dynamics of structures theory and applications to earthquake engineering _chopra

Hall, Editor Au and Christiano, Fundamentals of Structural Analysis Au and Christiano, Structural Analysis Barson and Rolfe, Fracture and Fatigue Control in Structures, 2/e Bathe, Finit

DYNAMICS OF STRUCTURES

PRENTICE-HALL INTERNATIONAL SERIES

IN CIVIL ENGINEERING AND ENGINEERING MECHANICS

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Au and Christiano, Fundamentals of Structural Analysis

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DYNAMICS OF STRUCTURES

Theory and Applications to

Earthquake Engineering

Anil K Chopra University of California at Berkeley

PRENTICE HALL

Englewood Cliffs, New Jersey 07632

Library of Congress Cataloging-in-Publication Data

Chopra, Anil K

Dynamics of structures : theory and applications to earthquake

engineering / Anil K Chopra

Acquisitions Editor: Bill Stenquist

Production Editor: Kurt Scherwatzky

Production Coordinator: Bayani Mendoza de Leon

Editorial-Production Service: Electronic Publishing Services, Inc

Buyer: Bill Scazzero

Cover Designer: Douglas DeLuca

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 of Transamerica Corporation.)

> © 1995 by Prentice-Hall, Inc

= A Simon & Schuster Company

Englewood Cliffs, New Jersey 07632

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 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 of these programs

All rights reserved No part of this book may be

reproduced, in any form or by any means, without

permission in writing from the publisher

Printed in the United States of America

ISBN 0-13-855214-2

Prentice-Hall International (UK) Limited, London

Prentice-Hall of Australia Pty Limited, Sydney

Prentice-Hall Canada Inc., Toronto

Prentice-Hall Hispanoamericana, S.A., Mexico

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Simon & Schuster Asia Pte Ltd., Singapore

Editora Prentice Hali do Brasil, Ltda., Rio de Janeiro

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, especially the past year and a half Their

