DYNAMICS OF STRUCTURES

DYNAMICS OF STRUCTURES


PRENTICE-HALL INTERNATIONAL SERIES
IN CIVIL ENGINEERING AND ENGINEERING MECHANICS
William J. Hall, Editor
Au and Christiano, Structural Analysis
Bathe, Finite Element Procedures
Biggs, Introduction to Structural Engineering
Chopra, Dynamics of Structures: Theory and Applications to Earthquake
Engineering, 4/e
Cooper and Chen, Designing Steel Structures
Cording et al., The Art and Science of Geotechnical Engineering
Hendrickson and Au, Project Management for Construction, 2/e
Higdon et al., Engineering Mechanics, 2nd Vector Edition
Hultz and Kovacs, Introduction in Geotechnical Engineering
Johnston, Lin, and Galambos, Basic Steel Design, 3/e
Kelkar and Sewell, Fundamentals of the Analysis and Design of Shell
Structures
Kramer, Geotechnical Earthquake Engineering
MacGregor, Reinforced Concrete: Mechanics and Design, 3/e
Melosh, Structural Engineering Analysis by Finite Elements
Nawy, Prestressed Concrete: A Fundamental Approach, 3/e
Nawy, Reinforced Concrete: A Fundamental Approach, 4/e
Ostwald, Construction Cost Analysis and Estimating
Pfeffer, Solid Waste Management
Popov, Engineering Mechanics of Solids, 2/e
Popov, Mechanics of Materials, 2/e
Schneider and Dickey, Reinforced Masonry Design, 3/e
Wang and Salmon, Introductory Structural Analysis

Weaver and Johnson, Structural Dynamics by Finite Elements
Wolf, Dynamic Soil–Structure Interaction
Young et al., The Science and Technology of Civil Engineering Materials


DYNAMICS OF STRUCTURES
Theory and Applications to
Earthquake Engineering

Anil K. Chopra
University of California at Berkeley

Fourth Edition

Prentice Hall


Vice President and Editorial Director, ECS:
Marcia J. Horton
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Cover Photo: Transamerica Building, San Francisco, California. The motions shown are accelerations recorded

during the Loma Prieta earthquake of October 17, 1989 at basement, twenty-ninth floor, and forty-ninth floor.
Courtesy Transamerica Corporation.
Credits and acknowledgments for material from other sources and reproduced, with permission, in this
textbook appear on appropriate page within text.

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The author and publisher of this book have used their best efforts in preparing this book. These efforts
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Library of Congress Cataloging-in-Publication Data on File

10 9 8 7 6 5 4 3 2 1

ISBN 10:
0-13-285803-7
ISBN 13: 978-0-13-285803-8

Dedicated to Hamida and Nasreen with gratitude for suggesting the idea of
working on a book and with appreciation for patiently enduring and sharing
these years of preparation with me. Their presence and encouragement
made this idea a reality.



