A Practical Guide to 

Soil-Structure Interaction 

FEMA P-2091 / December 2020 

FEMA P-2091 / December 2020

A Practical Guide

to Soil-Structure Interaction

Prepared by 

APPLIED TECHNOLOGY COUNCIL

201 Redwood Shores Parkway, Suite 240 

Redwood City, California 94065 

www.ATCouncil.org

Prepared for

FEDERAL EMERGENCY MANAGEMENT AGENCY

Michael Mahoney, Project Officer

Robert D. Hanson, Subject Matter Expert

Washington, D.C. 

APPLIED TECHNOLOGY COUNCIL

Jon A. Heintz, Program Executive, Program Manager 

Ayse Hortacsu, Project Manager

PROJECT TECHNICAL COMMITTEE

Bret Lizundia (Project Technical Director) 

C.B. Crouse 

Steve Harris

Boris Jeremic

Jonathan P. Stewart 

Michael Valley

PROJECT REVIEW PANEL

Peter Lee

Sissy Nikolaou* 

Robert Pekelnicky 

Payman Khalili Tehrani 

*ATC Board Representative

WORKING GROUP MEMBERS

Timothy D. Ancheta

Laurel Jiang

Ricardo Henoch

Gyimah Kasali

Lisa M. Star

Christos Tokas 

Jiejing Zhou 
Notice

Any opinions, findings, conclusions, or recommendations expressed in this publication do not necessarily 

reflect the views of the Applied Technology Council (ATC), the Department of Homeland Security 

(DHS), or the Federal Emergency Management Agency (FEMA). Additionally, neither ATC, DHS, 

FEMA, nor any of their employees, makes any warranty, expressed or implied, nor assumes any legal 

liability or responsibility for the accuracy, completeness, or usefulness of any information, product, or 

process included in this publication. Users of information from this publication assume all liability 

arising from such use. The contents of this document do not have the force and effect of law and are not 

meant to bind the public in any way. This document is intended only to provide clarity to the public 

regarding existing requirements under the law or agency policies. 

Cover image – Photo of project site showing soil conditions and foundation and superstructure under construction.
FEMA P-2091 Preface iii

Preface 

Modern building codes include criteria for how ground motions enter a 

structure’s foundation and affect the building response. This soil-structure 

interaction (SSI) can make a substantial difference in how buildings behave 

during earthquake shaking and how they should be designed, yet there is 

relatively little implementation of SSI effects by practicing structural 

engineers. Provisions are available in ASCE/SEI 7-16 and in ASCE/SEI 

41-17 that can be used to address SSI. In addition, a research report prepared 

by the National Institute of Standards and Technology (NIST) published in 

2012 is also a valuable resource. 

In 2019, the Applied Technology Council (ATC), with funding from FEMA 

under Task Order Contract HSFE60-17-D-1005, commenced the ATC-144 

project. The objective of this project was to develop this document, A 

Practical Guide to Soil-Structure Interaction, to present information 

regarding SSI as implemented in code provisions but in an easy-tounderstand, concise format targeted toward practicing engineers to help them 

determine when SSI effects are of importance and show them examples of 

how to implement them in design.

ATC is indebted to the leadership of Bret Lizundia, Project Technical 

Director, and to the members of the ATC-144 Project Team for their efforts 

in developing this Guide. The Project Technical Committee, consisting of 

C.B. Crouse, Steve Harris, Boris Jeremic, Bret Lizundia (Chair), Jonathan 

Stewart, and Michael Valley, performed the technical development efforts 

and served as primary authors. Tim Ancheta, Laurel Jiang, Gyimah Kasali, 

and Lisa Star assisted in the development of the design examples, and 

Ricardo Henoch and Jiejing Zhou provided detailed review of the design 

examples as members of the Project Working Group. Chris Tokas developed 

many of the figures in the document. The Project Review Panel, consisting 

of Peter Lee, Sissy Nikolaou, Robert Pekelnicky, and Payman Khalili 

Tehrani provided technical review, advice, and consultation at key stages of 

the work. The names and affiliations of all who contributed to this report are 

provided in the list of Project Participants.
iv Preface FEMA P-2091

ATC also gratefully acknowledges Michael Mahoney (FEMA Project 

Officer), and Robert D. Hanson (FEMA Subject Matter Expert) for their 

input and guidance in the preparation of this document. Carrie J. Perna 

(ATC) provided report production services. 

Ayse Hortacsu Jon A. Heintz 

ATC Director of Projects ATC Executive Director
FEMA P-2091 Table of Contents v 

Table of Contents 

Preface ........................................................................................................... iii

List of Figures ............................................................................................... ix 

List of Tables ................................................................................................xv 

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

1.1 Overview .................................................................................. 1-1 

1.2 Types of Soil-Structure Interaction Modeling .......................... 1-1 

1.3 Purpose of the Guide ................................................................ 1-2 

1.4 Scope of the Guide ................................................................... 1-2 

1.5 Target Audience for the Guide ................................................. 1-2

1.6 Key SSI Terminology ............................................................... 1-3 

1.7 Tips for Understanding and Implementing SSI ........................ 1-4 

1.8 Organization of the Guide ........................................................ 1-5 

2. Situations Where SSI is Important ................................................. 2-1 

2.1 Overview .................................................................................. 2-1 

2.2 Large Building Footprint Reduces Design Forces ................... 2-1 

2.3 Substantial Foundation Embedment Reduces Design 

Forces ....................................................................................... 2-2 

2.4 High Structure-to-Soil Stiffness Ratios Will Lengthen Period 

and Change Design Forces ....................................................... 2-2 

2.5 Foundation Rocking Impacts Superstructure Behavior ............ 2-4 

3. Rule of Thumb Test for Inertial SSI Significance ......................... 3-1 

3.1 Overview .................................................................................. 3-1 

3.2 Key Equation ............................................................................ 3-1 

3.3 Determining Average Effective Shear Wave Velocity ............. 3-2 

3.4 Determining Fundamental Period of the Structure ................... 3-3 

3.5 Example .................................................................................... 3-4 

3.5.1 Building Description ................................................... 3-4 

3.5.2 Average Effective Shear Wave Velocity ..................... 3-6 

3.5.3 Fundamental Period of the Structure ........................... 3-7 

3.5.4 Structure-to-Soil Stiffness Ratio ................................. 3-7 

3.6 Comparisons Using Site Class ................................................. 3-7 

4. Base Slab Averaging ........................................................................ 4-1 

4.1 Overview .................................................................................. 4-1 

4.2 Key Equations .......................................................................... 4-2 

4.3 Determining Effective Foundation Size ................................... 4-3 

4.4 Determining whether Base Slab Averaging Applies ................ 4-3 

4.5 Limitations and Issues .............................................................. 4-4 

4.6 Example .................................................................................... 4-5 
vi Table of Contents FEMA P-2091

5. Embedment Effects .......................................................................... 5-1 

5.1 Overview................................................................................... 5-1 

5.2 Key Equation ............................................................................ 5-1 

5.3 Determining Depth of Embedment ........................................... 5-2 

5.4 Limitations and Issues .............................................................. 5-6 

5.5 Example .................................................................................... 5-7 

6. Foundation and Soil Flexibility ....................................................... 6-1 

6.1 Overview................................................................................... 6-1 

6.2 Soil Properties for Flexibility Calculations .............................. 6-2 

6.3 Vertical and Rotational Springs ................................................ 6-3 

6.3.1 Method 1 – Rigid Foundation and Flexible Soil .......... 6-3 

6.3.2 Method 2 – Flexible Foundation and Nonlinear 

Flexible Soil ................................................................. 6-8 

6.3.3 Method 3 – Flexible Foundation and Linear Flexible 

Soil ............................................................................... 6-8 

6.4 Horizontal Springs .................................................................... 6-9 

6.4.1 Method 1 .................................................................... 6-10 

6.4.2 Resistance due to Passive Pressure ............................ 6-12 

6.4.3 Resistance due to Friction or Cohesion ..................... 6-13 

6.5 Bounding Analyses ................................................................. 6-14 

6.6 Vertical and Rotational Spring Example ................................ 6-14 

6.6.1 Method 1 .................................................................... 6-15 

6.6.2 Method 3 .................................................................... 6-15 

6.7 Horizontal Spring Example .................................................... 6-16 

6.7.1 Method 1 .................................................................... 6-16 

6.7.2 Passive Pressure and Friction .................................... 6-17 

7. Period Lengthening .......................................................................... 7-1 

7.1 Overview................................................................................... 7-1 

7.2 Key Equation ............................................................................ 7-2 

7.3 Examples to Illustrate Dynamic Response ............................... 7-5 

7.4 Example Using Code Equations ............................................. 7-11 

7.4.1 Calculations for the Fixed-Base Condition ................ 7-12 

7.4.2 Calculations for the Flexible-Base Condition ............ 7-13 

7.4.3 Summary of Design Demands ................................... 7-15 

8. Foundation Damping ........................................................................ 8-1 

8.1 Overview................................................................................... 8-1 

8.2 Foundation Damping Requirements ......................................... 8-2 

8.3 Effective Damping Ratio .......................................................... 8-4 

8.4 Soil Damping ............................................................................ 8-5 

8.4.1 Discussion .................................................................... 8-5 

8.4.2 Soil Damping Example ................................................ 8-6 

8.5 Radiation Damping ................................................................... 8-8 

8.5.1 Discussion .................................................................... 8-8 

8.5.2 Radiation Damping Example ....................................... 8-9 

8.6 Limitations and Issues ............................................................ 8-16 

8.6.1 Limitations on the Influence of Radiation 

Damping .................................................................... 8-16 

8.6.2 Potential Code Changes Related to Foundation 

Damping .................................................................... 8-16 
FEMA P-2091 Table of Contents vii

9. How to Model a Basement ............................................................... 9-1 

9.1 Overview .................................................................................. 9-1 

9.2 Past Analytical Comparisons of Different Models ................... 9-1 

9.2.1 Models Studied ............................................................ 9-1 

9.2.2 Recommended Models ................................................ 9-4 

9.3 Models for Buildings on Relatively Level Grade ..................... 9-5 

9.3.1 Example Building ........................................................ 9-5 

9.3.2 Model 1 for Example Building on Level Grade .......... 9-5 

9.3.3 Model 2 for Example Building on Level Grade .......... 9-7 

9.4 Models for Buildings on Sloping Grade ................................... 9-9 

9.4.1 Example Building ........................................................ 9-9 

9.4.2 Model 2 for Example Building .................................. 9-10 

10. Conclusions and Recommendations ............................................. 10-1 

10.1 Overview ................................................................................ 10-1 

10.2 Conclusions ............................................................................ 10-1 

10.3 Other Helpful Resources ........................................................ 10-2 

10.4 Comparison of SSI Provisions ............................................... 10-2 

10.5 Recommendations .................................................................. 10-2 

10.5.1 Recommendations for Code Developers ................... 10-2 

10.5.2 Recommendations for Further Study ........................ 10-6 

Appendix A: Two-Story Building Example ........................................... A-1 

A.1 Overview ................................................................................. A-1 

A.1.1 Outline of the Example............................................... A-2 

A.1.2 Limitations in ASCE/SEI 7-16 Chapter 19 ................ A-2 

A.2 Site Description ....................................................................... A-2 

A.3 Building Description ............................................................... A-5 

A.3.1 Gravity Framing ......................................................... A-5 

A.3.2 Lateral Framing .......................................................... A-5 

A.3.3 Gravity Loads and Seismic Weight ............................ A-6 

A.3.4 Foundation System ..................................................... A-7 

A.4 Initial Design ........................................................................... A-9 

A.4.1 Base Shear per ASCE/SEI 7-16 Chapter 12 

Equivalent Lateral Force Procedure ........................... A-9 

A.4.2 Brace Design ............................................................ A-10 

A.4.3 Foundation Design ................................................... A-11 

A.4.4 Fixed-base Modal Analysis ...................................... A-15 

A.5 Flexible-base Analysis .......................................................... A-15 

A.5.1 Vertical Springs ........................................................ A-16 

A.5.2 Lateral Springs ......................................................... A-17 

A.5.3 Flexible-base Modal Analysis .................................. A-20 

A.6 Foundation Damping ............................................................. A-21 

A.6.1 Soil Hysteretic Damping .......................................... A-22 

A.6.2 Radiation Damping .................................................. A-22 

A.6.3 Adjusted Base Shear ................................................. A-26 

A.6.4 Code-limited Adjusted Base Shear ........................... A-27 

A.7 Design for SSI-Reduced Base Shear per ASCE/SEI 7-16 

Chapter 19 ............................................................................. A-27 

A.7.1 Brace Design ............................................................ A-27 

A.7.2 Foundation Design ................................................... A-28 

A.7.3 Fixed-base Modal Analysis ...................................... A-28 

A.7.4 Flexible-base Modal Analysis .................................. A-29 
viii Table of Contents FEMA P-2091

A.7.5 Check of Base Shear ................................................. A-29 

A.8 Design for Full Theoretical SSI-Reduced Base Shear ........... A-30 

A.8.1 Base Slab Averaging................................................. A-30 

A.8.2 Brace Design ............................................................. A-32 

A.8.3 Foundation Design .................................................... A-33 

A.8.4 Fixed-base Modal Analysis ...................................... A-33 

A.8.5 Flexible-base Modal Analysis .................................. A-33 

A.8.6 Check of Base Shear ................................................. A-34 

A.8.7 Redesign (Iteration) .................................................. A-34 

A.8.8 Reanalysis ................................................................. A-34 

A.8.9 Check of Base Shear ................................................. A-35 

Appendix B: Twelve-Story Building Example ....................................... B-1 

B.1 Overview.................................................................................. B-1 

B.1.1 Outline of the Example ............................................... B-1 

B.2 Site Description ....................................................................... B-2 

B.2.1 Geotechnical Properties .............................................. B-2 

B.2.2 Seismic Ground Motion Parameters ........................... B-3 

B.3 Building Description ................................................................ B-3 

B.3.1 Framing Systems ........................................................ B-3 

B.3.2 Material Properties and Modeling Criteria ................. B-8 

B.3.3 Gravity Loads and Seismic Mass ............................... B-9 

B.4 Initial Design for Model 1 ..................................................... B-10 

B.4.1 Base Shear per ASCE/SEI 7-16 Chapter 12 

Equivalent Lateral Force Procedure ......................... B-10 

B.4.2 Fixed-base Modal Analysis ...................................... B-13 

B.4.3 Story Shears and Moments ....................................... B-13 

B.4.4 Story Drifts ............................................................... B-13 

B.4.5 Foundation Design .................................................... B-13 

B.5 Spectral Modifications for Kinematic SSI ............................. B-17 

B.5.1 Base Slab Averaging................................................. B-17 

B.5.2 Embedment ............................................................... B-18 

B.6 SSI Adjusted Structural Demands ......................................... B-20 

B.6.1 Period Elongation Due to Flexible Base ................... B-20 

B.6.2 Foundation Damping ................................................ B-21 

B.6.3 ELF Base Shear Adjusted for SSI ............................ B-23 

B.6.4 Story Shears, Moments, and Drifts ........................... B-24 

B.7 Design Variations .................................................................. B-27 

B.7.1 Building Without Basement, Fixed Base7 

(Model 3) .................................................................. B-27 

B.7.2 Building Without Basement, Flexible Base 

(Model 4) .................................................................. B-27 

B.7.3 Sensitivity to Lower-Bound and Upper-Bound 

Spring Stiffnesses ..................................................... B-29 

Appendix C: Helpful Resources .............................................................. C-1 

References .................................................................................................. D-1 

Glossary ...................................................................................................... E-1 

Symbols ...................................................................................................... F-1 

Project Participants ................................................................................... G-1 
FEMA P-2091 List of Figures ix

List of Figures

Figure 1-1 Types of soil-structure interaction modeling and scope 

of the Guide ....................................................................... 1-2 

Figure 1-2 Illustration of free-field motion and its relationship to 

kinematic interaction and inertial interaction..................... 1-3 

Figure 2-1 The building on the left with the larger footprint size 

will have lower design forces than the building with the 

smaller footprint on the right ............................................. 2-1 

Figure 2-2 The building on the left with a deeper foundation 

embedment will typically have a greater reduction in 

design forces than the building on the right....................... 2-2 

Figure 2-3 A structure where soil flexibility will have a significant 

impact on the lateral displacement and fundamental 

period of the structure ........................................................ 2-3 

Figure 2-4 Significant impacts of period lengthening and 

foundation damping on spectral response. The solid 

black line is the fixed-based spectral response; the 

dashed black line is the reduction due to the addition of 

foundation damping to the overall damping ratio.............. 2-4 

Figure 2-5 The significant impact of soil flexibility on a coupled 

braced frame system. Utilization ratio is the acceptance 

ratio as defined in FEMA (2018) and is similar to the 

demand-to-capacity ratio ................................................... 2-5 

Figure 2-6 The significant impact of soil flexibility on a reinforced 

concrete shear wall system................................................. 2-6 

Figure 3-1 Example building for rule of thumb test............................ 3-4 

Figure 3-2 Low strain shear wave velocity at example building 

site...................................................................................... 3-5 

Figure 4-1 Example building base configurations, showing 

differing degrees of interconnectivity................................ 4-4 

Figure 4-2 Example building foundation plan, representative of 

Condition b in Figure 4-1 ................................................... 4-5 

Figure 4-3 RRSbsa reduction as a function of period ............................ 4-7 
x List of Figures FEMA P-2091

Figure 4-4 Example resulting response spectrum for SDS =1.0 and 

SD1 = 0.6 ............................................................................. 4-7 

Figure 5-1 Depth of embedment for a building on a mat 

foundation .......................................................................... 5-2 

Figure 5-2 Depth of embedment for a building on grade beams and 

spread footings................................................................... 5-3 

Figure 5-3 Depth of embedment for a building with a basement on 

perimeter strip footings and interior spread footings ......... 5-3 

Figure 5-4 Maximum depth of embedment ......................................... 5-4 

Figure 5-5 Depth of embedment for a smaller partial basement.......... 5-5 

Figure 5-6 Depth of embedment for a larger partial basement ............ 5-5 

Figure 5-7 Depth of embedment on a sloping site ............................... 5-6 

Figure 5-8 Example building for embedment effects .......................... 5-7 

Figure 5-9 Shear wave velocity at example building........................... 5-8 

Figure 5-10 Site-specific response spectral acceleration 

modification factor for foundation embedment, RRSe...... 5-11

Figure 5-11 Response spectra for example building, with and 

without embedment SSI................................................... 5-11

Figure 6-1 Three methods for foundation modeling approaches 

with vertical and rotational springs presented in 

ASCE/SEI 41 ..................................................................... 6-4 

Figure 6-2 Definitions and orientations of dimensions and axes......... 6-5 

Figure 6-3 Variation of vertical stiffness with foundation 

dimensions.......................................................................... 6-5 

Figure 6-4 Variation of rotational stiffness with foundation 

dimensions.......................................................................... 6-6 

Figure 6-5 Variation of the embedment modifier vertical stiffness 

with foundation dimensions............................................... 6-7 

Figure 6-6 Variation of embedment modifiers for rotational 

stiffness with foundation dimensions................................. 6-7 

Figure 6-7 Vertical stiffness per unit length computed via Method 1, 

for various foundation half-lengths and half-widths, with 

stiffness computed by Method 3 for comparison ............... 6-9 

Figure 6-8 Variation of horizontal stiffness with foundation 

dimensions........................................................................ 6-11 
FEMA P-2091 List of Figures xi

Figure 6-9 Variation of the embedment modifier horizontal 

stiffness with foundation dimensions............................... 6-12 

Figure 6-10 Passive pressure mobilization curve ................................ 6-13 

Figure 7-1 Example acceleration and displacement spectra................ 7-2 

Figure 7-2 Example acceleration and displacement spectra, 

including points for each example building....................... 7-6 

Figure 7-3 Story shears for example buildings.................................... 7-8 

Figure 7-4 Story moments for example buildings............................... 7-9 

Figure 7-5 Story drift ratios for example buildings........................... 7-10

Figure 7-6 Displaced shape for flexible-base, 10-story RCSW 

building ............................................................................ 7-11

Figure 7-7 Design response spectrum for the example building ....... 7-12

Figure 8-1 Example of a foundation and superstructure where 

foundation damping is not permitted because the 

unconnected footings are too close .................................... 8-3 

Figure 8-2 Determining slab-on-grade in-plane flexibility.................. 8-4 

Figure 9-1 Illustration of an embedded building configuration with 

a basement surrounded by soil and a level grade on all 

sides. ug represents the free field motion .......................... 9-1 

Figure 9-2 Modeling approaches for buildings with basements. 

ug represents the free field motion, and uFIM represents 

the foundation input motion. In Model 3 and the full 

substructure model, the little boxcars are sliders to which 

the free-field motion is applied. They do not have mass. 

The mass of the soil is included in the calculation of the 

free-field motions (and, as applicable, its depthdependence) ....................................................................... 9-2 

Figure 9-3 Simplified version of Model 4 in Figure 9-2. ug 

represents the free field motion, and uFIM represents the 

foundation input motion..................................................... 9-2 

Figure 9-4 Example building with basement and relatively level 

grade .................................................................................. 9-5 

Figure 9-5 Model 1 distribution of forces for example building ......... 9-7 

Figure 9-6 Model 2 distribution of forces for example building ......... 9-9 

Figure 9-7 Example building on a sloping site.................................. 9-10

Figure 9-8 Model 2 distribution for example building on a slope ..... 9-11
xii List of Figures FEMA P-2091

Figure A-1 Generalized soil profile for the Loma Linda Site.............. A-4 

Figure A-2 Design acceleration response spectrum per 

ASCE/SEI 7-16 Chapters 11 and 12 ................................. A-5 

Figure A-3 Overall view...................................................................... A-7 

Figure A-4 Typical floor framing plan (roof framing similar) ............ A-8 

Figure A-5 Foundation plan................................................................. A-8 

Figure A-6 Free-body diagram of a typical transverse frame 

showing the 25%-reduced overturning demands and 

the maximum factored gravity loads............................... A-13

Figure A-7 Passive pressure mobilization curve ............................... A-17

Figure A-8 First-mode displaced shape of a typical frame with 

foundation flexibility included........................................ A-21

Figure A-9 Reductions for base slab averaging relative to building 

period for three values of effective foundation size ........ A-32

Figure B-1 Design response spectra for the example building............ B-3 

Figure B-2 Typical floor plan of the example building....................... B-4 

Figure B-3 Typical sections of the example building.......................... B-5 

Figure B-4 Analytical model for base case building (Model 1) .......... B-6 

Figure B-5 Foundation plan for the example building ........................ B-7 

Figure B-6 Arrangement of piles and columns or boundary 

elements at each pile group ............................................... B-8 

Figure B-7 Story shears and moments for Model 1, ELF vs. 

MRSA ............................................................................. B-14

Figure B-8 Story drifts for Model 1, ELF vs. MRSA........................ B-15

Figure B-9 Lateral pile analysis results vs. depth.............................. B-17

Figure B-10 Pile group lateral stiffness for Models 1 and 2................ B-18

Figure B-11 Kinematic SSI spectral modification factors and design 

response spectra for the example building ...................... B-19

Figure B-12 Analytical model for the example building, with spring 

supports (Model 2).......................................................... B-21 

Figure B-13 Story shears and moments for Models 1 and 2 ............... B-25 

Figure B-14 Story drifts for Models 1 and 2 ....................................... B-26 
FEMA P-2091 List of Figures xiii

Figure B-15 Analytical model of the example building, with 

basement removed and fixed base (Model 3) .................. B-27 

Figure B-16 Analytical model of the example building, with 

basement removed and flexible base (Model 4) .............. B-28 

Figure B-17 Pile group lateral stiffness for Models 3 and 4 ................ B-29

FEMA P-2091 List of Tables xv

List of Tables

Table 3-1 Measured Shear Wave Velocity, vsi, at Example 

Building ............................................................................. 3-5 

Table 3-2 Structure-to-Soil Stiffness Ratio, h′ / (vs T), for 

r = e + zp = 33 ft................................................................. 3-9 

Table 3-3 Structure-to-Soil Stiffness Ratio, h′ / (vs T), for 

r = e + zp = 52.5 ft............................................................ 3-10

Table 4-1 Values of RRSbsa at Selected Periods for be = 200 ft.......... 4-6 

Table 5-1 Measured Shear Wave Velocity at Example Building....... 5-8 

Table 5-2 Response Spectral Ordinates with and without 

Embedment Effects.......................................................... 5-10

Table 7-1 Fundamental Mode Dynamic Properties for Example 

Buildings............................................................................ 7-5 

Table 7-2 Summary of Design Demands for Example Building ..... 7-16

Table 8-1 Soil Hysteretic Damping Ratio, βs ..................................... 8-6 

Table 9-1 Model 1 Distribution of Forces.......................................... 9-6 

Table 9-2 Model 2 Distribution of Forces.......................................... 9-8 

Table 10-1 Comparison of SSI Provisions......................................... 10-3 

Table A-1 Flat Loads on Roof ........................................................... A-6 

Table A-2 Flat Loads on Second Floor.............................................. A-6 

Table A-3 Summary of Seismic Weights........................................... A-7 

Table A-4 Vertical Distribution of Seismic Forces.......................... A-10

Table A-5 Transverse Brace Core Areas.......................................... A-10

Table A-6 Longitudinal Brace Core Areas ...................................... A-10

Table A-7 Resisting Moments ......................................................... A-13

Table A-8 Soil Pressures.................................................................. A-14 
xvi List of Tables FEMA P-2091

Table A-9 Resisting Moments.......................................................... A-14

Table A-10 Soil Pressures.................................................................. A-15 

Table A-11 Total Lateral Resistance - Transverse Direction at 

Displacement of 0.015 in ................................................ A-19

Table A-12 Total Lateral Resistance - Longitudinal Direction at 

Displacement of 0.035 in ................................................ A-19

Table A-13 Spring Values.................................................................. A-20

Table A-14 Fundamental Periods for Fixed- and Flexible-Base 

Models............................................................................. A-21

Table A-15 Tabulation of Reduced Seismic Response ...................... A-27

Table A-16 Transverse Brace Core Areas.......................................... A-28

Table A-17 Longitudinal Brace Core Areas....................................... A-28

Table A-18 Updated Spring Values.................................................... A-29

Table A-19 Updated Fundamental Periods for Fixed-base and 

Flexible-base Models...................................................... A-29

Table A-20 Tabulation of Reduced Seismic Response ...................... A-29

Table A-21 Tabulation of Reduced Seismic Response with and 

without Base Slab Averaging.......................................... A-31

Table A-22 Transverse Brace Core Areas.......................................... A-33 

Table A-23 Longitudinal Brace Core Areas....................................... A-33

Table A-24 Updated Fundamental Periods for Fixed-base and 

Flexible-base Models...................................................... A-33

Table A-25 Tabulation of Reduced Seismic Response ...................... A-34

Table A-26 Updated Fundamental Periods for Fixed-base and 

Flexible-base Models...................................................... A-34

Table A-27 Tabulation of Reduced Seismic Response ...................... A-35

Table B-1 Site Characteristics............................................................ B-2 

Table B-2 Response Modification Coefficient, Overstrength 

Factor, and Deflection Amplification Factor for 

Structural Systems Used.................................................... B-6 

Table B-3 Story Weights and Masses .............................................. B-10
FEMA P-2091 List of Tables xvii

Table B-4 Fundamental Periods for Model 1 (Fixed-Base) and 

Model 2 (Flexible-Base) .................................................. B-21 

Table B-5 Design Base Shear for Model 1 (Fixed-Base) and 

Model 2 (Flexible-Base), Using Best-Estimate Springs.. B-24 

Table B-6 Summary of Periods and Base Shears for All Models, 

E-W Direction.................................................................. B-31 

FEMA P-2091 1: Introduction 1-1 

Chapter 1 

Introduction

1.1 Overview

Soil-structure interaction (SSI) can make a substantial difference in how 

buildings behave during earthquake shaking and how they should be 

designed, and yet there is relatively little implementation of SSI effects by 

practicing engineers. Provisions are available in Chapter 19 of ASCE/SEI

7-16 Minimum Design Loads and Associated Criteria for Buildings and 

Other Structures (ASCE, 2017a) and in Chapter 8 of ASCE/SEI 41-17 

Seismic Evaluation and Retrofit of Existing Buildings (ASCE, 2017b) that 

can be used to address SSI. Another valuable resource that was developed to 

synthesize fundamental principles and procedures of SSI for application to 

buildings is Soil-Structure Interaction for Building Structures (NIST, 2012a). 

The overall goal of this Guide is to present information regarding SSI as 

implemented in code provisions but in an easy-to-understand, concise format 

targeted towards practicing engineers to help them determine when SSI 

effects are of importance and show them examples of how to implement 

them in design. Relevant recent research findings that help clarify 

implementation are provided.

1.2 Types of Soil-Structure Interaction Modeling

There are two common approaches to modeling the interrelationships 

between a structure, its foundation, and the soil that supports it, including the 

flexibility and damping of the soil. 

One approach is called the substructure approach where the soil is 

represented with springs. The springs are typically vertically oriented to 

capture foundation rotations, which are often the dominant contributor to SSI 

effects. Often, the foundation is fixed against horizontal translation. 

However, horizontal springs can be used to capture the ability of the 

foundation to displace horizontally relative to the free-field, and dampers can 

be included to capture foundation damping. 

A second approach, called the direct analysis approach, is where the soil 

and the structure are both modeled using finite elements. The soil modeling 

extends sufficiently around and beneath the building to account for site 

properties, and seismic waves are imparted at the boundary of the soil and 
1-2 1: Introduction FEMA P-2091

excite the soil elements which in turn excite the structure. The structure, 

with its inertial weight and other properties, will in turn affect the behavior of 

the soil. In current practice, the direct method approach is typically only 

undertaken on large, critical projects like nuclear power plants or large 

infrastructure projects such as major bridges, tunnels, subway stations, tanks 

and marine structures, and requires specialist expertise. 

This Guide covers the substructure approach with equivalent springs. Some 

simple SSI techniques do not require development and modeling with springs 

but rather use calculations of building and site properties to make adjustments 

in seismic demands. The Guide also covers these topics. Figure 1-1 provides 

a summary of modeling approaches and what is covered by the Guide.

Figure 1-1 Types of soil-structure interaction modeling and scope of the Guide. 

1.3 Purpose of the Guide

The purpose of the Guide is (1) to help practicing engineers know when 

incorporating SSI would be important and (2) to show examples of how to 

implement different SSI techniques.

1.4 Scope of the Guide

The Guide covers the SSI topics in ASCE/SEI 7-16 Section 12.13 and 

Chapter 19, including base slab averaging, foundation embedment, 

foundation and soil flexibility, period lengthening, and foundation damping. 

It also provides advice on modeling of basements and gives easy-to-use 

rules-of-thumb on when SSI is likely to be of significance. The focus is on 

techniques that practicing engineers can use. As such, soil flexibility is 

addressed through the use of springs, rather than by finite element modeling. 

1.5 Target Audience for the Guide

The primary target audience for the Guide is practicing engineers who are 

familiar with seismic design using ASCE/SEI 7 but who have little to no 

experience with SSI. A secondary audience is engineers who have some 
FEMA P-2091 1: Introduction 1-3 

experience with some SSI techniques, such as using springs, but may need 

advice on other SSI techniques they have not utilized. 

1.6 Key SSI Terminology

It is helpful to understand some key SSI terms that are used in ASCE/SEI

7-16 and ASCE/SEI 41-17. The terms are also illustrated in Figure 1-2. 

Motion at the ground surface in the absence of a structure and its foundation 

is called free-field motion. It is typically larger than the foundation input 

motion that effectively excites the structure and its foundation. The 

modification of free-field motion into the foundation input motion comes 

from kinematic interaction. The foundation input motion can be viewed as 

the motion applied to the ends of horizontal foundation springs; it differs 

from the actual motion of the foundation due to the inertial response of the 

structure and the deflections that response produces in foundation springs. 

That response is referred to as inertial interaction. 

ug

θ FIM

uFIM

Incident 

Waves

ug

uFIM uFIM

Actual system: 

embedded building 

experiencing freefield motion, ug

Kinematic interaction 

modifies free-field motion, 

ug, into foundation input 

motion, uFIM

Inertial interaction is the dynamic 

interaction between the structure, 

its foundation, and the surrounding 

soil caused by the foundation input 

motion, uFIM

Figure 1-2 Illustration of free-field motion and its relationship to kinematic interaction and inertial 

interaction (modified from Figure 1-2 in NIST, 2012a). 

Kinematic interaction modifies the free-field ground motion to a foundation 

input motion as a result of spatial variability in the free-field ground motions. 

Those spatially variable motions are averaged within the building envelope 

and over the foundation depth within the foundation footprint due to the 

stiffness and strength of the foundation system. The two kinematic SSI 

effects are (1) base slab averaging of shallow (nonembedded) foundations 

caused primarily by wave incoherence over the base area and secondarily by 

incoming seismic waves at non-vertical incidence; and (2) embedment
1-4 1: Introduction FEMA P-2091

effects in which foundation-level motions are reduced as a result of ground 

motion reduction with depth below the free surface in structures with 

embedded foundations. Base slab averaging and embedment effects can be 

combined.

Inertial interaction is the dynamic interaction between the structure, its 

foundation, and the surrounding soil caused by the foundation input motion, 

and includes: (1) period lengthening, the increase in the building period due 

to foundation flexibility; (2) radiation damping, the damping in the soilstructure system caused by the generation and propagation of waves away 

from the foundation, which are caused by dynamic displacements of the 

foundation relative to the free-field displacements; and (3) soil damping, the 

hysteretic (material) damping of the soil, similar to inherent viscous damping 

in the superstructure, but it is independent of the flexible-base period of the 

structure. 

1.7 Tips for Understanding and Implementing SSI

Based on experience in performing SSI analyses, the following general 

observations are offered. These observations are discussed in detail in the 

Guide. Blue comment boxes are used to highlight useful tips and to 

summarize key points in the Guide. 

Blue comment boxes are 

used in the Guide to 

highlight useful tips and 

summarize key points. 

• SSI is not that difficult to implement.

• SSI is typically iterative, so it may require additional rounds of analysis, 

as compared to a fixed base analysis, to converge on the final solution. 

• SSI typically reduces the seismic demands that are used for design, but 

there are unusual cases with site-specific response spectra where 

demands can increase because period elongation may lead to climbing up 

the response spectrum with increasing levels of spectral acceleration. 

• Adding foundation flexibility to a model can change deformation 

patterns in the superstructure, which can stress particular elements in a 

different way than in a fixed base model. For example, adding 

foundation flexibility can lead to rocking of shear walls or braced frames 

and increased demands on couple beams connecting rocking elements. 

• Effective shear wave velocity, vs, is a key parameter in several SSI 

equations and techniques. The effective shear wave velocity differs from 

the low-strain shear wave velocity, vso, used for site classification in 

ASCE/SEI 7-16 Chapter 20. The parameter, vs, depends on soil type, 

site spectral acceleration, and the depth of importance. The Guide 

provides detailed guidance on the subtle, but important, differences 

between vs and vso. 
FEMA P-2091 1: Introduction 1-5 

• There are a number of code provisions, both in ASCE/SEI 7-16 Chapter 

12 and Chapter 19, that can limit the extent of SSI reductions that can be 

utilized.

• Although ASCE/SEI 7-16 is the standard that is referenced and used in 

the Guide’s design examples, ASCE/SEI 41-17 has a similar set of SSI 

provisions. In some cases, ASCE/SEI 41-17 has more relaxed 

requirements and limitations regarding the use of SSI. The Guide

highlights these differences, typically in a green comment box. 

ASCE/SEI 41-17

Although ASCE/SEI 7-16 is 

the standard that is 

referenced and used in the 

design examples in the 

Guide, key differences 

between the SSI provisions 

in ASCE/SEI 41-17 and 

those in ASCE/SEI 7-16 are 

noted in green comment 

boxes.

1.8 Organization of the Guide

The remainder of the Guide is organized into the following chapters. 

• Chapter 2, Situations Where SSI is Important, provides a series of 

example images and related discussion of situations that engineers 

commonly encounter where SSI can impact the forces used in design and 

the way the structure responds to earthquake shaking. 

• Chapter 3, Rule-of-Thumb Test for SSI Importance, describes a 

simple test that can be used at the start of a project when only very 

limited information is available to help determine if using SSI will likely 

make a difference in results. The rule of thumb is targeted at inertial 

interaction and does not provide information about the potential 

significance of kinematic interaction. 

• Chapter 4, Base Slab Averaging, addresses how the interconnectivity 

of the foundation can help reduce the demands into the structure. It 

provides examples of common foundation situations and whether base 

slab averaging can be used.

• Chapter 5, Embedment Effects, discusses how foundation embedment 

can reduce the demands on the structure. 

• Chapter 6, Foundation and Soil Flexibility, reviews different methods 

for adding vertical and horizontal springs to represent soil flexibility. 

• Chapter 7, Period Lengthening, covers provisions for how soil 

flexibility leads to period lengthening in the structural response and the 

resulting impact on seismic demands.

• Chapter 8, Foundation Damping, shows how and when to apply two 

types of foundation damping—called radiation damping and soil 

damping—that can reduce demands on the structure.

• Chapter 9, How to Model a Basement, discusses different simple but 

accurate analytical approaches to modeling basements.
1-6 1: Introduction FEMA P-2091

• Chapter 10, Conclusions and Recommendations, summarizes key 

points regarding SSI discussed in the Guide and provides 

recommendations on revisions needed to code SSI provisions and further 

SSI studies that should be undertaken. 

• Appendix A, Two-Story Building Example, provides a detailed 

example of applying different SSI techniques for a two-story steel 

buckling-restrained braced frame building supported by spread footings. 

SSI topics include implementation of soil springs and reduction in 

seismic demands due to foundation and soil flexibility, foundation

damping, and base slab averaging. 

• Appendix B, Twelve-Story Building Example, provides a detailed 

example of applying different SSI techniques for a 12-story concrete 

building that has a moment frame in one direction and a dual system with 

a moment frame and shear wall in the other direction. It is pile supported 

and includes variations with and without a basement. SSI topics covered 

include reduction in design demands due to base slab averaging and 

foundation embedment, adjustments to demands from period elongation 

and foundation damping, and impacts of limitations imposed by 

ASCE/SEI 7-16 provisions. 

The Guide also includes lists of helpful resources for SSI and references 

cited, as well as a glossary of key terms, a list of symbols used, and a list of 

project participants.

Computer output is shown in some design examples. FEMA, the authors, 

and the project participants do not endorse any particular computer software 

program or vendor. 
FEMA P-2091 2: Situations Where SSI is Important 2-1 

Chapter 2 

Situations Where SSI is Important

2.1 Overview

In some situations, soil-structure interaction (SSI) can make a substantial 

difference in how buildings behave during earthquake shaking and in the 

forces used in their design. This chapter shows qualitative, graphic examples 

of such situations that can arise in practice. Quantitative, detailed discussion 

is covered in subsequent chapters.

Topics Covered

This chapter shows graphic 

examples of situations that 

arise in practice where SSI 

can make a substantial 

difference in how the 

building behaves and in the 

forces used in their design. 

Examples include:

• Large building footprint. 

• Substantial foundation 

embedment. 

• Period lengthening 

changes design forces. 

• Soil flexibility induces 

rocking of the 

superstructure and 

impacts behavior.

2.2 Large Building Footprint Reduces Design Forces

Building footprint size has been shown to correlate with spectral demands, 

primarily in the shorter period range. The larger the building footprint, the 

greater the reduction in short period spectral response. This is due to the 

kinematic interaction effects of base slab averaging. Chapter 4 of this Guide 

discusses base slab averaging in detail.

Figure 2-1 shows an example of a large building where the reduction will be 

greater and a smaller building where the reduction will be smaller. 

Plan A-A Plan B-B

Elevation Elevation

A A B B

Figure 2-1 The building on the left with the larger footprint size will have 

lower design forces than the building with the smaller footprint 

on the right. 
2-2 2: Situations Where SSI is Important FEMA P-2091

2.3 Substantial Foundation Embedment Reduces Design 

Forces 

Foundation embedment has been shown to correlate with spectral demands, 

primarily in the shorter period range. Typically, the deeper the embedment, 

the greater the reduction in short period spectral response. This is due to the 

decrease of ground motion amplitudes with depth, which is a typical feature

of site response. Chapter 5 of this Guide discusses kinematic interaction 

effects from embedment effects in detail.

Combining SSI Effects

The reduction in design 

forces from large building 

footprints and foundation 

embedment can be and 

often are combined.

Figure 2-2 shows an example of a building with a basement where the 

reduction in demand is expected to be greater and a building without a 

basement where there is no reduction. 

Figure 2-2 The building on the left with a deeper foundation embedment 

will typically have a greater reduction in design forces than the 

building on the right.

2.4 High Structure-to-Soil Stiffness Ratios Will Lengthen 

Period and Change Design Forces

When the structure is relatively stiff compared to the soil, foundation rotation 

and translation relative to the free-field motion can occur adding to structural 

displacements and increasing or lengthening the fundamental period of the 

structure. The increase in period can affect the associated spectral 

accelerations used in design. This effect commonly occurs in buildings with 

a concentrated lateral force-resisting system, such as reinforced concrete 

shear walls and steel braced frames that are supported on localized 

foundation elements on flexible soils. Conversely, for a building with wide, 

stiff foundations on very stiff soils and relatively flexible superstructures, the 

impact of soil flexibility on the building response is typically relatively 

small. 

Figure 2-3 shows an example of a concentrated cantilever shear wall and 

foundation system where including soil flexibility will increase the 

fundamental period of vibration. The roof displacement of the shear wall 
FEMA P-2091 2: Situations Where SSI is Important 2-3

itself is shown in the left as Δw. In the figure on the right, the vertical 

flexibility of the soil is represented by a set of springs. During lateral 

displacement of the superstructure, there will be vertical displacement of the 

springs and foundation rotation. The drift from foundation rotation is shown 

in the figure on the right as Δr. The increase in displacement correlates with 

an increase in fundamental period for the structure. Figure 2-3 is shown at 

the shear wall level of detail to provide a scale to illustrate the impact of 

rotation more clearly and easily. If there are additional walls and other 

structural elements, they can impact the overall fundamental period. 

Development of springs is covered in detail in Chapter 6 of this Guide. 

∆w ∆w

∆TOT

∆r

Figure 2-3 A structure where soil flexibility will have a significant impact on 

the lateral displacement and fundamental period of the 

structure.

Figure 2-4 shows the potential impact of changing the period in a response

spectrum analysis. The period with the fixed base model is denoted as T, and 

the period with soil flexibility is denoted as T ̃

. Two cases are shown. In the 

short period case, the structure is very stiff, and the increase from T to T ̃

results in climbing up the response spectrum and an increase in spectral

acceleration. In the long period case, the structure is more flexible, and the 

increase in period results in a reduction in spectral acceleration. Period

lengthening is discussed in detail in Chapter 7 of this Guide. Figure 2-4 also

shows how the addition of foundation damping to the overall damping ratio 

reduces the spectral acceleration at most periods. Foundation damping is

discussed in detail in Chapter 8 of the Guide. 
2-4 2: Situations Where SSI is Important FEMA P-2091

Figure 2-4 Significant impacts of period lengthening and foundation 

damping on spectral response. The solid black line is the fixedbased spectral response; the dashed black line is the reduction 

due to the addition of foundation damping to the overall 

damping ratio. (Adapted from Figure 2-3 in NIST, 2012a). 

