. This section 

reviews various conceptual approaches that have been 

taken and discusses empirical observations that bear on 

the hypothesis that viscous damping is a suitable 

surrogate for the energy dissipated by hysteretic 

behavior in nonlinear systems. More discussion of 

equivalent linearization is included in Chapter 6.

Empirical Methods. Equivalent linearization requires 

that the stiffness and viscous damping of the equivalent 

linear system be established. A nonlinear system having 

μ=4, for example, can be represented by a linear system 

having stiffness equal to the secant stiffness and 

sufficient damping to cause the peak displacement 

response to equal the peak displacement of the 

nonlinear system. Iwan observes that the secant 

stiffness is a lower bound to the stiffnesses that could 

potentially be selected, and that for each admissible 

stiffness, there is an associated damping level that 

results in the desired peak displacement. Thus, the 

challenge is to identify the optimal combination of 

stiffness and damping that results in a least biased 

estimate of the peak response and minimizes the 

dispersion in the estimates.

Tagawa and MacRae (2001) identify effective damping 

values by adjusting the damping of an elastic system, 

having a period based on the secant stiffness, to obtain 

peak displacements equal to the peak displacement of 

the inelastic system. MacRae reports that negative 

values of substitute damping are required for some 

combinations of oscillator characteristics and ground 

motion records to match the peak displacement of an 

inelastic oscillator, although the mean values tend to be 

somewhere between the ATC-40 and Japanese Building 

Standard Law versions of CSM.

The Building Research Institute of Japan studied the 

CSM for adoption into the Japanese building code. The 

Japanese implementation (Otani, 2000) uses a lower 

amount of damping, heq, than the ATC-40 

implementation. The effective damping is a function of 

ductility, μ. For reinforced concrete and steel members 

in flexure, 

(A-1)

and for reinforced concrete members with pinching 

or slip related to shear cracking or bar slip,

(A-2)

Energy Methods. Judi et al. (2000) summarize the 

concepts of equivalent damping and substitute 

damping. Equivalent damping comes from Jacobsen 

(1930) and is applicable to nonlinear systems subjected 

to sinusoidal displacement histories. Substitute 

damping was defined by Gulkan and Sozen (1974) as 

the viscous damping needed in an elastic structure to 

dissipate the same amount of energy input to a structure 

responding inelastically to an earthquake ground 

motion, where the elastic system has a fundamental 

period based on the secant stiffness of the inelastic 

structure at its peak displacement. Gulkan and Sozen 

worked with stiffness-degrading systems. Hudson, 

working with bilinear systems responding to earthquake 

ground motions reportedly found (in 1965) that 

substitute damping values were approximately a third of 

the counterpart equivalent damping values. This 

observation may reflect the smaller amount of energy 

dissipation associated with the relatively few cycles of 

1 1 1 0.05

4

heq

μ

⎛ ⎞

=− + ⎜ ⎟

⎝ ⎠

1 1 1 0.05

5 eq h

μ

⎛ ⎞

=− + ⎜ ⎟

⎝ ⎠
Appendix A: Summary of Research on Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-7

earthquake shaking oscillations at or near the peak 

displacements compared to a uniform sinusoidal 

displacement history.

The Capacity-Spectrum Method assumes that the 

energy dissipated by nonlinear behavior can be equated 

to the energy dissipated by a linear elastic system 

undergoing simple harmonic oscillations at the peak 

displacement response. The stiffness of the linear elastic 

system is set equal to the secant stiffness of the 

nonlinear system at the peak displacement.

Following the same concept employed for evaluating 

nonlinear site response in the SHAKE program, Powell 

suggests that the secant stiffness be evaluated at 70-

80% of the maximum displacement, since there may be 

only one or two cycles at or near the peak displacement, 

with most cycles having peak displacements that are 

substantially less than the peak displacement. Powell 

suggests this would result in a smaller period and 

smaller effective damping relative to the ATC-40 

Capacity-Spectrum Method, and therefore would result 

in larger displacement estimates. The Perform-2D

computer program allows the equivalent linear stiffness 

to be set to a proportion of the peak displacement. 

Several observations diminish the hypothesis that 

equivalent damping should be obtained by equating 

hysteretic energy dissipation to viscous energy 

dissipation: 

1. Oscillators with different hysteretic properties can 

have the same peak displacement (Kowalsky).

2. As the post-yield stiffness changes from positive to 

negative, there is a disproportionate increase in displacement response amplitudes (Aschheim).

3. Nonlinear elastic systems (e.g. rocking walls) have 

no hysteretic energy dissipation, yet peak displacements are not much greater than systems with full 

energy dissipation (Miranda).

4. Initially undamaged and initially damaged oscillators were found to have nearly identical peak displacements, indicating that differences in the 

energy dissipated through hysteretic losses has little 

effect on peak displacement response (Aschheim 

and Black, 2000).

5. Changes in stiffness associated with nonlinear 

response interrupts the build up of resonance that 

drives the elastic spectral ordinates to their peaks 

(Paret, 2002). 

Accuracy of Effective Damping Relationships. Iwan 

notes there may be some sensitivity of the optimal 

effective stiffness and damping values to the suite of 

ground motions used. Miranda reports that he is finding 

that Iwan and Gates (1979) relationships are very good, 

and the Gulkan and Sozen damping is not very accurate. 

Fenves, however, reports that the Gulkan and Sozen 

damping is good for reinforced concrete structures. 

Kowalsky indicates that effective damping may differ 

for near-field motions containing significant velocity 

pulses. R. Goel reports that the Japanese Building 

Standard Law underestimates damping and leads to 

overestimates of displacements. Stanton expressed 

concern that the baseline value of damping, to which 

the equivalent or substitute damping is added, is not 

necessarily 5% and might be lower. MacRae has 

recalibrated the Capacity-Spectrum Method and finds 

the scatter is similar to a recalibrated version of the 

Displacement Coefficient Method, except for periods 

above 1.5 sec, where the recalibrated CSM has greater 

scatter.

Spectral Reduction Factors. Reduction factors to be 

applied to smoothed elastic design spectra to establish 

spectral amplitudes for larger amounts of effective 

damping are tabulated for the Capacity-Spectrum 

Method. Tagawa and MacRae (2001) find the actual 

reduction associated with elastic response for a given 

damping level is not as large as is determined in ATC40. While the Capacity-Spectrum Method specifies (a) 

a ductility-effective damping relationship and (b) an 

effective damping-spectral reduction factor 

relationship, it is possible to establish a direct 

relationship between ductility and the spectral reduction 

factor (MacRae and Tagawa, 2001). MacRae reports 

that less scatter results when this single relationship is 

used, in the so-called Direct Capacity Spectra Method. 

A.3.2.2 Displacement Modification

The coefficient C1 of the Displacement Coefficient 

Method as currently is derived from R-μ-T

relationships. Such relationships are usually determined 

by statistical analysis of the computed response of a 

large number of SDOF oscillators having prescribed 

load-deformation relationships to actual ground motion 

records. There is general agreement on the form of the

R-μ-T relationship (e.g. Miranda and Bertero, 1994), 

although there are some differences in the relationships 

determined by various investigators. There is 

significant variability in the R-factors determined for 

individual records. Larger R-factors generally can be 

expected for long-duration motions that allow 

resonance to build up the elastic response, and smaller
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

R-factors can generally be expected for systems 

subjected to predominantly pulse-type motions. 

Most investigators have determined R-μ-T relationships 

based on statistics computed on R, and have further 

determined displacement ratios such as C1 by algebraic 

manipulation of R-μ-T relationships. Miranda observes 

that when required strengths are determined by 

applying R factors to elastic spectral amplitudes, the 

parameter of interest is R-1. Furthermore if one 

assumes a given ductility for an oscillator, then the 

“required” R can be determined statistically from the 

analysis of the oscillator response to a suite of ground 

motions. If one then assumes the “required” R and 

calculates the ductility demand, it is not necessarily 

equivalent to that assumed in the first instance. Since 

the expected R factor for a given ductility level is not 

precisely correlated to the expected ductility that results 

from a given R factor, coefficients such as C1 should be 

determined from the oscillator response data and not by 

algebraic manipulation of the R-μ-T relationship. This 

is further complicated by the non-monotonic nature of 

the strength-ductility relationship, in which different 

strengths may result in the same ductility response. 

Recommended R factors are usually based on the 

largest strength associated with a desired ductility level. 

Cuesta et al. (2001) minimize the error in estimated 

strengths, and find that R-factors should be expressed in 

relation to a characteristic period of the ground motion 

(i.e. R-μ-T/Tg relationships). This is an observation also 

made by Vidic et al. (1994). Even so, there is some 

ambiguity in the identification of the characteristic 

period of a site, because ground motions recorded in 

different horizontal directions or in different 

earthquakes may display different characteristic 

periods.

A.3.2.3 Choosing Between Equivalent 

Linearization and Displacement 

Modification

While some investigators find the computation of Rfactors to be more direct than the use of equivalent 

linearization, Fenves observed that the averaging of Rfactors over many ground motions to obtain R-μ-T

relationships separates the relationships from the actual 

dynamics, and effective damping relationships may be 

as good. Fajfar noted that both approaches involve 

approximations, but R-μ-T relationships are easier to 

use, in part because no iteration is required, and most 

people accept the equal displacement approximation 

that is expressed in many R-μ-T relationships for 

periods greater than Tg. Fajfar (1999) and Chopra and 

Goel (1999a, 1999b) have recommended R-μ-T

relationships be used for reducing the elastic response 

spectra in the CSM. Aschheim and Black (2000) have 

also recommended the use of these relationships with 

smoothed elastic design spectra or the display of the 

actual, jagged, constant ductility spectra of a suite a 

ground motions, in the Yield Point Spectra format.

The choice between the two procedures is largely a 

matter of personal preference as opposed to relative 

accuracy. The comparisons of relative accuracy prior to 

the ATC 55 project are very limited. There have been 

no comprehensive studies that simultaneously 

incorporate the relevant scope (strength, period, 

hysteretic characteristics, site conditions, etc.) and 

range of parameters required to make conclusive 

judgements or recommendations for improvement. 

This fact is the motivation for the comparisons made in 

ATC 55 as summarized in Section 5.

A.3.3 Near-Field Effects on SDOF Systems

Near field motions are those that contain one or more 

large velocity pulses, usually originating from the 

superposition of waves emanating from the fault as the 

rupture progresses towards a site (i.e. directivity effect). 

Short-period systems experience the near-field effects 

as impulses. The large velocity pulses can cause the 

elastic spectra to be larger. The R factors associated 

with such pulses are smaller, in general, than those 

associated with motions in which resonance contributes 

to the elastic spectral amplitudes. It is now appreciated 

that structures with periods less than the characteristic 

period of the pulse may be severely affected 

(Krawinkler). Long-period structures may experience 

large interstory drifts associated with the large 

amplitude ground motion reversals

Iwan et al. (2000) observe that larger displacement 

amplification factors and smaller strength reduction 

factors are indicated for structures having fundamental 

periods less than the predominant period of the near 

field ground motion, relative to far-field cases. Baez 

and Miranda (2000) find that displacement 

amplification factors (the peak displacement of an 

inelastic system having a specified ductility divided by 

the peak displacement of an elastic system having the 

same initial stiffness) are up to about 20% larger for 

near field sites, with fault normal amplifications being 

larger than fault parallel amplifications. MacRae and 

Tagawa (2001) recommend an R-μ-T relation for near 

field motions that changes with directivity.
Appendix A: Summary of Research on Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-9

A.3.4 Equivalent SDOF Systems

Both the Capacity-Spectrum Method and the 

Displacement Coefficient Method use an “equivalent” 

SDOF model to represent the resistance of the structure 

to deformation as it responds in its predominant 

“mode.” Various techniques have been recommended 

for determining the properties of the “equivalent” 

SDOF system. Specifically, the relationships used in 

FEMA 356 and ATC-40 result in different SDOF 

systems. In many cases the period of the SDOF 

analogue differs from the natural period of vibration of 

the structure (Aschheim). For structures in which the 

predominant mode of response involves a change in the 

shape vector (e.g. once a plastic hinge develops at the 

base of a slender structural wall), further adjustments in 

the post-yield capacity curve of the SDOF analogue 

may be warranted, although such refinements may be 

overly precise given the approximate nature of the 

displacement estimate. Several alternative “equivalent” 

SDOF techniques have been proposed including those 

by Rothe and Sozen (1983), Miranda and Bertero, Qi 

and Moehle (1991), Fajfar and Gaspersic, (1996), 

Villaverde (1996), Han and Wen (1997), and Chopra 

and Goel (2001b).

A.3.5 Behavior Mode Effects 

Many structures will not exhibit the full hysteretic loops 

that are often used in analytical research studies. The 

presence of stiffness degradation, pinching, strength 

deterioration, and foundation rocking may influence 

peak displacement estimates. The general consensus 

appears to be that moderate levels of stiffness 

degradation and pinching will cause peak displacements 

of short period systems (below 0.3 to 0.5 sec) to 

increase slightly above those determined for bilinear 

systems. Negative values of post-yield stiffness, arising 

either due to the load-deformation behavior of the 

component or the presence of P-Delta effects, can cause 

increases in peak displacement, as can rapid strength 

deterioration.

Gupta and Kunnath (1998) investigated stiffness 

degradation and pinching, and found that “severe” 

degradation causes only structures with periods less 

than about 0.5 sec to have mean displacements 

substantially greater than elastic values. Gupta and 

Krawinkler (1998) find that peak displacements 

increase with the introduction of pinching, with the 

increase becoming larger with decreasing initial period. 

The ratio of peak displacement for a pinching model to 

the peak displacement without pinching seems to be 

nearly independent of the ductility of the system. This 

study finds that peak displacements increase 

substantially as the post-yield stiffness becomes 

increasingly negative, and the increase is larger for 

weaker systems. However, the effect of negative postyield stiffness on increasing peak displacement is 

reduced in the presence of pinching. 

Song and Pincheira (2000) discuss effects of pinching, 

stiffness degradation, and negative post-yield stiffness 

on displacement response. They find that the equal 

displacement approximation is generally applicable to 

degrading systems for periods greater than a 

characteristic period (Tg) of the ground motion. Peak 

displacements were generally larger than those of nondegrading systems for periods less than Tg. For systems 

on rock or firm soil, displacement amplification factors 

of 2 were found at T=0.3 sec, with even larger values 

possible at shorter periods or on soft soils. Fischinger 

reports that the shape of the hysteretic loop is important 

for short-period systems and for cases with negative 

post-yield stiffness or strength deterioration. Otani 

agreed that loop shape will not affect peak response 

amplitudes in the displacement-controlled portion of the 

spectrum.

Miranda reports that the period Tc that represents a 

breakpoint in the R-μ-T relationship depends on μ, as 

provided for in the Vidic et al. (1994). R factor 

relationship. Miranda reports that the shifting of the 

period at which the R factors change is moderate for 

cases of pinching and stiffness degradation, but is large 

for strength deterioration. Miranda (2000) has 

developed C1 values for oscillators having bilinear 

load-deformation relations located on firm sites, and 

has developed C1 values for stiffness degrading 

systems. Sause reportedly is determining similar 

parameters for nonlinear elastic systems.

A.3.6 MDOF and Inelastic Mechanism Effects

The use of “equivalent” SDOF systems to characterize 

the nonlinear response of multistory structures 

potentially may be misleading if higher modes play a 

significant role in the response or if inelastic 

mechanisms develop that were not identified in the 

nonlinear static (pushover) analysis. Higher modes 

may influence the mechanisms that develop, and 

different excitations potentially may cause different 

mechanisms to develop. This may be more pronounced 

in buildings in which mechanism strengths or modes of 

failure are not well separated in a dynamic sense. An 

example of the latter case would be the development of 

shear failures in beams or columns due to higher mode 

forces, in a structure that developed a ductile 
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

mechanism in a pushover analysis. Analytical studies 

have focused on only a limited number of case study 

buildings. Sufficient research to address these 

mechanism issues conclusively is not presently 

available.

Gupta and Krawinkler (2000a) relate the peak inelastic 

drifts observed in steel frame buildings to the elastic 

response of a SDOF oscillator through a series of 

factors that account separately for roof drift relative to 

SDOF response, the development of inelasticity, the 

presence of P-delta effects, the ratio of interstory to 

average roof drift, and the relation between element 

deformations and interstory drift. Three-, nine-, and 

twenty story steel moment frame buildings were 

subjected to the components of the SAC ground 

motions having a 2% probability of exceedance (PE) in 

50 years, a 10% PE in 50 years and a 50% PE in 50 

years, oriented at 45 degrees to the fault-normal and 

fault-parallel directions. They report that a good 

estimate of the ratio of elastic roof drift to the first mode 

spectral displacement is given by the first mode 

participation factor, but for structures with periods 

greater than 2 sec they advise use of 1.1 times the first 

mode participation factor. The effects of inelasticity on 

roof drift for the MDOF structures are consistent with 

and similar to the effects observed for SDOF systems. 

For the period range considered, inelasticity tended to 

cause peak drifts to be about 70 to 80% of the elastic 

values, at the median level. The MDOF inelasticity 

factor tended to become smaller with an increase in roof 

drift. This was explained as possibly being the result of 

a concentration of interstory drift demand in a few 

stories leading to a reduction in roof drift. Gupta and 

Krawinkler (2000b) find that P-Delta is a relatively 

benign phenomenon except in the instance when the 

ground motion drives the structure into the range of 

global negative post-yield stiffness, at which point large 

increases in displacement may occur. The ratio of peak 

story drift to peak roof drift is strongly dependent on the 

ground motion and structure characteristics. Median 

values of this ratio increase from about 1.2 for low-rise 

structures to 2.0 for mid-rise structures to about 2.5 to 

3.0 for tall structures, for the structures and motions 

considered. The drift patterns observed for these 

structures suggest that a common drift distribution that 

can be generalized does not exist. 

Foutch and Shi (1998) report the results of nonlinear 

dynamic analyses of steel frame buildings for the SAC 

steel program in which the beam plastic hinges were 

modeled with different load-deformation models. Steel 

moment frame buildings were analyzed that ranged 

from 3 to 9 stories in height. Eight hysteretic models 

were considered for the beam plastic hinges: bilinear 

(with and without strength degradation), stiffness 

degrading (with and without strength degradation), 

pinched stiffness degrading (with and without strength 

degradation), fracturing, and bilinear elastic. The 

plastic hinge model did affect response histories at the 

connections. The effect on the maximum story ductility 

demand relative to the maximum story ductility demand 

for the non-degrading bilinear model is as follows: a 

maximum increase of 10-20% for the non-pinching 

hysteretic models, a maximum increase of 20-30% for 

the pinching hysteretic models, and a maximum 

increase of 30-50% for the bilinear elastic model (which 

has no hysteretic energy dissipation). The Foutch and 

Shi results may be applicable to buildings that develop 

desirable mechanisms. Aschheim expects that a weakstory system having degrading column hinges would 

have much worse performance than the buildings 

described above.

Naeim et al (2000) also investigated the response of 

steel moment frame buildings for the SAC steel project. 

Three-, nine-, and twenty-story steel moment frame 

structures were investigated using a variety of hysteretic 

models. Stiffness degradation with slip or pinching was 

modeled in addition to bilinear response. Strength 

deterioration was modeled but results were not 

described. Severe stiffness deterioration increased 

interstory drifts and residual interstory drifts, with 

interstory drifts often increasing in the upper stories, 

and sometimes reducing in the lower stories. The 

authors suggest this may be attributed to higher modes 

causing the upper stories to go through many more 

cycles of sufficient amplitude to be affected by 

deterioration of the load-deformation response. The 

observed increases tended to be larger than those 

observed by Foutch and Shi. Stiffness degradation 

generally reduced force demands. Slip often reduced 

lower- and upper-story interstory drift demands, 

although increases were observed for some 

combinations of building height, city, and ground 

motion intensity level. Slip tended to decrease story 

shears and overturning moments. 

Iwan et al. (2000) used a shear beam building model to 

compare the Capacity-Spectrum Method with nonlinear 

dynamic analysis for near-field motions. They find that 

for structural periods shorter than the ground pulse 

duration, the MDOF model exhibits a fundamental 

mode type of response, and higher mode contributions 

to drift and shear demands are negligible. For long 

period structures, the large displacement and velocity 

pulses of near field records cause greater participation 

of the higher modes, resulting in the potential for very 
Appendix A: Summary of Research on Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-11

misleading results if a single mode analysis is used. 

Large interstory drifts were observed at the base, during 

the forward movement of the ground motion, and were 

again observed at the upper stories during a large 

reversal of the ground motion, associated with wave 

propagation through the structure. This latter case is 

not associated with the development of the maximum 

roof drift and does not correspond to a first mode shape. 

Thus, a fundamental mode analysis would not detect 

these effects. The investigators conclude that the use of 

a single-mode “equivalent” system provided a 

reasonable estimate of the maximum roof displacement 

regardless of building period, degree of nonlinearity, or 

distribution of stiffness, even for pulse type motions; 

but estimates of interstory drift for tall buildings 

(fundamental period significantly greater than the 

ground pulse duration) were poor, particularly in the 

upper stories.

A.3.7 Pushover Analysis

Nonlinear static (pushover) analysis is used to quantify 

the resistance of the structure to lateral deformation and 

to gauge the mode of deformation and intensity of local 

demands. Various techniques have been recommended, 

including the use of constant lateral force profiles and 

the use of adaptive and multimodal approaches. 

Pushover techniques provide useful information on the 

overall characteristics of the structural system and can 

be used to identify some (but not necessarily all) of the 

likely mechanisms. Because the prescribed loading 

used in pushover analyses can not represent the 

potential range of loading experienced in dynamic 

response, the results obtained by pushover analyses at 

best represent an approximation of the nonlinear 

behavior expected to develop in the response to 

earthquake ground motions. The applicability of 

pushover analyses is less clear for systems having 

discontinuities in strength and stiffness. Results may be 

misleading where multiple collapse mechanisms 

potentially may develop because mechanism strengths 

are not well separated, or where different modes of 

behavior potentially may develop (higher modes cause 

demands to approach or exceed the capacities of 

strength-controlled components). 

Pushover techniques are useful to estimate peak 

displacement response in conjunction with the use of 

“equivalent” SDOF systems. While higher modes 

typically have a small or negligible contribution to 

displacements, higher modes can significantly affect 

interstory drifts, plastic hinge rotations, story shears, 

and overturning forces. The contribution to interstory 

drifts stems directly from the higher mode shapes being 

more tortuous and therefore having a greater 

contribution to interstory drift. Consequently, estimates 

of interstory drift based on a first mode pushover 

analysis is prone to be inaccurate as the number of 

stories and period increases. Pushing to a target 

displacement will not necessarily develop the maximum 

interstory drifts in each story because the maximum 

values in each story do not occur simultaneously, and 

the sum of the individual maximum interstory drifts 

may be twice the peak roof displacement, depending on 

the mechanism that develops (Krawinkler). Some 

evidence suggests that pushovers tend to overestimate 

weak story drifts.

The application of lateral forces in a pushover analysis 

is preferred to applying a prescribed displacement 

pattern because the former allows softening of the 

structure to develop and allows story collapse 

mechanisms to develop. Many techniques involving 

application of lateral forces have been used. The 

simplest technique uses a fixed lateral force profile, 

with lateral forces being proportional to the mass and 

mode shape amplitude at each floor. An updated load 

vector would be more likely to identify concentrations 

of damage, although this presumes that first mode 

response is dominant. These techniques update the 

lateral force profile to adapt to the softening structure 

by using a step-wise lateral force profile that is 

modified from the first mode pattern to account for 

higher modes or by combining the results obtained from 

independent pushover analyses in each of several 

modes. Adaptive techniques that update the lateral load 

vector can make the updated load vector be proportional 

to the current displaced shape or to the current first 

mode (based on the current stiffness properties of the 

structure) or may make the increment in lateral loads 

proportional to the current displaced shape or mode 

shape. The displaced shape changes more quickly than 

does the mode shape (Valley). Inconsistencies can be 

introduced if the load vector is updated without 

updating the mode participation and mass participation 

factors used for determining the properties of the 

“equivalent” SDOF system. Methods that consider 

higher modes must contend with uncertainty in the 

amplitudes and algebraic signs of the higher modes, 

along with their timing relative to the first mode peaks. 

The question of how simple or complex a pushover 

technique to use depends on one’s analysis objectives. 

Simple techniques can provide very valuable but 

incomplete information, while techniques that are more 

complex are still unable to represent the full range of 

response that potentially may develop.
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-12 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Valley and Harris (1998) describe the development of a 

static pushover curve by repeated elastic analyses, with 

members removed sequentially as deformations exceed 

the member yield or ultimate capacities, and with loads 

reapplied in accordance with updated Ritz vectors. 

Reinhorn describes multimodal procedures (1997) that 

rely on updated modal properties. Bracci et al. (1997) 

also determined demand estimates based on the 

instantaneous dynamic properties of the structure. 

Gupta and Kunnath (2000) coupled the use of the 

instantaneous dynamic properties and the elastic 

spectral ordinates of the ground motion to determine 

incremental lateral forces to be applied in the pushover 

analysis. More recently, Kunnath has looked at sums 

and differences of modes. Elnashai (2000) also has 

applied adaptive techniques that make use of the 

instantaneous modal properties, and is able to follow 

the Sa vs Sd plot obtained in Incremental Dynamic 

Analysis reasonably well. Reinhorn suggests that the 

multi-mode pushover force distribution can be 

simplified to a linear distribution that is unique for each 

structure. Kunnath reports that even adaptive pushover 

techniques fail to capture the response of some stories 

in some buildings.

Sasaki et al. (1998) perform pushover analyses 

independently in each of several modes using invariant 

lateral force distributions, to identify the potential for 

higher modes to cause mechanisms to develop. Black 

and Aschheim (2000) combined the peak displacements 

and interstory drifts determined independently for the 

first two modes using square-root-of-the-sum-of-the 

squares (SRSS) combinations, and observed significant 

disparities between the peak interstory drifts and the 

SRSS estimates. This procedure is termed a Modal 

Pushover Analysis (MPA) by Chopra and Goel (2001b), 

who consider up to three or five modes. Chopra 

suggests that SRSS combination rules may be used for 

all computed quantities (e.g. member forces and 

moments), not just displacements and interstory drifts. 

Chopra and Goel (2001c) demonstrate that median 

estimates of interstory drift of the SAC buildings are 

improved by the use of three modes for the 9-story 

buildings and five modes for the 20-story buildings, 

with baseline values established by nonlinear dynamic 

analysis. Errors in the interstory drift estimates were 

larger, in general, than the errors associated with 

response spectrum analysis of linear elastic buildings 

and were largest for the “Los Angeles” buildings, which 

generally had larger interstory drift responses than the 

“Seattle” and “Boston” buildings. Patterns of the 

distribution of median interstory drifts of the “Los 

Angeles” and “Seattle” 9- and 20- story buildings 

differed. Chopra and Goel (2001b) also put forward an 

Uncoupled Modal Response History Analysis 

(UMRHA), in which dynamic response histories 

determined for each “equivalent” SDOF system are 

summed algebraically in time, and maximum values are 

determined from the summed response history. If 

nonlinearities are absent, the MPA and UMRHA 

approaches are equivalent to the traditional response 

spectrum and linear dynamic analysis methods, 

respectively. 

Iwan is working on load profiles to better predict spatial 

distribution of damage. Carr also is reported to be 

working on improved pushover techniques. Bracci 

recently has been doing pushovers on frames one story 

at a time. Deierlein suggested that one could apply a 

perturbation to a first mode load pattern, consisting of 

an additional force that is allowed to change its location 

over the height of the structure, to identify sensitivity in 

the development of the mechanism. 

A.4 Nonlinear Dynamic Procedures

A.4.1 Simplified Models

Nakashima has described the use of simplified models 

for nonlinear dynamic analysis. Much like the “notional 

frames” used by Sozen and Lepage, the generic frame 

or “fishbone” model consists of a single column with 

beams at every floor level extending halfway towards 

an adjacent column, with a roller supporting each beam 

at midspan. The model allows beam plastic hinges and 

story mechanisms to develop, much as they can in 

complete frames. The generic frame model, however, 

does not determine actions on individual members of 

the frame (Otani).

A.4.2 Incremental Dynamic Analysis

Incremental Dynamic Analysis (IDA) determines peak 

response quantities (e.g. roof drift) by a series of 

independent nonlinear dynamic analyses of a structure 

subjected to one or more scaled ground motions. The 

scale factor is increased successively from a small 

initial value, and peak response quantities are plotted 

against a measure of the ground motion intensity. Data 

from such analyses has been represented in two ways. 

Cornell and Krawinkler plot peak interstory drift as a 

function of the scaled Sa at the fundamental period of 

the building (Sa(T1)). Elnashai plots the peak roof 

displacement versus Sa(T1). Both investigators plot Sa

on the vertical axis and the response measure on the 

horizontal axis, to be consistent with the conventional 

plotting of deformations and displacements on the 
Appendix A: Summary of Research on Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-13

abscissa and forces on the ordinate. Cornell has 

developed relationships between the nonlinear static 

pushover capacity curve and the IDA curve that are 

implemented in a spreadsheet, allowing an engineer to 

observe the influence of changes in the capacity curve 

on response parameters, based on relationships 

embedded in the spreadsheet. Because these 

relationships reflect the behavior of the structure, it 

seems they must change as the relative distributions of 

strength, stiffness, and mass of the structure vary.

The strength of the IDA is that it captures aspects of the 

dynamic behavior of the system, and when done for a 

sufficient number of ground motions, reflects a range of 

response that may result. Although investigators 

apparently have not used the technique to characterize 

the range of mechanisms that may potentially occur 

under different excitations, the ability to recover this 

information seems to be another benefit of this form of 

analysis. Interstory drifts are observed to increase 

dramatically when the intensity measure is large 

enough. Such an increase suggests the structure has 

reached its “capacity”. This capacity might be viewed 

as the collapse limit as defined essentially by loss of 

dynamic stability as the intensity measure reaches 

higher levels. This is analogous to the capacity of a 

steel bar being measured by its ultimate strength, except 

that the capacity is expressed in terms of the spectral 

acceleration of a specific excitation waveform. 

Typically, there is substantial scatter in the capacities 

determined in this way, reflecting variability in the 

response of the structure to different excitations.

Concern has been expressed regarding (1) the validity 

of scaling the ground motion amplitude uniformly 

(because high frequencies attenuate more rapidly as 

distance from the fault increases), (2) the uncertainty in 

establishing an accurate structural model, (3) ambiguity 

in the definition of “capacity,” with Cornell focusing on 

the interstory drifts and Krawinkler now focusing on the 

value of the intensity measure (Sa(T1)) at which the 

response parameter seems to increase without limit, and 

(4) whether interstory drift is an appropriate parameter 

to monitor collapse, when collapse may be due to 

gravity loads acting, for example, after columns have 

failed in shear. 

The IDA curves are interesting because of the peculiar 

dynamic response characteristics that are apparent in 

this representation, and may be useful for identifying 

variability in demands, the “capacity” of the structure, 

as well as the onset of collapse, subject to limitations on 

modeling. The usefulness of IDA for design verification 

has not been investigated as yet.

A.5 Modeling Limitations

Accuracy in the estimate of response of a given 

structural model is of little value if the structural model 

itself is inaccurate. Issues relating to the accuracy of 

mathematical models used for estimating response 

include:

1. Evaluation of initial stiffness and strength.

2. Uncertainty and variation in the actual material 

properties and dimensions of the as-built structure.

3. Variation of the actual component strengths from 

calculated estimates.

4. The complexity of behaviors to be represented.

5. Limitations in the understanding and modeling of 

response to complex, inelastic loading histories.

Uncertainty in the initial stiffness and strength of a 

structure leads to further dispersion in the accuracy of 

the displacements estimated using the Capacity 

Spectrum and Displacement Coefficient Methods 

(Miranda). For reinforced concrete structures, there is 

ambiguity in how the period of vibration of the structure 

should be computed (Otani). An additional difficulty 

relates to actual compressive strengths exceeding the 

specified strength, leading to likely increases in the 

modulus of elasticity (Otani). Valley noted that of three 

tuned-mass-damper buildings that his firm designed, 

the estimate of period for one was sufficiently off that 

they had to redesign the tuned mass damper after 

construction. The Mexico City Building Code of 1976 

reportedly considered a range of possible periods, in 

response to concerns raised by Rosenblueth (Chopra). 

Choices made by structural engineers in modeling of a 

structure can affect computed response. Krawinkler 

recalled that in the SAC project, a centerline model of a 

20-story building was found to collapse in the presence 

of P-Delta, but had drifts of no more than about 5% 

when panel zones and gravity columns were modeled. 

Krawinkler noted that different investigators using 

different computer codes obtained very different results 

when first modeling buildings for the SAC project. 

Only when assumptions were made consistent were the 

results more or less identical. Diaphragm flexibility 

generally has not been incorporated into simplified 

inelastic procedures; an approximate method is 

described by Nakaki (2000).
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-14 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

There are relatively few instances in which models have 

been developed of instrumented buildings that were 

heavily damaged by ground shaking. Kunnath et al. 

(2000) considered four instrumented buildings, of 

which two were moderately damaged. He finds that 

calibrating structural models to observed response is 

sensitive to mass and stiffness modeling assumptions. 

Kunnath reports that linear and nonlinear static 

procedures did not adequately predict interstory drift 

estimates, and no one procedure consistently gave good 

results. Islam et al. (1998) modeled the 7-story 

instrumented reinforced concrete building in Van Nuys, 

and found that extensive flexural cracking in the beams 

observed in the pushover analysis at the measured roof 

drift did not occur; the actual building had only minor 

flexural cracking at the lower level beams. Browning et 

al. (2000) report on the ability of various analysis 

procedures to estimate peak drifts and interstory drifts 

of this building, and the difficulty in matching locations 

of column shear failure. 

Multiple actions (e.g. axial, shear, and flexural) result in 

inelastic behaviors that are not well-understood and 

represented poorly in analysis software. Modeling of 

collapse requires careful attention to component 

degradation and may require that the assumptions of 

small displacement theory be supplanted by large 

displacement theory. The accuracy of computed 

predictions of collapse has not been established; even 

the definition of a collapse limit state is ambiguous. 

A.6 Demand Characterization

The lack of an accepted and clearly-defined relationship 

between smoothed design spectra and the actual 

motions they ostensibly represent creates difficulties in 

(1) evaluating the accuracy of inelastic procedures, (2) 

assessing variability in response estimates, and (3) 

establishing design ground motions for use in 

performance-based earthquake engineering. 

Traditionally, smoothed design spectra were fit by 

judgment to the jagged elastic response spectra 

computed for real ground motions. Current approaches 

fit a smoothed design spectrum at T = 1 sec and at 

“short” periods, using values determined from a seismic 

hazard curve. The degree to which actual spectra may, 

and should, depart from a smoothed spectrum is not 

defined, yet the degree of variability surely affects the 

statistical distribution of peak displacements relative to 

estimates based on smoothed elastic response spectra. 

Scaling ground motions to precisely match a target 

design spectrum has been found to result in a systematic 

underestimate of inelastic response, because response 

amplitudes to the stronger ground motions are often 

disproportionately higher than those to weaker ground 

motions (Wen). 

Cornell notes that demand is not a design spectrum but 

a set of earthquake events that cannot be collapsed into 

a single spectrum. Particularly for uniform hazard 

spectra, there does not seem to be a clear answer on 

how to choose records (Cornell). However, to represent 

record-to-record variability, it appears to be necessary 

to use recorded ground motions rather than synthetic 

motions. For design applications, Wen and Wu (2001) 

suggests using records based on regional seismicity—

perhaps a Magnitude (M) 8 earthquake at 40 km, a few 

M7.5 earthquakes at 20 km, a few M6 earthquakes at 

closer distances, etc. 

A.7 Applicability for Performance-Based 

Earthquake Engineering and Design

A.7.1 Role for Inelastic Procedures

Many researchers have focused on improving 

simplified analysis procedures with the goal of 

accurately representing response quantities determined 

in nonlinear dynamic analyses, with some operating 

under the notion that analysis and design are so 

intertwined that they cannot be separated. Other 

researchers view the role of analysis is to enable good 

design, acknowledging that even the best analyses are 

approximate and that approximate analyses are 

sufficient. Given uncertainty in the accuracy of the 

mathematical model of a structure and uncertainties in 

future ground motions, engineers often must rely on 

their judgment to interpret analytical results. There is a 

fundamental uncertainty in response amplitudes that 

applies to all analysis techniques because of variability 

in the R-μ-T relationship from one motion to another, 

and variability in the elastic spectral ordinates, timing, 

and algebraic signs of the higher modes. Because even 

the best analysis techniques are prone to uncertainty 

with regard to performance under future earthquakes, 

there may be a role to be played by simplified analysis 

techniques. 

Simplified inelastic procedures can be used for 

preliminary proportioning and may also be useful for 

characterizing performance. Simple inelastic 

procedures can give good estimates of peak roof 

displacement, at least for regular structures in which 

response is dominated by the first mode under 

conditions where P-Delta effects are negligible. 

Estimates of interstory drift indices, story shears, and 

plastic rotations in relatively flexible buildings are 

prone to be inaccurate, due to higher mode 
Appendix A: Summary of Research on Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-15

contributions. Therefore, inelastic analysis procedures 

may be useful as a first approximation and to indicate 

when analyses of higher precision are needed. Elastic 

analysis procedures also can serve this purpose, 

although one would expect inelastic procedures to 

provide higher fidelity. Inelastic analysis procedures 

could be used to encourage capacity design approaches 

in new design. 

The profession is in the midst of a transition from forcebased design approaches to displacement-based design 

approaches. A complete implementation of a 

displacement-based approach involves (1) determining 

displacement demand, (2) breaking down overall 

demand into local components, and (3) comparing local 

capacity to demand (Bonacci). Simplified inelastic 

procedures can be used to move from force-based 

approaches (which are very imprecise but were useful 

for proportioning structures) to displacement-based 

approaches. Bonacci urges caution in rushing too 

rapidly to compare local demands and capacities, and 

cites as an example the difficulty in evaluating whether 

a stiffener will buckle when we may be 50% off on Tg

and PGA (peak ground acceleration). The desire for 

accurate analytical results must be balanced against the 

significant uncertainties in deformation capacities 

(Krawinkler). Furthermore, complicated techniques 

may be misused by engineers that are unfamiliar with 

them (Krawinkler).

Foutch suggested that if inelastic analysis techniques 

are used, they should be simple enough to be useful for 

conceptual design. Reinhorn suggested that one might 

use a simple technique to proportion the structure, then 

iteratively adjust the relative distributions of strength to 

ensure undesirable mechanisms will not form, and then 

follow with a more complex procedure to develop 

statistics on response. Miranda suggested that a 

simplified static procedure would be useful with 

estimates of dispersion, followed by nonlinear dynamic 

analyses to assess simulated response statistics. 

Aschheim suggests a simple inelastic analysis technique 

could be used for preliminary design, with nonlinear 

dynamic analysis being used to develop response 

statistics only for those structures where this 

comparison is deemed necessary (e.g. substantial 

irregularities, high importance, or to satisfy client 

requirements).

Because of uncertainties in the effects of higher modes, 

any simple procedure will require that prescriptive 

provisions are used to ensure that (1) desirable 

mechanisms form, with plastic hinges having sufficient 

ductility capacity to absorb uncertainties in plastic 

rotation demands arising from the presence of higher 

modes, (2) undesirable mechanisms (e.g. weak story 

mechanisms) will not form, even under the influence of 

higher modes, and (3) force-controlled components or 

modes of behavior have sufficient strength that forces 

associated with higher modes do not cause brittle 

failures to result. Variability due to higher modes can 

be expected to be a function of the number of stories as 

well as the spectral amplitudes at the higher mode 

periods. The separation of strengths required to prevent 

brittle modes of failure and undesirable mechanisms 

depends in part on the variability of material strengths 

in the as-built structure and the variability of actual 

strengths relative to calculated estimates.

Otani expressed concern about safety in view of the 

scatter in displacement estimates. The Japanese are 

using a modified form of the Capacity-Spectrum 

Method for checking the performance of designs that 

satisfy other criteria. Wilson expressed concern that 

nonlinear response spectra are 1not applicable to multidegree-of-freedom (MDOF) systems; response of 2D 

and 3D structures can and should be determined by 

nonlinear dynamic analysis, in part because only 

nonlinear dynamic analysis can really inform the 

engineer about the behavior of the systems being 

designed.

Lepage suggested that an iterative procedure could be 

used, wherein a variety of load patterns are used to 

determine a variety of deflected shapes and possibly a 

number of different mechanisms. If similar deflected 

shapes result, then the deflected shape would be used to 

determine the “equivalent” SDOF system for each of a 

suite of ground motions, recognizing that iteration will 

be required to identify the right shape to be used for 

different drift levels. Lepage also suggested an 

alternate approach in which linear estimates of roof 

displacement are coupled with a collapse mechanism 

analysis—if drifts concentrate in just a few stories, then 

all of the estimated drift would be assigned to those 

stories.

Otani notes that one could use a nonlinear static 

procedure to get design moments for beam hinge 

regions, and then apply a factor of safety to design the 

columns to prevent or limit the development of plastic 

hinges in the columns. 