Overview

PART I

SINGLE-DEGREE-OF-FREEDOM SYSTEMS
1

Equations of Motion, Problem Statement, and Solution
Methods

1

3

2

Free Vibration

39

3

Response to Harmonic and Periodic Excitations

65

4

Response to Arbitrary, Step, and Pulse Excitations

125

5

Numerical Evaluation of Dynamic Response

165

6

Earthquake Response of Linear Systems

197

7

Earthquake Response of Inelastic Systems

257

8

Generalized Single-Degree-of-Freedom Systems

307

vii


viii

Overview

PART II MULTI-DEGREE-OF-FREEDOM SYSTEMS
9

345

Equations of Motion, Problem Statement, and Solution
Methods

347

10

Free Vibration

403

11

Damping in Structures

447

12

Dynamic Analysis and Response of Linear Systems

467

13

Earthquake Analysis of Linear Systems

513

14

Analysis of Nonclassically Damped Linear Systems

617

15

Reduction of Degrees of Freedom

657

16

Numerical Evaluation of Dynamic Response

673

17

Systems with Distributed Mass and Elasticity

697

18

Introduction to the Finite Element Method

729

PART III EARTHQUAKE RESPONSE, DESIGN, AND EVALUATION
OF MULTISTORY BUILDINGS

755

19

Earthquake Response of Linearly Elastic Buildings

757

20

Earthquake Analysis and Response of Inelastic Buildings

775

21

Earthquake Dynamics of Base-Isolated Buildings

809

22

Structural Dynamics in Building Codes

835

23

Structural Dynamics in Building Evaluation Guidelines

863

APPENDIX A

FREQUENCY-DOMAIN METHOD OF RESPONSE
ANALYSIS

883

APPENDIX B

NOTATION

905

APPENDIX C

ANSWERS TO SELECTED PROBLEMS

917

Index

933


Contents

Foreword

xix

Preface

xxi

Acknowledgments
PART I

xxix

SINGLE-DEGREE-OF-FREEDOM SYSTEMS
1

1

Equations of Motion, Problem Statement, and Solution
Methods
1.1

Simple Structures

1.2

Single-Degree-of-Freedom System

1.3

Force–Displacement Relation

1.4

Damping Force

1.5

Equation of Motion: External Force

1.6

Mass–Spring–Damper System

1.7

Equation of Motion: Earthquake Excitation

1.8

Problem Statement and Element Forces

3

3
7

8

12
14

19
23

26
ix


x

Contents

1.9

Combining Static and Dynamic Responses

28

1.10

Methods of Solution of the Differential Equation

1.11

Study of SDF Systems: Organization

28

33

Appendix 1: Stiffness Coefficients for a Flexural
Element 33
2

3

Free Vibration

39

2.1

Undamped Free Vibration 39

2.2

Viscously Damped Free Vibration 48

2.3

Energy in Free Vibration

2.4

Coulomb-Damped Free Vibration

56
57

Response to Harmonic and Periodic Excitations
Part A: Viscously Damped Systems: Basic Results 66
3.1

Harmonic Vibration of Undamped Systems

66

3.2

Harmonic Vibration with Viscous Damping

72

Part B: Viscously Damped Systems: Applications

85

3.3

Response to Vibration Generator

85

3.4

Natural Frequency and Damping from Harmonic
Tests 87

3.5

Force Transmission and Vibration Isolation 90

3.6

Response to Ground Motion and Vibration
Isolation 91

3.7

Vibration-Measuring Instruments

3.8

Energy Dissipated in Viscous Damping

3.9

Equivalent Viscous Damping

95
99

103

Part C: Systems with Nonviscous Damping

105

3.10

Harmonic Vibration with Rate-Independent
Damping 105

3.11

Harmonic Vibration with Coulomb Friction

109

65


Contents

xi

Part D: Response to Periodic Excitation
3.12

Fourier Series Representation

3.13

Response to Periodic Force
Appendix 3:
Paper 118

4

113

114
114

Four-Way Logarithmic Graph

Response to Arbitrary, Step, and Pulse Excitations
Part A: Response to Arbitrarily Time-Varying Forces
4.1

Response to Unit Impulse

4.2

Response to Arbitrary Force

125

126
127

Part B: Response to Step and Ramp Forces
4.3

Step Force

4.4

Ramp or Linearly Increasing Force

4.5

Step Force with Finite Rise Time

129

129

Part C: Response to Pulse Excitations

5

125

131
132

135

4.6

Solution Methods

135

4.7

Rectangular Pulse Force

4.8

Half-Cycle Sine Pulse Force

4.9

Symmetrical Triangular Pulse Force

4.10

Effects of Pulse Shape and Approximate Analysis for
Short Pulses 151

4.11

Effects of Viscous Damping

154

4.12

Response to Ground Motion

155

137
143
148

Numerical Evaluation of Dynamic Response
5.1

Time-Stepping Methods

5.2

Methods Based on Interpolation of Excitation

5.3

Central Difference Method

5.4

Newmark’s Method

5.5

Stability and Computational Error 180

165

165

171

174

167


xii

6

Contents

5.6

Nonlinear Systems: Central Difference Method

5.7

Nonlinear Systems: Newmark’s Method

183

Earthquake Response of Linear Systems

197

6.1

Earthquake Excitation

6.2

Equation of Motion

6.3

Response Quantities

6.4

Response History 205

6.5

Response Spectrum Concept

6.6

Deformation, Pseudo-velocity, and Pseudo-acceleration
Response Spectra 208

6.7

Peak Structural Response from the Response
Spectrum 217

6.8

Response Spectrum Characteristics

6.9

Elastic Design Spectrum

6.10

Comparison of Design and Response Spectra

6.11

Distinction between Design and Response
Spectra 241

6.12

Velocity and Acceleration Response Spectra
Appendix 6:

7

183

197

203
204

207

222

230
239

242

El Centro, 1940 Ground Motion

246

Earthquake Response of Inelastic Systems

257

7.1

Force–Deformation Relations

258

7.2

Normalized Yield Strength, Yield Strength Reduction
Factor, and Ductility Factor 265

7.3

Equation of Motion and Controlling Parameters

7.4

Effects of Yielding

7.5

Response Spectrum for Yield Deformation and Yield
Strength 274

7.6

Yield Strength and Deformation from the Response
Spectrum 278

7.7

Yield Strength–Ductility Relation

266

267

278


Contents

8

xiii

7.8

Relative Effects of Yielding and Damping

7.9

Dissipated Energy

7.10

Supplemental Energy Dissipation Devices

7.11

Inelastic Design Spectrum

7.12

Applications of the Design Spectrum

7.13

Comparison of Design and Response
Spectra 302

280

281
284

289
296

Generalized Single-Degree-of-Freedom Systems

307

8.1

Generalized SDF Systems

307

8.2

Rigid-Body Assemblages

309

8.3

Systems with Distributed Mass and Elasticity 311

8.4

Lumped-Mass System: Shear Building

8.5

Natural Vibration Frequency by Rayleigh’s
Method 330

8.6

Selection of Shape Function
Appendix 8:

323

334

Inertia Forces for Rigid Bodies

338

PART II MULTI-DEGREE-OF-FREEDOM SYSTEMS
9

345

Equations of Motion, Problem Statement, and Solution
Methods
9.1

Simple System: Two-Story Shear Building

347

9.2

General Approach for Linear Systems

9.3

Static Condensation

9.4

Planar or Symmetric-Plan Systems: Ground
Motion 372

9.5

One-Story Unsymmetric-Plan Buildings

377

9.6

Multistory Unsymmetric-Plan Buildings

383

9.7

Multiple Support Excitation

9.8

Inelastic Systems

9.9

Problem Statement

352

369

392
392

387

347


xiv

10

Contents

9.10

Element Forces

393

9.11

Methods for Solving the Equations of Motion:
Overview 393

Free Vibration

403

Part A: Natural Vibration Frequencies and Modes
10.1

Systems without Damping

10.2

Natural Vibration Frequencies and Modes

10.3

Modal and Spectral Matrices

10.4

Orthogonality of Modes

10.5

Interpretation of Modal Orthogonality

10.6

Normalization of Modes

10.7

Modal Expansion of Displacements

404
406

408

409
410

410

Part B: Free Vibration Response

420

421

10.8

Solution of Free Vibration Equations: Undamped
Systems 421

10.9

Systems with Damping

10.10

Solution of Free Vibration Equations: Classically
Damped Systems 425

424

Part C: Computation of Vibration Properties

11

404

428

10.11

Solution Methods for the Eigenvalue Problem

428

10.12

Rayleigh’s Quotient

10.13

Inverse Vector Iteration Method

10.14

Vector Iteration with Shifts: Preferred Procedure

10.15

Transformation of kφ = ω mφ to the Standard
Form 440

430
430
435

2

Damping in Structures

447

Part A: Experimental Data and Recommended Modal
Damping Ratios 447
11.1

Vibration Properties of Millikan Library Building

11.2

Estimating Modal Damping Ratios

452

447


Contents

xv

Part B: Construction of Damping Matrix

12

454

11.3

Damping Matrix

454

11.4

Classical Damping Matrix 455

11.5

Nonclassical Damping Matrix 464

Dynamic Analysis and Response of Linear Systems
Part A: Two-Degree-of-Freedom Systems