2.5 Foundation Rocking Impacts Superstructure

Behavior 

Structures with concentrated coupled vertical lateral force-resisting systems 

can behave much differently when soil flexibility is introduced. Figure 2-5 

shows a nonlinear static (pushover) analysis example. In the fixed-base 

model shown on top, the mechanism is buckling of the compression brace in 

the lowest story, and the brace is highly overstressed. In the flexible-base 

model shown on the bottom where vertical springs are located under each 

column, the braced frames rock, and the system has sufficient capacity to 

resist the demands. Note that, in the flexible-base model, the ends of the 

beams linking the frames have higher rotations than they do in the fixed-base 

model.
FEMA P-2091 2: Situations Where SSI is Important 2-5 

Figure 2-5 The significant impact of soil flexibility on a coupled braced 

frame system (from FEMA, 2019). Utilization ratio is the 

acceptance ratio as defined in FEMA (2018) and is similar to the 

demand-to-capacity ratio.

Figure 2-6 shows a reinforced concrete cantilever shear wall coupled to an

adjacent gravity frame of beams and columns. The upper figure shows that a 

stiff, strong foundation system (represented as a fixed-base condition) can 

reduce superstructure displacement and protect the adjacent frame from 

damage, but damage will concentrate in the shear wall base. The lower 

figure has a more flexible foundation system, which leads to rocking of the 

shear wall. This increases displacement and damage in the adjacent frame 

but potentially reduces damage to the shear wall. The response in the lower 

figure could be modeled with vertical soil springs, similar to those in Figure 

2-3. Soil flexibility modeling is discussed in detail in Chapter 6 of this

Guide. 
2-6 2: Situations Where SSI is Important FEMA P-2091

Flexible/Weak Foundation

Stiff/Strong Foundation

Stiff/strong is not always favorable; nor 

is flexible/weak always conservative.

Small displacements 

protect against frame 

damage

Foundation stiffness and strength affect 

various structural components differently.

High forces cause 

shear wall damage

Foundation yielding 

and rocking protects 

shear wall

∆small

∆large

Large displacements 

cause frame damage

Figure 2-6 The significant impact of soil flexibility on a reinforced concrete shear wall 

system (from ATC, 1996). 
FEMA P-2091 3: Rule of Thumb Test for Inertial SSI Significance 3-1 

Chapter 3 

Rule of Thumb Test for Inertial 

SSI Significance

3.1 Overview

A rule of thumb test to determine if inertial soil-structure interaction (SSI) 

inertial effects are likely to be significant was developed in Soil-Structure 

Interaction for Building Structures (NIST, 2012a). This test combines three 

simple parameters that are relatively easy to obtain—building height, 

fundamental period of the structure, and shear wave velocity—into a term 

called the structure-to-soil stiffness ratio and correlates the ratio with the 

likelihood that SSI inertial effects will be significant. This test is not 

contained in ASCE/SEI 7-16 (ASCE, 2017a) or ASCE/SEI 41-17 (ASCE, 

2017b). SSI inertial effects include period lengthening addressed in Chapter 

7 and foundation damping addressed in Chapter 8 of this Guide. 

Rule of Thumb Test

Description: Provides an 

indication before substantial 

design effort of whether SSI 

inertial effects may be 

significant.

Requirements:

• Building height, hn

• Fundamental fixed-based 

period of the structure, T

• Average low strain shear 

wave velocity over the 

rocking depth, vso

• Footprint dimensions for 

the width, 2B, and length, 

2L of the building

• Depth of embedment of 

the foundation, e

• Design spectral response 

acceleration parameter at 

short periods, SDS

• Site class

3.2 Key Equation

The structure-to-soil stiffness ratio is defined as h′/ (vsT). The equation for 

the rule of thumb test is the following inequality:

If h′ / (vsT) > 0.1, then inertial SSI effects are likely to be significant. 

where:

h' = effective structure height, measured from base of foundation to the 

center of mass of the fundamental mode (in ft). It can be 

approximated as 2/3 hn, where hn is the structure height as defined 

in ASCE/SEI 7-16 Section 11.2, plus the depth from grade to the 

bottom of the foundation.

vs = the average effective shear wave velocity (ft/s) for site soil 

conditions, taken as the average value of velocity over the effective 

depth for foundation rotation determined using vso and a velocity 

reduction factor from ASCE/SEI 7-16 Table 19.3-1 or a site-specific 

study. Calculation of vso is discussed in Section 3.3 of this Guide.

T = fundamental mode of the structure assuming a fixed base at grade 

(in seconds). Calculation of T is discussed in Section 3.4 of this 

Guide.
3-2 3: Rule of Thumb Test for Inertial SSI Significance FEMA P-2091

3.3 Determining Average Effective Shear Wave Velocity

Site classification is based on the average low strain shear wave velocity, vso, 

over the top 100 ft (or 30 m) of soil from grade. The average effective shear 

wave velocity, vs, is a reduced value that accounts for the larger strains that 

occur during earthquake shaking. For the rule of thumb test, the depth of 

significance is not the same as that used for site soil classification.

Calculation of the average low strain shear wave velocity over the effective 

depth for foundation rotation is determined as follows.

Step 1: Determine footing embedment depth, e. 

The footing embedment depth, e, is the depth to from grade to the bottom of 

the footing. 

Step 2: Determine the effective profile depth, zp. 

The effective profile depth per Equation 2-18c of NIST (2012a) is the 

parameter, zp, as determined from the footing width and length as follows. 

zp = (B3L)

0.25

where:

B = overall foundation half width (ft). B is measured parallel to the 

direction of loading

L = overall foundation half width (ft). L is measured perpendicular to 

the direction of loading

Step 3: Determine the effective depth for foundation rotation, e + zp. 

The depth of interest for calculation of the average effective shear wave 

velocity is the effective depth for foundation rotation which is the sum of the 

foundation embedment depth and the effective profile depth or e + zp.

Step 4: Determine the average low strain shear wave velocity, vso, over 

the effective depth for foundation rotation. 

Low strain shear wave velocity measurements are those measured for site 

classification. Typically, they are reported or graphed in layers. The average 

is determined using the same formula as ASCE/SEI 7-16 Equation 20.4-1. 

vso = Σdi / Σ(di / vsi) 

where:

di = thickness of any layer between grade and the effective depth for 

foundation rotation

vsi = low strain shear wave velocity in ft/s
FEMA P-2091 3: Rule of Thumb Test for Inertial SSI Significance 3-3 

Step 5: Determine the effective shear wave velocity ratio, vs / vso. 

Low strain shear wave velocities are converted to the effective shear wave 

velocities using the effective shear wave velocity ratio in ASCE/SEI 7-16 

Table 19.3-1. It varies by site class and short period design spectral 

acceleration, SDS.

Step 6: Determine the average effective shear wave velocity, vs. 

Overburden Corrections

NIST (2012a) Section 2.2.2 

also includes a procedure 

for modifying the effective 

shear wave velocity to 

account for the increase due 

to the building weight on the 

soil. This calculation 

involves several steps. With 

an increase in the shear 

wave velocity, the structureto-soil stiffness ratio would

go down and may be less 

likely to exceed the 

threshold value for the rule 

of thumb test. The 

refinement is typically 

relatively minor and not 

necessary for the rule of 

thumb calculation.

The average effective shear wave velocity from the ground surface to depth 

e + zp is computed as follows: 

vs = [vs / vso] (vso) 

where:

vs = average effective shear wave velocity (ft/s) for site soil 

conditions

vso = average low strain shear wave velocity (ft/s) for site soil 

conditions, computed as described in Step 4 

[vs / vso] = effective shear wave velocity ratio from ASCE/SEI 7-16 

Table 19.3-1, as identified in Step 6.

Note that in ASCE/SEI 7-16 Section 19.3.3, the definition of vs uses a depth 

measured from ground surface to depth e, not e + zp.

3.4 Determining Fundamental Period of the Structure

The fundamental period of the structure can be obtained from a computer 

model or by using the approximate fundamental period from ASCE/SEI 7-16 

Equation 12.8-7. 

T = Ta = Ct hn

x

where:

hn = structure height as defined in ASCE/SEI 7-16 Section 11.2 plus 

depth of footing

Ct = coefficient from ASCE/SEI 7-16 Table 12.8-2 that depends on the 

structural system

x = coefficient from ASCE/SEI 7-16 Table 12.8-2 that depends on the 

structural system
3-4 3: Rule of Thumb Test for Inertial SSI Significance FEMA P-2091

3.5 Example

3.5.1 Building Description

Figure 3-1 shows an example building to illustrate the rule of thumb test. It 

is a three-story special concentric braced frame building with one bay of 

chevron bracing on each perimeter line supported by spread footings linked 

by grade beams. It is on Site Class D soil, with SDS = 1.0g and SD1 = 0.6g 

from a site-specific investigation. The rule of thumb test will be evaluated in 

the short direction of the building or parallel to the 2B dimension. 2B = 57' - 0"

6'×6'×3'H, Typ. 6'×6'×3'H 

Typ. at B.F.

25' - 0" 25' - 0"

25' - 0" 25' - 0" 25' - 0"

2L = 83' - 0"

12'

12'

12'

4.5'

6" slabon-grade

B.F.

B.F.

B.F.

B.F.

B

B

L L

Plan A-A

Elevation

A A

Figure 3-1 Example building for rule of thumb test.

For site classification, the low strain shear wave velocity has been measured 

at the site and is shown in Figure 3-2. The measured velocity for each layer 

is shown in Table 3-1. 
FEMA P-2091 3: Rule of Thumb Test for Inertial SSI Significance 3-5 

Figure 3-2 Low strain shear wave velocity at example building site. 

Shear Wave Velocity

There are several similar 

shear wave velocity 

parameters. ASCE/SEI 

7-16 is not consistent with 

the terms: 

In Chapter 20, is the 

“average shear wave 

velocity.” It is taken over the 

top 100 ft of soil. This is 

often designated as for 

30 m. Both and Vs30 are 

the low strain shear wave 

velocity, though this is never 

stated in Chapter 20.

In Chapter 19, vso, for inertial 

SSI is the “average low 

strain shear wave velocity 

over a depth of B below the 

base of the structure.” If the 

depth B were 100 ft it would 

be the same as . 

In Chapter 19, the low strain 

shear wave velocity needs 

to be modified by Table 

19.3-1 to obtain the average 

effective shear wave 

velocity, vs. 

Table 3-1 Measured Shear Wave Velocity, vsi , at Example Building

Depth to Top of Layer 

(ft)

Layer Thickness 

(ft)

Shear Wave Velocity, vsi 

(ft/s)

0.0 3.28 676

3.28 6.56 817

9.84 19.69 1,319

29.53 39.37 1,269

68.90 59.10 1,253

128.0 98.40 1,503
3-6 3: Rule of Thumb Test for Inertial SSI Significance FEMA P-2091

3.5.2 Average Effective Shear Wave Velocity

Since shear wave velocity data exist for the site, then the average effective 

shear wave velocity can be calculated directly as follows.

Step 1: Determine footing embedment depth, e. 

The footing embedment depth, e, is calculated per ASCE/SEI 7-16 Section 

19.4.2 as described in Chapter 5 of this Guide. As the area of the bottom of 

the grade beams and spread footings does not exceed 75% of the footprint 

area, the bottom of the grade beams and spread footings does not qualify as 

the embedment depth, so the embedment depth is taken at the bottom of the 

slab-on-grade or 0.5 ft.

Step 2: Determine the effective profile depth, zp. 

zp = (B3

L)

0.25 = (28.53 × 41.5)0.25 = 31.31 ft

where:

B = overall foundation half width, 57 ft /2 = 28.5 ft. B is in the parallel 

to the direction of loading 

L = overall foundation half width, 83 ft / 2 = 41.5 ft. L is perpendicular 

to the direction of loading 

Step 3: Determine the effective depth for foundation rotation, r. 

The effective depth for foundation rotation is r = e + zp = 0.5 + 31.31 = 31.81 

ft.

Step 4: Determine the average low strain shear wave velocity, vso, over 

the rocking depth. 

vso = Σdi / Σ(di / vsi) 

= (31.81 / [(3.28 / 676) + (6.56 / 817) + (19.69 / 1,319) + ((31.81 – 

29.53) / 1,269)] 

= 1,074 ft/s

Step 5: Determine the effective shear wave velocity ratio, vs / vso. 

With SDS / 2.5 = 1.00 /2.5 = 0.4 and Site Class D, then the effective shear 

wave velocity ratio, [vs / vso], is 0.71 per ASCE/SEI 7-16 Table 19.3-1. 

Step 6: Determine the average effective shear wave velocity, vs. 

Thus, the average effective shear wave velocity over the depth of embedment 

is vs = vso × [vs / vso] = 1,074 ft/s × 0.71 = 763 ft/s.
FEMA P-2091 3: Rule of Thumb Test for Inertial SSI Significance 3-7 

3.5.3 Fundamental Period of the Structure

A computer model has not yet been created for the structure. ASCE/SEI

7-16 Equation 12.8-7 is used to determine the fundamental period of the 

structure.

T = Ta = Ct hn

x = 0.020 × 360.75 = 0.294 s 

where:

hn = 36 ft

Ct = 0.020 (coefficient from ASCE/SEI 7-16 Table 12.8-2 for all other 

structural systems)

x = 0.75 (coefficient from ASCE/SEI 7-16 Table 12.8-2 for all other 

structural systems)

3.5.4 Structure-to-Soil Stiffness Ratio

The structure-to-soil stiffness ratio is then calculated as follows.

 h' / (vs T) = 24.3 ft / (763 ft/s × 0.294 s) = 0.11 > 0.10 

where:

h′ = This is approximated as 2/3 hn = 2/3 × (36 ft + e) = 2/3 × (36 + 

0.5) 

= 24.3 ft

vs = 763 ft/s, as shown in Section 3.5.2 

T = 0.294 s, as shown in Section 3.5.3 

Since h′ / (vs T) = 0.11 > 0.10, SSI inertial effects are likely to have 

significance. 

3.6 Comparisons Using Site Class

When shear wave velocity measurements are not available, the ASCE/SEI

7-16 site classifications can be used instead. Site classifications are defined 

by the low strain shear wave velocity over the top 100 ft of soil. Table 3-2 

compares four common seismic force-resisting systems: steel momentresisting frame (SMRF), buckling-restrained braced frame (BRBF), special 

steel concentrically braced frame (SCBF), and reinforced concrete shear wall 

(RCSW). For each system, five heights are evaluated: 12 ft (for a one-story 

building), 36 ft (three-story), 60 ft (five-story), 120 ft (ten stories), and 240 ft

(20 stories). The approximate fundamental period is calculated for each 

system at each height using ASCE/SEI 7-16 Equation 12.8-7 (as shown in 

Section 3.4). 
3-8 3: Rule of Thumb Test for Inertial SSI Significance FEMA P-2091

The low shear strain shear wave velocity values, vso, in ASCE/SEI 7-16 

Table 20.3-1 for each of the five site classifications are assumed as follows. 

For Site Class A, the minimum value of 5,000 ft/s is assigned. For Site 

Classes B, C, and D, the average of the range in Table 20.3-1 is used. For 

Site Class D, for example, it is (600 ft/s + 1,200 ft/s) /2 = 900 ft/s. For Site 

Class E, the maximum value of 600 ft/s is used. Shear wave velocity 

typically increases with depth. Thus, the velocity at the base for effective 

depth for foundation rotation will be lower than the low shear strain shear 

wave velocity used in ASCE/SEI 7-16 Table 20.3-1 which is based on a 30 m 

depth weight average. Boore et al. (2011) provided regression equations to 

estimate shear wave velocities at depths shallower than 30 m. These have 

been calculated and added into Table 3-2 as the ratio of vso_r/vso, where vso_r

is the vso at the effective depth for foundation rotation, r = e + zp. The 

average low shear strain shear wave velocity at the effective depth is then

converted to the average effective shear wave velocity using the ratios in 

ASCE/SEI 7-16 Table 19.3-1. For simplicity, a short period spectral 

acceleration of SDS = 1.0g is assumed. Table 3-2 is for the shallow 

embedment depth, e = 0.5 ft, in the example. Table 3-3 shows the difference 

for a much deeper embedment depth of e = 20 ft, which could be for a 16 ft 

deep basement on a 4 ft mat. Values in red in the tables have h′ / (vs T) > 0.1 

indicating SSI inertial effects are likely to be significant. 

Tables 3-2 and 3-3 show several trends. 

• The structure-to-soil stiffness ratio, h′ / (vsT), is higher for stiffer 

structures, so that the RCSW and SCBF structures have the highest ratio, 

then the BRBF, and then the SMRF with the lowest ratio. 

• The structure-to-soil stiffness ratio is higher with the softer site classes 

because they have the lower average effective shear wave velocity.

• The structure-to-soil stiffness ratio increases with height because of the 

scaling of period with height in the code equations.

• The structure-to-soil ratio only goes over the 0.10 threshold (indicating 

inertial SSI effects are potentially significant) for Site Classes D and E 

with the assigned shear wave velocity, except for taller SCBF and RCSW 

buildings on Site Class C. 
FEMA P-2091 3: Rule of Thumb Test for Inertial SSI Significance 3-9 

Table 3-2 Structure-to-Soil Stiffness Ratio, h′ / (vs T), for r = e + zp = 33 ft

Example Inputs

Site Class, Shear Wave Velocity and 

Structure-to-Soil Stiffness Ratio

e = 0.5 ft Amin Bave Cave Dave Emax

zp = 32.5 ft vso: 5000 3750 1750 900 600

r = e + zp = 33 ft vso_r / vso: 0.86 0.76 0.63 0.60 0.60

vs / vso_r: 1.0 0.97 0.87 0.71 0.22

System

hn

(ft)

h′=2/3(hn+e)

(ft)

T 

(s)

vs : 4300 2765 959 383 79

(ft/s) (ft/s) (ft/s) (ft/s) (ft/s)

SMRF 12 8 0.20 0.009 0.015 0.043 0.106 0.515

36 24 0.49 0.011 0.018 0.052 0.129 0.624

60 40 0.74 0.013 0.020 0.057 0.142 0.687

 120 80 1.29 0.014 0.023 0.065 0.162 0.786

 240 160 2.25 0.017 0.026 0.074 0.186 0.902

BRBF 12 8 0.19 0.010 0.016 0.045 0.112 0.544

 36 24 0.44 0.013 0.020 0.058 0.144 0.697

 60 40 0.65 0.015 0.023 0.065 0.163 0.787

 120 80 1.09 0.017 0.027 0.077 0.193 0.933

 240 160 1.83 0.020 0.032 0.091 0.229 1.107

SCBF 12 8 0.13 0.015 0.023 0.067 0.169 0.816

 36 24 0.29 0.019 0.030 0.086 0.216 1.045

 60 40 0.43 0.022 0.034 0.098 0.244 1.181

 120 80 0.73 0.026 0.040 0.116 0.289 1.399

 240 160 1.22 0.031 0.048 0.137 0.343 1.660

RCSW 12 8 0.13 0.015 0.023 0.067 0.169 0.816

 36 24 0.29 0.019 0.030 0.086 0.216 1.045

 60 40 0.43 0.022 0.034 0.098 0.244 1.181

 120 80 0.73 0.026 0.040 0.116 0.289 1.399

 240 160 1.22 0.031 0.048 0.137 0.343 1.660

Notes: 

1. vso = average shear wave velocity for small strains over upper 100 ft (30 m) of the site profile from ASCE/SEI 7-16 Table 20.3-1. 

2. vs_r / vso = reduction ratio at effective depth for foundation rotation, r = e + zp. 

3. vs / vso_r = ASCE/SEI 7-16 Table 19.3-1 reduction ratio from low strain shear wave velocity to effective shear wave velocity. 

4. vs = vso × (vs_r / vso) × (vs / vso_r) = effective shear wave velocity at the effective depth for foundation rotation.

5. SDS = 1.0g, so SDS/2.5 = 0.4.

6. T is per ASCE/SEI 7-16 Equation 12.8-7, T = Ct hn

x

. 

7. For SMRF, Ct = 0.028 and x = 0.80, per ASCE/SEI 7-16 Table 12.8-2.

8. For BRBF, Ct = 0.030 and x = 0.75, per ASCE/SEI 7-16 Table 12.8-2.

9. For SCBF and RCSW, Ct = 0.020 and x = 0.75, per ASCE/SEI 7-16 Table 12.8-2.
3-10 3: Rule of Thumb Test for Inertial SSI Significance FEMA P-2091

Table 3-3 Structure-to-Soil Stiffness Ratio, h′ / (vs T), for r = e + zp = 52.5 ft

Example Inputs

Site Class, Shear Wave Velocity and 

Structure-to-Soil Stiffness Ratio

e = 20 ft Amin Bave Cave Dave Emax

zp = 32.5 ft vso: 5000 3750 1750 900 600

r = e + zp = 52.5 ft vso_r / vso: 0.88 0.83 0.76 0.75 0.76

vs / vso_r: 1.0 0.97 0.87 0.71 0.22

System

hn

(ft)

h′=2/3(hn+e)

(ft)

T 

(s)

vs : 4400 3019 1157 479 100

(ft/s) (ft/s) (ft/s) (ft/s) (ft/s)

SMRF 12 8 0.20 0.024 0.014 0.035 0.085 0.406

 36 24 0.49 0.017 0.016 0.043 0.103 0.493

 60 40 0.74 0.016 0.018 0.047 0.114 0.543

 120 80 1.29 0.016 0.021 0.054 0.130 0.621

 240 160 2.25 0.018 0.024 0.062 0.149 0.712

BRBF 12 8 0.19 0.025 0.014 0.037 0.090 0.429

 36 24 0.44 0.019 0.018 0.048 0.115 0.550

 60 40 0.65 0.019 0.021 0.054 0.130 0.622

 120 80 1.09 0.020 0.024 0.064 0.154 0.736

 240 160 1.83 0.022 0.029 0.076 0.183 0.874

SCBF 12 8 0.13 0.038 0.021 0.056 0.135 0.644

 36 24 0.29 0.029 0.027 0.072 0.173 0.825

 60 40 0.43 0.028 0.031 0.081 0.195 0.932

 120 80 0.73 0.029 0.037 0.096 0.231 1.104

 240 160 1.22 0.032 0.044 0.114 0.274 1.311

RCSW 12 8 0.13 0.038 0.021 0.056 0.135 0.644

 36 24 0.29 0.029 0.027 0.072 0.173 0.825

 60 40 0.43 0.028 0.031 0.081 0.195 0.932

 120 80 0.73 0.029 0.037 0.096 0.231 1.104

 240 160 1.22 0.032 0.044 0.114 0.274 1.311

Notes: 

1. vso = average shear wave velocity for small strains over upper 100 ft (30 m) of the site profile from ASCE/SEI 7-16 Table 20.3-1.

2. vs_r / vso = reduction ratio at effective depth for foundation rotation, r = e + zp. 

3. vs / vso_r = ASCE/SEI 7-16 Table 19.3-1 reduction ratio from low strain shear wave velocity to effective shear wave velocity.

4. vs = vso × (vs_r / vso) × (vs / vso_r) = effective shear wave velocity at the effective depth for foundation rotation.

5. SDS = 1.0g, so SDS/2.5 = 0.4.

6. T is per ASCE/SEI 7-16 Equation 12.8-7, T = Ct hn

x

. 

7. For SMRF, Ct = 0.028 and x = 0.80, per ASCE/SEI 7-16 Table 12.8-2.

8. For BRBF, Ct = 0.030 and x = 0.75, per ASCE/SEI 7-16 Table 12.8-2.

9. For SCBF and RCSW, Ct = 0.020 and x = 0.75, per ASCE/SEI 7-16 Table 12.8-2.
FEMA P-2091 4: Base Slab Averaging 4-1 

Chapter 4 

Base Slab Averaging

4.1 Overview

Seismic input motions for buildings with large base areas are reduced due to 

incoherence of the ground motions that occur over the base area. Incoherent 

motions are those that vary from each other in some way. These variations 

can be due simply to the time of seismic wave arrival across the building 

base; they may also be due to local variations in soil properties that modify 

the waves as they travel across the site. The result of this incoherence is that 

various portions of the building will experience disparate ground motions. 

When the ground under one corner of a building is being moved in a 

particular direction, the ground at the opposite corner may be moving in 

another direction – these motions will partially offset one another, resulting 

in a smaller net motion. This effect is most pronounced at short periods, 

since longer period motions are more coherent (i.e., less likely to vary across 

the site). As a result, stiff buildings with large base areas and rigidly 

connected foundations will experience reductions in the ground motions 

transmitted to their superstructures.

Base Slab Averaging

Description: Kinematic SSI 

reduction of demands on 

structure due to variations in 

ground motion across large 

foundations. 

Requirements:

• Foundation base area, 

Abase. 

• Average effective shear 

wave velocity, vs. 

Base slab averaging is a kinematic interaction effect – these are effects that

depend only on the building and foundation geometry and the frequency 

content and spatial variability of the earthquake wave field. These effects are 

distinct from foundation damping and period lengthening effects, which also 

depend upon the local soil properties under the foundations and the dynamic 

characteristics of the seismic system and its foundations. 

Base slab averaging results in modified foundation input motions, the effects 

of which are most accurately described in the frequency domain (i.e., through 

the Fourier amplitudes and phase angles of the ground motions). However, 

to make the models more useful to practitioners, NIST (2012a) presents a 

relationship that is only a function of the building period and the average 

shear wave velocity, vs. In ASCE/SEI 7-16 (ASCE, 2017a), the functional 

dependence on vs was eliminated. This relationship reduces the ordinates of 

the response spectrum and is computed using formulas that are dependent 

only on the effective size of the building footprint. 

The only parameters needed are the foundation base area, Abase, and the 

average effective shear wave velocity, vs, over an effective profile depth 
4-2 4: Base Slab Averaging FEMA P-2091

below the foundation. Section 2.2.2 of NIST (2012a) indicates that this 

depth may be taken as equal to half of the equivalent building footprint size, 

computed as Abase 2 . Although the average effective shear wave velocity 

is not required explicitly to compute the reductions, the pertinent equations 

are only applicable for vs in the range of 650 to 1,650 ft/s. This is not to say 

that base slab averaging is not applicable to sites outside of this range, only 

that the equations in ASCE/SEI 7-16 Section 19.4 have not been 

demonstrated as applicable outside of this range. 

4.2 Key Equations 

ASCE/SEI 41-17 Equations

The equations for base slab 

averaging in ASCE/SEI 

41-17 Section 8.5.1.1 

(ASCE, 2017b) are the 

same as those in ASCE/SEI 

7-16, with the exception of 

ASCE/SEI 41-17 Equation 

8-17 for b0, which 

incorporates a factor of 

0.0001 × 2π, in lieu of the 

value of 0.00071 in 

ASCE/SEI 7-16. 

ASCE/SEI 41-17 includes 

only the equation for 

foundation size in feet, not a 

companion equation in 

meters. 

The starting point for base slab averaging is determination of the equivalent 

foundation size, be, according to ASCE/SEI 7-16 Equation 19.4-4, as

b A e = base , which shall not be taken as more than 260 ft. The dimensional 

limit occurs because the base slab averaging equation has been calibrated 

using data from actual sites, and the limit corresponds to the maximum 

foundation dimension at those sites (Kim and Stewart, 2003). A conservative 

estimate of base slab averaging for larger foundation footprints can be 

obtained by using the limiting distance.

The next step is to compute b0 according to ASCE/SEI 7-16 Equation 19.4-3 

as follows: 

 b b0 = 0.00071 e 

 , where be is in feet and T is in seconds  T 

 b (or b0 = 0.0023 e 

 , where be is in meters and T is in seconds) 

This quantity is a function of the response spectrum period, T, which shall 

not be taken as less than 0.2 sec. With b0 determined, Bbsa is computed 

according to ASCE/SEI 7-16 Equation 19.4-2 as follows: 

b6 b b 8 10

Bbsa =1+++ b b 2 4 000

0 0 + + for b

2 4 12

0 ≤ 1 

  1 1   B b bsa = exp(2 2

0 )  1− > 1

b 2  for b0 

  π   0  16b0 

When considering the upper limit on the effective foundation size (260 ft) 

and the lower limit on the period (0.2 s), the value of b0 will never exceed 

1.0, so the second equation will not apply. (However, if these limits are 

relaxed in future editions of ASCE/SEI 7, the second equation may apply.) 

The modification factor for base slab averaging, RRSbsa is computed 

according to ASCE/SEI 7-16 Equation 19.4-1 as follows:
FEMA P-2091 4: Base Slab Averaging 4-3 

( ) ( ) 2

2 0

0

1 0.25 0.75 1 exp 2 RRSbsa bsa b B

b

 

=+ − −        

Note that the theoretical reduction (within the braces) is multiplied by 0.75 

and added to an unreduced factor of 0.25. This code-specific language 

diminishes the computed reduction relative to the value provided by the 

underlying semi-empirical model in Chapter 3 of NIST (2012a). This 

limitation was applied because the reduction due to RRSbsa can become quite 

significant and because studies of these phenomena have indicated variability 

between the theoretically predicted modifications and actual measured 

modifications.

4.3 Determining Effective Foundation Size

The key parameter needed to apply base slab averaging SSI effects is the 

effective foundation size, be. ASCE/SEI 7-16 provides no guidance on this 

parameter other than defining it as the square root of the area of the base of 

the structure. The effective foundation size is then simply the equivalent 

width of the building base area, considering that the foundation was square in 

plan. 

The definition of the base area should be considered as the out-to-out 

dimensions of the extent of the building foundation, or it may be taken 

conservatively as the building extent based on the overall column-grid 

dimensions.

4.4 Determining whether Base Slab Averaging Applies

ASCE/SEI 7-16 Section 19.4.1 allows base slab averaging to be employed 

only when the foundations of the structure are interconnected. This 

condition can be met with a continuous mat foundation, or with individual

footings that are interconnected by slabs or grade beams providing sufficient 

stiffness so as not to be classified as a flexible diaphragm. The building base 

must be sufficiently stiff to permit the base to move as a unit, which allows 

the filtering of high-frequency motions to occur. Such filtering may still 

occur in structures with a more flexible base, but this has not yet been studied 

in sufficient detail. 

Figure 4-1 shows four different building base conditions, illustrating various 

degrees of interconnectivity. 

• Condition a, with a slab-on-grade that is not positively connected to the 

footings would not generally develop interconnectivity sufficient to 

cause the building base to move as a unit. Base slab averaging should 

not be considered for this case.
4-4 4: Base Slab Averaging FEMA P-2091

• Condition b, with the slab thickened and connected to the footings with 

reinforcing dowels, would likely produce sufficient interconnectivity, 

provided that the thickness of the slab provides sufficient axial and shear 

stiffness. Base slab averaging can be considered for this case if the slab 

is sufficiently thick.

• Condition c has a slab similarly constructed to Condition a, but it also 

includes a grid of tie beams. This condition would likely produce 

sufficient interconnectivity but only if there are diagonal tie beams to 

provide shear transfer between the parallel lines of footings. Base slab 

averaging can be considered for this case if diagonal tie beams are 

included.

• Condition d, with a continuous mat foundation, almost certainly provides 

sufficient interconnectivity, but its in-plane stiffness should still be 

verified. The mat must be stiff enough not to be characterized as a 

flexible diaphragm according to ASCE/SEI 7-16 Section 12.3.1.3. If this 

condition is met, then base slab averaging can be considered.

Individual spread footings and 

slab-on-grade unconnected

 a.

 c. Individual spread footings 

with a grid of tie beams

Continuous mat foundation

Individual spread footings and 

slab-on-grade thickened at 

foundations and connected with 

reinforcing dowels

 b.

 d.

Figure 4-1 Example building base configurations, showing differing degrees 

of interconnectivity. 

4.5 Limitations and Issues

There are several limitations to using base slab averaging effects in 

ASCE/SEI 7-16. They include:

• ASCE/SEI 7-16 Section 19.1.1: SSI does not apply on Site Class A or B. 

The analytical model for the structure must include foundation 

flexibility. 

• ASCE/SEI 7-16 Section 19.4.1: The pertinent equations are only 

applicable for sites with vs measured over the pertinent depth within the 
FEMA P-2091 4: Base Slab Averaging 4-5 

range of 650 to 1,650 ft/sec. Note that a code change proposal for the 

next edition of ASCE/SEI 7 is under review to eliminate this provision.

• ASCE/SEI 7-16 Section 19.4: Modifications of the response spectrum 

for kinematic SSI, including base slab averaging, are only permitted 

when nonlinear response history analysis is used per ASCE/SEI 7-16 

Chapter 16. 

• ASCE/SEI 7-16 Section 19.4: There is a limit on the amount of 

reduction that can be taken from kinematic effects, such that the product 

of RRSbsa × RRSe shall not be less than 0.7. 

• ASCE/SEI 7-16 Section 19.2.3: The site-specific response spectra 

modified for kinematic interaction shall not be taken as less than 0.7 of 

the design earthquake and MCER spectra determined from ASCE/SEI 

7-16 Sections 11.4.6. 

• ASCE/SEI 7-16 Section 19.2.1 limits the reduced base shear to be no less 

than α times the base shear as computed according to ASCE/SEI 7-16 

Section 12.8.1 for the fixed-base structure. The value of α is 0.7 for R ≤ 

3, is 0.9 for R ≥ 6 and varies linearly between. 

ASCE/SEI 41-17 

Limitations

The limitations in ASCE/SEI 

41-17 Section 8.5.1 are 

generally more relaxed than 

those in ASCE/SEI 7-16. 

• Kinematic SSI 

modifications, including 

base slab averaging, are 

permitted for any analysis 

method. However, for 

linear procedures, the 

reduction must be 

computed based on 

flexible-base building 

periods multiplied by 1.5.

• There is no limitation on 

the shear wave velocity for 

the site.

• The product of RRSbsa × 

RRSe shall not be less 

than 0.5.

• Foundations must be 

stronger than the vertical 

elements of the lateralforce-resisting system.

4.6 Example 

An example application is used to illustrate the response spectrum reductions 

due to base slab averaging per ASCE/SEI 7-16 Section 19.4.1. The example 

building has base-level plan dimensions of 100 ft by 400 ft and is shown in 

Figure 4-2. The site is classified as Site Class D with a soil shear wave 

velocity of 900 ft/s, which is within the specified range for applicability of 

the equations. 

Effective Foundation Size

200'-0"

400'-0"

100'-0"

6" Conc. Slab With 

Overpour & Dowels 

To Each Footing

Figure 4-2 Example building foundation plan, representative of Condition b in Figure 4-1. 
4-6 4: Base Slab Averaging FEMA P-2091

The area of the structure base is taken from the grid-to-grid dimensions as 

Abase = (100 ft)(400 ft) = 40,000 ft 2

The effective foundation size is then computed in accordance with 

ASCE/SEI 7-16 Equation 19.4-4:

2 40,000 ft 200 ft e base b A = = =

This dimension is less than 260 ft and therefore may be used as computed. 

For equivalent dimensions exceeding 260 ft, the following computations 

must be based on 260 ft.

The response spectrum reductions are period-dependent and most easily 

computed in a spreadsheet. The following calculations pertain to the 

minimum period allowed to be considered, 0.2 s, in accordance with 

ASCE/SEI 7-16 Equation 19.4-3: 

Then, the response spectral modification factor, RRSbsa, is computed in 

accordance with ASCE/SEI 7-16 Equation 19.4-1:

0

200 ft 0.00071 0.00071 0.71

0.2 sec

e b b

T

    = =   =      

This value is confirmed to be less than 1.0, so the applicable equation for Bbsa

is ASCE/SEI 7-16 Equation 19.4-2(a): 

6 8 10

2 4 000

0 0 1 1.841

2 4 12 bsa

bbb B bb =+ + + + + =

( ) ( ) 2

2 0

0

1 0.25 0.75 1 exp 2 0.855 RRSbsa bsa b B

b

 

=+ − − =        

This value is greater than the required lower limit of 0.7. 

Table 4-1 provides values of RRSbsa, the response spectral modification 

factor, at several periods of interest. 

Table 4-1 Values of RRSbsa at Selected Periods for be = 200 ft

Response Spectrum Period, T (s) RRSbsa for be=200 ft

0.2 0.855

0.3 0.925

0.4 0.956

0.5 0.971

1.0 0.993
FEMA P-2091 4: Base Slab Averaging 4-7 

Figure 4-3 shows RRSbsa for be = 200 ft plotted vs. building period. Figure 

4-4 shows example resulting response spectra (with and without the 

reduction applied) for SDS = 1.0 and SD1 = 0.6. Note that the spectrum shown 

includes the increased value of the corner period from Ts to 1.5Ts for Site 

Class D, per ASCE/SEI 7-16, Section 11.4.8. 

Note that when Method 1 of ASCE/SEI 7-16 Chapter 16 is used, which 

requires an MCER response spectrum, RRSbsa can be applied to a smooth sitespecific MCER response spectrum derived from ASCE/SEI 7-16 Chapter 21 

or 2/3 of that spectrum for the Design Earthquake.

Figure 4-3 RRSbsa reduction as a function of period. 

Figure 4-4 Example resulting response spectrum for SDS = 1.0 and SD1 = 0.6.

FEMA P-2091 5: Embedment Effects 5-1 

Chapter 5 

Embedment Effects

5.1 Overview

Chapter 4 of this Guide discussed base slab averaging, which is one 

kinematic soil-structure interaction (SSI) effect. This chapter discusses the 

second kinematic SSI effect, which is related to the reduction of ground 

motions with depth, and is known as embedment effects. Similar to the 

requirements for base slab averaging, embedment effects can only be 

considered for soil sites and do not apply to Site Classes A or B. Foundation 

embedment modifies the free field motion, assumed to be developed at the 

ground surface, and reduces the foundation input motion and the resulting 

demands on the structure. This is characterized by a reduction in the 

ordinates of the response spectrum. The deeper the foundation is embedded, 

the greater the reduction in response spectrum values. Soil flexibility 

modeling is not required. The parameters needed are the foundation 

embedment depth, e, and the average low strain shear wave velocity over the 

embedment depth, vso, and parameters used to reduce vso for the effects of 

nonlinearity (site class and SDS). If the free field motion is determined at the 

embedded foundation level, then reduction of this motion for embedment 

effects described below cannot be taken. If the free field motion is defined at 

some depth between the ground surface and the embedment depth, then 

embedment effects can be considered provided the embedment depth is 

computed from the depth where the free field motion is defined.

Embedment Effects

Description: Kinematic SSI 

reduction of demands on 

structure due to embedment 

of building foundation.

Requirements:

• Foundation embedment 

depth, e

• Average low strain shear 

wave velocity over the 

embedment depth, vso

• Design spectral response 

acceleration parameter at 

short periods, SDS

• Site class

5.2 Key Equation

ASCE/SEI 7-16 Equation 19.4-5 determines the modification for 

embedment, RRSe.

RRSe = 0.25 + 0.75 × cos (2πe / Tvs)

where:

e = foundation embedment depth, not greater than 20 ft. A minimum 

of 75% of the foundation footprint shall be present at the 

embedment depth. The foundation embedment for structures 

located on sloping sites shall be the shallowest embedment.

vs = the average effective shear wave velocity for site soil conditions, 

taken as the average value of velocity over the embedment depth 
5-2 5: Embedment Effects FEMA P-2091

of the foundation determined using vso and ASCE/SEI 7-16 Table 

19.3-1 or a site-specific study. It shall not be less than 650 ft/s.

vso = the average low strain shear wave velocity over the embedment 

depth of the foundation.

T = response spectra ordinate period, which shall not be taken as less 

than 0.20 s. 

5.3 Determining Depth of Embedment

The key parameter needed to apply embedment effects is the foundation 

embedment depth, e. ASCE/SEI 7-16 (ASCE, 2017a) provides no guidance 

on this parameter other than to say in Section 19.4.2: “A minimum of 75% of 

the foundation footprint shall be present at the embedment depth. The 

foundation embedment for structures located on sloping sites shall be the 

shallowest embedment.”

ASCE/SEI 41-17 

Embedment

ASCE/SEI 41-17 Section 

8.5.1.2 (ASCE, 2017b) adds 

more specific requirements 

to use embedment effects 

than ASCE/SEI 7-16.

• Structures must have

structural mats or 

foundation elements 

interconnected with 

concrete slabs or that are

continuously connected 

with grade beams or other 

foundation elements of 

sufficient lateral stiffness 

so as not to be 

characterized as a flexible 

diaphragms. 

• The foundation elements 

are stronger than the 

vertical elements of the 

lateral force-resisting 

system. 

Figure 5-1 shows a building supported by a mat. The depth of embedment 

goes to the bottom of the mat, as the area of the foundation at this elevation 

covers 100% of the foundation footprint, or more than 75% of the foundation 

footprint threshold in ASCE/SEI 7-16 Section 19.4.2. 

e

Figure 5-1 Depth of embedment for a building on a mat foundation.

Figure 5-2 shows a building with an orthogonal set of grade beams with 

spread footings at the intersections supporting a steel moment frame system. 

At the base of the footings, or e1 below the top of slab, the footing area is 

only 32% of the total footprint area or well under the 75% threshold in 

ASCE/SEI 7-16 Section 19.4.1. Thus, the embedment depth, e, is the depth 

of the slab-on-grade, e2, which is negligible. This example shows it is 

typically not possible for the depth of embedment to go to the base of the 

foundation, except for a mat. 
FEMA P-2091 5: Embedment Effects 5-3 

20' Typ.

6'×6'×3'H, Typ.

3'W×3'H, Typ.

Footing Area 

4×4×6'×6' = 576

2×4×3×14'×3' = 1,008

1,584 sf

Footprint Area 

66'×66'- 4×3×14'×1.5' = 4,104 sf 

Footing Area

Footprint Area

1,584

4,104 = 0.39

39% < 75%

e = e1

=

66' Typ.

e = e2

A A

Plan A-A

Elevation

Figure 5-2 Depth of embedment for a building on grade beams and spread 

footings. 

Figure 5-3 shows a building with a one-story basement with a perimeter strip 

footing and interior strip or spread footings. The depth of embedment, e = e2, 

is from grade to the top of the basement slab, because, like the situation in

Figure 5-2, the area at the base of the footings is much less than the 75% 

threshold in ASCE/SEI 7-16 Section 19.4.1. 

e ≠ e1 e = e2

Figure 5-3 Depth of embedment for a building with a 

basement on perimeter strip footings and 

interior spread footings. 
5-4 5: Embedment Effects FEMA P-2091

Figure 5-4 shows a situation with a two-story basement where the depth of 

the lowest basement floor is 24 ft below grade, or more than the 20 ft limit of 

ASCE/SEI 7-16 Section 19.4.1. In this case, the embedment depth, e, is 

capped at 20 ft. 