A.7.2 Design Formats

Design procedures have been formulated for use with 

three different types of spectral representations. Direct 
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-16 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Displacement Based Design uses the concept of 

effective damping to establish response spectra that are 

plotted on the same axes (ADRS) used in the CapacitySpectrum Method. The period of vibration (or stiffness) 

required to satisfy a performance objective is 

determined, along with a required strength. The use of 

effective damping is supplanted in design procedures 

recommended by Fajfar and by Chopra and Goel, who 

use inelastic spectra (based on R-μ-T relations) plotted 

on the same axes used in the Capacity-Spectrum 

Method to estimate peak displacements and to 

determine required strengths. Black and Aschheim 

(2000) used Yield Point Spectra (based on R-μ-T

relations or the actual jagged spectra associated with 

design ground motions) to determine the strength 

required to satisfy multiple performance objectives 

using admissible design regions. An iterative approach 

was suggested in which nonlinear static analyses are 

avoided entirely by relying only on design strengths and 

elastic properties.

A.7.3 Quantities to be Determined and 

Measures of Performance

There is uncertainty in estimates of both demands and 

capacities. Rather than compare very approximate 

values of local demands and capacities, some suggest 

that it may be preferable to focus on quantities that are 

of a more global nature, such as interstory drift (Goel). 

Estimates of deformation capacity are fairly crude. 

Krawinkler observed that the best measure of inelastic 

deformation capacity (e.g., total or plastic rotation, 

curvature ductility) has not even been identified yet.

One approach is to estimate peak interstory drifts as a 

factor times the average global roof drift. For regular 

buildings, the factor varies with the number of stories 

and may not follow a consistent pattern over the height 

of the building (Gupta and Krawinkler, 2000a), and 

may depend on the ground motion (MacRae). Uetani 

and Tagawa (1998) reportedly find that interstory drifts 

concentrate less in structures in which the eigenvalues 

obtained during the nonlinear response are more 

positive. Fenwick reportedly has introduced into the 

New Zealand Code an estimate of interstory drift equal 

to twice the drifts determined by elastic analysis. 

Interstory drifts for near-field motions appear to be 

related to ground motion reversals (Iwan), and might be 

better estimated using concepts of wave propagation 

theory rather than conventional modal response 

approaches.

The actual shears in a building can be significantly 

higher that those associated with development of 

capacity in the predominant mode. Dynamic shears, 

therefore, may be significantly higher than estimated by 

pushover analysis. Rodriguez, Restrepo, and Carr 

(2002) reportedly found the second and higher modes 

respond essentially elastically, contributing to the 

shears associated with inelastic first mode response. 

Forces in reinforced concrete collectors may be poorly 

estimated by typical procedures because their larger 

stiffness in compression causes greater force to be 

carried in compression than in tension.

Kunnath notes that plastic hinge rotation demands are 

calculated differently in different software programs. 

The post-yield stiffness, hinge lengths, and use of 

distributed or concentrated plasticity affects the values, 

as do the different solution strategies used by the 

programs. Estimates of yield and plastic rotation are 

often based on assuming points of inflection occur at 

midspan, leading to errors of 50 to 100%. The 

approximate nature of the demands estimated by any 

procedure makes comparison with estimated capacities 

less certain; significant improvements are needed to 

improve the reliability of estimates of local demands 

and capacities, to make their comparison more 

meaningful.

A.7.4 Statistical Measures and Treatment of 

Uncertainty

Performance may be evaluated in different ways and 

may include or exclude various types of uncertainties. 

For example, Wen determines the annual probability of 

exceeding drifts of various levels. Cornell’s work for 

SAC focuses on the level of confidence in the 

hypothesis that the structure will satisfy a given 

performance objective, for ground motions that have a 

stated probability of exceedance. Uncertainty in the 

hazard is neglected in the SAC work, although Cornell 

et al. (2000) presents a theoretical formulation that 

accounts for uncertainty in the hazard. While it stands 

to reason that variations in capacities (strengths, 

deformation capacities) should have an influence on 

demands, current formulations neglect such interaction. 

A.8 References and Bibliography

Akkar, S., and Gulkan, P., 2000, “Comparative performance evaluation of displacement based design 

procedures for near field earthquakes,” Proceedings 

of the 12th World Conference on Earthquake Engineering, New Zealand Society for Earthquake Engineering, Upper Hutt, New Zealand.
Appendix A: Summary of Research on Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-17

Albanesi, T., Nuti, C., and Vanzi, I., 2000, “A simplified 

procedure to assess the seismic response of nonlinear structures,” Earthquake Spectra, Vol. 16, No. 4, 

Earthquake Engineering Research Institute, Oakland, California, pp. 715-734.

ASCE, 2000, Prestandard and Commentary for the 

Seismic Rehabilitation of Buildings, FEMA 356 

Report, prepared by the American Society of Civil 

Engineers for the Federal Emergency Management 

Agency, Washington, D.C.

Aschheim, M., and Black, E.F., 2000, “Yield Point 

Spectra for seismic design and rehabilitation,” 

Earthquake Spectra, Vol. 16, No. 2, Earthquake 

Engineering Research Institute, Oakland, California, pp. 317-335.

Aschheim, M.A., Maffei, J., and Black, E.F., 1998, 

“Nonlinear static procedures and earthquake displacement demands,” 6th US National Conference 

on Earthquake Engineering, Seattle, Washington, 

Earthquake Engineering Research Institute.

ATC, 1996, Seismic Evaluation and Retrofit of Concrete 

Buildings, ATC-40 Report, Volumes 1 and 2, 

Applied Technology Council, Redwood City, California.

ATC/BSSC, 1997, NEHRP Guidelines for the Seismic 

Rehabilitation of Buildings, FEMA 273 Report 

(Guidelines) and FEMA 274 Report (Commentary), 

prepared by the Applied Technology Council for 

the Building Seismic Safety Council, published by 

the Federal Emergency Management Agency, 

Washington, D.C.

Baez, J.I., and Miranda, E., 2000, “Amplification factors to estimate inelastic displacement demands for 

the design of structures in the near field,” Proceedings of the 12th World Conference on Earthquake 

Engineering, New Zealand Society for Earthquake 

Engineering, Upper Hutt, New Zealand.

Black, E.F., and Aschheim, M., 2000, Seismic Design 

and Evaluation of Multistory Buildings using Yield 

Point Spectra, CD Release 00-04, Mid-America 

Earthquake Center, University of Illinois, Urbana. 

Blume, J.A., Newmark, N.M., and Corning, L.H., 1961, 

Design of Multistory Reinforced Concrete Buildings 

for Earthquake Motions, Portland Cement Association, Skokie, Illinois.

Bracci, J.B., Kunnath, S.K., and Reinhorn, A.M., 1997, 

“Seismic performance and retrofit evaluation of 

reinforced concrete structures,” Journal of Structural Engineering, Vol. 123, American Society of 

Civil Engineers, pp. 3-10.

Browning, J., 2000, “Implications of a proportioning 

method for earthquake-resistant RC frames subjected to strong ground motion,” Proceedings of the 

12th World Conference on Earthquake Engineering, 

New Zealand Society for Earthquake Engineering, 

Upper Hutt, New Zealand.

Browning, J., Li, R., Lynn, A., and Moehle, J.P., 2000, 

“Performance assessment for a reinforced concrete 

frame building,” Earthquake Spectra, Vol. 16, 

No. 3, Earthquake Engineering Research Institute, 

Oakland, California, pp. 541-555.

Chopra, A.K., and Goel, R.K., 1999a, CapacityDemand Diagram Methods for Estimating Seismic 

Deformation of Inelastic Structure: SDF Systems, 

Report No. PEER-1999/02, Pacific Earthquake 

Engineering Center, University of California, Berkeley.

Chopra, A.K., and Goel, R.K., 1999b, “Capacitydemand diagram methods based on inelastic design 

spectrum,” Earthquake Spectra, Vol. 15, No. 4, 

Earthquake Engineering Research Institute, Oakland, California, pp. 637-656.

Chopra, A.K., and Goel, R.K., 2001a, “Direct displacement based design: use of inelastic vs. elastic 

design spectra,” Earthquake Spectra, Vol. 17, 

No. 1, Earthquake Engineering Research Institute, 

Oakland, California, pp. 47-64.

Chopra, A.K., and Goel, R.K., 2001b, A Modal Pushover Analysis Procedure to Estimate Seismic 

Demands for Buildings: Theory and Preliminary 

Evaluation, Report No. PEER 2001/03, Pacific 

Earthquake Engineering Research Center, University of California, Berkeley.

Chopra, A.K., and Goel, R.K., 2001c, “Modal pushover 

analysis of SAC buildings,” Third U.S.-Japan 

Workshop on Performance-Based Earthquake 

Engineering Methodology for Reinforced Concrete 

Building Structures, August 16-18, Seattle, Washington.

Cornell, C.A., Vamvatsikos, D., Jalayer, F., and Luco, 

N., 2000, “Seismic reliability of steel frames,” Proceedings of the 9th IFIP WG 7.5 Working Conference on Reliability and Optimization of Structural 

Systems, Ann Arbor, Michigan.

Cuesta, I., and Aschheim, M., 2001, Using Pulse RFactors to Estimate Structural Response to Earthquake Ground Motions, CD Release 01-03, Mid-
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-18 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

America Earthquake Center, University of Illinois, 

Urbana.

Cuesta, I., Aschheim, M., and Fajfar, P., 2003, “Simplified R-factor relationships for strong ground 

motions,” Earthquake Spectra, Vol. 19, No. 1, 

Earthquake Engineering Research, pp. 22-45

Elnashai, A.S., 2000, “Advanced inelastic static (pushover) analysis for seismic design and assessment,” 

G. Penelis International Symposium on Concrete 

and Masonry Structures, Aristotle University of 

Thessaloniki, Thessaloniki, Greece, pp. 23-34.

Fajfar, P., 1999, “Capacity-Spectrum Method based on 

inelastic demand spectra,” Earthquake Engineering 

and Structural Dynamics, Vol. 28, pp. 979-993.

Fajfar, P., and Gaspersic, P., 1996, “The N2 method for 

the seismic damage analysis of RC buildings,” 

Earthquake Engineering and Structural Dynamics, 

Vol. 25, No. 1, pp. 31-46.

Filiatrault, A., and Folz, B., 2001, “Performance-based 

seismic design of wood framed buildings,” 

accepted for publication in Journal of Structural 

Engineering, American Society of Civil Engineers. 

Foutch, D.A, and Shi, S., 1998, “Effects of hysteresis 

type on the seismic response of buildings,” Proceedings of 6th U.S. National Conference on Earthquake Engineering, Seattle, Washington, 

Earthquake Engineering Research Institute, Oakland, California.

Gulkan P., and Sozen, M.A., 1974, “In-elastic responses 

of reinforced concrete structures to earthquake 

motions,” Proceedings of the American Concrete 

Institute, Vol. 71, No. 12, pp. 605-610.

Gupta, A., and Krawinkler, K., 1998, “Effect of stiffness degradation on deformation demands for 

SDOF and MDOF structures,” 6th US National 

Conference on Earthquake Engineering, Seattle, 

Washington, Earthquake Engineering Research 

Institute, Oakland, California.

Gupta, A., and Krawinkler, K., 2000a, “Estimation of 

seismic drift demands for frame structures,” Earthquake Engineering and Structural Dynamics, 

Vol 29, pp. 1287-1305.

Gupta, A., and Krawinkler, K., 2000b, “Dynamic pdelta effects for flexible inelastic structures,” Journal of Structural Engineering, Vol. 126, No. 1, 

American Society of Civil Engineers.

Gupta, B., and Kunnath, S.K., 1998, “Effect of hysteretic model parameters on inelastic seismic 

demands,” 6th US National Conference on Earthquake Engineering, Seattle, Washington, Earthquake Engineering Research Institute.

Gupta, B., and Kunnath, S.K., 2000, “Adaptive spectrabased pushover procedure for seismic evaluation of 

structures,” Earthquake Spectra, Vol. 16, No. 2, 

Earthquake Engineering Research Institute, Oakland, California, pp. 367-391.

Han, S.W., and Wen, Y.K., 1997, “Method of reliabilitybased seismic design. I: Equivalent nonlinear system,” Journal of Structural Engineering, Vol. 123, 

American Society of Civil Engineers, pp. 256-265.

Hudson, D.E., 1965, “Equivalent viscous friction for 

hysteretic systems with earthquake-like excitations,” 3rd World Conference on Earthquake Engineering, New Zealand, pp. II:185-201. 

Islam, M.S., Gupta, B., and Kunnath, S., 1998, “A critical review of state-of-the-art analytical tools and 

acceptance criterion in light of observed response 

of an instrumented nonductile concrete frame building,” Proceedings of 6th U.S. National Conference 

on Earthquake Engineering, Seattle, Washington, 

Earthquake Engineering Research Institute, Oakland, California.

Iwan, W.D., and Gates, N.C., 1979, “Estimating earthquake response of simple hysteretic structures,” 

Journal of the Engineering Mechanics Division, 

Vol. 105, EM3, American Society of Civil Engineers, pp. 391-405.

Iwan, W.D., Huang, C.T., and Guyader, A.C., 2000, 

“Important features of the response of inelastic 

structures to near-field ground motion,” Proceedings of the 12th World Conference on Earthquake 

Engineering, New Zealand Society for Earthquake 

Engineering, Upper Hutt, New Zealand.

Jacobsen, L.S., 1930, “Steady forced vibrations as influenced by damping,” Transactions ASME, Vol. 52 

(APM), pp. 169-181.

Judi, H.J., Davidson, B.J., and Fenwick, R.C., 2000, 

“The direct displacement based design method: a 

damping perspective,” Proceedings of the 12th

World Conference on Earthquake Engineering, 

Auckland, New Zealand Society for Earthquake 

Engineering, Upper Hutt, New Zealand.

Krawinkler, H., and Alavi, B, 1998, “Development of 

improved design procedures for near fault ground 

motions,” Proceedings, SMIP98 Seminar on Utilization of Strong-Motion Data, Oakland, California.
Appendix A: Summary of Research on Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures A-19

Kunnath, S. K., and Gupta, B., 2000, “Validity of deformation demand estimates using nonlinear static 

procedures,” Proceedings of the U.S.-Japan Workshop on Performance-Based Engineering of Reinforced Concrete Building Structures, Sapporo, 

Hokkaido, Japan.

Kunnath, S.K., Nghiem, Q., John Jr., A., and El-Tawil, 

S., 2000, “Validation of evaluation methods and 

acceptance criteria in evolving performance-based 

seismic codes,” Proceedings, SMIP2000 Seminar 

on Utilization of Strong-Motion Data: Sacramento, 

California, September 14, 2000, California Division 

of Mines and Geology, Sacramento, pp. 39-63.

Lepage, A., 1998, “Nonlinear drift of multistory RC 

structures during earthquakes,” Proceedings, 6th

U.S. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, 

Oakland, California.

Lew, H.S., and Kunnath, S.K., 2000, “Evaluation of 

analysis procedures for performance-based seismic 

design of buildings,” Proceedings of the 12th World 

Conference on Earthquake Engineering, Auckland, 

New Zealand Society for Earthquake Engineering, 

Upper Hutt, New Zealand.

MacRae, G., and Tagawa, H., 2001, Methods to Estimate Displacements of PG&E Structures, draft 

report on research conducted under PGE/PEER 

Task No. 505, University of Washington.

Miranda, E., 2000, “Inelastic displacement ratios for 

structures on firm sites,” Journal of Structural 

Engineering, Vol. 126, No. 10, American Society of 

Civil Engineers, pp. 1150-1159.

Miranda, E., 2001, “Estimation of Inelastic Deformation Demands of SDOF Systems,” Journal of Structural Engineering, Vol. 127, No. 9, American 

Society of Civil Engineers, pp, 1005-1012.

Miranda, E., and Bertero, V., 1994, “Evaluation of 

strength reduction factors for earthquake-resistant 

design”, Earthquake Spectra, Vol. 10, No. 2, Earthquake Engineering Research Institute, Oakland, 

California, pp. 357-379.

Mwafy, A.M., and Elnashai, A.S., 2001, “Static pushover versus dynamic collapse analysis of RC buildings,” Engineering Structures, Vol 23, pp. 407-424.

Naeim, F., Skliros, K., and Reinhorn, A.M., 2000, 

“Influence of hysteretic deteriorations in seismic 

response of multistory steel frame buildings,” Proceedings of the 12th World Conference on Earthquake Engineering, Auckland, New Zealand 

Society for Earthquake Engineering, Upper Hutt, 

New Zealand.

Nakaki, S., 2000, Developing Design Guidelines for 

Precast Concrete Diaphragms, Earthquake Engineering Research Institute, Oakland, California.

Nakashima, M., Ogawa, K., and Inoue, K., 2002, 

“Generic frame model for simulation of earthquake 

responses of steel moment frames,” Earthquake 

Engineering and Structural Dynamics.

Otani, S., 2000, “New seismic design provisions in 

Japan,” Proceedings of the U.S.-Japan Workshop 

on Performance-Based Earthquake Engineering for 

Reinforced Concrete Building Structures, Sapporo, 

Japan.

Paret, T.F., Freeman, S.A., and Dameron, R.A, 2002, 

“Rethinking the earthquake engineering paradigm: 

from response reduction to response suppression,” 

Proceedings, Seventh U.S. National Conference on 

Earthquake Engineering, Earthquake Engineering 

Research Institute, Oakland, California.

Qi, X., and Moehle, J.P., 1991, Displacement Design 

Approach for Reinforced Concrete Structures Subjected to Earthquakes, Report No. UCB/EERC-91/

02, Earthquake Engineering Research Center, University of California, Berkeley.

Reinhorn, A., 1997, “Inelastic techniques in seismic 

evaluations,” in Seismic Design Methodologies for 

the Next Generation of Codes, Fajfar, P., and 

Krawinkler, H., eds., Bled, Slovenia.

Rodriguez, M.E., Restrepo, J.I., and Carr, A.J., 2002, 

“Earthquake-induced acceleration in buildings,” 

Earthquake Engineering and Structural Dynamics, 

Vol. 31, pp. 693-718.

Rothe, D., and Sozen, M.A., 1983, A SDOF Model to 

Study Nonlinear Dynamic Response of Large- and 

Small-Scale R/C Test Structures, Structural 

Research Series No. 512, Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois.

Sasaki, K.K. Freeman, S.A., and Paret, T.F., 1998, 

“Multi-Mode pushover procedure (MMP) -- a 

method to identify the effects of higher modes in a 

pushover analysis,” Proceedings, Sixth U.S. 

National Conference on Earthquake Engineering, 

Seattle, Washington, Earthquake Engineering 

Research Institute, Oakland, California.

Song, J.K., and Pincheira, J.A., 2000, “Spectral displacement demands of stiffness and strength 

degrading systems,” Earthquake Spectra, Vol. 16, 
Appendix A: Summary of Research on Inelastic Analysis Procedures

A-20 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

No. 4, Earthquake Engineering Research Institute, 

Oakland, California, pp. 817-851.

Tagawa, H., and MacRae, G., 2001, “Capacity spectra 

method for estimating SDOF oscillator demands,” 

Proceedings of the 2001 Structures Congress & 

Exposition, American Society of Civil Engineers, 

Washington, D.C.

Tsopelas, P., Constantinou, M.C., Kircher, C.A., and 

Whittaker, A.S., 1997, Evaluation of Simplified 

Methods of Analysis of Yielding Structures, Technical Report NCEER-97-0012, National Center for 

Earthquake Engineering Research, State University 

of New York, Buffalo, New York. 

Uetani, K., Tagawa, H., 1998, “Criteria for suppression 

of deformation concentration of building frames 

under severe earthquakes,” Engineering Structures, 

Vol. 20, No. 4-6, pp. 372-383.

Vidic, T., Fajfar, P., and Fischinger, M., 1994, “Consistent inelastic design spectra: strength and displacement,” Earthquake Engineering and Structural 

Dynamics, Vol. 23, pp. 507-521.

Villaverde, R., 1996, “Simplified response spectrum 

analysis of nonlinear structures,” Journal of Engineering Mechanics, Vol. 122, American Society of 

Civil Engineers, pp. 282-285.

Valley, M.T., and Harris, J.R., 1998, “Application of 

modal techniques in a pushover analyses,” Proceedings, Sixth U.S. National Conference on Earthquake Engineering, Seattle, Washington, 

Earthquake Engineering Research Institute, Oakland, California.

Vamvatsikos, D., and Cornell, C.A., 2001, “Incremental 

dynamic analysis,” submitted to Earthquake Engineering and Structural Dynamics.

Wen, Y. K., and Wu, C.L., 2001, “Uniform Hazard 

Ground Motions for Mid-America Cities,” Earthquake Spectra, Vol. 17, No. 2, Earthquake Engineering Research Institute, Oakland, California, 

pp. 359-384.
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures B-1 

B. Summary of Practice using Inelastic Analysis 

Procedures

B.1 Introduction

The state of practice inquiry conducted under Phase I of 

the ATC-55 Project sought information about the use of 

inelastic analysis procedures for a broad sample of 

building applications from different practicing structural engineering firms. Respondees were asked to provide information on the following:

• types of buildings and structural systems for which 

the procedures are used;

• procedures used;

• software used for analysis, if any; and

• Engineers’ thoughts about the implementation of 

procedures, including problems encountered.

This appendix summarizes the information obtained 

from practicing engineers who responded to the state of 

practice inquiry, and the relation of their responses to 

the issues identified by the ATC-55 project team. The 

information was solicited through three primary means. 

First, a project web page was established and advertised 

to practitioners through e-mail and notices in professional newsletters. The website contained a “Summary 

of Practice Building Data” questionnaire form for completion by each respondent for each building example to 

be submitted. Second, the ATC-55 project team 

appealed directly by e-mail to a number of engineering 

firms, some of whom were known to have experience 

with inelastic analysis procedures. Finally, the e-mail 

requests were followed with personal telephone calls. 

The solicitations took place during the spring and summer of 2001. Information on over 60 examples was 

obtained from 23 respondents in 12 different structural 

engineering offices. 

B.2 Typical Buildings and Structural 

Systems

The example buildings submitted by engineers who 

responded to the ATC solicitation encompassed a broad 

range of building types, structural systems, and foundation systems. Following are the percentages of the total 

example building population by ownership type, purpose of analysis, year of construction, height, and floor 

area. The percentages of example buildings by seismic 

(lateral-force resisting) system, by gravity force resisting system, and by foundation type are provided in 

Table B-1, Table B-2, and Table B-3, respectively.

Ownership Type

Private: 42%

Institutional 52%

Unspecified 6%

Purpose of Analysis

Evaluation only 32%

Upgrade or Evaluation/Upgrade 41%

New 27%

Year of Construction

Earliest 1916

Latest 2001

Mean 1935

Height (stories)

1-2 15%

3-6 36%

7-11 28%

12+ 21%

Floor Area (sf)

< 10,000 3%

10,000 – 50,000 14%

50,000 – 100,000 19%

100,000 – 200,000 60%

200,000 – 500,000 2%

> 500,000 2% 

B.3 Inelastic Analysis Procedures

The procedures used by the respondents included the 

following:

• FEMA 273/356 (Coefficient Method)

• Nonlinear response history analysis 
Appendix B: Summary of Practice using Inelastic Analysis Procedures

B-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

• FEMA 351/SAC method

• ATC-40 (Capacity-Spectrum Method)

• Unspecified nonlinear static

A number of the respondents used two or more procedures for the same building example. The number of 

primary uses of each procedure (i.e., not accounting for 

secondary procedures), and the respective percentage of 

the total number of buildings are listed in Table B-4.

B.4 Software

With one exception, the inelastic procedures were 

implemented with the aid of computer analysis software. Table B-5 lists the programs used, the number of 

listings of each program name, and the percentage of 

listings of the total number of program usage listings.

Several examples utilized multiple linear elastic analyses with sequential stiffness modification to represent 

progressive yielding and degradation. These applications utilized extensive spreadsheet bookkeeping to sum 

member forces and check member demands against 

capacities from one analysis to the next.

Table B-1 Seismic Systems of Example Buildings 

Submitted by Respondees

Seismic System(s) Number Percentage of Total

Concrete shear walls 17 29%

Concrete moment 

frame 9 15%

Concrete frame/brick 

infill 6 10%

Steel CBF (concrete 

braced frame) 3 5%

Steel EBF (eccentric 

braced frame) 4 7%

Steel BRBF (unbonded 

braced frame) 2 3%

Steel moment frame 6 10%

Steel frame/brick infill 3 5%

Steel truss moment 

frame 1 2%

Plywood or OSB shear 

walls 3 5%

Reinforced masonry 

walls 1 2%

Passive damped frame 1 2%

Other 3 5%

Table B-2 Gravity Systems of Example Buildings 

Submitted by Respondees

Gravity System(s) Number Percentage of Total

Concrete columns/

beams and/or slab 22 37%

Concrete bearing walls 2 3%

Wood frame 5 8%

Steel frame /wood infill 2 3%

Steel frame/concrete 

slab 24 41%

Other 4 7%

Table B-3 Foundation Systems of Example 

Buildings Submitted by Respondees

Foundation System(s) Number Percentage of Total

Spread Footings 32 55%

Mat 9 15%

Piles 13 22%

Drilled Piers 3 5%

Unknown/Other 2 3%

Table B-4 Inelastic Analysis Procedures

Primary Procedure Number Percentage of Total

FEMA 273/356 

(Coefficient Method) 21 36%

Nonlinear response 

history analysis 12 20%

FEMA 351/SAC 

method 3 5%

ATC-40 (CapacitySpectrum Method) 8 14%

Unspecified nonlinear 

static1 15 25%

Multiple of above 

procedures 8 14%

1. The heading “Unspecified nonlinear static” indicates 

entries such as “NSP,” “CSM,” and “Equal Displacement.” 

It was noted that most such entries were associated with 

analyses that were implemented prior to the publication 

of ATC-40 and FEMA 273. Several examples cited the use 

of the Miranda-Bertero procedure, and one example cited 

the Army TM-5-809-10-1 document. 
Appendix B: Summary of Practice using Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures B-3

B.5 Implementation Issues

A total of 65 comments were submitted relating to the 

implementation of the inelastic procedures. A synopsis 

of the respondents’ comments on major issues follows:

The majority of the comments submitted were related to 

the relative accuracy of procedures. Engineers’ preoccupation with the topic of relative accuracy was indicated by the techniques used, such as variation of 

parameters (or “bounding”), by comments about the 

sensitivity of procedures to various assumptions, and by 

the implementation of comparative analyses using multiple procedures for the same building. Significantly, 

the large variation in ground motion parameters was not 

mentioned in any of the practitioners’ comments, 

although one respondent expressed doubt in the validity 

of using a static procedure to represent the effect of 

ground motion at a near-field location. Three of the 

example buildings were full-scale test specimens of 

wood buildings that were shaken on a simulator and 

evaluated using a nonlinear response history analysis 

procedure, for the purpose of research and comparison. 

Several of the respondents commented on the difficulty 

of reasonably accounting for cyclic degradation and Pdelta effects with existing procedures and/or software. 

Also, difficulty in establishing a suitable target displacement or ultimate drift was mentioned in two examples.

It is evident that some respondents question the appropriateness of procedures to determine the target displacement. One respondent wrote “FEMA 273 shear 

strain ratios [were] exceeded in local areas – deemed 

not to be hazardous.” Another wrote “Immediate Occupancy provisions [of FEMA 273] are too conservative.” 

A third wrote “Analysis was straightforward. Determination of target displacement was problematic.” 

There were several comments regarding the complexity 

of the procedures. For example, one respondent wrote 

“The most troublesome problem in implementing the 

FEMA [273] procedures was developing nonlinear 

hinge properties (strength and ductility).” Another 

wrote: “The shear capacity of the concrete columns 

was difficult to evaluate by the FEMA 273 methods 

(Eq. 6-4) due to constantly changing parameters.” A 

third wrote: “Convergence was difficult to achieve even 

for a relatively simple model and depended greatly on 

the method of solution used.” 

Several respondents commented during verbal discussions that the established analysis procedures did not 

allow the evaluation of behavior in the range of severe 

damage prior to collapse, such as damage to many 

structures observed in postearthquake reconnaissance.

Several respondents indicated that the results of inelastic procedures are very sensitive to assumptions regarding such parameters as initial stiffness, and pushover 

loading profile. There is also recognition among 

respondents that the dynamic and multi-degree-of-freedom (MDOF) effects that would be captured in a nonlinear response history analysis procedure could be 

quite different from the results of a nonlinear static procedure. Several respondents attempted to account for 

dynamic behavior, yielding, and MDOF effects by such 

techniques as adapting pushover loading profiles and 

use of simplified dynamic analysis. 

One respondent discussed the inability of static procedures to represent the response of structures to nearfield earthquake pulse-type motions.

Two other respondents collaborated in a comparison of 

the Capacity Spectrum approach for a single-degree-offreedom system with nonlinear response history analysis. They identified that the differences in results 

between the two methods could be largely explained by 

the dynamic response of the structure to the predominant velocity pulses in the time-history records. They 

developed a simplified technique to calculate the singleTable B-5 Computer Program Usage

Program Name

Number of 

Listings

Percentage of 

Listings

ABAQUS 1 1%

ANSYS 3 5%

CASHEW / 

RUAUMOKO 3 5%

(Custom software) 6 9%

DRAIN 2D 9 14%

DRAIN 2DX 8 12%

ETABS 3 5%

FEM-I 2 3%

FEM-II 1 1%

SAP 90 6 9%

SAP 2000 24 35%

Other 1 1%

Total Listings 66
Appendix B: Summary of Practice using Inelastic Analysis Procedures

B-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

degree-of-freedom (SDOF) dynamic displacement 

response for a single velocity pulse, and applied this 

technique to several structures, evaluating the response 

of each structure to various pulses for site-specific 

ground motion records. 

One respondent questioned the validity of static procedures for high-rise buildings that would experience significant higher mode components in their response. 

There seems to be a lack of understanding among practitioners about how to represent MDOF effects for static 

procedures. Only one respondent commented about the 

sensitivity of the static solution to such parameters as 

initial period and pushover profile. Another used an 

adaptive load pattern based on modal response at each 

significant step in the analysis process. A third simply 

assumed that all stories experienced equal drift. 

B.6 Use of Limitations on Coefficient C1

in FEMA 356

FEMA 356 currently contains arbitrary limitations 

(caps) on the maximum value of the coefficient C1. This 

cap tends to reduce the predicted inelastic displacement 

of relatively short period structures. At an early stage 

of the second phase of the ATC-55 project, it became 

apparent that the cap might influence the accuracy of 

the Coefficient Method. While there may be valid reasons that the response of short-period structures varies 

from that predicted by current analysis procedures, it 

seemed that the arbitrary nature of the cap conflicted 

with the goals of the project. In an effort to gauge qualitatively how this issue affected current practice, the 

project team contacted twelve practicing engineers from 

seven different firms from the respondents to the Phase 

I Practice Study. These individuals and firms are representative of a relatively high level of seismic expertise 

among practitioners. Three basic questions were posed:

a. Do you use the cap?

b. Why, or why not?

c. What are your thoughts and understanding on this 

choice?

All but a few engineers follow the same procedure. 

First they calculate C1 using the empirical equation. If 

this value is less than 1.5, they use it. If it is higher, 

they use 1.5. Thus the practice is to neglect the interpolation allowed between 0.10 sec and Ts. The pervasive 

attitude was that they use the cap because it is allowed. 

Very few were aware of the discussion on this issue in 

FEMA 273/274/356/357. These were the same few 

who tended not to use the cap.

B.7 Practical Guidance and Education

Respondents provided feedback (either in writing, verbally, or implicitly) about the following topics or questions related to the issue of practical guidance and 

education:

• The various methods lead to different results. Why?

It is evident that practicing engineers do not necessarily 

know why the various NSP methods result in different 

answers, or why the answers may differ significantly 

from those resulting from the use of nonlinear response 

history analysis. Consequently, engineers may lack a 

way to answer the next item:

• Which method is the most effective for a given 

project?

Respondents indicated that there is a general lack of 

understanding about how to select a method. In numerous cases, the methods had been dictated by the owner/

client. For instance, FEMA 273 is quickly being 

adopted as the governing guideline by government 

agencies and is therefore required for evaluations and 

design of government-funded retrofits. 

• Certain guidelines or evaluation techniques require 

an impractical amount of effort. 

Some engineers indicated that they chose to adopt an 

approach using sequential elastic analyses to developing a “backbone” resistance curve for their pushover 

analysis. In some cases a cumbersome amount of 

“bookkeeping” was required to keep track of individual 

member stresses, and to compare these stresses with 

estimated stress or strain capacities as they changed the 

model to simulate yielding or degradation. 

• What is the most efficient way to compute results for 

a given method?

Based on responses received, it is evident that practicing engineers have searched for efficient ways to handle 

the large amount of computational effort required for 

nonlinear analysis. 

• More effective software tools are needed.

Some respondents indicated that the software programs 

they currently use for inelastic analysis are sometimes 
Appendix B: Summary of Practice using Inelastic Analysis Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures B-5

difficult to use, or do not allow the user to model important aspects of the structure, such as degradation.

Other issues identified related to practical guidance and 

education, including the following:

• Clients who require these evaluations need to be 

educated about effort and fees required. Normally, 

this information comes from the engineer. However, 

without sufficient experience, the engineer would 

not be able to accurately estimate the required effort. 

This relates to the next issue:

• Some practicing engineers have embraced these 

methods as an improvement. Others have avoided 

them as requiring a steep learning curve and more 

effort, with an uncertain outcome. The methods are 

therefore more risky for the owner as well as the 

engineer.
Appendix B: Summary of Practice using Inelastic Analysis Procedures

B-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-1 

C. Supplemental Data on the Evaluation of 

Current Procedures

This appendix supplements Chapter 3 on the evaluation 

of current nonlinear static procedures. The contents are 

summarized as follows: Section C.1 tabulates the 

ground-motion data used for the evaluation; Section C.2 

presents the results of the response history analyses of 

the oscillators; Section C.3 presents data on the results 

of the evaluation of the ATC-40 version of the Capacity 

Spectrum Method; and Section C.4 presents data on the 

results of the evaluation of the Coefficient Method of 

FEMA 356.

C.1 Ground Motions 

Table C-1 Ground Motions Recorded on Site Class B

Date Earthquake Name

Magnitude 

(Ms) Station Name

Station 

Number

Component 

(deg) PGA (cm/s2)

06/28/92 Landers 7.5 Silent Valley, Poppet Flat 12206 0 48.9

06/28/92 Landers 7.5 Twentynine Palms Park 

Maintenance Bldg

22161 0 78.7

06/28/92 Landers 7.5 Amboy 21081 90 146.0

10/17/89 Loma Prieta 7.1 Point Bonita 58043 297 71.4

10/17/89 Loma Prieta 7.1 Piedmont, Piedmont Jr. High 

Grounds

58338 45 81.2

10/17/89 Loma Prieta 7.1 San Francisco, Pacific Heights 58131 270 60.2

10/17/89 Loma Prieta 7.1 San Francisco, Rincon Hill 58151 90 88.5

10/17/89 Loma Prieta 7.1 San Francisco, Golden Gate 

Bridge

1678 360 228.6

10/17/89 Loma Prieta 7.1 Hollister-SAGO vault 1032 360 60.1

10/17/89 Loma Prieta 7.1 South San Francisco, Sierra Point 58539 205 102.7

10/17/89 Loma Prieta 7.1 Berkeley, Lawrence Berkeley 

Lab.

58471 90 114.8

10/17/89 Loma Prieta 7.1 Coyote Lake Dam, Downstream 57504 285 175.6

01/17/94 Northridge 6.8 Mt Wilson, CIT Seismic Station 24399 90 228.5

01/17/94 Northridge 6.8 Antelope Buttes 24310 90 99.7

01/17/94 Northridge 6.8 Los Angeles, Wonderland 90017 185 168.7

01/17/94 Northridge 6.8 Wrightwood, Jackson Flat 23590 90 54.5

01/17/94 Northridge 6.8 Littlerock-Brainard Can 23595 90 7.2

01/17/94 Northridge 6.8 San Gabriel, E. Grand Ave. 90019 180 256.0

10/01/87 Whittier Narrows 6.1 Los Angeles, Gritfith Park 

Observatory

141 0 133.8

10/15/79 Imperial Valley 6.8 Superstition Mountain 286 135 189.2
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Table C-2 Ground Motions Recorded on Site Class C

Date Earthquake Name

Magnitude 

(Ms) Station Name

Station 

Number

Component 

(deg) PGA (cm/s2)

10/15/79 Imperial Valley 6.8 El Centro, Parachute Test Facility 5051 315 200.2

02/09/71 San Fernando 6.5 Pasadena, CIT Athenaeum 80053 90 107.9

02/09/71 San Fernando 6.5 Pearblossom Pump 269 21 133.4

06/28/92 Landers 7.5 Yermo, Fire Station 12149 0 167.8

10/17/89 Loma Prieta 7.1 APEEL 7, Pulgas 58378 0 153.0

10/17/89 Loma Prieta 7.1 Gilroy #6, San Ysidro 

Microwave site

57383 90 166.9

10/17/89 Loma Prieta 7.1 Saratoga, Aloha Ave. 58065 0 494.5

10/17/89 Loma Prieta 7.1 Gilroy, Gavilon College Phys Sch 

Bldg

47006 67 349.1

10/17/89 Loma Prieta 7.1 Santa Cruz, University of 

California

58135 360 433.1

10/17/89 Loma Prieta 7.1 San Francisco, Diamond Heights 58130 90 110.8

10/17/89 Loma Prieta 7.1 Fremont, Mission San Jose 57064 0 121.6

10/17/89 Loma Prieta 7.1 Monterey, City Hall 47377 0 71.60

10/17/89 Loma Prieta 7.1 Yerba Buena Island 58163 90 66.70

10/17/89 Loma Prieta 7.1 Anderson Dam, Downstream 1652 270 239.40

04/24/84 Morgan Hill 6.1 Gilroy, Gavilon College Phys Sci 

Bldg

47006 67 95.0

04/24/84 Morgan Hill 6.1 Gilroy #6, San Ysidro 

Microwave Site

57383 90 280.4

07/08/86 Palmsprings 6.0 Fun Valley 5069 45 129.0

01/17/94 Northridge 6.8? Littlerock, Brainard Canyon 23595 90 70.60

01/17/94 Northridge 6.8 Castaic, Old Ridge Route 24278 360 504.2

01/17/94 Northridge 6.8 Lake Hughes #1, Fire station 

#78

24271 0 84.9
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-3

Table C-3 Ground Motions Recorded on Site Class D

Date Earthquake Name

Magnitude 

(Ms) Station Name

Station 

Number

Component 

(deg) PGA (cm/s2)

6/28/92 Landers 7.5 Yermo, Fire Station 22074 270 240.0

6/28/92 Landers 7.5 Palm Springs, Airport 12025 90 87.2

6/28/92 Landers 7.5 Pomona, 4th and Locust, Free 

Field

23525 0 65.5

01/17/94 Northridge 6.8 Los Angeles, Hollywood Storage 

Bldg.

24303 360 381.4

01/17/94 Northridge 6.8 Santa Monica City Hall 24538 90 866.2

01/17/94 Northridge 6.8 Los Angeles, N. Westmoreland 90021 0 393.3

10/17/89 Loma Prieta 7.1 Gilroy 2, Hwy 101 Bolsa Road 

Motel

47380 0 394.2

10/17/89 Loma Prieta 7.1 Gilroy 3, Sewage Treatment Plant 47381 0 531.7

10/17/89 Loma Prieta 7.1 Hayward, John Muir School 58393 0 166.5

10/17/89 Loma Prieta 7.1 Agnews, Agnews State Hospital 57066 0 163.1

10/01/87 Whittier Narrows 6.1 Los Angeles, 116th St School 14403 270 288.4

10/01/87 Whittier Narrows 6.1 Downey, County Maintenance 

Bldg

14368 180 193.2

10/15/79 Imperial Valley 6.8 El Centro #13, Strobel 

Residence

5059 230 136.2

10/15/79 Imperial Valley 6.8 Calexico, Fire Station 5053 225 269.6

04/24/84 Morgan Hill 6.1 Gilroy #4, 2905 Anderson Rd 57382 360 341.4

04/24/84 Morgan Hill 6.1 Gilroy #7, Mantnilli Ranch, 

Jamison Rd

57425 0 183.0

04/24/84 Morgan Hill 6.1 Gilroy #2, Keystone Rd. 47380 90 207.9

04/24/84 Morgan Hill 6.1 Gilroy #3 Sewage Treatment 

Plant

47381 90 189.8

02/09/71 San Fernando 6.5 Los Angeles, Hollywood Storage 

Bldg.

