DYNAMICS OF STRUCTURES

Hall, Editor Au and Christiano, Structural Analysis Bathe, Finite Element Procedures Biggs, Introduction to Structural Engineering Chopra, Dynamics of Structures: Theory and Applications

DYNAMICS OF STRUCTURES

INCIVILENGINEERING ANDENGINEERINGMECHANICS

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

Marcia J Horton Art Editor: Greg Dulles

Executive Editor: Holly Stark Cover Design: Bruce Kenselaar

Vice President, Production: Vince O’Brien Manufacturing Buyer: Lisa McDowell

Senior Managing Editor: Scott Disanno Executive Marketing Manager: Tim Galligan

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.

Upper Saddle River, NJ 07458 All rights reserved Manufactured in the United States of America This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means,

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The author and publisher of this book have used their best efforts in preparing this book These efforts include the development, research, and testing of the theories and programs to determine their effectiveness The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book The author and publisher shall not be liable in any event for the incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use

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.

vii

PART II MULTI-DEGREE-OF-FREEDOM SYSTEMS 345

PART III EARTHQUAKE RESPONSE, DESIGN, AND EVALUATION

1.7 Equation of Motion: Earthquake Excitation 23

1.8 Problem Statement and Element Forces 26

ix

1.9 Combining Static and Dynamic Responses 28

1.10 Methods of Solution of the Differential Equation 28

1.11 Study of SDF Systems: Organization 33

Appendix 1: Stiffness Coefficients for a Flexural

Element 33

2.1 Undamped Free Vibration 39

2.2 Viscously Damped Free Vibration 48

2.3 Energy in Free Vibration 56

2.4 Coulomb-Damped Free Vibration 57

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 95

3.8 Energy Dissipated in Viscous Damping 99

3.9 Equivalent Viscous Damping 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

Part D: Response to Periodic Excitation 113

3.12 Fourier Series Representation 114

3.13 Response to Periodic Force 114

Appendix 3: Four-Way Logarithmic Graph

Paper 118

Part A: Response to Arbitrarily Time-Varying Forces 125

4.1 Response to Unit Impulse 126

4.2 Response to Arbitrary Force 127

Part B: Response to Step and Ramp Forces 129

4.3 Step Force 129

4.4 Ramp or Linearly Increasing Force 131

4.5 Step Force with Finite Rise Time 132

Part C: Response to Pulse Excitations 135

4.6 Solution Methods 135

4.7 Rectangular Pulse Force 137

4.8 Half-Cycle Sine Pulse Force 143

4.9 Symmetrical Triangular Pulse Force 148

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

5.1 Time-Stepping Methods 165

5.2 Methods Based on Interpolation of Excitation 167

5.3 Central Difference Method 171

5.4 Newmark’s Method 174

5.5 Stability and Computational Error 180

5.6 Nonlinear Systems: Central Difference Method 183

5.7 Nonlinear Systems: Newmark’s Method 183

6.1 Earthquake Excitation 197

6.2 Equation of Motion 203

6.3 Response Quantities 204

6.4 Response History 205

6.5 Response Spectrum Concept 207

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 222

6.9 Elastic Design Spectrum 230

6.10 Comparison of Design and Response Spectra 239

6.11 Distinction between Design and Response

Spectra 241

6.12 Velocity and Acceleration Response Spectra 242

Appendix 6: El Centro, 1940 Ground Motion 246

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 266

7.8 Relative Effects of Yielding and Damping 280

7.9 Dissipated Energy 281

7.10 Supplemental Energy Dissipation Devices 284

7.11 Inelastic Design Spectrum 289

7.12 Applications of the Design Spectrum 296

7.13 Comparison of Design and Response

Spectra 302

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 323

8.5 Natural Vibration Frequency by Rayleigh’s

Method 330

8.6 Selection of Shape Function 334

Appendix 8: Inertia Forces for Rigid Bodies 338

9.1 Simple System: Two-Story Shear Building 347

9.2 General Approach for Linear Systems 352

9.3 Static Condensation 369

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 387

9.8 Inelastic Systems 392

9.9 Problem Statement 392

9.10 Element Forces 393

9.11 Methods for Solving the Equations of Motion:

Overview 393

Part A: Natural Vibration Frequencies and Modes 404

10.1 Systems without Damping 404

10.2 Natural Vibration Frequencies and Modes 406

10.3 Modal and Spectral Matrices 408

10.4 Orthogonality of Modes 409

10.5 Interpretation of Modal Orthogonality 410

10.6 Normalization of Modes 410

10.7 Modal Expansion of Displacements 420

Part B: Free Vibration Response 421

10.8 Solution of Free Vibration Equations: Undamped

Systems 421

10.9 Systems with Damping 424