12'

emax = 20'

12'

12'

12'

12'

Figure 5-4 Maximum depth of embedment.

Figure 5-5 shows a situation with a partial basement. The area of the 

basement is less than 75% of the area of the overall building footprint, so the 

depth of embedment is limited to the slab-on-grade depth which is negligible. 

Figure 5-6, on the other hand, shows a situation where the partial basement is 

large enough to exceed the 75% threshold, and the depth of embedment is the 

from grade to the top of the basement slab.
FEMA P-2091 5: Embedment Effects 5-5 

Basement Footing Area 

23'×40' = 920 sf 

Footprint Area 

40'×40' = 1,600 sf

Basement Footing Area

Footprint Area

920

1,600= 0.58

58% < 75%

=

23'

40'

17'

e ≠ e1

e = e2

Plan A-A

Elevation

A

A

Figure 5-5 Depth of embedment for a smaller partial basement. 

Basement Footing Area 

63'×40' = 2,520 sf 

Footprint Area 

80'×40' = 3,200 sf 

Basement Footing Area

Footprint Area

2,520

3,200

= 0.79

79% > 75%

= 63'

40'

17'

e = e1

e ≠ e2

Plan A-A

Elevation

A

A

Figure 5-6 Depth of embedment for a larger partial basement. 
5-6 5: Embedment Effects FEMA P-2091

Figure 5-7 shows a building with a full basement on a sloping site. Per 

ASCE/SEI 7-16 Section 19.4.1, the depth of embedment is taken at the 

shallowest point of embedment, where e = e1. 

e ≠ e2

e = e1

Figure 5-7 Depth of embedment on a sloping site.

5.4 Limitations and Issues

There are several limitations to using embedment effects in ASCE/SEI 7-16

in addition to those already noted. They include: ASCE/SEI 41-17 

Limitations

The limitations in ASCE/SEI 

41-17 Section 8.5.1 are 

more relaxed than those in 

ASCE/SEI 7-16: 

• Kinematic SSI 

modifications, including 

those for embedment 

effects, are permitted for 

any analysis method.

• The product of RRSbsa ×

RRSe shall not be less 

than 0.5.

• ASCE/SEI 41-17 does not 

have minimum base shear 

requirements.

• ASCE/SEI 7-16 Section 19.4: Modifications of the response spectrum 

for kinematic SSI, including embedment, are only permitted when 

nonlinear response history analysis is used per ASCE/SEI 7-16 Chapter 

16. ASCE/SEI 41-17 does not have this restriction.

• ASCE/SEI 7-16 Section 19.4: There is a floor on the amount of 

reduction that can be taken for kinematic effects, such that the product of 

RRSbsa × RRSe shall not be less than 0.7. ASCE/SEI 41-17 has a less 

restrictive floor of 0.5.

• ASCE/SEI 7-16 Section 19.4.2: There is a maximum limit of 20 ft on 

the depth embedment, e, that can be used. NIST (2012a) does not have a 

similar limit. 

• ASCE/SEI 7-16 Section 19.2.3: The site-specific response spectra 

modified for kinematic interaction shall not be taken as less than 0.7 of 

the design earthquake and MCER spectra determined from ASCE/SEI

7-16 Section 11.4.6. The modified spectra shall not be taken as less than 

0.8 of the site-specific spectra are determined per ASCE/SEI 7-16 

Section 21.3.

Besides limitations in ASCE/SEI 7-16 Chapter 19, there are also minimum 

base shear limitations in ASCE/SEI 7-16 Chapter 12 that may govern. 
FEMA P-2091 5: Embedment Effects 5-7 

5.5 Example 

An example application is used to illustrate the ASCE/SEI 7-16 Section 

19.4.2 embedment effect equation. The example building is a three-story 

concrete shear wall building with a two-story basement. It is shown in 

Figure 5-8. The foundation uses perimeter strip footings and interior spread 

footings. The building is on a sloped site. Embedment on the high side of 

grade is 24 ft from grade to the top of the basement slab, and it is 22 ft on the 

low side of grade from grade to the top of the basement slab. The shallower 

embedment is used. However, embedment depth is capped at 20 ft per 

ASCE/SEI 7-16 Section 19.4.2. 

e1 = 22'

e2 = 24'

e = emax = 20' < 22' < 24'

Figure 5-8 Example building for embedment effects.

For site classification, the shear wave velocity has been measured at the site 

and is shown in Figure 5-9. The measured velocity for each layer is shown 

in Table 5-1.
5-8 5: Embedment Effects FEMA P-2091

Figure 5-9 Shear wave velocity at example building.

Shear Wave Velocity

There are several similar 

shear wave velocity 

parameters. ASCE/SEI 

7-16 is not consistent with 

the terms. 

In Chapter 20, is the 

“average shear wave 

velocity.” It is taken over the 

top 100 ft of soil. This is 

often designated as for 

30 m. Both and are 

the low strain shear wave 

velocity, though this is never 

stated in Chapter 20.

In Section 19.4.2 of Chapter 

19, vso, is the “average low 

strain shear wave velocity 

over the embedment depth.” 

If the embedment depth 

were 100 ft it would be the 

same as . 

In Chapter 19, the low strain 

shear wave velocity needs 

to be modified by Table 

19.3-1 to obtain the average 

effective shear wave 

velocity, vs, which cannot be 

less than 650 ft/sec. 

Table 5-1 Measured Shear Wave Velocity at Example Building

Depth to 

Top of 

Layer 

(m) 

Layer 

Thickness 

(m) 

Shear 

Wave 

Velocity 

(m/s)

Depth to 

Top of 

Layer 

(ft)

Layer 

Thickness 

(ft)

Shear 

Wave 

Velocity 

(ft/s)

0.0 0 206 0.0 3.28 676

1 2 249 3.28 6.56 817

3 6 402 9.84 19.69 1,319

9 12 387 29.53 39.37 1,269

21 18 382 68.90 59.10 1,253

39 30 458 128.0 98.40 1,503

69 >6 673 226.38 >19.68 2,207

The average shear wave velocity vs is calculated per ASCE/SEI 7-16 

Equation 20.4.1 for the top 30 m or 100 ft. In typical current geotechnical 

practice, measurements are usually made in depths using meters and 

calculated using metric layer depths and then converted to imperial units at 

the end. Since 30 m and 100 ft are not identical, this means calculations 
FEMA P-2091 5: Embedment Effects 5-9 

using layers up to 30 m and layers up to 100 ft would result in slightly 

different average shear wave velocities. The typical practice is followed 

here: 

vs = Σdi / Σ(di / vsi) 

where:

di = the thickness of any layer between 0 and 100 ft

vsi = the shear wave velocity in ft/s

vs = 30 / [(1 / 206) + (2 / 249) + (6 / 402) + (12 / 387) + ((30 – 21) / 

382)] = 365 m/s or 1,195 ft/s

Per ASCE/SEI 7-16 Table 20.3-1, this is just within the 600 ft/s to 1,200 ft/s 

range for Site Class D, Stiff Soil.

For embedment effects, the shear wave velocity calculation is not the same it 

is for site classification. Per ASCE/SEI 7-16 Section 19.4.2, the shear wave 

velocity is taken over the depth of the embedment, which is 20 ft in this case. 

ASCE/SEI 7-16 Section 19.4.2 does not provide explicit guidance on the 

formula to use for calculating the average low strain shear wave velocity 

over the depth of embedment, but the formulation in ASCE/SEI 7-16 

Equation 20.4.1 is used here.

vso = 20 / [(3.28 / 676) + (6.56 / 817) + ((20 – 9.84) / 1,319)] = 972 ft/s

The building is located in a high seismic zone with SDS = 1.06 and SD1 = 1.45.

The average low strain shear wave velocity is converted to the average 

effective shear wave velocity per ASCE/SEI 7-16 Table 19.3-1. With 

SDS / 2.5 = 1.06 /2.5 = 0.42 and Site Class D, and then using linear 

interpolation, the effective shear wave velocity ratio, [vs / vso], is 0.687. Thus, 

the average effective shear wave velocity over the depth of embedment is

vs × [vs / vso] = 972 ft/s × 0.687 = 668 ft/s.

The modification factor for embedment, RSSe, can now be calculated. Per 

ASCE/SEI 7-16 Section 19.4.2, the lowest response spectra period shall not 

be taken as less than 0.20. For this example calculation, T = 0.20 s is used.

RRSe = 0.25 + 0.75 × cos (2πe / Tvs)

RRSe = 0.25 + 0.75 × cos [2π × 20 ft / (0.20s × 668 ft/s)] = 0.692

The maximum reduction for RRSbsa × RRSe = 0.70. For this example, we will

assume RRSbsa = 1.0. Thus, RRSe cannot go below 0.70. 
5-10 5: Embedment Effects FEMA P-2091

Table 5-2 provides the remaining spectral ordinates for comparison, and 

Figure 5-10 plots the site-specific response spectral modification factor, RRSe

(with the floor of 0.70). Figure 5-11 plots the response spectra, with and 

without the modification for embedment. For example, for a period of 0.30

s, the embedment effects would permit reduction of the spectral acceleration 

down to 0.86 of the value without SSI.

Table 5-2 Response Spectral Ordinates with and without Embedment 

Effects

Period 

(s)

Sa without SSI 

(g) RRSe

Sa with SSI 

(g) 

0 0.42 0.700 0.30

0.05 0.54 0.700 0.38 

0.1 0.66 0.700 0.46

0.16 0.80 0.700 0.56 

0.2 0.89 0.700 0.62

0.225 0.95 0.753 0.71

0.27 1.06 0.829 0.88 

0.3 1.06 0.857 0.91 

0.4 1.06 0.919 0.97

0.5 1.06 0.948 1.00

0.6 1.06 0.963 1.02

0.8 1.06 0.979 1.04 

1 1.06 0.987 1.05 

1.2 1.06 0.991 1.05

1.37 1.06 0.993 1.05

1.4 1.04 0.993 1.03 

1.5 0.97 0.994 0.96

2 0.73 0.997 0.72

2.5 0.58 0.998 0.58 

3 0.48 0.999 0.48

3.5 0.41 0.999 0.41

4 0.36 0.999 0.36

4.5 0.32 0.999 0.32

5 0.29 0.999 0.29
FEMA P-2091 5: Embedment Effects 5-11 

Figure 5-10 Site-specific response spectral acceleration modification factor 

for foundation embedment, RRSe. 

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 0.5 1 1.5 2 2.5 3 3.5 4

Sa

Period, T

Figure 5-11 Response spectra for example building, with and without 

embedment SSI.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.5 1 1.5 2 2.5 3 3.5 4

Sa

Period, T

Sa - No SSI

Sa - With SSI

FEMA P-2091 6: Foundation and Soil Flexibility 6-1 

Chapter 6 

Foundation and Soil Flexibility

6.1 Overview

Conventional building design is most often performed based on the 

assumptions that the building is fixed to the ground at its base and the ground 

itself is rigid. These assumptions are not strictly true but are often 

considered to be acceptable and conservative. The flexibility of the 

foundation system, including both the foundation’s structural components

and the supporting soil, can have a significant effect on the building’s 

dynamic properties and its overall response. When incorporating soilstructure interaction (SSI) in the design of a building, ASCE/SEI 7-16 

Section 19.1.1 (ASCE, 2017a) requires that the analytical model of the 

structure incorporate vertical, horizontal, and rotational flexibility of the 

foundation and the soil. The flexibility of the foundation elements 

themselves (usually reinforced concrete) can be determined using 

conventional methods. However, ASCE/SEI 7-16 does not provide specific 

guidance on determining the finite stiffness of the soil itself. This chapter is 

intended to provide that guidance, based on the provisions included in 

ASCE/SEI 41-17 (ASCE, 2017b) and NIST (2012a). 

Foundation and Soil 

Flexibility 

Description: Computation 

of the flexibility of soil

supporting building 

foundations, which must be 

incorporated in analyses 

when SSI is considered. 

Requirements:

• Site class

• Design spectral response 

acceleration parameter at 

short periods, SDS

• The foundation geometry

• The effective soil shear 

modulus, G, computed 

from the measured shear 

wave velocity, vs0

• Poisson’s ratio for the soil, 

ν

The flexibility of soil in the analytical model of a building is generally 

accounted for through modeling the connection of structural elements to 

fixed supports with spring elements. These may be point springs, modeling 

each degree of freedom at the base with a single spring, or may be distributed 

springs representing the soil support as a discretized continuous medium, 

known as a Winkler foundation. 

Vertical and rotational springs can model similar behavior. They affect the 

structure rocking on its base due to elastic vertical compression of the soil—

either through vertical movement of the ends of a frame, or rotation at the 

base of a wall or core. This behavior can have a dramatic effect of the 

building’s fundamental period and displaced shape. 

Horizontal springs model the foundation displacement relative to the freefield soil displacement or resistance of the soils against basement walls or 

other vertical faces. Spring stiffness is limited by frictional and passive 

pressure resistance. 
6-2 6: Foundation and Soil Flexibility FEMA P-2091

ASCE/SEI 7-16 Section 12.13.3 requires consideration of both upper and 

lower bound estimates of the foundation soil flexibility by requiring a 50% 

increase and decrease in soil stiffness. This is to account for the inherent 

uncertainty and variability of soil, as well as a recognition that representing 

the soil’s actual nonlinear behavior as linear is an approximation. 

Chapter 2 of NIST (2012a) presents a discussion of inertial interaction, 

including both foundation flexibility and damping. The foundation system 

flexibility sections include two similar techniques, one developed by Pais and 

Kausel (1988) and one by Gazetas (1991), expanded by Mylonakis et al. 

(2006). Both of these techniques are applicable to rigid foundation elements 

supported by flexible soils. Both present equations to model six degrees of 

potential flexibility (three translations and three rotations) based on the 

foundation geometry and the soil stiffness, as expressed by the shear 

modulus, G. Both techniques include the effects of increased stiffness due to 

embedment of the foundation in the soil. Because these techniques address 

the degrees of flexibility assigned to a single point, they do not lend 

themselves well to conditions where the structural foundation elements are 

also a source of flexibility without enhancements that address such 

flexibility, as included in NIST (2012a) Section 2.2.3. 

ASCE/SEI 41-17 

NIST (2012a) defines L as 

half the foundation length 

and B as half the foundation 

width, which is consistent 

with the way these 

dimensions are treated in 

the source documents. 

ASCE/SEI 41-17 defines L

as the full foundation length 

and B as the full foundation 

width for springs, which is 

consistent with the 

nomenclature used by most 

structural engineers. 

As a result of this 

discrepancy, the equations 

in the two documents 

incorporate different 

coefficients to produce the 

same results. 

ASCE/SEI 41-17 also provides recommendations for computing the effects 

of foundation flexibility. These include the methods outlined in NIST as 

well as methods that allow consideration of the flexibility of the structural 

foundation elements. 

6.2 Soil Properties for Flexibility Calculations

In addition to foundation geometry, soil properties are required to compute 

the flexibility of the soil. These properties include the measured shear wave 

velocity, vso, the effective soil shear modulus, G, and Poisson’s ratio for the 

soil, ν. 

The shear wave velocity, vso, is determined based on the average measured 

value over an effective depth interval. The depth interval is defined slightly 

differently in the various documents. Both ASCE/SEI 41-17 and NIST 

describe a depth below the footing-soil interface of the half-dimension of an 

equivalent rectangular foundation matching the area of the actual foundation. 

NIST goes on to indicate that for a rocking foundation it may be more 

appropriate to consider the half-dimension of an equivalent square 

foundation matching the moment of inertia of the actual foundation. 

ASCE/SEI 41-17 notes that the soil that is one or two footing widths below 

the soil-footing interface should be considered. 
FEMA P-2091 6: Foundation and Soil Flexibility 6-3 

The small-strain soil shear modulus, Go, is computed according to the 

ASCE/SEI 41-17 Equation 8-4, which is also repeated in the definitions of 

ASCE/SEI 7-16 Section 19.3.3 as follows:

2

so

o

v G

g

γ =

where γ is the soil unit weight and g is the acceleration of gravity. 

Note that the shear wave velocity here, vso, is not the effective value 

including the modification per ASCE/SEI 7-16 Table 19.3-1, but the 

measured value. The effective shear modulus, G, is then determined by 

modifying Go by the applicable factor from ASCE/SEI 7-16 Table 19.3-2. 

ASCE/SEI 7-16 and NIST indicate that Poisson’s ratio, ν, may be taken as 

0.3 for sandy soils and 0.45 for clayey soils. ASCE/SEI 41-17 indicates that 

ν may be taken as 0.5 for saturated clays and 0.25 for other soils. 

6.3 Vertical and Rotational Springs

In this section, the discussion of vertical springs pertains to vertical stiffness 

of foundations that affect the rotation of a frame or wall about its base, not to 

rigid-body vertical movement of a building or portion of a building excited 

by vertical ground motion. 

ASCE/SEI 7-16 provides no guidance regarding the development of vertical 

and rotational springs to compute the rotational stiffness of a foundation 

about an axis perpendicular to its length. There is rotational spring formula 

in ASCE/SEI 7-16 Section 19.3 as part of the radiation damping calculations. 

However, this formula is for rotation about an axis parallel to its length. 

ASCE/SEI 41-17 Chapter 8 presents provisions for three methods, described 

in the sections below. The three methods are depicted graphically in Figure 

6-1. 

6.3.1 Method 1 – Rigid Foundation and Flexible Soil

This method defines point springs, modeling each degree of freedom at the 

base of a foundation with an uncoupled spring. This method is 

recommended for foundations considered to be rigid relative to the soil. The 

equations in ASCE/SEI 41-17 include all six degrees of freedom and are 

taken from NIST, but with the foundation length and width parameters 

defined differently, as discussed in the margin box above. The definitions of 

L (half-length) and B (half-width) in the NIST document are the same as 

those in ASCE/SEI 7-16 Chapter 19. 
6-4 6: Foundation and Soil Flexibility FEMA P-2091

Method 1 

Method 2 

Method 3 

Figure 6-1 Three methods for foundation 

modeling approaches with vertical 

and rotational springs presented in 

ASCE/SEI 41 from FEMA (2018).

NIST (2012a) provides two sources for these elastic solutions for spring 

values, as discussed above. The coordinate system employed in these 

equations specifies the x-axis as parallel to the longer footing dimension, the 

y-axis parallel to the shorter dimension, and the z-axis vertically down (righthand rule). The more widely used equations from Pais and Kausel (1988) for 

foundations at the ground surface for vertical (z-axis) stiffness and strongaxis (yy) rotation are presented below. The definitions and orientations of the 

foundation dimensions and axes are shown in Figure 6-2. 

GB     L 0.75

Kz,sur =   3.1  +1.6

1− ν       B

GB3     L 2.4

Kyy,sur =   3.73  + 0.27

1− ν       B
FEMA P-2091 6: Foundation and Soil Flexibility 6-5 

Figure 6-2 Definitions and orientations of 

dimensions and axes. 

Figure 6-3 shows the variation of vertical stiffness, Kz,sur, with foundation 

dimensions. The figure includes stiffness computed for example values of L

(half-length) = 10 ft, 15 ft, and 20 ft, B (half width) = 1 ft to 10 ft, G = 500 

kip/ft2

 and ν = 0.3. As the width increases by 10, the stiffness increases by 

approximately 2. As the length increases by a 2, the stiffness increases by 

approximately 1.5. Note that these particular values are dependent on the 

value of Poisson’s ratio. 

Figure 6-3 Variation of vertical stiffness with foundation dimensions.

Figure 6-4 shows the variation of strong-axis rotational stiffness Kyy,sur, with 

foundation dimensions, for the same parameters. As the width increases by 

10, the stiffness increases by approximately 4. As the length increases by 2, 

the stiffness increases by approximately 5. As would be expected, the 

rotational stiffness is much more sensitive to the length than to width. 
6-6 6: Foundation and Soil Flexibility FEMA P-2091

Figure 6-4 Variation of rotational stiffness with foundation dimensions.

Pais and Kausel (1988) provide modification factors to increase the values of 

stiffness for each degree of freedom based on the embedment of the 

foundation in the soil. These are presented below, for vertical (z-axis) 

stiffness and strong-axis (yy) rotation. 

   0.25   D 0.8 

η =z 1.0 +   0.25 +       L B   B 

  D D   1.6  2

η = yy   1.0 + +      B 0.35 +   (L B)

4   B  

Here, D is the depth from the ground surface to the bottom of the foundation. 

The net stiffness in each degree of freedom is the product of the applicable K

and η values above. 

Figure 6-5 shows the variation of the embedment modifier for vertical 

stiffness, ηz, with foundation dimensions, for example values of L (halflength) = 15 ft, B (half width) = 1 ft to 10 ft, D (depth) = 2 ft, 4 ft, and 6 ft. 

The modifier is essentially the same for values of L of 10 ft and 20 ft as it is 

for L of 15 ft. For small values of B and large values of D, the modifier can 

exceed 2, while as B increases, the modifier becomes less significant. 
FEMA P-2091 6: Foundation and Soil Flexibility 6-7 

Figure 6-5 Variation of the embedment modifier vertical stiffness with 

foundation dimensions.

Figure 6-6 shows the variation of the embedment modifier for strong-axis 

rotational stiffness ηyy, with foundation dimensions, for the same parameters. 

The modifier is similar for values of L of 10 ft and 20 ft as it is for L of 15 ft. 

For small values of B and large values of D, the modifier can be as high as 7 

for the cases illustrated here, while as B increases, the modifier becomes less 

significant. The rotational modifier is highly dependent on the term D/B. 

Figure 6-6 Variation of embedment modifiers for rotational stiffness with 

foundation dimensions.
6-8 6: Foundation and Soil Flexibility FEMA P-2091

6.3.2 Method 2 – Flexible Foundation and Nonlinear Flexible 

Soil

This method defines distributed springs representing the soil support as a 

discretized continuous medium, with the springs at each end of the footing 

increased in stiffness. This approach is recommended for nonlinear 

procedures where explicit coupling of axial and rotational stiffness is 

modeled with nonlinear springs. This method is explored in detail in Section 

2.2.3 of NIST (2012a) and in Chapter 5 of FEMA P-2006 (FEMA, 2018), 

and is not discussed in detail here.

6.3.3 Method 3 – Flexible Foundation and Linear Flexible Soil

This method defines distributed springs representing the soil support as a 

discretized continuous medium, with a uniform value for the springs along the 

length of the footing. This method is best used when the flexibility of the 

structural elements of the foundation are modeled explicitly. ASCE/SEI 41-17 

Equation 8-11 provides a recommended unit subgrade spring coefficient. 

1.3

(1 ) sv

f

G k

B = − ν

Here, G is the effective shear modulus, described in Section 6.1, Bf is the full 

width of the foundation (not the half-width), and ν is Poisson’s ratio, also 

described in Section 6.1. The subgrade spring coefficient, ksv, has units of 

force per length cubed, or vertical stiffness per unit area. Where the 

foundation element is modeled as a line, the vertical stiffness per area is 

multiplied by the foundation width, cancelling the Bf quantity in the 

denominator, resulting in a stiffness per unit length that is independent of the 

foundation width. This is applicable if the foundation width was determined 

based on the total foundation load. A wider foundation having a higher total 

load results in a greater depth to which the soil stress extends. More soil 

under stress means more deformation and therefore lower stiffness. 

Using the properties presented in Section 6.1, the value of this stiffness 

would be: 

2 1.3(500 kip/ft ) 929 kip/ft/ft (1 0.3) sv f k B = = −

It is also possible to employ the vertical stiffness from Section 6.3.1 divided 

by the foundation length as an alternative value of this quantity. Figure 6-7 

presents examples of the resulting values for the properties discussed above, 

which range in general from 1 to 3 times the stiffness of that computed via 

Method 3 and is not independent of the foundation width. 
FEMA P-2091 6: Foundation and Soil Flexibility 6-9 

Figure 6-7 Vertical stiffness per unit length computed via Method 1, for 

various foundation half-lengths and half-widths, with stiffness 

computed by Method 3 for comparison. 

The effect of the Method 3 stiffness may be compared to a rotational spring 

through the transformation below, adapted from Yim and Chopra (1984). 

2 3

3

K k BL yy sv f =

For the case of L = 15 ft, the equivalent rotational stiffness is then

3 6

2

2 kip kip-ft (929 )(15 ft) 2.09 10

3 ft rad

Kyy = = ×

This value is on the lower end of the curve for L = 15 ft in Figure 6-4. 

While Method 1 includes specific modifiers to account for the embedment 

depth, Method 3 does not. However, it may be appropriate to increase the 

Winkler stiffness developed in Method 3 in a manner similar to Method 1. 

6.4 Horizontal Springs

ASCE/SEI 7-16 requires that the analytical model of the structure 

incorporate the horizontal flexibility of the soil, as well as the vertical and 

rotational flexibility. Generally, the horizontal flexibility will affect the 

behavior of the structure far less significantly than will the vertical and 

rotational flexibility. Nonetheless, all sources of flexibility must be 

considered. 
6-10 6: Foundation and Soil Flexibility FEMA P-2091

Note that, unlike the vertical and rotational stiffness, which pertain only to 

the foundations supporting the individual seismic load resisting elements, the 

horizontal stiffness will usually include resistance from all foundation 

elements, including gravity-only footings. This will apply in most cases but 

may not apply if a building does not include a relatively rigid slab-on-grade 

or system of grade beams. 

As is the case with the vertical and rotational springs, ASCE/SEI 7-16 

provides no guidance regarding the development of horizontal springs 

parallel to the foundation length. There is a horizontal spring formula in 

ASCE/SEI 7-16 Section 19.3 as part of the radiation damping calculations. 

However, this formula is for horizontal movement parallel to the width of the 

foundation, not parallel to its length. Again, ASCE/SEI 41-17 Chapter 8 

provides some guidance. 

6.4.1 Method 1

Method 1 in ASCE/SEI 41-17 Chapter 8 provides recommendations for 

horizontal springs as well as for the vertical and rotational springs discussed 

in Section 6.3.1. The equations are again taken from NIST, with two 

alternative sets of equations included. The equation from Pais and Kausel 

(1988) for foundations at the ground surface for horizontal (x-axis) stiffness 

is presented below. This x-axis is parallel to L, the longer foundation halfdimension. 

0.65

, 6.8 2.4

2 x sur

GB L K

B

    =     +

− ν    

Note that while the equations for foundation flexibility in Method 1 are 

indicated as applicable to conditions where the foundation is rigid relative to 

the soil, and the foundation’s structural flexibility is not modeled explicitly, 

these limitations do not necessarily apply to the horizontal degrees of 

freedom. This is simply because there is essentially no structural flexibility 

of the foundation applicable to this direction of movement. 

Figure 6-8 shows the variation of horizontal stiffness, Kx,sur, for the same 

foundation parameters and dimensions discussed above in Section 6.3.1. As 

the width increases by 10, the stiffness increases by approximately 2 3⁄4. As 

the length increases by a 2, the stiffness increases by approximately 1.5. 
FEMA P-2091 6: Foundation and Soil Flexibility 6-11

Figure 6-8 Variation of horizontal stiffness with foundation dimensions.

The Pais and Kausel (1988) modification factor to increase the value of 

horizontal stiffness based on the embedment of the foundation in the soil is 

presented below. 

0.8 1.34 1.0 0.33

1 x

D

LB B

     η= + +        +    

Again, D is the depth from the ground surface to the bottom of the 

foundation. The net stiffness is the product of Kx and ηx. 

Figure 6-9 shows the variation of the embedment modifier for horizontal 

stiffness, ηx, for the same foundation parameters and dimensions discussed 

above in Section 6.3.1. Similar to the modifier for vertical stiffness, this 

modifier is essentially the same for values of L of 10 ft and 20 ft as it is for L

of 15 ft. For small values of B and large values of D, the modifier can 

exceed 2, while as B increases, the modifier becomes less significant. 

The horizontal spring values are limited by the maximum passive pressure 

and friction that can be developed. Section 6.4.2 of the Guide discusses 

passive pressure; Section 6.4.3 of the Guide discusses friction. 
6-12 6: Foundation and Soil Flexibility FEMA P-2091

Figure 6-9 Variation of the embedment modifier horizontal stiffness 

with foundation dimensions.

6.4.2 Resistance due to Passive Pressure

Rather than using the horizontal stiffness formulae in Method 1 of ASCE/SEI 

41-17 and in NIST (2012a), it is also possible to compute the horizontal 

stiffness by considering passive pressure on the sides of foundations based on 

the recommendations from the geotechnical engineer. Note that ultimate 

passive pressure is primarily applicable for assessment of the ultimate 

strength of the resistance provided by the soil, rather than the stiffness 

directly. A foundation’s horizontal stiffness may be computed by dividing 

the passive pressure developed by the deformation at which it is developed, 

which can be estimated via a passive pressure mobilization curve. 

ASCE/SEI 41-17 provides a mobilization curve for passive pressure, which 

is reproduced in Figure 6-10. This figure presents the ratio of the passive 

pressure mobilized to the ultimate pressure (P/Pult) as a function of the ratio 

of the lateral deformation to the depth of the footing (δ/H). This 

mobilization relationship can be approximated by the following equation, 

which has been fitted to the curve in the figure. 

0.43

0.15 2.88

ult

P

P H

  δ

= +    
FEMA P-2091 6: Foundation and Soil Flexibility 6-13

Figure 6-10 Passive pressure mobilization curve (reproduced from ASCE/SEI 

41-17, Figure 8-6).

Note, however, that it may actually be more appropriate to use the stiffness 

from Section 6.4.1 for the initial part of the curve, then develop the largerstrain response as shown here. 

6.4.3 Resistance due to Friction or Cohesion

Unlike the resistances discussed above, resistance due to friction between the 

structure and the soil depends on the weight of the supported structure and 

the foundation. Frictional resistance at the base of the foundation depends on 

the angle of internal friction of the soil and may be taken as ranging from 

0.67 to 1 times the tangent of the angle of internal friction (Bowles, 2001). 

Where foundation soils are cohesive, resistance may be developed by 

cohesion between the concrete foundation and the soil, which may be taken 

as 0.5 to 0.75 times the internal cohesion value of the soil (Bowles, 2001). 

Resistance due to friction or cohesion occurs at very small deformations, so 

the precise computation of the resulting stiffness can be difficult. It is 

reasonable to consider that the frictional resistance will develop at a rate 

similar to that of the initial, steep portion of the passive pressure mobilization 

curve. Thus, the resistance due to friction can be added to that due to passive 

pressure for small deformations.
6-14 6: Foundation and Soil Flexibility FEMA P-2091

6.5 Bounding Analyses

ASCE/SEI 7-16 Sections 12.13.3 and 19.1.1 require the use of bounding 

analyses when considering the flexibility of foundations. ASCE/SEI 7-16 

Section 12.13.3 requires consideration of a 50% increase and decrease in the 

computed soil stiffness for the use in dynamic analyses. ASCE/SEI 7-16 

Section 19.1.1 references these bounds when soil-structure interactions are 

performed. ASCE/SEI 41-17

ASCE/SEI 41-17 Section 

8.4.2 includes a similar 

bounding requirement. The 

upper bound is stated as 

twice the calculated value 

and the lower bound to be 

half the calculated value. 

See ASCE/SEI 41-17 Figure 

8-1 for a graphical 

representation of this 

requirement. Note that the 

upper bound is higher than 

that indicated in ASCE/SEI 

7-16.

The requirement for bounding analyses reflects recognition of the spatial 

variability and uncertainty of the soil’s mechanical properties. In addition, 

while linear structural analyses may include foundation flexibility, soil 

behavior is highly nonlinear, which introduces further uncertainty in its 

behavior. 

Another source of potential added flexibility and nonlinearity is the potential 

loss of contact between the structure and the soil due to rocking. In cases 

where such loss of contact is possible, the spring stiffness calculations 

discussed above will result in stiffer spring values that would be present if 

the loss of contact occurs. In nonlinear analyses, the loss of contact can be 

modeled explicitly and therefore more precisely. The degree to which the 

nonlinearity due to rocking can affect the rotational stiffness of a shallow 

foundation is addressed in Yim and Chopra (1984). In nonlinear analysis, it 

is important to include the loss of soil-structure contact in the elements at the 

base of the resisting elements so that this increased flexibility is considered. 

6.6 Vertical and Rotational Spring Example

This section includes illustrative examples of vertical and rotational springs, 

based on the foundation of the transverse frame of the building in Appendix 

A. The foundation and soil in that example have the following dimensions 

and properties: 

• Length, 2L = 38 ft; L = 19 ft

• Width, 2B = 8 ft; B = 4 ft

• Depth, D = 5 ft

• Effective shear modulus, G = 538 kip/ft2

• Poisson’s ratio, ν = 0.3 

The examples below will cover Methods 1 and 3 from ASCE/SEI 41-17. See 

FEMA P-2006 for an example of flexibility calculations according to Method 2. 
FEMA P-2091 6: Foundation and Soil Flexibility 6-15

6.6.1 Method 1

Following the Pais and Kausel (1988) equations presented in Section 6.3.1, 

the surface rotational stiffness is computed as follows:

GB3     L 2.4

Kyy,sur =   3.73  + 0.27

1− ν       B

  kip 538 3

    ft 2.4 2 (4 ft)    19 ft  kip-ft Kyy,sur = 3.73  + = 0.27 7.733×106

1− 0.3    4 ft  rad

The surface stiffness is then modified to account for the foundation depth. 

  D D   1.6  2

η = yy   1.0 + +  

B 4       +   B     0.35 (L B)

  5 ft  1.6   5 ft 2

η = yy   1.0 + +  4   = 2.25   4 ft  +   4 ft    0.35 (19 ft 4 ft) 

The stiffness of a single spring representing the rotational stiffness of the 

entire foundation is then computed as follows:

  6 6 kip ft kip-ft Kyy = Kyy,surη = yy   7.733×10 (2.25) =17.399×10   rad rad

6.6.2 Method 3

The uniform vertical spring providing rotational resistance at the base of the 

foundation is computed by ASCE/SEI 41-17 Method 3 as follows:

1.3G k Bsv f = (1− ν)

1.3(538 kip / ft 2) k Bsv f = =1,000 kip/ft/ft (1− 0.3)

A comparative uniform spring, based on the Method 1 vertical stiffness 

spread over the foundation length, is computed as follows. 

GB     L 0.75

Kz,sur =   3.1  +1.6

1− ν       B

  kip   538

ft 0.75   2 (4 ft)     19 ft kip Kz,sur =   3.1  +1.6 = 35,583 1− 0.3       4 ft ft
6-16 6: Foundation and Soil Flexibility FEMA P-2091

This can be converted to a uniform surface spring per unit length by dividing 

by the total foundation length, 2L. 

( )

,

, ,

kip 35,583 ft 936 kip/ft/ft

2 2 19 ft

z sur

z unif sur

K

K

L = = =

The value of this Winkler spring is within 5% of that computed by Method 3. 

Note that neither of these computations include a modification for the depth 

of the foundation, as this is not addressed in Method 3. 

6.7 Horizontal Spring Example

This section includes illustrative examples of horizontal springs, also based 

on the foundation of the transverse frame of the building in Appendix A. 

The properties of the foundation and soil are as described in Section 6.6, 

above. 

As above, the examples below will cover Method 1 from ASCE/SEI 41-17 as 

well as resistance due to passive pressure on the face of the footing and 

friction at the base of the footing. 

6.7.1 Method 1

Following the Pais and Kausel (1988) equations presented in Section 6.3.1, 

the surface rotational stiffness is computed as follows:

0.65

, 6.8 2.4

2 x sur

GB L K

B

    =     +

− ν    

( ) 0.65 2

,

kip 538 4 ft

ft 19 ft 6.8 2.4 26,738 kip/ft 2 0.3 4 ft

Kx sur

          =   + =   −    

The surface stiffness is then modified to account for the foundation depth. 

0.8 1.34 1.0 0.33

1 / x

D

LB B

     η= + +         +   

( )

0.8 1.34 5 ft 1.0 0.33 1.67

1 19 ft 4 ft 4 ft x

     η= + +      =

  +      

The stiffness of a single spring representing the horizontal stiffness 

foundation is then computed as follows: 

, ( ) kip kip kip 26,738 1.67 44,735 3,728 ft ft in

K K x x sur x

 

= η=   = =  
FEMA P-2091 6: Foundation and Soil Flexibility 6-17

6.7.2 Passive Pressure and Friction

The following structural information is also required to compute the 

horizontal stiffness of the transverse frame from the building in Appendix A: 

• Imposed gravity load (two columns), W = 152 kips. The gravity load 

used here is based on the load combination D + 0.25L, where L is the 

unreduced live load, as described in ASCE/SEI 41-17 Section 7.2.2. 

• Passive pressure = 1.6 kip/ft2, acting on the leading face of the footing.

• Friction coefficient = 0.35. 

Note that, for some sites, the geotechnical engineer will recommend that the 

passive pressure be computed based on a triangular distribution or may 

recommend that the upper portion of the footing face be neglected for this 

calculation. 

The frictional resistance at the base of the foundation is 

(0.35)(152 kip) 53.2 kip =

The fully mobilized passive pressure resistance is 

2

kip 1.6 (8 ft)(5 ft) 64 kip ft

  =    

In this example, the total horizontal force to be resisted by this single footing 

is 68 kips. The frictional resistance is developed at a very small deformation, 

resulting in a net force to be resisted by passive pressure of 

68 kip – 53.2 kip = 14.8 kip 

The deformation at which this level of resistance occurs can be determined 

by solving the fitted equation to the passive-pressure mobilization curve. 

( )

0.43

1/0.43

14.8 kip 0.23 0.15 2.88

64 kip

0.34 0.15 0.00025

2.88

0.00025 60 in 0.015 in

ult

P

P H

H

H

  δ

= = = +    

  − δ   = =  

δ= = =

The horizontal stiffness of the soil-foundation interface computed in this way 

is then 

Kx = 68 kip / 0.015 in 4,550 kip/in =

This value is 22% higher than that computed by Method 1. Please note that 

the stiffness computed in this manner is highly dependent on the magnitude 

of the force resisted, due to the shape of the passive pressure mobilization 

curve. 

FEMA P-2091 7: Period Lengthening 7-1 

Chapter 7 

Period Lengthening

7.1 Overview

Chapter 6 discusses how vertical and horizontal springs are used to model 

soil flexibility. Consideration of that flexibility in lateral analysis produces 

longer fundamental periods and generally increases the degree of mass 

participation in fundamental modes. The shift to a longer fundamental 

period always increases displacement response and can increase or decrease 

acceleration response.

The key equation used to address period lengthening also includes the effect 

of foundation damping. Consideration of foundation damping is discussed in 

Chapter 8; this chapter uses the typical damping ratio of 0.05 so that the 

period lengthening effect can be isolated.

These effects of period lengthening on acceleration and displacement 

response are a consequence of the basic shape of acceleration and 

displacement spectra. Figure 7-1 shows a smooth acceleration design 

spectrum and the corresponding displacement spectrum. The spectral 

displacement ordinates, Sd, are computed from the spectral acceleration 

ordinates, Sa, using a following fundamental equation of structural dynamics. 

2

2 4 d a

T SS g = π

For increasing period, spectral acceleration increases to a peak value (at 

approximately 0.5 seconds in this example) and then decreases for longer 

periods. Spectral displacement generally increases with increasing period. 

As a result, period lengthening will result in corresponding changes to the 

acceleration and displacement response. Figure 7-1 shows the possible 

effects for Structure “X” and Structure “Y.” For Structure X, increasing the 

period increases both spectral acceleration and spectral displacement. For 

Structure Y, increasing the period decreases spectral acceleration and 

increases spectral displacement.

Period Lengthening

Description: Soil flexibility 

leads to period lengthening 

in the superstructure, 

changing seismic demands.

Requirements:

• Two models must be 

prepared: one that 

considers the structure to 

be fixed at its base, and 

another where the loaddeformation 

characteristics of the 

foundation-soil system 

(including lower- and 

upper-bound variations) 

are represented explicitly.

• Seismic response 

coefficients Cs and 

computed using the fixedbase fundamental period, 

T, and flexible-base 

fundamental period, , 

respectively.

• Seismic weight, which can 

be taken as either the 

effective seismic weight, 

W, or the weight 

corresponding to the 

mass participating in the 

fundamental mode, . 
7-2 7: Period Lengthening FEMA P-2091

Figure 7-1 Example acceleration and displacement spectra.

7.2 Key Equation

The effect of period lengthening on the design base shear is determined by 

comparing the fundamental periods and corresponding seismic response 

coefficients for the fixed-base and flexible-base models. ASCE/SEI 7-16 

Equation 19.2-2 is used to compute the base shear adjustment due to SSI, ∆V.

s

s

SSI

VC W

B

  ∆= −    

C ̃
FEMA P-2091 7: Period Lengthening 7-3

where:

Cs = the seismic response coefficient determined in accordance with

ASCE/SEI 7-16 Section 12.8.1.1 assuming a fixed structural base

at the foundation-soil interface.

C ̃

s = the seismic response coefficient determined in accordance with 

ASCE/SEI 7-16 Section 12.8.1.1 assuming flexibility of the 

structural base at the foundation-soil interface in accordance with 

ASCE/SEI 7-16 Section 19.1.1 using T ̃ as the fundamental period 

of the structure in lieu of the fundamental period of the structure, 

T, as determined by ASCE/SEI 7-16 Section 12.8.2. 

BSSI= the factor to adjust the design response spectrum and MCER

response spectrum in accordance with ASCE/SEI 7-16 Sections 

11.4.6 and 11.4.7 or a site-specific response spectrum for damping 

ratios other than 0.05. (BSSI is taken as 1.0 in this chapter, i.e., the 

typical, default damping ratio of 0.05 is assumed.)

W = weight caused by the effective modal mass in the fundamental 

mode, alternatively taken as the effective seismic weight in 

accordance with ASCE/SEI 7-16 Section 12.7.2.

T = fundamental period of the structure determined in accordance with

ASCE/SEI 7-16 Section 12.8.2 based on a mathematical model 

with a fixed base condition. The upper bound limitation of CuTa

on the fundamental period from ASCE/SEI 7-16 Section 12.8.2 

shall not apply, and the approximate structural period, Ta, shall not 

be used.

T ̃ = fundamental period of the structure using a model with a flexible

base in accordance with ASCE/SEI 7-16 Section 19.1.1. The 

upper bound limitation of CuTa on the fundamental period from 

ASCE/SEI 7-16 Section 12.8.2 shall not apply, and the 

approximate structural period, Ta, shall not be used.

The key equation for period lengthening calls for computation of seismic 

response coefficients in accordance with ASCE/SEI 7-16 Section 12.8.1.1 for 

both fixed-base and flexible-base models. The intent is to better reflect the 

changes in dynamic characteristics of the system (due to period elongation 

and foundation damping). ASCE/SEI 7-16 Equations 12.8-2, 12.8-3, and 

12.8-4 define seismic response coefficients that correspond to the constant 

acceleration, constant velocity, and constant displacement portions of the 

design response spectrum, respectively. To illustrate the important changes 

in dynamic response due to period lengthening as clearly as possible, the 
7-4 7: Period Lengthening FEMA P-2091

examples in Section 7.3 use the smooth spectra directly—modified by 1/R

for strength demands and by Cd/R for drift demands. The example in Section 

7.4 uses the equations in ASCE/SEI 7-16 Sections 12.8.1.1 and 19.2.1, which

produce results that differ from the pure dynamic calculations.