135 90 207.0

02/09/71 San Fernando 6.5 Vernon, Cmd Terminal Building

4814 Loma Vista

288 277 104.6
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Table C-4 Ground Motions Recorded on Very Soft Soil Sites Used in This Study

Date Earthquake Name

Magnitude 

(Ms) Station Name

Station 

Number

Component 

(deg) PGA (cm/s2)

10/17/89 Loma Prieta 7.1 Foster City (APEEL 1; Redwood 

Shores)

58375 90 277.6

10/17/89 Loma Prieta 7.1 Foster City (APEEL 1; Redwood 

Shores)

58375 360 63.0

10/17/89 Loma Prieta 7.1 Larkspur Ferry Terminal 1590 

(USGS)

270 134.7

10/17/89 Loma Prieta 7.1 Larkspur Ferry Terminal 1590 

(USGS)

360 94.6

10/17/89 Loma Prieta 7.1 Redwood City (APEEL Array 

Stn. 2)

1002 

(USGS)

43 270.0

10/17/89 Loma Prieta 7.1 Redwood City (APEEL Array 

Stn. 2)

1002 

(USGS)

133 222.0

10/17/89 Loma Prieta 7.1 Treasure Island (Naval Base Fire 

Station)

58117 0 112.0

10/17/89 Loma Prieta 7.1 Treasure Island (Naval Base Fire 

Station)

58117 90 97.9

10/17/89 Loma Prieta 7.1 Emeryville, 6363 Christie Ave. 1662 

(USGS)

260 254.7

10/17/89 Loma Prieta 7.1 Emeryville, 6363 Christie Ave. 1662 

(USGS)

350 210.3

10/17/89 Loma Prieta 7.1 San Francisco, International 

Airport

58223 0 231.5

10/17/89 Loma Prieta 7.1 San Francisco, International 

Airport

58223 90 322.7

10/17/89 Loma Prieta 7.1 Oakland, Outer Harbor Wharf 58472 35 281.4

10/17/89 Loma Prieta 7.1 Oakland, Outer Harbor Wharf 58472 305 265.5

10/17/89 Loma Prieta 7.1 Oakland, Title & Trust Bldg. 

(2-story)

58224 180 191.3

10/17/89 Loma Prieta 7.1 Oakland, Title & Trust Bldg. 

(2-story)

58224 270 239.4

10/15/79 Imperial Valley 6.8 El Centro Array 3, Pine Union 

School

5057 140 260.9

10/15/79 Imperial Valley 6.8 El Centro Array 3, Pine Union 

School

5057 230 216.8

04/24/84 Morgan Hill 6.1 Foster City (APEEL 1; Redwood 

Shores)

58375 40 45.1

04/24/84 Morgan Hill 6.1 Foster City (APEEL 1; Redwood 

Shores)

58375 310 66.7
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-5

Table C-5 Near-Fault Records with Forward Directivity Used in this Study

Date Earthquake Name

Magnitude 

(Ms) Station Name

NEHRP Site 

Class PGA (cm/s2)

10/17/89 Loma Prieta 7.1 Los Gatos D 704.0

10/17/89 Loma Prieta 7.1 Lexington Dam D 673.0

01/16/95 Hyogo-Ken-Nanbu, Kobe 6.9 Takatori Station D 771.0

01/16/95 Hyogo-Ken-Nanbu, Kobe 6.9 Kobe Station D 1067.0

01/16/95 Hyogo-Ken-Nanbu, Kobe 6.9 Port Island D 426.0

03/13/92 Erzican, Turkey 6.9 Erzican Station D 424.0

01/17/94 Northridge 6.8 Rinaldi Receiving Station D 873.0

01/17/94 Northridge 6.8 Sepulveda D 715.0

01/17/94 Northridge 6.8 Sylmar County Hospital D 718.0

01/17/94 Northridge 6.8 Newhall, LA County Fire Station D 709.0

10/15/79 Imperial Valley 6.8 Meloland D 372.0

10/15/79 Imperial Valley 6.8 El Centro Array 6 D 424.0

04/24/84 Morgan Hill 6.1 Coyote Dam D 712.0

04/24/84 Morgan Hill 6.1 Anderson Dam D 436.0

08/17/99 Kocaeli, Turkey 7.8 YPT D 311.5

08/17/99 Kocaeli, Turkey 7.8 DZC1 D 390.1

11/12/99 Duzce, Turkey 7.8 DZC2 D 404.2

08/17/99 Kocaeli, Turkey 7.8 IZT1 AB 164.3

11/12/99 Duzce, Turkey 7.8 BOL2 D 755.9

08/17/99 Kocaeli, Turkey 7.8 GBZ1 AB 240.2
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.2 Response History Results

C.2.1 Effect of Site Class on C1 of SDOF Systems with Elastoplastic Perfectly Plastic (EPP) Hysteretic 

Behavior

S IT E C L A S S E S B

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,E P P

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S E S C

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 P E R IO D 

C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S E S D

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,E P P

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S E S E

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 T / T g

C R,E P P

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

NEAR FAULT SET 

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,E P P

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

S IT E C L A S S E S B

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T p

C R,E P P

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-7

C.2.2 Effect of Site Class on C1 of SDOF Systems with Stiffness Degrading (SD) Hysteretic Behavior

S IT E C L A S S B

(m e an of 20 ground m otio ns )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,S D

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S C

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,S D

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S D

(m e an of 20 ground m otio ns )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,S D

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S E

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T g

C R,SD

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

NEAR FAULT SET

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T p

C R,S D

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

NEAR FAULT SET

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,S D

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.2.3 Effect of Site Class on C1 of SDOF Systems with Strength and Stiffness Degrading (SSD) 

Hysteretic Behavior

S IT E C L A S S B

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,S SD

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S C

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,SS D

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S D

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,S SD

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S E

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T g

C R,SS D

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

NEAR FAULT SET 

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T p

C R,SSD

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

NEAR FAULT SET 

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,SSD

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-9

C.2.4 Effect of Site Class on C1 of SDOF Systems with Nonlinear Elastic Hysteretic Behavior

S IT E C L A S S B

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,NLE

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S C

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D 

C R,NLE

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S D

(m ea n of 20 gro un d m otio ns )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,NLE

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S E

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T g

C R,NLE

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

NEAR FAULT SET

(m ean of 20 ground m otions )

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T p

C R,NLE

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.2.5 Evaluation of Coefficient C2 for Site Class B

SITE CLASS B

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,SD/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS B

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,SSD/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS B

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,NLE/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-11

C.2.6 Evaluation of Coefficient C2 for Site Class C

SITE CLASS C

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD 

CR,SD/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS C

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD 

CR,SSD/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS C

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD 

CR,NLE/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-12 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.2.7 Evaluation of Coefficient C2 for Site Class D

SITE CLASS D

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,MC/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS D

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,SSD/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS D

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,NLE/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-13

C.2.8 Evaluation of Coefficient C2 for Site Class E

SITE CLASS E

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR,SD/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS E

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR,SSD/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS E

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR,NLE/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-14 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.2.9 Evaluation of Coefficient C2 for Near Fault Set

NEAR FAULT SET 

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tp

CR,SD/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

NEAR FAULT SET 

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tp

CR,SSD/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

NEAR FAULT SET

(mean of 20 ground motions)

0.2

0.6

1.0

1.4

1.8

2.2

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tp

CR,NLE/CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-15

C.2.10 Effect of Site Class on Coefficient C2 (Stiffness Degrading Hysteretic Behavior)

S IT E C L A S S B

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,SD /C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S C

(m ean of 20 ground m otions )

0.2

0.6

1.0

1.4

1.8

2.2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D 

C R,SD /C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S D

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,SD /C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S E

(m ean of 20 ground m otions )

0.2

0.6

1.0

1.4

1.8

2.2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T g

C R,SD /C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

NEAR FAULT SET 

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T p

C R,SD /C R,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-16 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.2.11 Effect of Site Class on Coefficient C2 (Strength-Stiffness Degrading Hysteretic Behavior)

S IT E C L A S S B

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,SSD /C R,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

S IT E C L A S S C

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D 

C R,SSD /C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S D

(m ean o f 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,SSD /C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S E

(m ean of 2 0 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T g

C R,SSD /C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

NEAR FAULT SET 

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T p

C R,SSD /C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-17

C.2.12 Effect of Site Class on Coefficient C2 (Nonlinear Elastic Hysteretic Behavior)

S IT E C L A S S B

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

2 .6

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,NLE/C R,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

S IT E C L A S S C

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

2 .6

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D 

C R,NLE/C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

S IT E C L A S S D

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

2 .6

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D

C R,NLE/C R,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

S IT E C L A S S E

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

2 .6

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T g

C R,NLE/C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

NEAR FAULT SET

(m ean of 20 ground m otions )

0 .2

0 .6

1 .0

1 .4

1 .8

2 .2

2 .6

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

T / T p

C R,NLE/C R,EPP

R = 6 .0

R = 5 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-18 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.2.13 Effect of Hysteretic Behavior on C1 of SDOF Systems (Site Class B)

SITE CLASS B

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS B

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,SD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS B

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,SSD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS B

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,NLE

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-19

C.2.14 Effect of Hysteretic Behavior on C1 of SDOF Systems (Site Class C)

SITE CLASS C

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,SSD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS C

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,SD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS C

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD 

CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS C

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD 

CR,NLE

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-20 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.2.15 Effect of Hysteretic Behavior on C1 of SDOF Systems (Site Class D)

SITE CLASS D

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS D

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,SD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS D

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,SSD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS D

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD

CR,NLE

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-21

C.2.16 Effect of Hysteretic Behavior on C1 of SDOF Systems (Site Class E)

SITE CLASS E

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS E

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR,SD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS E

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR,SSD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS E

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR,NLE

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-22 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.2.17 Effect of Hysteretic Behavior on C1 of SDOF Systems (Near Fault Set) 

NEAR FAULT SET

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tp

CR,EPP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

NEAR FAULT SET 

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tp

CR,SSD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

NEAR FAULT SET

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tp

CR,NLE

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

NEAR FAULT SET

(mean of 20 ground motions)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tp

CR,SD

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-23

C.3 Evaluation of ATC-40 Version of Capacity Spectrum Method: Summary Results

C.3.1 Comparisons for Site Class B: 

MEAN ERROR ERROR DISPERSION 

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING

SITE CLASS B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-24 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.3.2 Comparisons for Site Class C: 

MEAN ERROR ERROR DISPERSION 

SITE CLASS C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

SITE CLASS C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

SITE CLASS C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

SITE CLASS C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

SITE CLASS C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING

SITE CLASS C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-25

C.3.3 Comparisons for Site Class D: 

MEAN ERROR ERROR DISPERSION 

SITE CLASS D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

SITE CLASS D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

SITE CLASS D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

SITE CLASS D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

SITE CLASS D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING

SITE CLASS D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-26 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.3.4 Comparisons for Site Class E: 

MEAN ERROR ERROR DISPERSION 

SITE CLASS E (Ts=1.0 s)

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

SITE CLASS E (Ts=1.0 s)

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

SITE CLASS E (Ts = 1.0 s)

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

SITE CLASS E (Ts=1.0 s)

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

SITE CLASS E (Ts=1.0 s)

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING

SITE CLASS E (Ts=1.0 s)

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-27

C.3.5 Comparisons for Near-Fault Ground Motions: 

MEAN ERROR ERROR DISPERSION 

NEAR-FAULT (Tp = 1.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Tp

E[(∆i)app/(∆ie)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

NEAR-FAULT (Tp = 1.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Tp

σ[(∆i)app/(∆ie)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE A

EXACT: ELASTO PLASTIC

NEAR-FAULT (Tp = 1.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Tp

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

NEAR-FAULT (Tp = 1.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Tp

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE B

EXACT: STIFFNESS DEGRADING

NEAR-FAULT (Tp = 1.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Tp

E[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING

NEAR-FAULT (Tp = 1.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Tp

σ[(∆i)app/(∆i)ex]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

APPROXIMATE: ATC40 - TYPE C

EXACT: STRENGTH AND STIFFNESS DEGRADING
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-28 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.4 Evaluation of the Coefficient Method of FEMA 356: Summary Results

C.4.1 FEMA 356 Nonlinear Static Procedure (NSP) C1 Values for Different Ts Values:

S IT E C L AS S B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 .0 0.5 1.0 1.5 2.0 2 .5 3 .0

P E R IO D [s]

C 1

R = 8 .0

R = 6 .0

R = 4 .0

R = 3 .0

R = 2 .0

R = 1 .5

T s = 0 .4 s

W IT H O U T C A P P IN G

S IT E C L AS S B

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 .0 0 .5 1 .0 1 .5 2.0 2.5 3.0

P E R IO D [s]

C 1

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

T s = 0 .4 s

W IT H C A P P IN G

S IT E C L AS S C

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

0 .0 0 .5 1 .0 1 .5 2.0 2.5 3.0

P E R IO D [s]

C 1

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

T s = 0 .5 5 s

W IT H O U T C A P P IN G

S IT E C L AS S C

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D [s]

C 1

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

T s = 0 .5 5 s

W IT H C A P P IN G

S IT E C L AS S D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D [s]

C 1

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

T s = 0 .6 0 s

W ITH O U T C A P P IN G

S IT E C L AS S D

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

0 .0 0 .5 1 .0 1 .5 2.0 2 .5 3 .0

P E R IO D [s]

C 1

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

T s = 0 .6 0 s

W IT H C A P P IN G

S IT E C L AS S E

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D [s]

C 1

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

T s = 1 .0 0 s

W ITH O U T C A P P IN G

S IT E C L A S S E

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D [s]

C 1

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

T s = 1 .0 0 s

W IT H C A P P IN G
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-29
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-30 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

C.4.2 FEMA 356 NSP C2 Values for Different Ts Values:

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D [s]

C 2

T s = 0 .4 0 s (S IT E C L A S S B )

T s = 0 .5 5 s (S IT E C L A S S C ) 

T s = 0 .6 0 s (S IT E C L A S S D )

T s = 1 .0 0 s (S IT E C L A S S E ) F r a m in g T y p e I

L ife S a fe ty p e rfo rm a n c e le ve l

C 2 = 1 .3 fo r T < 0 .1 s

C 2 = 1 .1 fo r T > T s

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

P E R IO D [s]

C 2

T s = 0 .40 s (S IT E C LA S S B )

T s = 0 .55 s (S IT E C LA S S C )

T s = 0 .60 s (S IT E C LA S S D )

T s = 1 .00 s (S IT E C LA S S E )

F r a m in g T y p e I

C o lla p s e P r e v e n t io n P e r f o r m a n c e le v e l

C 2 = 1 .5 fo r T < 0 .1 s

C 2 = 1 .2 fo r T > T s
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-31

C.4.3 Mean Error of FEMA 356 NSP (Mean of Approximate to Exact Maximum Inelastic 

Displacements):

C.4.3.1 Comparison with Elastic Perfectly Plastic Hysteretic Behavior:

W IT H O U T C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P E R IO D [s]

E[(∆ i

) app / (∆ i) e x ]

R= 8.0

R= 6.0

R= 4.0

R= 3.0

R= 2.0

R= 1.5

S IT E C LAS S B

T s = 0.4 s

( C 2 = 1.0)

W ITH C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P ER IO D [s]

E[(∆ i) ap p / (∆ i) e x ]

R= 8.0

R= 6.0

R= 4.0

R= 3.0

R= 1.5

R= 1.5

S IT E C L AS S B

T s = 0.4 s

( C 2 = 1.0)

W ITH O U T C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P E R IO D [s]

E[(∆ i) app / (∆ i ) e x ]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

S IT E C L ASS C

T s = 0.55 s

( C 2 = 1.0)

W ITH C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P E R IO D [s]

E[(∆ i) ap p / (∆ i) e x ]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 1.5

R = 1.5

SIT E C LAS S C

T s = 0.55 s

( C 2 = 1.0)

W ITH O U T C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P E R IO D [s]

E[(∆ i) app / (∆ i ) e x ]

R= 8.0

R= 6.0

R= 4.0

R= 3.0

R= 2.0

R= 1.5

S IT E C LAS S D

T s = 0.60 s

( C 2 = 1.0)

W ITH C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P E R IO D [s]

E[(∆ i) ap p / (∆ i) e x ]

R= 8.0

R= 6.0

R= 4.0

R= 3.0

R= 1.5

R= 1.5

S IT E C LAS S D

T s = 0.60 s

( C 2 = 1.0)
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-32 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆ i)app/(∆ i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

Ts = 1.00 s

(C2 = 1.0)

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆ i)app/(∆ i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

SITE CLASS E

Ts = 1.00 s

(C2 = 1.0)

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆ i)app/(∆ i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

NEAR-FAULT

Ts = 1.00 s

(C2 = 1.0)

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆ i)app/(∆ i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

NEAR-FAULT

Ts = 1.00 s

(C2 = 1.0)
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-33

C.4.3.2 Comparison with Stiffness Degrading Hysteretic Behavior:

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS B

Ts = 0.40 s 

C2 as for Life Safety performance 

level

C2=1.3 for T < 0.1 s

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

SITE CLASS B

Ts = 0.40 s 

C2 as for Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as for Life Safety performance 

level

C2=1.3 for T < 0.1 s

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as for Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS D

Ts = 0.60 s

C2 as for Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

SITE CLASS D

Ts = 0.60 s

C2 as for Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-34 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

Ts = 1.0 s

C2 as Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

SITE CLASS E

Ts = 1.0 s

C2 as Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

NEAR-FAULT

Ts = 1.0 s

C2 as for Life Safety performance 

level

C2=1.3 for T < 0.1 s

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

NEAR-FAULT

Ts = 1.0 s

C2 as for Life Safety performance 

level

C2=1.3 for T < 0.1 s
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-35

C.4.3.3 Comparison with Stiffness and Strength Degrading Hysteretic Behavior:

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS B

Ts = 0.40 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

SITE CLASS B

Ts = 0.40 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS D

Ts = 0.60 s 

C2 as Collapse Prevention performance level 

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS D

Ts = 0.60 s 

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-36 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

Ts = 1.0 s

C2 as collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

1.0

2.0

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

Ts = 1.0 s

C2 as collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

NEAR-FAULT

Ts = 1.0 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

E[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=1.5

R=1.5

NEAR-FAULT

Ts = 1.0 s

C2 as Collapse Prevention performance level 

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-37

C.4.4 Dispersion of the Error in FEMA 356 NSP (Standard Deviation of Approximate to Exact 

Maximum Inelastic Displacements):

C.4.4.1 Comparison with Elastic Perfectly Plastic Hysteretic Behavior:

W IT H O U T C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P E R IO D [s ]

σ [ (∆ i) app / (∆ i e ) e x ]

R= 8.0

R= 6.0

R= 4.0

R= 3.0

R= 2.0

R= 1.5

S IT E C LAS S B

T s = 0.4 s

( C 2 = 1.0)

W ITH C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P E R IO D [s ]

σ [ (∆ i) app / (∆ i) e x ]

R= 8.0

R= 6.0

R= 4.0

R= 3.0

R= 2.0

R= 1.5

S IT E C LAS S B

T s = 0.4 s

( C 2 = 1.0)

W IT H O U T C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P ER IO D [s]

σ [ (∆ i) app / (∆ i) e x ]

R= 8.0

R= 6.0

R= 4.0

R= 3.0

R= 2.0

R= 1.5

S IT E C LAS S C

T s = 0.55 s

( C 2 = 1.0)

W ITH C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P ER IO D [s]

σ [ (∆ i) ap p / (∆ i) e x ]

R= 8.0

R= 6.0

R= 4.0

R= 3.0

R= 2.0

R= 1.5

S IT E C L AS S C

T s = 0.55 s

( C 2 = 1.0)

W IT H O U T C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P E R IO D [s]

σ [ (∆ i) app / (∆ i) e x ]

R= 8.0

R= 6.0

R= 4.0

R= 3.0

R= 2.0

R= 1.5

S IT E C LAS S D

T s = 0.60 s

( C 2 = 1.0)

W ITH C A P P IN G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P E R IO D [s]

σ [ (∆ i) app / (∆ i) e x ]

R = 8.0

R = 6.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

S IT E C LAS S D

T s = 0.60 s

( C 2 = 1.0)
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-38 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ [(∆ i)app/(∆ i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

Ts = 1.00 s

(C2 = 1.0)

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ [(∆ i)app/(∆ i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

Ts = 1.00 s

(C2 = 1.0)

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆ i)app/(∆ i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

NEAR-FAULT

Ts = 1.00 s

(C2 = 1.0)

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆ i)app/(∆ i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

NEAR-FAULT

Ts = 1.00 s

(C2 = 1.0)
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-39

C.4.4.2 Comparison with Stiffness Degrading Hysteretic Behavior:

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS B

Ts = 0.40 s 

C2 as for Life Safety performance 

level

C2=1.3 for T < 0.1 s

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS B

Ts = 0.40 s 

C2 as for Life Safety performance 

level

C2=1.3 for T < 0.1 s

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as for Life Safety performance level 

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as for Life Safety performance 

level

C2=1.3 for T < 0.1 s

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS D

Ts = 0.60 s

C2 as for Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS D

Ts = 0.60 s

C2 as for Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-40 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

To = 1.0 s

C2 as Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > To

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

Ts = 1.0 s

C2 as Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

NEAR-FAULT

Ts = 1.0 s

C2 as for Life Safety performance level

C2=1.3 for T < 0.1 s

C2=1.1 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆ie)app/(∆ie)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

NEAR-FAULT

Ts = 1.0 s

C2 as for Life Safety performance 

level

C2=1.3 for T < 0.1 s
Appendix C: Supplemental Data on the Evaluation of Current Procedures

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures C-41

C.4.4.3 Comparison with Stiffness and Strength Degrading Hysteretic Behavior:

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS B

Ts = 0.40 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS B

Ts = 0.40 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS C

Ts = 0.55 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS D

Ts = 0.60 s 

C2 as Collapse Prevention performance level 

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS D

Ts = 0.60 s 

C2 as Collapse Prevention performance level 

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts
Appendix C: Supplemental Data on the Evaluation of Current Procedures

C-42 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

Ts = 1.0 s

C2 as collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

1.0

2.0

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

SITE CLASS E

Ts = 1.0 s

C2 as collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITHOUT CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

NEAR-FAULT

Ts = 1.0 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > Ts

WITH CAPPING

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T/Ts

σ[(∆i)app/(∆i)ex]

R=8.0

R=6.0

R=4.0

R=3.0

R=2.0

R=1.5

NEAR-FAULT

To = 1.0 s

C2 as Collapse Prevention performance level

C2=1.5 for T < 0.1 s

C2=1.2 for T > To
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures D-1 

D. Supplementary Information and Data on 

Equivalent Linearization

D.1 Introduction

This appendix provides material supporting the 

improved procedures contained in Chapter 6. It 

provides a basic discussion of structural capacity and 

seismic demand for use with equivalent linearization 

procedures. It then reviews the theoretical 

underpinnings and past development of equivalent 

linearization procedures. It describes the new 

methodology for developing the improved effective 

period and damping equations found in Sections 6.2.1 

and 6.2.2.

D.2 Capacity-Spectrum Method 

The Capacity-Spectrum Method (CSM) has rapidly 

grown in acceptance as a tool for determining the 

displacement response of structures in the inelastic 

response range. One of the attributes that makes the 

CSM appealing is its intuitive nature. It is relatively 

straightforward to think of an earthquake as presenting 

a demand on a structure and the structure as possessing 

a certain capacity to resist this demand. When the 

capacity and maximum demand are equal, the system 

can be considered to be in a state of “Equilibrium” that 

defines the expected performance of the structure. 

The use of effective or equivalent linear parameters in 

the CSM is also appealing from an intuitive point of 

view. First, it seems reasonable that the period of a 

structure lengthens as it loses stiffness. Second, it 

seems logical that inelastic behavior and damage 

produce increased damping. The use of equivalent 

linear parameters also allows the CSM procedure to be 

applied with equal ease to cases where the earthquake 

demand is specified by a smooth design spectrum, a 

uniform hazard spectrum, some other form of sitespecific design spectrum. Additionally, an extension of 

the conventional CSM solution procedure can provide 

the designer and researcher with useful information 

about the nature of the response beyond just the 

projected maximum amplitude of response. 

The CSM approach was initially conceived using the 

secant stiffness as the effective linear stiffness along 

with various formulas or rules for effective viscous 

damping. However, from nonlinear vibration theory it is 

know that the secant stiffness is not an optimal 

equivalent linear stiffness parameter for defining the 

response of inelastic systems subjected to random-like 

excitations. Therefore, this and related elements of the 

CSM approach deserve re-examination with the goal of 

developing improved linearization procedures.

Herein, optimal equivalent linear stiffness and damping 

parameters are determined through a statistical analysis 

that minimizes the extreme occurrences of the 

difference (i.e., error) between the response of an actual 

inelastic system and its equivalent linear counterpart. 

The linear parameters are determined as functions of 

response ductility. Ductility is defined as the maximum 

inelastic response displacement divided by the yield 

displacement. A variety of different inelastic systems 

have been studied including bilinear hysteretic, stiffness 

degrading, and strength degrading behavior. It is found 

that the proposed linearization parameters provide a 

significant improvement over those employed in ATC40, as judged by either response amplitude or 

performance point error measures.

D.2.1 Structural Capacity: Inelastic 

Pushover

Nonlinear static procedures generally employ a 

“pushover” analysis to develop a representation of 

structural capacity. The ability to perform a nonlinear 

static analysis is based on the fundamental requirement 

that accurate information is obtainable about the 

structure, components, connections and material 

properties. These techniques are summarized in 

Section 2.4 and covered in detail in ATC-40 and FEMA 

356.

The pushover curve is a structural surrogate for the 

actual multi-degree-of-freedom building model. The 

pushover curve characterizes the load versus 

deformation of cyclic structural response. It is 

generally taken to represent the backbone curve of the 

load-deformation hysteresis loops. From a pushover 

curve, the value of the initial elastic stiffness (elastic 

period) can be determined, as well as an estimate of the 

post- elastic stiffness.

The structural response behavior may also be 

categorized by hysteresis loop category. The backbone 

curve of response (from the push-over curve) does not 

fully specify how the building will respond to 

earthquake excitation. The hysteresis loop shape may 

be roughly bilinear and stable for subsequent cycles of 

response, or there may exhibit stiffness only or stiffness 
Appendix D: Supplementary Information and Data on Equivalent Linearization

D-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

and strength degradation. Another type of hysteretic 

behavior is the “pinching” of the hysteresis loops that is 

associated with many concrete structures. However, this 

latter type of behavior has not been addressed in this 

document. Categorizing the hysteretic behavior as 

being of a certain type is left to the discretion of the 

engineer and usually requires considerable engineering 

judgment. 

D.2.2 Seismic Demand: Response Spectra

Traditional linear analysis methods use lateral forces to 

represent a design condition. For nonlinear methods it is 

easier and more direct to use a set of lateral 

displacements as a design condition. For a given 

structure and ground motion, the displacement demand 

is the maximum expected response of the building 

during the ground motion.

The differential equation of motion for the system in 

Figure D-1 is expressed as:

(D-1)

Rearranging the equation and dividing through by the 

mass, m, results in:

(D-2)

where β is the fraction of critical damping and is the 

natural frequency which is related to the natural period 

by T = 2π /ω. Equation D-2 can be rearranged as:

(D-3)

Define Spectral Displacement (SD) and PseudoSpectral Acceleration (PSA) as follows:

SD (D-4)

PSA (D-5)

where has the mathematical meaning, “for all 

time.” Although Spectral Acceleration (SA) may also be 

defined from Equation D-3 it is assumed in this 

document that SA is interchangeable with PSA. 

Therefore, no distinction will be made between SA and 

PSA and for consistency, only SA will be used in the 

remainder of this appendix.

In the ADRS format, a radial line represents a constant 

structural period, T, independent of the amount of 

damping present in the system. The relationship 

between SA, SD and T may be determined using 

Equations D-4 and D-5 which results in:

(D-6)

An example of this is shown in Figure D-2 for 5% 

damping. 

D.3 Theoretical Basis for Equivalent 

Linearization

The general form of a SDOF oscillator equation of 

motion for the system shown in Figure D-3 can be 

expressed as

(D-7)

where m is the mass of the system and is the 

acceleration time history imparted to the oscillator. The 

term can have many forms. For this 

formulation, a linear system will be expressed as 

 where keff is the 

effective linear stiffness and ceff is the constant of 

proportionality for the effective viscous damping force, 

Figure D-1 SDOF oscillator model subjected to 

ground motion, u(t).

mx t cx t kx t mu t && & && () () () () + + =−

2

( ( ) ( )) 2 ( ) ( ) 0 && && & xt ut xt xt ++ + = βω ω

ω

2 && && & x() () 2 () () t ut xt xt + =− − βω ω

= ∀ max ( ) txt

2

= ∀ max ( ) ωn t xt

∀t

= 2π T SD

SA

mx t && & && ( ) ( ( ), ( )) ( ) + f x t x t mu t = −

u t &&( )

f ( ( ), ( )) xt xt &

f ( ( ), ( )) ( ) ( ) xt xt k xt c xt & & = eff eff +
Appendix D: Supplementary Information and Data on Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures D-3

. Hysteretic systems, possess history dependent 

restoring forces that are non-separable functions of both 

displacement and velocity. 

Equivalent linearization is an approximate technique for 

solving nonlinear differential equations. Equation D-7 

may be rewritten as 

(D-8)

where

(D-9)

The objective of equivalent linearization is to somehow 

select optimal values for the linear coefficients ceff and 

keff such that the quantity is in some 

sense minimized. Then, is ignored and 

Equation D-8 is solved as an ordinary linear differential 

equation. The approximate linear system is shown in 

Figure D-4. 

One possible approach is to minimize the mean square 

value of ε. The minimization criteria can be written as

(D-10)

(D-11)

where the over bar represents an averaging operation.

Figure D-2 Components of the ADRS format for representing Seismic Demand - PSA versus SD; left plot shows SD as 

a function of period, T; middle plot shows PSA as a function of period, T; right plot (ADRS format) is a 

compilation of the left and middle plots showing PSA versus SD, with period T defined by radial lines 

stemming from the origin.

Figure D-3 SDOF oscillator model represented by 

Equation D-7.

&( ) eff c xt

&& & & && ( ) ( ) ( ) ( ( ), ( )) ( ) + + + =− ε mx t c x t k x t x t x t mu t eff eff

ε ( ( ), ( )) ( ( ), ( )) ( ) ( ) & && = −− eff eff x t xt f xt xt c xt k xt

ε ( ( ), ( )) x t xt & ε ( ( ), ( )) x t xt &

2

0 ∂ε = ∂K

2

0 ε

γ

∂ = ∂
Appendix D: Supplementary Information and Data on Equivalent Linearization

D-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

If the excitation, , is a harmonic function of time, 

the steady-state solution can be assumed to be of the 

form

 where (D-12)

Analyzing a single cycle of the steady state response 

leads to the following equation

(D-13)

Applying the minimization criteria in Equations D-10 

and D-11 to Equation D-13 results in

(D-14)

and

(D-15)

This approach leads to an effective stiffness similar to 

what is seen in Figure D-5 where the effective stiffness 

is less than the secant stiffness, Ksec. 

Another way to determine equivalent linear damping 

parameter is through energy balance. The energy 

dissipated by the hysteretic system may be equated to 

the energy dissipated by an equivalent viscous damper. 

Assume the response to be of a harmonic form over one 

full cycle of response expressed as

 where (D-16)

Then, energy dissipated by a viscous damper over one 

cycle of response, E, can be expressed as

(D-17)

where T is the period of cyclic motion.

For the bilinear hysteretic model seen in Figure D-5, the 

energy dissipated over one cycle of response, E, can be 

expressed as

(D-18)

Figure D-4 Linear SDOF oscillator model with 

effective linear parameters as represented 

by Equation D-8.

u t &&( )

max xt x ( ) cos( ) = θ θ = − ω φ t

2 2 2

max max max

0

1 ( ( , ) sin cos ) 2

π

ε θ ω θ θθ

π = −− ∫ eff eff f x c x kx d

2

max

max 0

1

( , ) sin

π

θ θ θ

ωπ = − eff ∫ c fx dx

2

max

max 0

1 ( , )cos

π

θ θ θ

π = eff ∫ k fx d

x

Figure D-5 Bilinear hysteretic system.

max xt x ( ) cos( ) = θ θ = − ω φ t

2 2

max E = 2 / π eff cx T

max E =− − 4 ( )( ) xk k x x yo o y α
Appendix D: Supplementary Information and Data on Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures D-5

Equating energies from Equations D-17 and D-18 leads 

to

(D-19)

In Figure D-5, the secant stiffness is labeled and 

can be expressed as 

(D-20)

If the secant period, , is assumed to be the period of 

structural response, then

(D-21)

The secant stiffness can be related to the secant period 

and substituting Equation D-20 into Equation D-19 

leads to the following expression for .

(D-22)

Equation D-22 corresponds to the equivalent viscous 

damping equation found in ATC-40 (Section 8.2.2.1). 

However, the response of inelastic systems to 

earthquake ground motions is not the same as the 

steady-state response to a constant amplitude sinusoidal 

forcing function as assumed in the above equivalent 

linearization formulation. Repeated full hysteresis loops 

with constant amplitude occur infrequently for inelastic 

systems subjected to earthquake time histories. 

Furthermore, partial and one-sided loops are likely to 

occur.

D.4 Starting Point For Optimization

The ductility demand, μ, is defined as the maximum 

displacement of the inelastic system divided by its yield 

displacement. For the bilinear hysteretic system shown 

in Figure D-5, the ductility demand is xmax/xy. In 1980, 

an optimal set of equivalent linear parameters for 

earthquake excitation was defined based on making an 

adjustment to the linear response spectrum. In that 

study (Iwan, 1980) ductility dependent inelastic 

response spectra were compared with elastic response 

spectra, and displacement preserving shifts of the 

inelastic spectra were determined which minimized the 

average absolute value difference between the inelastic 

and equivalent linear spectra over a range of periods. A 

family of hysteresis behavior was considered including 

bilinear hysteretic as well as pinching hysteretic 

models. Figure D-6 shows a typical set of inelastic and 

elastic response spectra and indicates the manner in 

which the spectra were adjusted. 

Using the stated procedure, the following relationships 

were obtained for the optimal effective linear 

parameters:

(D-23)

(D-24)

It is easily shown that the optimal effective period 

defined by the above relationship is significantly less

than the period associated with the secant stiffness that 

is employed in the conventional CSM approach found 

in ATC-40 (or, the optimal stiffness is significantly 

greater than the secant stiffness). Indeed, the secant 

stiffness may overestimate the optimal effective period 

by more than 50% for larger values of ductility. It is also 

observed that the damping value used in the 

conventional CSM approach is significantly greater 

than the optimal damping parameter in Equation D-24. 

Thus, the conventional CSM approach significantly 

overestimates both the effective period and damping of 

inelastic systems. In some regions of response, these 

two overestimates combine in such a manner that the 

resultant response prediction is not much different from 

the prediction obtained from the optimal parameters. 

However, these two overestimates do not always 

counteract each other to produce reliable estimates of 

displacement.

2 2 c k kxx xT x eff o o y y =− − 4( ) ( ) /(2 ) α max max π

Ksec

sec max max K kx x x x =+ − oy y ( ( )) / α

Tsec

eff = sec k K

eff c

2

max max

sec

eff o o y y =− − 4( ) ( ) /( ) α π

m c k kxx x x

K

Figure D-6 Early effort to define optimal equivalent 

linear parameters (Iwan, 1980).

( )

0.939

0 T T eff / 1 0.121 1 −= − μ

0.371

0 ββ μ −= − 5.87( 1) eff
Appendix D: Supplementary Information and Data on Equivalent Linearization

D-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Even though the conventional and some more optimal 

set of equivalent linear parameters may predict about 

the same average response for some range of cases, that 

does not mean they are equally effective. This may be 

illustrated by considering the distribution of the error 

for the two different approaches. Figure D-7shows the 

distribution of the Performance Point displacement 

error that is obtained using the early optimal parameters 

and the conventional CSM parameters for an 

elastoplastic system. It is evident from the figure that 

there is only a modest difference between the mean 

values of the error for the two approaches. The optimal 

parameters give a mean Performance Point error of -

4.4% while the conventional CSM approach gives a 

mean error of -9.5%. However, there is a very 

substantial difference in the standard deviation of the 

error for the two approaches. The optimal approach 

error has a standard deviation of 21.2% while that of the 

conventional CSM approach is 68.7%. As seen from the 

figure, the conventional CSM approach has a much 

greater probability of exhibiting extreme over 

prediction errors than does the optimal approach, even 

though the mean error of the conventional approach is 

less conservative than the optimal approach. Hence, it is 

clear that it is inadequate to merely minimize the mean 

value of the displacement error when defining an 

optimal set of effective linear parameters. It is necessary 

to simultaneously minimize both the mean and standard 

deviation in some sense.

D.5 Alternative Statistical Analysis

The results of the early optimization study described 

above provide the motivation for a more comprehensive 

study of equivalent linearization for earthquake 

response prediction. In the earlier study, the error 

measure used for optimization was the mean of the 

absolute value of the displacement error. This is an 

intuitive error measure which leads to reasonable 

results. However, this error measure may not be as 

directly meaningful as other possible measures. In 

practice, it would seem to be more appropriate to have 

the measure of goodness of the optimal effective linear 

parameters based on some measure of engineering 

acceptability. This is the approach used in this study.

Recall the equation of motion for the single-degree-offreedom system in Figure D-3. When 

represents a linear viscous damped system, the equation 

of motion may be expressed as

(D-25)

where and are the viscous damping coefficient 

and spring stiffness, respectively. For a given ground 

excitation, , the solution, xlin(t), may be computed 

using any numerical solution procedure. For an inelastic 

system, the restoring force, , may take a variety 

of forms. The solution for the inelastic system will be 

designated as xinel (t).

Many different approaches are available for making a 

comparison between the displacement time histories 

xinel (t) and xlin(t). These include, but are not limited to, 

a point by point comparison of the displacement, 

velocity or acceleration time histories, comparing the 

number of zero displacement crossing or comparison of 

the amplitude spectra from a Fourier Transform. 

However, to quantify a comparison, there must be a 

value assigned to the amount of similarity or difference. 

Within the framework of performance-based 

engineering, the key performance variable is the 

maximum relative displacement amplitude that a 

structure experiences from the demand earthquake. The 

relative displacements for the inelastic and linear 

single-degree-of-freedom systems are xinel (t) and 

xlin(t), respectively.

The effective linear parameters obtained based on a 

comparison of displacement values would not be 

appropriate to be used in a velocity or force-based 

f (,) x x&

( ) mx c x k x mu t && & && lin eff lin eff lin + + =−

Figure D-7 Distribution of percent error in 

Performance Point displacement. Bilinear 

system with alpha=0, T0 = 0.1-2.0 sec 

(0.1 sec increments), μ=2, 28 far-field 

earthquakes.

eff c eff k

u t &&( )

f (,) x x&
Appendix D: Supplementary Information and Data on Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures D-7

design procedure. For example, the maximum velocities 

or accelerations from the linear solution should not be 

used as estimates for the maximum values of or 

. The maximum acceleration or maximum 

pseudo-acceleration would be a much better 

comparison parameter for effective linear parameters 

intended for use in a force-based approach.

The maximum displacement amplitude of the nonlinear 

time history xinel (t) will be designated as Dinel and the 

maximum displacement amplitude of the linear time 

history xlin(t) will be designated as Dlin. The effective 

linear parameters developed in this study will be used 

for estimating the response of structures subjected to 

earthquake excitations. Therefore, using real earthquake 

time histories as the model inputs is most logical.

The methodology developed in this study employs a 

search over a two-dimensional parameter space related 

to the linear system coefficients ceff and keff in 

Equation D-25. One can expect to find a combination or 

combinations of ceff and keff that give the best “match” 

with an inelastic system, in some sense. The terms ceff

and keff will be replaced by the fraction of critical 

damping, βeff, and the natural period of oscillation, Τeff. 

Equation D-25 can be expressed as

(D-26)

The system parameters βeff and Τeff completely describe 

the linear single-degree-of-freedom system. 

D.5.1 Error Measure

In order to compare the maximum displacements, Dinel

and Dlin, an error measure must be defined. In 

engineering design, unconservative displacement 

predictions are generally less desirable than 

conservative predictions. Therefore, a fundamental 

requirement of any error measure is that it distinguish 

between a conservative displacement prediction and a 

non-conservative displacement prediction. An error 

measure that uses an absolute value of the difference 

between Dinel and Dlin would not satisfy this 

requirement.

A simple error measure satisfying the above 

requirement is the ratio of the difference between the 

linear system maximum displacement, Dlin, and the 

inelastic system maximum displacement, Dinel, to the 

inelastic system maximum displacement.

(D-27)

Using this definition, a negative value of εD reflects an 

unconservative displacement prediction while a positive 

value reflects a conservative displacement prediction. 

εD might be considered to have a positive bias as it 

ranges from -1 to ∞. However, for the range of systems 

and excitations considered in this study, the slight 

positive bias in the statistical distribution of εD is 

inconsequential.

For a given inelastic system and ground excitation, 

there will be a certain topology associated with the 

error, εD, as a function of linear system parameters Τeff

and βeff as shown in Figure D-8. Note that there exists a 

nearly diagonal contour of zero error. For any 

combination of Τeff and βeff lying along this contour 

there will be a perfect match between Dlin and Dinel.

For any specified ensemble of inelastic systems and 

ground excitations, distributions of εD can be obtained 

for every combination of Τeff and βeff. This is illustrated 

in Figure D-9.

& ( ) inel x t && ( ) inel x t

2

4 2

( ) πβ π+ + =−

⎛ ⎞ ⎜ ⎟

⎝ ⎠

&& & && eff

eff eff

x x x ut

T T

Figure D-8 Contour values of εD over the twodimensional parameter space of Teff and 

βeff for a single combination of inelastic 

system and ground excitation.