10.10 Solution of Free Vibration Equations: Classically

Damped Systems 425

Part C: Computation of Vibration Properties 428

10.11 Solution Methods for the Eigenvalue Problem 428

10.12 Rayleigh’s Quotient 430

10.13 Inverse Vector Iteration Method 430

10.14 Vector Iteration with Shifts: Preferred Procedure 435

10.15 Transformation of kφ = ω2mφ to the Standard

Form 440

Part A: Experimental Data and Recommended Modal

Damping Ratios 447

11.1 Vibration Properties of Millikan Library Building 447

11.2 Estimating Modal Damping Ratios 452

Part B: Construction of Damping Matrix 454

11.3 Damping Matrix 454

11.4 Classical Damping Matrix 455

11.5 Nonclassical Damping Matrix 464

Part A: Two-Degree-of-Freedom Systems 467

12.1 Analysis of Two-DOF Systems Without Damping 467

12.2 Vibration Absorber or Tuned Mass Damper 470

Part B: Modal Analysis 472

12.3 Modal Equations for Undamped Systems 472

12.4 Modal Equations for Damped Systems 475

12.5 Displacement Response 476

12.6 Element Forces 477

12.7 Modal Analysis: Summary 477

Part C: Modal Response Contributions 482

12.8 Modal Expansion of Excitation Vector

p(t) = sp(t) 482

12.9 Modal Analysis for p(t) = sp(t) 486

12.10 Modal Contribution Factors 487

12.11 Modal Responses and Required Number of Modes 489

Part D: Special Analysis Procedures 496

12.12 Static Correction Method 496

12.13 Mode Acceleration Superposition Method 499

12.14 Mode Acceleration Superposition Method: Arbitrary

Excitation 500

Part A: Response History Analysis 514

13.1 Modal Analysis 514

13.2 Multistory Buildings with Symmetric Plan 520

13.3 Multistory Buildings with Unsymmetric Plan 540

13.4 Torsional Response of Symmetric-Plan Buildings 551

13.5 Response Analysis for Multiple Support

Excitation 555

13.6 Structural Idealization and Earthquake Response 561

Part B: Response Spectrum Analysis 562

13.7 Peak Response from Earthquake Response

Spectrum 562

13.8 Multistory Buildings with Symmetric Plan 567

13.9 Multistory Buildings with Unsymmetric Plan 579

13.10 A Response-Spectrum-Based Envelope for

Simultaneous Responses 587

13.11 Response to Multicomponent Ground

Motion 595

Part A: Classically Damped Systems: Reformulation 618

14.1 Natural Vibration Frequencies and Modes 618

14.2 Free Vibration 619

14.3 Unit Impulse Response 620

14.4 Earthquake Response 621

Part B: Nonclassically Damped Systems 622

14.5 Natural Vibration Frequencies and Modes 622

14.6 Orthogonality of Modes 623

14.7 Free Vibration 627

14.8 Unit Impulse Response 632

14.9 Earthquake Response 636

14.10 Systems with Real-Valued Eigenvalues 638

14.11 Response Spectrum Analysis 646

14.12 Summary 647

Appendix 14: Derivations 648

15 Reduction of Degrees of Freedom 657

15.1 Kinematic Constraints 658

15.2 Mass Lumping in Selected DOFs 659

15.3 Rayleigh–Ritz Method 659

15.4 Selection of Ritz Vectors 663

15.5 Dynamic Analysis Using Ritz Vectors 668

16.1 Time-Stepping Methods 673

16.2 Linear Systems with Nonclassical Damping 675

16.3 Nonlinear Systems 681

17.1 Equation of Undamped Motion: Applied Forces 698

17.2 Equation of Undamped Motion: Support

Excitation 699

17.3 Natural Vibration Frequencies and Modes 700

17.4 Modal Orthogonality 707

17.5 Modal Analysis of Forced Dynamic Response 709

17.6 Earthquake Response History Analysis 716

17.7 Earthquake Response Spectrum Analysis 721

17.8 Difficulty in Analyzing Practical Systems 724

Part A: Rayleigh–Ritz Method 729

18.1 Formulation Using Conservation of Energy 729

18.2 Formulation Using Virtual Work 733

18.3 Disadvantages of Rayleigh–Ritz Method 735

Part B: Finite Element Method 735

18.4 Finite Element Approximation 735

18.5 Analysis Procedure 737

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 743

18.10 Comparison of Finite Element and Exact

Solutions 747

18.11 Dynamic Analysis of Structural Continua 748

PART III EARTHQUAKE RESPONSE, DESIGN, AND EVALUATION

19.1 Systems Analyzed, Design Spectrum, and Response

Quantities 757

19.2 Influence of T1andρ on Response 762

19.3 Modal Contribution Factors 763

19.4 Influence of T1on Higher-Mode Response 765

19.5 Influence ofρ on Higher-Mode Response 768

19.6 Heightwise Variation of Higher-Mode Response 769

19.7 How Many Modes to Include 771

Part A: Nonlinear Response History Analysis 776

20.1 Equations of Motion: Formulation and Solution 776

20.2 Computing Seismic Demands: Factors

To Be Considered 777

20.3 Story Drift Demands 781

20.4 Strength Demands for SDF and MDF Systems 787

Part B: Approximate Analysis Procedures 788

20.5 Motivation and Basic Concept 788

20.6 Uncoupled Modal Response History Analysis 790

20.7 Modal Pushover Analysis 797

20.8 Evaluation of Modal Pushover Analysis 802

20.9 Simplified Modal Pushover Analysis

for Practical Application 807

21.1 Isolation Systems 809

21.2 Base-Isolated One-Story Buildings 812