ASCE/SEI 7-16 Equations 12.8-5 and 12.8-6 set forth code minima that are 

independent of both the dynamic characteristics of the system and the basic 

spectral shape. Although the adjustment of structural demands to reflect SSI 

in accordance with ASCE/SEI 7-16 Section 19.2 is expressed as 

modifications to the base shear for the fixed-base structure computed in 

accordance with ASCE/SEI 7-16 Section 12.8.1, it is not intended that SSI 

modify the code minima reflected in ASCE/SEI 7-16 Equations 12.8-5 and 

12.8-6. 

ASCE/SEI 7-16 Equations 19.2-1 and 19.2-3 are used to limit the reduction 

in base shear due to SSI. While earlier editions of ASCE/SEI 7 limit the base 

shear reduction to 30%, ASCE/SEI 7-16 limits the reduction as a function of 

the response modification factor, R. According to the Commentary this 

reflects “the limited understanding of how the effects of SSI interact with the 

R factor” and the concern that damping caused by structural yielding could 

make foundation damping less effective. The limits are computed as 

follows: 

V = V−∆V  ̃ ≥ αV (ASCE 7-16 Eq. 19.2-1) 

0.7 for R ≤ 3 

α = 0.5 + R R 15 for 3 < < 6 (ASCE 7-16 Eq. 19.2-3) 



0.9 for R ≥ 6

where:

V ̃ = the base shear adjusted for soil-structure interaction

V = the fixed-base structure base shear computed in accordance with 

ASCE/SEI 7-16 Section 12.8.1

α = the coefficient that accounts for the reduction in base shear caused 

by foundation damping SSI

R = the response modification coefficient

Unfortunately, this approach does not distinguish between period lengthening 

and foundation damping effects. It would be possible to recast ASCE/SEI 

7-16 Section 19.2.1 so that the α coefficient applies only to the portion of

SSI reduction that results from foundation damping. See Section 10.5.1 of

this Guide for a recommended code change.

Limited Reduction Due to 

Foundation Damping

The intent of ASCE/SEI 7-16 

Equation 19.2-3 is to limit 

base shear reductions 

caused by foundation 

damping, but as formulated 

in ASCE/SEI 7-16 

reductions due to period 

lengthening are also 

affected. A change proposal 

for the next edition of 

ASCE/SEI 7 has been 

submitted to address this 

issue. 
FEMA P-2091 7: Period Lengthening 7-5

7.3 Examples to Illustrate Dynamic Response

To illustrate important changes in dynamic response due to period 

lengthening, calculations are performed for three example buildings, as 

follows: 

• 3-story special reinforced concrete shear wall building

• 10-story special reinforced concrete shear wall building

• 10-story steel special moment frame building

The examples in this section reflect only the dynamic effects computed using 

the best available tools—without consideration of limitations imposed by 

ASCE/SEI 7-16. The fixed-base buildings, which do not have basements, 

have been realistically proportioned to satisfy strength and drift requirements 

of ASCE/SEI 7-16. Table 7-1 shows the fundamental mode dynamic 

properties computed for both fixed-base models and flexible base models.

Table 7-1 Fundamental Mode Dynamic Properties for Example 

Buildings

Example Building

Fixed-base Condition Flexible-base Condition

Period, T

(s)

Fundamental 

Mode Mass 

Participation

Period, T

 ̃

(s)

Fundamental 

Mode Mass 

Participation

3-story reinforced

concrete shear wall 0.193 76.5% 0.492 91.%

10-story reinforced

concrete shear wall 1.32 65.0% 2.28 76.0%

10-story steel special

moment frame 2.04 80.3% 2.40 82.1%

As expected, introducing foundation flexibility (represented by vertical and 

horizontal springs) results in period lengthening and increase in fundamental 

mode mass participation. The introduction of a concentrated source of 

(foundation) flexibility at the base of a structure has an effect similar to 

seismic isolation, which allows more of the superstructure mass to participate 

in the adjusted fundamental mode. Figure 7-2 highlights the changes in 

fundamental mode properties and response on the acceleration and 

displacement spectra.
7-6 7: Period Lengthening FEMA P-2091

Figure 7-2 Example acceleration and displacement spectra, including 

points for each example building. 

Key lessons that can be drawn from Figure 7-2 are as follows:

• Because the fixed-base period for the 3-story reinforced concrete shear 

wall building is less than the period with maximum spectral acceleration, 

period lengthening leads to an increase in the spectral acceleration 

(which would be associated with increase strength demands). 

• Because the fixed-base periods for the 10-story example buildings are 

greater than the period with maximum spectral acceleration, period 

lengthening leads to a decrease in spectral acceleration (which would be 

associated with reduced strength demands).

• Period lengthening always results in increased spectral displacement 

(which would be associated with increased story drifts). 
FEMA P-2091 7: Period Lengthening 7-7 

Although fundamental mode response is dominant, as can be seen from the 

mass participation factors in Table 7-1, the example buildings have 

additional modes of dynamic response. Figures 7-3, 7-4, and 7-5 show the 

global results of modal response spectrum analysis (MRSA) for the example 

buildings. The story shears, story moments, and story drifts follow the same 

trends that appear in the fundamental mode response illustrated in Figure 7-2. 

Figure 7-5 shows story drift ratios greater than zero at the base for the 

flexible-base conditions. Because foundation springs are introduced in those 

models, the displacement at the base is non-zero. The base displacement is 

equal to the base “drift.” Since Figure 7-5 shows story drift ratios, it is 

necessary to divide the story drift by a story height; story drift ratios are 

effectively a measure of story rotation angle. Taking a story height of zero at 

the base would result in undefined drift ratios (due to a divide by zero error), 

so the figure assumes a fictitious story height of 12’-6” below grade. As a 

result, for this building with uniform story heights, the drift ratio plot exactly 

mirrors the story drift plot, which would show non-zero base drift. This 

method also illustrates the increased base rotation due to foundation (soil) 

flexibility.
7-8 7: Period Lengthening FEMA P-2091

Figure 7-3 Story shears for example buildings.

The effects of adding foundation flexibility on global structural demands 

(story shear and moment) are more pronounced for the inherently stiffer 

reinforced concrete shear wall structures. The changes in global structural 

demands for the inherently flexible steel moment frame structure are more 

modest. 
FEMA P-2091 7: Period Lengthening 7-9 

Figure 7-4 Story moments for example buildings.
7-10 7: Period Lengthening FEMA P-2091

Figure 7-5 Story drift ratios for example buildings.

A few lessons can be learned from Figure 7-5, as follows:

• Foundation (soil) flexibility leads to displacement at the base.

• Rocking response of the 10-story reinforced concrete shear wall building

produces greater changes in story drift ratio in the lower half of the

building.

• Where story drift ratios for the fixed-base 10-story steel moment frame

structure are less than the allowable story drift (2%) set forth in

ASCE/SEI 7-16 Table 12.12-1, the flexible-base structure slightly

exceeds the allowable story drift at Level 2.
FEMA P-2091 7: Period Lengthening 7-11

• Although the story drift ratios for the fixed-base 10-story reinforced

concrete shear wall building are below the allowable story drift ratio,

consideration of foundation flexibility results in story drifts that exceed

the limit considerably—due to rotation at the base. Since the rotation

about the base of the wall is largely rigid body rotation, there is very

little shear distortion in the upper levels of the wall. However, as

illustrated in Figure 7-6 the gravity framing and nonstructural systems

are subject to shear distortion corresponding to the full story drift.

Figure 7-6 Displaced shape for flexible-base, 10-story RCSW building. 

7.4 Example Using Code Equations

Application of several provisions in ASCE/SEI 7-16 affect the way that soil 

structure interaction impacts design—beyond the basic dynamic effects 

discussed above. These include short period spectral minima, upper limits on 

the fundamental period, minimum base shear equations, scaling of MRSA 

results to achieve the base shear values from equivalent lateral force 

procedure (ELF), and limits on foundation damping. 

An example application is used to illustrate the calculation of the period 

lengthening effect in accordance with ASCE/SEI 7-16 Section 19.2 as 
7-12 7: Period Lengthening FEMA P-2091

modifications to the ELF procedure of Section 12.8 and the modal response 

spectrum analysis design requirements of Section 12.9.1. 

The example building has plan dimensions of 120 ft in both directions, ten 

stories at 12 ft 6 in story height, no basement, and perimeter steel special 

moment frames. The seismic weight tributary to each perimeter moment 

frame, W, is 7,000 kip. The response modification factor, R, is 8. The 

seismic importance factor, Ie, is 1.0. 

The building is located in a high seismic area with SDS = 1.39 and SD1 = 1.16. 

Figure 7-7 shows the design response spectrum. 

Figure 7-7 Design response spectrum for the example building. 

0.0

0.5

1.0

1.5

0.0 1.0 2.0 3.0

Spectral Acceleration, 

Sa (g)

Period, T (s)

Design response spectrum

Short period minimum

10SMF, fixed

10SMF, flexible

7.4.1 Calculations for the Fixed-Base Condition 

The fundamental period from modal analysis for the model with pin supports

at the base of each column is 2.04 seconds (with 80.3% of the system mass 

participating in this mode).

The approximate period is computed using ASCE/SEI 7-16 Equation 12.8-7, 

with values of Ct and x taken from Table 12.8-2 for steel moment-resisting 

frames resisting 100% of the required seismic force, as: 

( )0.8 0.028 125 1.33 s x T Ch a tn = = =

The upper limit period is computed in accordance with ASCE/SEI 7-16 

Section 12.8.2 as: 

1.4 1.33 1.87 s C Tu a =× =
FEMA P-2091 7: Period Lengthening 7-13

Since this period is shorter than that computed in the analysis, the upper limit 

period is used in subsequent calculations for the fixed-base condition. 

The seismic response coefficient (for the constant acceleration portion of the 

design spectrum) is computed using ASCE/SEI 7-16 Equation 12.8-2 as 

follows: 

1.39 0.174

8

1.0

DS

s

e

S C

R

I

= = =          

Note that the portion of the design spectrum extending from zero period to 

the plateau may not be used in computing the base shear. Instead, ASCE/SEI 

7-16 Equation 12.8-2 enforces the short period minimum shown in Figure

7-7.

However, the seismic response coefficient need not be taken greater than the 

value computed using ASCE/SEI 7-16 Equation 12.8-3 or 12.8-4, as 

applicable. Since T = 1.87 seconds < TL, Equation 12.8-3 applies as follows: 

1 1.16 0.0775

8 1.87

1.0

D

s

e

S C

R T

I

= = =           

The minimum seismic response coefficient is computed using ASCE/SEI 

7-16 Equation 12.8-5 as follows:

C SI s = = 0.044 0.044 1.39 1.0 0.0612 DS e ( )( ) =

ASCE/SEI 7-16 Equation 12.8-3 controls, so the seismic base shear (for each 

frame) is computed using ASCE/SEI 7-16 Equation 12.8-1 as follows:

V CW = = s 0.0775 7,000 543 kip ( ) =

The base shear from modal analysis for the fixed-base condition is Vt = 433 

kips. Design force values from modal analysis would require scaling by the 

factor V/Vt = 543/433 = 1.25 in accordance with ASCE/SEI 7-16 Section 

12.9.1.4.1. 

7.4.2 Calculations for the Flexible-Base Condition

The fundamental period from modal analysis for the model with vertical and 

horizontal spring supports at the base of each column is 2.40 seconds (with

82.1% of the system mass participating in this mode). This reflects a period 

lengthening ratio, T ̃ T = 2.40 2.04 =1.18 . 
7-14 7: Period Lengthening FEMA P-2091

As noted in the definitions of T and T ̃in ASCE/SEI 7-16 Section 19.1.3, 

where the provisions of Chapter 19 are used for SSI, the upper bound 

limitation of CuTa does not apply. Instead, the computed periods, T and T ̃

are used directly.

The seismic response coefficient for the fixed-base condition (where that

condition will be adjusted in accordance with ASCE/SEI 7-16 Section 19.2) 

is controlled by ASCE/SEI 7-16 Equation 12.8-3 computed using T = 2.04 s

as follows:

1 1.16 0.0711

8 2.04

1.0

D

s

e

S C

R T

I

= = =           

The corresponding fixed-base base shear (using T, since CuTa doesn’t apply), 

is computed as 0.0711 times 7,000 kips, which equals 498 kips. 

The seismic response coefficient for the flexible-base condition computed 

using ASCE/SEI 7-16 Equation 12.8-3 is as follows:

1 1.16 0.0604

8 2.40

1.0

D

s

e

S

R T

I

= = =           

C ̃

 ̃

However, since that flexible-base value of C ̃

s

is less than the minimum

seismic response coefficient computed using ASCE/SEI 7-16 Equation 12.8-5 

(0.0612, as shown above), the minimum value controls and the seismic base 

shear (for each frame) is computed using ASCE/SEI 7-16 Equation 12.8-1 as 

follows: 

V ̃ = C W ̃

s 0.0612 7,000 428 kip ( ) = =

Adjustment of the design base shear to account for SSI effects using linear 

analysis is in accordance with ASCE/SEI 7-16 Section 19.2.1. This example, 

which focuses on period lengthening, neglects reductions due to damping, 

which are discussed in Chapter 8; the value of BSSI is taken as 1.0 

(corresponding to 5% of critical damping). 

The base shear adjustment, ∆V, is computed using Equation 19.2-2. 

0.0612 0.0711 7,000 69.3 kip 1.0

s

s

SSI

VC W

B

    ∆= − = −   =      

C ̃

Note that this permitted reduction is affected by application of the minimum 

seismic response coefficient, which governs the design base shear.
FEMA P-2091 7: Period Lengthening 7-15

The value of effective seismic weight used in this calculation, W , could be 

taken as the “weight caused by the effective modal mass in the fundamental 

mode,” which would be 0.821W, based on the fundamental mode mass 

participation factor. However, ASCE/SEI 7-16 permits the direct use of the 

full effective seismic weight, W. The latter approach is taken as it allows a 

greater reduction in base shear, without being unconservative since the fixedbase base shear being reduced is computed using the full effective seismic 

weight. 

The limitation on base shear reduction is quantified by the α factor computed 

in accordance with ASCE/SEI 7-16 Equation 19.2-3. For this system R = 8, 

so α = 0.9. 

The final base shear adjusted for SSI is computed in accordance with 

ASCE/SEI 7-16 Equation 19.2-1 as follows:

V ̃ = V−∆V ≥ αV

V ̃

 = V−∆V = 498 − 69.3 = 429 kip 

≥ αV = 0.9(498) = 448 kip ← controls

The final design base shear computed in accordance with ASCE/SEI 7-16 

Chapter 19 reflects only a 10% reduction from the fixed-base base shear

(without the CuTa period limit). This reduction is controlled by application 

of the limitation for foundation damping, reflected by α, even though no 

reduction for foundation damping was included in the calculations. This

illustrates a flaw in the formulation of ASCE/SEI 7-16 Section 19.2.1, which 

does not distinguish between reductions due to period lengthening and 

reductions due to foundation damping. However, this is still an 18%

reduction of the base shear used for a truly fixed-base design, where the CuTa

period limit applies. 

The base shear from modal analysis for the flexible-base condition is 

V ̃

t = 384 kips . Design force values from modal analysis would require

scaling by the factor V ̃ V ̃

t = 448 384 =1.17 in accordance with ASCE/SEI 

7-16 Sections 19.2.2 and 12.9.1.4.1.

7.4.3 Summary of Design Demands 

Table 7-2 summarizes design demands for the example building. These 

demands differ from the pure spectral values in several ways. For the fixedbase condition the upper limit on period, CuTa, defined in ASCE/SEI 7-16 

Section 12.8.2 controls. For the flexible-base condition the period independent 

minimum base shear applies since it is larger than the base shear computed 
7-16 7: Period Lengthening FEMA P-2091

using the calculated period. However, the design base shear is governed by the 

maximum reduction permitted by ASCE/SEI 7-16 Section 19.2.1. 

Table 7-2 Summary of Design Demands for Example Building

Fixed-base 

condition

Flexible-base 

condition

Computed period T = 2.04 s T

 ̃= 2.40 s 

Upper limit on period CuTa = 1.87 s -- 

Base shear, ASCE/SEI 7-16 Equation

12.8-3 

V = 543 kips

(using CuTa) 

 ̃V = 423 kips

(using T

 ̃ ) 

Minimum base shear, ASCE/SEI 7-16 

Equation 12.8-5 

V = 428 kips  ̃V = 428 kips

Base shear limited by ASCE/SEI 7-16

Section 19.2.1

--  ̃V = 448 kips

Design base shear V = 543 kips  ̃V = 448 kips

Base shear from MRSA Vt = 433 kips Vt ̃ = 384 kips 

Scale factor for MRSA design forces, 

per ASCE/SEI 7-16 Section

12.9.1.4.1

1.25 1.17 
FEMA P-2091 8: Foundation Damping 8-1 

Chapter 8 

Foundation Damping

8.1 Overview

Inertial interaction is the dynamic interaction between the structure, its 

foundation, and the surrounding soil caused by the foundation input motion. 

Period lengthening is one inertial interaction effect. It results in an increase 

in the building period due to foundation flexibility. Period lengthening is 

covered in Chapter 7. This chapter covers foundation damping, a second 

inertial interaction effect. There are two contributors to foundation damping: 

soil damping and radiation damping. 

Foundation Damping

Description: Foundation 

damping is an inertial SSI 

effect. There are two types: 

soil damping and radiation

damping.

Requirements for Soil 

Damping

• Design spectral response 

acceleration parameter at 

short periods, SDS

• Site class

• Fundamental period in a 

fixed base model, T

• Fundamental period in a 

flexible base model, T̃

• Response modification 

coefficient, R

• Overstrength factor, Ω0

• Expected ductility 

demand, μ, which is 

permitted to be taken as 

R / Ω0

Additional Requirements

for Radiation Damping

• Foundation geometry

• Average effective shear 

wave velocity, vs

• Average low strain shear 

wave velocity, vso

• Average unit weight of 

soil, γ

• Poisson’s ratio, ν

Soil damping is the hysteretic (material) damping of the soil. It is similar to 

inherent viscous damping in the superstructure, but it is independent of the 

flexible-base period of the structure. 

Radiation damping is the damping in the soil-structure system caused by the 

generation and propagation of waves away from the foundation, which are 

caused by dynamic displacements of the foundation relative to the free-field 

displacements. Radiation damping is larger when the structure-to-soil 

stiffness ratio is larger. The structure-to-soil stiffness ratio is described in 

Chapter 3. 

Foundation damping is less commonly applied in practice than other types of 

SSI effects, such as period lengthening and kinematic interaction. NIST 

(2012a) and Givens et al. (2016) provide an analysis framework for 

foundation damping that clearly distinguishes the separate contributions from 

soil damping and radiation damping from different types of foundation 

vibration. This is convenient for application because the soil damping

component of foundation damping is easy to implement. The adiation 

damping component of foundation damping, on the other hand, is 

considerably more complex.

ASCE/SEI 7-16 Section 19.3 provides provisions, modified from NIST 

(2012a), for both the soil and radiation components of foundation damping. 

Foundation damping can be combined with the effective viscous damping in 

the superstructure, increasing the damping used, and leading to a reduction in 

spectral demands. More sophisticated models can incorporate damping 

effects as dashpots directly into the analytical model. Chapter 9 shows some 
8-2 8: Foundation Damping FEMA P-2091

conceptual examples of how dashpots can be incorporated in various 

modeling approaches; Section 4.3 of NIST (2012a) provides more details. 

This chapter focuses on the approach in ASCE/SEI 7-16 Section 19.3. 

8.2 Foundation Damping Requirements

In order to use foundation damping, the foundation system of the building 

has to meet the requirements in ASCE/SEI 7-16 Section 19.3.1. Foundation 

damping is not permitted for the following three cases.

[Case 1]: A foundation system consisting of discrete footings that are not 

interconnected and that are spaced less than the larger dimension of the 

supported lateral force-resisting element in the direction under 

consideration. 

[Case 2]: A foundation system consisting of, or including, deep 

foundations such as piles or piers.

[Case 3]: A foundation system consisting of structural mats 

interconnected by concrete slabs that are characterized as flexible in 

accordance with [ASCE/SEI 7-16] Section 12.3.1.3 or that are not 

continuously connected to grade beams or other foundation elements. 

For Case 1, the commentary in ASCE/SEI 7-16 Section C19.3 notes that: 

The radiation damping procedure ... could be potentially 

unconservative where wall and frame elements are close enough so 

that waves emanating from distinct foundation components 

destructively interfere with each other across the period range of 

interest. That is why the limit of spacing of the vertical lateral forceresisting elements is imposed on the use of these provisions. 

Figure 8-1 illustrates an example where a pair of discrete, unconnected shear 

walls and their foundations are too close together. The spacing between the 

two shear wall footings is s = 14 ft. Although this is greater than the 

narrower shear wall with width of bw2 = 10 ft, it is less than the wider shear 

wall with a width of bw1 = 18 ft. It is the larger wall that governs per the 

provision. There is additional discussion of this case in FEMA 440 Section 

8.3 (FEMA, 2005), which notes that such cases can “effectively decrease the 

energy dissipated in the soil material, and ... could overestimate the related 

damping.” 
FEMA P-2091 8: Foundation Damping 8-3 

12'

12'

12'

12'

21' bw1 = 18' 20' 21'

bw2

= 10'

24' s = 14' 16' No grade beams 

connecting footings

s = 14' < bw1 = 18'

> bw2 = 10'

Figure 8-1 Example of a foundation and superstructure where foundation 

damping is not permitted because the unconnected footings are 

too close.

For Case 2, the commentary in ASCE/SEI 7-16 Section C19.3 notes that the 

equations for foundation damping are for shallow foundations. 

This is not to say that radiation damping does not occur with deep 

(pile or caisson) foundation systems, but the phenomenon is more 

complex. Soil layering and group effects are important, and there 

are the issues of the possible contributions of the bottom structural 

slab and pile caps. Because the provisions are based on the 

impedance produced by a rigid plate in soil, these items cannot be 

easily taken into account. Therefore, more detailed modeling of the 

soil and the embedded foundations is required to determine the 

foundation impedances. The provisions permit such modeling but do 

not provide specific guidance for it. Guidance can be found for 

example in NIST GCR 12-917-21 [NIST, 2012a] and its references.

In Case 3, the concern is that a lack of adequate coupling, either through a 

rigid diaphragm or through continuous grade beams, means the vertical 

elements of the lateral force-resisting system, such as braced frames or shear 

walls, may be too weakly coupled and use of the provisions, which are based 

on the full width of the structure would be potentially unconservative. 

ASCE/SEI 7-16 Section 19.3.1.1 refers to ASCE/SEI 7-16 Section 12.3.1.3 

for evaluation of the flexibility of the slab-on-grade linking footings. 
8-4 8: Foundation Damping FEMA P-2091

ASCE/SEI 7-16 Section 12.3.1.3 is used in general to characterize slabs as 

rigid or flexible. A slab is characterized as flexible if δMDD / ΔADVE > 2, with 

terms defined as shown in Figure 8-2.

Drift ∆ADVE

Deflection δMDD

Equivalent tributary 

lateral load

Figure 8-2 Determining slab-on-grade in-plane flexibility (from ASCE/SEI 

7-16 Figure 12.3-1).

8.3 Effective Damping Ratio

Typically, seismic code demands for linear procedures are based on an 

equivalent viscous damping ratio for the structure of 5%. When foundation 

damping SSI is considered, the goal is to determine an effective damping 

ratio for the soil-structure system that incorporates contributions from both 

the structure and the foundation-soil interaction. This is captured in 

ASCE/SEI 7-16 Equations 19.3-1, 19.3.2, and 19.3.3 shown below. 

0 2 0.20 f

eff T

β β =β + ≤

     

T ̃

(ASCE/SEI 7-16 Eq. 19.3-1) 

where: 

β0 = effective damping ratio of the soil-structure system

βf = effective viscous damping ratio relating to foundation-soil 

interaction

β = effective viscous damping ratio of the structure, taken as 5% 

unless otherwise justified by analysis.

(T̃/T)eff = effective period lengthening ratio defined in ASCE/SEI 7-16 

Equation 19.3-2. 
FEMA P-2091 8: Foundation Damping 8-5

 0.5

T 

 

 ̃

2

 

 ̃   1  T

  =   1 1 + −      T T  μ   eff      

where: 

μ = effective ductility demand. For equivalent lateral force or modal 

response spectrum analysis procedures, μ is the maximum base 

shear divided by the elastic base shear capacity; alternatively, μ is 

permitted to be taken as R / Ω0, where R and Ω0 are per ASCE/SEI 

7-16 Table 12.2-1. This is based on the assumption that R can be

considered as the product of ductility and overstrength, so dividing

R by Ω0 results in an estimate of ductility. For response history

analysis procedures, μ is the maximum displacement divided by

the yield displacement of the structure measured at the highest

point above grade.

  ̃  T 

2    −1  T   βf =   T ̃

2 βs + βrd (ASCE/SEI 7-16 Eq. 19.3-3) 

    

  T   

where: 

βs = soil hysteretic damping ratio 

βrd = radiation damping ratio

The effective damping ratio of the soil-structure system, βo, is then used in 

lieu of β when applying damping factors. There is a limit of up to 20% for 

β0, or effectively a limit of 15% on βf. Even so, large values of βf and β0 can 

lead to large reductions in demand. 

Note that when a site has a deep softer surface layer over a very stiff layer 

with a shear wave velocity of twice that of the surface layer, then ASCE/SEI 

7-16 Equation 19.3-3 is replaced with Equation 19.3-4 (not reproduced here).

8.4 Soil Damping

8.4.1 Discussion

Soil damping, or soil hysteretic damping, occurs because of shearing within 

the soil and at the soil–foundation interface. Values of the soil hysteretic

damping ratio, βs, to model the soil hysteretic damping can be obtained from 

site response analysis as determined by a geotechnical engineer or directly 

from ASCE/SEI 7-16 Table 19.3-3 as shown in Table 8-1. 
8-6 8: Foundation Damping FEMA P-2091

Table 8-1 Soil Hysteretic Damping Ratio, βs (from ASCE/SEI 7-16 

Table 19.3-3)

Site Class

Effective Peak Acceleration, SDS/2.5(1)

SDS/2.5 = 0 SDS/2.5 = 0.1 SDS/2.5 = 0.4 SDS/2.5 ≥ 0.8

C 0.01 0.01 0.03 0.05

D 0.01 0.02 0.07 0.15

E 0.01 0.05 0.20 (2)

F (2) (2) (2) (2)

(1) Use straight-line interpolation for intermediate values of SDS/2.5.

(2) Site-specific geotechnical investigation and dynamic site response analyses shall be 

performed.

8.4.2 Soil Damping Example

As an example of soil damping, assume the following.

• The design spectral response acceleration parameter at short periods at 

the site is SDS = 1.0g; then the effective peak acceleration is SDS / 2.5 = 

0.4. 

• The site is characterized as Site Class E; then βs = 0.20 per Table 8-1. 

Review of the values in Table 8-1 indicates that softer soils and higher 

ground shaking levels lead to higher values of soil damping. 

• The building is Risk Category II, so Ie = 1.0. 

• The lateral force-resisting system is a special concentrically braced frame 

structure with a seismic response or base shear coefficient governed by 

ASCE/SEI 7-16 Equation 12.8-2 of V / W = Cs = SDS / (R / Ie) 

 = 1.0 / (6 / 1.0) = 0.167. 

• The effective viscous damping ratio of the structure is the typical 0.05. 

• The equivalent lateral force procedure is being used for design. 

In order to implement soil damping and determine its impact on design 

forces, several steps are required.

Step 1: The effective viscous damping ratio related to foundation-soil 

interaction, βf, must be calculated, and this depends on the ratio (T/T). This 

in turn requires modeling the building with soil springs to compare fixedbase and flexible-base periods. This is described in Chapters 6 and 7. Say 

that the results of this exercise lead to a ratio (T/T) = 1.2. Entering the values 

into ASCE/SEI 7-16 Equation 19.3-3 yields the following, assuming zero for 

radiation damping. 

̃

̃
FEMA P-2091 8: Foundation Damping 8-7

  ̃

2   

  T

  −1     ( 1.2)

2

−1  T (   ) β f = 2 2 β + β = (0.20) + =   0 0.06

 ̃

s rd     1.2     ( ) T

         T 

Step 2: Calculate the expected ductility demand, μ. For a special 

concentrically braced frame, R = 6 and Ω0 = 2. μ is permitted to be taken as 

R / Ω0 = 6 / 2 = 3.

Step 3: Calculate the effective period lengthening ratio per ASCE/SEI 7-16 

Equation 19.3-2. 

 0.5

1  2   T 

 ̃    T ̃    =   1 1 + −   

T T μ    eff      

    1

= + ( 0.5

  1   (1.2)

2

− = 1

3 ) 1.071

   

Step 4: Calculate the effective viscous damping ratio of the soil-structure 

system per ASCE/SEI 7-16 Equation 19.3-1. 

β   0.05 β =0 βf + 2 2 ≤ 0.20 = 0.06 +   = 0.06 + 0.04 = 0.10 < 0.20

 ̃

     (1.071 T 

)    T eff

Thus, in this example, soil damping doubles the effective damping from 0.05 

for the effective viscous damping ratio of the structure to 0.10 for the 

effective damping ratio of the soil-structure system.

Step 5: Determine the factor to adjust the design response spectrum, BSSI per 

ASCE/SEI 7-16 Equation 19.2.4. 

BSSI = 4 / [5.6 – ln (100 β0)] = 4 / [5.6 – ln (100 × 0.10)] = 1.21

Step 6: For the equivalent lateral force procedure, determine the reduction in 

base shear per a modified version of ASCE/SEI 7-16 Equation 19.2.2 to 

compare the base shear coefficient V / W = Cs directly. For this short 

structure, it is assumed that the flexible base period does not lengthen 

sufficiently to change the governing base shear, and Cs = C̃s = 0.167. 

 ̃

s 0.167 ∆C C s s = − C = 0.167 − = 0.167 − 0.138 = 0.029

BSSI 1.213
8-8 8: Foundation Damping FEMA P-2091

Step 7: Determine the ratio of the base shear adjusted for SSI vs. the base 

shear from a fixed base model per a modified version of ASCE/SEI 7-16 

Equation 19.2.1: 0.138 / 0.167 = 0.826. 

Step 8: Compare with the limits in ASCE/SEI 7-16 Equation 19.2-3. With 

R = 6, there is a floor on the reduction α = 0.9. This is higher than the 0.826 

ratio calculated in Step 7, so the α = 0.9 floor governs, and ASCE/SEI 7-16 

limits the reduction due to soil damping to 10%. 

8.5 Radiation Damping

8.5.1 Discussion

Radiation damping is discussed in NIST (2012a) and the equations in 

ASCE/SEI 7-16 are a simplification of those in NIST (2012a). Radiation 

damping provisions for rectangular footings are in ASCE/SEI 7-16 Section 

19.3.3. Section 19.3.4 provides provisions for circular footings, which have 

been a traditional focus of research, but are rarely needed in practice since 

circular footings are rarely found in actual buildings. This Guide focuses on 

rectangular footings. Even with the simplifications in ASCE/SEI 7-16, 

calculations for radiation damping are involved. In addition, the order in 

which the equations are given in ASCE/SEI 7-16 does not follow the 

sequence that is needed to perform the calculations. As such, a step by step 

example is given in Section 8.5.2 to illustrate the proper sequence. 

For rectangular footings, the long direction is defined as 2L, and the narrow 

direction is defined as 2B, in keeping with the traditional approach used in 

the geotechnical literature. Earthquake movement can occur parallel to either 

axis. Movement parallel to the long direction is called movement in the 

strong direction. Movement parallel to the narrow direction is called 

movement in the weak direction. NIST (2012a) provides equations to allow 

calculating radiation damping for translational and rotational movement in 

both the strong direction and weak direction.

In the development process for the SSI provisions in ASCE/SEI 7-16, the 

desire for simplification led code writers to include only the translational and 

rotational stiffness values that result in the lower values of damping which 

are associated with movement in the weak direction parallel to the narrow 

direction of the footing. They are purposely conservative. This is not stated 

in the ASCE/SEI 7-16 provisions or commentary. The engineer could, with 

more effort, use the more refined equations that are in NIST (2012a). In fact, 

it would be more rational in many cases since the long direction of the 

footing is typically associated with the primary direction of resistance of the 
FEMA P-2091 8: Foundation Damping 8-9 

vertical lateral-force resisting element that the footing supports. The long 

direction will also typically produce more radiation damping. 

8.5.2 Radiation Damping Example

The example of radiation damping here is taken from Appendix A.6, except 

that more background is provided here. The resulting impact on the base 

shear coefficient is determined, similar to the soil damping example in 

Section 8.4.2. In the appendix, the design implications on member sizes of 

the base shear reduction are studied. Assumptions for the example are as 

follows. 

• The design spectral response acceleration parameter at short periods at 

the site is SDS = 1.57g. 

• The site is characterized as Site Class D.

• The building is Risk Category II, so Ie = 1.0. 

• The lateral force-resisting system is a buckling-restrained braced frame 

structure on spread footings with a seismic response or base shear 

coefficient governed by ASCE/SEI 7-16 Equation 12.8-2 of V / W = Cs = 

SDS / (R / Ie) = 1.57 / (8 / 1.0) = 0.196. 

• The effective viscous damping ratio of the structure is the typical 0.05. 

• Spread footings under the braced frames are 38 ft long by 8 ft wide by 5 

ft deep, not connected to one another, and are far enough apart to satisfy 

ASCE/SEI Section 19.3.1.1. The top of the footing is 1’0” below the top 

of the slab-on-grade. 

• The slab-on-grade is 5” thick, and overpoured above the footings. 

• Equivalent lateral force (ELF) procedure is being used for design. 

• Figure A-1 presents the soil profile at the site.

In order to implement radiation damping and determine its impact on design 

forces, a number of steps are required. Initial steps focus on determining the 

appropriate geometry, followed by determining soil properties under the 

footings, including the shear wave velocity, then the stiffness properties, then 

the damping coefficients, then the foundation damping ratio, and finally the 

effective reduction in base shear. Note that these soil, stiffness, and damping 

properties are all particular to the radiation damping computations. 

Step 1: Determine the L and B dimensions to use. In a mat foundation, or a 

foundation connected by grade beams, the dimensions for entire structure 

footprint are used, so that L is half the larger dimension of the base of the 

structure and B is half the smaller dimension of the base of the structure. For 
8-10 8: Foundation Damping FEMA P-2091

this example, however, the braced frames are on discrete spread footings that 

are not connected to one another. The radiation damping comes in part from 

rotation of the individual footings in the soil. The commentary of ASCE/SEI 

7-16 Section C19.3 recommends use of the individual footing dimensions in 

this case. Thus 2L = 38 ft for the long direction of the footing parallel to the 

braced frame and 2B = 8 ft for the narrow direction of the footing 

perpendicular to the braced frame, so L = 19 ft and B = 4 ft. The value of B = 

4 ft is key in the next steps.

Step 2: Calculate the average low strain shear wave velocity, vso. Note that 

the shear wave velocity here is not vs , the value used to determine the site 

class. Here, vs is considered for the upper soil layer, over the depth from 

base of footing at 1’0” for the slab-on-grade and overpour plus the 5’0” 

footing at Elevation -6’0” to B = 4 ft below to Elevation -10’0”. The use of 

vs for site class determination is intended to account for the average soil 

stiffness in the upper 100 ft (30 m); the use of vs for the upper soil layer is 

intended to capture the effects of the upper soils that more directly affect the 

damping at the soil-structure interface. The shear wave velocity between 

Elevation -6’0” and Elevation -10’0” is 653 ft/s, per Figure A-1. Note that a 

profile of shear wave velocity vs. depth is thus needed in the geotechnical 

report, not simply the vs value. 

Step 3: Calculate the average effective shear wave velocity, vs. This value 

represents the reduction in velocity from the average low strain velocity as 

the soil softens and yields at higher strains during earthquake shaking. It is 

thus dependent on the earthquake shaking level. This calculation uses 

ASCE/SEI 7-16 Table 19.3-1 and the design spectral response acceleration 

parameter at short periods, SDS. For this site, SDS =1.57. Thus, the effective 

peak acceleration is SDS / 2.5 = 0.628. This is in between the columns for 

SDS / 2.5 = 0.4 and SDS / 2.5 ≥ 0.8 in Table 19.3-1, so interpolation is used. 

For Site Class D and SDS / 2.5 = 0.4, the effective shear wave velocity ratio, 

[vs / vso] = 0.71. For SDS / 2.5 ≥ 0.8, the effective shear wave velocity ratio, 

[vs / vso] = 0.32. Interpolating at SDS / 2.5 = 0.628 yields [vs / vso] = 0.488. 

Thus vs = vso [vs / vso] = 653 ft/s (0.488) = 318.664 ft/s = 319 ft/s. 

Step 4: Calculate the average shear modulus for the soils beneath the 

foundation, G0. This also applies to the soil from Elevation -6’0” to 

Elevation -10’0”. Figure A-1 shows the soil density in this region, γ, to be 

120 to 125 lb/ft3

. Use an average value of 122.5 lb/ft3. From the ASCE/SEI 

7-16 Section 19.3.3 definition, the soil shear modulus at small strain levels, 

G0, is computed as 
FEMA P-2091 8: Foundation Damping 8-11

( ) 2 2

0 2

653 kip 122.5 1,609 32.174 ft

so v G

g

 

=γ =   =

 

Step 5: Calculate the effective shear modulus, G. Like the average effective 

shear wave velocity in Step 3, G represents the reduction in shear modulus 

from the shear modulus based on average low strain velocity as the soil 

softens and yields at higher strains during earthquake shaking. This 

calculation uses ASCE/SEI 7-16 Table 19.3-2 and the design spectral 

response acceleration parameter at short periods, SDS. For this site, per above 

SDS / 2.5 = 0.628. This is in between the columns for SDS / 2.5 = 0.4 and 

SDS / 2.5 ≥ 0.8 in Table 19.3-2, so interpolation is used. For Site Class D and 

SDS / 2.5 = 0.4, the effective shear modulus ratio, [G/G0] = 0.50. For SDS / 

2.5 ≥ 0.8, the effective shear modulus ratio, [G/G0] = 0.10. Interpolating at 

SDS / 2.5 = 0.628 yields [G/G0] = 0.272. Thus G = G0 [G/G0] = 1,609 kip/ft2

(0.272) = 441.597 kip/ft2

 = 442 kip/ft2. 

Step 6: Calculate the period lengthening ratio, (T̃/T), where T̃is the flexiblebase period and T is the fixed-base period. These values come from models 

without and with springs. The value of (T̃/T) is computed as 0.532 s / 0.466 s 

= 1.142. 

Step 7: Calculate the expected ductility demand, μ. For a bucklingrestrained braced frame, R = 8 and Ω0 = 2.5. μ is permitted to be taken as 

R / Ω0 = 8 /2.5 = 3.2.

Step 8: Calculate the effective period lengthening ratio, (T̃/T)eff. It depends 

on the period lengthening ratio (T̃/T) =1.142 from Step 6 and the system 

ductility, μ = 3.2 from Step 7. The effective period lengthening ratio is then 

computed according to ASCE/SEI 7-16 Equation 19.3-2. 

0.5 2

1 1 1

eff T T

        =+ −      

μ          

T ̃ T ̃

( ) ( )

0.5

1 2

1 1.142 1 1.046

3 2.

    =+ − =        

Step 9: Calculate the translational foundation stiffness, Ky. This represents 

the horizontal stiffness of the foundation moving horizontally within the soil. 

It requires the effective shear modulus G = 442 kip/ft2

 from Step 5, B = 4 ft, 

a Poisson’s ratio of ν = 0.3 (for the sandy soil), and the plan dimensionless 

ratio (L/B) = (38 ft / 8 ft) = 4.75. The translational foundation stiffness is 

calculated per ASCE/SEI 7-16 Equation 19.3-8. 
8-12 8: Foundation Damping FEMA P-2091

GB    L 0.65   L 

Ky = 6.8  + + 0.8  1.6 2 − ν       B B 

(441,597)(4) =  [18.722 + 3.8 +1.6] = 2.506×107 lb/ft

 1.7 

Step 10: Calculate the rotational foundation stiffness, Kxx. This represents 

the rotational stiffness of the frame and footing rocking in the soil. The 

rocking foundation stiffness is calculated per ASCE/SEI Equation 19.3-9. 

GB3     L Kxx = 3.2  +   0.8

1− ν     B

  (441,597)(4)

3

[ ] 8   ft − lb 15.2 0.8 6.460 10

0.7 rad =   += ×        

Step 11: Calculate the dimensionless frequency, a0, per ASCE/SEI Equation 

19.3-11. 

( )( )( )

( )( ) 0

2 2 4

0.1489

s 0.532 319

B

a

π π   = = =     Tv ̃

Step 12: Calculate the dimensionless factor, ψ, per ASCE/SEI Equation 

19.3-13. 

( ) ( )( ) 0.5

2 1 2 1 0.3

1 2 1 0.6

−ν  − 

ψ= =  

−ν −  

= 1.871 ≤ 2.5, use 1.871

Step 13: Calculate the dimensionless factor, αxx, per ASCE/SEI Equation 

19.3-14. 

2

0

2

3 0

0.55 0.01 1

1.0

0.4 2.4

xx

L

a

B

a

a

L

B

 

 

      + −       = −    

  − +              

( ( ))( )

( ) ( )

2

2

0.55 0.01 1.936 0.148

1.0 = 0.995

2.4 0.0037 0.148

  +   = −   − +    
FEMA P-2091 8: Foundation Damping 8-13

Step 14: Calculate the effective structure height, h*. As indicated in the 

ASCE/SEI 7-16 Section 19.3.3 definitions, this is the vertical distance from 

the foundation to the centroid of the first mode shape for multistory 

structures. Alternatively, h* is permitted to be approximated as 70% of the 

total structural height for multistory structures or as the full height of the 

structure for one-story structures. The alternate formulation is 0.70 × (12.5 ft 

first story + 12.5 ft second story) = 17.5 ft. The more precise formulation 

comes from analyzing the first mode shape output from computer results. 

This is described in Appendix A and results in h* = 0.677 (25 ft) = 16.92 ft.

Step 15: Calculate the fundamental translational period, Ty. The 

fundamental translational period of the damping system is computed based 

on the computed stiffness, Ky, and the effective modal mass, M*. The 

effective modal mass comes from computer results. As described in 

Appendix A, the effective modal mass, M*, is the building seismic mass, 

multiplied by the first modal mass participation factor (in this case, 0.87), 

divided by the number of frames in the direction under consideration (4). 