ζeff

Teff

−0.4

−0.2

−0.2

0

0

0.2

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

lin inel

D

inel

D D

D

ε − =
Appendix D: Supplementary Information and Data on Equivalent Linearization

D-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

The importance of using the standard deviation as well 

as the mean of the error distribution is illustrated in the 

following example. Two hypothetical probability 

density functions are shown in Figure D-10. For the 

more widely spread error distribution, the mean error 

value is zero, while for the tighter distribution, the mean 

error value is -5%. Solely in terms of the mean value, 

the widely spread distribution is more accurate than the 

tighter distribution. However, a more insightful way to 

analyze the distributions would be in terms of an 

acceptable range of error values. In this example, an 

acceptable range of error values might be chosen to be 

from -20% to 20%. In this case, the distribution with the 

mean value of -5% would be both more “acceptable” 

compared to the distribution with a mean value of 0%. 

Let R be the probability that the error εD lies outside the 

range from a to b. Then, R may be expressed as

(D-28)

If the distribution of εD is assumed to be Normal, R can 

be expressed as

(D-29)

where m is the mean value and σ is the standard 

deviation of the distributions of εD values. It will be 

assumed that the desirable range of error values, εD, 

from an engineering design point of view is between -

10% and +20%. This assumption has been adopted after 

consultation with several members of the practicing 

structural engineering community. This range of error 

values will be referred to as the Engineering 

Acceptability Range (EAR). This range takes into 

account the general desire for a more conservative 

design rather than an unconservative design. That is, a 

20% error is more acceptable than a -20% error. 

D.5.2 Optimization Criterion

The optimum point in the Teff, βeff parameter space is 

chosen to be the point that minimizes the probability 

that the error, εD, will be outside the Engineering 

Acceptability Range. The Engineering Acceptability 

Criterion may therefore be defined as σ.

(D-30)

Figure D-11 shows contours of REAR as a function of 

Teff and βeff. Also shown is the optimal point over the 

Figure D-9 Illustration of assembling εD error 

distributions at every combination of Teff and βeff over an ensemble.

ζeff Teff

Distributions of ε

D

R= − < < 1 Pr( ) ε D a b

Figure D-10 Illustration of probability density 

functions of displacement error for a 

Normal distribution.

−30 −20 −10 0 10 20 30

0

0.05

0.1

0.15

0.2

Error (%)

Probability Density Function

2

2

( )

2 1 1

2

b x m

a

e dx σ

σ π

− −

R= − ∫

1 Pr( 0.1 0.2) min EAR D R ≡− − < < = ε
Appendix D: Supplementary Information and Data on Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures D-9

two-dimensional parameter space which is denoted by a 

square. 

The diagonal trend to the contours in Figure D-11 can 

be explained by the following physical reasoning. 

Consider the displacement response of a linear 

oscillator subjected to an earthquake excitation. 

Decreasing the system damping will always increase 

the displacement response. Generally speaking, 

decreasing the natural period will also decrease the 

displacement response. Although this is not true in all 

cases, especially for near-field ground motions, it is a 

general trend that by increasing period and damping in 

the correct proportion, a nearly constant maximum 

displacement can be achieved.

The size and shape of the contours in Figure D-11 give 

insight into the ramifications of using effective linear 

parameters that are different from the values at the 

optimal point. In Figure D-11, the contour closest to the 

optimum point has a value of 0.35 while the minimum 

value of REAR ( ) is 0.31. The gradient of the 

contours is more gradual along a line roughly from 

lower left to upper right. Therefore, if the effective 

period is under predicted, it is best to also have an 

under-predicted damping. If the effective period is over 

predicted, it is best to also have an over-predicted 

damping. In the general direction from lower right to 

upper left, the gradient of the contours is very large and 

the value of R quickly increases for relatively small 

changes in the effective parameters. Over predicting 

one parameter and under predicting the other can have 

serious repercussions on the reliability of the 

displacement prediction.

D.6 Effective Linear Parameters

The full explicit functional dependence of εD may be 

indicated as follows

(D-31)

The maximum displacement of the nonlinear system, 

Dinel, is a function of initial period, Τ0, linear viscous 

damping, β0, second slope ratio, α, response ductility, 

μ, and hysteretic model, denoted ``HYST''. The linear 

system response, Dlin, is a function of the two linear 

system parameters: period, Teff, and damping, βeff. It is 

desired to find effective linear parameters that are 

applicable over a range of Τ0 and β0 values. Therefore, 

multiple values of Τ0 and β0 will be included in the 

same ensemble. The two-dimensional Teff, βeff 

parameter space is transformed into the Τeff - Τ0, βeff - 

β0 parameter space.

The Engineering Acceptability Criterion is applied to 

the error distributions over the entire Teff / T0, βeff – β0

parameter space and the optimum combination of Teff / 

T0 and βeff – β0, is determined. Next, the ductility 

value is changed, and the entire process is repeated. The 

ductility values used in this study range from 1.25 to 6.5 

in increments of 0.25. Additionally, ductilities of 8 and 

10 are included. 

The optimum values of Teff / T0 and βeff – β0 may be 

graphed as functions of ductility. Then, these results can 

be fitted with an analytical expression. Figure D-12 

shows a typical example of the discrete optimum values 

of Teff / T0 and βeff – β0 graphed as a function of 

ductility along with a curve fit of the data. Information 

pertaining to the details of the curve fitting process may 

be found in Guyader, 2004.

Figure D-11 Contours of REAR over the Teff, βeff parameter space. The optimum point is 

marked by a square.

0 5 10 15 20 25

1

1.2

1.4

1.6

1.8

2

2.2

ζeff−ζ

o

Teff/T0

35

35

35

40

40

40

45

45

45

50

50

50

50

60

60

60

60

70

70

70

80

80

80

90

90

REARmin

( , ,,, )

( , ) (, ,,, )

(, ,,, )

ε β β αμ

β β αμ

β αμ

−

− =

eff

D eff o

o

lin eff eff inel o o

inel o o

T

HYST

T

D T D T HYST

D T HYST
Appendix D: Supplementary Information and Data on Equivalent Linearization

D-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

The new optimization criterion has been applied to the 

basic hysteretic models shown in Figure D-13. In each 

case, the basic model has been augmented by the 

addition of a linear spring element to create a non-zero 

second slope. α is the ratio of the post-yield stiffness to 

the elastic stiffness as seen in Figure D-6. The BLH and 

STDG models have been analyzed for alpha values of 

0,2,5 and 10% while the STRDG model has been 

computed for alpha values of -3 and -5%. Results for all 

models and certain combinations of models are 

presented and discussed in Chapter 6. The elastoplastic 

system represents the greatest challenge in application 

of equivalent linearization due to the existence of longperiod drifting displacement response in addition to the 

more quasi-harmonic motion (Paparizos and Iwan, 

1988). It is noted that in general the results for the 

systems with alpha greater than zero are more favorable 

than for the systems with alpha equal to or less than 

zero.

Figure D-12 Example of optimal effective linear parameters - discrete points and the curve fitted to the data

Figure D-13 Types of inelastic behavior considered. BLH=Bilinear hysteretic, STDG=Stiffness Degrading, and 

STRDG=Strength Degrading.

BLH (α=0%) STDG (α=0%) STRDG (α=−5%)
Appendix D: Supplementary Information and Data on Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures D-11

D.7 Performance Point Errors

The Capacity-Spectrum Method incorporates both 

structural capacity and seismic demand to determine a 

point where the demand and capacity are equal, referred 

to as the Performance Point. This point gives the 

expected displacement in the structure. The accuracy of 

the Capacity-Spectrum Method will be evaluated using 

a new error measure. For a given ground motion, the 

Performance Point Error, , is defined as the 

difference between the displacement at the Performance 

Point, as determined using equivalent linear parameters, 

and the actual maximum inelastic displacement 

response divided by the maximum inelastic 

displacement. This can be expressed as

(D-32)

Error statistics are created by combining all Τ0 and β0

values for a given hysteretic model, second slope ratio 

and ductility.

Several sources of error are introduced by the CapacitySpectrum Method. Errors may arise in both the 

determination of structural capacity and seismic 

demand. To evaluate the error from the equivalent 

linear parameters alone, all other sources of error must 

either be eliminated or shown to be negligible.

In determining structural capacity, two sources of error 

exist: the capacity spectrum calculation and the 

hysteretic classification. A large source of error may 

come from representing a multi-degree-of-freedom 

building model by a single-degree-of-freedom system. 

This source of error is eliminated herein by considering 

only single-degree-of-freedom structures.

The second source of error in determining the structural 

capacity is the hysteretic classification. In what follows, 

errors associated with determination of the hysteretic 

model are eliminated since the actual hysteretic model 

is assumed apriori. In this way, both sources of error 

associated with the structural capacity have been 

removed.

In determining seismic demand, errors may be 

introduced through smoothing of the demand spectrum. 

A design spectrum that represents the effects of many 

possible earthquake sources is generally smooth and 

conservative. However, the spectrum of any actual 

earthquake ground motion is generally quite irregular. 

The potential errors associated with using a design 

spectrum instead of an actual earthquake response 

spectrum are eliminated in what follows by using actual 

earthquake response spectra. Demand spectra are 

calculated using the appropriate effective linear 

parameters. The only remaining source of error in the 

Capacity-Spectrum Method are errors associated with 

the effective linear parameters.

Performance Point Error results are presented for the 

bilinear hysteretic (BLH) and strength degrading 

(STDG) models with second slope ratios of 0% and 5% 

in Figures D-14 and D-15. The results clearly show an 

improvement using the new effective parameters as 

compared to the effective parameters used in ATC-40. 

For all cases, the probability of the Performance Point 

Error lying within the range of Engineering 

Acceptability is much higher for the new approach than 

for the current Capacity-Spectrum Method, especially 

for lower ductilities. This would appear to validate the 

use of higher order curve fitting for lower ductilities to 

help capture important local variations in the effective 

parameters.

At low values of ductility, the conventional CapacitySpectrum Method approach is noticeably 

unconservative. Therefore, a building needing 

rehabilitated could be judged to not need an upgrade 

using the conventional Capacity-Spectrum Method 

approach. Within the framework of performance-based 

engineering, where Performance Objectives are very 

precise, accurate prediction at the lower ductility values 

can be quite important in terms of Immediate 

Occupancy and Operational building performance 

levels. 

D.8 References 

Iwan, W.D., 1980, “Estimating inelastic response spectra from elastic spectra,” Earthquake Engineering 

and Structural Dynamics, Vol. 8, pp. 375-388.

Guyader, A, 2004, A Statistical Approach to Equivalent 

Linearization with Application to PerformanceBased Engineering, EERL 2004-04 Report, Pasadena, California.

Paparizos, L.G., and W.D., Iwan, 1988, “Some observations on the random response of an elasto-plastic 

system,” Journal of Applied Mechanics, Paper No. 

88-WA/APM-64, American Society of Civil Engineers.

εDpp

(,, )

( ( , , ), ( , , ))

(, ,,, )

(, ,,, )

(, ,,, )

ε αμ

αμ β αμ

β αμ

β αμ

β αμ

=

−

DPP

lin eff o PP eff o PP

inel o o

inel o o

inel o o

HYST

DTT T

D T HYST

D T HYST

D T HYST
Appendix D: Supplementary Information and Data on Equivalent Linearization

D-12 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure D-14 Summary of Performance Point errors for bilinear hysteretic (BLH) model

1 2 3 4 5 6 7

0

20

40

60

80

100

μ

Prob[PP Error > 0%]

ATC−55 BLH 0%

ATC−55 BLH 5%

ATC−40 BLH 0%

ATC−40 BLH 5%

1 2 3 4 5 6 7

0

20

40

60

80

100

μ

Prob[PP Error > 20%]

1 2 3 4 5 6 7

0

20

40

60

80

100

μ

Prob[PP Error < −10%]

1 2 3 4 5 6 7

0

20

40

60

80

100

μ

Prob[−10% < PP Error < 20%]
Appendix D: Supplementary Information and Data on Equivalent Linearization

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures D-13

Figure D-15 Summary of Performance Point Errors for Strength Degrading (STDG) model

1 2 3 4 5 6 7

0

20

40

60

80

100

μ

Prob[PP Error > 0%]ATC−55 STDG 0%

ATC−55 STDG 5%

ATC−40 STDG 0%

ATC−40 STDG 5%

1 2 3 4 5 6 7

0

20

40

60

80

100

μ

Prob[PP Error > 20%]

1 2 3 4 5 6 7

0

20

40

60

80

100

μ

Prob[PP Error < −10%]

1 2 3 4 5 6 7

0

20

40

60

80

100

μ

Prob[−10% < PP Error < 20%]
Appendix D: Supplementary Information and Data on Equivalent Linearization

D-14 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-1 

E. Supplementary Information and Data on SoilStructure Interaction Effects 

E.1 Introduction

This appendix provides detailed information on soilstructure (SSI) effects. The intent is to provide 

background information supporting the simplified 

design procedures presented in Chapter 8. Section E.2 

describes kinematic interaction effects and engineering 

models used to describe these effects. Section E.3 

describes foundation damping effects and how these 

effects contribute to the system damping ratio, which is 

the damping ratio for which the initial seismic demand 

spectrum should be computed. Reference citations are 

provided in Section E.4.

The soil-structure interaction effects described in this 

appendix are relevant at the stage of defining the elastic 

response spectrum, to be used during application of the 

Capacity-Spectrum Method of ATC-40, or the 

Coefficient Method of FEMA 356 (ASCE, 2000). The 

elastic spectra used during these procedures pertain to a 

free-field condition and for 5% damping. The spectral 

ordinates of the elastic spectra can be modified for 

kinematic interaction and foundation damping using the 

procedures presented in Sections E.2.4 and E.3.3, 

respectively (and summarized in Sections 8.2 and 8.3 

of the main body of the report). The modified spectral 

ordinates can then be used in nonlinear static analyses 

of structural response and performance.

E.2 Kinematic interaction 

Kinematic interaction results from the presence of stiff 

foundation elements on or in soil, which causes 

foundation motions to deviate from free-field motions 

as a result of base slab averaging and embedment 

effects. The base slab averaging effect can be 

visualized by recognizing that the motion that would 

have occurred in the absence of the structure within and 

below the footprint of the building is spatially variable. 

Placement of a foundation slab across these variable 

motions produces an averaging effect in which the 

foundation motion is less than the localized maxima 

that would have occurred in the free-field. The 

embedment effect is simply associated with the 

reduction of ground motion that tends to occur with 

depth in a soil deposit. 

The information provided in this section on kinematic 

interaction covers simple models for the analysis of 

ground motion variations between the free-field and 

shallow foundations at the ground surface (in which 

case kinematic interaction is dominated by base slab 

averaging) and embedded shallow foundations (in 

which case kinematic interaction can result from both 

base slab averaging and embedment effects). 

Kinematic interaction for pile-supported foundations is 

not covered. Theoretical models for kinematic 

interaction effects are expressed as frequencydependent ratios of the Fourier amplitudes (i.e., transfer 

functions) of foundation input motion (FIM) to freefield motion. The FIM is the theoretical motion of the 

base slab if the foundation and structure had no mass, 

and is a more appropriate motion for structural response 

analysis than is the free-field motion. 

In the following subsections, formulations for transfer 

functions that account for base slab averaging and 

embedment effects are presented. Recommendations 

are then provided regarding how transfer functions can 

be used to modify a free-field response spectrum or 

time history suite to estimate foundation input motions 

(FIMs) for use in nonlinear static procedures.

E.2.1 Shallow Foundations at the Ground 

Surface

Base-slab averaging results from inclined or incoherent 

incident wave fields. Motions of surface foundations 

are modified relative to the free-field when incident 

waves impinge upon the foundation with an angle to the 

vertical axis, αv, or when the incident wave is 

incoherent. The first case is referred to as the wave 

passage effect and the second case as the ground motion 

incoherence effect. In the presence of these wave fields, 

translational base-slab motions are reduced relative to 

the free-field, and rotational motions are introduced. 

The reductions of base-slab translation, and the 

introduction of torsion and rotation in the vertical plane, 

are all effects that tend to become more significant with 

decreasing period. The period-dependence of these 

effects is primarily associated with the increased 

effective size of the foundation relative to the seismic 

wavelengths at low periods. In addition, ground 

motions are more incoherent at low periods.

Veletsos and Prasad (1989) and Veletsos et al. (1997) 

developed useful models for base slab averaging that 

combine an analytical representation of the spatial 

variation of ground motion with rigorous treatment of 
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

foundation-soil contact. The models evaluate the 

response of rigid, massless circular and rectangular 

foundations on the surface of an elastic halfspace to 

incoherent SH waves propagating either vertically or at 

an angle αv to the vertical. A result of the model is a 

transfer function between free-field motions and 

translational foundation motions (denoted with 

subscript ‘u’). 

The transfer function amplitudes computed by Veletsos 

and his co-workers are presented in Figure E-1 for 

circular and rectangular foundations subject to 

vertically incident incoherent SH waves. Similar curves 

are available for nonvertically incident coherent waves 

in the references. The transfer functions in Figure E-1

are plotted against the dimensionless frequency 

parameter , defined as follows for circular and 

rectangular foundations, respectively,

Circular

Rectangular (E-1)

where a0 = ωr/Vs,r, Vs,r denotes a strain-reduced shear 

wave velocity, r = radius of circular foundation, a × b = 

full footprint dimensions of rectangular foundation (b

being measured perpendicular to the direction of SH 

wave polarization), , and κ = a ground 

motion incoherence parameter (information on the 

selection of κ values are presented below).

Figure E-1 indicates that the transfer functions for 

circular and various rectangular geometries are similar 

to each other for small . As noted by Veletsos et al. 

(1997), the near equivalence of the results for different 

aspect ratios (a/b=1/4 - 4) of rectangular foundations 

suggests that translational transfer functions primarily 

depend on foundation area. Given this near 

equivalence, the transfer function is principally a 

function of dimensionless frequency . As shown in 

Equation E-1, is essentially the foundation 

dimension normalized by the wavelength (since 

wavelength is wave velocity divided by frequency), and 

then scaled by an incoherence/wave inclination term. 

When the foundation dimension is large relative to the 

wave length, is large and the transfer function 

ordinate is low. Conversely, small foundation 

dimensions relative to the wavelength produce transfer 

function ordinates near unity. 

Kim and Stewart (2003) calibrated the above analysis 

procedure against observed foundation / free-field 

ground motion variations as quantified by frequencydependent transmissibility function amplitudes, |H|. 

The above analytical models were fit to |H| for the 

assumed condition of a rigid base slab and a vertically 

propagating, incoherent incident wave field. Calibrated 

from the fitting process was a ground motion 

incoherence parameter, κ. Since the limiting 

assumptions of the model were not strictly satisfied for 

actual structures, the results of the identification were 

denoted apparent κ values (κa) that reflect not only 

incoherence effects, but also average foundation 

flexibility and wave inclination effects within the 

calibration data set. The foundation flexibility effects 

within the calibration data set generally correspond to 

shallow foundation conditions in which foundation 

components are inter-connected (i.e., continuous mats 

or footings inter-connected with grade beams). 

Parameter κa was found to be correlated to average soil 

shear wave velocity as shown in Figure E-2. These 

values of κa can be used with Figure E-1 (assuming αv

= 0) to define site-specific transfer functions given the 

foundation radius (r) and effective small-strain shear 

wave velocity (vs). In these procedures, effective 

foundation radius is defined as (where 

Figure E-1 Amplitude of transfer function between 

free-field motion and foundation input 

motion for vertically incident incoherent 

waves (Veletsos and Prasad, 1989; 

Veletsos et al., 1997).

a ̃

o

2 2

0 0 = + κ α sin v %a a

2

2 2

,

sin

2

ω

κ α

⎛ ⎞

= + ⎜ ⎟ ⎝ ⎠

e o v

sr e

b b

a

v b %

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Transfer Function Amplitude

Disk

a/b=1

a/b=1/4, 4

a0

~

e b ab =

a ̃

o

a ̃

o a ̃

o

a ̃

o

r A = f π
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-3

Af = foundation area) and the effective vs for the site is 

defined as r / (travel time for shear wave to travel from 

depth r to ground surface). Depth is measured down 

from the base of the foundation.

The model has not been validated for foundations with 

low in-plane stiffness, buildings with large footprint 

dimensions (> 200 ft), and pile-supported foundations 

in which the cap and soil are not in contact. However, 

the judgment of the project technical team that 

developed this report is that the model can provide a 

reasonable first order estimate of the kinematic 

interaction effect for those conditions. 

E.2.2 Embedded Shallow Foundations

Foundation “embedment” refers to a foundation base 

slab that is positioned at a lower elevation than the 

surrounding ground, which will usually occur when 

buildings have a basement. When subjected to 

vertically propagating coherent SH waves, embedded 

foundations experience a reduction in base-slab 

translational motions relative to the free-field, and 

rotations in the vertical plane are introduced. The 

rotations are caused by incompatible shear strains along 

the sides of the excavation and the free-field. 

Elsabee and Morray (1977) and Day (1978) have 

developed analytical transfer functions relating baseslab translational and rotational motions to free-field 

translations for an incident wave field consisting of 

vertically propagating, coherent SH waves. Base-slab 

averaging does not occur within this wave field, but 

foundation translations are reduced relative to the freefield due to ground motion reductions with depth and 

wave scattering effects. Day (1978) used finite element 

analyses to evaluate the base motions of a rigid 

cylindrical foundation embedded in a uniform elastic 

half space (β = 0, ν = 0.25) and subjected to vertically 

incident, coherent SH waves. Elsabee and Morray 

(1977) performed similar studies but for the case of a 

visco-elastic soil layer of finite depth over a rigid base 

(β = 0.05 and ν = 0.33). The amplitude of the halfspace 

and finite soil layer transfer functions are shown 

together in Figure E-3 for foundation embedment / 

radius ratio e/r = 1.0. The primary difference between 

the two solutions is oscillations in the finite soil layer 

case at high frequencies. Also shown in Figure E-3a is 

the following approximate transfer function amplitude 

model developed by Elsabee and Morray (1977):

[Hu must be ≥ 0.454] (E-2)

where a0 = ωr/vs and e = foundation embedment. 

Figure E-3b shows the transfer function amplitude 

model is a somewhat more convenient form in which it 

is plotted as a unique function of ωe/vs. 

Figure E-2 Relationship between effective incoherence parameter ka and small-strain shear wave velocity νs from 

case histories (from Kim and Stewart, 2003).

κ = -0.037 + 7.4E-04 Vs (m/s)

0 200 400 600

νs (m/s)

0.00

0.20

0.40

0.60

κa

Surface foundations

Shallowly emb.

σ = 0.55

90% confidence intervals

( ) 0 cos cos

ω

ω

⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ u

s

e e H a

r v
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

The results in Figure E-3 can be contrasted with the 

behavior of a surface foundation, which would have no 

reduction of translational motions and no rotational 

motions when subjected to vertically incident coherent 

shear waves. Transfer function amplitudes in the 

presence of more realistic incident wave fields can be 

estimated at each frequency by the product of the 

transfer function ordinates from the previous section 

(for base slab averaging) and those from this section at 

the corresponding frequency.

Elsabee and Morray (1977) found these transfer 

functions to also be applicable to nonhomogeneous soil 

profiles, provided vs,r is averaged across the 

embedment depth. Mita and Luco (1989) found that 

solutions for circular foundations can be extended to 

square foundations, provided the radius of the 

equivalent cylinder is the average of the radii necessary 

to match the area and moment of inertia of the square 

base.

The analysis procedure described herein has been 

verified against recorded motions from two relatively 

deeply embedded structures with circular foundations 

having e/r = 0.9 and 2.9 (Kim, 2001). Embedment 

effects dominated the kinematic interaction for these 

deeply embedded foundations; for foundations with e/r

< 0.5 Kim (2001) found that the embedment and base 

slab averaging models should be coupled by 

multiplying the respective transfer function ordinates 

from the two models to accurately simulate observed 

transfer functions. 

E.2.3 Application of Transfer Functions to 

Calculation of Foundation Motions

The analysis of free-field motions generally results in 

the specification of a design-basis acceleration response 

spectrum. Sometimes suites of time histories are 

specified that are compatible with this spectrum. The 

question addressed in this section is how this spectrum 

or time history suite should be modified once the 

transfer function amplitude for the site has been 

evaluated using the analysis procedures described 

above. 

When free-field motions are specified only as response 

spectral ordinates, the evaluation of a modified 

response spectrum consistent with the FIM is needed. 

Veletsos and Prasad (1989) evaluated ratios of 

foundation / free-field response spectral ordinates (at 

2% damping) for conditions where the corresponding 

transfer function ordinates could be readily determined. 

The transfer function ordinates and ratios of response 

spectra (RRS) were compared for an input motion with 

Figure E-3 (a) Transfer function amplitudes for embedded cylinders from Day (1978) and Elsabee and Morray (1977) 

along with approximate solution by Elsabee and Morray; (b) Transfer function amplitude model by Elsabee 

and Morray (1977).

024 6

a0=ωr/Vs

0.0

0.2

0.4

0.6

0.8

1.0

1.2

uFIM/ug

Approximation

Halfspace

Finite soil layer

e/r = 1

Translation (a)

0 0.4 0.8 1.2 1.6 2

eω/Vs

0

0.2

0.4

0.6

0.8

1

1.2

uFIM/ug

(b)
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-5

specified power spectral densities and random phase. 

The results indicated that transfer function ordinates 

provide a reasonable estimate of response spectral ratios 

for low frequencies (e.g., < 5 Hz), but that at high 

frequencies (≥ 10 Hz) transfer function ordinates are 

significantly smaller than response spectrum ratios. The 

inconsistency at high frequencies is attributed to the low 

energy content of free-field excitation at high 

frequencies and the saturation of spectral ordinates at 

these frequencies. 

The analytical results of Veletsos and Prasad were 

checked by (1) calculating the transfer function for a 

fixed set of conditions (surface foundation, r = 50 m, Vs

= 250 m/s), (2) using this transfer function to modify a 

set of recorded free-field time histories to 

corresponding foundation-level time histories, and (3) 

evaluating the RRS using the two time histories. 

Representative results of these analyses are presented in 

Figure E-4. The left frame shows results for a time 

Figure E-4 Comparison of transfer function amplitude to ratios of response spectra (RRS) at different damping ratios. 

Left figure applies for time histories in which the spectral response is dominated by high-frequency spikes 

in the accelerogram, whereas in the right figure the response is dictated by lower frequency spikes. 

0.01 0.1 1 10

 Period (s)

0

0.2

0.4

0.6

0.8

1

1.2

Tranfer Function Amplitude, RRS

CAP_fn (Tm = 0.51s)

Transfer Function

RRS, 2% damping

RRS, 5%

RRS, 10%

RRS, 20%

2 4 6 8 10 12

Time (s)

-0.4

-0.2

0

0.2

0.4

Acceleration (g)

Recorded

Filtered

0.01 0.1 1 10

 Period (s)

0

0.2

0.4

0.6

0.8

1

1.2

Tranfer Function Amplitude, RRS

NWH_fn (Tm = 0.70s)

Transfer Function

RRS, 2% damping

RRS, 5%

RRS, 10%

RRS, 20%

2 4 6 8 10 12

Time (s)

-0.8

-0.4

0

0.4

0.8

Acceleration (g)

Recorded

Filtered
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

history in which the spectral response is largely 

controlled by relatively high-frequency components of 

the waveform. The right frame shows results for a time 

history in which the peak response is associated with 

relatively low-frequency pulses. These types of lowfrequency pulses are common for sites located on soft 

soils, but are also found in some sites subjected to 

significant near-fault, forward rupture directivity 

effects. The results suggest that for ordinary ground 

motions, RRS over a wide range of damping ratios can 

be reasonably estimated by transfer function ordinates 

for T > 0.2 s, but that some caution should be exercised 

for soft soil sites and perhaps for near-fault ground 

motions. It should be noted that only a few ground 

motion time histories were used in these analyses, and 

additional research is needed to evaluate the 

relationship between RRS and transfer function 

ordinates as a function of ground motion characteristics 

and damping ratio. 

Based on the above, the following procedure is 

recommended for estimation of RRS from transfer 

function ordinates:

1. For periods > 0.2 s, estimate foundation response 

spectral ordinates as the product of free-field 

response spectral ordinates and the transfer function 

amplitude at the corresponding frequency.

2. For periods < 0.2 s, estimate foundation response 

spectral ordinates as the product of free-field 

response spectral ordinates and the transfer function 

amplitude at 0.2 s.

For structures on very soft soils (i.e., NEHRP Site 

Category E), no reductions of response spectra for 

kinematic interaction should be taken. 

When free-field motions are specified as time histories 

for use in nonlinear time history analyses of structures, 

modified time histories representing the FIM can be 

evaluated as follows:

1. Calculate the Fourier transforms of the free-field 

time histories.

2. Multiply the amplitude of the free-field motions by 

the transfer function amplitude.

3. Use the amplitudes from (2) along with the phase 

angles of the free-field motions, and perform 

reverse Fourier transforms to estimate FIM time 

histories.

4. If needed, a revised response spectrum that 

accounts for kinematic interaction effects could be 

calculated from the FIM time histories.

It should be noted that maintaining the free-field phase 

angles in Step 3 is not strictly correct, especially for 

embedded foundations. If desired, phase shifts of ωe/Vs

(in radians) could be introduced for motions of 

embedded foundations relative to ground surface 

motions. Models for phase adjustment are not available 

for kinematic interaction effects involving surface 

foundations, but the assumption of consistent phase 

should not significantly bias response spectral ordinates 

for estimated FIMs. 

E.2.4 Simplified Procedure for Design

The Kim and Stewart (2003) model for incoherence 

parameter κa (presented in Section E.2.1), along with 

the procedure for converting transfer function ordinates 

to RRS (presented in Section E.2.3), enables the 

development of simplified design charts for kinematic 

interaction effects for non-embedded base slabs 

founded on alluvial soils. A significant simplification 

results from the fact that κa is nearly proportional to vs (as seen in Figure E-2), which per Equation E-1 causes 

dimensionless frequency term to effectively reduce 

to a function of frequency and foundation size (be). This 

is shown below, written for vertically propagating 

waves (αv = 0):

(E-3)

where n1 ≈ 6.5 × 10-4 s/m and n2 is the square root of 

the soil modulus reduction factor, which can be 

estimated as shown in Table E-1. 

Figure E-5 shows the degree of approximation 

associated with taking κa as proportional to vs (using 

Table E-1 Approximate values of n

Peak Ground Acceleration (PGA)

0.10 0.15 0.20 0.30

n 0.90 0.80 0.70 0.65

a ̃

o

2

2 2

0

,

1 1

,2 2

sin

2

22 2

ω

κ α

ωκ ω ω

⎛ ⎞

= + ⎜ ⎟ ⎝ ⎠

≈≈ =

% e

v

sr e

e es e

sr s

b b

a

V b

b b nV b n

V nV n
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-7

proportionality term n1). In this and subsequent figures, 

parameter n2 = 0.65, which is the appropriate value for 

regions of high seismicity. The results for site classes 

C-D are shown for typical shear wave velocities within 

the categories based on borehole compilations. The 

difference between the simplified model and the result 

for individual site classes is small. Figure E-6 shows the 

resulting curves of RRS per Equation E-3 for 

foundations of various sizes. As expected, the kinematic 

interaction effect increases as the foundation size 

increases and as period decreases. 

As with the base slab averaging model for surface 

foundations, simplified design charts for the RRS of 

embedded foundations can also be developed. These 

charts are based on the simplified model of Elsabee and 

Morray (1977) shown in Figure E-3b, but with the RRS 

interpreted from the transfer function amplitude as 

described in Section E.2.3. Figure E-7a presents RRS as 

a function of period in site categories A-D for a 

relatively large embedment depth of 30 ft. As can be 

seen in the figure, embedment effects are negligible at 

practical levels of embedment for firm rock site 

conditions (Site Categories A and B). Accordingly, 

Figure E-7b presents RRS values at three levels of 

embedment (e = 10, 20, and 30 ft) only for Site Classes 

C and D.

Based on the above, the following simplified procedure 

is recommended for analysis of kinematic interaction 

effects:

1. Evaluate effective foundation size , 

where a and b are the foundation dimensions in 

plan view. 

2. Evaluate period-dependant RRS from base slab 

averaging (RRSbsa) using Figure E-6. An approximate equation to the curves in Figure E-6 is presented below:

(E-4)

where be = effective foundation size (from Step 1) 

in feet, and T = period in sec.

3. If the foundation is embedded a depth e from the 

ground surface, evaluate an additional RRS from 

embedment (RRSe) as a function of period due to 

embedment effects using Figure E-7. The equation 

of the curves in Figure E-7 is,

Figure E-5 RRS for foundation with be = 330 ft. 

Simplified model (κa /νs = n1) vs. exact 

solution for κa

0 0.2 0.4 0.6 0.8 1 1.2

Period (s)

0.4

0.5

0.6

0.7

0.8

0.9

1

/ noit adnuoF fr - eef S

R

Rdl ei

S

R

R( gni gar eva bal s esab m

orf asb )

be = 330 ft

C

D

Simplified Model

Figure E-6 RRS from simplified model as function of 

foundation size, be

0 0.2 0.4 0.6 0.8 1 1.2

Period (s)

0.4

0.5

0.6

0.7

0.8

0.9

1

S

R

Rdl eif- eerf/ noit adnuo

F

S

R

R( gni gar ev

a bal s esab m

orf asb )

Simplified Model

be = 65 ft

be = 130 ft

be = 200 ft

be = 330 ft

e b ab =

1.2

1 1

14100

e bsa

eq

b RRS

T

⎛ ⎞ = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ %
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

 larger of 0.453 or 

the value of RRSe at T = 0.2 sec (E-5)

where e = foundation embedment (in feet) and vs,r

= effective strain-degraded shear wave velocity in 

the soil (in ft/s). Factors that can be used to estimate 

vs,r from small-strain shear wave velocity vs are 

given in Table E-1. 

4. Evaluate the product RRSbsa times RRSe to obtain 

the total RRS. The spectral ordinates of the foundation input motion is the product of the free-field 

spectral ordinates and the total RRS.

Limitations associated with application of this approach 

include the following:

• Kinematic interaction effects should be neglected for 

soft clay sites, such as Site Class E. 

• Embedment effects can be neglected for foundations 

embedded in firm rock (Site Classes A and B). 

• The base slab averaging model has the following 

limitations:

a) Underestimates ground motion reduction for 

sites on rock (i.e., use of the above formulation is 

conservative).

b) The model has not been validated for foundations with low in-plane stiffness, buildings with 

large footprint dimensions (> 200 ft), and pilesupported foundations in which the cap and soil 

are not in contact. However, the judgment of the 

project technical team that developed this report 

is that the model can provide a reasonable first 

order estimate of the kinematic interaction effect 

for those conditions. 

c) The model should not be used for structures that 

have both foundations without significant connectivity of lateral elements and flexible floor 

and roof diaphragms. 

E.3 Foundation Damping

Inertia developed in a vibrating structure gives rise to 

base shear, moment, and torsion at the foundation soil 

interface, and these loads in turn cause displacements 

and rotations of the structure relative to the free field. 

These relative displacements and rotations are only 

possible because of compliance in the soil, which can 

significantly contribute to the overall structural 

flexibility in some cases. Moreover, the difference 

between the foundation input motion and free-field 

motion gives rise to energy dissipation via radiation 

damping and hysteretic soil damping, and this energy 

dissipation affects the overall system damping. Since 

these effects are rooted in the structural inertia, they are 

Figure E-7 (a) RRS for foundation embedded to depth e = 30 ft in different site categories; (b) RRS for foundations 

with variable depths in Site Classes C and D.

0 0.4 0.8 1.2 1.6 2

Period, T (s)

0

0.2

0.4

0.6

0.8

1

1.2

Foundation/Free-Field RRS

e = 30 ft

Site Class A

B

C

D

(a)

0 0.4 0.8 1.2 1.6 2

Period, T (s)

0

0.2

0.4

0.6

0.8

1

1.2

Foundation/Free-Field RRS

Site Classes C and D

e = 10 ft

e = 20 ft

e = 30 ft

(b)

C

D

(a) (b)

,

2

cos

⎛ ⎞ π = ≥ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ % e

eq s r

e RRS

T v
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-9

referred to as inertial interaction effects, in contrast to 

the kinematic interaction effects discussed in the prior 

section. 

The ATC-40 and FEMA 356 documents contain 

provisions for evaluating the properties of foundation 

springs (e.g., Sections 10.3 and 10.4 of ATC-40), and 

hence this aspect of inertial interaction is not 

emphasized here. Rather, the focus of this section is on 

the damping component of inertial interaction and the 

contribution of this damping to the overall system 

damping. 

In the SSI literature, foundation stiffness and damping 

effects are often described in terms of an impedance 

function. The impedance function should account for 

the soil stratigraphy and foundation stiffness and 

geometry, and is typically computed using equivalentlinear soil properties appropriate for the in situ dynamic 

shear strains. Impedance functions can be evaluated for 

multiple independent foundation elements, or (more 

commonly) a single 6×6 matrix of impedance functions 

is used to represent the complete foundation (which 

assumes foundation rigidity). 

In the following sub-sections, factors affecting 

foundation impedance functions are described, with an 

emphasis on those factors significantly affecting the 

damping component. The section is concluded with a 

discussion of how system damping ratios can be 

evaluated once factors affecting impedance functions 

and the fixed-based structural damping ratio are known. 

E.3.1 Analysis of Impedance Functions

E.3.1.1 Basic Case

Simplified impedance function solutions are available 

for rigid circular or rectangular foundations located on 

the ground surface and underlain by a uniform, viscoelastic halfspace. A thorough listing of impedance 

functions for these and other foundation shapes is 

provided in Gazetas (1991a, 1991b). A circular 

foundation shape with the above assumptions of 

foundation rigidity and soil uniformity comprise the 

“basic case” for foundation impedance considered here. 

Terms in the complex valued impedance function are 

expressed in the form 

(E-6)

where j denotes either deformation mode x (for 

translation) or θ (for rotation in the vertical plane), ω is 

angular frequency (radians/sec.), a0 is a dimensionless 

frequency defined by a0 = ωr/vs, r = foundation radius, 

vs = soil shear wave velocity, and υ = soil Poisson ratio. 

Foundation radii can be computed separately for 

translational and rotational deformation modes to match 

the area (Af

) and moment of inertia (If) of the actual 

foundation (i.e. , ). There 

are corresponding (a0)x and (a0)θ values as well.

The real stiffness and damping of the translational and 

rotational springs and dashpots are expressed, 

respectively, by

(a)

 (b) (E-7)

where αx, βx, αθ, and βθ express the frequency 

dependence of the impedance terms, and and 

are the static stiffness of a disk on a halfspace, 

(E-8)

where G = soil dynamic shear modulus. Additional 

solutions for Kx and Kθ that take into account the 

foundation geometry in plan are presented in Table 10-2 

of ATC (1996). Presented in Figure E-8 are the 

frequency-dependent values of αx, βx, αθ, and βθ for a 

rigid circular foundation on the surface of a viscoelastic halfspace with soil hysteretic damping ratio βs

(Veletsos and Wei, 1971; Veletsos and Verbic, 1973). 

Validation studies for the above and similar impedance 

function formulations have been conducted by Lin and 

Jennings (1984) and Crouse et al. (1990) for small 

foundations (< 10 ft plan dimension), and by Luco et al. 

(1988), Wong et al. (1988), and DeBarros and Luco 

(1995) for larger scale building foundations (up to 80 ft 

plan dimension). These studies have generally found 

reasonably good agreement between experimental 

observations and analytical predictions, although the fit 

is usually markedly better for rotation vibration modes 

than for translation. The improved fit for rotation likely 

results from the relative ease of identifying impedance 

functions from rotation data as compared to relative 

0 0 k k a ic a jj j = + ( ,) ( ,) υ ω υ

r A x f = π 4 4 r I θ = f π

x =α x x k K = β x x x x

S

K r

c

v

θ =αθ θ k K θ θ

θ θ = β

S

K r

c

v

Kx Kθ

8

2

K Gr x x υ = − ( )

8 3

3 1

K Gr θ θ

υ = −
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

foundation/free-field translations, which have weaker 

signals. 

The above solutions for rigid, circular foundations on a 

halfspace can provide reasonable estimates of 

foundation impedance in many cases. However, the 

potentially significant effects of non-uniform soil 

profiles, embedded foundations, non-circular 

foundation shapes, and flexible foundations should be 

accounted for in some cases. The following sections 

discuss the effects of these factors on the damping 

component of impedance functions.

E.3.1.2 Nonuniform Soil Profiles

Gazetas (1991b) provides solutions for the impedance 

of rigid foundations overlying soil for which the shear 

stiffness increases with depth according to prescribed 

functions. The damping components of these solutions 

are plotted in Figure E-9 in terms of the frequencydependent and dimensionless βx and βθ terms; actual 

dashpot coefficients can be computed using these terms 

in Equations E-7a and b. Also plotted for comparative 

purposes are the solutions for a halfspace presented 

previously in Figure E-8. Note that the damping values 

for non-uniform profiles are plotted for a zero hysteretic 

damping condition (radiation damping only) and that 

the normalizing shear modulus and shear wave velocity 

are the values at the ground surface (G0 and νs0, 

respectively). 