Analysis of Two-DOF Systems Without Damping

12.2

Vibration Absorber or Tuned Mass Damper

467

470

472

12.3

Modal Equations for Undamped Systems

12.4

Modal Equations for Damped Systems

12.5

Displacement Response

12.6

Element Forces

12.7

Modal Analysis: Summary

472
475

476

477
477

Part C: Modal Response Contributions

482

12.8

Modal Expansion of Excitation Vector
p(t) = s p(t) 482

12.9

Modal Analysis for p(t) = s p(t) 486

12.10

Modal Contribution Factors

12.11

Modal Responses and Required Number of Modes

487

Part D: Special Analysis Procedures

13

467

12.1

Part B: Modal Analysis

467

489

496

12.12

Static Correction Method

496

12.13

Mode Acceleration Superposition Method

12.14

Mode Acceleration Superposition Method: Arbitrary
Excitation 500

499

Earthquake Analysis of Linear Systems
Part A: Response History Analysis

513

514

13.1

Modal Analysis

514

13.2

Multistory Buildings with Symmetric Plan

520


xvi

Contents

13.3

Multistory Buildings with Unsymmetric Plan

13.4

Torsional Response of Symmetric-Plan Buildings

13.5

Response Analysis for Multiple Support
Excitation 555

13.6

Structural Idealization and Earthquake Response

Part B: Response Spectrum Analysis

14

540
551

561

562

13.7

Peak Response from Earthquake Response
Spectrum 562

13.8

Multistory Buildings with Symmetric Plan

13.9

Multistory Buildings with Unsymmetric Plan

13.10

A Response-Spectrum-Based Envelope for
Simultaneous Responses 587

13.11

Response to Multicomponent Ground
Motion 595

567
579

Analysis of Nonclassically Damped Linear Systems
Part A: Classically Damped Systems: Reformulation
14.1

Natural Vibration Frequencies and Modes

14.2

Free Vibration

14.3

Unit Impulse Response

14.4

Earthquake Response

619
620
621

Part B: Nonclassically Damped Systems

622

14.5

Natural Vibration Frequencies and Modes

14.6

Orthogonality of Modes

14.7

Free Vibration

14.8

Unit Impulse Response

14.9

Earthquake Response

14.10

Systems with Real-Valued Eigenvalues

14.11

Response Spectrum Analysis

14.12

Summary

622

623

627
632
636

646

647

Appendix 14:

618

Derivations

648

638

618

617

Contents

xvii

15

657

16

17

18

Reduction of Degrees of Freedom
15.1

Kinematic Constraints

658

15.2

Mass Lumping in Selected DOFs

15.3

Rayleigh–Ritz Method

15.4

Selection of Ritz Vectors

15.5

Dynamic Analysis Using Ritz Vectors

659

659
663
668

Numerical Evaluation of Dynamic Response

673

16.1

Time-Stepping Methods

673

16.2

Linear Systems with Nonclassical Damping

16.3

Nonlinear Systems

675

681

Systems with Distributed Mass and Elasticity

697

17.1

Equation of Undamped Motion: Applied Forces

17.2

Equation of Undamped Motion: Support
Excitation 699

17.3

Natural Vibration Frequencies and Modes

17.4

Modal Orthogonality

17.5

Modal Analysis of Forced Dynamic Response

17.6

Earthquake Response History Analysis

17.7

Earthquake Response Spectrum Analysis

721

17.8

Difficulty in Analyzing Practical Systems

724

698

700

707
709

716

Introduction to the Finite Element Method
Part A: Rayleigh–Ritz Method

729

729

18.1

Formulation Using Conservation of Energy

18.2

Formulation Using Virtual Work

18.3

Disadvantages of Rayleigh–Ritz Method

Part B: Finite Element Method

Finite Element Approximation

18.5

Analysis Procedure

737

733

735

18.4

729

735

735


xviii

Contents

18.6

Element Degrees of Freedom and Interpolation
Functions 739

18.7

Element Stiffness Matrix 740

18.8

Element Mass Matrix 741

18.9

Element (Applied) Force Vector

18.10

Comparison of Finite Element and Exact
Solutions 747

18.11

Dynamic Analysis of Structural Continua

743

748

PART III EARTHQUAKE RESPONSE, DESIGN, AND EVALUATION
OF MULTISTORY BUILDINGS
19

20

Earthquake Response of Linearly Elastic Buildings

755
757

19.1

Systems Analyzed, Design Spectrum, and Response
Quantities 757

19.2

Influence of T1 and ρ on Response

19.3

Modal Contribution Factors

19.4

Influence of T1 on Higher-Mode Response

19.5

Influence of ρ on Higher-Mode Response

19.6

Heightwise Variation of Higher-Mode Response

19.7

How Many Modes to Include

762

763
765
768
769

771

Earthquake Analysis and Response of Inelastic Buildings
Part A: Nonlinear Response History Analysis

776

20.1

Equations of Motion: Formulation and Solution

20.2

Computing Seismic Demands: Factors
To Be Considered 777

20.3

Story Drift Demands

20.4

Strength Demands for SDF and MDF Systems

776

781

Part B: Approximate Analysis Procedures

787

788

20.5

Motivation and Basic Concept

788

20.6

Uncoupled Modal Response History Analysis

790

775


Contents

21

22

xix

20.7

Modal Pushover Analysis

797

20.8

Evaluation of Modal Pushover Analysis

20.9

Simplified Modal Pushover Analysis
for Practical Application 807

802

Earthquake Dynamics of Base-Isolated Buildings
21.1

Isolation Systems

21.2

Base-Isolated One-Story Buildings

21.3

Effectiveness of Base Isolation 818

21.4

Base-Isolated Multistory Buildings

21.5

Applications of Base Isolation

809
812

822

828

Structural Dynamics in Building Codes

835

Part A: Building Codes and Structural Dynamics

836

22.1

International Building Code (United States), 2009

22.2

National Building Code of Canada, 2010

22.3

Mexico Federal District Code, 2004

22.4

Eurocode 8, 2004

22.5

Structural Dynamics in Building Codes

836

839

841

844

Part B: Evaluation of Building Codes

23

809

846

852

22.6

Base Shear

852

22.7

Story Shears and Equivalent Static Forces

22.8

Overturning Moments

22.9

Concluding Remarks

856

858
861

Structural Dynamics in Building Evaluation Guidelines
23.1

Nonlinear Dynamic Procedure: Current Practice

23.2

SDF-System Estimate of Roof Displacement

23.3

Estimating Deformation of Inelastic SDF Systems

23.4

Nonlinear Static Procedures

23.5

Concluding Remarks

880

874

864

865
868

863


xx

Contents

A

Frequency-Domain Method of Response Analysis

883

B

Notation

905

C

Answers to Selected Problems

917

Index

933


Foreword

The need for a textbook on earthquake engineering was first pointed out by the eminent
consulting engineer, John R. Freeman (1855–1932). Following the destructive Santa Barbara, California earthquake of 1925, he became interested in the subject and searched the
Boston Public Library for relevant books. He found that not only was there no textbook
on earthquake engineering, but the subject itself was not mentioned in any of the books
on structural engineering. Looking back, we can see that in 1925 engineering education
was in an undeveloped state, with computing done by slide rule and curricula that did not
prepare the student for understanding structural dynamics. In fact, no instruments had been
developed for recording strong ground motions, and society appeared to be unconcerned

about earthquake hazards.
In recent years books on earthquake engineering and structural dynamics have been
published, but the present book by Professor Anil K. Chopra fills a niche that exists between more elementary books and books for advanced graduate studies. The author is a
well-known expert in earthquake engineering and structural dynamics, and his book will
be valuable to students not only in earthquake-prone regions but also in other parts of
the world, for a knowledge of structural dynamics is essential for modern engineering. The
book presents material on vibrations and the dynamics of structures and demonstrates the
application to structural motions caused by earthquake ground shaking. The material in
the book is presented very clearly with numerous worked-out illustrative examples, so that
even a student at a university where such a course is not given should be able to study the
book on his or her own time. Readers who are now practicing engineering should have no
difficulty in studying the subject by means of this book. An especially interesting feature
of the book is the application of structural dynamics theory to important issues in the seismic response and design of multistory buildings. The information presented in this book
xxi


xxii

Foreword

will be of special value to those engineers who are engaged in actual seismic design and
want to improve their understanding of the subject.
Although the material in the book leads to earthquake engineering, the information
presented is also relevant to wind-induced vibrations of structures, as well as man-made
motions such as those produced by drophammers or by heavy vehicular traffic. As a textbook on vibrations and structural dynamics, this book has no competitors and can be recommended to the serious student. I believe that this is the book for which John R. Freeman
was searching.
George W. Housner
California Institute of Technology