The modal mass participation factor expresses the proportion of the total 

mass excited in the first natural mode of vibration. It can be found as part of 

the computer modal results. In this case, M* = 3.26×104 lb-sec2

/ft. 

The translational period, Ty, is computed according to ASCE/SEI 7-16 

Equation 19.3-6. Note again that the fundamental translational period (and 

the fundamental rotational period) is particular to the damping computations 

and is not directly related to the fixed base and flexible base periods from the 

structural analyses. 

( )( )

0.5 32,600 2 2 0.227 sec

25,060,000

*

y

y

M T

K

 

=π = π =    

Step 16: Calculate the fundamental rotational period, Txx. Note that this is 

incorrectly identified as the fundamental translational period in the 

ASCE/SEI 7-16 Section 19.1.3 definitions. The fundamental rotational 

period, Txx, is computed according to ASCE/SEI 7 Equation 19.3-7. 

( )( ) ( )( )

( )( )

0.5 2 2 32,600 16.92 2 2 0.757 sec

0.995 646,000,000

* *

xx

xx xx

M h T

K

 

=π = π   = α  

Step 17: Calculate the translational foundation damping coefficient, βy. per 

ASCE/SEI 7-16 Equation 19.3-10. 
8-14 8: Foundation Damping FEMA P-2091

    2 2

1 1 rd y xx

Ty Txx

β = β+ β

       

 T ̃  T ̃

( )( )

( )( )

0

4 4 4.75 0.148 0.099

2 2 25,060,000

441,597 4

y

y

L

B a

K

GB

     

            β =   =   =                  

Step 18: Calculate the rotational foundation damping coefficient, βxx. per 

ASCE/SEI 7-16 Equation 19.3-12. 

2

0

0

2

3 3 0

4

3

2

0.4 2 2

xx

xx

xx

L

a

B a

K

. a

GB L

B

 

 

   ψ         β =       α

       − +               

( ) ( )( ) (( ) ( )( ) ) ( ) ( ) ( )( )

2 32

3

1.871 0.4 0.148 4 4.75 0.148 646,000,000 441,597 4 2.2 0.148

3 4.75 2 0.995

/ /

          =         − +                  

= 0.000383 

Note that the translational damping is a much more significant effect than the 

rotational damping (βy = 0.0998 > βxx = 0.00383), although the elastic 

behavior of the building is more significantly affected by the rotation. 

Step 19: Calculate the effective radiation damping ratio, βrd, per ASCE/SEI 

7-16 Equation 19.3-5.

( ) ( ) 2 2

1 1 0.0998 + 0.000389

0.532 0.532

0.228 0.761

   

=

       

= 0.0183 + 0.00080 = 0.019

Step 20: Calculate the foundation damping ratio, βf. This can include the 

effect of both soil damping and radiation damping. The foundation damping 

ratio is calculated per ASCE/SEI 7-16 Equation 19.3-3. Since this example 

focuses on radiation damping, assume that soil damping is zero. Typically, 

they would be included together, and they are for the example in Appendix 
FEMA P-2091 8: Foundation Damping 8-15

A. For the assumption here that soil damping is zero, ASCE/SEI 7-16

Equation 19.3-3 simplifies to:

2

2

1

0 0.019 0.019 f s rd

T

T

     −     β = β +β = + =              



 T ̃ 

T ̃

Step 21: Calculate the effective viscous damping ratio of the soil-structure 

system per ASCE/SEI 7-16 Equation 19.3-1. 

0 2 0.20 f

eff T

β β =β + ≤

   

 T ̃ 

( )2

0.05 = 0.019 0.019 0.046 0.065 0.20

1.046

 

+ =+=<      

Thus, in this example, radiation damping increases the effective damping 

from 0.05 for the effective viscous damping ratio of the structure to 0.065 for 

the effective damping ratio of the soil-structure system.

Step 22: Determine the factor to adjust the design response spectrum, BSSI

per ASCE/SEI 7-16 Equation 19.2.4. 

BSSI = 4 / [5.6 – ln (100 β0)] = 4 / [5.6 – ln (100 × 0.065)] = 1.073 

Step 23: For the equivalent lateral force procedure, determine the reduction 

in base shear per a modified version of ASCE/SEI 7-16 Equation 19.2.2 to 

compare the base shear coefficient V / W = Cs directly. For this short 

structure, it is assumed that the flexible base period does not lengthen 

sufficiently to change the governing base shear, and Cs = C̃s = 0.196. 

0.196 0.196 0.196 0.183 0.013

1.073

s

s s

SSI

C C

B

∆= − = − = − = C ̃

Step 24: Determine the ratio of the base shear adjusted for SSI vs. the base 

shear from a fixed base model per a modified version of ASCE/SEI 7-16 

Equation 19.2.1: 0.183 / 0.196 = 0.933. 

Step 25: Compare with the limits in ASCE/SEI 7-16 Equation 19.2-3. With 

R = 8, there is a floor on the reduction of α = 0.9. This is lower than the 

0.933 ratio calculated in Step 23, so the 0.933 value governs, and radiation 

damping results in a 7% reduction in the base shear for this example. 
8-16 8: Foundation Damping FEMA P-2091

Step 26: Incorporate soil damping, if calculated. Soil damping and radiation 

damping are typically implemented together. In this example, if the 

reduction in base shear from soil damping were more than 3%, then the 

combined effect of soil damping and radiation damping would be more than 

10%, and the 10% floor of ASCE/SEI 7-16 Equation 19.2-3 would govern 

for this R = 8 building. 

8.6 Limitations and Issues

8.6.1 Limitations on the Influence of Radiation Damping

As the examples for soil damping and radiation damping show, there are 

several limitations that have been incorporated into the ASCE/SEI 7-16 

provisions. These include: 

• In ASCE/SEI 7-16 Section 19.3.2, the limit on the effective damping 

ratio including both the effective viscous damping ratio of the structure 

and the effective viscous damping ratio related to foundation-soil 

interaction is 20%. It is possible to reach this limit on soft soils, 

particularly when both soil damping and radiation damping are 

calculated. 

• In ASCE/SEI 7-16 Section 19.3.1, there is a 10% limit on the reduction 

in base shear for systems with a response modification coefficient, R ≥ 6. 

This will often govern when foundation damping is incorporated. The 

ASCE/SEI 7-16 Section C19.3 commentary notes that there is limited 

understanding of how the effects of SSI interact with the R-factor. It 

goes on to say:

For higher R factors systems, where significant damping caused by 

structural yielding is expected, the contribution of foundation 

damping is assumed to have little effect on the reduction of the 

response. Some reduction is permitted because of (1) an assumed 

period lengthening resulting from the incorporation of base 

flexibility, (2) potential reduction in mass participation in the 

fundamental mode because two additional degrees of freedom are 

present caused by translation and rotation of the base, and (3) 

limited foundation damping interacting with the structural damping. 

8.6.2 Potential Code Changes Related to Foundation Damping

As part of the code development process for the next edition of ASCE/SEI 7, 

editorial proposals are under consideration related to foundation damping. 

They cover two issues: 
FEMA P-2091 8: Foundation Damping 8-17

• Reordering equations: A minor recommended revision is to reorder 

equations such that they appear in the order in which they need to be 

calculated, similar to the step by step process illustrated by the design 

examples in Sections 8.4.2 and 8.5.2 above. 

• Shear wave velocity and shear modulus definitions: There are several 

similar shear wave velocity parameters in ASCE/SEI 7-16 in Chapters 19 

and 20. ASCE/SEI 7-16 is not consistent with the terms. An editorial 

proposal is under consideration to explicitly define the shear wave 

velocity and shear modulus to be used for radiation damping and make it 

clearer the soil depth over which it should apply. 

FEMA P-2091 9: How to Model a Basement 9-1 

Chapter 9 

How to Model a Basement

9.1 Overview

ASCE/SEI 7-16 (ASCE, 2017a) does not provide explicit direction on how to 

model a basement. There are several strategies that practicing engineers use. 

In Soil-Structure Interaction for Building Structures (NIST, 2012a), 

relatively simple approaches were evaluated against models with 

sophisticated soil-structure interaction (SSI) effects, and recommendations 

were developed. This chapter provides recommendations and examples for 

modeling of basements when grade is relatively level around the entire 

building and when grade slopes so that there is unbalanced soil pressure on 

the building. 

Modeling of Basements

This chapter provides 

recommendations for 

modeling of basements for 

two conditions: 

• Buildings on level grade

• Buildings on sloping grade

9.2 Past Analytical Comparisons of Different Models

9.2.1 Models Studied

NIST (2012a) documents surveys of practitioners regarding typical models 

they used and the analytical studies that were undertaken to compare various 

models against the most sophisticated substructure model. Figure 9-1 shows 

an example building with a one-story basement, and Figure 9-2 shows the 

different models that were studied. Figure 9-3 shows a more detailed image 

of Model 4 in Figure 9-2.

Figure 9-1 Illustration of an embedded building 

configuration with a basement surrounded by 

soil and a level grade on all sides [from Figure 

6-1 in NIST (2012a)]. ug represents the free 

field motion.
9-2 9: How to Model a Basement FEMA P-2091

Figure 9-2 Modeling approaches for buildings with basements [from Figure 

6-2 in NIST (2012a)]. ug represents the free field motion, and 

uFIM represents the foundation input motion. In Model 3 and 

the full substructure model, the little boxcars are sliders to 

which the free-field motion is applied. They do not have mass. 

The mass of the soil is included in the calculation of the freefield motions (and, as applicable, its depth-dependence). 

Figure 9-3 Simplified version of Model 4 in Figure 9-2 [from Figure 4-6d in 

PEER (2017)]. ug represents the free field motion, and uFIM

represents the foundation input motion.

A summary of the models is as follows:

• Model 1. In this approach, the building is assumed to be fixed at the 

ground level, and no SSI effects are considered. Reactions are calculated 

at the base of the model at grade, and then applied to the foundation in a 

separate model. This approach is used occasionally by some engineers, 

and frequently by others. Some believe this model is more appropriate 

for moment frame and braced frame buildings, and less appropriate for 

shear wall buildings. Others would only use this model if there were no 

“back-stay effects.” Back-stay effects occur when the lateral system of 

the superstructure does not align with the full footprint of the foundation. 
FEMA P-2091 9: How to Model a Basement 9-3 

An example of such a configuration is a superstructure with a concrete 

shear wall core system founded on a wider basement podium structure 

with perimeter walls. Backstay effects involve the transfer of lateral 

forces in the superstructure into additional elements that exist within the 

basement, typically through one or more floor diaphragms. This force 

transfer helps tall buildings resist overturning effects and is referred to as 

the backstay effect because of its similarity to the back-span of a 

cantilever beam. The added stiffness of the retaining walls, and the 

relative rigidity of the walls and diaphragms comprising the basement, 

can lead to larger shear forces in the basement levels than in the 

superstructure. Modeling of the in-plane flexibility of the basement 

diaphragms can provide more rational results for this effect. When the 

lateral force-resisting system of the superstructure covers the full 

footprint of the basement, the backstay effect is reduced or eliminated. 

• Model 2. In this approach, the soil on the sides of the retaining walls is 

ignored, but the basement structure is explicitly included in the model. 

The lowest basement level is taken as the base of the model. It is argued 

that the amount of movement required to develop passive pressure of any 

significance far exceeds the amount of movement anticipated in the 

basement retaining walls, so the retained earth can be conservatively 

ignored. One variation of this model (Model 2A) has a fixed base. A 

second variation of this model (Model 2B) has vertical springs under the 

foundation. Model 2A is occasionally used, though limited by some to 

the preliminary design phases of their projects. Model 2B is used more 

frequently, though some use both Model 1 and Model 2B to bound their 

analyses for design. 

• Model 3. In this approach, horizontal springs are used to capture the 

effect of the surrounding soil. The ends of the springs are fixed against 

translation, and ground motion is input at the base of the model. When 

this modeling approach is used, it is used in pushover analyses. One 

variation of this model (Model 3A) is fixed vertically. A second 

variation of this model (Model 3B) includes vertical springs. Model 3A 

is used occasionally by some, while Model 3B is used more frequently. 

Some engineers never use either version of this modeling approach. 

• Model 4. In this approach, horizontal springs are attached to rigid walls 

(referred to as a “bathtub”), and the ground motion is applied to the 

bathtub so that the ends of the horizontal springs all move together with 

the input motion. This modeling approach is rarely used, although it is 

recommended in the Guidelines for Performance-Based Seismic Design 

of Tall Buildings, Version 2.03 (PEER, 2017). A simplified version of 
9-4 9: How to Model a Basement FEMA P-2091

Model 4 taken from PEER (2017) is shown in Figure 9-3. As noted in 

PEER (2017): 

The bathtub model [Model 4] applies the same ground motion to the 

ends of soil foundation interaction elements at the foundation level 

and across the embedded depth of the structure. To apply the same 

ground motion at these depths requires the use of rigid elements (for 

the ends of interaction elements opposite the foundation), which can 

create numerical difficulties in some structural engineering software 

packages. An option that can be considered to overcome this 

problem is to use interaction elements at the foundation level only 

[Figure 9-3]. When interaction elements are used only at the 

foundation level, they should in aggregate reproduce the cumulative 

stiffness of the embedded foundation as given in NIST [2012a] or 

similar documents. (This stiffness is higher than that of the base slab 

alone as a result of embedment effects.)

• Full Substructure Model (Baseline Model). This represents a 

comprehensive modeling approach in which dashpots are used to address 

soil damping variation, foundation rotation is considered, and multisupport excitation is applied through the horizontal springs so that the 

inputs vary up the height of the basement walls. This model is not 

currently used in practice.

9.2.2 Recommended Models

Based on the analyses done for the NIST (2012a) report, the following 

findings were observed for two detailed case studies. These findings are 

based on comparisons of dynamic response analysis results (e.g., story drift 

profiles) between simplified models and the full substructure model

(Baseline Model) shown in Figure 9-2. 

• For design of a building with a moment frame superstructure, Model 1 

and Model 2 lead to comparable or slightly conservative results to the 

Baseline Model and thus represent reasonable and practical design 

alternatives.

• For design of buildings with shear wall superstructures, Model 4 

provides more consistent results to the Baseline Model, though Model 2 

typically conservatively bounds the Baseline Model results and is thus a 

reasonable and practical design alternative. Model 1 results are more 

variable, less of a match, and conservative for several parameters, but not 

all.

• Model 3 showed the least agreement with the Baseline Model. 
FEMA P-2091 9: How to Model a Basement 9-5 

Based on these findings, Model 1, Model 2, and the simplified version of 

Model 4 as shown in Figure 9-3 are recommended.

9.3 Models for Buildings on Relatively Level Grade

9.3.1 Example Building

A section cut through an example building is shown in Figure 9-4. It is a 

four-story buckling-restrained braced frame building above grade with a onestory basement. 

14'

17'

17'

17'

17'

3'

R

25' 25' 25'

4

3

2

1

B

Figure 9-4 Example building with basement and 

relatively level grade.

The example building is 75 feet in the transverse direction (between column 

centerlines) as shown in Figure 9-4 and 346.5 feet in the longitudinal 

direction. It has concrete fill on metal deck floors and a roof on steel 

framing. The steel columns continue down to the mat foundation, but the 

braced frames stop at grade at Level 1. Lateral forces are transferred out 

from the braced frames through the Level 1 diaphragm to concrete shear 

walls in the basement.

9.3.2 Model 1 for Example Building 

The first approach for seismic analysis of the example building in Figure 9-4 

is to use the fixed based Model 1 approach shown schematically in Figure 

9-2(1) and the equivalent lateral force procedure of ASCE/SEI 7-16 Section 

12.8. The dynamic base of the building is set at grade which aligns with 

Level 1. Table 9-1 shows the story force distribution from an equivalent 

lateral force procedure static analysis. The top portion of Figure 9-5 shows 
9-6 9: How to Model a Basement FEMA P-2091

free body diagrams for the forces Table 9-1. Those forces are then applied to 

the basement story in a separate model for design of the basement and 

foundation. The seismic weight of the building, W, is 24,498 kip. The 

spectral acceleration, Sa, is 1.11g. The response modification coefficient, R, 

is 8 for the buckling-restrained braced frames. The building is a Risk 

Category III structure, so the Seismic Importance Factor, Ie, is 1.25. The 

governing equation for base shear is ASCE/SEI 7-16 Equation 12.8-1 which 

yields V = CsW = (Sa / R) × Ie = (1.11 / 8) × 1.25W = 0.173W = 

0.173(24,498 kip) = 4,240 kip. The base shear of 4,240 kip is the story shear 

between Level 1 and Level 2. 

Table 9-1 Model 1 Distribution of Forces

Diaphragm

Story 

(ft)

Height, hi

(ft)

Weight, 

wi 

(kip) 

Static Analysis

Force

(kip) 

Story 

Shear

(kip) 

Roof 68 7,620 2,047

4 2,047

4 51 5,626 1,113

3 3,160

3 34 5,626 726

2 3,886

2 17 5,626 354

1 4,240

1 (Base) 0 

Total 24,498 4,240 4,240

Notes: 

1. For Model 1, the dynamic base is set at grade (Level 1).

2. The ASCE/SEI 7-16 Equation 12.8-1 base shear is V = CsW = 0.173W. 
FEMA P-2091 9: How to Model a Basement 9-7 

2,047 k

2,047 k

Fi

1,113 k

354 k

726 k

vi

3,160 k

3,886 k

4,240 k

2,120 k 2,120 k

2,120 k 2,120 k

9,066 k 9,066 k

1,060 k 1,060 k 1,060 k 1,060 k

R

4

3

2

1

B

1

3,737 k

1,246 k 1,246 k

3,737 k

Figure 9-5 Model 1 distribution of forces for example building.

9.3.3 Model 2 for Example Building 

The second approach to modeling the example building uses Model 2A from 

Figure 9-2(2) and model response spectrum analysis of ASCE/SEI 7-16 

Section 12.9.1. The soil flexibility from using vertical springs in Model 2B 

is not included. It would have lengthened the building fundamental period 

and might have impacted the resulting design forces. In Model 2A, the 
9-8 9: How to Model a Basement FEMA P-2091

dynamic base is set at the bottom of the building, which is the basement 

level. More specifically, for this example, it is set at the bottom of the mat 

where it contacts the soil. The soil surrounding the basement is ignored. The 

mass of Level 1 is included in the model. A dynamic analysis is run. Due to 

the stiffness of the retaining walls in the basement, relatively little of the 

Level 1 mass impacts the forces at the upper floors. Table 9-2 shows how 

the initial dynamic analysis forces are scaled. Per ASCE/SEI 7-16 Section 

12.9.1.4.1, the dynamic analysis forces from the modal response spectrum 

analysis are scaled to the full value from the equivalent lateral force 

procedure. For this example, this is done for the story shear between Level 1 

and Level 2. The story shear for the dynamic analysis is 3,660 kip, and the 

story shear (or base shear) for the static analysis is 4,240 k. The scale factor 

is thus 4,240 kip / 3,660 kip = 1.1585. The initial dynamic analysis values 

are multiplied by 1.1585 to produce the final scaled dynamic analysis values 

used in design. Figure 9-6 shows this graphically.

Table 9-2 Model 2 Distribution of Forces

Diaphragm

Story

(ft)

Height, 

hi 

(ft)

Weight, 

wi

(kip) 

Dynamic Analysis Static Analysis

Scaled Dynamic 

Analysis

Force 

(kip)

Story 

Shear

(kip) 

Force

(kip) 

Story 

Shear

(kip) 

Force

(kip) 

Story 

Shear

(kip) 

Roof 85 7,620 1,809 2,047 2,096 

 4 1,809 2,047 2,096

4 68 5,626 886 1,113 1,026 

 3 2,695 3,160 3,122

 3 51 5,626 602 726 697 

 2 3,297 3,886 3819

 2 34 5,626 363 354 421 

 1 3,660 4,240 4,240

1 17 9,685 144 167 

 Basement 3,804 4,407

Basement 0 

Total 34,183 3,804 3,804 4,240 4,240 4,407 4,407

Notes: 

1. For Model 2, the dynamic base is set at the basement floor.

2. The ASCE/SEI 7-16 Equation 12.8-1 equivalent lateral force procedure base shear is V = Cs W = 0.173W. 

3. The scale factor is taken at the bottom story for the static analysis (Story 1) where the story shear for the dynamic analysis is 

3,660 kip and the story shear (or base shear) for the static analysis is 4,240 kip. This produces a scale factor of 4,240 kip / 

3,660 kip = 1.1585.

4. The dynamic analysis values are scaled by the scale factor to produce the scaled dynamic analysis values. 
FEMA P-2091 9: How to Model a Basement 9-9 

1,102 k

3,608 k

1,203 k

1,102 k 1,102 k 1,102 k

1,203 k

3,608 k

R

4

3

2

1

B

1,809 k

Fi

886 k

362 k

602 k

144 k

1,809 k

vi

2,695 k

3,297 k

3,660 k

3,804 k

2,047 k

2,047 k

Fi

1,113 k

354 k

726 k

vi

3,160 k

3,886 k

4,240 k

Fi

1,026 k

421 k

697 k

167 k

2,096 k

2,096 k

vi

3,122 k

3,819 k

4,240 k

4,407 k

Initial 

Dynamic Static Scaled 

Dynamic

Scale Factor

4,240 k

3,660 k = 1.1585

Figure 9-6 Model 2 distribution of forces for example building.

9.4 Models for Buildings on Sloping Grade

9.4.1 Example Building

When buildings are on a sloping grade that creates a retaining wall, soil 

pressure is exerted on the building. In addition, during an earthquake, the 

soil exerts an additional “seismic increment” of pressure on the structure. 

These pressures are typically provided by the project geotechnical engineer. 

To illustrate how to model a building on a sloping grade, the example 

building in Section 9.3 on a level site has been moved to a sloping site as 

shown in Figure 9-7. The basement floor aligns with the lower grade level 

on the left; the Level 1 floor aligns with the higher grade on the right. 
9-10 9: How to Model a Basement FEMA P-2091

R

4

3

2

1

B

Figure 9-7 Example building on a sloping site.

9.4.2 Model 2 for Example Building

In this approach, seismic loads from the design of the superstructure are 

combined with those from the soil by linear superposition. The dynamic 

base is set at the bottom of the mat. Figure 9-8 shows the scaled dynamic 

forces from Figure 9-6 applied to the building. It also shows the derivation 

of the earth pressure loads. For this example, there is the “static” or “active” 

soil pressure component of 40 pcf of equivalent fluid pressure plus the 

seismic increment of 20 pcf of equivalent fluid pressure. Both are given at 

the factored level so they can be directly combined with the factored loads 

from the superstructure. The values to be used for actual projects will vary 

depending on the soil conditions and geotechnical engineer’s approach. It is 

important to make sure there is a clear understanding of the assumed load 

level that underlies the geotechnical engineer’s design recommendations. 

They are sometimes given for the allowable stress design level, rather than 

the factored level. For this simple example, the resistance that may exist 

from passive pressure due to shallow embedment of the mat in Figure 9-7 is 

not included in the free body diagram in Figure 9-8. 

Note that this seismic pressure increment should be a consequence of the SSI 

that is the subject of this Guide, not a prescribed quantity. This is a 

significant logic gap in analysis procedures currently used in practice. The 

way that this pressure should be derived is to provide springs, conduct a 

response analysis, and then obtain the spring forces. FEMA (2020b) 

provides a resource paper current thinking in retaining wall pressures.

For this example, it is interesting to note that the base shear that the basement 

and foundation must resist is substantially increased due to the unbalanced 
FEMA P-2091 9: How to Model a Basement 9-11

soil pressure. It rises from 4,407 kip from the building seismic inertial 

weight alone to 4,407 kip + 1,001 kip + 2,002 kip = 7, 410 kip or a 1.68 

increase. On the other hand, because the centroid of the applied soil loading 

is relatively low, the overturning load at the ends of the structure of the only 

rises from 3,608 kip to 3,812 kip or an increase of 1.06. 

R

4

3

2

1

B

Fi

1,026 k

421 k

697 k

167 k

2,096 k

2,096 k

vi

3,122 k

3,819 k

4,240 k

4,407 k

Scaled 

Dynamic

1,853 k

1,271 k

1,853 k 1,853 k 1,853 k

1,271 k

3,812 k 3,812 k

1,001 k

2,002 k

FT

FB

Unbalanced Earth Pressure 

Including Seismic Increment

Building Length = 346.5 ft

FT = (346.5 × 1⁄2 × 40 × 17 2

∕3

+ 346.5 × 1⁄2 × 20 × 17 2

∕3)

= 668 k + 334 k = 1,001 k

FB= 1,001 k × 2 = 2,002 k

40 EFP 20 EFP

= +

Static

Pressure

Seismic

Increment

Passive pressure 

resistance ignored 

in this example

Figure 9-8 Model 2 distribution for example building on a slope.

FEMA P-2091 10: Conclusions and Recommendations 10-1 

Chapter 10 

Conclusions and 

Recommendations

10.1 Overview

Soil-structure interaction (SSI) can make a substantial difference in how 

buildings behave during earthquake shaking and how they should be 

designed, and yet there is relatively little implementation of SSI effects by 

practicing engineers. Provisions are available in Chapter 19 of ASCE/SEI

7-16 Minimum Design Loads and Associated Criteria for Buildings and 

Other Structures (ASCE, 2017a) and in Chapter 8 of ASCE/SEI 41-17

Seismic Evaluation and Retrofit of Existing Buildings (ASCE, 2017b) that 

can be used to address SSI. The overall goal of this Guide is to present 

information regarding SSI as implemented in code provisions but in an easyto-understand, concise format. It is directed towards practicing engineers to 

help them determine when SSI effects are of importance and to show them 

examples of how to implement SSI in design.

10.2 Conclusions

• SSI is not that difficult to implement.

• SSI is typically iterative, so it may require additional rounds of analysis, 

as compared to a fixed base analysis, to converge on the final solution.

• Inertial SSI typically reduces the seismic force demands that are used for 

design, but there are unusual cases with site-specific response spectra 

where force demands can increase because period elongation may lead to 

climbing up the response spectrum with increasing levels of spectral 

acceleration at short periods. Displacement demands—in particular, 

drift—always increase when SSI is considered as compared to when it is 

not.

• Adding foundation flexibility to a model can dramatically affect how the 

building behaves in some situations, particularly where foundation 

flexibility would lead to rocking of shear walls or braced frames in the 

superstructure.

• Effective shear wave velocity, vs, is a key parameter in several SSI 

equations and techniques. The effective shear wave velocity differs from 
10-2 10: Conclusions and Recommendations FEMA P-2091

the low-strain shear wave velocity, vso, used for site classification in 

ASCE/SEI 7-16 Chapter 21. Modifications are made based on site class, 

site spectral acceleration, and the depth of importance. 

• There are several code provisions, both in ASCE/SEI 7-16 Chapter 12

and Chapter 19, that can limit the extent of SSI reductions that can be 

utilized.

• Although ASCE/SEI 7-16 is the standard that is referenced and used in 

this Guide’s design examples, ASCE/SEI 41-17 has a similar set of SSI 

provisions. In some cases, ASCE/SEI 41-17 has more relaxed 

requirements and limitations regarding the use of SSI. 

10.3 Other Helpful Resources 

In addition to the examples in this Guide in Chapters 3 through 9, Appendix 

A, and Appendix B, there are other resource documents that provide practical 

SSI advice and worked examples. These are summarized in Appendix C. 

10.4 Comparison of SSI Provisions 

It is useful to compare the SSI provisions in ASCE/SEI 7-16 Chapter 19 with 

those in ASCE/SEI 41-17 Chapter 8. The provisions in the two standards are 

similar, but the ASCE/SEI 41-17 provisions have fewer limitations, except in 

a couple of cases. Table 10-1 provides a comparison.

10.5 Recommendations 

Based on project studies and the design example development process for 

this Guide, revisions to code provisions and further studies regarding SSI 

provisions are needed.

10.5.1 Recommendations for Code Developers

General recommendations are made here to code developers, such as the 

Building Seismic Safety Council Provisions Update Committee that develops 

the NEHRP Recommended Seismic Provisions for New Buildings and Other 

Structures and the ASCE Minimum Design Loads on Buildings and Other 

Structures Standards Committee that develops ASCE/SEI 7. Both technical 

and editorial revisions are recommended. At the time of writing of this 

Guide in October 2020, several code change proposals had been developed 

and were under consideration in the code development process. Other 

proposals were considered, but they did not find sufficient support to 

proceed. Both are indicated below.
FEMA P-2091 10: Conclusions and Recommendations 10-3 

Table 10-1 Comparison of SSI Provisions

Issue ASCE/SEI 7-16 ASCE/SEI 41-17 Comment/Justification

Site Class No Class A or B for any SSI 

(§19.1)

Class C, D, E for kinematic 

(§8.5.1.1)

SSI effects in Class A and B 

assumed minimal 

(ASCE/SEI 7-16 §C19.1)

Kinematic SSI (base slab 

averaging and embedment)

Only permitted for 

Nonlinear Response History 

Analysis (NRHA)

Permitted for any method No equivalent lateral force 

procedure or response 

spectrum studies 

(ASCE/SEI 7-16 §C19.2)

Kinematic reduction floor RRSbsa × RRSe ≥ 0.7 (§19.4) RRSbsa × RRSe ≥ 0.5 (§8.5.1.1) To be conservative 

(ASCE/SEI 7-16 §C19.4)

Kinematic connectivity Mat, grade beam, connected 

slab (§19.4.1)

Mat, grade beam, connected 

slab (§8.5.1.1)

Further study is needed

Kinematic restrictions T ≥ 0.2 and be < 260 ft 

(§19.4.1)

be < 260 ft (§8.5.1.1) No studies of larger be

Embedment depth limit 20 ft (§19.4.2) 20 ft (§8.5.1.2) No case study over 20 ft

Equivalent lateral force 

procedure reduction floor

Capped at 0.9 if R > 6 

(§19.2)

Not applicable Limited studies on 

interaction of SSI and 

R-factor 

(ASCE/SEI 7-16 §C19.2)

Overall SSI reduction floor None 0.7 x non-SSI results for 

procedures besides NRHA

No justification given

Minimum base shear Chapter 12 minimums No minimum Limits SSI in tall buildings

Upper and lower bounds Required (§19.1.1, 12.13.3) Required (§8.4.2) Further study is needed

When SSI is required Optional (§19.1.1) Required if longer period 

increases Sa (§7.2.7)

Provisions are typically 

ignored in practice

Notes: 

RRSbsa = Ratio of response spectra factor for base slab averaging

RRSe = Ratio of response spectra factor for embedment 

T = Fundamental period of the building

be = Effective foundation size

R = Response modification coefficient

Technical Revisions

• Require SSI unless it can be shown not including SSI is conservative: 

There are some cases, such as short period structures with site-specific 

response spectra that are not capped at short periods all the way to the 

zero period, where the increase in period from inclusion of foundation 

flexibility will lead to climbing up the response spectrum and an increase 

in design forces. This situation can occur for short period structures 

when a site-specific response spectrum is used. This is already required 

by ASCE/SEI 41-17 Section 7.2.7. A proposal was considered to change 

ASCE/SEI 7-16 Section 12.7.1 as follows:

o Current Text: Section 12.7.1 Foundation Modeling. For the 

purposes of determining seismic loads, it is permitted to consider the 
10-4 10: Conclusions and Recommendations FEMA P-2091

structure to be fixed at the base. Alternatively, where foundation 

flexibility is considered, it shall be in accordance with Section 

12.13.3 or Chapter 19.

o Proposed Text: Section 12.7.1 Foundation Modeling. For the 

purposes of determining seismic loads and analytical results, 

foundation flexibility shall be considered in accordance with Section 

12.13.3 or Chapter 19, unless it can be shown that this flexibility 

does not substantially change the analytical results.

The proposal did not find sufficient support to proceed.

• Remove limit without charging language: ASCE/SEI 7-16 Equation 

19.4-4 has a limitation on the average effective shear wave velocity of 

200 ≤ vs ≤ 500 m/s in the base slab averaging provisions, but since this 

parameter is not used in the equations, it is unclear how it should be 

interpreted. Can base slab averaging only be used if vs is within those 

limits? At the time of writing of this Guide in October 2020, a proposal 

was under review to remove this requirement, as base slab averaging is 

already restricted to Site Classes C, D, and E, and a vs value less than 200 

m/s should not reduce the effects of base slab averaging.

• Revise the limitation on the equivalent lateral force procedure: 

ASCE/SEI 7-16 Section 19.2.1 included a revision in the limits for SSI 

reductions that unintentionally limited the reduction in the equivalent 

lateral force procedure base shear due to period elongation as a function 

of the structure’s response modification coefficient, when the original 

intent was to apply the limit only to foundation damping. At the time of 

writing of this Guide in October 2020, a code change proposal was under 

review to correct this. A maximum reduction of 30% is imposed due to 

the effects of period elongation. The effect of the revision does not alter

the results for foundation damping reductions.

Editorial Revisions

• Shear wave velocity definitions: There are several similar shear wave 

velocity parameters in ASCE/SEI 7-16 in Chapters 19 and 20. 

ASCE/SEI 7-16 is not consistent with the terms. This should be 

clarified.

o In ASCE/SEI 7-16 Chapter 20, vs is the “average shear wave 

velocity.” It is taken over the top 30 m of soil. This is often 

designated as Vs30 for 30 m. Both vs and Vs30 are the low strain shear 

wave velocity, though this is never stated in ASCE/SEI 7-16 Chapter 

20. This should be clarified.
FEMA P-2091 10: Conclusions and Recommendations 10-5 

o In ASCE/SEI 7-16 Chapter 19, vso, is the “average low strain shear 

wave velocity over a depth of B (half the smaller dimension of the 

base of the structure)” for damping considerations and the “average 

low strain shear wave velocity over the embedment” for embedment 

considerations. If the embedment depth were 30 m, it would be the 

same as vs . When the embedment depth is shallower, they are not 

the same. Clarification is also needed that vso is determined using 

ASCE/SEI 7-16 Equation 20.4-1, and that the sum of di in Equation 

20.4-1 is the depth in question noted above. 

o In ASCE/SEI 7-16 Chapter 19, the low strain shear wave velocity 

needs to be modified by ASCE/SEI 7-16 Table 19.3-1 to obtain the 

average effective shear wave velocity, vs. 

A code change proposal for ASCE/SEI 7-16 Chapter 19 is under review 

to use specific variables to define the terms to help address the above 

inconsistencies. Similar revisions are proposed for the shear modulus to 

clarify the appropriate depth of soil to apply to consider for different SSI 

effects.

• Clarifying when SSI can be used for linear procedures: The current 

provisions have overlapping requirements in ASCE/SEI 7-16 Chapters 

12, 16, and 19 regarding which SSI provisions can be used with linear 

procedures. Explicit statements and reordering of text are being 

considered to assist in clarity. An example is the linear analysis that is 

required to accompany a nonlinear response history analysis (NRHA) 

covered by ASCE/SEI 7-16 Section 16.1.2. The linear analysis is to be 

conducted in accordance with one of the linear methods in ASCE/SEI 

Chapter 12 (equivalent lateral force procedure, modal response spectrum 

analysis, and linear response history analysis). The ASCE/SEI Chapter 

19 spectral modifications for kinematic effects (base slab averaging and 

embedment) are not permitted for linear analyses, including the Chapter 

16 required linear analysis. The effects of inertial SSI (foundation 

flexibility and foundation damping) can be used for linear analyses, 

including the Chapter 16 required linear analyses. A code change 

proposal is under review to make this intent clearer.

• Reordering equations: A minor recommended revision is to reorder 

equations such that they appear in the order in which they need to be 

calculated. A code change proposal with the appropriate reordering is 

under review. 
10-6 10: Conclusions and Recommendations FEMA P-2091

10.5.2 Recommendations for Further Study

As part of the project that included development of this Guide, detailed case

studies were performed to investigate various SSI issues. The case study 

analyses used what is termed the “direct analysis” approach where the soil 

and building are both modeled with nonlinear finite elements. The ground 

motion inputs included spatial variability so that the effects of this variability 

on foundation input motions could be compared with semi-empirical models 

used in current procedures such as NIST (2012a). 

Case studies covered both instrumented buildings and archetype buildings. 

Recordings from instrumented buildings were compared with results from 

analytical models to help validate and calibrate the modeling techniques and 

assumptions. Since there are very limited recorded data from ground shaking 

at the design earthquake or maximum considered earthquake levels, 

archetype buildings, based on those in Appendices A and B, were used 

together with records scaled to large shaking levels. Archetype building 

parameter studies included varying the level and spatial variability of ground 

shaking, the soil-to-structure interface assumptions, and nonlinearity in the 

soil and superstructure. 

The studies revealed a variety of issues with the assumptions and methods 

used in direct analyses and the challenges in simulating realistic ground 

motions throughout the soil column profile. The direct analysis case studies

generally show a reduction in short period spectral response of the 

foundation as compared to free field motion, but results were highly sensitive 

to modeling assumptions, soil properties, and soil nonlinearity. Building 

response appears to be more sensitive to soil nonlinearity than to spatial 

variability of ground motion. Further checking and validation of these 

findings is needed before the results could be used to inform

recommendations for application. Although definitive conclusions were not 

reached regarding potential revisions to code SSI provisions, such as the 

response spectral modification factor for kinematic interaction, the work 

provides an excellent starting point that includes a large body of preliminary 

results, an outline of the approach and issues, and recommendations on the

next steps that are needed for further study. Jeremic, et al. (2020) provide a 

summary of the case studies that were conducted.

This project was limited in terms of time and funding and could not address 

all of the issues that had been originally raised. Because of this, further study 

is recommended for the following topics: 

• Kinematic interaction models: The database used to develop current 

kinematic interaction models is approximately 25 years old. This work 
FEMA P-2091 10: Conclusions and Recommendations 10-7

should be revisited because (1) the scope of the database (e.g., range of 

conditions, numbers of recordings) will be broader; and (2) the quality of 

modern recordings is much better than recordings from the 1980s-1990s, 

which can significantly improve the resolution with which kinematic 

effects can be resolved. 

• Direct analysis: Direct analyses conducted for this study generally show 

a reduction in short period spectral response of the foundation as 

compared to free field motion, but results are highly sensitive to 

modeling assumptions, soil properties, and soil nonlinearity. Study is 

needed for a greater range of soil conditions, building structural systems 

and geometries, and ground motion variability including spatial 

variability, and soil nonlinearity and contact properties with the 

foundation. Future work should begin by continuing the validation of

direct analyses against available data, and then extend to conditions for 

which data is not available, such as higher ground motion levels.

• Deconvolution: More detailed investigation is needed of what is called 

“deconvolution,” which involves the process of simulating ground 

motions at depth to produce the desired free field motions at the ground 

surface.

• Shear wave conversions: A study should evaluate whether the 

conversions from average low strain shear wave velocity to effective 

shear wave velocity in ASCE/SEI 7-16 Table 19.3-1 should always be 

based on SDS, or whether SD1 be used for longer periods. Regardless of 

the decision, the revisions must account for the differences between the 

Design Earthquake and MCER ground motion levels. A code change 

proposal has been submitted to correct this error.

• Limitations on use of kinematic SSI provisions: ASCE/SEI 41-17 

permits use of kinematic SSI provisions for all analysis methods; 

ASCE/SEI 7-16 limits their use to only NRHA. Analysis is needed to 

investigate whether there is analytical justification for relaxing the 

ASCE/SEI 7-16 provisions. This should include the limits on 

embedment depth (20 ft), effective foundation size, be (260 ft), and 

average effective shear wave velocity (650 ft/s ≤ vs) that are now 

somewhat dated. 

• Limitations on the reduction in design forces from SSI: ASCE/SEI 

7-16 Section 19.2.1 correlates the amount of reduction in base shear 

permitted with SSI to the response modification coefficient, R, for the 

lateral force-resisting system of the superstructure. Systems with higher 

R factors and more ductility have more significant limitations because 
10-8 10: Conclusions and Recommendations FEMA P-2091

SSI research has primarily been based on linear elastic superstructure 

models. The Section 19.2.1 provisions should be studied to determine if 

they are justified by carefully examining the interaction of superstructure 

nonlinearity and SSI effects.

• Foundation interconnectivity for use of kinematic SSI provisions:

ASCE/SEI 7-16 Section 19.4.1 limits the use of base slab averaging for 

structures with a full mat foundation, a grid of grade beams, or a slab-ongrade connected to the footings by overpours and dowels. Detailed case 

studies are needed to investigate differences in response to foundations 

for those cases and those where the slab-on-grade is separated from the 

top of the footings by soil but is either poured tight to the columns or 

separated by a typical sealant gap for crack control. 

• Bounding of soil properties: ASCE/SEI 7-16 Sections 12.13.3 and 

19.1.1 require bounding of soil stiffnesses when flexible base models are 

used, with a 50% increase and decrease in soil stiffness. The worst case 

results are to be used. This leads to substantial analytical work. Further 

study of this requirement is recommended to determine the variability of 

the results on design and on performance to determine if the effort is 

warranted.

• Seismic earth pressures on basement walls: Current procedures for 

analysis of seismic earth pressures on basement walls do not consider the 

inertial and kinematic SSI effects that are the subject of this Guide. 

Recent research has provided a more rational SSI-based framework for 

characterizing the seismic increment of earth pressure as discussed in 

FEMA (2020b). These new procedures should be vetted through case 

studies of buildings, with participation of practicing geotechnical and 

structural engineers. The aim of these studies would be to advance the 

detailed implementation of the procedures to an extent where code 

change proposals could be considered in future cycles.
FEMA P-2091 A: Two-Story Building Example A-1 

Appendix A

Two-Story Building Example

A.1 Overview

This chapter provides an example of application of the foundation damping 

provisions and base slab averaging provisions of ASCE/SEI 7-16 Chapter 19. 

The example applies the provisions to a two-story steel-frame building with 

buckling-restrained braced frames and shallow foundations. The example 

building is based on the three-story steel braced-frame building evaluated in 

Chapter 9 of FEMA P-2006 (FEMA, 2018). The gravity framing and plan 

dimensions of this example are the same as those in the FEMA P-2006 

example. The site conditions, ground motions, number of stories, and 

bracing type have been changed for this example to better illustrate the 

provisions. 

Example Summary

Building Type: 2-story 

steel-frame building with 

buckling-restrained braced 

frames

Foundation Type: Spread 

footings

Analysis Method: 

Equivalent Lateral Force 

(ELF) method, with modal 

analysis as required for 

damping computations.

SSI Topics Covered: 

• Foundation flexibility 

(vertical and lateral)

• Reductions in loading due 

to soil damping, radiation 

damping, and base slab 

averaging

• Impacts of limitations 

imposed by ASCE/SEI 

7-16 provisions

The soil-structure interaction (SSI) computations illustrated in this example 

depend on the foundation geometry and the dynamic properties of the 

structure; it is therefore necessary to begin by preparing a conventional 

design as a starting point. This starting point is based on ASCE/SEI 7-16, 

employing the equivalent lateral force (ELF) method, without consideration 

of soil-structure interaction (Section A.4). 