21.3 Effectiveness of Base Isolation 818

21.4 Base-Isolated Multistory Buildings 822

21.5 Applications of Base Isolation 828

Part A: Building Codes and Structural Dynamics 836

22.1 International Building Code (United States), 2009 836

22.2 National Building Code of Canada, 2010 839

22.3 Mexico Federal District Code, 2004 841

22.4 Eurocode 8, 2004 844

22.5 Structural Dynamics in Building Codes 846

Part B: Evaluation of Building Codes 852

22.6 Base Shear 852

22.7 Story Shears and Equivalent Static Forces 856

22.8 Overturning Moments 858

22.9 Concluding Remarks 861

23.1 Nonlinear Dynamic Procedure: Current Practice 864

23.2 SDF-System Estimate of Roof Displacement 865

23.3 Estimating Deformation of Inelastic SDF Systems 868

23.4 Nonlinear Static Procedures 874

23.5 Concluding Remarks 880

A Frequency-Domain Method of Response Analysis 883

The need for a textbook on earthquake engineering was first pointed out by the eminentconsulting engineer, John R Freeman (1855–1932) Following the destructive Santa Bar-bara, California earthquake of 1925, he became interested in the subject and searched theBoston 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 educationwas in an undeveloped state, with computing done by slide rule and curricula that did notprepare the student for understanding structural dynamics In fact, no instruments had beendeveloped for recording strong ground motions, and society appeared to be unconcernedabout earthquake hazards

In recent years books on earthquake engineering and structural dynamics have beenpublished, but the present book by Professor Anil K Chopra fills a niche that exists be-tween more elementary books and books for advanced graduate studies The author is awell-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 ofthe world, for a knowledge of structural dynamics is essential for modern engineering Thebook presents material on vibrations and the dynamics of structures and demonstrates theapplication to structural motions caused by earthquake ground shaking The material inthe book is presented very clearly with numerous worked-out illustrative examples, so thateven a student at a university where such a course is not given should be able to study thebook on his or her own time Readers who are now practicing engineering should have nodifficulty 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 mic response and design of multistory buildings The information presented in this book

seis-xxi

will be of special value to those engineers who are engaged in actual seismic design andwant to improve their understanding of the subject.

Although the material in the book leads to earthquake engineering, the informationpresented is also relevant to wind-induced vibrations of structures, as well as man-mademotions such as those produced by drophammers or by heavy vehicular traffic As a text-book on vibrations and structural dynamics, this book has no competitors and can be rec-ommended to the serious student I believe that this is the book for which John R Freemanwas searching

George W Housner California Institute of Technology

PHILOSOPHY AND OBJECTIVES

This book on dynamics of structures is conceived as a textbook for courses in civil neering It includes many topics in the theory of structural dynamics, and applications ofthis theory to earthquake analysis, response, design, and evaluation of structures No priorknowledge of structural dynamics is assumed in order to make this book suitable for thereader learning the subject for the first time The presentation is sufficiently detailed andcarefully integrated by cross-referencing to make the book suitable for self-study This fea-ture of the book, combined with a practically motivated selection of topics, should interestprofessional engineers, especially those concerned with analysis and design of structures

engi-in earthquake country

In developing this book, much emphasis has been placed on making structural namics easier to learn by students and professional engineers because many find this sub-ject to be difficult To achieve this goal, the presentation has been structured around severalfeatures: The mathematics is kept as simple as each topic will permit Analytical proce-dures are summarized to emphasize the key steps and to facilitate their implementation bythe reader These procedures are illustrated by over 120 worked-out examples, includingmany comprehensive and realistic examples where the physical interpretation of results isstressed Some 500 figures have been carefully designed and executed to be pedagogicallyeffective; many of them involve extensive computer simulations of dynamic response ofstructures Photographs of structures and structural motions recorded during earthquakesare included to relate the presentation to the real world