From Figure E-9, radiation damping in translation for a 

nonuniform profile is seen to be less than that for a 

halfspace at low frequencies. For rotation, a small 

reduction can occur at low frequencies, but the effect is 

less significant than for translation. At large 

frequencies, the radiation damping for nonuniform 

profiles exceeds that for the halfspace. 

The low-frequency reduction in damping is due to 

reflections of body waves emanating from the 

foundation; the frequency dependence of the reduction 

is related to the depth over which the shear modulus 

increases relative to wavelength. For short-period (and 

short-wavelength) body waves, the nonuniform soil 

medium is “seen” as being effectively uniform, whereas 

Figure E-8 Foundation stiffness and damping factors for elastic and viscoelastic halfspaces (υ = 0.4). After Veletsos 

and Verbic, 1973.

0 2 4 6 8 10

0

1

2

βx

0 2 4 6 8 10

a0=ωr/vs

0

1

2

βθ

0 2 4 6 8 10

a0

-2

-1

0

1

αx

βs=0

βs=0.1

0 2 4 6 8 10

a0

-2

-1

0

1

αθ

TRANSLATION ROCKING

a0=ωr/vs
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-11

long-period waves (with long wavelengths) “see” a 

much more nonuniform medium and wave transmission 

into the medium is impeded. The increase of radiation 

damping at high frequencies is due to the higher νs of 

the nonuniform profiles at depth as compared to the 

velocity of the halfspace model (for which νs was taken 

as νs0). 

The above effects on low-frequency damping can be 

contrasted with the effect of soil nonuniformity on 

foundation stiffness. A number of researchers (e.g., 

Roesset, 1980; Gazetas, 1991b; Stewart et al., 2003) 

have suggested that effective foundation stiffnesses can 

be computed from an average vs value over a specified 

depth range. For the common case in which soil 

stiffness increases with depth, this averaging will result 

in an effective vs value that is larger than vs0. The use of 

this averaged vs would be unconservative in the case of 

damping, as the low-frequency damping is 

overpredicted by a halfspace model even when the 

halfspace velocity is taken as vs0. Thus, different 

effective velocities of nonuniform soil profiles should 

be used for calculations of foundation stiffness and 

damping. 

Figure E-9 Foundation damping factors for halfspace with and without hysteretic damping (Veletsos and Verbic, 

1973) and for soil profiles with indicated shear modulus profiles and no hysteretic damping (Gazetas, 

1991b).

012

0.0

0.5

1.0

012

0.00

0.15

0.30

TRANSLATION ROCKING

Half., β=0.1

Half., β=0

α=0.025

α=0.2

n

3

=0.5

n=1

α=0.025

α=0.23

Half., β=0.1

Half., β=0

02468

G(z)/G0

6

4

2

0

r/ z

α=0.23

α

=

0.0

2

5

02468

G(z)/G0

6

4

2

0

r/ z

n=1

2/3 1/2

z

G(z)

υ, ρ

2r

a0=ωr/vs0 a0=ωr/vs0

βx

βθ
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-12 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

For the case of a finite soil layer overlying a much 

stiffer material, the key issue is a lack of radiation 

damping at periods greater than the fundamental period 

of the finite soil layer, Ts = 4H/vs. Halfspace damping 

ratios can be used for periods less than the soil layer 

period. Above this period in materials with hysteretic 

damping, Elsabee and Morray (1977) developed the 

following damping recommendations: 

for a0/a01 ≤ 1 (E-9)

for a0/a01 ≤ vp/vs (E-10)

where a01 = 0.5πr/H, r = foundation radius, and 

H = finite soil layer thickness. 

In terms of practical application of the above results, the 

following observations are noted:

• For translational damping, profile non-uniformity is 

not significant for a0=ωr/vs > 1. Case history studies 

suggest that inertial soil-structure interaction is 

generally not important for h/(vsT) < 0.1 (Stewart et 

al., 1999). Hence, for sites where SSI is important, 

profile non-uniformity need not be considered if h/r

< 2πh/(vsT). The value on the right hand side of the 

inequality will generally be more than 2/3 for cases 

where inertial SSI is important, which is larger than 

the aspect ratios for many short-period buildings. 

Accordingly, it is often justified to treat the 

nonuniform soil as a halfspace, taking the halfspace 

velocity as the in situ value immediately below the 

foundation. Note that the above inequality to allow 

profile non-uniformity effects to be neglected can be 

re-written as vsT/r < 2π.

• Rotational damping for a non-uniform profile can 

generally be reasonably well estimated by a 

halfspace model, with the halfspace velocity taken 

as the in situ value immediately below the 

foundation.

• The use of halfspace models is unconservative for 

sites with a finite soil layer overlying a very stiff 

layer, if the structural system period is greater than 

the soil layer period. Alternative dashpot 

coefficients for such cases can be developed using 

Equation E-10. 

E.3.1.3 Embedded Foundations

Foundation “embedment” refers to a foundation base 

slab that is positioned at a lower elevation than the 

surrounding ground, which will usually occur when 

buildings have a basement. The impedance of 

embedded foundations differs from that of shallow 

foundations in several important ways. First, the static 

stiffness of embedded foundations is increased, which is 

accounted for with the embedment factors given in 

Table 10.3 of ATC-40 (ATC, 1996). For circular 

foundations, these embedment terms are as follows: 

(E-11)

where e = embedment depth. The second important 

difference between embedded and surface foundations 

is that embedded foundations can produce much larger 

damping due to the greater foundation-soil contact area. 

An approximate and generally conservative approach 

for estimating the damping of embedded foundations 

consists of using the modified static stiffness terms 

from Equation E-11 coupled with the dynamic 

modifiers for a surface foundation in Figure E-8. This 

approach has been found to provide reasonable 

estimates of observed foundation damping in actual 

structures for embedment ratios e/ru < 0.5 (Stewart et 

al., 1999). As short-period structures are seldom deeply 

embedded, this approximate approach will often suffice 

for practical applications. For more deeply embedded 

foundations, alternative formulations can be used such 

as Bielak (1975) or Apsel and Luco (1987). However, 

caution should also be exercised in the application of 

these approaches for embedded foundations with poor 

quality backfill against basement walls. For such 

foundations, gapping is likely and impedance functions 

should probably be formulated using the shallow 

foundation approach noted previously.

E.3.1.4 Foundation Shape

Impedance functions for foundations of arbitrary shape 

are commonly analyzed as equivalent circular mats 

(BSSC, 2001). As described previously, an equivalent 

( )( )

0 01

2

0 01

0.65

1 12 x

a a

a a

β β

β

≈

− −

( )( )

0 01 0

2 2

0 01 0

0.35 0.50

1 12 1

aa a

a a a

β β θ

β

≈ ≤

− − + ( )

( )

2 1

3

1 2

x x E u

E

e K K

r

e K K

r

θ θ

θ

⎛ ⎞

= + ⎜ ⎟ ⎝ ⎠

⎛ ⎞

= + ⎜ ⎟ ⎝ ⎠
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-13

radius for translational stiffness is derived by equating 

the areas of the mats, while an equivalent radius for 

rotational stiffness is derived by equating the moments 

of inertia of the mats. The issue addressed in this 

section is the adequacy of this assumption for oblong 

foundations. 

Combining a number of analytical impedance function 

solutions from the literature for foundations of arbitrary 

shape, Dobry and Gazetas (1986) found that the use of 

equivalent circular mats is acceptable for aspect ratios 

less than 4:1, with the notable exception of dashpot 

coefficients in the rotation mode. As shown in 

Figure E-10, dimensionless radiation damping 

coefficients crx and cry (for longitudinal and transverse 

rotations, respectively) are seen to be underestimated by 

the equivalent disk assumption at low frequencies. This 

is a consequence of the tendency for rotational 

vibrations to be dissipated into the soil primarily via the 

ends of the foundation. Hence, as aspect ratio increases, 

the two ends act increasingly as independent wave 

sources with reduced destructive interference between 

waves emanating from the foundation. For the case of 

longitudinal rotations, damping can be underpredicted 

by a factor of two or more for aspect ratios of L/B ≈ 4. 

For higher frequencies (a0 > 3-4, not shown in figure), 

the results for the various aspect ratios converge to crx, 

cry = ~1. This occurs because these high frequency 

waves have short wavelengths relative to the foundation 

dimension regardless of L/B, so destructive interference 

between the waves is small in all cases. 

The use of dashpot coefficients for disk-shaped 

foundations can be used to provide conservative (lowerbound) estimates of the damping of oblong foundations. 

This approximation may be sufficient for many 

practical applications, especially given the relatively 

small influence of damping from rotations on system 

damping (damping from horizontal vibrations often 

contribute more significantly, see Section E.3.2). If 

more refined analysis of rotational damping is needed, 

rotational radiation dashpot coefficients for oblong, 

non-circular foundations can be calculated using 

procedures given in Gazetas (1991a, b).

E.3.1.5 Foundation Flexibility

This section addresses flexibility in the foundation 

structural system (i.e., the base mat, or assemblage of a 

base mat and grade beams/footings). The foundation 

flexibility referred to here is not associated with the 

soil. 

Impedance functions for flexible circular foundation 

slabs supporting shear walls have been evaluated for a 

number of wall configurations, including: (1) rigid core 

walls (Iguchi and Luco, 1982), (2) thin perimeter walls 

(Liou and Huang, 1994), and (3) rigid concentric 

interior and perimeter walls (Riggs and Waas, 1985). 

These studies focused on the effects of foundation 

flexibility on rotation impedance; the horizontal 

impedance of flexible and rigid foundations are similar 

(Liou and Huang, 1994). Foundation flexibility effects 

on rotation impedance were found to be most 

significant for a rigid central core with no perimeter 

walls. For this case, the flexible foundation has 

significantly less stiffness and damping than the rigid 

foundation. The reductions are most significant for 

narrow central cores and large deviations between soil 

and foundation slab rigidity. 

Significant additional work remains to be done on 

foundation flexibility effects on impedance functions 

because the existing research generally has investigated 

wall/slab configurations that are seldom encountered in 

practice for building structures. Nonetheless, based on 

the available studies and engineering judgment, the 

following preliminary recommendations are provided:

1. The rigid foundation assumption is probably generally acceptable for the analysis of damping associated with horizontal vibrations of reasonably stiff, 

inter-connected foundation systems.

2. For buildings with continuous shear walls or braced 

frames around the building perimeter, and continuous footing or mat foundations beneath these walls, 

the rigid foundation approximation can likely be 

used to provide a reasonable estimate of damping 

from rotation vibrations. In this case, the effective 

foundation radius (rθ) would be calculated using the 

full building dimensions. This recommendation 

also applies if continuous basement walls are 

present around the building perimeter. This case is 

referred to as stiff rotational coupling. 

3. For buildings with a core of shear walls within the 

building, but no shear walls outside of this core, a 

conservative estimate of foundation damping can 

be obtained by calculating the effective foundation 

radius (rθ) using the dimensions of the wall foundations (which, in this case, would be smaller than the 

overall building plan dimensions). This is an example of soft rotational coupling between the shear 

walls and other load bearing elements. 
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-14 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

4. For buildings with distributed shear walls at various 

locations around the building plan, the key issues 

are (1) the rotational stiffness of the building system 

as a whole (i.e., does the building tend to rotate as a 

single rigid block due to significant rotational stiffness coupling between adjacent elements, or do 

individual vertical components such as shear walls 

rotate independently of each other?), and (2) the 

degree to which destructive interference occurs 

between waves emanating from rotation of distinct 

foundation components. 

Figure E-10 Dashpot coefficients for radiation damping vs. normalized frequency for different foundation shapes (after 

Dobry and Gazetas, 1986).

0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

c yr c = θ y, (β (/) 0

=

ρ

v aL I

y

)

(a) rocking around x-axis

(b) rocking around y-axis

L/B > 10

L/B = 5

range for L/B = 1 - 2

and circles

L/B = 4-5

L/B = 3

L/B = 2

range for L/B = 1

and circles

Footing

2L

2B

L/B→ ∞

y

x

 3.4v s

π(1- ν)

v a = 

a0=ωB/vs

c xr c = θ x, (β (/) 0

=

ρ

v aL I

x

)
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-15

In practice, it may sometimes be difficult to decide on 

the degree of rotational coupling between foundation 

elements. However, pushover analyses of the building 

with foundation springs utilized below foundation 

elements incorporate rotational coupling between 

foundation elements in a natural way. Hence, the results 

of such analyses can be used to infer the effective 

foundation size associated with the building’s rotational 

impedance. This process is described in the following. 

The derivation begins with the relationship between 

period lengthening and foundation spring stiffness 

values by Veletsos and Meek (1974):

(E-12)

In Equation E-12, the following quantities are known or 

can be estimated reliably:

• T is the fixed base first mode period, and can be 

evaluated from the model of the structure utilized in 

pushover analyses, but with foundation spring 

stiffnesses set to infinity. 

• is the flexible base first mode period, and can be 

evaluated from the model of the structure utilized in 

pushover analyses including foundation springs. The 

foundation spring stiffness should reflect straindegraded soil properties. 

• Stiffness parameter K*

fixed is the stiffness of the 

fixed-base structure, and can be evaluated as 

(E-13)

where M* is the effective mass for the first mode 

calculated as the total mass times the effective mass 

coefficient (see ATC-40, Equation 8-21). 

• Foundation stiffness parameter kx represents the 

horizontal stiffness of the foundation system, and 

can be evaluated as described previously (Sections 

Sections E.3.1.1 – E.3.1.3). For the present 

application, a good approximation of kx is Kx. 

• Height h is the effective structure height taken as the 

full height of the building for one-story structures, 

and as the vertical distance from the foundation to 

the centroid of the first mode shape for multi-story 

structures. In the latter case, h can often be well 

approximated as 70% of the total structure height.

Equation E-12 can then be re-arranged to estimate Kθ as 

follows:

(E-14)

In the above, it has been assumed that ku ≈ Ku and kθ ≈

Kθ, which is generally a reasonable approximation. The 

evaluation of an effective foundation radius from Kθ

can be accomplished using Equation E-8, with the 

following result:

(E-15)

The value rθ will decrease as the degree of rotational 

coupling decreases. For very stiff rotational coupling rθ

will approach the value that would be calculated from 

the moment of inertia derived from the full foundation 

dimension ( ). 

A potential complication to the above may occur when 

foundations are closely spaced, and destructive 

interference occurs between waves emanating from 

adjacent foundation elements. If this occurred, the 

above formulation would be unconservative. 

Unfortunately, this topic has not been researched, and 

thus what footing separation distances constitute 

“close” and “widely spaced” is unknown, which in turn 

precludes the development of recommendations for the 

analysis of rotation damping for distributed walls. 

Finally, it should be noted that for buildings with only 

moment resisting frames (no walls or braced frames), 

foundation rotation is not likely to be significant, and 

hence foundation flexibility effects on rotation damping 

are also likely insignificant. 

E.3.2 Analysis of System Damping Ratios

The effect of foundation flexibility on the response of a 

structure can be visualized using the single-degree-offreedom oscillator depicted in Figure E-11. In the 

figure, displacement ug denotes the free-field ground 

motion, uf

 denotes foundation translation relative to the 

* *2

1 fixed fixed

x

T K Kh

T k k θ

=+ +

%

T%

2

* * 2 K M fixed T

⎛ ⎞ π = ⎜ ⎟ ⎝ ⎠

* 2

2 *

1

fixed

fixed

x

K h

K

T K

T K

θ =

⎛ ⎞ ⎜ ⎟ − −

⎝ ⎠

%

( ) 1

3 1 3

8

θ

θ

−υ = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠K

r

G

4 4 f r I θ = π
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-16 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

free-field, θ denotes foundation rotation, and u denotes 

displacement resulting from deformation within the 

structure with stiffness K*

fixed and dashpot coefficient c. 

SSI effects are manifested by a lengthening of the 

building period from the fixed-base case (T) to the 

flexible-base case ( ), and by a change in the damping 

ratio (from βi

 to βo). These effects have been evaluated 

as closed-form expressions for the simple oscillator 

configuration shown in Figure E-11. In this case, the 

impedance function is represented by complex-valued 

terms for the translation ( ) and vertical plane rotation 

( ) foundation vibration modes. A vertical foundation 

degree-of-freedom also exists (impedance term ), 

but does not affect or βo.

The flexible-base damping ratio of the oscillator has 

contributions from the viscous damping in the structure 

as well as radiation and hysteretic damping in the 

foundation. Jennings and Bielak (1973), Bielak (1975, 

1976) and Veletsos and Nair (1975) expressed the 

flexible-base damping βo as

(E-16)

where βf

 is referred to as the foundation damping and 

represents the damping contributions from foundationsoil interaction (with hysteretic and radiation 

components), and and Teq represent the flexibleand fixed-base period of the structure accounting for the 

effects of yielding in the superstructure. From 

Equation E-16, it can be readily seen that the flexiblebase damping, which is the damping ratio for which 

response spectra should be evaluated, is a function of 

fixed-base damping (βi

), the period lengthening ratio 

( ), and βf

. Parameter βi is generally assumed 

to be 5%. The period lengthening can be evaluated 

using the structural model used in pushover analyses as 

follows:

1. Evaluate the first-mode vibration period of the 

model, including foundation springs. This period is 

. The period can be calculated using initial stiffness values for the structure and strain-degraded 

soil stiffness values. 

2. Evaluate the first-mode vibration period of the 

model with the foundation springs removed (or 

their stiffness and capacity set to infinity). This 

period is T. As before, this period should correspond to pre-yield conditions. 

3. Calculate the ratio , which is the period 

lengthening under small-deformation (elastic) conditions. 

4. Calculate using the following equation:

(E-17)

where μsys is the peak system ductility (including 

structure and soil effects). 

With βi

 and known, the estimation of βo reduces to an analysis of foundation damping βf

. 

Graphical solutions and closed-form expressions for βf

Figure E-11 Oscillator model for analysis of inertial interaction under lateral excitation.

kx kθ

θ

ug uf hθ u

m

h K*

fixed, c

T%

x k

θ k

v k

T%

( ) 0 3

i

f

T T eq eq

β β β = + %

Teq

%

/ %T T eq eq

T%

/ T T %

/ %T T eq eq

0.5 2

1 1 1 eq

eq sys

T T

T T μ

⎧ ⎡ ⎤⎫ ⎪ ⎛ ⎞ ⎪ =+ − ⎢ ⎥ ⎨ ⎜ ⎟ ⎬ ⎢⎝ ⎠ ⎥ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦

% %

/ %T T eq eq
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-17

are available for the simple case of a circular foundation 

with radius r on a uniform halfspace with velocity vs

and hysteretic damping ratio βs (Veletsos and Nair, 

1975). The expression for βf

 given by Veletsos and 

Nair is reproduced below,

(E-18)

where , , 

, , and

The first term within the brackets in Equation E-18 is 

related to damping from foundation vibration in 

translation whereas the second term is related to 

foundation rotation. To develop approximate solutions 

for βf

 for non-circular foundations, Equation E-18 can 

be implemented with the radius and a0 values in the 

first term taken as rx and (a0)x, respectively, and in the 

second term as rθ and (a0)θ. 

Parameters σ and γ in Equation E-18 represent the ratio 

of the soil-to-structure stiffness and structure-to-soil 

mass, respectively. Most conventional short-period 

building structures have σ <10 and γ ≈ 0.1 to 0.2 [a 

representative value of γ = 0.15 is recommended by 

Veletsos and Meek (1974)]. 

Due to the availability of these βf

 formulations for rigid 

circular foundations on a halfspace, it is convenient to 

idealize actual foundation and site conditions in terms 

of representative values of velocity and foundation 

radius. As described in Section E.3.1, this can generally 

be accomplished by taking the representative site shear 

wave velocity as the soil velocity immediately beneath 

the foundation (vs0), and by calculating effective 

foundation radii for translational and vertical plane 

rotation vibration modes (i.e. , 

). As noted in Section E.3.1.5, special 

consideration may be required for oblong foundations 

and for the analysis of rθ when shear walls or braced 

frames are distributed across the foundation plan.

Figure E-12 presents a customization of the Veletsos 

and Nair (1975) solution for βf

 in which different ratios 

of rθ/rx are used (the original solution applied for true 

circular foundations in which rθ/rx = 1.0) for three fixed 

values of h/rθ. Note that terms vs,r and Teq are used in 

the labeling of the horizontal axis to emphasize that 

strain-reduced shear wave velocities (evaluated from 

small-strain shear wave velocities using Table E-1) and 

ductility-reduced periods should be used in the analysis 

of βf

. Figure E-13 presents an identical set of plot to 

Figure E-12, but for a shallowly embedded foundation 

using the simplified approach for estimating 

embedment effects discussed in Section E.3.1.3. 

In Figure E-12, βf

 is seen to increase with h/(vs,rTeq)

and to decrease with h/rθ. The decrease of βf

 with h/rθ

indicates that lateral movements of the foundation 

(which dominate at low h/rθ) dissipate energy into soil 

more efficiently than foundation rotation (which 

dominates at high h/rθ). For a given h/rθ, the suites of 

curves within each frame indicate that βf

 increases with 

decreasing rθ /rx for h/(vs,rTeq) < ~ 0.2. This occurs 

because decreasing rθ/rx implies increasing foundation 

area (rx), which provides additional damping from 

translational vibration. Note also the significantly 

higher damping when hysteretic damping is included 

(βs=0.1) as opposed to radiation damping only (βs =0). 

Finally, a comparison of βf

 in Figures E-12 and E-13 

indicates that additional foundation damping occurs for 

embedded foundations, as expected. 

The above analysis procedure for βf

 has been found to 

reproduce reasonably accurately SSI effects on firstmode vibration properties of actual structures, as 

inferred from system identification analyses of recorded 

motions (Stewart et al., 1999). These case history 

studies revealed that the single most important 

parameter controlling the significance of inertial 

interaction is h/(vs,rTeq), and that inertial SSI effects are 

generally small for h/(vs,rTeq) < 0.1. This condition 

occurs for flexible structures such as moment frame 

buildings located on competent soil or rock. 

Conversely, SSI effects tend to be significant for stiff 

structures such as shear wall or braced frame buildings, 

particularly when located on soft soil. 

To simplify the evaluation of foundation damping ratios 

in engineering practice, the fact that both βf and 

 are strongly dependent on h/(vs,rTeq) is 

( )

( )

( )

( )

3 4

3

2

2

0 0

2

2 31

eq

f

eq

x

x x

T

T

r

ia ia h

θ

θθ θ

π γ β

σ

υ β υ ββ

αα β α α β

⎛ ⎞

= × ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

⎡ ⎤ − − ⎢ ⎥ +

+ + ⎣ ⎦

%

S σ = vT h ( ) 2 γ ρπ = m r h

( )

2

0 1( ) α α= −x Tf% f K 0 = x f m (2π )

( )

2

0 1( )

1 eq

x ox

T f

ia β

α β = − −

%

r A x f = π

4 4 r I θ = f π

/ %T T eq eq
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-18 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure E-12 Foundation damping for single degree-of-freedom structures on elastic halfspace with various aspect ratios 

(h/rθ) and foundation shapes (rθ/rx), non-embedded foundation case (e/rx = 0).

0 0.1 0.2 0.3

h/(vs,rTeq)

0

5

10

15

20

25

βf(%)

h/rθ = 0.5, e/rx = 0

βs=0

βs=0.1

rθ/rx=0.5

1

2

0 0.1 0.2 0.3

h/(vs,rTeq)

0

5

10

15

20

25

βf(%)

h/rθ = 1, e/rx = 0

βs=0

βs=0.1

rθ/rx=0.5

1 2

0 0.1 0.2 0.3

h/(vs,r

Teq)

0

5

10

15

20

25

βf(%)

h/rθ = 2, e/rx = 0

βs=0

βs=0.1

rθ/rx=0.5, 1, 2
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-19

Figure E-13 Foundation damping for single degree-of-freedom structures on elastic halfspace with various aspect ratios 

(h/rθ) and foundation shapes (rθ/rx), small foundation embedment case (e/rx = 0.5).

0 0.1 0.2 0.3

h/(vs,rTeq)

0

5

10

15

20

25

βf(%)

h/rθ = 0.5, e/rx = 0.5

βs=0

βs=0.1

rθ/rx=0.5

1

2

0 0.1 0.2 0.3

h/(vs,rTeq)

0

5

10

15

20

25

βf(%)

h/rθ = 1, e/rx = 0.5

βs=0

βs=0.1

rθ/rx=0.5 1 2

0 0.1 0.2 0.3

h/(vs,r

Teq)

0

5

10

15

20

25

βf(%)

h/rθ = 2, e/rx = 0.5

βs=0

βs=0.1

rθ/rx=0.5, 1, 2
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-20 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

leveraged to generate curves relating βf to 

without the use of h/(vs,rTeq). The relationship 

was developed by also assuming equi-dimensional 

foundations (rθ/rx = 1.0). Although results are shown 

for significant hysteretic soil damping (βs=0.1) and zero 

hysteretic damping (βs=0), use of the βs=0 results is 

recommended because ductility in soil springs is 

already included as part of structural pushover analysis. 

The result is shown in Figure E-14, and requires the 

user only to know (easily obtained from a 

structural model, as described above) as well as h/rθ, 

and e/rx. The damping ratios in Figure E-14 are 

conservative for rθ/rx < 1.0, which is generally the case 

for buildings.

Another point that should be made in connection with 

the use of Figure E-14 is that the foundation spring 

stiffnesses used in the analysis of are based on 

average shear wave velocities to a depth of 

approximately rx, whereas the velocity that should be 

used for the analysis of foundation damping at a nonuniform site is vs0 (the velocity immediately below the 

foundation, which is typically smaller than the average 

vs over a depth range). The fact that βf is evaluated in 

terms of therefore introduces a bias, although 

the bias will generally result in underprediction of βf

, 

which is conservative and thus acceptable for a 

simplified design procedure. 

Flexible base damping βo can actually increase or 

decrease relative to βi

 depending on and 

foundation damping βf

. The effect of the change in 

damping from βi

 to βo on spectral acceleration can be 

estimated using the procedures in Section 6.3. 

E.3.3 Simplified Procedure for Design

1. Evaluate the linear periods for the structural model 

assuming a fixed base, T, and a flexible-base using 

appropriate foundation modeling assumptions, . 

Guidelines for the evaluation of soil spring stiffnesses are provided in FEMA 356 and ATC-40.

2. Calculate the effective structural stiffness for fixed 

base conditions, K*

max using Equation E-13. 

3. Determine the equivalent foundation radius for 

translation, , where Af is the area of 

the foundation footprint if the foundation components are inter-connected laterally.

4. Calculate the translational stiffness of the foundation, Kx. This can be evaluated using the proceFigure E-14 Foundation damping factor βf

 expressed as a function of period lengthening for different 

building aspect ratios (h/rθ) and embedment ratios (e/rx). 

1 1.5 2

Period Lengthening, Teq/Teq

0

10

20

30

Foundation Damping, βf (%)

e/rx = 0

βs = 0.1

βs = 0

h/rθ = 0.5

1.0

2.0

∼ 1 1.5 2

Period Lengthening, Teq/Teq

0

10

20

30

Foundation Damping, βf (%)

e/rx = 0.5

βs = 0.1

βs = 0

h/rθ = 0.5

1.0

2.0

∼

/ %T T eq eq

/ %T T eq eq

/ %T T eq eq

/ %T T eq eq

/ %T T eq eq

/ %T T eq eq

T%

r A x f = π
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-21

dures in FEMA 356 (Chap. 4) or ATC-40 (Chap. 

10). For many applications, Kx can be estimated 

using Equations E-8 and E-11. 

5. Calculate the equivalent foundation radius for rotation, rθ, using Equations E-14 and E-15. 

6. Determine the foundation embedment, e, if applicable. 

7. Estimate the effective period lengthening ratio, 

, for the nonlinear structure using 

Equation E-17.

8. Evaluate the initial fixed base damping ratio for the 

structure (βi

), which is often taken as 5%.

9. Using Figure E-14, determine foundation damping 

(βf

) based on , e/rx, and h/rθ. Intermediate 

values may be interpolated from these figures. An 

approximation to those curves is given by the following for the case without hysteretic soil damping 

(denoted βs = 0):

(E-19)

where βf

 is in percent and

(E-20)

(E-21)

(E-22)

The above equations are most applicable for 

 < 1.5, and generally provide conservative 

(low) damping estimates for higher .

10. Evaluate the flexible base damping ratio (β0) from 

βi, βf, and using Equation E-16.

11. Evaluate the effect on spectral ordinates of the 

change in damping ratio from βi

 to β0 in accordance 

with Section 6.3 then modify the spectrum for freefield motion or that for the foundation input motion 

if kinematic effects are being included.

The change in spectral ordinate computed above can be 

combined with the change in spectral ordinate from 

kinematic interaction. 

Limitations on the damping analysis described above 

include the following:

• The analysis approach should not be used when 

shear walls or braced frames are spaced sufficiently 

closely that waves emanating from distinct 

foundation elements will destructively interfere with 

each other across the period range of interest. 

Further discussion is presented in Section E.3.1.5. 

• The analysis can be conservative (under-predicting 

the damping) when foundation aspect ratios exceed 

about 2:1. Further discussion is presented in 

Section E.3.1.4. 

• The analysis is conservative when foundations are 

deeply embedded, e/rx > 0.5. Further discussion is 

presented in Section E.3.1.3.

• The analysis is unconservative (over-predicting the 

damping) if vsT/rx > 2π (where vs = average shear 

wave velocity to a depth of about rx) and the 

foundation soils have significant increases of shear 

stiffness with depth. Further discussion is presented 

in Section E.3.1.2. 

• The analysis is unconservative if the foundation soil 

profile consists of a soil layer overlying a very stiff 

material (i.e., there is a pronounced impedance 

contrast within the soil profile), and if the system 

period is greater than the first-mode period of the 

layer. Further discussion is presented in 

Section E.3.1.2. 

E.4 References

ASCE, 2000, Prestandard and Commentary for the 

Seismic Rehabilitation of Buildings, FEMA 356 

Report, prepared by the American Society of Civil 

Engineers for the Federal Emergency Management 

Agency, Washington, D.C.

ATC, 1996, Seismic Evaluation and Retrofit of Concrete 

Buildings, ATC-40 Report, funded by the Seismic 

Safety Commission, published by the Applied 

Technology Council, Redwood City, California.

Apsel, R.J. and Luco, J.E., 1987, “Impedance functions 

for foundations embedded in a layered medium: an 

integral equation approach,” J. Earthquake Engrg. 

Struct. Dynamics, Vol. 15, No. 2, pp. 213-231.

Bielak, J., 1975, “Dynamic behavior of structures with 

embedded foundations,” J. Earthquake Engrg. 

Struct. Dynamics, Vol. 3, pp. 259-274.

/ %

eff eff T T

/ %T T eq eq

2

1 2 1 1 eq eq

f

eq eq

T T

a a

T T β

⎛ ⎞⎛ ⎞

= −+ − ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

% %

a c hr 1 = − e exp 4.7 1.6 / ( θ )

a c hr 2 = − e ⎡ ⎤ 25ln / 16 ( θ ) ⎣ ⎦

c er e x = + 1.5 / 1 ( )

/ %

eff eff T T

/ %

eff eff T T

/ %

eff eff T T
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-22 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Bielak, J., 1976, “Modal analysis for building-soil interaction,” J. Engrg. Mech., Vol. 102, American Society of Civil Engineers, pp. 771-786.

BSSC, 2001, NEHRP Recommended Provisions for 

Seismic Regulations for New Buildings and Other 

Structures, FEMA 368 Report, prepared by the 

Building Seismic Safety Council for the Federal 

Emergency Management Agency, Washington, 

D.C.

Crouse, C.B., Hushmand, B., Luco, J.E., and Wong, 

H.L., 1990, “Foundation impedance functions: Theory versus Experiment,” J. Geotech. Engrg., 

Vol. 116, No. 3, American Society of Civil Engineers, pp. 432-449.

Day, S.M.,1978, “Seismic response of embedded foundations,” Proc. ASCE Convention, Chicago, Illinois, October, Preprint No. 3450.

de Barros, F.C.P. and Luco, J.E., 1995, “Identification 

of foundation impedance functions and soil properties from vibration tests of the Hualien containment 

model,” J. Soil Dyn. Earthquake Eng., Vol. 14, 

pp. 229-248.

Dobry, R. and Gazetas, G., 1986, “Dynamic response of 

arbitrarily shaped foundations,” J. Geotech. Engrg., 

Vol. 112, No. 2, American Society of Civil Engineers, pp. 109-135.

Elsabee, F. and Morray, J.P., 1977, Dynamic Behavior 

of Embedded Foundations, Rpt. No. R77-33, Dept. 

of Civil Engrg., MIT, Cambridge, Massachusetts.

Gazetas, G., 1991a, “Formulas and charts for impedances of surface and embedded foundations,” J. 

Geotech. Engrg., Vol. 117, No. 9, pp. 1363-1381.

Gazetas, G., 1991b, Foundation Vibrations, Foundation 

Engineering Handbook, Chapter 15, H.-Y. Fang, 

ed., 2nd Edition, Chapman and Hall, New York, 

New York.

Iguchi, M. and Luco, J.E., 1982, “Vibration of flexible 

plate on viscoelastic medium,” J. Engrg. Mech., 

Vol. 108, No. 6, American Society of Civil Engineers, pp. 1103-1120.

Jennings, P.C. and Bielak, J., 1973, “Dynamics of building-soil interaction,” Bull. Seism. Soc. Am., Vol. 63, 

pp. 9-48.

Kim, S., 2001, Calibration of Simple Models for Seismic Soil Structure Interaction from Field Performance Data, Ph.D. Dissertation, University of 

California, Los Angeles.

Kim, S. and Stewart, J.P., 2003, “Kinematic soil-structure interaction from strong motion recordings,” J. 

Geotech. & Geoenv. Engrg., Vol. 129, No. 4, 

American Society of Civil Engineers, pp. 323-335. 

Lin, A.N. and Jennings, P.C., 1984, “Effect of embedment on foundation-soil impedances,” J. Engrg. 

Mech., Vol. 110, No. 7, American Society of Civil 

Engineers, pp. 1060-1075.

Liou, G.-S. and Huang, P.-H., 1994, “Effect of flexibility on impedance functions for circular foundations,” J. Engrg. Mech., Vol. 120, No. 7, American 

Society of Civil Engineers, pp. 1429-1446.

Luco, J.E., Trifunac, M.D., and Wong, H.L., 1988, “Isolation of soil-structure interaction effects by fullscale forced vibration tests,” J. Earthquake Engrg. 

Struct. Dynamics, Vol. 16, No. 1, pp. 1-21.

Mita, A. and Luco, J.E., 1989, “Dynamic response of a 

square foundation embedded in an elastic halfspace,” J. Soil Dynamics and Earthquake Engrg., 

Vol. 8, No. 2, pp. 54-67.

Riggs, H.R. and Waas, G., 1985, “Influence of foundation flexibility on soil-structure interaction,” J. 

Earthquake Engrg. Struct. Dynamics, Vol. 13, No. 

5, pp. 597-615.

Roesset, J.M., 1980, “A review of soil-structure interaction,” in Soil-Structure Interaction: The Status of 

Current Analysis Methods and Research, J.J. 

Johnson, ed., NUREG/CR-1780 Report and UCRL53011 Report, U.S. Nuclear Regulatory Com., 

Washington D.C., and Lawrence Livermore Lab., 

Livermore, California.

Stewart, J.P., Seed, R.B., and Fenves, G.L., 1999, “Seismic soil-structure interaction in buildings. II: 

Empirical findings,” J. Geotech. & Geoenv. Engrg., 

Vol. 125, No. 1, American Society of Civil Engineers, pp. 38-48. 

Stewart, J.P., Kim, S., Bielak, J., Dobry, R., and Power, 

M., 2003, “Revisions to soil structure interaction 

procedures in NEHRP design provisions,” Earthquake Spectra, Vol. 19, No. 3, pp. 677-696.

Veletsos, A.S. and Meek, J.W., 1974, “Dynamic behavior of building-foundation systems,” J. Earthquake 

Engrg. Struct. Dynamics, Vol. 3, No. 2, 

pp. 121-138.
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures E-23

Veletsos, A.S. and Nair V.V., 1975, “Seismic interaction of structures on hysteretic foundations,” J. 

Struct. Engrg., Vol. 101, No. 1, American Society 

of Civil Engineers, pp. 109-129.

Veletsos, A.S. and Prasad, A.M., 1989, “Seismic interaction of structures and soils: stochastic approach,” 

J. Struct. Engrg., Vol. 115, No. 4, American Society 

of Civil Engineers, pp. 935-956.

Veletsos, A.S. and Verbic, B., 1973, “Vibration of viscoelastic foundations,” J. Earthquake Engrg. Struct. 

Dynamics, Vol. 2, No. 1, pp. 87-102.

Veletsos, A.S. and Wei, Y.T., 1971, “Lateral and rocking vibrations of footings,” J. Soil Mech. and Foundations Div., Vol. 97, No. 9, American Society of 

Civil Engineers, pp. 1227-1248.

Veletsos, A.S., Prasad, A.M., and Wu, W.H., 1997, 

“Transfer functions for rigid rectangular foundations,” J. Earthquake Engrg. Struct. Dynamics, 

Vol. 26, No. 1, pp. 5-17.

Wong, H.L., Trifunac, M.D., and Luco, J.E., 1988, “A 

comparison of soil-structure interaction calculations with results of full-scale forced vibration 

tests,” Soil Dyn. & Earthquake Engrg., Vol. 7, 

pp. 22-31.
Appendix E: Supplementary Information and Data on Soil-Structure Interaction Effects

E-24 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440
FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-1 

F. Supplementary Information and Data on MultiDegree-of-Freedom Effects

F.1 Introduction

F.1.1 Objectives

The primary objective of the multi-degree-of-freedom 

(MDOF) studies is to illustrate the application of 

current nonlinear static procedures (proposed within 

ATC-40 and FEMA 356 and elsewhere) for estimating 

peak response quantities (floor and roof displacements, 

interstory drifts, story shears, and overturning 

moments) for a range of structural models/behaviors 

and for both ordinary and near-fault ground motions. 

The intention is to illustrate the range of results 

obtained with these procedures and their relationship to 

the results of nonlinear dynamic analysis. Secondary 

objectives include the identification of potential 

limitations of these procedures and the identification of 

possible improvements. The examples provide a 

uniform basis for evaluating the strengths and 

weaknesses of the various procedures in a single study, 

in which comparisons are made using a consistent 

framework and methodology. The study has a limited 

scope, and must be considered together with other, 

more detailed studies on the specific procedures. 

F.1.2 Scope

The evaluation of MDOF effects is divided into two 

portions. The first compares the response quantities 

determined in dynamic analyses with those estimated 

using various pushover procedures, for five building 

models subjected to both ordinary and near-fault ground 

motions. These comparisons are made assuming that 

the peak roof drift (or target displacement) is 

determined accurately by the pushover procedures. The 

second portion assesses the accuracy of the estimates of 

peak roof drift determined using “equivalent” singledegree-of-freedom (SDOF) systems. 

Section F.2 describes the example buildings, analytical 

models and properties, and ground motions. Section F.3 

describes the simplified inelastic analysis procedures. 

Section F.4 addresses the accuracy of the estimates of 

response quantities made using the simplified 

procedures for both regular and near-field ground 

motions. Section F.5 addresses the accuracy of 

“equivalent” SDOF estimates of peak roof displacement 

using relationships provided in ATC-40 and FEMA 

356. Section F.6 provides information relating to a new 

approach for using scaled ground motion records in 

nonlinear response history analysis. Section F.7 reports 

some results obtained using an energy-based pushover 

procedure. Section F.8 contains (1) detailed descriptions 

of the ground motions used in the study, (2) detailed 

plots comparing the distributions of response quantities 

observed in the dynamic analyses with the deterministic 

values determined in the pushover analyses, (3) 

summary error statistics, and (4) plots of the observed 

coefficients of variation of the peak dynamic response 

quantities. 

The ordinary ground motions were scaled to achieve 

predetermined values of peak roof drift. The peak roof 

drifts were selected to represent elastic response and 

two levels of nonlinear response. Thus, while the roof 

displacements achieved in the dynamic and static 

analyses were equal to the predetermined target values, 

the frequency content and timing of the ground motion 

records differed and introduced variability to the other 

peak response quantities. Of primary interest is the 

comparison of the deterministic estimates of the 

response quantities obtained using the simplified 

inelastic procedures to the statistical distributions of the 

peaks of these quantities determined by dynamic 

analyses.

The near-fault ground motions were applied at their 

natural (unscaled) intensities because of a concern that 

scaled near-fault records may be unrealistic. Thus, the 

target displacement used for each simplified inelastic 

procedure was set equal to the peak roof displacement 

observed for each near-fault ground motion. These 

results allow for comparisons of estimated and actual 

response quantities for individual records, as well as the 

determination of normalized errors. The small number 

of near-fault ground motions used and the lack of 

consistency in target displacements make the data less 

statistically meaningful than in the case of the ordinary 

ground motions. Consequently, the evaluation of the 

inelastic procedures for the near-fault motions is more 

qualitative, while the evaluation for the ordinary 

motions has a stronger quantitative basis. 

F.2 Example Buildings and Demand 

Parameters

Five example buildings were selected for study. These 

consist of two steel moment-resistant frame buildings 

designed as part of the FEMA-funded SAC joint 
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-2 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

venture project, modifications to each of these frames to 

induce weak story behavior, and a reinforced concrete 

shear wall building described in ATC-40.