Preface

PHILOSOPHY AND OBJECTIVES
This book on dynamics of structures is conceived as a textbook for courses in civil engineering. It includes many topics in the theory of structural dynamics, and applications of
this theory to earthquake analysis, response, design, and evaluation of structures. No prior
knowledge of structural dynamics is assumed in order to make this book suitable for the
reader learning the subject for the first time. The presentation is sufficiently detailed and
carefully integrated by cross-referencing to make the book suitable for self-study. This feature of the book, combined with a practically motivated selection of topics, should interest
professional engineers, especially those concerned with analysis and design of structures
in earthquake country.
In developing this book, much emphasis has been placed on making structural dynamics easier to learn by students and professional engineers because many find this subject to be difficult. To achieve this goal, the presentation has been structured around several
features: The mathematics is kept as simple as each topic will permit. Analytical procedures are summarized to emphasize the key steps and to facilitate their implementation by
the reader. These procedures are illustrated by over 120 worked-out examples, including
many comprehensive and realistic examples where the physical interpretation of results is
stressed. Some 500 figures have been carefully designed and executed to be pedagogically
effective; many of them involve extensive computer simulations of dynamic response of
structures. Photographs of structures and structural motions recorded during earthquakes
are included to relate the presentation to the real world.

xxiii


xxiv

Preface

The preparation of this book has been inspired by several objectives:
• Relate the structural idealizations studied to the properties of real structures.
• Present the theory of dynamic response of structures in a manner that emphasizes
physical insight into the analytical procedures.

• Illustrate applications of the theory to solutions of problems motivated by practical
applications.
• Interpret the theoretical results to understand the response of structures to various
dynamic excitations, with emphasis on earthquake excitation.
• Apply structural dynamics theory to conduct parametric studies that bring out several
fundamental issues in the earthquake response, design, and evaluation of multistory
buildings.
This mode of presentation should help the reader to achieve a deeper understanding
of the subject and to apply with confidence structural dynamics theory in tackling practical problems, especially in earthquake analysis, design, and evaluation of structures, thus
narrowing the gap between theory and practice.

EVOLUTION OF THE BOOK
Since the book first appeared in 1995, it has been revised and expanded in several ways,
resulting in the second edition (2001) and third edition (2007). Prompted by an increasing
number of recordings of ground motions in the proximity of the causative fault, Chapter 6 was expanded to identify special features of near-fault ground motions and compare them with the usual far-fault ground motions. Because of the increasing interest in
seismic performance of bridges, examples on dynamics of bridges and their earthquake
response were added in several chapters. In response to the growing need for simplified dynamic analysis procedures suitable for performance-based earthquake engineering,
Chapter 7 was expanded to provide a fuller discussion relating the earthquake-induced deformations of inelastic and elastic systems, and to demonstrate applications of the inelastic
design spectrum to structural design for allowable ductility, displacement-based design,
and seismic evaluation of existing structures. Chapter 19 (now Chapter 20) was rewritten
completely to incorporate post-1990 advances in earthquake analysis and response of inelastic buildings. Originally limited to three building codes—United States, Canada, and
Mexico—Chapter 21 (now Chapter 22) was expanded to include the Eurocode. The addition of Chapter 22 (now Chapter 23) was motivated by the adoption of performance-based
guidelines for evaluating existing buildings by the structural engineering profession.
In response to reader requests, the frequency-domain method of dynamic analysis
was included, but presented as an appendix instead of weaving it throughout the book.
This decision was motivated by my goal to keep the mathematics as simple as each topic
permits, thus making structural dynamics easily accessible to students and professional
engineers.