dy-xxiii

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 multistorybuildings

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 cal problems, especially in earthquake analysis, design, and evaluation of structures, thusnarrowing the gap between theory and practice

practi-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 increasingnumber of recordings of ground motions in the proximity of the causative fault, Chap-ter 6 was expanded to identify special features of near-fault ground motions and com-pare them with the usual far-fault ground motions Because of the increasing interest inseismic performance of bridges, examples on dynamics of bridges and their earthquakeresponse were added in several chapters In response to the growing need for simpli-fied dynamic analysis procedures suitable for performance-based earthquake engineering,Chapter 7 was expanded to provide a fuller discussion relating the earthquake-induced de-formations of inelastic and elastic systems, and to demonstrate applications of the inelasticdesign spectrum to structural design for allowable ductility, displacement-based design,and seismic evaluation of existing structures Chapter 19 (now Chapter 20) was rewrittencompletely to incorporate post-1990 advances in earthquake analysis and response of in-elastic buildings Originally limited to three building codes—United States, Canada, andMexico—Chapter 21 (now Chapter 22) was expanded to include the Eurocode The addi-tion of Chapter 22 (now Chapter 23) was motivated by the adoption of performance-basedguidelines for evaluating existing buildings by the structural engineering profession

In response to reader requests, the frequency-domain method of dynamic analysiswas 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 topicpermits, thus making structural dynamics easily accessible to students and professionalengineers

WHAT’S NEW IN THIS EDITION

Dynamics of Structures has been well received in the 16 years since it was first

pub-lished It continues to be used as a textbook at universities in the United States and manyother countries, and enjoys a wide professional readership as well Translations in Japanese,Korean, Chinese, Greek, and Persian have been published Preparation of the fourth editionprovided me with an opportunity to improve, expand, and update the book

Chapter 14 has been added, requiring renumbering of Chapters 14 to 22 as 15 to 23(the new numbering is reflected in the rest of the Preface); Chapters 5 and 16 underwentextensive revision; Chapters 12 and 13 have been expanded; and Chapters 22 and 23 havebeen updated Specific changes include:

• Chapter 14 on nonclassically damped systems has been added This addition has

been motivated by growing interest in such systems that arise in several cal situations: for example, structures with supplemental energy-dissipating sys-tems or on a base isolation system, soil–structure systems, and fluid-structuresystems

practi-• Chapters 5 and 16 on numerical evaluation of dynamic response have been rewritten

to conform with the ways these numerical methods are usually implemented in puter software, and to offer an integrated presentation of nonlinear static analysis—also known as pushover analysis—and nonlinear dynamic analysis

com-• A section has been added at the end of Chapter 12 to present a general version of the

mode acceleration superposition method for more complex excitations, such as waveforces on offshore drilling platforms

• Chapter 13 has been extended to include two topics that so far have been

con-fined to the research literature, but are of practical interest: (1) combining peakresponses of a structure to individual translational components of ground motion toestimate its peak response to multicomponent excitation; and (2) response-spectrum-based equations to determine an envelope that bounds the joint response trajectory

of all simultaneously acting forces that control the seismic design of a structuralelement

• Chapters 22 and 23 have been updated to reflect the current editions of building codes

for designing new buildings, and of performance-based guidelines and standards forevaluating existing buildings

• The addition of Chapter 14 prompted minor revision of Chapters 2, 4, 6, 10,

and 12

• Several new figures, photographs, worked-out examples, and end-of-chapter

prob-lems have been added

Using the book in my teaching and reflecting on it over the years suggested ments The text has been clarified and polished throughout, and a few sections have beenreorganized to enhance the effectiveness of the presentation

improve-SUBJECTS COVERED

This book is organized into three parts: I Single-Degree-of-Freedom Systems; II Degree-of-Freedom Systems; and III Earthquake Response, Design, and Evaluation ofMultistory Buildings

Multi-Part I includes eight chapters In the opening chapter the structural dynamics lem is formulated for simple elastic and inelastic structures, which can be idealized assingle-degree-of-freedom (SDF) systems, and four methods for solving the differentialequation governing the motion of the structure are reviewed briefly We then study the dy-namic response of linearly elastic systems (1) in free vibration (Chapter 2), (2) to harmonicand periodic excitations (Chapter 3), and (3) to step and pulse excitations (Chapter 4).Included in Chapters 2 and 3 is the dynamics of SDF systems with Coulomb damping, atopic that is normally not included in civil engineering texts, but one that has become rel-evant to earthquake engineering, because energy-dissipating devices based on friction arebeing used in earthquake-resistant construction After presenting numerical time-steppingmethods for calculating the dynamic response of SDF systems (Chapter 5), the earthquakeresponse of linearly elastic systems and of inelastic systems is studied in Chapters 6 and

prob-7, respectively Coverage of these topics is more comprehensive than in texts presentlyavailable; included are details on the construction of response and design spectra, ef-fects of damping and yielding, and the distinction between response and design spectra.The analysis of complex systems treated as generalized SDF systems is the subject ofChapter 8