The 3- and 9-story steel frames were designed and 

modeled in Drain-2DX as part of the SAC joint venture 

effort. The frames adopted in this study had been 

designed for Los Angeles using the 1994 Uniform 

Building Code, and employed “pre-Northridge” special 

moment-resistant frame connections along the building 

perimeter. The weak stories were introduced into the 

lowest story of the 3- and 9-story frames by reducing 

the flexural strengths of the columns, without changing 

their stiffnesses. This was done in order to affect the 

mechanism while keeping the elastic properties (e.g. 

periods of vibration) the same as for the regular frames. 

The lowest story column strengths were determined by 

trial and error to ensure that a weak story mechanism 

developed in dynamic analyses using records scaled to 

cause peak roof drifts equal to 4% of the building 

height.

The 8-story reinforced concrete wall building was based 

on the Escondido Village building that is described in 

ATC-40.

Detailed descriptions of these buildings follow.

F.2.1 Prototype Buildings

F.2.1.1 Regular 3-Story Frame

The 3-story steel frame, as shown in Figure F-1, is the 

north-south lateral force-resisting system of a 

benchmark building for the SAC project. The building 

is 120 ft by 180 ft in plan and 39 ft in elevation, with a 

2-ft extension from the perimeter column lines to the 

building edge. Typical floor-to-floor height is 13 ft. The 

building consists of four bays in the north-south 

direction and six bays in the east-west direction. As 

shown in Figure F-1, all connections are momentresistant for the three left-most bays of the frame under 

consideration. The “gravity columns” on the right, 

labeled as such in the figure, are used to model P-delta 

effects, as described in Section F.2.2. The assumed 

gravity loading for the building is shown in Table F-1. 

F.2.1.2 Regular 9-Story Frame

The 9-story steel frame is part of the lateral forceresisting system of another SAC model building. The 

building is 150 ft by 150 ft in plan and 122 ft in 

elevation with a 2 ft extension from the perimeter 

column lines to the building edge. As shown in 

Figure F-1 Elevation view of the 3-story (regular and weak-story) steel frames used in the study.

13’-0”

30’-0” 30’-0” 30’-0” 30’-0”

Roof

3rd Floor

2nd Floor

1st Floor

W14x257

W33x118

W14x311

W30x116

W24x68

W33x118

W14x311

W14x257

W30x116

W24x68

W33x118

W30x116

W24x68

W18x35

W18x35

W16x26

W14x68

13’-0”

13’-0”

A B C D E

(Gravity Columns)

Moment Resisting Connection Simple (hinged) Connection

Table F-1 Assumed Loading for the 3- and 9-Story 

Buildings

Type Intensity (psf)

Floor dead load for weight calculations 96

Floor dead load for mass calculations 86

Roof dead load excluding penthouse 83

Penthouse dead load 116

Reduced live load per floor and for roof 20
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-3

Figure F-2, the frame considered in this study consists 

of five 30-ft long bays. The typical floor-to-floor height 

is 13 ft. The 1st story and basement floor-to-floor 

heights are 18 ft and 12 ft, respectively. The assumed 

gravity loading for this building is the same as that for 

the 3-story building.

F.2.1.3 3-Story Weak Story Frame

The strengths of the lowest story columns of the regular 

3-story frame (Section F.2.1.1) were reduced to create 

the 3-story weak story frame. Based on the response 

observed in dynamic analyses, the lowest story column 

strength were reduced to 50% of their original values in 

order to develop weak-story behavior at a roof drift of 

4% of the building height. No other changes were made.

F.2.1.4 9-Story Weak Story Frame

The strengths of the lowest story columns of the regular 

9-story frame (Section F.2.1.2) were reduced to create 

the 9-story weak story frame. Based on dynamic 

response, the lowest story column strengths were 

reduced to 60% of their original values in order to 

develop weak-story behavior at a roof drift of 4% of the 

building height. No other changes were made.

F.2.1.5 8-Story Shear Wall

The 8-story shear wall represented in this study is one 

of the two longitudinal walls of the midrise building at 

Escondido Village, located at line A and between lines 

4 and 6 of the as-built drawings of June 10, 1964. The 

wall is 8 stories in height, with a basement below. This 

structural wall was selected because it plays a 

substantial role in the lateral force resisting system for 

Figure F-2 Elevation view of the 9-story (regular and weak-story) steel frames used in the study.

12’-0”

18’-0”

30’-0” 30’-0” 30’-0” 30’-0”

3rd Floor

2nd Floor

Basement

W14x370 W14x370

W14x455 W14x500

W36x160

W36x160

W36x160

4th Floor

5th Floor

6th Floor

7th Floor

8th Floor

9th Floor

W36x135

W36x135

W36x135

W36x135

W30x99

W27x84

W24x68

30’-0”

W14x283 W14x257 W14x233W14x370 W14x283 W14x257

W14x455 W14x500 W14x370 W14x283 W14x257

W14x455 W14x500 W14x370 W14x283 W14x257W14x455 W14x500 W14x370 W14x283 W14x257

W14x370 W14x370 W14x283 W14x257 W14x233

13’-0”

W36x160

W36x160

W36x160

W36x135

W36x135

W36x135

W36x135

W30x99

W27x84

W24x68

W36x160

W36x160

W36x160

W36x135

W36x135

W36x135

W36x135

W30x99

W27x84

W24x68

W36x160

W36x160

W36x160

W36x135

W36x135

W36x135

W36x135

W30x99

W27x84

W24x68

W36x160

W36x160

W36x160

W36x135

W36x135

W36x135

W36x135

W30x99

W27x84

W24x68

A B C D E F

13’-0”

13’-0”

13’-0”

13’-0”

13’-0”

13’-0”

13’-0”

1st Floor

Roof

(Weak Axis) (Weak Axis) (Weak Axis) (Weak Axis) (Weak Axis)

Moment Resisting Connection Simple (hinged) Connection
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-4 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

the building and its location suggests that the degree of 

coupling is negligible. The information for this wall 

was gathered from ATC-40 Vol. 2 and the as-built 

drawings. Table F-2 summarizes the properties of the 

wall used in this study. Figure F-3 shows the elevation 

and cross sections of the wall. The assumed gravity 

loading is shown inTable F-3.

F.2.2 Modeling

Two-dimensional models of the frame and wall 

buildings were prepared using standard elements that 

are available in Drain-2DX version 1.10. The models.. 

were subjected to horizontal excitation or lateral forces 

after the application of gravity loads. Inertial mass 

resisted horizontal excitations only. P-Delta effects 

were considered for all building models, using dead 

loads in combination with 40% of the design live loads. 

For the frame models, these loads were applied to a 

separate gravity column that was connected to the 

lateral force resisting system. The gravity column was 

pinned at each story, providing a “truss-bar” 

approximation of the effect of P-Delta on the global 

stiffness matrix. For the wall model, gravity loads 

attributary to the walls were applied. This induced 

compression in the concrete and steel fibers of the 

model, causing the wall to have an initial stiffness 

approximately equal to the gross section stiffness. 

Fixed and variable time step solution schemes were 

employed, in all cases with events monitored. The 

default overshoot tolerances were used for members 

modeled with a plastic hinge beam-column element 

(Type 02). The overshoot tolerances for the members 

Table F-2 Properties of the 8-Story Reinforced Concrete Structural Wall

Item

Gathered Information from ATC-40 Vol. 2 (ATC) and As-Built 

Drawings (DWG) Wall Used in this Study

Typical floor height 9 ' − 1" (Source: ATC and DWG) 9 ' − 1" 

Basement height 12' − 7" (Source: ATC and DWG) 12' − 7"

Wall length 25' − 5" (Source: ATC and DWG) 25'

Wall thickness Typical floor: 10" (Source: ATC)

9 3/4" (Source: DWG)

Basement 12" (Source: ATC and DWG)

10"

12"

Boundary reinforcement 7th floor to roof 3 #6 (Source: DWG)

5th floor to 7th floor 3 #11 (Source: DWG)

3rd floor to 5th floor 6 #11 (Source: DWG)

Foundation to 3rd floor 9 #11 (Source: DWG)

3 #6

3 #11

6 #11

9 #11

Confinement reinforcement 

at boundary

#3 @ 12" with 135° hook (Source: ATC) #3 @ 12" with 135° 

hook

Distributed vertical 

reinforcement

Typical floor ρ = 0.0023 (Source: ATC)

2 #4 @ 18" (Source: DWG, 

ρ = 0.0023)

Basement 2 #5 @ 18" (Source: DWG, 

ρ=0.00287)

2 #4 @ 18" (ρ = 0.0023)

2 #5 @ 18" (ρ=0.00287)

Horizontal reinforcement Typical floor ρ = 0.0023 (Source: ATC)

2 #4 @ 18" (Source: DWG, 

ρ = 0.0023)

Basement 2 #4 @ 12" (Source: DWG, 

ρ = 0.00278)

2 #4 @ 18" (ρ = 0.0023)

2 #4 @ 12" (ρ=0.00278)

Concrete strength

3000 psi (Source: DWG)

3000 psi (Design) and 2470 psi (Test) (Source: ATC)

3000 psi

Reinforcing steel yield 

strength 40 ksi (Source: ATC and DWG) 40 ksi
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-5

modeled with a fiber beam-column element (Type 15) 

were set to be 0.01% of the yield strengths of the fibers.

The steel frames were modeled using beam-column 

elements (Type 02), as illustrated in Figures F-4 and 

F-5. The SAC M1 model was used in this study, in 

which beams and columns span between nodes located 

at the intersections of the beam and column centerlines; 

rigid end offsets were set to zero and beam column 

joints were not modeled. Material yield strengths were 

49.2 and 57.6 ksi, for the beams and columns 

respectively, and the post-yield stiffness of the momentcurvature relationship was set to 3% of the initial 

stiffness, as assumed in the SAC models. Gravity loads 

were applied along the frame elements as well as at the 

ends of these elements, at the nodes located where the 

Figure F-3 Elevation and plan views of the 8-story 

reinforced concrete shear wall used in 

the study

8 @ 9’-1” = 72’-8”

3rd Floor

2nd Floor

1st Floor

Basement

4th Floor

5th Floor

6th Floor

7th Floor

8th Floor

Roof

12’-7”

3’

Soil Level

25’-0”

(a) Wall elevation 

3" 4 @ 6" 6"

150"

End Reinforcement

9 #11 Bars

with Single Hoops

#3 @ 12" o.c.

12"

Distributed Reinforcement

2 Curtains of #5 @ 18" o.c.

Horizontal Reinforcement

2 Curtains of #4 @ 12" o.c. CL

(b) Plan view of wall cross section: basement to 1st floor 

Distributed Reinforcement

2 Curtains of #4 @ 18" o.c.

Horizontal Reinforcement

2 Curtains of #4 @ 18" o.c.

150"

3" 4 @ 6" 6" 10"

CL

End Reinforcement

9 #11 Bars

with Single Hoops

#3 @ 12" o.c.

(c) Plan view of wall cross section: 1st floor to 3rd floor 

150"

Horizontal Reinforcement

2 Curtains of #4 @ 18" o.c.

Distributed Reinforcement

2 Curtains of #4 @ 18" o.c.

3"

2 @ 6"

End Reinforcement

6 #11 Bars

with Single Hoops

#3 @ 12" o.c.

10"

CL

18"

(d) Plan view of wall cross section: 3rd floor to 5th floor 

Figure F-3 Elevation and plan views of the 8-story 

reinforced concrete shear wall used in the 

study (continued)

Table F-3 Assumed Loading for the 8-Story 

Building

Type Intensity (psf)

Floor dead load for weight calculations 165

Floor dead load for mass calculations 165

Roof dead load 145

Reduced live load per floor and for roof 20

Distributed Reinforcement

2 Curtains of #4 @ 18" o.c.

Horizontal Reinforcement

2 Curtains of #4 @ 18" o.c.

150"

End Reinforcement

3 #11 Bars

with Single Hoops

#3 @ 12" o.c.

3"10"

CL

6"

6"

(e) Plan view of wall cross section: 5th floor to 7th floor 

150"

Horizontal Reinforcement

2 Curtains of #4 @ 18" o.c.

Distributed Reinforcement

2 Curtains of #4 @ 18" o.c.

End Reinforcement

3 #6 Bars

with Single Hoops

#3 @ 12" o.c. 10"

CL

3"

6"

6"

(f) Plan view of wall cross section: 7th floor to roof 
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-6 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

beams and columns intersect. To avoid the possibility 

that overturning effects might influence the formation 

of column hinges, an effect that would be highly 

specific to the geometry and specific details of these 

particular frames, the flexural strengths of both beam 

and column elements were modeled to be independent 

of axial force. The strength and stiffness of the gravity 

column framing is neglected in the M1 model.

The reinforced concrete shear wall was modeled using a 

fiber element (Type 15), as illustrated in Figure F-6. 

The wall consists of nine elements, one element per 

story. Each element is divided into four segments along 

the element axis. The cross section of each segment is 

divided into 12 fibers. The base of the wall is assumed 

to be fixed, and a horizontal restraint is provided at the 

1st floor. Inelasticity in flexure was modeled; it was 

assumed that the wall would have sufficient shear 

strength and that only elastic shear deformations needed 

to be represented. While the fibers had zero tensile 

strength, preloading by gravity ensured that all fibers 

contributed to the initial stiffness of the wall.

Because the degree of confinement at the wall 

boundaries is considered low, an unconfined concrete 

stress-strain relationship shown in Figure F-7(a) was 

used. For the longitudinal steel, a bilinear stress-strain 

relationship was employed [Figure F-7(b)].

The mass is lumped at the ends of the element. The 

mass contribution (assumed uniform) for the wall is 

calculated in such a way that the resulting fundamental 

period, based on effective stiffness, matches the elastic 

period reported in ATC-40, resulting in a mass 

contribution of 19.2% of the total floor mass. The 

resulting base shear coefficient at yield, obtained from a 

pushover analysis, is 0.129.

For the frame models, a Rayleigh damping ratio of 2% 

was applied to the first mode period and a period of 

0.2 s, as assumed in the SAC models. For the wall 

model, 5% Rayleigh damping was set for the first and 

fourth mode periods corresponding to gross-section 

stiffness. In all models, the modal periods used to 

determine the damping ratios were those computed 

before gravity loading was applied.

F.2.2.1 Dynamic Characteristics of Models

The first three periods and mode shapes for the frame 

and wall buildings are provided in Table F-4. Because 

flexural cracking of the reinforced concrete wall was 

modeled, modal properties reported for the 8-story 

reinforced concrete shear wall (part b of Table F-4) are 

the properties determined based on the tangent stiffness 

of the cracked wall, determined in a first mode 

pushover analysis at a base shear equal to 60% of the 

base shear corresponding to yield of a bilinear curve 

that was fitted to the capacity curve.

To illustrate basic characteristics of each building 

model, capacity curves are presented for the five 

Figure F-4 Drain model of the 3-story (regular and weak story) steel frames.

Y = 156”

Y = 312”

Y = 468”

Y = 0”

156”

360” 360” 360” 360”

Roof

3rd

2nd

1st

X = 0” X = 360” X = 720” X = 1080” X = 1440” W14x257 W14x257 W14x257 W14x311 W14x311 W14x311

W33x118

W30x116

W24x68

W14x311 W14x311 W14x311

W14x257 W14x257 W14x257

1

234

5

6

W33x118

W30x116

W24x68

W33x118

W30x116

W24x68

W18x35

W18x35

W16x26

W14x68 W14x68 W14x68

101 102 103 104 105 106

201 202 203 204 205 206

301 302 303 304 305 306

401 402 403 404 405 406

1 2 3 4

5 6 7 8

9 10 11 12

156”

131415161718

156”

79

10

11

12

8

A B C D E

Node Number

Column Element Number

Beam Element Number Rigid Link

Dummy Element

Rigid Connection

Element Pin Connection
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-7

Figure F-5 Drain model of the 9-story (regular and weak-story) steel frames. 216”

144”

360” 360” 360” 360”

3rd

2nd

1st

Basement

W14x370 W14x370 W14x370

W14x455 W14x500 W14x500 W36x160 W36x160 W36x160

12

7 8 6 9 10

29

30

31

32

33

34

35

50

51

52

53

54

55

56

71

72

73

74

75

76

77

92

93

94

95

96

97

98

16 17 18 19 2021 22 23 24 2526 27 28 29 3031 32 33 34 3536 37 38 39 4041 42 43 44 4546 47 48 49 50

9th8th7th6th5th4th

Roof

101 102 103 104 105 106 107301 302 303 304 305 306 307

201 202 203 204 205 206 207

11

2

1

3

4

7

6

5

W24x68W27x84W30x99

W36x135W36x135W36x135W36x135

15

16

17

18

19

20

21

22

23

24

25

26

27

28

36

37

38

39

40

41

42

43

44

45

46

47

48

49

57

58

63

62

61

60

59

64

65

70

69

68

67

66

78

79

80

81

82

83

84

85

86

87

88

89

90

91

9901 9902 9903 9904 9905 9906 9907

1 3 4 2 5

9

8

10

11

14

13

12

501 502 503 504 505 506 507

4501 4502 4503 4504 4505 4506 45072301 2302 2303 2304 2305 2306 2307

601 602 603 604 605 606 607701 702 703 704 705 706 707

6701 6702 6703 6704 6705 6706 6707

12 13 11 14 15

360”

X = 0” X = 360” X = 720” X = 1080” X = 1440” X = 1800”

801 802 803 804 805 806 807901 902 903 904 905 906 907

1001 1002 1003 1004 1005 1006 1007

401 402 403 404 405 406 407

W14x283 W14x257 W14x233

W14x370 W14x283 W14x257

W14x455 W14x500 W14x500 W14x370 W14x283 W14x257

W14x455 W14x500 W14x500 W14x370 W14x283 W14x257

W14x455 W14x500 W14x500 W14x370 W14x283 W14x257

W14x370

(Weak Axis)

W14x370

(Weak Axis)

W14x370

(Weak Axis)

W14x283

(Weak Axis)

W14x257

(Weak Axis)

W14x233

(Weak Axis) 156” Y = 1464” Y = 1308”

Y = 1152”

Y = 996”Y = 840”

Y = 684”

Y = 528”

Y = 372”Y = 216”

Y = 0”

Y = -144”

W24x68

W27x84

W30x99

W36x135W36x135W36x135W36x135W36x160W36x160W36x160

W24x68W27x84W30x99

W36x135W36x135W36x135W36x135W36x160W36x160W36x160

W24x68W27x84W30x99

W36x135W36x135W36x135W36x135W36x160W36x160W36x160

W24x68W27x84W30x99

W36x135W36x135W36x135W36x135W36x160W36x160W36x160

8901 8902 8903 8904 8905 8906 8907

A B C D E F

Y = 1224”Y = 912”Y = 600”Y = 288”

156”156”156”156”156”156”156”

Node NumberColumn Element Number

Beam Element Number Rigid Link

Dummy Element

Rigid ConnectionPin Connection Element
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-8 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

models in Figure F-8. The capacity curve represents the 

sum of the applied lateral forces at a given 

displacement,1 as determined in a nonlinear static 

(pushover) analysis. In this case, the lateral forces were 

proportional to the amplitude of the first mode and mass 

at each floor level, where the modal amplitudes were 

determined with P-Delta effects considered. Based on a 

bilinear fit to the capacity curves, the base shear 

coefficient at yield and the drift at yield are as given in 

Table F-5.

F.2.2.2 Model Verification

Because the SAC frames have been the subject of prior 

research, the models could be validated by comparison 

to published research. The periods of the regular 

frames, reported in Table F-4, match those reported in 

FEMA 355C (SAC, 2000). The 9-story frame capacity 

curves (computed without P-Delta considered) are 

nearly the same as those published by Chopra and Goel 

(2002). 

F.2.3 Ground Motions and Demand 

Intensities

It was desired to assess the accuracy of the pushover 

procedures with respect to the results obtained from 

Figure F-6 Drain model of the 8-story reinforced 

concrete shear wall.

300”

1

2

3

5

6

7

12 Concrete Fibers per Element

(Typical)

3rd

2nd

1st

Basement

4th

5th

6th

7th

2

9

4

8

9 10

8th

Roof

3 4 5 6 7 8

Y = 0”

Y = -151”

Y = 109”

Y = 218”

Y = 327”

Y = 436”

Y = 545”

Y = 654”

Y = 763”

Y = 872”

4 Segments per Fiber

(Typical)

1

Node Number

Element Number

Node

Element

1. When P-Delta is modeled, the element shear forces and 

base shear are increased by the P-Delta shears. The applied 

lateral force may be obtained equivalently as (1) the load factor times the sum of the components of the load vector or (2) 

the sum of the lowest story shears minus P∆/h, where P = the 

gravity load on the lowest story, ∆ = the displacement of the 

lowest story, and h = the height of the lowest story.

Figure F-7 Idealized material stress-strain relationships used in drain model of the 8-story reinforced concrete shear 

wall

0

0.5

1

1.5

2

2.5

3

3.5

0 0.001 0.002 0.003 0.004 0.005 0.006

Strain

No tension properties 

are defined.

Note:

Compression Stress (ksi)

0

10

20

30

40

50

60

0 0.02 0.04 0.06 0.08 0.1

Strain

Stress (ksi)

Tension and compression 

properties are the same.

Note:

(a) Concrete (b) Reinforcing Steel
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-9

Table F-4 Periods and Mode Shapes for the Frame and Wall Buildings

Modal Properties

Mode

1 2 3

without P-∆ with P-∆ without P-∆ with P-∆ without P-∆ with P-∆

Part a: 3-Story (Regular and Weak-Story) Frames

Period (sec) 1.01 1.03 0.33 0.33 0.17 0.17

Participation factor 1.27 1.27 0.33 0.33 0.06 0.06

Modal mass coefficient 0.83 0.83 0.14 0.13 0.04 0.04

Mode shape 

amplitude

2nd floor 0.27 0.28 1.20 1.21 4.07 4.04

3rd floor 0.66 0.66 1.14 1.14 -3.34 -3.33

Roof 1.00 1.00 -1.00 -1.00 1.00 1.00

Part b: 8-Story Reinforced Concrete Shear Wall

Period (sec) 0.71 0.71 0.13 0.13 0.05 0.05

Participation factor 1.49 1.49 0.71 0.71 0.34 0.34

Modal mass coefficient 0.64 0.64 0.20 0.20 0.08 0.08

Mode shape 

amplitude

 2nd floor 0.03 0.03 0.15 0.15 0.44 0.44

3rd floor 0.09 0.09 0.42 0.42 0.98 0.98

4th floor 0.19 0.19 0.68 0.68 1.06 1.06

5th floor 0.31 0.31 0.84 0.84 0.47 0.47

6th floor 0.46 0.46 0.77 0.77 -0.48 -0.48

7th floor 0.63 0.63 0.42 0.42 -1.09 -1.09

8th floor 0.81 0.81 -0.21 -0.21 -0.52 -0.52

Roof 1.00 1.00 -1.00 -1.00 1.00 1.00

Part c: 9-Story (Regular and Weak-Story) Frames

Period (sec) 2.27 2.34 0.85 0.87 0.49 0.50

Participation factor 1.37 1.36 0.53 0.52 0.24 0.24

Modal mass coefficient 0.83 0.84 0.11 0.11 0.04 0.04

Mode shape 

amplitude

 2nd floor 0.17 0.17 0.38 0.39 0.80 0.81

3rd floor 0.28 0.29 0.59 0.60 1.04 1.04

4th floor 0.39 0.40 0.72 0.72 0.84 0.83

5th floor 0.51 0.52 0.74 0.75 0.26 0.23

6th floor 0.62 0.63 0.64 0.63 -0.49 -0.51

7th floor 0.72 0.73 0.40 0.39 -1.04 -1.04

8th floor 0.83 0.83 -0.01 -0.02 -0.96 -0.94

9th floor 0.93 0.93 -0.54 -0.54 -0.14 -0.12

Roof 1.00 1.00 -1.00 -1.00 1.00 1.00
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-10 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

nonlinear dynamic analyses and whether these 

procedures are suitable for the special case of near-fault 

ground motions. Accordingly, two sets of ground 

motions were used. The first set of motions was 

selected to represent the range of motion that may be 

expected at a specific building site. This range was 

established by selecting strong-motion records having a 

limited range of source distance for a specified site soil 

type. Site Class B motions had been proposed 

originally, but Site Class C motions were used because 

these soil conditions are more typical. The 11 motions 

in this set are referred to as “ordinary” motions in this 

report. The second set of motions consists of motions 

recorded close to the epicenter that contain very strong 

velocity pulses. The four motions in this set are referred 

to as “near-fault” motions in this report. The records are 

summarized in Table F-6; their acceleration, velocity, 

and displacement time histories are plotted in 

Section F.8.1. The characteristic periods identified in 

Table F-6 correspond approximately to the corner 

period at the intersection of the “constant acceleration” 

and “constant velocity” portions of the spectrum for the 

ordinary motions, and were computed as

Table F-5 Base Shear Coefficient and Drift At Yield for Each Building Model

Idealized Capacity Curve Properties

Building

3-Story 3-Story Weak-Story 8-Story 9-Story 9-Story Weak-Story

Yield drift (%) 1.11 0.833 0.250 1.06 0.936

Base shear coefficient 0.329 0.247 0.129 0.177 0.156

Table F-6 Ground Motions

# Identifier Earthquake Date Magnitude Station Location (Number Component

PGA 

(g)

PGV 

(cm/

sec)

Char. 

Period 

(sec) Source

Ordinary

1 ICC000 Superstitn 11-24-87 Ms = 6.6 El Centro Imp. Co. Cent (01335) 000 0.358 46.4 0.60 CDMG

2 LOS000 Northridge 1-17-94 Ms = 6.7 Canyon Country − W Lost Cany (90057) 000 0.41 43 0.59 USC

3 G02090 Loma Prieta 10-18-89 Ms = 7.1 Gilroy Array #2 (47380) 090 0.322 39.1 0.69 CDMG

4 TCU122N Chi-Chi, Taiwan 9-20-99 Ms = 7.6 (TCU122) N 0.261 34 0.85 CWB

5 G03090 Loma Prieta 10-18-89 Ms = 7.1 Gilroy Array #3 (47381) 090 0.367 44.7 0.40 CDMG

6 CNP196 Northridge 1-17-94 Ms = 6.7 Canoga Park − Topanga Can (90053) 196 0.42 60.8 0.61 USC

7 CHY101W Chi-Chi, Taiwan 9-20-99 Ms = 7.6 (CHY101) W 0.353 70.6 1.27 CWB

8 ICC090 Superstitn 11-24-87 Ms = 6.6 El Centro Imp. Co. Cent (01335) 090 0.258 40.9 1.03 CDMG

9 CNP106 Northridge 1-17-94 Ms = 6.7 Canoga Park − Topanga Can (90053) 106 0.356 32.1 0.45 USC

10 E02140 Imperial Valley 10-15-79 Ms = 6.9 El Centro Array #2 (5115) 140 0.315 31.5 0.29 USGS

11 E11230 Imperial Valley 10-15-79 Ms = 6.9 El Centro Array #11 (5058) 230 0.38 42.1 0.27 USGS

Near-Field (Maximum Velocity Direction)

1 ERZMV1 Erzincan 3-13-92 Ms = 6.9 Erzincan Station NA 0.442 126 1.13 EERL Caltech

2 RRSMV1 Northridge 1-17-94 Ms = 6.7 Rinaldi Receiving Station 213 0.891 186 0.92 EERL Caltech

3 LUCMV1 Landers 6-28-92 Ms = 7.3 Lucerne Valley Station 280 0.732 147 0.52 EERL Caltech

4 SCHMV1 Northridge 1-17-94 Ms = 6.7 Sylmar County Hospital Parking Lot 190 0.865 138 0.51 EERL Caltech

CDMG: California Division of Mines and Geology

CWB: Central Weather Bureau, Taiwan

EERL Caltech: Earthquake Engineering Research Laboratory, California Institute of Technology

USC: University of Southern California

USGS: U.S. Geological Survey
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-11

Figure F-8 Capacity curves for the five model building examples.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

01234

Base Shear/Weight

Roof Displacement/Height (%)

3-Story

Yield Drift = 1.11%

Base Shear Coefficient

= 0.329

0

0.05

0.1

0.15

0.2

0.25

0.3

01234

Base Shear/Weight

Roof Displacement/Height (%)

3-Story Weak-Story

Yield Drift = 0.833%

Base Shear Coefficient

= 0.247

(a) 3-Story Steel Frame (b) 3-Story Weak-Story Steel Frame 

(c) 8-Story Reinforced Concrete Shear Wall 

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

01234

Base Shear/Weight

Roof Displacement/Height (%)

9-Story

Yield Drift = 1.06%

Base Shear Coefficient

= 0.177

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

01234

Base Shear/Weight

Roof Displacement/Height (%)

9-Story Weak-Story

Base Shear Coefficient

= 0.156

Yield Drift = 0.936%

(d) 9-Story Regular Steel Frame (e) 9-Story Weak-Story Steel Frame 

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

Base Shear/Weight

Roof Displacement/Height (%)

8-Story

Base Shear Coefficient

= 0.237Yield Drift = 0.25%
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-12 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

(F-1)

where Sv and Sa are the elastic pseudo-velocity and 

pseudo-acceleration spectra, respectively, for linear 

elastic systems having β = 5%, as described by Cuesta 

and Aschheim (2001).

F.2.3.1 Ordinary (Site Class C) Motions

The Pacific Earthquake Engineering Research (PEER) 

Center strong-motion database1

 produced a set of over 

50 strong ground motions in response to multiple 

queries in which magnitudes were restricted to the 

range 5.5 < Ms < 8.0, the closest distance to fault 

rupture was restricted to 8 to 20 km, and site 

classification was restricted to Site Class C. Of these 

records, those with the largest elastic spectral 

displacements at a period of 1 second were retained, 

producing a set of 17 motions. Six of these were 

excluded based on (a) the presence of an identifiable 

strong velocity pulse early in the record, (b) the 

identification of the record as being “near-fault” or 

“near-field” in some research reports, and (c) a 

preliminary elimination of those records that would 

require the largest amplitude scale factors in order to 

achieve the drift levels described in Section 9.3. The 11 

records that remain, listed in Table F-6, were generated 

from a number of events, with no event contributing 

more than 3 records.

F.2.3.2 Near-Field Motions

A variety of near-field motions was considered 

important for the example analyses. Recognizing that 

special processing is often necessary to accurately 

recover the record (Iwan and Chen, 1994), records were 

obtained from the Earthquake Engineering Research 

Laboratory at the California Institute of Technology (A. 

Guyader, personal communication). The component of 

near field motion used is oriented in the horizontal 

plane in the direction in which the maximum ground 

velocity occurs. These records are identified in 

Table F-6, and are not always aligned in the fault 

normal direction. The ground motion velocity histories 

(Section F.8.1) do show large velocity pulses. The nearfield records were used without any further processing, 

and were applied at their natural intensities, that is, 

without scaling.

F.2.3.3 Drift Levels

The drift levels used for the ordinary ground motions 

were set to 0.5%, 2.0%, and 4.0% of the height of the 

building for the steel frames, and 0.2%, 1.0%, and 2.0% 

of the height of the building for the reinforced concrete 

structural wall building. These drift levels are referred 

to as “low,” “moderate,” and “high” in subsequent 

sections of this report. The low drift level results in 

elastic response. Because the regular 3- and 9-story 

steel frames have an effective yield drift of 0.83% and 

1.1% of the height of the frame, the high drift levels 

cause system ductility demands of about 4.8 and 3.6, 

respectively, if response is predominantly in the first 

mode. The reinforced concrete wall building has a yield 

drift of approximately 0.25% and thus has system 

ductility demands of about 4 and 8 at the moderate and 

high drift levels, respectively. 

These drift levels were used to illustrate the influence of 

yielding on the accuracy of the estimates obtained from 

the inelastic procedures for a range of response that is 

relevant for many buildings. For example, roof drifts of 

2.5% and 5% of the height of the frame buildings and 

1% and 2% of the height of the wall building 

correspond to the nominal Life Safety and Collapse 

Prevention performance limits, respectively, given in 

FEMA 356.

F.2.3.4 Ground Motion Scaling

The scale factors required to cause the peak roof drifts 

to be equal to the predetermined target values are 

reported for each building, drift level, and ground 

motion in Table F-7. These scale factors were used to 

determine mean elastic spectra for each building and 

drift level in order to determine relative contributions 

for the 2nd and 3rd modes as required for the squareroot-of-the-sum-of-the-squares (SRSS) load vector and 

multimode pushover analysis (MPA) methods. This 

implementation detail is described more fully in 

Section F.3.1.6 and Section F.3.2.

F.2.4 Extensions to Address P-Delta

Nearly all the pushover procedures have been presented 

in the literature without explicit treatment of P-Delta 

effects. Only in the Displacement Coefficient Method 

are P-Detla effects addressed, by modification of the 

SDOF displacement response using the term C3. 

P-Delta affects elastic and inelastic response. Elastic 

response is affected because the geometric stiffness 

causes an increase in the period of vibration and a 

1. Available at http://peer.berkeley.edu/smcat/. change in the elastic mode shape. For inelastic systems, 

( )

( )max

max

= 2π v

g

a

S

T

S
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-13

the geometric stiffness reduces the post-yield stiffness 

and may result in large increases in peak displacement 

or collapse. The predominant mechanism may change 

as well. 

Chopra and Goel (2001) show for elastic response that 

independent pushovers in each mode are equivalent to 

conventional modal analysis, and that superposition in 

time, or by combination of individual peaks, is 

equivalent to modal superposition and response spectral 

analysis, respectively. A structure responding elastically 

in the presence of P-Delta responds with altered periods 

of vibration and mode shapes relative to those 

determined without P-Delta. Thus, theory indicates that 

the elastic portions of response should be determined 

using the modal properties considering P-Delta. Thus, 

for the pushover techniques that use elastic mode 

shapes (first mode, SRSS, and MPA), the mode shapes 

employed were those determined with P-Delta effects 

present.

The slope of the post-yield portion of the capacity curve 

determined by pushover analysis is reduced in the 

presence of P-Delta. This reduced slope, particularly 

when negative, can be expected to cause an increase in 

peak displacement response. Because ground-motion 

scale factors were adjusted to achieve predetermined 

target roof drifts, as described in Section F.2.3, this 

effect is subdued for the dynamic analyses using the 

ordinary ground motions.

F.3 Simplified Techniques

Seven pushover methods were applied. In all methods, 

lateral forces are applied incrementally in a nonlinear 

static analysis to determine a capacity curve. The 

capacity curve represents the relationship between the 

applied lateral force and the displacement at the roof. 

The applied lateral force at any floor is proportional to 

the mass and displacement associated with a shape 

vector at the floor under consideration. The pushover 

methods differ in whether the shape vector remains 

proportional to an initial shape (which may be the first 

mode or another displacement pattern) or evolves as the 

onset of material nonlinearity causes softening of the 

structure, and in whether one or multiple modes are 

considered. These methods used in this study are 

summarized in the following sections. In addition, 

where specific assumptions or adaptations were 

required to implement these methods in this study, these 

implementation-specific details are also described.

F.3.1 Single Load Vectors

F.3.1.1 First Mode

The first mode technique applies forces in proportion to 

the amplitude of the elastic first mode and mass at each 

Table F-7 Scale Factors Applied to Each of the Ordinary Ground Motions for the Dynamic Analyses

Building Drift

Ground Motion

1

ICC000

2

LOS000

3

G02090

4

TCU122n

5

G03090

6

CNP196

7

CHY101w

8

ICC090

9

CNP106

10

E02140

11

E11230

3-story 0.5% 0.5281 0.4499 0.3675 0.4416 0.3934 0.2739 0.2827 0.6529 0.5243 0.5551 0.6335

2% 1.8493 1.6607 1.2930 1.5955 1.4885 1.4912 1.4255 2.9310 2.6645 2.8917 2.3071

4% 2.4626 2.6943 2.3852 2.2346 2.4672 2.0994 2.9252 3.6492 4.9953 6.2765 3.0257

3-story 

weak-story

0.5% 0.5281 0.4499 0.3690 0.4416 0.3934 0.2739 0.2827 0.6529 0.5243 0.5551 0.6335

2% 1.6962 1.4968 1.5223 1.4177 1.3330 1.3269 1.4034 2.3378 2.7180 2.7006 2.0535

4% 4.0050 2.4302 2.2434 2.1137 2.6378 2.6051 2.1937 3.4655 4.6072 5.3996 4.4179

8-story 0.5% 0.476 0.323 0.690 0.811 0.750 0.267 0.542 0.936 0.421 0.606 0.581

1% 1.957 1.858 2.246 2.480 2.396 2.162 2.530 3.889 2.084 3.069 3.412

2% 3.213 3.947 3.329 3.349 3.401 3.318 4.172 4.641 3.586 6.220 6.540

9-story 0.5% 0.4871 0.3133 0.5683 0.4237 0.3736 0.2348 0.2372 0.3243 0.4369 0.4844 0.8223

2% 3.0281 2.5204 4.0783 3.0244 3.2888 1.5376 1.4722 2.7696 3.9072 4.4352 2.9019

4% 5.4384 4.6774 6.3237 4.2556 5.6177 3.2536 3.2282 5.3885 9.4438 6.8838 7.6710

9-story 

weak-story

0.5% 0.4871 0.3133 0.5683 0.4237 0.3736 0.2348 0.2372 0.3243 0.4369 0.4844 0.8223

2% 2.8924 2.5527 4.0639 2.2949 3.2723 1.4242 1.8111 1.7457 2.7755 2.9933 4.5670

4% 3.6906 4.4874 6.3830 2.7551 4.9506 1.9801 1.9313 4.2350 3.4149 5.8639 5.0538
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-14 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

floor. The mode shapes of the five building models 

(Table F-4) are illustrated in Figure F-9, and the 

corresponding capacity curves, determined using this 

pushover technique, are illustrated in Figure F-8. 

The capacity curve for the 8-story shear wall model 

(Figure F-8(c)) shows softening as cracks develop at the 

base of the wall. The shape vector used for this model 

was the elastic mode shape determined using the initial 

properties of the structure, after loading by gravity 

loads.

F.3.1.2 Inverted Triangular

The inverted triangular pattern uses a shape vector that 

increases linearly with height. This shape vector is also 

shown in Figure F-10 for the three building 

configurations.

F.3.1.3 Rectangular

The rectangular (or uniform) pattern uses a shape vector 

that is uniform with height. This shape vector is shown 

in Figure F-11 for the three building configurations.

F.3.1.4 Code Force Distribution

The “code” load pattern appears in many documents, 

including FEMA 368. The pattern varies from an 

inverted triangular shape for periods less than 0.5 s to a 

parabolic shape for periods greater than 2.5 s as a means 

to account for higher mode effects. The lateral force 

coefficient for floor x, Cvx, is given by

(F-2)

where wx and wi

 are the weights of floor i or x, hi and hx

are the height of floor i or x above the base, and k is an 

exponent that varies linearly with period from 1 for T1 < 

0.5 s to 2 for T1 ≥ 2.5 s. Since these forces are equal to 

the product of floor mass and the amplitude of a shape 

vector at each floor, the corresponding shape vectors are 

proportional to hk

. These shape vectors are illustrated in 

Figure F-12 for the three building configurations.1 For 

the 8-story shear wall building, k was determined based 

on the initial stiffness of the building, after gravity 

loading was applied.

Figure F-9 Shape vectors of the 1st mode shape load pattern.

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

0.93

0.83

0.73

0.63

0.52

0.40

0.29

0.17

0.00

3rd

2nd

1st

4th

5th

6th

7th

8th 0.83

0.04

0.51

0.67

0.36

0.23

0.12

0.00

Roof 1.00

3rd

2nd

1st

Roof 1.00

0.66

0.28

0.00

(a) 3-Story (Regular and WeakStory) Steel Frames (b) 8-Story Reinforced 

Concrete Wall 

(c) 9-Story (Regular and WeakStory) Steel Frames 

(a) 3-story (Regular and WeakStory) Steel Frames

(b) 8-Story Reinforced

Concrete Wall

(c) 9-Story (Regular and WeakStory) Steel Frames

=1

=

∑

k x x vx n k

i i

i

w h C

w h
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-15

Figure F-10 Shape vectors of the triangular load pattern.

Figure F-11 Shape vectors of the rectangular load pattern.

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

0.89

0.79

0.68

0.57

0.47

0.36

0.25

0.15

0.00

3rd

2nd

1st

4th

5th

6th

7th

8th 0.88

0.12

0.62

0.75

0.50

0.38

0.25

0.00

Roof 1.00

3rd

2nd

1st

Roof 1.00

0.67

0.33

0.00

(a) 3-Story (Regular and WeakStory) Steel Frames (b) 8-Story Reinforced 

Concrete Wall 

(c) 9-Story (Regular and Weak3-story (Regular and Weak- Story) Steel Frames 

Story) Steel Frames

8-Story Reinforced

Concrete Wall

9-Story (Regular and WeakStory) Steel Frames

(a) (b) (c)

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.00

3rd

2nd

1st

4th

5th

6th

7th

8th 1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.00

Roof 1.00

3rd

2nd

1st

Roof 1.00

1.00

1.00

0.00

(a) 3-Story (Regular and WeakStory) Steel Frames (b) 8-Story Reinforced 

Concrete Wall 

(c) 9-Story (Regular and WeakStory) Steel Frames 3-story (Regular and WeakStory) Steel Frames

8-Story Reinforced

Concrete Wall

9-Story (Regular and WeakStory) Steel Frames

(a) (b) (c)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-16 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

F.3.1.5 Adaptive First Mode

The adaptive first mode procedure recognizes that 

softening of the capacity curve reflects a reduction in 

stiffness, which, in turn, causes a change in the mode 

shape. Thus, lateral forces are applied proportional to 

the amplitude of an evolving first-mode-shape 

amplitude and mass at each floor.