In Sections A.5 and A.6, the dynamic properties and foundation geometry 

determined based on the conventional analysis are employed to compute the 

foundation springs, flexible-base analysis and reductions in demand due to 

foundation damping, as permitted in ASCE/SEI 7-16. Section A.7 presents 

an updated design with smaller brace areas and foundation sizes, which is 

verified to produce dynamic behavior and demand reductions consistent with 

the design input. 

Section A.8 then explores the soil-structure interaction reductions without 

consideration of the limitations in ASCE/SEI 7-16, allowing more significant 

reductions due to foundation damping and including the additional reduction 

due to base slab averaging, which is not permitted for linear analysis methods 

in ASCE/SEI 7-16. 

This example is not a complete design of the structure. The example focuses 

on selected elements of the design that are essential to the application of the 

provisions being illustrated.
A-2 A: Two-Story Building Example FEMA P-2091

A.1.1 Outline of the Example

This example includes the following:

• Section A.2: Site description, including ground motion parameters and 

soil properties

• Section A.3: Building description, including geometry, gravity framing, 

seismic force-resisting system and foundations 

• Section A.4: Initial design, including buckling-restrained brace core 

areas, column sizes, and foundation sizes

• Section A.5: Flexible-base analysis, including vertical springs under 

foundations that allow the frames to rock as well as lateral springs 

• Section A.6: Foundation damping, including the computations for 

reductions in seismic demand according to ASCE/SEI 7-16 Section 19.3 

• Section A.7: Design for SSI-reduced base shear per ASCE/SEI 7-16 

Chapter 19, including reduced brace and foundation sizes

• Section A.8: Alternative design for full theoretical SSI-reduced base 

shear, including reductions due to base slab averaging 

A.1.2 Limitations in ASCE/SEI 7-16 Chapter 19

This example employs the ELF procedure for design, employing dynamic 

modal analysis as a tool to assess the damping, which depends on period 

lengthening due to foundation flexibility. ASCE/SEI 7-16 Chapter 19 allows 

the application of the foundation damping provisions to buildings designed 

with the ELF procedure, but only allows application of base slab averaging 

(and other kinematic effects) with the nonlinear response history analysis. 

ASCE/SEI 7-16 Chapter 19 also places limits on the permissible reduction in 

lateral forces due to soil-structure interaction. Sections A.4 through A.7 

present a design complying strictly with the requirements in ASCE/SEI 7-16 

Chapter 19 (limiting the design force reduction and not including base slab 

averaging). Section A.8 presents an alternative (non-compliant) design based 

on the design force reductions computed without the limitations and 

including the reductions due to base slab averaging. 

A.2 Site Description

For the purpose of this example, this fictitious building is located at the site 

of the VA Loma Linda Healthcare System building in San Bernardino 

County, California (Latitude 34.049601, Longitude -117.250073). The site 

soils comprise sands and gravels, loose at the surface and increasing in 

density with depth. Figure A-1 shows the generalized soil profile. The 
FEMA P-2091 A: Two-Story Building Example A-3 

average shear wave velocity in the upper 100 feet is 950 ft/s, resulting in a 

site classification as Site Class D. The ground motion parameters according 

to the ATC Hazards by Location web application (available at 

https://hazards.atcouncil.org/

• SS = 2.355g 

• S1 = 0.943g 

• SMS = 2.355g 

• SDS = 1.57g 

The values of SM1 and SD1 are not specified by the ATC Hazards by Location 

web application, which points to ASCE/SEI 7-16 Section 11.4.8, which 

requires a site response analysis. An exception in ASCE/SEI 7-16 Section 

11.4.8 allows consideration of a conservative spectral shape with CS taken as 

the value determined by ASCE/SEI 7-16 Equation 12.8-2 for periods up to 

1.5TS (1.02 s) and taken as 1.5 times the value computed by ASCE/SEI 7-16 

Equation 12.8-3 for longer periods considering Fv = 1.7. The effect of the 

modified spectral shape is to elongate the short-period plateau, pushing the 

descending branch of the spectrum to the right. The design response 

spectrum is shown in Figure A-2, along with the spectrum based on the 

mapped values without the increases required by ASCE/SEI 7-16 Section 

11.4.8 for comparison. Note that the short period of the example building 

will not require consideration of the descending portion of the response 

spectrum. The longest period computed in this example (for 50% decreased 

soil springs and reduced brace sizes) is 0.62 s, which remains less than Ts = 

0.68 s and considerably less than 1.5 Ts = 1.02 s. 

, last accessed October 15, 2020) are as follows: 
A-4 A: Two-Story Building Example FEMA P-2091

Figure A-1 Generalized soil profile for the Loma Linda Site, taken from Stewart and 

Stewart (1997) UCB/EERC-97/01. 
FEMA P-2091 A: Two-Story Building Example A-5 

Figure A-2 Design acceleration response spectrum per ASCE/SEI 7-16 

Chapters 11 and 12. 

A.3 Building Description

The example building is a two-story office building. (For the purpose of this 

example, the original FEMA P-2006 example was reduced from three stories 

to two.) An overall building view, a typical floor plan, and the foundation 

plan are shown in Figures A-3, A-4, and A-5, respectively. The building is 

rectangular in plan, with five 30-foot bays in the north-south (transverse) 

direction and seven 30-foot bays in the east-west (longitudinal) direction. 

Each story height is 12’-6”. Discussion of the loading in these two directions 

uses the terminology “transverse” and “longitudinal.” 

A.3.1 Gravity Framing

The roof and second floor are constructed of 31⁄4 in structural lightweight 

concrete fill over 3 in composite steel deck, supported by steel wide-flange 

beams at 10-foot spacing and wide-flange girders at 30-foot spacing. The 

gravity framing is supported on W14 columns at the grid intersections. 

A.3.2 Lateral Framing

The building resists lateral loading with a system of single-bay bucklingrestrained braced frames, arranged symmetrically on the perimeter, with two 
A-6 A: Two-Story Building Example FEMA P-2091

frames on each of the short sides acting in the transverse direction and three 

frames on each of the longer sides acting in the longitudinal direction. 

A.3.3 Gravity Loads and Seismic Weight

Dead loads and seismic weights are given in Table A-1 for the roof and 

Table A-2 for the typical floor. Live loads are 20 lb/ft2 for the roof and 50 

lb/ft2 for the typical floor. 

The total seismic weight at the roof level is 2,385 kip at the roof and 2,449 

kip at the second floor, resulting in a total seismic weight of 4,834 kip, as 

indicated in Table A-3. 

Table A-1 Flat Loads on Roof

Component

Dead Load 

(lb/ft2) 

Seismic Weight 

(lb/ft2) 

Roofing 5.0 5.0

Insulation 3.0 3.0

Steel deck and concrete fill 46.4 46.4 

Steel framing (beams and columns) 7.0 7.0

Ceiling 3.0 3.0

Partitions 0.0 5.0

Exterior walls, MEP, miscellaneous 

components, extra concrete for deck 

deflection

4.6 4.6 

Total 69.0 74.0

Table A-2 Flat Loads on Second Floor

Component

Dead Load 

(lb/ft2) 

Seismic Weight 

(lb/ft2) 

Floor Covering 1.0 1.0

Steel deck and concrete fill 46.4 46.4 

Steel framing (beams and columns) 11.0 11.0

Ceiling 3.0 3.0

Partitions 15.0 10.0

Exterior walls, MEP, miscellaneous 

components, extra concrete for deck 

deflection

4.6 4.6 

Total 81.0 76.0
FEMA P-2091 A: Two-Story Building Example A-7 

Table A-3 Summary of Seismic Weights

Level

East-West 

Dimension 

(ft) 

NorthSouth 

Dimension 

(ft) 

Area

(ft2) 

Flat Load

(lb/ft2) 

Total 

Weight 

(kip)

Roof 212 152 32,224 74 2,385

Second 

Floor

212 152 32,224 76 2,449

Total 4,834

A.3.4 Foundation System

The building is supported on a system of spread and continuous footings, 

with square spread footings at each gravity column and continuous strip 

footings at each braced frame, encompassing both frame columns and 

extending beyond each column to provide overturning resistance. 

The slab-on-grade is designed as an interconnecting diaphragm, with Zdowels cast into each footing. The slab is underlain by a moisture barrier. 

Frictional resistance between the slab and the soil is neglected. 

The foundation design is based on the LRFD (strength) design provisions in 

ASCE/SEI 7-16 Section 12.13.5. The geotechnical engineer has provided a 

nominal bearing strength of 10 kip/ft2

, to which a resistance factor of 0.45 

applies per ASCE/SEI 7-16 Table 12.13-1 for a design strength of 4.5 kip/ft2

. 

Figure A-3 Overall view. 
A-8 A: Two-Story Building Example FEMA P-2091

BF-8 BF-9 BF-10

BF-5 BF-6 BF-7

BF-2 BF-1

BF-4 BF-3

A B C G H

31⁄4" lightweight concrete 

over 3" deep 18 gage steel 

deck (61⁄4" total depth)

7 at 30'-0" = 210'-0"

D E

4

3

2

1

6

5

5 at 30'-0" = 150'-0"

1'-0", typ. 

edge of 

deck

F

Figure A-4 Typical floor framing plan (roof framing similar).

A B C D E F G H

4

3

2

1

5

6

7 at 30'-0" = 210'-0"

BF-5 BF-6 BF-7

BF-4 BF-3

BF-2

5 at 30'-0" = 150'-0"

BF-1

Spread footing at gravity columns typ. 

(T.O. FTG. 1'-0" below T.O. slab) 

Strip footing at braced frame typ.

(T.O. FTG. 1'-0" below T.O. slab) 

BF-8 BF-9 BF-10

4'-0"

5" slab-on-grade thickened to 1'-0" above 

spread footings and connected with Z-dowels

4'-0"

Figure A-5 Foundation plan. 
FEMA P-2091 A: Two-Story Building Example A-9 

A.4 Initial Design 

Design incorporating soil-structure interaction requires a more iterative 

approach than traditional fixed-base design, because determination of the 

design-force reductions applicable to the building depend on the foundation 

sizes and on both the fixed-base and flexible-base building periods. This 

section presents the design of the structure and foundations without 

consideration of any reductions in seismic loading due to soil-structure 

interaction. The SSI reductions are then computed in subsequent sections, 

based on the structure dynamics and foundation geometry of the initial 

design. 

A.4.1 Base Shear per ASCE/SEI 7-16 Chapter 12 Equivalent 

Lateral Force Procedure 

The initial design is based on the ELF procedure of ASCE/SEI 7-16 Section 

12.8. In this procedure, the base shear is computed considering the building 

to be fixed at the base, and forces are distributed over the height to find the 

story shears used to compute the required brace sizes and foundation 

overturning demands. 

The fundamental period of the building for determination of the base shear, 

Ta, is computed according to ASCE/SEI 7-16 Section 12.8.2.

T = C hx 0.75

a tn = (0.03)(25) = 0.34 s (ASCE/SEI 7-16 Eq. 12.8-7) 

Where hn is 25 feet and Ct and x are determined from ASCE/SEI 7-16 Table 

12.8-2. 

Because Ta is less than 1.5 Ts, the base shear, V, is calculated according to 

ASCE/SEI 7-16 Equations 12.8-1 and 12.8-2 as follows:

V = CSW (ASCE/SEI 7-16 Eq. 12.8-1) 

where: 

S C DS

S = (ASCE/SEI 7-16 Eq. 12.8-2)   R

    Ie

R = 8 for Buckling-restrained braced frames

Ie = 1.0 for Risk Category II 

V = (Cs=0.196)(W=4,834 kip) = 949 kip

Distribute the seismic forces over the building height according to ASCE/SEI 

7-16 Section 12.8.3, as indicated in Table A-4. 
A-10 A: Two-Story Building Example FEMA P-2091

Table A-4 Vertical Distribution of Seismic Forces

Level

wX

(kip) 

hX 

(ft) CVX

FX 

(kip) 

Roof 2,385 25 0.66 627

Second Floor 2,449 12.5 0.34 322 

Total 4,834 949 

A.4.2 Brace Design 

For the purpose of this example, application of accidental torsion according 

to ASCE/SEI 7-16 Section 12.8.4.3 is not considered. This simplifies the 

computations that are not related to the concepts being illustrated. 

Determination of the required brace areas for the buckling restrained braces 

requires that the total story shears be divided by the number of braces in each 

direction, multiplied by the ratio of the brace length to the half-bay-width

(1.30), and divided by the design yield strength of the brace core material, 

which in this case is (0.9) 42 ksi = 37.8 ksi. The computed brace areas are 

then rounded up to the nearest quarter of a square inch for fabrication 

convenience. The brace core areas are shown in Tables A-5 and A-6. Note 

that the brace demands in the longitudinal direction are two thirds of those in 

the transverse direction, since the only difference between the two 

calculations is the number of braces in each direction.

Table A-5 Transverse Brace Core Areas

Level

Number of 

braces

Brace 

Force 

(kip) 

Computed 

Core Area 

(in2) 

Specified 

Core Area 

(in2) 

Below Roof 8 102 2.98 3.00

Below Second Floor 8 154 4.51 4.75

Table A-6 Longitudinal Brace Core Areas

Level

Number of 

braces

Brace 

Force 

(kip) 

Computed 

Core Area 

(in2) 

Specified 

Core Area 

(in2) 

Below Roof 12 68 1.99 2.00

Below Second Floor 12 103 3.01 3.25

The columns in each frame are designed to resist the amplified factored

overturning demands, which are 461 kip for the transverse frames and 351 

kip for the longitudinal direction. The smallest seismically compact W10 

column section, W10×68, is used with an axial compression capacity of 687 

kip for the building’s story height. 
FEMA P-2091 A: Two-Story Building Example A-11

A.4.3 Foundation Design 

The foundations for the frames are designed to resist gravity loads along with 

seismic overturning demands. Two LRFD load combinations are considered 

to capture minimum and maximum gravity loads that ASCE/SEI 7-16 

considers applicable in concert with earthquake demands. The load 

combinations from ASCE/SEI 7-16 Section 2.3.6 are employed below: 

1.2D E + v h + E + 0.5L + 0.2S (ASCE/SEI 7-16 Sec. 2.3.6, Eq. 6) 

0.9D E − v h − E (ASCE/SEI 7-16 Sec. 2.3.6, Eq. 7)

where Ev is 0.2SDSD, as defined in ASCE/SEI 7-16 Section 12.4.2.2. With 

SDS for the site equal to 1.57g, this is an added or reduced dead load of 

0.314D. Note that Exception 2b of this section indicates that the vertical 

seismic load effect, Ev, may be taken as zero in (ASCE/SEI 7-16 Sec. 2.3.6 

Equation 7, where determining overturning demands on the soil-structure 

interface of foundations, so the reduction of 0.314D need not be considered 

in in the equation when determining the foundation stability and soil 

pressures.

Although the frame columns are designed to incorporate the amplified 

loading due to overstrength, there is no requirement to consider the amplified 

loading for the foundation design. 

Since the frames and their loading are symmetrical, the total gravity loading 

can be considered to act at the frame centerline. The weight of the 

foundation is included in the dead load, D. The snow load, S, is zero. The 

live load, L, is the reduced live loading acting on the second floor. The 

factor of 0.5 on the live load is according to Exception 1 in ASCE/SEI 7-16 

Section 2.3.6. Roof live load is not considered to act with the earthquake 

load. 

Consider a continuous foundation with a depth of 5 ft, extending four feet 

beyond the center of each column, resulting in a foundation length of 38 ft. 

Determine foundation width by trial and adjustment in order to provide 

overturning stability and soil pressures remaining less than the design 

strength of 4.5 kip/ft2

. Because the tributary areas are the same between the 

two frame types, the superstructure gravity loads on each frame column are 

the same for both types of frames. 

The live load reduction is computed according to ASCE/SEI 7-16 Section 

4.7.2, with the tributary area, AT of 480 ft2

 and KLL taken as 4, per ASCE/SEI 

7-16 Table 4.7-1. 
A-12 A: Two-Story Building Example FEMA P-2091

  15 L L = ( 2 2

0 0   0.25 + = L = 50 lb/ft )(0.59) = 29.6 lb/ft     K A LL T

(ASCE/SEI 7-16 Eq. 4.7-1) 

The gravity column loads at the foundation are as follows: 

D = (480 ft 2

)(69 lb/ft 2 2 + = 81 lb/ft ) 72 kip

L = (480 ft 2 2 )(29.6 lb/ft ) =14.2 kip

The overturning demand is reduced by 25% for stability and soil pressure 

design, as permitted by ASCE/SEI 7-16 Section 12.13.4. 

A.4.3.1 Transverse Frame Foundation 

The foundation dead load for a foundation width of 8 feet is computed as 

(5 ft)(38 ft)(8 ft)(0.15 k/ft3

) = 228 kip. When the weight of the slab overpour 

of 1 ft is added for (1 ft)(38 ft)(8 ft)(0.15 k/ft3

) = 45.6 kip, the total 

foundation weight is 273.6 kip. 

As indicated in Section A.5.2, below, in computation of the lateral soil 

springs, resistance to shear at the base of the frames is mobilized by all of the 

building footings acting together and interconnected by the slab-on-grade. It 

is necessary to confirm this assumption by verifying the capacity of the slabon-grade and its connections to the footings. Grade beam ties may be 

necessary in addition to the slab. About 90% of the resistance to any single 

frame’s lateral loading acts through the slab-on-grade. The remaining 10% 

of the resistance acts on the face of the frame footing through passive 

pressure or at its base through friction. The resultant resistance acts at or 

near the top of the footing, which is also the point from which the bottom 

story height is defined. Accordingly, the story heights employed in the 

overturning computations below are the same as the nominal story heights 

defined above. Moments are summed about point “A” as indicated in Figure 

A-6. 

The reduced overturning demand on each of the four frames in the transverse 

direction is: 

MOT = (1/4) [ (626.7 kip)(25 ft) + (321.8 kip)(12.5 ft) ] (1-0.25) = 3,692 kip-ft

The resisting moment is computed separately for each factored load 

combination. Table A-7 shows the sum of gravity loads for each 

combination, along with the resisting moment and the factor of safety against 

overturning. 
FEMA P-2091 A: Two-Story Building Example A-13

Figure A-6 shows a free-body diagram of the transverse frame and foundation, 

indicating the 25%-reduced overturning demands and the maximum gravity 

loads on the frame (ASCE/SEI 7-16 Section 2.3.6 Equation 6). 

The factored foundation weight shown in Figure A-6, per ASCE/SEI 7-16 

Section 2.3.6 Equation 6 is: 

P S u =1.2 + = [0.2( DS 1.57)]273.6 kip = 414.2 kip

13'- 31⁄2"

50.1 k 50.1 k

117.5 k

60.4 k

66.0 k 66.0 k

12'- 6" 12'- 6"

5'- 0"

414.2 k

177.9 k

646.5 k

38'- 0"

30'- 0"

Point “A”

Figure A-6 Free-body diagram of a typical transverse frame showing the 

25%-reduced overturning demands and the maximum factored 

gravity loads. 

Table A-7 Resisting Moments

ASCE/SEI 7-16 Section 2.3.6 

Equation

Total 

Weight (kip) 

Resisting 

Moment 

(kip-ft) 

Factor of Safety 

Against 

Overturning

Eq. 6 (Maximum gravity load) 647 12,282 3.3

Eq. 7 (Minimum gravity load) 376 7,141 1.9 

Table A-8 shows the resulting distance from the end of the footing to the 

centroid of the soil reaction, a, along with the computed maximum and 

minimum soil pressures for each case.

The controlling case is the maximum gravity load combination, which results 

in a nearly triangular soil pressure distribution with almost no pressure at the 

back (uplifting) end of the foundation. The maximum pressure is less than 
A-14 A: Two-Story Building Example FEMA P-2091

the design soil bearing pressure (4.0 kip/ft2 < 4.5 kip/ft2), so this foundation 

size is acceptable. 

Table A-8 Soil Pressures

ASCE/SEI 7-16 Section 2.3.6 

Equation No.

a

(ft) 

Maximum Soil 

Pressure (kip/ft2) 

Minimum Soil 

Pressure 

(kip/ft2) 

Eq. 6 (Maximum gravity load) 13.3 4.0 0.21 

Eq. 7 (Minimum gravity load) 9.2 3.4 0.0

A.4.3.2 Longitudinal Frame Foundation 

There are six longitudinal frames sharing the same seismic load as the four 

transverse frames, while the gravity loading is the same. As a result, 

narrower foundations are possible. The foundation length and depth are kept 

the same. 

Foundation dead load for a foundation width of 5.5 feet is computed as 

(5 ft)(38 ft)(5.5 ft)(0.15 kip/ft3

) = 157 kip. When the weight of the slab 

overpour of 1 ft (1 ft)(38 ft)(5.5 ft)(0.15 kip/ft3

) = 31 kip is added the total 

foundation weight is 188 kip. 

The reduced overturning demand considering six frames in the longitudinal 

direction is: 

MOT = (1/6) [ (626.7 kip)(25 ft) + (321.8 kip)(12.5 ft) ] (1-0.25) = 2,461 kip-ft

The resisting moment about the foundation tip is computed separately for 

each load combination. Table A-9 shows the sum of gravity loads for each 

combination, along with the resisting moment and the factor of safety against 

overturning. 

Table A-9 Resisting Moments

ASCE/SEI 7-16 Section 2.3.6 

Equation

Total 

Weight (kip) 

Resisting 

Moment 

(kip-ft)

Factor of Safety 

Against 

Overturning

Eq. 6 (Maximum gravity load) 517 9,823 4.0

Eq. 7 (Minimum gravity load) 299 5,679 2.3

Table A-10 shows the resulting distance from the end of the footing to the 

centroid of the soil reaction, a, along with the computed maximum and 

minimum soil pressures for each case.
FEMA P-2091 A: Two-Story Building Example A-15

Table A-10 Soil Pressures

ASCE/SEI 7-16 Section 2.3.6 

Equation No.

a

(ft)

Maximum 

Soil Pressure 

(k/ft2) 

Minimum Soil 

Pressure (k/ft2) 

Eq. 6 (Maximum gravity load) 14.2 4.3 0.62

Eq. 7 (Minimum gravity load) 10.8 3.4 0.00

The controlling case is again the maximum gravity load combination, which 

results in a nearly triangular soil pressure distribution very low pressure at 

the back (uplifting) end of the foundation. The maximum pressure is less 

than the design soil bearing pressure, so this narrower foundation size is 

acceptable. 

A.4.4 Fixed-base Modal Analysis

Although the analyses in this section employ the ELF method, it is necessary 

to compute the periods of vibration for the structure for the fixed-base

condition. This allows comparison of the dynamic behavior of the fixed-base 

and flexible-base conditions, as will be required to compute radiation 

damping. 

The braces are modeled in ETABS 2016 (CSI, 2016)as prismatic rectangular 

members, extending from node to node, with stiffness modification factors of 

1.28 to account for the larger stiffness of the connection elements that occur 

near the brace ends. 

The fixed-base modal analysis results in primary modes of vibration of 

0.466 s and 0.459 s due to movement in the transverse direction and 

longitudinal direction, respectively. The transverse direction is slightly more 

flexible than the longitudinal direction. While the brace areas in the two 

directions are proportional to the story shear divided by the number of frames 

(rounded), the column sizes are the same. The longitudinal frames are 

effectively stiffer because there are six columns resisting overturning as 

opposed to four. 

A.5 Flexible-base Analysis

The soil-structure interaction calculations require determination of the 

behavior of the building with a flexible base. This is for two reasons. First, 

it is necessary to consider the change in dynamic behavior between the fixedbase and the flexible-base conditions in order to compute the radiation 

damping reductions, which rely on the ratio of the fixed-base to flexible-base 

periods. Second, ASCE/SEI 7-16 Section 19.1 requires that all elements of 

foundation flexibility must be considered to be permitted to consider soil 
A-16 A: Two-Story Building Example FEMA P-2091

structure interaction. This includes horizontal, vertical, and rotational 

foundation and soil flexibility. 

ASCE/SEI 7-16 Section 12.13.3 requires a 50% increase and decrease in the 

values of the soil springs for analysis. This requires that the maximum and 

minimum spring values be computed as 1.5 times and 0.5 times the best 

estimates. 

For the flexible-base analysis, it is necessary to include vertical springs under 

each of the strip footings under the frames, as well as lateral springs 

representing resistance to sliding due to passive pressure on the foundation 

faces and friction at the bottoms of the footings under gravity load. 

A.5.1 Vertical Springs

Vertical soil springs are computed according to ASCE/SEI 41-17 Section 

8.4.2.5 Method 3. This method employs a uniform spring along the length of 

the footing based on some flexibility of the concrete foundation element 

relative to the soil. The method is considered applicable because the 

continuous footings extend as cantilevers beyond the frame columns, 

producing flexibility in the foundation structure, and the flexibility of the 

footings is included explicitly in the model. See Chapter 6 for a detailed 

discussion of foundation flexibility, including the effects of footing 

embedment in the soil. The spring values are based on the shear modulus of 

the soil, G, and Poisson’s ratio, ν. These may be estimated by the 

geotechnical engineer or determined according to the definitions in 

ASCE/SEI 7-16 Section 19.3.3. For the purpose of the vertical soil spring 

computations, the geotechnical engineer has recommended the use of ν = 0.3 

and G=538 kip/ft2

, which are employed in ASCE/SEI 41-17 Equation 8-11 to 

compute the vertical soil springs.

1.3G 1.3(538 kip/ft 2

) 1,000 ksv = = = kip/ft3

( ) 1 ( ) 1 0.3 BB B f f −ν − f

 (ASCE/SEI 41-17 Eq. 8-11) 

The soil stiffness intensity is expressed as kip/ft3

, which is stiffness (kip/ft) 

per unit area (ft2

). Note that in this formulation the stiffness intensity by area 

is inversely related to the foundation width—as the footing gets wider, its 

stiffness intensity decreases, due to a larger volume of soil being subject to 

compression. As a result, the stiffness intensity per unit length is 

independent of the foundation width. So, multiplying through by a unit 

footing width, the linear spring value in kip/ft2 is the same as the spring value 

per area in kip/ft3

. The unit spring value of 1,000 kip/ft per linear ft is then 
FEMA P-2091 A: Two-Story Building Example A-17

employed for any foundation width. This is modeled in ETABS as a vertical 

line spring with the stiffness of 1,000 / (12)2 = 6.94 kip/in per linear inch. 

The approach outlined in ASCE/SEI 41-17 is only one method of developing 

the soil springs. Other methods, such as that in NIST GCR 12-917-21 

(NIST, 2012a) may also be applicable. This method results in spring values 

that are not independent of the footing width. A geotechnical engineer

should be consulted regarding the appropriateness of these and other 

methods. 

A.5.2 Lateral Springs

The lateral resistance of the building at the base includes two components: 

passive pressure on the sides of footings, and friction at the base of the 

footings. Because the building has a continuous slab-on-grade that is 

connected to the braced frame footings and the individual gravity footings, 

the entire building base is considered to act as a unit, such that all footings 

are mobilized to resist lateral loads. 

The geotechnical engineer has provided a recommended value for the passive 

pressure that can be developed at each footing, which is 1.6 kip/ft2

, spread 

uniformly over the face of the footing which is developed according to the 

relative displacement of the footing face and the soil according to ASCE/SEI 

41-17 Figure 8-6, which is reproduced in Figure A-7. The proportion of the 

ultimate passive pressure that is developed, P/Pult, is a function of δ/H, where 

δ is the lateral displacement and H is the depth of the footing. This 

mobilization relationship can be approximated by the following equation, 

which has been fitted to the curve in the figure. 

0.43

0.15 2.88

ult

P

P H

  δ

= +    

The geotechnical engineer has also provided a friction coefficient of 0.35, to 

be applied to the gravity load at each footing. This friction is developed at 

very small displacement. For the purpose of these calculations the 

mobilization of friction is presumed to occur at zero displacement. 

The friction is proportional to the gravity load on the footings, which can 

vary depending on which load combination is selected. Since the springs 

developed in this manner are intended to be best estimates, the gravity load 

used here is based on the load combination D + 0.25L, where L is the 

unreduced live load, as described in ASCE/SEI 41-17 Section 7.2.2. 
A-18 A: Two-Story Building Example FEMA P-2091

Figure A-7 Passive pressure mobilization curve (reproduced from ASCE/SEI 

41-17, Figure 8-6).

Because of the shape of the passive pressure mobilization curve, the lateral 

springs at the base of the building are nonlinear. In order to perform a 

flexible-base linear analysis, a secant stiffness is developed based on the 

expected value of base shear resistance. This is not the nominal base shear 

computed in Section A.4.1, which includes the response modification factor, 

R. Since the passive pressure resistance is nonlinear (and the frictional 

resistance is effectively constant), it is appropriate to base this calculation on 

an estimate of the actual demand rather than the demand reduced for design 

purposes. ASCE/SEI 7-16 uses the overstrength factor, Ω0, as a multiplier 

on the nominal base shear to approximate the real seismic load. 

Accordingly, the best estimate for the base shear to be resisted is taken as 

Ω0V, which is (Ω0 = 2.5)(V = 949 kip) = 2,371 kip. 

The total lateral resistance is computed by trial and adjustment to find the 

displacement that produces the desired resistance of 2,371 kip. For ease of 

spreadsheet calculations, Tables A-11 and A-12 show the total resistance 

developed at each column type, with each column in the braced frames 

considered separately (with foundations of half the total length). In each of 

these tables, the passive resistance for the frames parallel to the load 

considers only one passive face per frame (not one per column), while the 

passive pressure developed in the frame footings perpendicular to the load 

engages the half-length of the foundation at both columns in the frame. 
FEMA P-2091 A: Two-Story Building Example A-19

Table A-11 Total Lateral Resistance - Transverse Direction at Displacement 

of 0.015 in

Column 

Type

No. 

Each

Depth

(ft)

Width

(ft)

Axial 

Load 

(kip)

Friction 

Each

(kip)

Passive 

Developed 

Each 

(kip)

Total 

Resistance 

(kip)

Corner 4 2 4 38 13 3 67 

Interior 24 3 8 151 53 10 1,500

Transverse 

Frame 

friction

8 76 27 213

Transverse 

Frame 

passive

4 5 8 15 59

Longitudinal 

Frame 

12 5 19 76 27 35 741

TOTAL 2,580

Table A-12 Total Lateral Resistance - Longitudinal Direction at Displacement 

of 0.035 in

Column 

Type

No. 

Each

Depth

(ft)

Width

(ft)

Axial 

Load 

(kip)

Friction

Each

(kip)

Passive 

Developed

Each

(kip)

Total 

Resistance 

(kip)

Corner 4 2 4 38 13 4 70

Interior 24 3 8 151 53 12 1541

Transverse 

Frame 

8 5 19 76 27 21 375 

Longitudinal 

Frame 

friction

12 76 27 319

Longitudinal 

Frame 

passive

6 5 5.5 6 71 

TOTAL 2,376

An example computation for the passive resistance at a typical interior 

footing, which is 8 ft by 8 ft in plan and 3 ft deep at 0.015 inches 

displacement is shown below:

Ultimate passive resistance = (3 ft)(8 ft)(1.6 kip/ft2

) = 38.4 kip. 

δ/H = 0.015/36 = 0.00042 

The value P/Pult may be computed according to the curve-fit equation above 

as follows:

( )0.43 0.15 2.88 0.00042 0.25

ult

P

P = + =
A-20 A: Two-Story Building Example FEMA P-2091

Passive pressure developed = (38.4)(0.25) = 10 kip 

These example values are shaded in Table A-11. 

The total displacement required in the transverse direction is lower than that 

required for the longitudinal direction, because of the broader total passive 

width provided by the longitudinal frame footings when they are pushed 

perpendicular to their length. 

The lateral spring stiffness is computed as the total resistance provided 

divided by the displacement at which it is developed. The best estimate 

spring value computed for the transverse direction is computed as (2,580 

kip)/(0.015 in) = 172,026 kip/in, which is divided by 4 (43,006 kip/in) to be 

applied in the model at each corner of the slab. The longitudinal lateral 

spring is computed similarly. 

There will be some additional flexibility in the ground floor slab as it and 

deforms in flexure and shear due to the resistance developed within the span 

of the slab between the frames, which are on the edges of the building. There 

will also be additional stiffness developed due to the friction between the 

slab and the ground, although this friction will be mitigated by the presence 

of the vapor barrier between the slab and the ground. These two refinements 

are not considered in this example. They are small and will produce 

opposing effects. 

A.5.3 Flexible-base Modal Analysis

As required by ASCE/SEI 7-16 Section 12.13.3, flexible-base analyses 

incorporating three estimates for each spring must be performed, these 

include a 50% increase and a 50% decrease in the best estimate spring 

values. Table A-13 shows the values of spring stiffnesses employed in the 

ASCE/SEI 41 three flexible-base models.

Note that in ASCE/SEI 41, 

sensitivity analysis 

requirements differ: the 

upper bound spring values 

are twice the best estimate 

values. The lower bound 

values in both standards are 

the same.

Table A-13 Spring Values

Spring Lower Bound Best Estimate Upper Bound

Vertical linear springs 3.47 kip/in/in 6.94 kip/in/in 10.42 kip/in/in

Transverse lateral springs 

(each building corner)

21,503 kip/in 43,006 kip/in 64,510 kip/in

Longitudinal lateral springs 

(each building corner) 

8,485 kip/in 16,970 kip/in 25,456 kip/in

The lateral springs are very stiff and do not significantly affect the behavior, 

but the vertical springs under each frame foundation produce a rotation of 

each frame and its foundation and the primary mode of behavior. 
FEMA P-2091 A: Two-Story Building Example A-21

Table A-14 presents the fundamental modes for the fixed-base model and the 

three versions of the flexible-base model. Figure A-8 shows the first-mode 

displaced shape of one transverse frame. Note the vertical displacement of 

the ends of the footings due to the vertical foundation springs. 

A

5

A

4

Figure A-8 First-mode displaced shape of a typical frame 

with foundation flexibility included. 

Table A-14 Fundamental Periods for Fixed- and Flexible-Base Models

Model

Transverse first 

mode period (s) 

Longitudinal first 

mode period (s) 

Fixed base (for comparison) 0.466 0.459

Flexible-base upper bound 0.510 0.493

Flexible-base best estimate 0.532 0.509

Flexible-base lower bound 0.592 0.554

The first mode periods increase by 27% in the transverse direction and 21% 

in the longitudinal direction between the fixed-base and the lower bound 

flexible-base models. 

A.6 Foundation Damping

With the analyses complete, soil-structure interaction reductions due to 

foundation damping are computed. There are two categories of damping to 

be considered: soil hysteretic damping and radiation damping. These are 

addressed in the following sections. Note that foundation damping is not 

permitted per ASCE/SEI 7-16 Section 19.3.1.1 when the foundation system 

consists of discrete footings that are both (1) not interconnected and that 

(2) are spaced less than the larger dimension of the supported lateral forceresisting element in the direction under consideration. In this example 
A-22 A: Two-Story Building Example FEMA P-2091

building, the slab-on-grade is reinforced and thickened over footings and 

connected to them as shown in Figure A-5. This is considered to provide 

sufficient interconnectivity to engage all footings to move together. Thus, 

foundation damping is permitted. However, if the footings were not 

interconnected, they would be too closely spaced to use the foundation 

damping provisions: the braced frames are 30’0” wide and the spread 

footings supporting them are 38’0” long parallel to the brace, leaving a space 

of 22’0” between footings. More details are in Chapter 8. However, in order 

to illustrate the application of the foundation damping provisions, this 

limitation is ignored in the following calculations.

A.6.1 Soil Hysteretic Damping

This effect is simple to address; it depends only on the site class and the 

value of SDS for the site. The applicable value is taken directly from 

ASCE/SEI 7-16 Table 19.3.3. For Site Class D, the tabular values of 0.07 for 

SDS/2.5 = 0.4 and 0.15 for SDS/2.5 = 0.8 are tabulated. For this site, with SDS

of 1.57, SDS/2.5 is 0.628, the soil hysteretic damping ratio, βs, is 0.116. 

A.6.2 Radiation Damping

Radiation damping is considerably more complex. The initial inputs pertain 

to the soil at the site: the soil density and the shear wave velocity. Note that 

the shear wave velocity here is not vs , the value used to determine the site 

class. Here, vs is considered for the upper soil layer, over a depth B from the 

base of the foundation. Here, B is half the width of the foundation under 

consideration. The use of vs for site class determination is intended to

account for the average soil stiffness in the upper 100 ft (30m); the use of vs

for the upper soil layer is intended to capture the effects of the upper soils 

that more directly affect the damping at the soil-structure interface.

Figure A-1 shows the soil density in the upper soils, γ, to be 120 to 125 lb/ft3

. 

Use an average value of 122.5 lb/ft3

. Scale the shear wave velocity from the 

profile in Figure A-1 as 653 ft/s. This is vso. Then use ASCE/SEI 7-16 

Table 19.3-1 to compute the effective shear wave velocity, vs. The 

modification factor is interpolated in a manner like that used for the soil 

hysteretic damping ratio, as 0.488, which results in an effective shear wave 

velocity, vs, of 319 ft/s. 

The soil shear modulus at small strain levels, Go, is computed as: 

v2

G =γ = so

o 1,624 kip ft 2

g

 (ASCE/SEI 7-16 Sec. 19.3.3 definition) 
FEMA P-2091 A: Two-Story Building Example A-23

where g is the acceleration of gravity.

ASCE/SEI 7-16 Table 19.3-2 is used to compute the effective shear modulus, 

G. The modification factor is interpolated in a manner like that used for the

soil hysteretic damping ratio, as 0.272, which results in an effective shear

modulus, G, of 442 kip/ft2

. This is slightly lower than the value

recommended by the geotechnical engineer for the vertical soil spring

calculation. The value computed according to ASCE/SEI 7-16 Section 19.3

is used here for consistency with the other formulations within this section.

The following computations are then followed for each loading direction, for 

the three estimates of the soil spring values. The values shown in the 

paragraphs below are for the best estimate in the transverse direction. The 

results for both directions and all three spring value estimates are tabulated at 

the end of this section. 

Since the foundations under consideration behave independently, compute 

the radiation damping for one frame in each direction. For the transverse 

direction, the foundation plan dimensions are 2L = 38 ft by 2B = 8 ft. The 

plan dimensional ratio, L/B is 4.75. 

~

The first computation is the period lengthening ratio,  ̃ T / T , where T is the 

flexible-base period and T is the fixed-base period. The value of  ̃ T / T is 

computed as 0.532/0.466 = 1.142. 

The effective period-lengthening ratio, which depends on the system 

ductility, μ, is computed. ASCE/SEI 7-16 indicates in the definition of μ that 

it may be taken as the ratio of the response modification factor, R, and the

overstrength factor, Ω0, to be 8/2.5 = 3.2. R can be considered as the product

of ductility and overstrength, so dividing R by the overstrength results in an 

estimate of the ductility. The effective period lengthening ratio is then 

computed according to ASCE/SEI 7-16 as 

0.5 2 ~ ~

1 1 1 1.046

eff

T T

T T

            =+ − =     μ          

(ASCE/SEI 7-16 Eq. 19.3-2) 

The translational stiffness, Ky, and rotational stiffness, Kxx, are then 

computed according to ASCE/SEI 7-16 Equation 19.3-8 and 19.3-9, 

respectively. Here, ν is Poisson’s ratio, taken as 0.3 for sandy soil, according 

to definitions in ASCE/SEI 7-16 Section 19.3. Note that these stiffness 

equations are particular to the damping computations. See Chapter 6 for 

further information regarding foundation flexibility. The geometry and 
A-24 A: Two-Story Building Example FEMA P-2091

direction of movement implicit in these equations are based on the weak-axis 

movement of a rectangular foundation, which in this case results in a 

conservative assessment of the damping. These stiffness equations are not 

directly related to the stiffness from the structural analyses in ETABS. 

GB    L 0.65   L  7 6.8 0.8 1.6 2.506 10 lb/ft

2

Ky B B =    + += × 

− ν      

(ASCE/SEI 7-16 Eq. 19.3-8)

( ) 3

8 3.2 0.8 6.46 10 ft-lb rad

1 xx

GB L K

B

    = +=×     − ν    

 (ASCE/SEI 7-16 Eq. 19.3-9)

In order to compute the damping values, three dimensionless constants, a0, 

Ψ, and αxx, are computed according to ASCE/SEI 7-16 Equations 19.3-11, 

19.3-13, and 19.3-14, respectively.

0 ~

2 0.148

s

B

a

T v

π = = (ASCE/SEI 7-16 Eq. 19.3-11) 

2 1( ) 1.871

1 2

− ν

ψ= =

− ν 

(ASCE/SEI 7-16 Eq. 19.3-13)

(ψ must not exceed 2.5) 

( ( ) )

( )

2

0

2

3 0

0.55 0.01 1

1.0 0.995

0.4 2.4

xx

LB a

a

L B

 

 

+ −

α= − =     − +

   

(ASCE/SEI 7-16 Eq. 19.3-14) 

The fundamental translational period and fundamental rotational period of 

the damping system are computed based on the computed stiffness, the 

effective modal mass, M*, and the effective structure height, h*. The 

effective modal mass, M*, is the building seismic mass, multiplied by the 

first modal mass participation factor (in this case, 0.87), divided by the 

number of frames in the direction under consideration (4). The modal mass 

participation factor expresses the proportion of the total mass excited in the 

first natural mode of vibration. It can be found as part of the ETABS (or 

similar analysis program) modal results. In this case, M* = 3.26×104 lbsec2

/ft. The effective structure height, h*, is computed from the displaced 

shape of the first mode or may be taken as 0.7 times the building height. A 

more explicit computation based on the actual first mode shape results in 

0.68 times the building height, 16.92 ft. The first mode shape for this two-
FEMA P-2091 A: Two-Story Building Example A-25

story building is defined by two points—the lateral displacements of the 

second story and of the roof in the first mode, with each displacement 

considered at the height of the associated diaphragm. The effective height 

may be computed as the centroid of the polygon defined by these two points 

and the ground. For the two-story building under consideration, the effective 

height may be computed as follows: 

( )

( )

21 2 1

2 1 2 1 12 1 1

*

1 1 1 22

2 2 ( )

3 2 2 32 16.92 ft

2 2

h hh h h h h h

h

h h

    δ −δ    δ   +    + + δ+           = = δ δ +δ

+

Where h1 and h2 are the first and second story heights, respectively, and δ1

and δ2 are the lateral displacements of the second story and roof in the first 

mode, which in this case are 0.48 and 1.0. The translational period, Ty, and 

rotational period, Txx, are then computed according to ASCE/SEI 7-16 

Equations 19.3-6 and 19.3-7, respectively. Note again that these values are 

particular to the damping computations and are not directly related to the 

periods from our structural analyses. 

*

2 0.227sec y

y

M T

K

=π = (ASCE/SEI 7-16 Eq. 19.3-6)

* *2

2 0.757sec xx

xx xx

M h T

K

=π =

α

 (ASCE/SEI 7-16 Eq. 19.3-7) 

The translational damping coefficient, βy and the rotational damping 

coefficient, βxx, are now computed according to ASCE/SEI 7-16 Equations 

19.3-10 and 19.3-12, respectively. 