Part II includes Chapters 9 through 18 on the dynamic analysis of freedom (MDF) systems In the opening chapter of Part II the structural dynamics problem

multi-degree-of-is formulated for structures idealized as systems with a finite number of degrees of freedomand illustrated by numerous examples; also included is an overview of methods for solvingthe differential equations governing the motion of the structure Chapter 10 is concernedwith free vibration of systems with classical damping and with the numerical calculation

of natural vibration frequencies and modes of the structure Chapter 11 addresses severalissues that arise in defining the damping properties of structures, including experimentaldata—from forced vibration tests on structures and recorded motions of structures duringearthquakes—that provide a basis for estimating modal damping ratios, and analytical pro-cedures to construct the damping matrix, if necessary Chapter 12 is concerned with thedynamics of linear systems, where the classical modal analysis procedure is emphasized.Part C of this chapter represents a “new” way of looking at modal analysis that facilitatesunderstanding of how modal response contributions are influenced by the spatial distribu-tion and the time variation of applied forces, leading to practical criteria on the number ofmodes to include in response calculation In Chapter 13, modal analysis procedures forearthquake analysis of classically damped systems are developed; both response historyanalysis and response spectrum analysis procedures are presented in a form that providesphysical interpretation; the latter procedure estimates the peak response of MDF systemsdirectly from the earthquake response or design spectrum The procedures are illustrated

by numerous examples, including coupled lateral-torsional response of unsymmetric-planbuildings and torsional response of nominally symmetric buildings The chapter ends

with response spectrum-based procedures to consider all simultaneously acting forces thatcontrol the design of a structural element, and to estimate the peak response of a structure

to multicomponent earthquake excitation The modal analysis procedure is extended inChapter 14 to response history analysis; of nonclassically damped systems subjected toearthquake excitation For this purpose, we first revisit classically damped systems and re-cast the analysis procedures of Chapters 10 and 13 in a form that facilitates their extension

to the more general case

Chapter 15 is devoted to the practical computational issue of reducing the number

of degrees of freedom in the structural idealization required for static analysis in order

to recognize that the dynamic response of many structures can be well represented bytheir first few natural vibration modes In Chapter 16 numerical time-stepping methodsare presented for MDF systems not amenable to classical modal analysis: systems withnonclassical damping or systems responding into the range of nonlinear behavior Chap-ter 17 is concerned with classical problems in the dynamics of distributed-mass systems;only one-dimensional systems are included In Chapter 18 two methods are presented fordiscretizing one-dimensional distributed-mass systems: the Rayleigh–Ritz method and thefinite element method The consistent mass matrix concept is introduced, and the accuracyand convergence of the approximate natural frequencies of a cantilever beam, determined

by the finite element method, are demonstrated

Part III of the book contains five chapters concerned with earthquake response sign, and evaluation of multistory buildings, a subject not normally included in structuraldynamics texts Several important and practical issues are addressed using analytical pro-cedures developed in the preceding chapters In Chapter 19 the earthquake response oflinearly elastic multistory buildings is presented for a wide range of two key parameters:fundamental natural vibration period and beam-to-column stiffness ratio Based on theseresults, we develop an understanding of how these parameters affect the earthquake re-sponse of buildings and, in particular, the relative response contributions of the variousnatural modes, leading to practical information on the number of higher modes to include

de-in earthquake response calculations Chapter 20 is concerned with the important subject ofearthquake response of multistory buildings deforming into their inelastic range Part A ofthe chapter presents rigorous nonlinear response history analysis; identifies the importantinfluence of modeling assumptions, key structural parameters, and ground motion details

on seismic demands; and determines the strength necessary to limit the story ductilitydemands in a multistory building Recognizing that rigorous nonlinear response historyanalysis remains an onerous task, the modal pushover analysis (MPA) procedure—an ap-proximate analysis procedure—is developed in Part B of the chapter In this procedure,seismic demands are estimated by nonlinear static analyses of the structure subjected tomodal inertia force distributions Base isolation is the subject of Chapter 21 Our goal is

to study the dynamic behavior of buildings supported on base isolation systems with thelimited objective of understanding why and under what conditions isolation is effective inreducing the earthquake-induced forces in a structure In Chapter 22 we present the seis-