The procedure was implemented as follows: the lateral 

load vector was adjusted at drift increments of 0.5% of 

the height of the building. The increment in lateral load 

for each interval (0% to 0.5%, 0.5% to 1.0%, 1.0% to 

1.5% and so on) was based on the mode shape 

computed at the end of the preceding interval. The 

initial mode shape was used for the first interval.

The capacity curve of the 9-story weak story building 

develops a negative tangent stiffness as the roof 

displacement increases. Drain-2DX is not able to 

provide solutions to determine the free-vibration mode 

shapes when this occurs. Thus, the last computed mode 

shape was used for the subsequent steps 

F.3.1.6 SRSS

The SRSS technique is based on a consideration of 

elastic modal responses. Associated with the response 

in each mode is a lateral force pattern, which can be 

summed to obtain story shears associated with each 

mode. An SRSS combination of the modal story shears 

results in a particular shear profile. The lateral forces 

required to generate the SRSS story-shear profile are 

applied in this pushover technique. The elastic spectral 

amplitudes are used to determine the modal story 

shears, even when nonlinear response is anticipated. A 

sufficient number of modes to represent at least 90% of 

the mass is included.

The first three modes were used for each building 

model. Because a single spectrum typically would be 

used for design, the SRSS procedure was applied to the 

mean of the scaled spectra used to achieve each 

predetermined drift level for each building. The mean 

elastic spectrum differed for each building and each 

drift level, because the scale factors used with each 

Figure F-12 Shape vectors of the code load pattern.

3rd

2nd

1st

4th

5th

6th

7th

8th 0.88

0.13

0.63

0.75

0.50

0.38

0.25

0.00

Roof 1.00

3rd

2nd

1st

Roof 1.00

0.60

0.25

0.00

k = 1.27 k = 1

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

0.81

0.63

0.48

0.34

0.23

0.14

0.07

0.03

0.00

k = 1.92

(a) 3-Story (Regular and WeakStory) Steel Frames (b) 8-Story Reinforced 

Concrete Wall 

(c) 9-Story (Regular and WeakStory) Steel Frames 3-story (Regular and WeakStory) Steel Frames

8-Story Reinforced

Concrete Wall

9-Story (Regular and WeakStory) Steel Frames

(a) (b) (c)

1. Because the weak story frames have the same elastic 

properties as their SAC counterparts, the load vectors that are 

based on elastic properties are identical for the 3-story and 9-

story frames.
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-17

record were specific to each building and drift level. 

Thus, a different SRSS load vector was determined for 

each structure and drift level. Because the load vectors 

are proportional to the product of the amplitude of a 

shape vector and mass at each floor level, the shape 

vectors corresponding to the SRSS distribution can be 

derived. These shape vectors are shown in Figure F-13 

for the three building configurations and three drift 

levels. 

For the near-fault motions, spectral amplitudes and 

SRSS combinations were determined for each near fault 

record. For the 8-story wall building, the periods used 

for the SRSS combinations were based on the initial 

stiffness, after gravity loading was applied.

F.3.2 Multiple Mode Pushover Analysis

One approach to represent the influence of higher 

modes on response quantities combines peak response 

quantities determined in separate pushover analyses for 

the first several modes using an SRSS combination. The 

procedure proposed by Chopra and Goel (2001) 

considers the potential for nonlinear response in each 

Figure F-13 Shape vectors of the SRSS load pattern.

3rd

2nd

1st

Roof 1.00

0.36

0.36

0.00

2% Drift Level

3rd

2nd

1st

Roof 1.00

0.35

0.37

0.00

0.5% Drift Level

3rd

2nd

1st

Roof 1.00

0.36

0.36

0.00

4% Drift Level

(a) 3-Story Regular Steel Frame 

3rd

2nd

1st

Roof 1.00

0.35

0.36

0.00

4% Drift Level

3rd

2nd

1st

Roof 1.00

0.35

0.36

0.00

0.5% Drift Level

3rd

2nd

1st

Roof 1.00

0.36

0.36

0.00

2% Drift Level

(b) 3-Story Weak-Story Steel Frame 

3rd

2nd

1st

4th

5th

6th

7th

8th 0.79

0.06

0.47

0.61

0.35

0.24

0.14

0.00

Roof 1.00

0.2% Drift Level

3rd

2nd

1st

4th

5th

6th

7th

8th 0.79

0.06

0.47

0.61

0.35

0.24

0.14

0.00

Roof 1.00

2% Drift Level

3rd

2nd

1st

4th

5th

6th

7th

8th 0.79

0.06

0.47

0.61

0.35

0.24

0.14

0.00

Roof 1.00

1% Drift Level

(c) 8-Story Reinforced Concrete Wall 
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-18 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

independent modal analysis. According to their 

procedure, known as Multimode Pushover Analysis 

(MPA), capacity curves are developed for each “modal” 

response by applying lateral forces proportional to the 

mode shape amplitude and mass at each floor. 

In the MDOF studies, capacity curves were determined 

by pushover analysis for each of the first three modes 

for each building model. In some cases the higher mode 

capacity curves displayed softening behavior similar to 

that observed in first mode pushover analyses. 

However, in other cases, the higher mode force patterns 

caused the roof displacement to reverse as inelasticity 

developed in the structure. Such reversals were 

observed for the 3rd mode pushover of the regular 3-

story building (Figure F-14) and for the 2nd mode 

Figure F-13 Shape vectors of the SRSS load pattern (continued).

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

0.62

1.07

0.90

0.41

0.31

0.25

0.25

0.21

0.00

2% Drift Level

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

0.61

1.04

0.88

0.40

0.30

0.25

0.25

0.21

0.00

0.5% Drift Level

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

0.64

1.12

0.95

0.43

0.32

0.26

0.25

0.21

0.00

4% Drift Level

(d) 9-Story Regular Steel Frame 

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

0.61

1.04

0.88

0.40

0.30

0.25

0.25

0.21

0.00

0.5% Drift Level

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

0.62

1.08

0.92

0.42

0.31

0.25

0.25

0.21

0.00

4% Drift Level

3rd

2nd

1st

4th

5th

6th

7th

8th

9th

Roof 1.00

0.61

1.05

0.88

0.40

0.30

0.25

0.25

0.21

0.00

2% Drift Level

(e) 9-Story Weak-Story Steel Frame 
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-19

pushover of the 3-story weak story building. Such 

reversals have also been observed by M. Aschheim and 

A. Chopra in previous analyses. 

It is difficult to rationalize the use of a capacityspectrum type procedure to identify a target 

displacement for capacity curves that do not display the 

usual softening behavior. Rather than making an ad hoc

adjustment to the MPA method for just the 3-story 

buildings, a modified version of the MPA procedure 

capable of representing higher mode contributions was 

used for all five buildings. In the “modified MPA” 

procedure, elastic contributions associated with the 2nd

and 3rd modes are combined with contributions from 

the 1st mode, which may be inelastic, using an SRSS 

combination. 

For both the ordinary motions and the near-field 

motions, the target displacement used to determine the 

first mode contribution was the predetermined peak 

roof displacement, just as for the first mode pushover. 

In order to determine the higher mode contributions for 

the ordinary ground motions, mean spectra were 

computed for each building and drift level, using the 

ground motions as scaled to achieve the predetermined 

drifts. This allowed the contributions of the 2nd and 3rd

modes to be determined directly from the mean spectra 

for the ordinary motions. For the near-field records, the 

elastic higher mode contributions were determined from 

the jagged elastic spectra associated with each unscaled 

near-field record. The higher-mode contributions were 

determined using spectra computed for viscous 

damping ratios of 2% and 5% for the frame and wall 

buildings, respectively.

For the 8-story shear wall building, the first mode 

capacity curve was determined by applying lateral 

forces proportional to the initial mode shape; a bilinear 

approximation to the capacity curve was then 

determined, having an intersection at 0.6Vy, where Vy is 

the yield strength of the fitted bilinear curve. The 

structure was pushed to the displacement corresponding 

to this point of intersection to determine the effective 

modal properties. The periods and modal participation 

factors associated with the tangent stiffness of the 

cracked wall at a base shear equal to 0.6Vy were used in 

the modified MPA procedure described above. 

F.4 Accuracy of Estimates Made Using 

Simplified Procedures

This section compares the estimates of response 

quantities obtained using the simplified inelastic 

procedures with the values obtained by nonlinear 

dynamic analysis. Comparisons for the ordinary 

motions are presented in Section F.4.2 and those for 

near-field motions are presented in Section F.4.3, after a 

discussion of error evaluation in Section F.4.1.

F.4.1 Error Measurement

Two measures of error are used. For the ordinary 

motions, which were scaled to achieve predetermined 

target drifts, the error measure E1 is defined as

Figure F-14 First, second, and third mode pushover results for the 3-story regular steel frame.

0

0.1

0.2

0.3

0.4

0.5

-1 0 1 2 3 4

Base Shear/Weight

Roof Displacement/Height (%)

Mode 1

Mode 2 Mode 3

SAP2000

DRAIN-2X
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-20 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

(F-3)

where = pushover response quantity i at 

story/floor j for drift level k, and = mean 

overall responses to the ordinary ground motions of the 

dynamic response quantity i at story/floor j at drift level 

k. 

For the near-field motions, the peak roof drifts varied 

because the motions were used without scaling. For 

these motions, a second error measure, E2, was defined 

as

(F-4)

where = mean overall near-fault responses of 

pushover response quantity i at story/floor j for drift 

level k, and = mean overall near-fault 

responses of dynamic response quantity i at story/floor j

at drift level k for record l.

F.4.2 Results for Ordinary Ground Motions

Complete results for the five example buildings are 

provided in Sections F.8.2 and F.8.3.

The minimum, maximum, mean, median, and mean 

plus and minus one standard deviation values of the 

dynamic response quantities to the ordinary ground 

motions are plotted for each floor or story in 

Section F.8.2 (Figure F-32 through F-46). These plots 

include the deterministic results obtained using the 

various pushover techniques, plotted using various line 

styles. Errors in the pushover estimates relative to mean 

dynamic response quantities, calculated using the error 

measure E1, are plotted as a function of drift level and 

pushover technique for each of the buildings in 

Section F.8.3 (Figures F-47 through F-51). The mean 

error is the mean of the errors calculated over the height 

of the building, and the maximum error is the maximum 

of the errors over the height of the building. 

A summary of the results relating to peak responses, 

their estimates, and the errors of these estimates, is 

presented in the following subsections.

F.4.2.1 Floor Displacements

The peak displacement response showed the smallest 

variance relative to that observed for the other response 

quantities. Coefficients of variation are plotted in 

Section F.8.6 (Figures F-77 through F-81). The variance 

for the 8-story wall building was appreciably less than 

that of the frame buildings. The low variance in the 

displacement response of the five buildings is attributed 

to the relatively small contribution of higher modes to 

displacements, coupled with methodology of the study, 

in which the ordinary ground motions were scaled to 

obtain peak roof displacements equal to predetermined 

values. 

The displacement response of the buildings generally 

followed an approximate first mode pattern. Exceptions 

to this pattern were noted as follows: (1) the response of 

the regular 9-story building to the CNP106 record 

appeared to be affected by higher modes at 4% drift; (2) 

substantial variability in the displacement pattern was 

observed for the 9-story weak story building at 2% 

drift, with a weak story developing for some motions 

but not others. 

Peak displacement estimates were generally quite good 

for all load patterns. Differences occurred primarily 

with the rectangular load pattern, and for the 9-story 

frames, the code load pattern. 

F.4.2.2 Interstory Drift Ratios

The peak interstory drifts of the 8-story shear wall 

building displayed little variance, in contrast to the 

larger variance evident in the interstory drifts of the 

frame buildings. All pushover techniques provided 

good estimates of the interstory drifts of the shear wall 

building at all drift levels, with slightly larger 

discrepancies occurring for the code and rectangular 

load patterns. This indicates that even the interstory 

drifts were dominated by response in a quasi-first mode 

for the shear-wall building.

The largest mean interstory drift ratios for the regular 3-

story frame were about 20% greater than the average 

roof drift ratio at 0.5% drift, and decreased to about 

10% greater than the average roof drift ratio at 4% drift. 

For the regular 9-story frame, the largest mean 

interstory drift ratios were about 35% greater than the 

average roof drift ratio at 0.5% drift, and increased to 

about 65% greater at roof drifts of 2% and 4%. Thus, 

while it appears that higher modes make larger relative 

contributions to the interstory drifts of frame structures, 

particularly longer period frame structures, the relative 

contribution may increase or decrease as drift levels 

increase and inelasticity develops. 

Of the seven pushover methods considered, only the 

modified MPA procedure explicitly accounts for higher 

( ) ( ) ( )

( ) 1

,, ,, , , , ,− = P D

D

Q i jk Q i jk E i jk Q i jk

Q i jk P ( , , )

Q i jk D ( , , )

( ) ( ) ( )

( )

,

2

,

,, ,, , , , ,− = P Dl

D l

Q i jk Q i jk E i jk Q i jk

Q i jk P ( , , )

Q i jk D l, ( ) , ,
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-21

modes. For the frame structures, this procedure often 

provided more accurate estimates of peak interstory 

drifts relative to the other pushover procedures. 

However, even the multiple mode estimates of frame 

interstory drifts were less than the mean dynamic values 

for the upper stories of the regular 9-story frame at the 

low and moderate drift levels and for several stories 

above the weak story of the 9-story weak-story frame at 

drifts of 2% and 4%.

The original MPA procedure and the modified MPA 

procedure used in the MDOF studies both use SRSS 

combinations and assume that there is no interaction of 

the modes, as is the case for elastic response. The SRSS 

combinations of response peaks, using the first three 

modes, nearly always underestimated the peak 

interstory drifts of the 3- and 9-story frames at 0.5% 

drift, suggesting that the SRSS estimates are not 

necessarily an upper bound to the mean dynamic 

peaks,1 and that, in the case of the 9-story frames, 

consideration of additional modes may be required for 

improved estimates of interstory drift. Randomness in 

the timing of “modal” peaks generates variability in 

interstory drifts, limiting the potential accuracy of 

prospective estimates of interstory drift for individual 

events.

Interestingly, for each of the single mode load vectors 

except the rectangular load vector, the maximum of the 

interstory drifts determined over the height of the five 

buildings was a good estimate of the interstory drift that 

occurred in the nonlinear dynamic analyses at that story 

in the building. Furthermore, this maximum interstory 

drift provided a reasonable estimate of the largest 

interstory drifts that developed over the height of each 

structure. The weak story frames provide one 

illustration of this finding—each of these pushover 

methods provided good estimates of the interstory drifts 

that occurred at the weak stories of the 3- and 9-story 

weak story frames at drift levels of 2% and 4%. Of 

interest, this observation held even though the story at 

which the maximum interstory drift is calculated may 

vary with the choice of pushover load vectors.

F.4.2.3 Story Shears

The peak dynamic story shears begin with 

disproportionately large values at the uppermost story 

of all five buildings and increase monotonically towards 

the base. For the regular frames, peak dynamic story 

shears often exceeded pushover estimates, at all stories. 

For the regular 3-story frame, the modified MPA 

procedure underestimated story shears at 0.5% drift and 

significantly overestimated story shears at 4% drift. For 

the regular 9-story frame, the modified MPA procedure 

underestimated story shears over most of the building 

height at 0.5% drift, and overestimated story shears and 

2% and 4% drift. The other pushover techniques 

generally had larger error. 

For the weak story frames, all pushover techniques 

except for the modified MPA procedure provided good 

estimates of the lowest story shear at 2% and 4% drift. 

The dynamic shears in the remaining stories greatly 

exceeded the single-mode estimates at these drift levels. 

Estimates for these stories using the modified MPA 

procedure could be substantially more or less than the 

dynamic values, with the tendency to overestimate story 

shears becoming more pronounced with increasing drift 

levels. A possible improvement may be to include more 

modes, with each modal contribution reduced in some 

way as drift levels increase.

For the shear-wall building, the pattern of peak story 

shears changed with increasing drift. Response at 0.2% 

drift was marked by a quasi-first mode pattern, with a 

disproportionately large shear at the uppermost story. 

With increasing drift, the shears at the uppermost two 

stories and lowest three stories increased 

disproportionately, particularly for the lowest two 

stories. At 0.2% drift, the pattern of the story shears was 

offset from the code pattern by a more or less constant 

amount that resembles the Ft

 force used in the 

Equivalent Lateral Force procedure of earlier codes2. At 

1% drift, the modified MPA procedure typically 

underestimated the story shears for the lowest stories, 

although the estimates improved at 2% drift. Estimates 

with the other procedures were not as good at the 1% 

and 2% drift levels. The rectangular load pattern, often 

used to bound wall shears for design, underestimated 

the shears over the entire height of the wall at the 1% 

and 2% drift levels, and underestimated the shears over 

the upper four stories at the 0.2% drift level.

F.4.2.4 Overturning Moments

The peak dynamic overturning moments displayed less 

variance than the story shears. The overall pattern of 

overturning moments was captured with the single-load 

1. This may be because the modal peaks were estimated with a 2% damped spectrum, but the damping 

present in the nonlinear dynamic analyses may have 

been lower for some modes, thus leading to underestimates of higher mode contributions to drift. 

2.Perhaps a better correlation would have been 

observed if the exponent k had been determined 

based on the effective period of the structure rather 

than the initial period.
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-22 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

vector techniques except for the 9-story weak story 

building, and the single-load vector techniques gave 

similar estimates, with the exception of the rectangular 

and code load vectors. Although the overall pattern was 

captured, the dynamic values were substantially 

underestimated in some cases, particularly at the upper 

stories and for the higher drift levels. The modified 

MPA procedure was inconsistent, sometimes providing 

accurate estimates and sometimes severely 

overestimating or underestimating the peak overturning 

moments. 

F.4.3 Results for Near Field Motions

Sections F.8.4 and F.8.5 (Figures F-52 through F-71, 

and Figures F-72 through F-76) provide complete 

results of the analyses of the response of the examples 

to the near-field ground motions. Peak roof drift ratios 

obtained with the unscaled near-field motions were 

generally within or close to the ranges of predetermined 

drifts used with the ordinary ground motions 

(Table F-8), ranging between 1.79% and 4.96% for the 

3-story frames, 1.69% and 1.88% for the 9-story 

frames, and 0.64% and 2.06% for the 8-story wall 

building. 

For the near field motions, peak interstory drifts for the 

regular frames were at most about 25% higher than the 

average roof drift for the 3-story frame and were at most 

about 110% higher than the average roof drift for the 9-

story frame. While these values are larger than the mean 

values reported in Section F.3.1 for the ordinary 

motions, they are not inconsistent with the peak values 

of interstory drift observed for the ordinary motions.

Errors in the estimates of the response quantities for 

both the ordinary and near fault ground motions are 

presented in Figures F-72 through F-76 for all buildings 

and pushover load vectors. The mean of the values of 

the error measure E2 over the height of the buildings is 

plotted according to the drift level for the ordinary 

motions and is also plotted for the near-fault motions.1

Maximum values of this error measure over the height 

of each building are also plotted. Note that the error 

measure E2, when applied to the ordinary motions, is 

equivalent to the error measure E1, because the mean of 

the pushover estimates is simply the single estimate 

obtained at a predetermined drift level.

A review of Figures F-72 through F-76 shows that the 

quality of the estimates is, in general, as good or better 

than the estimates made for the ordinary ground 

motions, with the exception of displacement estimates 

of the 9-story weak-story building. In some cases the 

weak-story frame showed a clear weak story response 

(RRSMV1 and LUCMV1 ground motions), while in 

other cases, peak interstory drifts in the weak story were 

only moderately elevated over those occurring in the 

regular frames for the same motion (ERZMV1 and 

SCHMV1 ground motions). These differences occurred 

even though the peak roof drifts of the 9-story weakstory frame (1.71% to 2.13%) were similar to those of 

the regular 9-story frame (1.69% to 1.91%). For those 

cases in which weak story responses occurred, the 

pushover methods provided good estimates of 

displacement response. Only for the two near-fault 

motions that did not generate a weak-story response 

were the estimates poor.

F.5 Equivalent SDOF Estimates of Peak 

Roof Displacement Response

The preceding analyses of the example buildings for 

MDOF effects focused on the accuracy of estimates 

made using various load vectors for building models 

subjected to prescribed peak drift levels. Underlying 

this approach was the assumption that accurate 

estimates of the peak roof displacement can be obtained 

using the simplified inelastic procedures. Work by 

many researchers, including Chopra et al. (2003), 

Miranda (1991), Collins et al. (1995), Seneviratna and 

Krawinkler (1997), and Cuesta and Aschheim (2001), 

indicates a tendency for the equivalent SDOF models to 

overestimate peak roof displacements of inelastic 

structures, by up to 20% or more, depending on the 

level of nonlinearity in the system. To illustrate this 

Table F-8 Peak Roof Drift Ratios for the Five 

Building Models (%)

Building 

Model Near-Fault Record

ERZMV1 RRSMV1 LUCMV1 SCHMV1

3-story frame 

(regular)

4.07 4.96 1.79 2.62

3-story frame 

(weak story)

2.95 3.62 2.13 2.12

8-story wall 1.24 2.06 0.64 0.73

9-story frame 

(regular)

1.91 1.84 1.69 1.82

9-story frame 

(weak story)

1.88 1.85 1.71 2.13

1. The E2 error measure is defined in Section F.4.1.
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-23

tendency, estimates were made for the five example 

buildings based on their first mode capacity curves. The 

estimates were made using the ATC-40 and FEMA 356 

relationships for establishing the yield-strength 

coefficient of the equivalent SDOF system.

F.5.1 Analysis Details

Peak roof displacement estimates were made for the 

five example buildings subjected to the 11 ordinary 

ground motion records scaled to achieve the 

predetermined drift levels and for the unscaled nearfield ground motions. Estimates were made for cases in 

which P-Delta effects were included as well as for cases 

in which P-Delta effects were excluded. Results are 

reported in detail for cases in which the bilinear curve, 

fitted to the capacity curve obtained from a first mode 

pushover analysis, displayed a positive post-yield 

stiffness. P-Delta effects were included for the 3-story 

frames and the 8-story wall (Figure F-8). However, to 

avoid a negative post-yield stiffness for the 9-story 

frames, P-Delta effects had to be excluded for the 

analyses of these frames.

In cases in which P-Delta effects were considered, the 

mode shape and the nonlinear static analysis were 

determined with P-Delta effects included, and the 

applied lateral force is plotted rather than the base shear 

(which is amplified due to P-Delta). In cases in which 

P-Delta effects were not considered, the mode shape 

and nonlinear static analysis were determined without 

considering P-Delta effects. For each case considered, 

ground motions were scaled to obtain the predetermined 

target drift levels for the MDOF models. If P-Delta 

effects were included, they were included in the 

nonlinear MDOF dynamic analyses and were 

represented using bilinear hysteretic models for the 

steel frame buildings and a stiffness degrading model 

for the concrete shear-wall building, with initial- and 

post-yield stiffnesses adjusted to reflect the effects of PDelta on the MDOF capacity curves. Similarly, if PDelta effects were not considered, they were excluded 

from both the MDOF and SDOF analyses.

“Equivalent” SDOF (ESDOF) systems were determined 

according to the methods of ATC-40 and FEMA 356. 

For both methods, the yield displacement ∆y of the 

ESDOF system is determined as 

(F-5)

where ∆y, roof = the roof displacement at yield, and Γ1= 

the first mode participation factor (given by φTM1/

φTMφ)1. 

In the ATC-40 method, the yield strength coefficient of 

the ESDOF system is given by

(F-6)

where Sa = the pseudo-acceleration associated with 

yield of the ESDOF system, g = the acceleration of 

gravity, Vy, mdof = the yield strength of the MDOF 

system, W = the weight of the MDOF system, and α1 = 

the modal mass coefficient (given by Γ1(φTM1/1TM1)).

In the FEMA 356 method, the yield strength coefficient 

of the ESDOF system is approximated as

(F-7)

which relies on the approximation Γ1 ≈ 1/α1. 

If φ is set equal to an elastic mode shape, the ATC-40 

method produces an ESDOF system that has a period of 

vibration equal to the period associated with the mode 

shape. However, the approximation in Equation F-7 

causes the period of vibration and the yield strength 

coefficient of the FEMA 356 ESDOF system to deviate 

slightly from the corresponding period of the MDOF 

system.2

The ESDOF systems resulting from the ATC-40 and 

FEMA 356 methods were subjected to the scaled 

ground motion records, determined for each building 

model and drift level. A bilinear hysteretic model was 

used for the frames, and a simple stiffness degrading 

model was used for the ESDOF system representing the 

8-story wall building. The resulting peak displacement 

was scaled by Γ1 to obtain the estimated peak roof 

displacement. 

,

1

∆

∆ =

Γ

y roof

y

1. For simplicity in presentation, this document presumes that 

φ has been normalized to unit amplitude at the roof.

2. Note that if higher modes are to be considered, Γi

 is a poor 

approximation to 1/αi

, for i > 1.

,

1

/

α = = a y mdof

y

S V W

C

g

, = = Γ1 a y mdof

y

S V

C

g W
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-24 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

F.5.2 Analysis Results

The ratio of the roof displacement estimated with the 

ESDOF system and the peak roof displacement that 

developed in the nonlinear dynamic analysis of the 

MDOF system was determined for each building model 

and each ground motion record. Statistics of this ratio, 

termed the “displacement ratio,” were determined for 

each building model at each drift level and for the set of 

near fault motions. The minimum, maximum, mean, 

median, and standard deviation of this ratio were 

computed. Detailed results for the ordinary motions are 

summarized for the ATC-40 formulation (Equation F-6) 

in Figure F-15. 

Mean displacement ratios for the ordinary motions were 

between approximately 0.95 and 1.25 for the five 

buildings (Figure F-15), with a tendency to increase 

with increasing roof drift. Similar means were obtained 

with the FEMA 356 formulation (Equation F-7), 

although dispersions were larger for this formulation. 

Accuracy was similar for the near-field motions. The 

ATC-40 formulation is preferred because it resulted in 

smaller dispersions, accurately reflects the frequency 

content of the excitation for elastic response, and is 

consistent with common derivations of “equivalent” 

SDOF systems. 

Accuracy was compromised in cases in which the postyield stiffness of the ESDOF system was negative. In 

such cases, there was a tendency for the displacements 

of the equivalent SDOF systems to be exaggerated, with 

some SDOF systems collapsing, although the MDOF 

systems simply reached their predetermined drifts. For 

such cases, nonlinear dynamic analyses may be 

preferred, given the potential for ESDOF systems to 

significantly overestimate the roof displacement.

F.6 Scaled NDP Analysis Method

F.6.1 Background

The ATC-55 MDOF studies were conducted to illustrate 

the accuracy of several available pushover methods for 

estimating peak response quantities, by comparison 

with results obtained from nonlinear dynamic analyses. 

The pushover techniques were not consistently able to 

provide accurate estimates of response quantities 

(interstory drifts, story shears, and overturning 

moments) for many of the example buildings. The 

difference between the pushover estimates and the 

results from nonlinear dynamic analyses is attributed 

primarily to the presence of higher modes or MDOF 

effects. Although scatter is to be expected in the results 

from nonlinear dynamic analyses, the dispersion in the 

peak dynamic values of interstory drifts, story shears, 

and overturning moments was relatively small, and the 

values of these quantities could differ significantly from 

the estimates obtained using the various pushover 

methods. 

The Scaled NDP is based on the idea that a relatively 

small number of nonlinear dynamic analyses can be 

used to estimate the response quantities of interest. In 

the Scaled NDP (described in Section 9.5.2), the ground 

motion records are scaled so that the peak roof 

displacement matches the target displacement 

determined from nonlinear static analysis. Each 

dynamic analysis contributes positively to the estimate 

of the central tendency and range of dynamic response 

values. In contrast, none of the pushover methods was 

able to consistently provide reliable estimates of the 

peak interstory drifts, story shears, and overturning 

moments, for the example buildings considered. 

F.6.2 Elaboration of Step 3 and Examples

Step 3 of the basic procedure described in Section 9.5.2 

suggests that estimates of a response quantity at the 

mean plus κ standard deviation level can be determined 

by multiplying the mean of the response quantity 

observed in the n dynamic analyses by a coefficient. Let 

the sample mean of the response quantity of interest be 

designated by . As shown in Section F.6.3, if the 

response quantities are normally distributed, the 

quantity exceeds the true mean plus κ

standard deviations with confidence level α. In the 

preceding, c is given by

(F-8)

where represents the value of the variate of 

the Student’s t-distribution with n-1 degrees of freedom 

at a confidence level of α, and COV represents the 

coefficient of variation determined for the sample of n

observations of the response quantity x.

Equation F-8 simplifies to c = 1 for a confidence level, 

α, of 50%. For a confidence level of 90%, Equation 

Equation F-8 can be solved to obtain the values of c

given in Table F-9.

Thus, the quantity is said to exceed the 

true mean plus κ standard deviation value with 

confidence level α. The quantity κ assumes a value of 

zero where estimates of the true mean are sought.

nx

(1 ) + κ n c COV x

( ) 1

1

1 () α −

=

− Φ

c

COV

n

1

, 1 ( ) t n α − Φ −

(1 ) + κ n c COV x
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-25

Figure F-15 Example statistical distributions of displacement ratios for the ordinary ground motions.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.5% 2% 4%

Drift Level

∆ESDOF/∆MDOF ATC-40

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.5% 2% 4%

Drift Level

∆ESDOF/∆MDOF ATC-40

(a) 3-Story Regular Steel Frame (b) 3-Story Weak-Story Steel Frame 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2% 1% 2%

Drift Level

∆ESDOF/∆MDOF ATC-40

 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.5% 2% 4%

Drift Level

∆ESDOF/∆MDOF ATC-40

(c) 8-Story Shear Wall (d) 9-Story Regular Steel Frame 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.5% 2% 4%

Drift Level

∆ESDOF/∆MDOF ATC-40

(e) 9-Story Weak-Story Steel Frame 

Min Max Median

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-26 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Table F-9 also can be used to indicate the number of 

analyses to run—that is, the point at which additional 

analytical data are of negligible benefit. The derivation 

of Equation F-8 is provided in Section F.6.3 below.

F.6.2.1 Example Illustrations

Interstory Drift Estimate: The sample mean of the peak 

values of interstory drift at the lowest story of the 9-

story frame at a predetermined roof drift of 4% is 

. The true COV is estimated from the 11 

peak dynamic responses to be 0.16. For this COV, 

Equation F-8 results in c = 1.05. The true mean value of 

peak interstory drift is estimated to not exceed = 

1.05(6.5%) = 6.8% at the 90% confidence level. That is, 

there is a 90% probability that the true mean peak 

interstory drift at the lowest story is less than 6.8% at 

the hazard level that produces a roof drift of 4%.

Story Shear Estimate: The sample mean of the peak 

story shears at the lowest story of the 8-story wall at a 

predetermined roof drift of 2% is = 1070 kips. To 

guard against the potential for shear failure, an “upper 

bound” limit on shear demands is desired. Based on the 

11 analyses, the true COV of the peak story shears is 

estimated to be 0.22. Using Equation F-8, c = 1.10. 

Therefore, there is a 90% probability that the true mean 

plus one standard deviation peak story shear is less than 

(1 + κCOV) = 1.10(1 + 0.22)(1070 kips) = 1440 

kips, for the hazard that produces a roof drift of 2%.

Note that the numerical values determined by this 

approach should be considered estimates rather than 

exact values. The estimates are considered to be more 

reliable, in general, than those determined using only 

static analysis techniques. 

F.6.3 Statistical Basis

Equation F-8 was derived assuming that the response 

quantities are normally distributed, an assumption that 

is approximately applicable even for log-normally 

distributed data if the dispersion in the data is not large. 

Problem Statement: A response quantity X has peak 

values x1, x2, ...xn in n dynamic analyses of a structure. 

The mean of the n responses is . The responses are 

assumed to be normally distributed, with mean μ and 

standard deviation σ. What is the scale factor c’ such 

that exceeds μ + κσ with a specified level of 

confidence α? 

Answer: 

1. X is normally distributed with true mean μ and standard deviation σx. That is, X ~ N(μ, σ).

2. The sample mean of X is given by where 

(F-9)

3. If the standard deviation σ is estimated by the sample standard deviation s, then the sample mean has 

the Student’s t-distribution with n-1 degrees of freedom:

Table F-9 Values of c at the 90% Confidence Level

n Coefficient of Variation

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

3 1.886 1.06 1.12 1.20 1.28 1.37 1.48 1.62 1.77 1.96 2.19

5 1.533 1.04 1.07 1.11 1.16 1.21 1.26 1.32 1.38 1.45 1.52

7 1.440 1.03 1.06 1.09 1.12 1.16 1.20 1.24 1.28 1.32 1.37

10 1.372 1.02 1.05 1.07 1.10 1.12 1.15 1.18 1.21 1.24 1.28

20 1.328 1.02 1.03 1.05 1.06 1.08 1.10 1.12 1.13 1.15 1.17

50 1.299 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10

100 1.290 1.01 1.01 1.02 1.03 1.03 1.04 1.05 1.05 1.06 1.07

1

, 1( ) t n α − Φ −

xn = 6.5%

n cx

n x

n cx

n x

′ n c x

nx

~ , n x N

n

σ

μ

⎛ ⎞ ⎜ ⎟ ⎝ ⎠
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-27

(F-10)

This can be expressed as

(F-11)

4. We seek to establish c’ such that

(F-12)

which can be restated as

(F-13)

5. Given Equation F-10, this probability can be reexpressed as 

(F-14)

where is the cumulative distribution function 

for the Student’s t-distribution with n-1 degrees of 

freedom. Thus,

(F-15)

6. Algebraic manipulation allows c’ to be expressed as

(F-16)

or equivalently as 

(F-17)

where σ/μ is approximated by the sample coefficient 

of variation, COV. For convenience, we may express 

where

(F-18)

where is the value of the variate of the 

Student’s t-distribution with n-1 degrees of freedom 

at a confidence level of α.

F.6.4 Observed Coefficients of Variation

The coefficients of variation (COV) of the response 

quantities determined in the MDOF studies are plotted 

in Section F.8.6 (Figures F-77 through F-81). The 

COVs are plotted for each response quantity at each 

floor or story for each of the five building models, at 

each of the three drift levels. In general, the COVs are 

highest at the upper stories and near the base of each 

model, and differ for each response quantity. The COVs 

for floor displacements diminish to zero at the top, due 

to the methodology employed in the study.

Approximate upper bounds to the COVs are tabulated 

in Table F-10, where “approximate” indicates that the 

limit was exceeded by a small amount at a limited 

number of locations. The COVs appear to increase with 

the number of stories (or period) and appear to be larger 

~ , −1 μ

⎛ ⎞ ⎜ ⎟ ⎝ ⎠ n n

s

x t

n

~ 0,1 1 ( ) /

μ − n − nx

t

s n

Pcx (' ) n >+ = μ κσ α

1 '

μ κσ α

⎛ ⎞ +

< =− ⎜ ⎟ ⎝ ⎠ P xn c

, 1 ' 1

/ t n c

s n

μ κσ μ

Φ α −

⎛ ⎞ + − ⎜ ⎟

⎜ ⎟ = −

⎜ ⎟

⎝ ⎠

Φt n, 1−

1

, 1

' (1 ) t nc

s

n

μ κσ

μ

Φ α −

−

+ −

− =

⎛ ⎞ ⎜ ⎟

⎜ ⎟

⎜ ⎟ ⎝ ⎠

1

, 1

1 / ' / 1 (1 ) t n

c

s

n

κσ μ

μ Φ α −

−

+ =

+ −

1

, 1

1 '

1 () t n

COV

c

COV

n

κ

Φ α −

−

+ =

−

c c COV ' (1 ) = + κ

1

, 1

1

1 () t n

c

COV

n Φ α −

−

=

−

1

, 1 ( ) t n α − Φ −

Table F-10 Approximate Upper Bounds to the COVs over the Height of each Building Model

Building Model Interstory Drift Story Shear Overturning Moment

3-story frame 0.15 0.15 0.15

3-story frame (weak story) 0.20 0.15 0.15

8-story wall 0.10 0.20 0.15

9-story frame 0.20 0.20 0.20

9-story frame (weak story) 0.30 0.25 0.25
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-28 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

for buildings with weak story behavior. It is suggested 

that a COV of 0.25 to 0.30 may be appropriate for all 

quantities in cases where a sufficient number of 

analyses are not available for establishing an accurate 

estimate of the COV.

F.7 Energy-based Approaches for 

Pushover Analysis

Motivated by the difficulties associated with reversals 

of the higher mode capacity curves (see Figure F-14), 

Hernández Montes et al. (2003) formulated an energybased pushover analysis approach, in which a 

displacement is derived that represents the work done 

by the lateral forces acting through the floor 

displacements during the pushover analysis. The 

energy-based displacement is derived to coincide with 

the spectral displacement of conventional pushover 

approaches in the elastic portion of the response. 

However, the resulting capacity curves do not display 

the reversals observed in some conventional higher 

mode pushover analyses. The energy-based pushover 

approach was applied to the five example buildings to 

estimate (1) roof displacements based on response in 

the first mode and (2) other response quantities using a 

multiple mode procedure. Results are reported in the 

following.

F.7.1 Peak Displacement Response

As described in Section F.5, many research studies have 

reported a tendency for conventional procedures to 

overestimate peak roof displacement of structures 

responding inelastically by up to 20% or more. This 

tendency is illustrated for the five example buildings in 

Figure F-15. These results are compared with estimates 

made using the energy-based first mode capacity curve 

in Table F-11. 

The data in Table F-11 suggest that the energy-based 

approach provides an improvement in roof 

displacement estimates for the four frames. Note that 

the same hysteretic model was used for the frame 

elements and for the corresponding equivalent SDOF 

systems. Results for the wall building are more difficult 

to interpret because SDOF responses were computed 

using a simple stiffness-degrading model, while the 

MDOF responses were computed using a fiber element. 

Nevertheless, the energy-based displacement may be 

capable of providing improved estimates of roof 

displacement response.

F.7.2 Multiple Mode Estimates of Response 

Quantities

As described in Section F.3.2, the contributions of 

higher modes to response quantities were evaluated 

using a modified MPA procedure in which only elastic 

contributions of the higher modes were used, because 

the conventional multimode pushover procedure could 

not be used due to reversals of some higher mode 

capacity curves. The development of the energy-based 

pushover method provides a way to account for the 

potentially inelastic response associated with each of 

the modes, because the pushover curves obtained with 

the energy-based approach display the usual softening 

behavior. One concern with such a procedure is that 

responses in each mode are determined independently, 

and hence, estimates of response quantities computed 

by SRSS combinations can exceed capacity limits on 

force and moment. Results obtained with the modified 

Table F-11 Means of the Ratio of Roof Displacements: SDOF Estimate / Actual MDOF 

Building

Roof Drift 

Level

Conventional 

First Mode

EnergyBased First 

Mode

3-story frame 

(with P-Delta)

0.5%

2%

4%

1.00

1.05

1.04

1.00

1.03

1.02

3-story weak 

story frame 

(with P-Delta)

0.5%

2%

4%

1.00

1.07

1.13

1.00

0.94

0.98

8-story wall 

(with P-Delta)

0.2%

1%

2%

1.13

1.23

1.25

1.16

1.20

NA1

9-story frame 

(without 

P-Delta)

0.5%

2%

4%

0.94

1.14

1.14

0.95

1.03

1.02

9-story weakstory frame 

(without 

P-Delta)

0.5%

2%

4%

0.94

1.18

1.16

0.95

1.05

1.03

1. NA: the nonlinear static analysis failed to converge.
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-29

MPA procedure, used in the first portion of the MDOF 

studies, are compared in this section with the results 

obtained using a modified MPA procedure in which 

potentially inelastic responses are considered using the 

energy-based pushover curves.

In the energy-based modification to the multimode 

pushover analysis procedure, independent pushover 

analyses are done in each of the first three modes. For 

consistency, the target displacement for the first mode 

was set equal to the predetermined roof drift. The target 

displacements of the second and third modes were 

determined by application of an R-C1-T relationship to 

the mean spectrum, which was determined for the 

scaled ground motions used for each building and drift 

level. The R-C1-T relationship that was applied is given 

by Equation 5-1. Response quantities were determined 

for each of these modal pushover analyses at the 

corresponding target displacements. SRSS 

combinations of these quantities were then taken.

Figure F-16 compares results obtained with the 

modified MPA and energy-based multiple mode 

procedures against the backdrop of results obtained in 

the dynamic responses. A sampling of results are 

plotted that correspond to those presented in 

Section F.8.2. In some cases the results obtained with 

the energy-based multiple mode procedure are 

improvements, but the estimates still are not 

consistently reliable. 

Figure F-16 Example comparisons of energy-based and conventional multiple mode calculations.