  4(L B)   a β = 0

y   =   0.099

    Ky GB   2

 (ASCE/SEI 7-16 Eq. 19.3-10) 

(4ψ 3)(L B) a2

0  a0  βxx =   = 3.827×10−4

  2α  K    

 xx 0.4  xx

  −  3 2.2 2

( B)

3 + a  GB   L 0

   

 (ASCE/SEI 7-16 Eq. 19.3-12) 

Note that the translational damping is a much more significant effect than the 

rotational damping (βy = 0.099 > βxx = 3.287 × 10-4), although the elastic 

behavior of the building is more significantly affected by the rotation. The 

radiation damping ratio, βrd, is computed according to ASCE/SEI 7-16 

Equation 19.3-5. 
A-26 A: Two-Story Building Example FEMA P-2091

1 1 4

2 2 ~ ~

0.018 7.75 10 0.019 rd y xx

TT TT y xx

− β = β+ β = + × =

        

 (ASCE/SEI 7-16 Eq. 19.3-5) 

Next, the foundation damping ratio, βf, is computed and includes both the 

radiation damping computed here and the hysteretic damping (Section 

A.6.1), according to ASCE/SEI 7-16 Equation 19.3-4, and the effective 

damping ratio, β0, according to ASCE/SEI 7-16 Equation 19.3-1. 

2 ~

2 ~

1

0.046 f s rd

T T

T T

        −   β =  β +β =            

 

 (ASCE/SEI 7-16 Eq. 19.3-4) 

0 2 ~

0.091 f

eff

T T

β β =β + =

     

 

 (ASCE/SEI 7-16 Eq. 19.3-1) 

(must not exceed 0.20) 

where β is the effective viscous damping ratio of the structure, taken as 5% 

as recommended in ASCE/SEI 7, and the effective period lengthening ratio is 

computed above. 

A.6.3 Adjusted Base Shear

With this effective damping ratio, the response adjustment factor, BSSI, is 

computed from ASCE/SEI 7-16 Equation 19.2-4, the reduction in base shear, 

∆V, from ASCE/SEI 7-16 Equation 19.2-2, and the adjusted base shear from 

ASCE/SEI 7-16 Equation 19.2-1. 

( ) 0

4 1.181

5.6 ln 100

BSSI = =  − β  

 (ASCE/SEI 7-16 Eq. 19.2-4) 

For ease of comparison, the reduction in seismic response coefficient, ∆Cs, is 

computed rather than computing the reduction in base shear, ∆V. 

~

0.196 0.196 0.196 0.166 0.03

1.181

s

s S

SSI

C C C

B

∆= − = − = − = 

 (modified ASCE/SEI 7-16 Eq. 19.2-2) 

Here Cs is the fixed-base response coefficient, computed in Section A.4.1, ~

and Cs is the value of the response coefficient computed based on the 
FEMA P-2091 A: Two-Story Building Example A-27

flexible-base period. In this case, the value is the same as that computed at 

the fixed-base period, since all fundamental periods are within the flat top 

(acceleration controlled) region of the response spectrum. 

Accordingly, the updated value of Cs is computed as 0.196 - 0.03 = 0.166. 

This value is 87% of the ASCE/SEI 7-16 Chapter 12 value of 0.196. Similar 

reductions are computed for the other directions and stiffness values, as 

indicated in Table A-15. 

Table A-15 Tabulation of Reduced Seismic Response

Model Cs Transverse (g) Cs Longitudinal (g) 

Fixed base (for comparison) 0.196

Upper bound springs 0.168 0.175

Best estimate springs 0.166 0.172

Lower bound springs 0.162 0.167

A.6.4 Code-limited Adjusted Base Shear

The largest value of Cs computed is for the upper bound soil springs in the 

longitudinal direction, which is 0.175g. In the transverse direction, the 

largest value is 0.168g. ASCE/SEI 7-16 Section 19.2.1 places limitation on 

the reduction in demand. For structures with R factors greater than or equal 

to 6, the adjusted response coefficient must be at least 90% of the ASCE/SEI 

7-16 Chapter 12 value: (0.9) (0.196g) = 0.177g. 

A.7 Design for SSI-Reduced Base Shear per ASCE/SEI 7-16 

Chapter 19

Now that the reduced base shear is computed, the structure is redesigned 

employing the lower design forces. This will result in smaller brace areas, 

and narrower foundations. Following that redesign, the reductions in seismic 

response coefficient are recomputed to compare with the starting point of 

0.177. If the resulting value is smaller than this value, the reduced sizes 

remain applicable. If the resulting seismic response coefficient is larger than 

0.177, a further iteration is required with the larger coefficient and larger 

resulting sizes. 

A.7.1 Brace Design

Brace areas are computed in the same manner as in Section A.4.2. Tables 

A-16 and A-17 show the brace sizes required for the reduced seismic 

demands. Column sizes remain unchanged, as they were already the smallest 

compact W10 sections available. 
A-28 A: Two-Story Building Example FEMA P-2091

Table A-16 Transverse Brace Core Areas

Level

Number of 

braces

Brace 

Force 

(kip) 

Computed 

Core Area 

(in2) 

Specified 

Core Area 

(in2) 

Below roof 8 92 2.69 2.75

Below second floor 8 139 4.07 4.25

Table A-17 Longitudinal Brace Core Areas

Level

Number of 

braces

Brace 

Force 

(kip) 

Computed 

Core Area 

(in2) 

Specified 

Core Area 

(in2) 

Below Roof 12 61 1.79 2.00

Below Second Floor 12 93 2.71 2.75

In general, the brace areas are reduced in rough proportion to the seismic 

demand. However, due to rounding, the braces below the roof in the 

longitudinal direction remain the same size.

A.7.2 Foundation Design

With the reduced demands, the overturning moment reduces, but the gravity 

load remains the same, so the effect on the foundation sizes is not a linear 

reduction. The foundation sizes are determined in the same manner as in 

Section A.4.3, above. The lengths and depths of the foundations remain the 

same. Trial and adjustment result in width reductions from 8 ft to 7 ft for the 

transverse frame and from 5.5 ft to 4 ft for the longitudinal frame. These 

reduced widths are the minimum sizes (rounded up to the nearest half foot) 

that produce acceptable soil pressures.

A.7.3 Fixed-base Modal Analysis

The smaller brace sizes result in a softer structure. The primary modal 

period in the transverse direction increases from 0.466 s to 0.489 s. The 

primary modal period in the longitudinal direction increases from 0.459 s to 

0.483 s. 

A.7.4 Flexible-base Modal Analysis

The vertical linear springs are not dependent on the footing width, so those 

values do not change with the narrower foundations. The lateral spring 

values include the passive pressure, which does depend on the foundation 

sizes, but the widths of the frame foundations are not a significant factor. 

The base shear is lower, however, and this affects the lateral spring 

properties by reducing the anticipated displacement. Table A-18 shows the 
FEMA P-2091 A: Two-Story Building Example A-29

updated spring values used in the flexible-base analysis. Table A-19 shows 

the updated flexible-base fundamental periods.

Table A-18 Updated Spring Values

Spring Lower Bound Best Estimate Upper Bound

Vertical linear springs 3.47 kip/in/in

(unchanged) 

6.94 kip/in/in

(unchanged) 

10.42 kip/in/in

(unchanged) 

Transverse lateral springs 

(each building corner)

32,079 kip/in 64,158 kip/in 96,238 kip/in

Longitudinal lateral springs 

(each building corner)

21,406 kip/in 42,811 kip/in 64,217 kip/in

Table A-19 Updated Fundamental Periods for Fixed-base and Flexiblebase Models

Model

Transverse first 

mode period (s) 

Longitudinal first 

mode period (s) 

Fixed base (for comparison) 0.489 0.483

Flexible-base upper bound 0.531 0.515

Flexible-base best estimate 0.552 0.530

Flexible-base lower bound 0.610 0.573

A.7.5 Check of Base Shear

Based on the updated foundation sizes and dynamic properties, recompute 

the damping properties and resulting seismic response coefficient using the 

same approach outlined in Section A.6, above. The resulting seismic 

response coefficients are shown in Table A-20. 

Table A-20 Tabulation of Reduced Seismic Response

Model

Reduced Cs

Transverse 

(g) 

Reduced Cs

Longitudinal 

(g) 

Upper bound springs 0.171 0.177

Best estimate springs 0.171 0.175

Lower bound springs 0.164 0.169

The highest value of the seismic response coefficient computed with the 

reduced brace sizes and the reduced foundation widths is the same as our 

starting value of 0.177g, indicating that no further iteration is required and 

the design including soil-structure interaction is acceptable.
A-30 A: Two-Story Building Example FEMA P-2091

A.8 Design for Full Theoretical SSI-Reduced Base Shear 

This section explores extending the reductions in design loads beyond those 

permitted by the ASCE/SEI 7. This allows larger reductions due to 

foundation damping as well as the further reduction due to base slab 

averaging, which is not permitted by ASCE/SEI 7-16 for linear analyses. 

ASCE/SEI 41 

Note that the restrictions in 

ASCE/SEI 7 are not present 

in ASCE/SEI 41. 

Accordingly, the reductions 

computed in this section 

would be applicable for an 

ASCE/SEI 41 analysis of an 

existing structure. 

A.8.1 Base Slab Averaging

Base slab averaging is one of the two kinematic soil-structure interaction 

effects covered in ASCE/SEI 7-16 Section 19.4. The embedment effect, 

addressing buildings with basements, is not addressed in this example, but is 

covered in Appendix B. 

Base slab averaging is an effect that addresses the incoherence of ground 

motions over a building’s footprint. If the building base is large enough and 

the foundation system has sufficient interconnection and lateral rigidity, the 

effect of earthquake ground motions will be reduced. This effect is more 

pronounced for buildings with shorter periods of vibration and for those with 

larger base extents. These two quantities alone determine the reduction due 

to base slab averaging, termed RRSbsa. 

The effective foundation size, be, depends on the building base area Abase, 

which is the extent of the ground-level slab. 

Abase = (150 ft)(210 ft) = 31,500 ft 2

b A e = base =177 ft

 (ASCE/SEI 7-16 Equation 19.4-4) 

The computed value of be shall not exceed 260 ft. ASCE/SEI 7-16 requires 

that the shear wave velocity of the soil, vs, be in the range of 650 to 1,650 

ft/s. In this case, vs is taken as the average value over a depth equal to the 

effective foundation size, be. Referring back to Figure A-1, the average 

shear wave velocity for this site is clearly within that range. 

The reduction due to base slab averaging is dependent on be and on the 

building period. The reduction is computed according to ASCE/SEI 7-16 

Equations 19.4-1, 19.4-2, and 19.4-3. The values shown below pertain to the 

flexible-base period in the transverse direction for the best estimate soil 

springs (T = 0.532 s). 

  b b e

0 = 0.00071  = 0.236   T

 (ASCE/SEI 7-16 Equation 19.4-3) 
FEMA P-2091 A: Two-Story Building Example A-31

Where the period, T, shall not be taken as less than 0.2 s. ASCE/SEI 7-16 

Equation 19.4-2 for Bbsa, has two forms, depending on the value of b0. The 

first form (for b0 ≤ 1) is applicable:

6 8 10

2 4 000

0 0 1 1.06

2 4 12 bsa

bbb B bb =+ + + + + =

(ASCE/SEI 7-16 Eq. 19.4-2(a))

The second form, (for b0 > 1) is 

( ) 2

0 2

0 0

1 1 exp 2 1 16

B b bsa b b

    =     −

    π  

(ASCE/SEI 7-16 Eq. 19.4-2(b)) 

The reduction is computed as follows.

( ) ( ) 2

2 0

0

1 0.25 0.75 1 exp 2 0.98 RRSbsa bsa b B

b

 

=+ − − =        

(ASCE/SEI 7-16 Eq. 19.4-1) 

This reduction is applied to the value of Cs in Section A.6.3 of 0.166g. The 

updated value of Cs is then (0.98)(0.166g) = 0.163g. Table A-21 presents the 

values of Cs for each direction and each soil spring estimate for with and 

without the base slab averaging reduction. 

Table A-21 Tabulation of Reduced Seismic Response with and without 

Base Slab Averaging

Model

Cs 

Transverse 

(No BSA) (g) 

Cs 

Transverse 

(w/ BSA) (g) 

Cs

Longitudinal 

(No BSA) (g) 

Cs

Longitudinal 

(w/ BSA) (g) 

Fixed base

(for comparison) 0.196

Upper bound 

springs 0.168 0.164 0.175 0.164

Best estimate 

springs 0.166 0.163 0.172 0.164 

Lower bound 

springs 0.162 0.159 0.167 0.159

Note that the reductions due to base slab averaging are more significant in 

the longitudinal direction, due to the shorter periods. The upper bound 

spring estimates result in the largest values of Cs, 0.164g. 
A-32 A: Two-Story Building Example FEMA P-2091

Figure A-9 shows the base slab averaging reductions for three values of be: 

100 ft, 177 ft (this example), and 260 ft (maximum permitted). Note that the 

reductions are limited to the values computed at periods of 0.2 s. 

Figure A-9 Reductions for base slab averaging relative to building period for 

three values of effective foundation size. 

As in Section A.7, above, redesigning the structure employing the lower 

design forces will result in smaller brace areas, and narrower foundations. 

Following that redesign, the reductions in seismic response coefficient are 

recomputed to compare with our starting point of 0.164g. If the resulting 

value is smaller than this value, the reduced sizes remain applicable. If the 

resulting seismic response coefficient is larger than 0.164g, a further iteration 

is required with the larger coefficient and larger resulting sizes. 

A.8.2 Brace Design

Brace areas are computed in the same manner as in Section A.4.2. Tables 

A-22 and A-23 show the brace sizes required for the reduced seismic 

demands, with the original (unreduced) sizes and the code-limited reduced 

sizes shown for comparison. 

In general, brace areas are reduced in rough proportion to the seismic 

demand. However, due to rounding, the braces below the second floor in the 

longitudinal direction remain the same size.
FEMA P-2091 A: Two-Story Building Example A-33

Table A-22 Transverse Brace Core Areas

Level

Unreduced 

Core Area 

(in2) 

Code-Limited 

Reduced Core 

Area (in2) 

Fully Reduced 

Core Area (in2) 

Below Roof 3.00 2.75 2.50

Below Second Floor 4.25 4.25 4.00

Table A-23 Longitudinal Brace Core Areas

Level

Unreduced 

Core Area 

(in2) 

Code-Limited 

Reduced Core 

Area (in2) 

Fully Reduced 

Core Area (in2) 

Below Roof 2.00 2.00 1.75

Below Second Floor 3.25 2.75 2.75

A.8.3 Foundation Design

With the reduced demands, the overturning moment reduces, but the gravity 

load remains the same, so the effect on the foundation sizes is not a linear 

reduction. The foundation sizes are determined in the same manner as in 

Sections A.4.3 and A.7.2, above. The foundation plan dimensions remain the 

same as those computed in Section A.7.2 for the code-limited reductions. 

The foundation depths are reduced from 5 feet to 4 feet. 

A.8.4 Fixed-base Modal Analysis

The smaller brace sizes result in a softer structure. The primary modal 

period in the transverse direction increases further from 0.489 s. to 0.506 s. 

The primary modal period in the longitudinal direction increases further from 

0.483 s to 0.496 s. 

A.8.5 Flexible-base Modal Analysis

The lateral spring values are again adjusted based on the lower base shear 

and resulting lower anticipated displacement. Table A-24 shows the updated 

flexible-base fundamental periods.

Table A-24 Updated Fundamental Periods for Fixed-base and Flexiblebase Models

Model

Transverse first 

mode period (s) 

Longitudinal first 

mode period (s) 

Fixed base (for comparison) 0.506 0.496

Flexible-base upper bound 0.545 0.524

Flexible-base best estimate 0.564 0.538

Flexible-base lower bound 0.618 0.577
A-34 A: Two-Story Building Example FEMA P-2091

A.8.6 Check of Base Shear

The seismic response coefficient is recomputed based on the updated 

dynamic properties using the same approach outlined above. The resulting 

seismic response coefficients are shown in Table A-25. 

Table A-25 Tabulation of Reduced Seismic Response

Model

Reduced Cs

Transverse (g) 

Reduced Cs

Longitudinal (g) 

Upper bound springs 0.169 0.168

Best estimate springs 0.167 0.166

Lower bound springs 0.163 0.162

The highest values of the seismic response coefficients computed are 0.168g 

and 0.169g, compared to the starting value of 0.164g, indicating that the 

structure designed for the fully reduced loads no longer produces the same 

amount of reduction in loads. Accordingly, a redesign using the higher 

computed loads is required. 

A.8.7 Redesign (Iteration)

The affected structural elements are redesigned for the higher value of Cs = 

0.169g. Small changes in some elements of the design are required. These 

are an increase in the transverse brace size below the roof to 2.75 in2

 and an 

increase in the transverse foundation depth to 5 ft. (These are the same 

values of these elements as in the SSI-reduced design as limited by 

ASCE/SEI 7.) 

A.8.8 Reanalysis

The fixed-base and flexible-base models are reanalyzed incorporating the 

updated design. The periods in the longitudinal direction are essentially 

unchanged, since the design modifications did not affect these frames. Table 

A-26 shows the updated dynamic properties. 

Table A-26 Updated Fundamental Periods for Fixed-base and Flexiblebase Models

Model

Transverse first 

mode period (s) 

Longitudinal first 

mode period (s) 

Fixed base (for comparison) 0.498 0.496

Flexible-base upper bound 0.537 0.524

Flexible-base best estimate 0.556 0.538

Flexible-base lower bound 0.611 0.578
FEMA P-2091 A: Two-Story Building Example A-35

A.8.9 Check of Base Shear

The seismic response coefficient is recomputed based on the updated 

dynamic properties using the same approach outlined above. The resulting 

seismic response coefficients are shown in Table A-27. 

Table A-27 Tabulation of Reduced Seismic Response

Model

Reduced Cs

Transverse (g) 

Reduced Cs

Longitudinal (g) 

Upper bound springs 0.168 0.168

Best estimate springs 0.166 0.166

Lower bound springs 0.162 0.162

The highest values of the seismic response coefficients computed are 0.168g, 

just slightly lower than the starting value of 0.169g, indicating that the 

modified structure produces the same amount of reduction in loads as used in 

computing the member and foundation sizes. Therefore, this design is 

internally consistent, although it is not permitted by ASCE/SEI 7, as it 

includes base slab averaging reductions with the ELF procedure and does not 

consider the limitation on the amount of design force reduction. 

FEMA P-2091 B: Twelve-Story Building Example B-1

Appendix B

Twelve-Story Building Example

B.1 Overview

This example is based on the structure designed in Chapter 10 of FEMA 

P-1051 (FEMA, 2016), with deep foundations similar to those illustrated in

FEMA P-1051 Section 7.2. This is a 12-story reinforced concrete office

building in Antioch, California with some retail shops on the first floor. The

building is assigned to Risk Category II with seismic importance factor, Ie =

1.0.

Four variations of structural configuration and boundary conditions are 

considered in this example, as follows:

• Model 1: Base case building, with basement and fixed base

• Model 2: Base case building, with basement and flexible base

• Model 3: Building modified by removing the basement, fixed base

• Model 4: Building modified by removing the basement, flexible base

The aspects of soil-structure interaction considered in this example include 

spectral modifications for kinematic SSI (base slab averaging and 

embedment), adjustments to structural demands due to period elongation and 

foundation damping, and the impacts of limitations imposed by ASCE/SEI 

7-16 provisions, including the following:

• Minimum base shear per ASCE/SEI 7-16 Equations 12.8-5 and 

12.8-6

• Consideration of CuTa for T and T ̃

• Coefficient α that limits reduction in base shear caused by foundation

damping

• Analysis procedures permitted to include various aspects of SSI

• Foundation types permitted to use simple damping procedures in

ASCE/SEI 7-16 Section 19.3

• Consideration of bounding stiffnesses for foundation/soil system

Example Summary

Building Type: 12-story 

concrete building with 

moment frame in one 

direction and dual system in 

the other.

Foundation Type: Drilled 

piles.

SSI Topics Covered:

• Spectral modifications for

kinematic interaction,

including base slab

averaging and

embedment effects.

• Adjustments to demands

from period elongation

and foundation damping.

• Impacts of limitations

imposed by ASCE/SEI

7-16 provisions.

B.1.1 Outline of the Example

This example includes the following:
B-2 B: Twelve-Story Building Example FEMA P-2091

• Section B.2: Site description

• Section B.3: Building description

• Section B.4: Initial design (Model 1) 

• Section B.5: Spectral modifications for kinematic SSI

• Section B.6: SSI adjusted structural demands 

• Section B.7: Design variations 

B.2 Site Description

B.2.1 Geotechnical Properties

Table B-1 outlines the geotechnical characteristics of the example site.

Table B-1 Site Characteristics

Depth Geotechnical Parameters

Nominal Unit 

Capacities

0 to 5 ft

Loose sand/fill

effective unit weight, γ = 110 pcf

effective friction angle = 28 deg

soil modulus parameter, k = 25 pci

low-strain shear wave velocity, vso = 580 ft/s

neglect skin friction

neglect end bearing

5 to 30 ft

Soft clay

effective unit weight, γ = 110 pcf

undrained shear strength, su = 700 psf

soil modulus parameter, k = 25 pci

strain at 50% of maximum stress, ε50 = 0.01

low-strain shear wave velocity vso = 700 ft/s

skin friction = 0.40 ksf

neglect end bearing

30 to 100 ft

Medium dense sand

effective unit weight, γ = 120 pcf

effective friction angle = 36 deg

soil modulus parameter, k = 50 pci

low-strain shear wave velocity vso = 1,040 

ft/s

skin friction = 3.0 ksf

end bearing = 100 ksf

Pile cap 

resistance 300 pcf, ultimate passive pressure

Using the equations and definitions in ASCE/SEI 7-16 Section 20.4.1, the 

site is assigned to Site Class D based on vs = 896 ft/s, based on the following 

calculation: 

∑

n

di

i 1 5 25 70

v = + + s =

∑

n = = 896 ft/s di 5 25 70

+ +

i=1 vsi 580 700 1,040
FEMA P-2091 B: Twelve-Story Building Example B-3 

B.2.2 Seismic Ground Motion Parameters

For this example site, located at 38.0021 ̊N, 121.7976 ̊W, the mapped ground 

motion parameters are SS = 1.537, S1 = 0.525, and TL = 8 seconds. Based on 

ASCE/SEI 7-16 Sections 11.4 and 21.3, the short-period and long-period site 

coefficients are Fa = 1.0 and Fv = 2.5. 

Using the site-specific ground motion procedures of ASCE/SEI 7-16 Chapter 

21 (not shown in detail here), the following design acceleration parameters, 

which satisfy the minima prescribed in Section 21.3, are determined as SDS = 

1.0 and SD1 = 0.7. For the purpose of this example, the free-field, surface, 

design response spectrum is taken as the generalized spectral shape 

consistent with these values. Figure B-1 compares the design spectra 

computed in accordance with ASCE/SEI 7-16 Section 21.3 to the sitespecific, free-field spectrum. It may be seen that the site-specific, free-field 

spectrum does not fall below the minimum spectrum defined in Section 21.3. 

The site-specific spectrum is used for all designs in this appendix.

Figure B-1 Design response spectra for the example building.

B.3 Building Description

B.3.1 Framing Systems

Gravity. The basic structural configuration is shown in Figures B-2 and 

B-3. The building has 12 stories above grade and one basement level. The

section of a variation on this building, with no basement, is shown in Figure 

B-3b. The typical bays are 30 feet long in the north-south (N-S) direction 

and either 40 or 20 feet long in the east-west (E-W) direction.
B-4 B: Twelve-Story Building Example FEMA P-2091

Figure B-2 Typical floor plan of the example building (after FEMA P-1051, 

2016).
FEMA P-2091 B: Twelve-Story Building Example B-5 

(a) Section for building (b) Section for building 

with basement without basement

Figure B-3 Typical sections of the example building (after FEMA P-1051, 

2016).

The main gravity framing system consists of seven continuous 30-foot spans 

of pan joists. These joists are spaced at 36 inches on center and have an 

average web thickness of 6 inches and a depth below slab of 16 inches. Due 

to fire code requirements, a 4-inch-thick floor slab is used, giving the joists a 

total depth of 20 inches. The joists are supported by concrete beams running 

in the E-W direction. The building is constructed of normalweight concrete.

12-inch-thick concrete walls are located around the entire perimeter of the 

basement level.

Lateral. The seismic force-resisting system in the N-S direction consists of 

four 7-bay special moment-resisting frames. In the E-W direction, the 

seismic force-resisting system is a dual system composed of a combination of 

moment frames and walls integrated into a moment-resisting frame. Along 

Grids 1 and 8, the frames have five 20-foot bays. Along Grids 2 and 7, the 

frames consist of two exterior 40-foot bays and one 20-foot interior bay. At 

Grids 3, 4, 5, and 6, the interior bay consists of shear walls infilled between 

the interior columns. The exterior bays of these frames are similar to Grids 2 

and 7. Hereafter, frames are referred to by their gridline designation (e.g., 
B-6 B: Twelve-Story Building Example FEMA P-2091

Frame 1 is located on Grid 1). Table B-2 summarizes the response 

modification coefficient, R, overstrength factor, Ω0, and deflection 

amplification factor, Cd, for the lateral framing system. Figure B-4 shows 

the analytical model for the base building. 

Table B-2 Response Modification Coefficient, Overstrength Factor, and 

Deflection Amplification Factor for Structural Systems Used

Response 

Direction Building Frame Type R Ω0 Cd

N-S Special moment frame 8 3 5.5

E-W Dual system incorporating special moment 

frame and special shear wall 7 2.5 5.5

Figure B-4 Analytical model for base case building (Model 1). 

Foundation. Figure B-5 shows the foundation plan for this example 

building. The deep foundations consist of a 4-pile group at each column or 

shear wall boundary element. ASCE/SEI 7-16 Section 12.13.7.2 requires ties 

between individual pile groups, which are provided in this example by a 

system of tie beams; it would also be permitted to design the basement slab 

to perform that function. 

Each pile group consists of four 22-inch diameter drilled shafts, located 5’-6” 

on center in each direction, extending at least 3 inches into a 9’-2”×9’-2”×4’-
FEMA P-2091 B: Twelve-Story Building Example B-7 

0” deep pile cap, as illustrated in Figure B-6. The top of the pile cap is 

located 1’-0” below top of slab on grade.

Figure B-5 Foundation plan for the example building. 
B-8 B: Twelve-Story Building Example FEMA P-2091

Figure B-6 Arrangement of piles and columns or 

boundary elements at each pile group 

(after FEMA P-1051, 2016).

B.3.2 Material Properties and Modeling Criteria

Normalweight concrete of 5,000 psi strength is used everywhere. All 

reinforcement has a specified yield strength of 60 ksi. 

The building is analyzed with a three-dimensional mathematical model using 

the ETABS software (CSI, 2019). Typical practice is to use a reduced 

moment of inertia for the beams, columns, and walls based on the expected 

level of cracking. This example utilizes the following effective moments of 

inertia, Ieff, which are slightly different from those in ACI 318: 

• Beams: Ieff = 0.3Igross

• Columns: Ieff = 0.5Igross

• Walls: Ieff = 0.5Igross

The effective stiffness of the moment frame elements is based on the 

recommendations in NIST GCR 16-917-40 (NIST, 2016) and ASCE/SEI

41-17 Table 10-5 and account for the expected axial loads and reinforcement 

levels in the members. The value for the structural walls is based on the 

recommendations in NIST GCR 11-917-11 (NIST, 2011) and ASCE/SEI

41-17 Table 10-5 for cracked concrete structural walls.

The following are other significant aspects of the mathematical model that 

should be noted: 

• The structure is modeled with 12 levels above grade and one level below 

grade. The perimeter basement walls are modeled as shear panels as are 

the main structural walls. For the primary analyses, the walls are 

assumed to be fixed against translation at nodes located at the center of 

each boundary element at the basement level. Piles, pile caps, and tiebeams are not included in the analytical model. A second series of 

analyses are performed using foundation springs at the base. 
FEMA P-2091 B: Twelve-Story Building Example B-9 

• All floor diaphragms are modeled as infinitely rigid in plane and 

infinitely flexible out-of-plane, consistent with common practice for a 

regular-shaped concrete diaphragm.

• Beams, columns, and structural wall boundary members are represented 

by two-dimensional frame elements. The beams are modeled as 

T-beams using the effective slab width per ACI 318. 

• The structural walls are modeled as a combination of boundary columns 

and shear panels with composite stiffness.

• Beam-column joints are modeled in accordance with NIST GCR 

16-917-40, which references the procedure in ASCE/SEI 41-17. Both 

the beams and columns are modeled with end offsets based on the 

geometry, but the beam offset is modeled as 0 percent rigid, while the 

column offset is modeled as 100 percent rigid. This provides effective 

stiffness for beam-column joints consistent with the expected behavior of 

the joint: strong column-weak beam condition.

• P-delta effects are neglected in the analysis for simplicity. This 

assumption was verified, but it is not shown in this example.

• Although the base of the model is located at the basement level, the 

seismic base for determination of forces is assumed to be at the first 

floor, which is at the exterior grade. As a result, Table B-3 neglects the 

weight at Level 1. 

B.3.3 Gravity Loads and Seismic Mass

In addition to the building structural weight, the following superimposed 

dead loads are assumed:

• Roofing = 10 psf 

• Partition = 10 psf 

• Ceiling and M/E/P = 10 psf 

• Curtain wall cladding = 10 psf (on vertical surface area) 

Building mass includes contributions from the various members of the 

seismic force-resisting system, as sized in the source example in FEMA 

P-1051. It is assumed that all columns and structural wall boundary elements 

are 30 inches by 30 inches, beams are 24 inches wide by 32 inches deep 

(including slab thickness), and the panel of the structural wall is 16 inches 

thick. It has already been established that pan joists are spaced at 36 inches 

on center, have an average web thickness of 6 inches, and are 20 inches deep, 

inclusive of the 4-inch-thick slab. 
B-10 B: Twelve-Story Building Example FEMA P-2091

Based on the above member sizes and superimposed dead load, the 

individual story weights and masses are listed in Table B-3. 

Table B-3 Story Weights and Masses

Level Weight (kip) Mass (kip-s2/inch)

Roof 3,352 8.68 

12 3,675 9.52 

11 3,675 9.52 

10 3,675 9.52 

9 3,675 9.52 

8 3,675 9.52 

7 3,675 9.52 

6 3,675 9.52 

5 3,675 9.52 

4 3,675 9.52 

3 3,675 9.52 

2 3,817 9.88 

Total 43,919 113.75 

In the ETABS model, these masses are applied as uniform distributed masses 

across the extent of the floor diaphragms in order to provide a realistic 

distribution of mass in the dynamic model. The structural framing is 

modeled using massless elements since their mass is included with the floor 

mass.

B.4 Initial Design (Model 1)

B.4.1 Base Shear per Chapter 12 ELF

The seismic base shear for structural design is determined in accordance with 

ASCE/SEI 7-16 Section 12.8.1, and is taken as the product of the seismic 

weight, W, and the seismic response coefficient, Cs. That base shear is used 

directly for the equivalent lateral force (ELF) procedure and for scaling the 

results of modal response spectrum analysis (MRSA).

ASCE/SEI 7-16 Equations 12.8-2, 12.8-3, and 12.8-4 define seismic 

response coefficients that correspond to the constant acceleration, constant 

velocity, and constant displacement portions of the design response 

spectrum, respectively. ASCE/SEI 7-16 Equations 12.8-5 and 12.8-6 set 

forth code minima that are independent of both the dynamic characteristics of 

the system and the basic spectral shape.
FEMA P-2091 B: Twelve-Story Building Example B-11

The SSI procedures set forth in ASCE/SEI 7-16 Chapter 19 are intended to 

modify the shape of the design spectrum (due to base slab averaging and 

embedment) or better reflect the dynamic characteristics of the system (due 

to period elongation and foundation damping). Although the adjustment of 

structural demands to reflect SSI in accordance with ASCE/SEI 7-16 Section 

19.2 is expressed as modifications to the base shear for the fixed-base 

structure computed in accordance with ASCE/SEI 7-16 Section 12.8.1, it is 

not intended that SSI modify the code minima reflected in ASCE/SEI 7-16 

Equations 12.8-5 and 12.8-6. 

Base Shear in the N-S Direction. The fundamental period in the N-S 

direction from modal analysis is 2.18 seconds. 

The approximate period is computed using ASCE/SEI 7-16 Equation 12.8-7, 

with values of Ct and x taken from ASCE/SEI 7-16 Table 12.8-2 for concrete 

moment-resisting frames resisting 100% of the required seismic force, as: 

T x 0.016(161)

0.9

a = C ht n = =1.55 sec

The upper limit period is computed in accordance with ASCE/SEI 7-16 

Section 12.8.2 as: 

C Tu a =1.4× = 1.55 2.17 sec

Since this period is shorter than that computed in the analysis, the upper limit 

period is used in subsequent calculations.

The seismic response coefficient (for the constant acceleration portion of the 

design spectrum) is computed using ASCE/SEI 7-16 Equation 12.8-2 as 

follows: 

1.0 0.125

8

1.0

DS

s

e

S C

R

I

= = =          

However, it need not be taken greater than the value computed using 

ASCE/SEI 7-16 Equation 12.8-3 or 12.8-4, as applicable. Since T = 2.17 

seconds < TL: 

1 0.7 0.040

8 2.17

1.0

D

s

e

S C

R T

I

= = =           

The minimum seismic response coefficient is computed using ASCE/SEI 

7-16 Equation 12.8-5 as follows:

Cs = 0.044S I DS e = 0.044(1.0)(1.0) = 0.044
B-12 B: Twelve-Story Building Example FEMA P-2091

This minimum controls, so the seismic base shear in the N-S direction is 

computed using ASCE/SEI 7-16 Equation 12.8-1 as follows:

VN S− = C Ws = 0.044(43,919) =1,932 kip

Base Shear in the E-W Direction. The fundamental period in the E-W

direction from modal analysis is 1.49 seconds. 

The approximate period is computed using ASCE/SEI 7-16 Equation 12.8-7, 

with values of Ct and x taken from ASCE/SEI 7-16 Table 12.8-2 for “all 

other structural systems” since the concrete moment-resisting frames resist 

less than 100% of the required seismic force in this direction, as: 

T x = 0.020(161)

0.75

a = C ht n = 0.90 sec

The upper limit period is computed in accordance with ASCE/SEI 7-16 

Section 12.8.2 as: 

C Tu a =1.4× 0.090 =1.27 sec

Since this period is shorter than that computed in the analysis, the upper limit 

period is used in subsequent calculations.

The seismic response coefficient (for the constant acceleration portion of the 

design spectrum) is computed using ASCE/SEI 7-16 Equation 12.8-2 as 

follows: 

1.0 0.143

7

1.0

DS

s

e

S C

R

I

= = =          

However, it need not be taken greater than the value computed using 

ASCE/SEI 7-16 Equation 12.8-3 or 12.8-4, as applicable. Since T = 1.27 

seconds < TL: 

1 0.7 0.079

7 1.27

1.0

D

s

e

S C

R T

I

= = =           

The minimum seismic response coefficient is computed using ASCE/SEI 

7-16 Equation 12.8-5 as follows:

Cs = 0.044S I DS e = 0.044(1.0)(1.0) = 0.044

The seismic response coefficient (for the constant velocity portion of the 

design spectrum) from ASCE/SEI 7-16 Equation 12.8-3 controls, so the 
FEMA P-2091 B: Twelve-Story Building Example B-13

seismic base shear in the E-W direction is computed using ASCE/SEI 7-16 

Equation 12.8-1 as follows:

VE W− = C Ws = 0.079(43,919) = 3,470 kip

B.4.2 Fixed-Base Modal Analysis

Where ELF is used, the base shears computed above would be distributed 

vertically in accordance with ASCE/SEI 7-16 Section 12.8.3. Where MRSA

is used, these base shears are used for scaling design forces in accordance 

with ASCE/SEI 7-16 Section 12.9.1.4.1. That section requires scaling 

MRSA results to 100% of the ELF base shear; earlier editions of ASCE/SEI 

7-16 permitted scaling to 85% of the ELF base shear.

Although the base shear is identical for both methods, MRSA typically results 

in smaller overturning moments and design story drifts. The following sections 

compare ELF and MRSA story shears, story moments, and story drifts. 

B.4.3 Story Shears and Moments

Figure B-7 compares story shears and moments for the base building (Model 

1) analyzed using ELF and MRSA. The vertical force distribution prescribed 

for the ELF procedure tends to produce story shears at mid-height and story 

moments at the base that are slightly larger than those computed using MRSA.

B.4.4 Story Drifts

For drift checks, the minimum seismic response coefficient computed using 

ASCE/SEI 7-16 Equation 12.8-5 does not apply, and periods are not limited 

to CuTa. Design story drifts are taken as the drift from the elastic analysis 

multiplied by Cd. In the N-S (moment frame) direction, ELF produces drifts 

at mid-height that are slightly larger than those from MRSA. In the E-W 

(dual system) direction, ELF drifts are nearly 50% larger than those from 

MRSA. Proportioning for strength demands controls the design in both 

directions, so drifts are well below the drift limit.

B.4.5 Foundation Design

For typical deep foundation systems resistance to lateral loads is provided by 

both piles and pile cap. Although the behavior of foundation and 

superstructure are closely related, they typically are modeled independently. 

Earthquake loads are applied to a model of the superstructure, which is 

assumed to have fixed or spring supports. Then the support reactions are 

seen as demands on the foundation system. A similar substructure technique 

is usually applied to the foundation system itself, whereby the behavior of 

pile cap and piles are considered separately.
B-14 B: Twelve-Story Building Example FEMA P-2091

Figure B-7 Story shears and moments for Model 1, ELF vs. MRSA. 

The response of individual piles to lateral loads is highly nonlinear. It has 

become increasingly common to consider that nonlinearity directly. Based 

on extensive testing of full-scale specimens and small-scale models for a 

wide variety of soil conditions, researchers have developed empirical 

relationships for the nonlinear p-y response of piles that are suitable for use 

in design. The p-y response is sensitive to pile size; soil type and properties; 

and, in the case of sands, vertical stress, which increases with depth. Pile 

response to lateral loads is usually computed using computer programs such 

as LPile (ENSOFT, 2018). For this example, LPile is used to analyze 

22-inch diameter piles for the soil layering shown in Table B-1. 
FEMA P-2091 B: Twelve-Story Building Example B-15

Figure B-8 Story drifts for Model 1, ELF vs. MRSA. 

Pile caps contribute to the lateral resistance of a pile group in two important 

ways: directly as a result of passive pressure on the face of the cap that is 

being pushed into the soil mass and indirectly by producing a fixed head 

condition for the piles, which can significantly reduce displacements for a 

given applied lateral load, thus improving economy of the design. Similar to 

the p-y response of piles, the passive pressure resistance of the cap is 

nonlinear. 

The response of a group of piles to lateral loading will differ from that of a 

single pile due to pile-soil-pile interaction. In this example p-multipliers of 

0.5 are used for group effects on these shafts spaced 3 diameters center to 

center.
B-16 B: Twelve-Story Building Example FEMA P-2091

Figure B-9 shows the shear, moment, and displacement versus depth 

response for an applied lateral load of 32 kip. 

Figure B-9 Lateral pile analysis results vs. depth (applied lateral load is 32 

kip). 

Group stiffness includes four drilled shafts plus passive pressure on the 

leading face. Passive pressure mobilization is computed in accordance with 

ASCE/SEI 41-17 Figure 8-6, which can also be computed using the 

following expression, which is fitted to the curve in the figure: 

P   δ 0.43

= 0.15 + 2.88  Pult   H

As a point of reference, for a lateral displacement of 0.33 inches the total 

lateral load is 144 kip, including 32 kip in each of four drilled shafts and 16 

kip of passive pressure resistance. Figure B-10 shows the total lateral load 

versus head displacement for eight points in a range of loads. Pult is 

computed considering that the top of cap is 1 ft below the top of the slab on 

grade. Although the response is slightly nonlinear, a straight line provides a 

reasonable fit that can be used to estimate lateral stiffness. In this case the 

central value can be estimated as 425 kip/inch, with reasonable lower and 

upper bounds of 300 kip/inch and 600 kip/inch, based on field measurements 

of dynamic soil properties. These lower- and upper-bound variations, which 

are required by ASCE/SEI 7-16 Section 12.13.3, are discussed in more detail 

in Section B.7.3. 
FEMA P-2091 B: Twelve-Story Building Example B-17

Figure B-10 Pile group lateral stiffness for Models 1 and 2. 

B.5 Spectral Modifications for Kinematic SSI

The average effective shear wave velocity used in SSI calculations is based 

on a reduction of the average low strain shear wave velocity due to 

seismically induced demands. Using ASCE/SEI 7-16 Table 19.3-1 for Site 

Class D with SDS/2.5 = 1.0/2.5 = 0.4, the effective shear wave velocity ratio, 

νs/νso = 0.71. 

B.5.1 Base Slab Averaging

Spectral modifications due to base slab averaging depend on the area of 

interconnected foundation. For this example,

L = half the larger dimension of the base of the structure

 = 212.5 ft / 2 = 106.25 ft 

B = half the smaller dimension of the base of the structure

 = 102.5 ft / 2 = 51.25 ft 

Abase = the area of the base of the structure

 = 2L×2B = 202.5 × 102.5 = 21,781 ft2

Using ASCE/SEI 7-16 Equation 19.4-4, 260 ft 147.6 ft e base b A =≤= . 

The vs limits in ASCE/SEI 7-16 Section 19.4.1 should be based on the 

average effective shear wave velocity in Section 19.3 computed over a depth 
B-18 B: Twelve-Story Building Example FEMA P-2091

of B below the base of the structure. Using the values in Table B-1, the 

average low strain shear wave velocity over a depth of B = 51.25 ft below the 

base of the structure, νso, is 791 ft/s, which is computed using ASCE/SEI 

7-16 Equation 20.4-1 as follows:

1

1

5 25 21.25 791 ft/s 5 25 21.25

580 700 1,040

n

i

i

so n

i

i si

d

d

=

=

+ +

ν= = =

+ +

ν

∑

∑

The average effective shear wave velocity, νs = 0.71 × 791 ft/s = 562 ft/s. 

This value is less than the lower limit shown in the inequality immediately 

after ASCE/SEI 7-16 Equation 19.4-4 (650 ft/s). Although that inequality 

indicates the range of effective shear wave velocity considered in the source 

research on base slab averaging, since νs is not used in computing base slab 

averaging there is no consequence for this structure that otherwise satisfies 

the charging requirements of Section 19.4.1.