mic force provisions in four building codes—International Building Code (United States), National Building Code of Canada, Eurocode, and Mexico Federal District Code—together

with their relationship to the theory of structural dynamics developed in Chapters 6, 7, 8,

and 13 Subsequently, the code provisions are evaluated in light of the results of dynamicanalysis of buildings presented in Chapters 19 and 20 Performance-based guidelines andstandards for evaluating existing buildings consider inelastic behavior explicitly in esti-mating seismic demands at low performance levels, such as life safety and collapse pre-vention In Chapter 23, selected aspects of the nonlinear dynamic procedure and of thenonlinear static procedure in these documents—ATC-40, FEMA 356, and ASCE 41-06—are presented and discussed in light of structural dynamics theory developed in Chapters 7and 20.

A NOTE FOR INSTRUCTORS

This book is suitable for courses at the graduate level and at the senior undergraduate level

No previous knowledge of structural dynamics is assumed The necessary background isavailable through the usual courses required of civil engineering undergraduates Theseinclude:

• Static analysis of structures, including statically indeterminate structures and matrix

formulation of analysis procedures (background needed primarily for Part II)

• Structural design

• Rigid-body dynamics

• Mathematics: ordinary differential equations (for Part I), linear algebra (for Part II),

and partial differential equations (for Chapter 17 only)

By providing an elementary but thorough treatment of a large number of topics, thisbook permits unusual flexibility in selection of the course content at the discretion of theinstructor Several courses can be developed based on the material in this book Here are afew examples

Almost the entire book can be covered in a one-year course:

• Title: Dynamics of Structures I (1 semester)

Syllabus: Chapter 1; Sections 1 and 2 of Chapter 2; Parts A and B of Chapter 3;

Chapter 4; selected topics from Chapter 5; Sections 1 to 7 of Chapter 6; Sections 1

to 7 of Chapter 7; selected topics from Chapter 8; Sections 1 to 4 and 9 to 11 ofChapter 9; Parts A and B of Chapter 10; Sections 1 and 2 of Chapter 11; Parts A and

B of Chapter 12; Sections 1, 2, 7, and 8 (excluding the CQC method) of Chapter 13;and selected topics from Part A of Chapter 22

• Title: Dynamics of Structures II (1 semester)

Syllabus: Sections 5 to 7 of Chapter 9; Sections 3 to 5 of Chapter 11; Parts C and D

of Chapter 12; Sections 3 to 11 of Chapter 13; selected parts of Chapters 14, 15, 17,

19 to 21, and 23; and Appendix A

The selection of topics for the first course has been dictated in part by the need to providecomprehensive coverage, including dynamic and earthquake analysis of MDF systems, forstudents taking only one course.

Abbreviated versions of the outline above can be organized for two quarter courses.One possibility is as follows:

• Title: Dynamics of Structures I (1 quarter)

Syllabus: Chapter 1; Sections 1 and 2 of Chapter 2; Sections 1 to 4 of Chapter 3;

Sections 1 and 2 of Chapter 4; selected topics from Chapter 5; Sections 1 to 7 ofChapter 6; Sections 1 to 7 of Chapter 7; selected topics from Chapter 8; Sections 1

to 4 and 9 to 11 of Chapter 9; Parts A and B of Chapter 10; Part B of Chapter 12;Sections 1, 2, 7, and 8 (excluding the CQC method) of Chapter 13

• Title: Dynamics of Structures II (1 quarter)

Syllabus: Sections 5 to 7 of Chapter 9; Sections 3 to 9 of Chapter 13; and selected

topics from Chapters 19 to 23

A one-semester course emphasizing earthquake engineering can be organized asfollows:

• Title: Earthquake Dynamics of Structures

Syllabus: Chapter 1; Sections 1 and 2 of Chapter 2; Sections 1 and 2 of Chapter 4;

Chapters 6 and 7; selected topics from Chapter 8; Sections 1 to 4 and 9 to 11 ofChapter 9; Parts A and B of Chapter 10; Part A of Chapter 11; Sections 1 to 3 and

7 to 11 of Chapter 13; and selected topics from Chapters 19 to 23

Solving problems is essential for students to learn structural dynamics For thispurpose the first 18 chapters include 373 problems Chapters 19 through 23 do not includeproblems, for two reasons: (1) no new dynamic analysis procedures are introduced in thesechapters; (2) this material does not lend itself to short, meaningful problems However, thereader will find it instructive to work through the examples presented in Chapters 19 to 23and to reproduce the results The computer is essential for solving some of the problems,and these have been identified In solving these problems, it is assumed that the studentwill have access to computer programs such as MATLAB or MATHCAD Solutions tothese problems are available to instructors as a download from the publisher