(a) Story shears for 8-story wall at 1% drift (b) Overturning moments for 8-story wall at 

 1% drift

Modified MPA
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-30 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-16 Example comparisons of energy-based and conventional multiple mode calculations (continued).

(c) Interstory drifts for 9-story frame 

 at 4% drift

(d) Story shears for 9-story frame at 

 4% drift

(e) Overturning moments for 9-story frame 

 at 4% drift

(f) Overturning moments for 9-story weak 

 story frame at 4% drift

Modified MPA
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-31

F.8 Detailed Figure Sets for the MDOF 

Examples

This section contains detailed figure sets resulting from 

the analyses described earlier in this Appendix. 

Section F.8.1 provides details of the ground motions, 

and includes plots of spectral acceleration and spectral 

displacement in addition to acceleration, velocity, and 

displacement time histories for the unscaled ground 

motions.

Section F.8.2 presents plots that compare the 

deterministic response quantities obtained in the static 

pushover analyses with the statistical distributions 

obtained in the dynamic analyses for the 11 ground 

motions. Results are presented for the peak values of 

each response quantity over the height of each building 

model, for the five building models, each at three 

predetermined values of roof drift.

Section F.8.3 presents an evaluation of mean and 

maximum errors in the static analysis estimates of the 

mean dynamic results presented in Section F.8.2. Mean 

and maximum errors in these estimates over the height 

of each building model are plotted, for each static 

analysis method, for each response quantity, and for 

each building model at each of three predetermined drift 

levels. See Section F.4.1 for further information.

Section F.8.4 presents plots that compare the 

deterministic response quantities obtained in the static 

pushover analyses with the peak values obtained in the 

dynamic analyses, for each building model subjected to 

each near-field ground motion. The peak roof drift 

obtained in the dynamic analysis and used in the static 

pushover analysis is shown. 

Section F.8.5 presents an evaluation of mean and 

maximum errors in the static analysis estimates of the 

near-field response values, as described in Section F.4.1 

These errors are plotted together with those obtained for 

the ordinary ground motions for comparison purposes.

Section F.8.6 presents observed coefficients of variation 

of the response quantities determined for the ordinary 

(Site Class C) motions.
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-32 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

F.8.1 Ground Motion Details

Figure F-17 Characteristics of the ICC000 ground motion

-400

-200

0

200

400

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-25

0

25

50

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

-10

0

10

20

0 10 20 30 40 50

Time (sec)

Displacement (cm)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

10

20

30

40

50

60

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-33

Figure F-18 Characteristics of the LOS000 ground motion

-400

-200

0

200

400

600

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-50

-25

0

25

50

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

-10

0

10

0 10 20 30 40 50

Time (sec)

Displacement (cm)

0

0.5

1

1.5

2

2.5

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

10

20

30

40

50

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-34 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-19 Characteristics of the G02090 ground motion

-400

-200

0

200

400

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-50

-25

0

25

50

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

-10

0

10

20

0 10 20 30 40 50

Time (sec)

Displacement (cm)

0

0.3

0.6

0.9

1.2

1.5

1.8

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

10

20

30

40

50

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-35

Figure F-20 Characteristics of the TCU122N ground motion

-300

-200

-100

0

100

200

300

0 10 20 30 40 50 60 70 80 90 100

Acceleration (cm/sec2)

Time (sec)

-50

-25

0

25

50

0 10 20 30 40 50 60 70 80 90 100

Time (sec)

Velocity (cm/sec)

-20

0

20

40

0 10 20 30 40 50 60 70 80 90 100

Time (sec)

Displacement (cm)

0

0.2

0.4

0.6

0.8

1

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

10

20

30

40

50

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-36 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-21 Characteristics of the G03090 ground motion

-400

-200

0

200

400

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-50

-25

0

25

50

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

0

20

40

0 10 20 30 40 50

(sec)

Displacement (cm)

0

0.5

1

1.5

2

2.5

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

10

20

30

40

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-37

Figure F-22 Characteristics of the CNP196 ground motion

-600

-400

-200

0

200

400

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-50

-25

0

25

50

75

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

-10

0

10

20

0 10 20 30 40 50

(sec)

Displacement (cm)

0

0.5

1

1.5

2

2.5

3

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

10

20

30

40

50

60

70

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-38 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-23 Characteristics of the CHY101W ground motion

-400

-200

0

200

400

0 10 20 30 40 50 60 70 80 90 100

Acceleration (cm/sec2)

Time (sec)

-100

-50

0

50

100

0 10 20 30 40 50 60 70 80 90 100

Time (sec)

Velocity (cm/sec)

-30

-10

10

30

50

0 10 20 30 40 50 60 70 80 90 100

Time (sec)

Displacement (cm)

0

0.25

0.5

0.75

1

1.25

1.5

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

20

40

60

80

100

120

140

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-39

Figure F-24 Characteristics of the ICC090 ground motion

-300

-200

-100

0

100

200

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-50

-25

0

25

50

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

0

20

40

0 10 20 30 40 50

(sec)

Displacement (cm)

0

0.2

0.4

0.6

0.8

1

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

10

20

30

40

50

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-40 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-25 Characteristics of the CNP106 ground motion

-400

-200

0

200

400

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-50

-25

0

25

50

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-10

-5

0

5

10

0 10 20 30 40 50

(sec)

Displacement (cm)

0

0.4

0.8

1.2

1.6

2

2.4

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

10

20

30

40

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-41

Figure F-26 Characteristics of the E02140 ground motion

-400

-200

0

200

400

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-50

-25

0

25

50

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

-10

0

10

20

0 10 20 30 40 50

(sec)

Displacement (cm)

0

0.5

1

1.5

2

2.5

3

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

10

20

30

40

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-42 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-27 Characteristics of the E11230 ground motion

-600

-400

-200

0

200

400

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-50

-25

0

25

50

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

-10

0

10

20

0 10 20 30 40 50

Time (sec)

Displacement (cm)

0

0.5

1

1.5

2

2.5

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

5

10

15

20

25

30

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-43

Figure F-28 Characteristics of the ERZMV1 ground motion

-600

-400

-200

0

200

400

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-100

-50

0

50

100

150

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

0

20

40

60

0 10 20 30 40 50

(sec)

Displacement (cm)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

20

40

60

80

100

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-44 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-29 Characteristics of the RRSMV1 ground motion

-900

-600

-300

0

300

600

900

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-100

-50

0

50

100

150

200

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

0

20

40

60

80

0 10 20 30 40 50

Time (sec)

Displacement (cm)

0

0.5

1

1.5

2

2.5

3

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

20

40

60

80

100

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-45

Figure F-30 Characteristics of the LUCMV1 ground motion

-800

-400

0

400

800

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-80

-40

0

40

80

120

160

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-50

0

50

100

150

200

250

300

0 10 20 30 40 50

Time (sec)

Displacement (cm)

0

0.5

1

1.5

2

2.5

3

3.5

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

20

40

60

80

100

120

140

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-46 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-31 Characteristics of the SCHMV1 ground motion

-600

-400

-200

0

200

400

600

0 10 20 30 40 50

Acceleration (cm/sec2)

Time (sec)

-100

-50

0

50

100

150

0 10 20 30 40 50

Time (sec)

Velocity (cm/sec)

-20

0

20

40

60

0 10 20 30 40 50

Time (sec)

Displacement (cm)

0

0.5

1

1.5

2

2.5

3

3.5

01234

2% Damping

5% Damping

10% Damping

Pseudo Spectral Acceleration (g )

Period (sec)

0

20

40

60

80

100

120

01234

2% Damping

5% Damping

10% Damping

Spectral Displacement (cm)

Period (sec)
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-47

F.8.2 Responses to Ordinary (Site Class C) Motions

Figure F-32 Response quantities of the 3-story building for 0.5% drift level 0 0.2 0.4 0.6 0.8 1st Story 3rd 2nd Interstory Drift (%) 0.5% Drift

0 5000 10000 15000 20000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

0.5% Drift

0 0.5 1 1.5 2 2.5

3rd

2nd

Floor

Roof

Floor Displacement (in.)

0.5% Drift

0 200 400 600 800

1st

Story

3rd

2nd

Story Shear (kips)

0.5% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-48 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-33 Response quantities of the 3-story building for 2% drift level 0 0.5 1 1.5 2 2.5 3 1st Story 3rd 2nd Interstory Drift (%) 2% Drift

0 10000 20000 30000 40000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

2% Drift

0 2 4 6 8 10

3rd

2nd

Floor

Roof

Floor Displacement (in.)

2% Drift

0 500 1000 1500 2000

1st

Story

3rd

2nd

Story Shear (kips)

2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-49

Figure F-34 Response quantities of the 3-story building for 4% drift level 0123456 1st Story 3rd 2nd Interstory Drift (%) 4% Drift

0 10000 20000 30000 40000 50000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

4% Drift

0 5 10 15 20

3rd

2nd

Floor

Roof

Floor Displacement (in.)

4% Drift

0 500 1000 1500 2000 2500

1st

Story

3rd

2nd

Story Shear (kips)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-50 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-35 Response quantities of the 3-story weak-story building for 0.5% drift level 0 0.2 0.4 0.6 0.8 1st Story 3rd 2nd Interstory Drift (%) 0.5% Drift

0 5000 10000 15000 20000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

0.5% Drift

0 0.5 1 1.5 2 2.5

3rd

2nd

Floor

Roof

Floor Displacement (in.)

0.5% Drift

0 200 400 600 800

1st

Story

3rd

2nd

Story Shear (kips)

0.5% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-51

Figure F-36 Response quantities of the 3-story weak-story building for 2% drift level 012345 1st Story 3rd 2nd Interstory Drift (%) 2% Drift

0 10000 20000 30000 40000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

2% Drift

0 2 4 6 8 10

3rd

2nd

Floor

Roof

Floor Displacement (in.)

2% Drift

0 500 1000 1500

1st

Story

3rd

2nd

Story Shear (kips)

2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-52 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-37 Response quantities of the 3-story weak-story building for 4% drift level 0 2 4 6 8 10 12 1st Story 3rd 2nd Interstory Drift (%) 4% Drift

0 10000 20000 30000 40000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

4% Drift

0 5 10 15 20

3rd

2nd

Floor

Roof

Floor Displacement (in.)

4% Drift

0 500 1000 1500 2000 2500

1st

Story

3rd

2nd

Story Shear (kips)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-53

Figure F-38 Response quantities of the 8-story building for 0.2% drift level

0 0.1 0.2 0.3 0.4

1st

3rd

2nd

Story

8th7th6th5th4th

Interstory Drift (%)

0.2% Drift

0 0.5 1 1.5 2

3rd

2nd

8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

0.2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-54 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-38 Response quantities of the 8-story building for 0.2% drift level (continued)

0 5000 10000 15000 20000 25000

1st

3rd

2nd

Floor

8th7th6th5th4th

Overturning Moment (kips-ft)

0.2% Drift

0 100 200 300 400 500 600

1st

3rd

2nd

Story

8th7th6th5th4th

Story Shear (kips)

0.2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-55

Figure F-39 Response quantities of the 8-story building for 1% drift level

0 0.5 1 1.5

1st

3rd

2nd

Story

8th7th6th5th4th

Interstory Drift (%)

1% Drift

0 2 4 6 8 10

3rd

2nd

8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

1% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-56 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-39 Response quantities of the 8-story building for 1% drift level (continued)0 5000 10000 15000 20000 25000 30000 1st Floor 8th 7th 6th 5th 4th 3rd 2nd

Overturning Moment (kips-ft)

1% Drift

0 200 400 600 800 1000 1200

1st

Story

8th7th6th5th4th

3rd

2nd

Story Shear (kips)

1% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-57

Figure F-40 Response quantities of the 8-story building for 2% drift level

01234

1st

3rd

2nd

Story

8th7th6th5th4th

Interstory Drift (%)

2% Drift

0 5 10 15 20

3rd

2nd

8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-58 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-40 Response quantities of the 8-story building for 2% drift level (continued)0 10000 20000 30000 40000 1st 3rd 2nd Floor 8th 7th 6th 5th 4th

Overturning Moment (kips-ft)

2% Drift

0 500 1000 1500 2000

1st

3rd

2nd

Story

8th7th6th5th4th

Story Shear (kips)

2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-59

Figure F-41 Response quantities of the 9-story building for 0.5% drift level

0 0.2 0.4 0.6 0.8 1

1st

3rd

2nd

Story

9th8th7th6th5th4th

Interstory Drift (%)

0.5% Drift

02468

3rd

2nd

9th8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

0.5% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-60 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-41 Response quantities of the 9-story building for 0.5% drift level (continued)

0 20000 40000 60000 80000 100000

1st

3rd

2nd

Floor

9th8th7th6th5th4th

Overturning Moment (kips-ft)

0.5% Drift

0 400 800 1200 1600

1st

3rd

2nd

Story

9th8th7th6th5th4th

Story Shear (kips)

0.5% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-61

Figure F-42 Response quantities of the 9-story building for 2% drift level

012345

1st

3rd

2nd

Story

9th8th7th6th5th4th

Interstory Drift (%)

2% Drift

0 10 20 30 40

3rd

2nd

9th8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-62 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-42 Response quantities of the 9-story building for 2% drift level (continued)

0 50000 100000 150000 200000

1st

3rd

2nd

Floor

9th8th7th6th5th4th

Overturning Moment (kips-ft)

2% Drift

0 1000 2000 3000 4000

1st

Story

9th8th7th6th5th4th

3rd

2nd

Story Shear (kips)

2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-63

Figure F-43 Response quantities of the 9-story building for 4% drift level

0 5 10 15 20

1st

Story

9th8th7th6th5th4th

3rd

2nd

Interstory Drift (%)

4% Drift

0 20 40 60 80 100

9th8th7th6th5th4th

3rd

2nd

Floor

Roof

Floor Displacement (in.)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-64 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-43 Response quantities of the 9-story building for 4% drift level (continued)

0 50000 100000 150000 200000 250000

1st

Floor

9th8th7th6th5th4th

3rd

2nd

Overturning Moment (kips-ft)

4% Drift

0 1000 2000 3000 4000 5000

1st

Story

9th8th7th6th5th4th

3rd

2nd

Story Shear (kips)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-65

Figure F-44 Response quantities of the 9-story weak-story building for 0.5% drift level

0 0.2 0.4 0.6 0.8 1

1st

3rd

2nd

Story

9th8th7th6th5th4th

Interstory Drift (%)

0.5% Drift

02468

3rd

2nd

9th8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

0.5% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-66 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-44 Response quantities of the 9-story weak-story building for 0.5% drift level (continued) 0 20000 40000 60000 80000 100000 1st Floor 9th 8th 7th 6th 5th 4th 3rd 2nd Overturning Moment (kips-ft) 0.5% Drift 0 400 800 1200 1600 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Story Shear (kips) 0.5% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-67

Figure F-45 Response quantities of the 9-story weak-story building for 2% drift level

0 5 10 15 20

1st

Story

9th8th7th6th5th4th

3rd

2nd

Interstory Drift (%)

2% Drift

0 10 20 30 40

9th8th7th6th5th4th

3rd

2nd

Floor

Roof

Floor Displacement (in.)

2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-68 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-45 Response quantities of the 9-story weak-story building for 2% drift level (continued) 0 50000 100000 150000 200000 1st Floor 9th 8th 7th 6th 5th 4th 3rd 2nd

Overturning Moment (kips-ft)

2% Drift

0 1000 2000 3000 4000

1st

Story

9th8th7th6th5th4th

3rd

2nd

Story Shear (kips)

2% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-69

Figure F-46 Response quantities of the 9-story weak-story building for 4% drift level

0 10 20 30 40

1st

Story

9th8th7th6th5th4th

3rd

2nd

Interstory Drift (%)

4% Drift

0 20 40 60 80

9th8th7th6th5th4th

3rd

2nd

Floor

Roof

Floor Displacement (in.)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-70 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-46 Response quantities of the 9-story weak-story building for 4% drift level (continued) 0 50000 100000 150000 200000 1st Floor 9th 8th 7th 6th 5th 4th 3rd 2nd

Overturning Moment (kips-ft)

4% Drift

0 1000 2000 3000 4000 5000

1st

Story

9th8th7th6th5th4th

3rd

2nd

Story Shear (kips)

4% Drift

Median

Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular

Min Max

Mean

SD SD
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-71

F.8.3 Errors Associated with Ordinary (Site Class C) Motions

Figure F-47 Mean and maximum errors for the 3-story building

50

25201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Floor Displacement (%) 3-Story

50

40353025201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Interstory Drift (%)

50

25201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Floor Displacement (%)

50

40353025201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Interstory Drift (%)

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-72 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-47 Mean and maximum errors for the 3-story building (continued)

0

70605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Story Shear (%) 3-Story

0

70605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

70605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Story Shear (%)

0

70605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Overturning Moment (%)

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-73

Figure F-48 Mean and maximum errors for the 3-story weak-story building

50

25201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Floor Displacement (%) 3-Story Weak-Story

0

70605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Interstory Drift (%)

50

25201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Floor Displacement (%)

0

70605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Interstory Drift (%)

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-74 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-48 Mean and maximum errors for the 3-story weak-story building (continued)

0

8070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Story Shear (%) 3-Story Weak-Story

0

8070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

8070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Story Shear (%)

0

8070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Overturning Moment (%)

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-75

Figure F-49 Mean and maximum errors for the 8-story building

86420

201816141210

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Maximum Error in Floor Displacement (%)

Pushover Load Vector

8-Story

86420

201816141210

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Maximum Error in Interstory Drift (%)

Pushover Load Vector

86420

201816141210

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Mean Error in Floor Displacement (%)

Pushover Load Vector

86420

201816141210

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Mean Error in Interstory Drift (%)

Pushover Load Vector

0.2% Drift Level 1% Drift Level 2% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-76 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-49 Mean and maximum errors for the 8-story building (continued)

0

8070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Maximum Error in Story Shear (%)

Pushover Load Vector

8-Story

0

8070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

8070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Mean Error in Story Shear (%)

Pushover Load Vector

0

8070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Mean Error in Overturning Moment (%)

Pushover Load Vector

0.2% Drift Level 1% Drift Level 2% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-77

Figure F-50 Mean and maximum errors for the 9-story building

0

605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Floor Displacement (%) 9-Story

0

908070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Interstory Drift (%)

0

605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Floor Displacement (%)

0

908070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Interstory Drift (%)

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-78 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-50 Mean and maximum errors for the 9-story building (continued)

0

80604020

140120100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Story Shear (%) 9-Story

0

80604020

180160140120100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

80604020

140120100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Story Shear (%)

0

80604020

180160140120100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Overturning Moment (%)

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-79

Figure F-51 Mean and maximum errors for the 9-story weak-story building

50

353025201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Maximum Error in Floor Displacement (%)

Pushover Load Vector

9-Story Weak-Story

0

908070605040302010

100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Interstory Drift (%)

50

353025201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Floor Displacement (%)

0

908070605040302010

100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Interstory Drift (%)

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-80 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-51 Mean and maximum errors for the 9-story weak-story building (continued)

0

908070605040302010

100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Story Shear (%) 9-Story Weak-Story

0

908070605040302010

100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

908070605040302010

100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Story Shear (%)

0

908070605040302010

100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Overturning Moment (%)

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-81

F.8.4 Responses to Near Fault Motions

Figure F-52 Response quantities of the 3-story building under ERZMV1 ground motion 012345 1st Story 3rd 2nd Interstory Drift (%) 4.07% Drift

0 10000 20000 30000 40000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

4.07% Drift

0 5 10 15 20 25

3rd

2nd

Floor

Roof

Floor Displacement (in.)

4.07% Drift

0 500 1000 1500 2000

1st

Story

3rd

2nd

Story Shear (kips)

4.07% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-82 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-53 Response quantities of the 3-story building under RRSMV1 ground motion 0123456 1st Story 3rd 2nd Interstory Drift (%) 4.96% Drift

0 10000 20000 30000 40000 50000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

4.96% Drift

0 5 10 15 20 25

3rd

2nd

Roof

Floor

Floor Displacement (in.)

4.96% Drift

0 500 1000 1500 2000

1st

Story

3rd

2nd

Story Shear (kips)

4.96% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-83

Figure F-54 Response quantities of the 3-story building under LUCMV1 ground motion 0 0.5 1 1.5 2 2.5 1st Story 3rd 2nd Interstory Drift (%) 1.79% Drift

0 10000 20000 30000 40000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

1.79% Drift

0 2 4 6 8 10

3rd

2nd

Floor

Roof

Floor Displacement (in.)

1.79% Drift

0 500 1000 1500

1st

Story

3rd

2nd

Story Shear (kips)

1.79% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-84 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-55 Response quantities of the 3-story building under SCHMV1 ground motion 01234 1st Story 3rd 2nd Interstory Drift (%) 2.62% Drift

0 10000 20000 30000 40000

1st

Floor

3rd

2nd

Overturning Moment (kips-ft)

2.62% Drift

0 5 10 15

3rd

2nd

Floor

Roof

Floor Displacement (in.)

2.62% Drift

0 500 1000 1500 2000

1st

Story

3rd

2nd

Story Shear (kips)

2.62% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-85

Figure F-56 Response quantities of the 3-story weak-story building under ERZMV1 ground motion 02468 1st Story 3rd 2nd Interstory Drift (%) 2.95% Drift 0 10000 20000 30000 40000 1st Floor 3rd 2nd Overturning Moment (kips-ft) 2.95% Drift 0 5 10 15 3rd 2nd Floor Roof Floor Displacement (in.) 2.95% Drift 0 500 1000 1500 1st Story 3rd 2nd Story Shear (kips) 2.95% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-86 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-57 Response quantities of the 3-story weak-story building under RRSMV1 ground motion 0 2 4 6 8 10 1st Story 3rd 2nd Interstory Drift (%) 3.62% Drift 0 10000 20000 30000 40000 1st Floor 3rd 2nd Overturning Moment (kips-ft) 3.62% Drift 0 5 10 15 20 3rd 2nd Floor Roof Floor Displacement (in.) 3.62% Drift 0 500 1000 1500 2000 1st Story 3rd 2nd Story Shear (kips) 3.62% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-87

Figure F-58 Response quantities of the 3-story weak-story building under LUCMV1 ground motion 012345 1st Story 3rd 2nd Interstory Drift (%) 2.13% Drift 0 5000 10000 15000 20000 25000 30000 1st Floor 3rd 2nd Overturning Moment (kips-ft) 2.13% Drift 0 2 4 6 8 10 12 3rd 2nd Floor Roof Floor Displacement (in.) 2.13% Drift 0 500 1000 1500 1st Story 3rd 2nd Story Shear (kips) 2.13% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-88 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-59 Response quantities of the 3-story weak-story building under SCHMV1 ground motion 012345 1st Story 3rd 2nd Interstory Drift (%) 2.12% Drift 0 10000 20000 30000 40000 1st Floor 3rd 2nd Overturning Moment (kips-ft) 2.12% Drift 0 2 4 6 8 10 12 3rd 2nd Floor Roof Floor Displacement (in.) 2.12% Drift 0 500 1000 1500 2000 1st Story 3rd 2nd Story Shear (kips) 2.12% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-89

Figure F-60 Response quantities of the 8-story building under ERZMV1 ground motion

0 0.5 1 1.5 2

1st

Story

8th7th6th5th4th

3rd

2nd

Interstory Drift (%)

1.24% Drift

0 2 4 6 8 10 12

3rd

2nd

8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

1.24% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-90 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-60 Response quantities of the 8-story building under ERZMV1 ground motion (continued) 0 5000 10000 15000 20000 25000 30000 1st 3rd 2nd Floor 8th 7th 6th 5th 4th Overturning Moment (kips-ft) 1.24% Drift 0 200 400 600 800 1st 3rd 2nd Story 8th 7th 6th 5th 4th Story Shear (kips) 1.24% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-91

Figure F-61 Response quantities of the 8-story building under RRSMV1 ground motion

0 0.5 1 1.5 2 2.5 3

1st

3rd

2nd

Story

8th7th6th5th4th

Interstory Drift (%)

2.06% Drift

0 5 10 15 20

3rd

2nd

8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

2.06% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-92 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-61 Response quantities of the 8-story building under RRSMV1 ground motion (continued) 0 5000 10000 15000 20000 25000 30000 1st 3rd 2nd Floor 8th 7th 6th 5th 4th Overturning Moment (kips-ft) 2.06% Drift 0 500 1000 1500 1st 3rd 2nd Story 8th 7th 6th 5th 4th Story Shear (kips) 2.06% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-93

Figure F-62 Response quantities of the 8-story building under LUCMV1 ground motion

0 0.2 0.4 0.6 0.8 1

1st

Story

8th7th6th5th4th

3rd

2nd

Interstory Drift (%)

0.64% Drift

0123456

8th7th6th5th4th

3rd

2nd

Floor

Roof

Floor Displacement (in.)

0.64% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-94 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-62 Response quantities of the 8-story building under LUCMV1 ground motion (continued) 0 5000 10000 15000 20000 25000 30000 1st Floor 8th 7th 6th 5th 4th 3rd 2nd Overturning Moment (kips-ft) 0.64% Drift 0 200 400 600 800 1st 3rd 2nd Story 8th 7th 6th 5th 4th Story Shear (kips) 0.64% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-95

Figure F-63 Response quantities of the 8-story building under SCHMV1 ground motion

0 0.2 0.4 0.6 0.8 1

1st

3rd

2nd

Story

8th7th6th5th4th

Interstory Drift (%)

0.73% Drift

02468

3rd

2nd

8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

0.73% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-96 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-63 Response quantities of the 8-story building under SCHMV1 ground motion (continued) 0 5000 10000 15000 20000 25000 30000 1st Floor 8th 7th 6th 5th 4th 3rd 2nd Overturning Moment (kips-ft) 0.73% Drift 0 200 400 600 800 1st 3rd 2nd Story 8th 7th 6th 5th 4th Story Shear (kips) 0.73% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-97

Figure F-64 Response quantities of the 9-story building under ERZMV1 ground motion

012345

1st

3rd

2nd

Story

9th8th7th6th5th4th

Interstory Drift (%)

1.91% Drift

0 5 10 15 20 25 30

3rd

2nd

9th8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

1.91% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-98 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-64 Response quantities of the 9-story building under ERZMV1 ground motion (continued) 0 50000 100000 150000 200000 1st 3rd 2nd Floor 9th 8th 7th 6th 5th 4th Overturning Moment (kips-ft) 1.91% Drift 0 500 1000 1500 2000 2500 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Story Shear (kips) 1.91% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-99

Figure F-65 Response quantities of the 9-story building under RRSMV1 ground motion

012345

1st

3rd

2nd

Story

9th8th7th6th5th4th

Interstory Drift (%)

1.84% Drift

0 10 20 30 40

3rd

2nd

9th8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

1.84% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-100 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-65 Response quantities of the 9-story building under RRSMV1 ground motion (continued) 0 50000 100000 150000 200000 1st Floor 9th 8th 7th 6th 5th 4th 3rd 2nd Overturning Moment (kips-ft) 1.84% Drift 0 1000 2000 3000 4000 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Story Shear (kips) 1.84% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-101

Figure F-66 Response quantities of the 9-story building under LUCMV1 ground motion

01234

1st

3rd

2nd

Story

9th8th7th6th5th4th

Interstory Drift (%)

1.69% Drift

0 5 10 15 20 25 30

3rd

2nd

9th8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

1.69% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-102 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-66 Response quantities of the 9-story building under LUCMV1 ground motion (continued) 0 50000 100000 150000 200000 1st 3rd 2nd Floor 9th 8th 7th 6th 5th 4th Overturning Moment (kips-ft) 1.69% Drift 0 500 1000 1500 2000 2500 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Story Shear (kips) 1.69% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-103

Figure F-67 Response quantities of the 9-story building under SCHMV1 ground motion

01234

1st

Story

9th8th7th6th5th4th

3rd

2nd

Interstory Drift (%)

1.82% Drift

0 5 10 15 20 25 30

3rd

2nd

9th8th7th6th5th4th

Floor

Roof

Floor Displacement (in.)

1.82% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-104 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-67 Response quantities of the 9-story building under SCHMV1 ground motion (continued) 0 50000 100000 150000 200000 1st Floor 9th 8th 7th 6th 5th 4th 3rd 2nd Overturning Moment (kips-ft) 1.82% Drift 0 500 1000 1500 2000 2500 3000 1st Story 9th 8th 7th 6th 5th 4th 3rd 2nd Story Shear (kips) 1.82% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-105

Figure F-68 Response quantities of the 9-story weak-story building under ERZMV1 ground motion 0 2 4 6 8 10 1st Story 9th 8th 7th 6th 5th 4th 3rd 2nd Interstory Drift (%) 1.88% Drift 0 5 10 15 20 25 30 3rd 2nd 9th 8th 7th 6th 5th 4th Floor Roof Floor Displacement (in.) 1.88% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-106 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-68 Response quantities of the 9-story weak-story building under ERZMV1 ground motion (continued) 0 50000 100000 150000 200000 1st 3rd 2nd Floor 9th 8th 7th 6th 5th 4th Overturning Moment (kips-ft) 1.88% Drift 0 500 1000 1500 2000 2500 3000 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Story Shear (kips) 1.88% Drift Dynamic Analysis Code SRSS Adaptive Multimode Rectangular First Mode Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-107

Figure F-69 Response quantities of the 9-story weak-story building under RRSMV1 ground motion 0 2 4 6 8 10 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Interstory Drift (%) 1.85% Drift 0 10 20 30 40 9th 8th 7th 6th 5th 4th 3rd 2nd Floor Roof Floor Displacement (in.) 1.85% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-108 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-69 Response quantities of the 9-story weak-story building under RRSMV1 ground motion (Continued) 0 50000 100000 150000 200000 1st Floor 9th 8th 7th 6th 5th 4th 3rd 2nd Overturning Moment (kips-ft) 1.85% Drift 0 1000 2000 3000 4000 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Story Shear (kips) 1.85% Drift Dynamic Analysis Code SRSS Adaptive Multimode Rectangular First Mode Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-109

Figure F-70 Response quantities of the 9-story weak-story building under LUCMV1 ground motion0 2 4 6 8 10 1st Story 9th 8th 7th 6th 5th 4th 3rd 2nd

Interstory Drift (%)

1.71% Drift

0 5 10 15 20 25 30

9th8th7th6th5th4th

3rd

2nd

Floor

Roof

Floor Displacement (in.)

1.71% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-110 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-70 Response quantities of the 9-story weak-story building under LUCMV1 ground motion (continued) 0 50000 100000 150000 200000 1st Floor 9th 8th 7th 6th 5th 4th 3rd 2nd Overturning Moment (kips-ft) 1.71% Drift 0 500 1000 1500 2000 2500 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Story Shear (kips) 1.71% Drift Dynamic Analysis Code SRSS Adaptive Multimode Rectangular First Mode Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-111

Figure F-71 Response quantities of the 9-story weak-story building under SCHMV1 ground motion 0 2 4 6 8 10 12 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Interstory Drift (%) 2.13% Drift 0 10 20 30 40 3rd 2nd 9th 8th 7th 6th 5th 4th Floor Roof Floor Displacement (in.) 2.13% Drift

Dynamic Analysis Code SRSS

Adaptive

Multimode

Rectangular First Mode

Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-112 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-71 Response quantities of the 9-story weak-story building under SCHMV1 ground motion (continued) 0 50000 100000 150000 200000 1st 3rd 2nd Floor 9th 8th 7th 6th 5th 4th Overturning Moment (kips-ft) 2.13% Drift 0 500 1000 1500 2000 2500 3000 1st 3rd 2nd Story 9th 8th 7th 6th 5th 4th Story Shear (kips) 2.13% Drift Dynamic Analysis Code SRSS Adaptive Multimode Rectangular First Mode Inverted Triangular
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-113

F.8.5 Errors Associated with Near Fault Motions

Figure F-72 Mean and maximum errors for the 3-story building

50

25201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Floor Displacement (%) 3-Story

50

40353025201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

3-Story Maximum Error in Interstory Drift (%)

50

25201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Floor Displacement (%)

50

40353025201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Interstory Drift (%)

0.5% Drift Level 2% Drift Level 4% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-114 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-72 Mean and maximum errors for the 3-story building (continued)

0

70605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Story Shear (%) 3-Story

0

70605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Story Shear (%)

0

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Overturning Moment (%)

0.5% Drift Level 2% Drift Level 4% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-115

Figure F-73 Mean and maximum errors for the 3-story weak-story building

50

25201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Floor Displacement (%) 3-Story Weak-Story

0

70605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Interstory Drift (%)

50

25201510

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Floor Displacement (%)

0

70605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Interstory Drift (%)

0.5% Drift Level 2% Drift Level 4% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-116 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-73 Mean and maximum errors for the 3-story weak-story building (continued)

0

8070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Story Shear (%) 3-Story Weak-Story

0

8070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

8070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Story Shear (%)

0

8070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Overturning Moment (%)

0.5% Drift Level 2% Drift Level 4% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-117

Figure F-74 Mean and maximum errors for the 8-story building

840

24201612

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Floor Displacement (%) 8-Story

840

24201612

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Interstory Drift (%)

840

24201612

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Floor Displacement (%)

840

24201612

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Interstory Drift (%)

0.2% Drift Level 1% Drift Level 2% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-118 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-74 Mean and maximum errors for the 8-story building (continued)

0

8070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Story Shear (%) 8-Story

0

8070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Story Shear (%)

0

8070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Overturning Moment (%)

0.2% Drift Level 1% Drift Level 2% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-119

Figure F-75 Mean and maximum errors for the 9-story building

0

605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Floor Displacement (%) 9-Story

0

908070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Interstory Drift (%)

0

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Floor Displacement (%)

0

908070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Interstory Drift (%)

0.5% Drift Level 2% Drift Level 4% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-120 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-75 Mean and maximum errors for the 9-story building (continued)

0

80604020

140120100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Story Shear (%) 9-Story

0

80604020

180160140120100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

80604020

140120100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Story Shear (%)

0

80604020

180160140120100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Overturning Moment (%)

0.5% Drift Level 2% Drift Level 4% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-121

Figure F-76 Mean and maximum errors for the 9-story weak-story building

0

908070605040302010

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Floor Displacement (%) 9-Story Weak-Story

0

908070605040302010

100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Interstory Drift (%)

0

908070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Floor Displacement (%)

0

908070605040302010

100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Interstory Drift (%)

0.5% Drift Level 2% Drift Level 4% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-122 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-76 Mean and maximum errors for the 9-story weak-story building (continued)

0

908070605040302010

100

First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Story Shear (%) 9-Story Weak-Story

0

908070605040302010

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First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Maximum Error in Overturning Moment (%)

0

8070605040302010

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Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Story Shear (%)

0

908070605040302010

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First Mode Inverted

Triangular

Rectangular Code Adaptive SRSS Multimode

Pushover Load Vector

Mean Error in Overturning Moment (%)

0.5% Drift Level 2% Drift Level 4% Drift Level Near-Fault
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-123

F.8.6 Observed Coefficients of Variation of the Response Quantities Determined for the Ordinary (Site 

Class C) Motions 

Figure F-77 Observed COVs for the 3-story frame building. 0 5 10 15 20 1st 3rd 2nd Story COV (%) Interstory Drift 3-Story

0 5 10 15 20

1st

3rd

2nd

Floor

COV (%)

Overturning Moment

3-Story

0 5 10 15 20

3rd

2nd

Floor

Roof

COV (%)

Floor Displacement

3-Story

0 5 10 15 20

1st

3rd

2nd

Story

COV (%)

Story Shear

3-Story

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-124 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-78 Observed COVs for the 3-story weak story frame building 0 10 20 30 40 1st 3rd 2nd Story COV (%) Interstory Drift 3-Story Weak

0 10 20 30 40

1st

3rd

2nd

Floor

COV (%)

Overturning Moment

3-Story Weak

0 10 20 30 40

3rd

2nd

Floor

Roof

COV (%)

Floor Displacement

3-Story Weak

0 10 20 30 40

1st

3rd

2nd

Story

COV (%)

Story Shear

3-Story Weak

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-125

Figure F-79 Observed COVs for the 8-story wall building

0 10 20 30 40

1st

8th7th6th5th4th

3rd

2nd

Story

COV (%)

Interstory Drift

8-Story

0 10 20 30 40

8th7th6th5th4th

3rd

2nd

FloorRoof

COV (%)

Floor Displacement

8-Story

0.2% Drift Level 1% Drift Level 2% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-126 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-79 Observed COVs for the 8-story wall building (continued)

0 10 20 30 40

1st

8th7th6th5th4th

3rd

2nd

Floor

COV (%)

Overturning Moment

8-Story

0 10 20 30 40

1st

8th7th6th5th4th

3rd

2nd

Story

COV (%)

Story Shear

8-Story

0.2% Drift Level 1% Drift Level 2% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-127

Figure F-80 Observed COVs for the 9-story frame building

0 10 20 30 40

1st

3rd

2nd

9th8th7th6th5th4th

Story

COV (%)

Interstory Drift

0 10 20 30 40

3rd

2nd

9th8th7th6th5th4th

Floor

Roof

COV (%)

Floor Displacement

9-Story

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-128 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-80 Observed COVs for the 9-story frame building (continued)

0 10 20 30 40

1st

9th8th7th6th5th4th

3rd

2nd

Floor

COV (%)

Overturning Moment

0 10 20 30 40

1st

3rd

2nd

9th8th7th6th5th4th

Story

COV (%)

Story Shear

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-129

Figure F-81 Observed COVs for the 9-story weak story frame building

0 10 20 30 40

1st

3rd

2nd

9th8th7th6th5th4th

Story

COV (%)

Interstory Drift

9-Story Weak

0 10 20 30 40

3rd

2nd

9th8th7th6th5th4th

Floor

Roof

COV (%)

Floor Displacement

9-Story Weak

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-130 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440

Figure F-81 Observed COVs for the 9-story weak story frame building (continued)

0 10 20 30 40

1st

3rd

2nd

9th8th7th6th5th4th

Floor

COV (%)

Overturning Moment

9-Story Weak

0 10 20 30 40

1st

3rd

2nd

9th8th7th6th5th4th

Story

COV (%)

Story Shear

9-Story Weak

0.5% Drift Level 2% Drift Level 4% Drift Level
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

FEMA 440 Improvement of Nonlinear Static Seismic Analysis Procedures F-131

F.9 References

Chopra, A., and Goel, R., 2001, A Modal Pushover 

Analysis Procedure to Estimate Seismic Demands 

for Buildings: Theory and Preliminary Evaluation

PEER Report No. 2001/03, Pacific Earthquake 

Engineering Research Center, University of California, Berkeley.

Chopra, A.K., and Goel, R.K., 2002, “A modal pushover analysis procedure for estimating seismic 

demands for buildings,” Earthquake Engineering 

and Structural Dynamics, Vol. 31, pp. 561-582.

Chopra, A.K., Goel, R.K., and Chintanapakdee, C., 

2003, “Statistics of single-degree-of-freedom estimate of displacement for pushover analysis of 

buildings,” Journal of Structural Engineering, Vol. 

129, No. 4, pp. 459-469.

Collins, K.R., Wen, Y.K., and Foutch, D.A., 1995, 

Investigation of Alternative Seismic Design Procedures for Standard Buildings, UILU-ENG-95-2003 

Report, Civil Engineering Studies, Structural 

Research Series No. 600, Dept. of Civil Engineering, University of Illinois at Urbana-Champaign, 

187 pages.

Cuesta, I., and Aschheim, M.A., 2001, Using Pulse RFactors to Estimate Structural Response to Earthquake Ground Motions, CD release 01-03, MidAmerica Earthquake Center, Urbana, Illinois.

Hernández Montes, E., Kwon, O-S, and Aschheim, M., 

2004, “An energy-based formulation for first- and 

multiple-mode nonlinear static (pushover) analyses,” Journal of Earthquake Engineering, Vol. 8, 

No. 1, pp. 69-88.

Iwan, W.D., and Chen, X.D., 1995, “Important nearfield ground motion data from the Landers earthquake,” Proceedings 10th European Conference on 

Earthquake Engineering, Vol. 1, A.A. Balkema, 

Rotterdam, pp. 229-234.

Miranda, E., 1991, Seismic Evaluation and Upgrading 

of Existing Buildings, Ph.D. thesis, University of 

California at Berkeley.

SAC, 2000, State of the Art Report on Systems Performance of Steel Moment Frames Subject to Earthquake Ground Shaking, FEMA 355C Report, 

prepared by the SAC Joint Venture (a partnership of 

the Structural Engineers Association of California, 

the Applied Technology Council, and California 

Universities for Research in Earthquake Engineering) for the Federal Emergency Management 

Agency, Washington, D.C.

Seneviratna, G. D.P.K., and Krawinkler, H., 1997, Evaluation of Inelastic MDOF Effects for Seismic 

Design, BLUME-120 Report, John A. Blume 

Earthquake Engineering Center, Stanford, California, 177 pages.

FEMA Reports

FEMA 368, 2000, NEHRP Recommended Provisions 

for Seismic Regulations for New Buildings and 

Other Structures, Part 1 - Provisions, prepared by 

the Building Seismic Safety Council for the Federal 

Emergency Management Agency, Washington, 

D.C.
Appendix F: Supplementary Information and Data on Multi-Degree-of-Freedom Effects

F-132 Improvement of Nonlinear Static Seismic Analysis Procedures FEMA 440