ASCE/SEI 7-16 Equations 19.4-1 through 19.4-3 are used to compute the 

period-dependent modification factor RRSbsa. The period used in this 

calculation may not be taken less than 0.20 s per ASCE/SEI 7-16 Section 

19.4.1, which produces a floor on the spectral reduction. Chapter 4 of this 

Guide illustrates the process of computing modification factors for base slab 

averaging. 

B.5.2 Embedment

Spectral modifications due to embedment depend on the depth of embedment 

(subject to a maximum of 20 ft per ASCE/SEI 7-16 Section 19.4.2) and the 

average effective shear wave velocity over the embedment depth (which is 

the depth of basement for this condition). For this example,

 e = foundation embedment depth = 15 ft

Using the values in Table B-1, the average low strain shear wave velocity 

over the embedment depth of the foundation (e = 15 ft), νso, is 655 ft/s, 

which is computed using ASCE/SEI 7-16 Equation 20.4-1 as follows:

1

1

5 10 655 ft/s 5 10

580 700

n

i

i

so n

i

i si

d

d

=

=

+

ν= = =

+

ν

∑

∑

The average effective shear wave velocity, νs = 0.71 × 655 ft/s = 465 ft/s. 

This value is less than the lower limit of 650 ft/s, so a value of νs = 650 ft/ is 

used in computing the modification factor for embedment.
FEMA P-2091 B: Twelve-Story Building Example B-19

ASCE/SEI 7-16 Equation 19.4-5 is used to compute the period-dependent 

modification factor RRSe. The period used in this calculation may not be 

taken less than 0.20 s, which produces a floor on the spectral reduction. 

Chapter 5 of this Guide illustrates the process of computing modification 

factors for embedment.

Figure B-11 shows the kinematic SSI spectral modification factors RRSbsa

and RRSe as well as the free-field and modified design response spectra. As 

expected kinematic SSI has substantial effects only in the short period range.

Figure B-11 Kinematic SSI spectral modification factors and design response 

spectra for the example building. 

According to ASCE/SEI 7-16 Section 19.2.3 Item 3, the site-specific 

response spectrum modified for kinematic SSI cannot be taken less than 80% 

of the site-specific free-field response spectrum. Where the product of 

RRSbsa and RRSe is less than 0.8 (periods between 0 and 0.23 s), this limit 
B-20 B: Twelve-Story Building Example FEMA P-2091

controls. Although ASCE/SEI 7-16 Section 19.4 specifies that the product of 

RRSbsa and RRSe not be taken less than 0.7, Section 19.2.3 Item 3 (which 

limits that product to 0.8) always controls. ASCE/SEI 7-16 Section 19.2.3 

Item 4 requires that the site-specific response spectrum modified for 

kinematic interaction not be taken less than 70% of the response spectrum 

computed in accordance with ASCE/SEI 7-16 Chapter 11 (using SDS = 1.025 

and SD1 = 0.875 per ASCE/SEI 7-16 Section 21.3, for the mapped values and 

tabulated site response factors in this case). Figure B-11b shows this limit, 

confirming that the 70% criterion is satisfied.

Although the process illustrated above results in design spectra modified for 

kinematic SSI, those spectra may not be used for design using the ELF 

procedure, MRSA, or linear response history analysis. ASCE/SEI 7-16 

Section 19.4 clearly limits use of the design spectra modified for kinematic 

SSI to nonlinear response history analysis, where they may serve as target 

spectra for ground motion selection and scaling. 

B.6 SSI Adjusted Structural Demands

B.6.1 Period Elongation Due to Flexible Base

Since spectral acceleration generally reduces at longer periods (particularly

in the constant velocity and constant displacement portions of the spectrum), 

accounting for period elongation due to foundation flexibility can reduce 

seismic design loads. On the other hand, spectral displacements increase 

with increasing period (in the constant acceleration and constant velocity 

portions of the spectrum). Therefore, ASCE/SEI 7-16 Section 19.1.1 

requires that the analytical model directly incorporate horizontal, vertical, 

and rotational foundation, and soil flexibility where soil-structure interaction

effects are considered.

In recognition of more detailed modeling, ASCE/SEI 7-16 Section 19.1.3 

indicates that the upper bound limitation of CuTa on the fundamental period 

does not apply to T or T ̃ for the flexible base condition. The corresponding 

modeling direction provided in ASCE/SEI 7-16 Section 12.13.3 does not 

address whether the CuTa limitation is lifted where that section is applied. 

However, since it relies on and is consistent with Chapter 19 for various 

parameters and procedures, the limitation is interpreted to not apply. 

Section B.4.5 illustrates the estimation of lateral stiffness for the pile groups

typical of this example. Similarly, vertical stiffness can be computed 

considering the springs-in-series stiffness corresponding to axial shortening

of the drilled shafts and the displacement required to mobilize expected skin

friction and end bearing capacity. For the typical pile group in this example 
FEMA P-2091 B: Twelve-Story Building Example B-21

(with 50-ft-long piles), lower-bound, best estimate, and upper-bound pile 

group stiffnesses are estimated by the geotechnical consultant, based on field 

measurements of dynamic soil properties, as 2,00 kip/inch, 3,000 kip/inch, 

and 3,800 kip/inch. If lower and upper bounds are established without field 

measurements, ASCE/SEI 7-16 Section 12.13.3 requires a 50% decrease and 

increase in stiffness. Figure B-12 shows Model 2, which is a refinement of 

Model 1 to include lateral and vertical springs at each pile group. 

Figure B-12 Analytical model for the example building, with spring supports

(Model 2). 

Table B-4 compares the fundamental periods from the fixed-base and 

flexible-base models, T and T T , which is  ̃ , respectively and tabulates T ̃

used in subsequent calculations. 

Table B-4 Fundamental Periods for Model 1 (Fixed-Base) and Model 2 

(Flexible-Base) 

Response Direction T T

 ̃  ̃T / T

N-S 2.177s 2.259s 1.038

E-W 1.491s 1.765s 1.184

B.6.2 Foundation Damping

Once the period elongation effect has been quantified, foundation damping 

effects can be computed in accordance with ASCE/SEI 7-16 Section 19.3. 

Item 2 in ASCE/SEI 7-16 Section 19.3.1 indicates that the simple procedures 

of this section are not permitted to be used where a foundation system 
B-22 B: Twelve-Story Building Example FEMA P-2091

includes deep foundations. Instead, the section would require “direct 

incorporation of soil hysteretic damping and radiation damping in the 

mathematical model of the structure.” This interpretation is borne out by the 

Commentary in Section C19.3 of ASCE/SEI 7-16, which indicates, “The

provisions in this section provide simplified ways to approximate these 

effects. However, they are complex phenomena and there are considerably 

more detailed methods to predict their effects on structures. ... The equations 

in Sections 19.3.3 and 19.3.4 are for shallow foundations. This is not to say 

that radiation damping does not occur with deep (pile or caisson) foundation 

systems, but the phenomenon is more complex. Soil layering and group 

effects are important, and there are the issues of the possible contributions of

the bottom structural slab and pile caps. Because the provisions are based on

the impedance produced by a rigid plate in soil, these items cannot be easily

taken into account. Therefore, more detailed modeling of the soil and the 

embedded foundations is required to determine the foundation impedances. 

The provisions permit such modeling but do not provide specific guidance 

for it. Guidance can be found for example in NIST GCR 12-917-21 [NIST 

2012a] and its references.” However, for the purposes of this example, 

calculations are performed using Section 19.3.1. The calculations are 

presented in the required order of computation, which is the opposite of the 

order of presentation in ASCE/SEI 7-16. 

Table B-4 shows the period lengthening ratio,  ̃T / T . Since reductions are 

more pronounced for larger values of  ̃ T / T , the E-W direction is illustrated 

in the following calculations. 

Next, the soil hysteretic damping ratio, βs in determined. Using ASCE/SEI

7-16 Table 19.3-3 for Site Class D with SDS/2.5 = 0.4, βs = 0.07.

Since this section does not provide a method to compute radiation damping 

effects for deep foundations, βrd is conservatively taken as 0. 

The foundation damping ratio, βf, is computed using ASCE/SEI 7-16 Equation 

19.3-3. 

( )

( )

( )

( ) ( )

2 2

2 2

1 1.184 1

0.07 0.0 0.020

1.184 f s rd

T

T

  −  − 

β =  β +β =   + =        

T ̃

T ̃

As permitted in the notes following ASCE/SEI 7-16 Equation 19.3-2, the 

expected ductility demand, μ, is computed based on the values of R and Ω0

provided in ASCE/SEI 7-16 Table 12.2-1. 

μ= Ω = = R 0 7 2.5 2.8
FEMA P-2091 B: Twelve-Story Building Example B-23

The effective period lengthening ratio is computed using ASCE/SEI 7-16 

Equation 19.3-2. 

 ̃

 0.5

 T  1 1    T ̃ 

2

   0.5

  = 1+     − = 1 ( )2

 1+  1.184 − = 1 1.069

 T T  μ 

f 2.8 ef            

As permitted in the notes following ASCE/SEI 7-16 Equation 19.3-1, the 

effective viscous damping ratio of the structure, β, is taken as 0.05. 

The effective damping ratio of the soil-structure system is computed using 

ASCE/SEI 7-16 Equation 19.3-1. 

β 0.05 β =0 βf + = 0.020 + = 0.064 < 0.20

(T ̃

T )

2 2 (1.069) eff

B.6.3 ELF Base Shear Adjusted for SSI

Having quantified the period elongation effect and the effective damping 

ratio of the soil-structure system, the reduced ELF base shear is computed in 

accordance with ASCE/SEI 7-16 Section 19.2.1. 

The adjustment factor for damping ratios other than 0.05 is computed using 

ASCE/SEI 7-16 Equation 19.2-4. 

BSSI = 4 5.6 − 0   ln (100β ) = 4 5.6 − ln(100(0.064)) =    1.067

The seismic response coefficient is computed including soil flexibility, C ̃

s

, 

using the equations in ASCE/SEI 7-16 Section 12.8.1 but substituting T ̃ for 

T. With T ̃ =1.765 seconds per Table B-4, the controlling value, using

ASCE/SEI 7-16 Equation 12.8-3, is C ̃

s = 0.057 , calculated as follows: 

S 0.70 C ̃

s = D1 = = 0.057   R  7 

T ̃   1.765     Ie   1.0

The base shear adjustment, ∆V, is computed using ASCE/SEI 7-16 Equation 

19.2-2. 

  C

 ̃

  0.057 ∆ = V C  s

s W =   0.079 − 43,919 =1,123 kip

  BSSI   1.067

This would result in a 32% reduction of the fixed-base base shear from 3,470 

kip to 2,347 kip. It is worth noting that 966 kip of this reduction is due to 

period elongation (taking BSSI = 1.0) and 157 kip is due to the slight increase 

in damping. 
B-24 B: Twelve-Story Building Example FEMA P-2091

The limitation on base shear reduction quantified by the α factor computed in

accordance with ASCE/SEI 7-16 Equation 19.2-3 is checked. For this 

system with R = 7, so α = 0.9. 

The final base shear adjusted for SSI is computed in accordance with

ASCE/SEI 7-16 Equation 19.2-1. 

V ̃ = V – ΔV ≥ αV

V ̃ = V – ΔV = 3,470 – 1,123 = 2,347 

≥ αV = 0.9(3,470) = 3,123 kip ← controls

The final design base shear computed in accordance with ASCE/SEI 7-16 

Chapter 19 reflects only a 10% reduction from the fixed-base base shear.

Both the text and commentary indicate that the α limitation computed in

accordance with ASCE/SEI 7-16 Equation 19.2-3 reflects concern that 

damping caused by structural yielding could make foundation damping less 

effective. However, in this example, most of the computed reduction is due 

to period elongation, so the validity of that concern is questionable. 

By comparison, analysis that includes foundation load-deformation 

characteristics in accordance with ASCE/SEI 7-16 Section 12.13.3 would be 

permitted to realize all of the period elongation effect. (Sensitivity to the

required consideration of lower-bound and upper-bound foundation stiffness 

is discussed further in Section B.7.3.) Table B-5 compares the design base

shear from the fixed-base condition to that from the flexible-base condition 

determined in accordance with ASCE/SEI 7-16 Chapter 19 or Section 

12.13.3. Response in the N-S direction is controlled by minimum base shear

in ASCE/SEI 7-16 Equation 12.8-5, which is independent of period and 

damping, so no reduction due to SSI can be realized. 

Table B-5 Design Base Shear for Model 1 (Fixed-Base) and Model 2 

(Flexible-Base), Using Best-Estimate Springs

Response

Direction

Model 1

Fixed-base

Model 2

Flexible-base,

ASCE/SEI 7-16 

Chapter 19

Model 2

Flexible-base,

ASCE/SEI 7-16 

Section 12.13.3

N-S 1,932 kip 1,932 kip 1,932 kip

E-W 3,470 kip 3,123 kip 2,488 kip

B.6.4 Story Shears, Moments, and Drifts

Figure B-13 compares the story shears and moments for Model 1 (fixedbase) and Model 2 (flexible-base), where the flexible base model can satisfy 

either the limitations in ASCE/SEI 7-16 Chapter 12 or in Chapter 19. These 

results are from MRSA using the site-specific free-field spectrum, since
FEMA P-2091 B: Twelve-Story Building Example B-25

ASCE/SEI 7-16 Section 19.4 does not permit modification of the response 

spectrum for kinematic SSI effects for ELF or MRSA. As discussed above, 

no reductions due to SSI may be realized in the N-S direction. In the E-W

direction the process in ASCE/SEI 7-16 Chapter 12 results in base shear and 

base overturning moment that are 28% and 31% lower, respectively, than for 

Model 1; the reductions using the Chapter 19 process are only 10% and 13%, 

respectively.

Figure B-13 Story shears and moments for Models 1 and 2. 

Figure B-14 shows story drifts for Models 1 and 2, which are computed from 

MRSA using the site-specific free-field spectrum. Since the value of Cs used 

in design is not controlled by ASCE/SEI 7-16 Equation 12.8-6, drifts are not 

subject to scaling, so the base shear limitations defined in Section 19.2.1 do 
B-26 B: Twelve-Story Building Example FEMA P-2091

not apply to drift calculations. Although drifts are not subject to scaling, the 

drifts for Model 2 still differ slightly depending on whether ASCE/SEI 7-16 

Chapter 12 or Chapter 19 procedures are used since Section 19.2.2.2 allows 

the site-specific spectrum to be adjusted for the slightly beneficial effect of 

increased damping. Although the design forces for Model 2 are less than 

those for Model 1 in the E-W direction, story drifts are greater. This is due 

to period elongation and base rotation for the shear walls. It is possible for 

inclusion of SSI to increase drifts enough that the design must be revised to 

satisfy drift limits.

Figure B-14 Story drifts for Models 1 and 2. 
FEMA P-2091 B: Twelve-Story Building Example B-27

B.7 Design Variations

B.7.1 Building Without Basement, Fixed Base (Model 3)

To investigate further the sensitivity of the design seismic demands, the base 

building is reanalyzed with the basement removed and assuming a fixed base 

condition (Model 3). Figure B-15 shows this model.

Figure B-15 Analytical model of the example building, with basement 

removed and fixed base (Model 3). 

Because the flexibility in the basement is removed from this model, the 

fundamental periods are slightly shorter than for Model 1. However, since 

the base shear in the N-S direction is governed by the minimum base shear 

from ASCE/SEI 7-16 Equation 12.8-5 and the period used for design of the 

fixed-base structure in the E-W direction is controlled by the upper limit 

period, CuTa, the design base shears are unchanged from those for Model 1.

B.7.2 Building Without Basement, Flexible Base (Model 4)

The response of individual piles to lateral loads is highly nonlinear. It has 

become increasingly common to consider that nonlinearity directly. Based 

on extensive testing of full-scale specimens and small-scale models for a 

wide variety of soil conditions, researchers have developed empirical 

relationships for the nonlinear p-y response of piles that are suitable for use 

in design. The p-y response is sensitive to pile size; soil type and properties; 

and, in the case of sands, vertical stress, which increases with depth. Pile 

response to lateral loads is usually computed using computer programs such 
B-28 B: Twelve-Story Building Example FEMA P-2091

as LPile (ENSOFT, 2018). For this example, LPile is used to analyze 

22-inch diameter piles for the soil layering shown in Table B-1. 

Since the drilled shafts extend into the soft soils near the ground surface, the 

pile group lateral stiffness for Model 4 shown in Figure B-16 is considerably 

less than that for Model 2. As a point of reference, for a lateral displacement 

of 0.49 inches the total lateral load is 146 kip, including 32 kip in each of 

four drilled shafts and 18 kip of passive pressure resistance. Figure B-17 

shows the total lateral load versus head displacement for a range of loads. In 

this case the best estimate linear stiffness is estimated as 300 kip/inch, with 

reasonable lower and upper bounds of 215 kip/inch and 430 kip/inch. Axial 

stiffnesses are the same as those computed for the Model 2.

Figure B-16 Analytical model of the example building, with basement 

removed and flexible base (Model 4).
FEMA P-2091 B: Twelve-Story Building Example B-29

Figure B-17 Pile group lateral stiffness for Models 3 and 4. 

B.7.3 Sensitivity to Lower-Bound and Upper-Bound Spring 

Stiffnesses

Where foundation flexibility is included for linear analysis, ASCE/SEI 7-16 

Section 12.13.3 requires use of best estimate “equivalent linear stiffness using 

soil properties that are compatible with the soil strain levels associated with the 

design earthquake motion,” and consideration of lower-bound and upper-bound 

variations in stiffness, with the largest values of response being used in design. 

Similarly, ASCE/SEI 7-16 Section 19.1.1 requires that “both upper and lower 

bound estimates for the foundation and soil stiffnesses per Section 12.13.3 ... be 

considered.” The detailed example calculations and analyses above use best 

estimate soil springs. The lower- and upper-bound stiffnesses used in this 

appendix presume they were provided by a geotechnical consultant, based on 

field measurements of dynamic soil properties. If lower and upper bounds are 

established without field measurements, ASCE/SEI 7-16 Section 12.13.3 requires 

a 50% decrease and increase in stiffness. This differs from what ASCE/SEI

41-17 Section 8.4.3 requires (dividing and multiplying by 1 + the coefficient of 

variation, Cv, but not less than 1.5). Table B-6 summarizes periods and base 

shears in the E-W direction for all models, including spring stiffness variations 

for Models 2 and 4. For comparison it also tabulates base shears three variations 

of the flexible-base conditions: using ASCE/SEI 7-16 Chapter 19 (which reflects 

period elongation and foundation damping, with limits on maximum reductions 

per ASCE/SEI 7-16 Equation 19.2-3), using ASCE/SEI 7-16 Section 12.13.3 

(which reflects only period elongation), and using ASCE/SEI Chapter 19 but 

ignoring the limitation computed using ASCE/SEI 7-16 Equation 19.2-3. 
B-30 B: Twelve-Story Building Example FEMA P-2091

Table B-6 Summary of Periods and Base Shears for All Models, E-W 

Direction

Model Springs T (s) V (kip) VSSI (kip) VSSI/V

1 — 1.491 3,470

2, using ASCE/SEI 7-16 

Chapter 19

UB 1.717 3,123 90%

BE 1.765 3,123 90%

LB 1.839 3,123 90%

2, using ASCE/SEI 7-16 

Section 12.13.3

UB 1.717 2,558 74%

BE 1.765 2,488 72%

LB 1.839 2,388 69%

2, using ASCE/SEI 7-16 

Chapter 19, no α

UB 1.717 2,414 70%

BE 1.765 2,331 67%

LB 1.839 2,216 64%

3 — 1.366 3,470

4, using ASCE/SEI 7-16 

Chapter 19

UB 1.992 3,123 90%

BE 2.074 3,123 90%

LB 2.181 3,123 90%

4, using ASCE/SEI 7-16 

Section 12.13.3

UB 1.992 2,205 64%

BE 2.074 2,118 61%

LB 2.181 2,104 61%

4, using ASCE/SEI 7-16 

Chapter 19, no α

UB 1.992 1,993 57%

BE 2.074 1,907 55%

LB 2.181 1,807 52%

Although the upper-bound spring stiffnesses used in this example are 

approximately twice the lower-bound values, the periods and base shears vary 

by less than 10%. A wider range of bounds would increase this variation, but 

the work presented here demonstrates relative insensitivity of results.

As discussed above where ASCE/SEI 7-16 Section 12.13.3 is used to account 

for period elongation effects, design base shear is less than would be 

computed using ASCE/SEI 7-16 Chapter 19, which limits the total reduction 

due to SSI. In this example the period elongation effect dominates—making 

the Chapter 12 process result in more economical designs. 

Unfortunately, the manner in which ASCE/SEI 7-16 Section 19.2.1 limits the 

reduction due to SSI does not distinguish between period elongation and 

foundation damping. It would be possible to recast Section 19.2.1 so that the 

α coefficient applies only to the portion of SSI reduction that results from 

foundation damping. A code change proposal is under consideration to make 

this clarification.
FEMA P-2091 C: Helpful Resources C-1 

Appendix C

Helpful Resources

Several documents are recommended as helpful resources for practicing 

engineers to better understand soil-structure interaction (SSI) and to see 

additional examples. They include the following, in approximate order of 

likely interest and value to practicing engineers.

• NIST GCR 12-917-21, Soil-Structure Interaction for Building 

Structures, (NIST, 2012a): This is a comprehensive compilation of SSI 

information and forms much of the underlying basis of the SSI 

provisions in both ASCE/SEI 7-16 (ASCE, 2017a) and ASCE/SEI 41-17

(ASCE, 2017b). It discusses the underlying technical basis and includes 

several significant cases study examples and comparisons of different 

modeling techniques.

• ASCE/SEI 41-17, Seismic Evaluation and Retrofit of Existing Buildings, 

(ASCE, 2017b): Chapter 8 of this standard covers foundation evaluation 

and retrofit design. Procedures to develop vertical and horizontal springs 

are in ASCE/SEI 41-17 Section 8.4. SSI provisions are in ASCE/SEI 

41-17 Section 8.5. There are no design examples.

• FEMA P-2006, Example Application Guide for ASCE/SEI 41-13 Seismic 

Evaluation and Retrofit of Existing Buildings with Additional 

Commentary for ASCE/SEI 41-17, (FEMA, 2018): This document 

provides design examples for many of the provisions and building types

in ASCE/SEI 41-13 (ASCE, 2014) and ASCE/SEI 41-17 (ASCE, 

2017b). Chapter 5 discusses the foundation chapter of ASCE/SEI 41-13 

including the various methods for modeling the foundation and soil 

springs. Chapter 9 is a design example of a steel braced frame building, 

and the example includes a comparison of results between a fixed base 

and a flexible base model. Chapter 11 is a design example for a 

nonlinear static analysis of a concrete shear wall building, and the 

example addresses both kinematic interaction and foundation damping. 

• FEMA P-1051, 2015 NEHRP Recommended Seismic Design Provisions: 

Design Examples, (FEMA, 2016): This document provides a series of 

design examples for the 2015 NEHRP Provisions that form the basis of 

those adopted into ASCE/SEI 7. Chapter 8 covers SSI and includes 
C-2 C: Helpful Resources FEMA P-2091

detailed design examples for foundation damping, base slab averaging, 

and embedment effects.

• Guidelines for Performance-Based Seismic Design of Tall Buildings, 

Version 2.03 (PEER, 2017): Section 4.5 of this document addresses 

foundation modeling and soil-structure interaction. Although the 

document focuses on tall buildings in general, the discussion is relevant 

to buildings of all heights.

• ATC-40, The Seismic Evaluation and Retrofit of Concrete Buildings,

(ATC, 1996): This report addresses the capacity spectrum method for 

seismic evaluation and design. Chapter 10 covers foundation modeling 

and evaluation, including vertical and horizontal springs.

• NEHRP Seismic Design Technical Brief No. 7: Seismic Design of 

Reinforced Concrete Mat Foundations: A Guide for Practicing 

Engineers (NIST, 2012b): Sections 3 and 4 of this document provide a 

very brief overview of SSI issues as they apply to mat foundations. It 

references NIST (2012a) for more detailed information.
FEMA P-2091 References D-1 

References 

ACI, 2014, Building Code Requirements for Structural Concrete and 

Commentary, ACI 318-14, American Concrete Institute, Farmington 

Hills, Michigan.

ASCE, 2014, Seismic Evaluation and Retrofit of Existing Buildings, 

ASCE/SEI 41-13, Structural Engineering Institute of American 

Society of Civil Engineers, Reston, Virginia. 

ASCE, 2017a, Minimum Design Loads and Associated Criteria for Buildings 

and Other Structures, ASCE/SEI 7-16, Structural Engineering 

Institute of American Society of Civil Engineers, Reston, Virginia.

ASCE, 2017b, Seismic Evaluation and Retrofit of Existing Buildings, 

ASCE/SEI 41-17, Structural Engineering Institute of American 

Society of Civil Engineers, Reston, Virginia. 

ATC, 1996, The Seismic Evaluation and Retrofit of Concrete Buildings, 

Applied Technology Council, ATC-40 Report, Redwood City, 

California.

Boore, D., Thompson, E., and Cadet, H., 2011, “Regional correlations of VS30

and velocities averaged over depths less than and greater than 30 

meters,” Bulletin of the Seismological Society of America, Vol. 101, 

No. 6, pp. 3046–3059, December, doi: 10.1785/0120110071. 

Bowles, J.E., 2001, Foundation Analysis and Design, Fifth Edition,

McGraw-Hill, New York, New York. 

CSI, 2016, SAP2000 Version 18.2, Computers and Structures, Inc., Walnut 

Creek, California. 

CSI, 2019, ETABS, Computers and Structures, Inc., Walnut Creek, 

California

ENSOFT, 2018, LPile 2018 Using Data Format Version 10, developed by 

W.M. Isenhower, S-T. Wang, and L.G. Vasquez for ENSOFT, Inc, 

Austin, Texas. 

FEMA, 2005, Improvement of Nonlinear Static Seismic Analysis Procedure, 

FEMA 440, prepared by the Applied Technology Council for the 

Federal Emergency Management Agency, Washington, D.C.
D-2 References FEMA P-2091

FEMA, 2016, 2015 NEHRP Recommended Seismic Design Provisions: 

Design Examples, FEMA P-1051, prepared by the Building Seismic 

Safety Council of the National Institute for Standards and 

Technology for the Federal Emergency Management Agency, 

Washington, D.C.

FEMA, 2018, Example Application Guide for ASCE/SEI 41-13 Seismic 

Evaluation and Retrofit of Existing Buildings with Additional 

Commentary for ASCE/SEI 41-17, FEMA P-2006, prepared by the 

Applied Technology Council for the Federal Emergency 

Management Agency, Washington, D.C.

FEMA, 2019, FEMA P-2006 Training Steel Braced Frame, training 

presentation to accompany FEMA P-2006 Example Application 

Guide for ASCE/SEI 41-13 Seismic Evaluation and Retrofit of 

Existing Buildings with Additional Commentary for ASCE/SEI 

41-17, prepared by the Applied Technology Council for the Federal 

Emergency Management Agency, Washington, D.C.

FEMA, 2020a, 2020 NEHRP Recommended Seismic Design Provisions for 

New Buildings and Other Structures, Volume I: Part 1 Provisions, 

Part 2 Commentary, FEMA P-2082-1, prepared by the Building 

Seismic Safety Council for the Federal Emergency Management 

Agency, Washington, D.C.

FEMA, 2020b, 2020 NEHRP Recommended Seismic Design Provisions for 

New Buildings and Other Structures, Volume II: Part 3 Resource 

Papers, FEMA P-2082-2, prepared by the Building Seismic Safety

Council for the Federal Emergency Management Agency, 

Washington, D.C.

Gazetas, G., 1991, “Foundation vibrations,” Foundation Engineering 

Handbook, 2nd Edition, Chapter 15, H.-Y. Fang, ed., Chapman and 

Hall, New York, New York.

Givens, M.J., Mylonakis, G., and Stewart, J.P., 2016, “Modular analytical 

solutions for foundation damping in soil-structure interaction 

applications,” Earthquake Spectra, Vol. 32, No. 3, pp. 1749-1768. 

Jeremic, B., Yang, H., Wang, H., Lizundia, B., 2020, Direct Analysis 

Soil-Structure Interaction Case Studies for the ATC-144 Project.

Kim, S., and Stewart, J.P., 2003, “Kinematic soil-structure interaction from 

strong motion recordings,” Journal of Geotechnical and 

Geoenvironmental Engineering, Vol. 129, No. 4, pp. 323-335.
FEMA P-2091 References D-3

Mylonakis, G., Nikolaou, S., and Gazetas, G., 2006, “Footings under seismic 

loading: Analysis and design issues with emphasis on bridge 

foundations,” Soil Dynamics and Earthquake Engineering, Vol. 26, 

pp. 824-853.

NIST, 2011, NEHRP Seismic Design Technical Brief No. 4: Nonlinear 

Structural Analysis for Seismic Design: A Guide for Practicing 

Engineers, NIST GCR 11-917-11, prepared by the Applied 

Technology Council and the Consortium of Universities for 

Research in Earthquake Engineering for the National Institute of 

Standards and Technology, Gaithersburg, Maryland.

NIST, 2012a, Soil-Structure Interaction for Building Structures, NIST GCR 

12-917-21, prepared by the Applied Technology Council and the 

Consortium of Universities for Research in Earthquake Engineering 

for the National Institute of Standards and Technology, 

Gaithersburg, Maryland.

NIST, 2012b, NEHRP Seismic Design Technical Brief No. 7: Seismic Design 

of Reinforced Concrete Mat Foundations: A Guide for Practicing 

Engineers, NIST GCR 12-917-22, prepared by the Applied 

Technology Council and the Consortium of Universities for 

Research in Earthquake Engineering for the National Institute of 

Standards and Technology, Gaithersburg, Maryland.

NIST, 2016, NEHRP Seismic Design Technical Brief No. 1: Seismic Design 

of Reinforced Concrete Special Moment Frames: A Guide for 

Practicing Engineers, Second Edition, NIST GCR 16-917-40, 

prepared by the Applied Technology Council for the National 

Institute of Standards and Technology, Gaithersburg, Maryland.

Pais, A., and Kausel, E., 1988, “Approximate formulas for dynamic 

stiffnesses of rigid foundations,” Soil Dynamics and Earthquake 

Engineering, Vol. 7, No. 4, pp. 213-227.

PEER, 2017, Guidelines for Performance-Based Seismic Design of Tall 

Buildings, Version 2.03, Report No. 2017/06, developed by the 

Pacific Earthquake Engineering Research Center (PEER) as part of 

the Tall Buildings Initiative (TBI), Berkeley, California.

Yim, C-S, and Chopra, A.K., 1984, “Earthquake response of structures with 

partial uplift on Winkler foundation,” Earthquake Engineering and 

Structural Dynamics, Vol. 12, pp. 263-281.

FEMA P-2091 Glossary E-1 

Glossary

This Guide uses terminology consistent with the definitions in ASCE/SEI

7-16. Where ASCE/SEI 7-16 does not have explicit definitions, other 

sources are used and are identified below.

Base slab averaging: Kinematic SSI of a shallow (nonembedded) 

foundation caused by wave incongruence over the base area. 

Direct analysis approach: An approach to SSI modeling where the soil and 

structure are included within the same model and analyzed as a complete 

system. The soil is often represented as a continuum (e.g., finite elements) 

along with foundation and structural elements, transmitting boundaries at the 

limits of the soil mesh, and interface elements at the edges of the foundation 

(adapted from NIST, 2012a).

Embedment effects: Kinematic SSI embedment effects in which 

foundation-level motions are reduced as a result of ground motion reduction 

with depth below the free surface in structures with embedded foundations 

(adapted from NIST, 2012a).

Foundation and soil flexibility: The vertical, horizontal, and rotational 

flexibility of foundation elements and the soil. Incorporation of foundation 

and soil flexibility typically increases the flexible-base fundamental period of 

building as compared to the fixed-based period where foundation and soil 

flexibility are not included. 

Foundation input motion: Motion that effectively excites the structure and 

its foundation. 

Free-field motion: Motion at ground surface in absence of structure and its 

foundation. 

Inertial SSI (or inertial interaction): The dynamic interaction between the 

structure, its foundation, and the surrounding soil caused by the foundation 

input motion. 

Kinematic SSI (or kinematic interaction): The modification of free-field 

ground motion caused by nonvertical incident seismic waves and spatial 

incoherence; the modification yields the foundation input motion. 
E-2 Glossary FEMA P-2091

Per FEMA (2018), “it is important to distinguish between rigid (fixed) or 

flexible foundations and rigid or flexible footings... A rigid base (also 

known as fixed base) or flexible base foundation refers to the flexibility of 

the structural footing and soil system as a boundary condition of the building, 

whereas a rigid or flexible footing refers to the structural footing element 

itself relative to the supporting soil stiffness. Therefore, the foundation may 

be modeled and evaluated with a combination of foundation and footing 

flexibility. For instance, a structure can have a rigid footing with a flexible 

foundation where the flexibility of the soil is modeled with springs.” Figure 

5-1 of FEMA (2018) illustrates three potential combinations: (1) rigid 

footing with rigid (fixed base) foundation where there are no springs for soil 

flexibility; (2) rigid footing with a flexible foundation where there are springs 

for soil flexibility; and (3) flexible footing with a flexible foundation where 

there are also springs for soil flexibility.

Radiation damping: The damping in the soil-structure system caused by 

the generation and propagation of waves away from the foundation, which 

are caused by dynamic displacements of the foundation relative to the freefield displacements.

Shear wave velocity: The effective shear wave velocity, vs, is defined as 

vs = (G / ρs)

1/2, where G is the effective soil shear modulus and ρs is the soil 

mass density. Effective shear wave velocity is calculated over different 

depths in ASCE/SEI 7-16 Sections 19.3 and 19.4 depending on how the 

parameter is used, and it is influenced by the level of ground shaking. 

(Adapted from NIST, 2012a).

Shear wave: An S wave, or shear wave, is a seismic body wave that shakes 

the ground back and forth perpendicular to the direction the wave is moving. 

(From https://earthquake.usgs.gov/learn/glossary/?term=S%20wave, last 

accessed November 5, 2020).

Soil damping: The hysteretic (material) damping of the soil.

Structure-to-Soil Stiffness Ratio: A ratio defined as h′ / (vsT), where h′ is 

the effective height of the structure, vs is the effective shear wave velocity 

over the depth of interest, and T is the fundamental period of the structure. 

See Chapter 3 for details. (Adapted from NIST, 2012a).

Substructure Approach: An approach to SSI modeling where the SSI 

problem is partitioned into distinct parts that are combined to formulate the 

complete solution. Proper consideration of SSI effects in a substructure 

approach requires: (1) an evaluation of free-field soil motions and 

corresponding soil material properties; (2) an evaluation of transfer functions 
FEMA P-2091 Glossary E-3

to convert free-field motions to foundation input motions; (3) incorporation 

of springs and dashpots (or more complex nonlinear elements) to represent 

the stiffness and damping at the soil foundation interface; and (4) a response 

analysis of the combined structure spring/dashpot system with the foundation 

input motion applied. (Adapted from NIST, 2012a).

FEMA P-2091 Symbols F-1 

Symbols 

This Guide uses symbols and notations consistent with referenced standards, 

primarily ASCE/SEI 7-16. This section presents a list of symbols and 

notations used in this Guide and their definitions as presented in the 

referenced standards. The list is not exhaustive; for comprehensive 

definition of symbols and notation, please refer to the referenced standards

identified within the sections of the Guide. 

If a symbol or notation is referenced in ASCE/SEI 7-16, this is not indicated 

in the list below, as this constitutes most of the entries. If a symbol or 

notation is from a different reference document, a citation is provided. 

Abase Area of foundation footprint if the foundation components are 

interconnected laterally

AT Tributary area

B Width of footing, typically taken as the dimension perpendicular to 

the direction of seismic force unless noted otherwise

Bbsa Bessel function used to compute base slab averaging effects

Bf Width of footing, typically taken as the dimension perpendicular to 

the direction of seismic force D unless noted otherwise

BSSI Factor to adjust the design response spectrum and MCER response 

spectrum 

Cd Deflection amplification factor

Cs Seismic response coefficient

Ct Coefficient from ASCE/SEI 7-16 Table 12.8-2 that depends on the 

structural system

D Depth from the ground surface to the bottom of the foundation

Fa Factor to adjust spectral acceleration in the short period range for site 

class

Fv Factor to adjust spectral acceleration at 1 s for site class
F-2 Symbols FEMA P-2091

Fx Pseudo-seismic force applied at floor level x 

G Soil shear modulus

G0 Initial or maximum soil shear modulus

[G/G0] The effective shear modulus ratio beneath the foundation over a 

depth specified in ASCE/SEI 7-16 Section 19.3 and used in 

ASCE/SEI 7-16 Table 19.3-2 

Ie Importance Factor

Ksur Stiffness at ground surface

Kxx Rotational foundation stiffness 

Ky Translational foundation stiffness 

L Length

M* Effective mass for the first mode

MOT Total overturning moment induced on the element by seismic forces 

applied at and above the level under consideration

P Passive pressure

Pult Ultimate pressure

R Response modification coefficient

RRSbsa Ratio of response spectra factor for base slab averaging

RRSe Ratio of response spectra factor for embedment

S1 Spectral response acceleration parameter at a 1-second

period, obtained from response acceleration contour maps

Sa Spectral response acceleration

Sd Spectral displacement

SD1 Design, 5% damped, spectral response acceleration parameter at a 

period of 1 s 

SDS Design, 5% damped, spectral response acceleration parameter at 

short periods

SMS the MCER, 5% damped, spectra response acceleration parameter at 

short periods adjusted for site class effects
FEMA P-2091 Symbols F-3

T Fundamental period of the building in the direction under 

consideration

T ̃

Fundamental period of the building using a model with a flexible base

Ta Approximate fundamental period of the building

Txx Fundamental translational period of SSI system

Ty Fundamental translational period of SSI system

W Effective seismic weight of a building, including total dead load and 

applicable portions of other gravity loads

b0 Parameter relating effective foundation area to building period

be Effective foundation size, ft

di Thickness of any soil or rock layer i between 0 and 100 ft

e Foundation embedment depth

h′ Effective structure height, measured from base of foundation to the 

center of mass of the fundamental mode

h* Effective height of building

hn Height above base to roof level

hx Height from base to floor level x

k Soil modulus parameter

ksv Winkler spring stiffness in vertical direction, expressed as force/unit 

displacement/unit area

r Effective depth for foundation rotation

vs The average effective shear wave velocity over a depth specified in 

ASCE/SEI 7-16 Sections 19.3 and 19.4 specific to the SSI effect 

being computed determined using vso and ASCE/SEI 7-16 Table 

19.3-1 or a site-specific geotechnical study

vso The average shear wave velocity for soils beneath the foundation at 

small strain levels (10-3% or less) over a depth specified in 

ASCE/SEI 7-16 Sections 19.3 and 19.4 specific to the SSI effect 

being computed. Note that the small strain shear wave velocity in 

ASCE/SEI 7-16 Chapter 19 is the same as the shear wave velocity 

used in ASCE/SEI 7-16 Chapter 20 for site class determination
F-4 Symbols FEMA P-2091

[vs/vso] The effective shear wave velocity ratio beneath the foundation over a 

depth specified in ASCE/SEI 7-16 Sections 19.3 and 19.4 and used 

in ASCE/SEI 7-16 Table 19.3-1 

wi Portion of the effective seismic weight located on or assigned to 

floor level i 

zp Effective profile depth from NIST (2012a)

ΔADVE Average drift of adjoining vertical elements of the seismic forceresisting system over the story below the diaphragm under 

consideration, under tributary lateral load equivalent to that used in 

the computation of δMDD

Ψ Dimensionless factor, function of Poisson’s ratio

Ω0 Overstrength factor

α Coefficient that accounts for the reduction in base shear caused by 

foundation damping SSI

αxx Dimensionless factor, function of dimensionless frequency, a0 

β Effective viscous damping ratio of the structure, taken as 5% unless 

otherwise justified by analysis

β0 Effective viscous damping ratio of the soil-structure system

βf Effective viscous damping ratio relating to foundation-soil 

interaction

βrd Radiation damping ratio

βs Soil hysteretic damping ratio

βxx Rotational foundation damping coefficient

βy Translational foundation damping coefficient

γ Average unit weight of the soils over a depth of B below the base of 

the structure

δMDD Computed maximum in-plane deflection of the diaphragm under 

lateral load

μ Expected ductility demand

ν Poisson’s ratio
FEMA P-2091 Project Participants G-1 

Project Participants

FEMA Oversight

Mike Mahoney (Project Officer)

Federal Emergency Management Agency

500 C Street, SW

Washington, DC 20472

Robert D. Hanson (Subject Matter Expert) 

Federal Emergency Management Agency

500 C Street, SW

Washington, DC 20472

ATC Management and Oversight

Jon A. Heintz (Program Executive, Program Manager)

Applied Technology Council

201 Redwood Shores Parkway, Suite 240

Redwood City, California 94065

Ayse Hortacsu (Project Manager)

Applied Technology Council

201 Redwood Shores Parkway, Suite 240

Redwood City, California 94065

Project Technical Committee

Bret Lizundia (Project Technical Director)

Rutherford + Chekene

375 Beale Street, Suite 310

San Francisco, California 94105

C.B. Crouse

AECom

1111 3rd Avenue, Suite 1600 

Seattle, Washington 98101

Steve Harris

Simpson Gumpertz & Heger, Inc.

100 Pine Street, Suite 1600

San Francisco, California 94111

Boris Jeremic

Department of Civil and Environmental Engineering

University of California Davis

Davis, California 95616

Jonathan P. Stewart

Department of Civil and Environmental Engineering

University of California Los Angeles

5731 Boelter Hall

Los Angeles, California 90095

Michael Valley

Magnusson Klemencic Associates

909 Fifth Avenue, Unit 305

Seattle, Washington 98164

Project Review Panel

Peter Lee 

Skidmore, Owings, Merrill

One Maritime Plaza 

San Francisco, California 94111

Sissy Nikolaou

(ATC Board Representative)

WSP USA

One Penn Plaza

New York, New York 10119

Robert Pekelnicky

Degenkolb Engineers

375 Beale Street, Suite 500

San Francisco, California 94105

Payman Khalili Tehrani

SC Solutions 

1261 Oakmead Parkway

Sunnyvale, California 94085
G-2 Project Participants FEMA P-2091

Working Group Members

Timothy D. Ancheta

RMS 

7575 Gateway Blvd., Suite 300

Newark, California 94560

Laurel Jiang 

Rutherford + Chekene

375 Beale Street, Suite 310

San Francisco, California 94105

Ricardo Henoch

Skidmore, Owings, Merrill

One Maritime Plaza 

San Francisco, California 94111

Gyimah Kasali

Rutherford + Chekene

375 Beale Street, Suite 310

San Francisco, California 94105

Lisa M. Star

17664 Sarentina Court 

Bellflower, California 90706

Christos Tokas

920 El Dorado Way

Sacramento, California 95819

Jiejing Zhou

Skidmore, Owings, Merrill

One Maritime Plaza 

San Francisco, California 94111

FEMA P-2091