In my lectures at Berkeley, I develop the theory on the blackboard and illustrate it

by transparencies of the more complex figures in the book; enlarged versions of many

of the figures, which are suitable for making transparencies for use in the classroom, areavailable to instructors as a download from the publisher Despite requests for a completeset of PowerPoint slides, they have not been developed because I do not think this approach

is the most effective strategy for teaching dynamics of structures

A NOTE FOR PROFESSIONAL ENGINEERS

Many professional engineers encouraged me during 1980s to prepare a book more

com-prehensive than Dynamics of Structures, A Primer, a monograph published in 1981 by the

Earthquake Engineering Research Institute This need, I hope, is filled by the present book.Having been conceived as a textbook, it includes the formalism and detail necessary forstudents, but these features should not deter the professional from using the book, becauseits philosophy and style are aimed to facilitate learning the subject by self-study

For professional engineers interested in earthquake analysis, response, design, andevaluation of structures, I suggest the following reading path through the book: Chapters 1and 2; Chapters 6 to 9; Parts A and B of Chapter 10; Part A of Chapter 11; and Chapters 13and 19 to 23

REFERENCES

In this introductory text it is impractical to acknowledge sources for the information sented References have been omitted to avoid distracting the reader However, I haveincluded occasional comments to add historical perspective and, at the end of almost everychapter, a brief list of publications suitable for further reading

pre-YOUR COMMENTS ARE INVITED

I request that instructors, students, and professional engineers write to me (chopra@ce.berkeley.edu) if they have suggestions for improvements or clarifications, or if they identifyerrors I thank you in advance for taking the time and interest to do so

Anil K Chopra

I am grateful to the many people who helped in the preparation of this book

• Professor Rakesh K Goel, a partner from beginning to end, assisted in numerous

ways and played an important role His most significant contribution was to developand execute the computer software necessary to generate the numerical results andcreate most of the figures

• Professor Gregory L Fenves read the first draft of the original manuscript, discussed

it with me weekly, and provided substantive suggestions for improvement

• Six reviewers—Professors Luis Esteva, William J Hall, Rafael Riddell, C C Tung,

and the late George W Housner and Donald E Hudson—examined a final draft of theoriginal manuscript They provided encouragement as well as perceptive suggestionsfor improvement

• Professors Gregory L Fenves and Filip C Fillipou advised on revising Chapters 5

and 16, and commented on the final draft

• Dr Ian Aiken provided the resource material—including photographs—and advice

on revising Sections 7.10.1 and 7.10.2 on supplemental energy dissipation devices

• Dr Charles Menun, whose research results were the basis for new Section 13.10,

provided extensive advice on preparation of this section and reviewed several drafts

• Professor Oscar Lopez and Dr Charles Menun, whose research results were the basis

for new Section 13.11, provided advice and reviewed a final draft

xxxi

• Several reviewers—Professors Michael C Constantinou, Takeru Igusa, George C.

Lee, Fai Ma, and Carlos E Ventura—suggested improvements for the final draft ofChapter 14

• Six experts advised on the interpretation of the updated versions of the four building

codes in Chapter 22: Yousef Bozorgnia and Ronald O Hamburger (International Building Code); Jagmohan L Humar (National Building Code of Canada); Eduardo Miranda (Mexico Federal District Code); and Peter Fajfar and Michael N Fardis (Eurocode).

• Several professors who have adopted the book in their courses have over the years

suggested improvements Some of the revisions and additions in this edition wereprompted by recommendations from Professors Stavros A Anagnostopoulos,Michael C Constantinou, Kincho Law, and Jose M Roesset

• Many former students have over the years assisted in preparing solutions for the

worked-out examples and end-of-chapter problems and helped in other ways: AshrafAyoub, Ushnish Basu, Shih-Po Chang, Juan Chavez, Chatpan Chintanapakdee, JuanCarlos De la llera, Rakesh K Goel, Garrett Hall, Gabriel Hurtado, Petros Keshishian,Allen Kwan, Wen-Hsiung Lin, Charles Menun, and Tsung-Li Tai Han-Chen Tan didthe word processing and graphics for the original Solutions Manual for the 233 prob-lems in the first edition

• Several students, present and former, assisted in the preparation of new material in

the fourth edition: Juan Carlos Reyes solved examples and end-of-chapter lems in Chapters 5, 14, and 16; and prepared figures Yvonne Tsui generated thenumerical results for Section 13.10 and prepared preliminary figures Neal SimonKwong solved the examples and prepared the figures in Sections 12.14 and 13.11,and finalized the figures for Section 13.10 Eric Keldrauk developed the results forFigure 11.4.3

prob-• Charles D James, Information Systems Manager for NISEE at the University of

California, Berkeley, helped in selecting and collecting the new photographs

• Claire Johnson prepared the text for the revised and new parts of the manuscript, and

also assembled and edited the Solutions Manual

• Barbara Zeiders served as the copy editor for this edition, just as she did for the first

three

• Professor Joseph Penzien assumed my duties as Associate Editor of Earthquake

En-gineering and Structural Dynamics from June 1993 until August 1994 while I was

working on the original book

I also wish to express my deep appreciation to Professors Ray W Clough, Jr., JosephPenzien, Emilio Rosenblueth, and A S Veletsos for the influence they have had on myprofessional growth In the early 1960s, Professors Clough, Penzien, and Rosenbluethexposed me to their enlightened views and their superbly organized courses on structuraldynamics and earthquake engineering Subsequently, Professor Veletsos, through his re-search, writing, and lectures, influenced my teaching and research philosophy His work,

in collaboration with the late Professor Nathan M Newmark, defined the approach adopted

for parts of Chapters 6 and 7; and that in collaboration with Professor Carlos E Venturadefined the presentation style for Chapter 14.

This book has been influenced by my own research experience in collaboration with

my students Since 1969, several organizations have supported my research in earthquakeengineering, including the National Science Foundation, U.S Army Corps of Engineers,and California Strong Motion Instrumentation Program

This book and its revised editions were prepared during sabbatical leaves, a privilegefor which I am grateful to the University of California at Berkeley

Anil K Chopra

PART I

Single-Degree-of-Freedom

Systems

1

struc-1.1 SIMPLE STRUCTURES

We begin our study of structural dynamics with simple structures, such as the pergola

shown in Fig 1.1.1 and the elevated water tank of Fig 1.1.2 We are interested in standing the vibration of these structures when subjected to a lateral (or horizontal) force

under-at the top or horizontal ground motion due to an earthquake

We call these structures simple because they can be idealized as a concentrated or lumped mass m supported by a massless structure with stiffness k in the lateral direction.

Such an idealization is appropriate for this pergola with a heavy concrete roof supported

by light-steel-pipe columns, which can be assumed as massless The concrete roof is verystiff and the flexibility of the structure in lateral (or horizontal) motion is provided entirely

by the columns The idealized system is shown in Fig 1.1.3a with a pair of columns

supporting the tributary length of the concrete roof This system has a lumped mass m

3

Figure 1.1.1 This pergola at the Macuto-Sheraton Hotel near Caracas, Venezuela, was

damaged by earthquake on July 29, 1967 The Magnitude 6.5 event, which was centered

about 15 miles from the hotel, overstrained the steel pipe columns, resulting in a permanent

roof displacement of 9 in (From the Steinbrugge Collection, National Information Service

for Earthquake Engineering, University of California, Berkeley.)

equal to the mass of the roof shown, and its lateral stiffness k is equal to the sum of the

stiffnesses of individual pipe columns A similar idealization, shown in Fig 1.1.3b, isappropriate for the tank when it is full of water With sloshing of water not possible in a

full tank, it is a lumped mass m supported by a relatively light tower that can be assumed

as massless The cantilever tower supporting the water tank provides lateral stiffness k to

the structure For the moment we will assume that the lateral motion of these structures issmall in the sense that the supporting structures deform within their linear elastic limit

We shall see later in this chapter that the differential equation governing the lateral

displacement u (t) of these idealized structures without any external excitation—applied

force or ground motion—is

where an overdot denotes differentiation with respect to time; thus˙u denotes the velocity of

the mass and ¨u its acceleration The solution of this equation, presented in Chapter 2, will

show that if the mass of the idealized systems of Fig 1.1.3 is displaced through some initial

displacement u (0), then released and permitted to vibrate freely, the structure will oscillate

or vibrate back and forth about its initial equilibrium position As shown in Fig 1.1.3c, thesame maximum displacement occurs oscillation after oscillation; these oscillations con-tinue forever and these idealized systems would never come to rest This is unrealistic,

Figure 1.1.2 This reinforced-concrete tank on a 40-ft-tall single concrete column, located near the Valdivia Airport, was undamaged by the Chilean earthquakes

of May 1960 When the tank is full of water, the structure can be analyzed as a single-degree-of freedom system (From the Steinbrugge Collection, National Information Service for Earthquake Engineering, University of California, Berkeley.)

of course Intuition suggests that if the roof of the pergola or the top of the water tank werepulled laterally by a rope and the rope were suddenly cut, the structure would oscillate withever-decreasing amplitude and eventually come to rest Such experiments were performed

on laboratory models of one-story frames, and measured records of their free vibration

u m

k

(a)

k m

(b)

u

Masslesstower