presence and encouragement made this idea a reality

_—ÓOycryicv Bi

1 Equations of Motion, Problem Statement, and Solution

Methods

Free Vibration

Response to Harmonic and Periodic Excitations

Response to Arbitrary, Step, and Pulse Excitations

Numerical Evaluation of Dynamic Response

Earthquake Response of Linear Systems

Earthquake Response of Inelastic Systems

Generalized Single-Degree-of-Freedom Systems

viii Overview

9 Equations of Motion, Problem Statement, and Solution

_—_——_“ Contents ia

1 Equations of Motion, Problem Statement, and Solution

1.7 Equation of Motion: Earthquake Excitation 20

1.8 Problem Statement and Element Forces 23

1.9 Combining Static and Dynamic Responses 25

Contents 1.10 Methods of Solution of the Differential Equation 25

1.11 Study of SDF Systems: Organization 29

Appendix I: Stiffness Coefficients for a Flexural

Element 30

2.2 Viscously Damped Free Vibration 44

2.3 Energy in Free Vibration 52

Part A: Viscously Damped Systems: Basic Results 62

3.1 Harmonic Vibration of Undamped Systems 62

3.2 Harmonic Vibration with Viscous Damping 68

Part B: Viscously Damped Systems: Applications 80

3.3 Response to Vibration Generator 80

3.4 Natural Frequency and Damping from Harmonic

Tests 83

3.5 Force Transmission and Vibration Isolation 85

3.6 Response to Ground Motion and Vibration

Isolation 87

3.7 Vibration-Measuring Instruments 91

3.8 Energy Dissipated in Viscous Damping 94

3.9 Equivalent Viscous Damping 98

Part C: Systems with Nonviscous Damping 100

3.10 Harmonic Vibration with Rate-Independent

Damping 100

3.11 Harmonic Vibration with Coulomb Friction 104

Part D: Response to Periodic Excitation 108

3.12 Fourier Series Representation 109

3.13 Response to Periodic Force 109

Contents xi

Appendix 3: Four-Way Logarithmic Graph

Paper 113

Part A: Response to Arbitrarily Time-Varying Forces 119

4.1 Response to Unit Impulse 120

4.2 Response to Arbitrary Force 121

Part B: Response to Step and Ramp Forces 123

4.4 Ramp or Linearly Increasing Force 125

4.5 Step Force with Finite Rise Time 126

Part C: Response to Pulse Excitations 129

4.7, Rectangular Pulse Force 131

48 Half-Cycle Sine Pulse Force 137

4.9 Symmetrical Triangular Pulse Force 142

4.10 Effects of Pulse Shape and Approximate Analysis for

Short Pulses 144

4.11 Effects of Viscous Damping 147

4.12 Response to Ground Motion 149

5.2 Methods Based on Interpolation of Excitation 157

5.3 Central Difference Method 161

5.4 Newmark’s Method 164

5.5 Stability and Computational Error 170

5.6 Analysis of Nonlinear Response: Central Difference

Method 174

5.7 Analysis of Nonlinear Response: Newmark’s

6.6 Deformation, Pseudo-velocity, and Pseudo-acceleration

Response Spectra 198

6.7 Peak Structural Response from the Response

Spectrum 206

6.8 Response Spectrum Characteristics 211

6.9 Elastic Design Spectrum 217

6.10 | Comparison of Design and Response Spectra 225

6.11 Distinction between Design and Response

Spectra 227

6.12 Velocity and Acceleration Response Spectra 228

Appendix 6: El Centro, 1940 Ground Motion 232

7.2 Normalized Yield Strength, Yield Reduction Factor,

and Ductility Factor 248

7.3 Equation of Motion and Controlling Parameters 249

17 Design Yield Strength 261

7.8 Relative Effects of Yielding and Damping 263

7.10 Inelastic Design Spectrum 269

Contents xiii 7.11 Comparison of Design and Response Spectra 274

8.1 Generalized SDF Systems 277

8.3 Systems with Distributed Mass and Elasticity 281

8.4 Lumped-Mass System: Shear Building 292

8.5 Natural Vibration Frequency by Rayleigh’s

Method 298 8.6 Selection of Shape Function 302

Appendix 8: Inertia Forces for Rigid Bodies 306

9 Equations of Motion, Problem Statement, and Solution

9.1 Simple System: Two-Story Shear Building 313

9.2 General Approach for Linear Systems 318

9.3 Static Condensation 334

9.4 Planar or Symmetric-Plan Systems: Ground

Motion 337 9.5 Unsymmetric-Plan Buildings: Ground Motion 342

9.6 Symmetric-Plan Buildings: Torsional Excitation 350

9.7 Multiple Support Excitation 351

xiv

11

Contents 10.1 Systems without Damping 366

10.2 Natural Vibration Frequencies and Modes 368

10.3 Modal and Spectral Matrices 370

10.4 Orthogonality of Modes 371

10.5 Interpretation of Modal Orthogonality 372

10.6 Normalization of Modes 372

10.7 Modal Expansion of Displacements 382

Part B: Free Vibration Response 383

10.8 Solution of Free Vibration Equations: Undamped

Systems 383

10.9 Free Vibration of Systems with Damping 386

10.10 = Solution of Free Vibration Equations: Classically

Damped Systems 390

Part C: Computation of Vibration Properties 392

10.11 = Solution Methods for the Eigenvalue Problem 392

10.12 Rayleigh’s Quotient 394

10.13 Inverse Vector Iteration Method 394

10.14 ~~ Vector Iteration with Shifts: Preferred Procedure 399

10.15 Transformation of k@ = wm to the Standard

Form 404

Part A: Experimental Data and Recommended Modal

Damping Ratios 409

11.1 Vibration Properties of Millikan Library Building 409

11.2 Estimating Modal Damping Ratios 414

Part B: Construction of Damping Matrix 416

11.3 Damping Matrix 416

11.4 Classical Damping Matrix 417

11.5 Nonclassical Damping Matrix 425

Contents xv

12

13

Part A: Two-Degree-of-Freedom Systems 429

12.1 Analysis of Two-DOF Systems without Damping 429

12.2 Vibration Absorber or Tuned Mass Damper 432

Part B: Modal Analysis 434

12.3 Modal Equations for Undamped Systems 434

12.4 Modal Equations for Damped Systems 436

12.5 Displacement Response 438

12.6 Element Forces 438

Part C: Modal Response Contributions 444

12.8 Modal Expansion of Excitation Vector

ptt) = sp(t) 444

12.9 Modal Analysis for p(t) = sp(t) ; 447

12.10 Modal Contribution Factors 448

12.11 Modal Contributions to Response 449

Part D: Special Analysis Procedures 455

12.12 Static Correction Method 455

12.13 Mode Acceleration Superposition Method 458

12.144 = Analysis of Nonclassically Damped Systems 459

Part A: Response History Analysis 468

13.1 Modal Analysis 468

13.2 Multistory Buildings with Symmetric Plan 474

13.3 Multistory Buildings with Unsymmetric Plan 492

13.4 Torsional Response of Symmetric-Plan Buildings 503

13.5 Response Analysis for Multiple Support

Excitation 508

13.6 Structural Idealization and Earthquake Response 313

Part B: Response Spectrum Analysis 514

13.7 Peak Response from Earthquake Response

Spectrum 514

13.8 Multistory Buildings with Symmetric Plan 519

13.9 Multistory Buildings with Unsymmetric Plan 532

Reduction of Degrees of Freedom

14.1 Kinematic Constraints 550

14.2 Static Condensation 551

14.3 Rayleigh-Ritz Method 551

14.4 Selection of Ritz Vectors 554

14.5 Dynamic Analysis Using Ritz Vectors 560

Numerical Evaluation of Dynamic Response

15.1 Time-Stepping Methods 565

15.2 Analysis of Linear Systems with Nonclassi

Damping 567

15.3 Analysis of Nonlinear Systems 574

Systems with Distributed Mass and Elasticity

16.1 Equation of Undamped Motion: Applied Forces 586

16.2 Equation of Undamped Motion: Support

16.6 Earthquake Response History Analysis 600

16.7 Earthquake Response Spectrum Analysis 604

16.8 Difficulty in Analyzing Practical Systems

introduction to the Finite Element Method

Part A: Rayleigh-Ritz Method 613

Contents xvii

PART Iil

18

19

17.1 Formulation Using Conservation of Energy 613

17.2 Formulation Using Virtual Work 617

17.3 Disadvantages of Rayleigh-Ritz Method 618

Part B: Finite Element Method 619

17.4 Finite Element Approximation 619

17.5 Analysis Procedure 621

17.6 Element Degrees of Freedom and Interpolation

Functions 622 17.7 Element Stiffness Matrix 624

17.8 Element Mass Matrix 625

17.9 Element (Applied) Force Vector 626

17.10 Comparison of Finite Element and Exact

Solutions 630 17.11 Dynamic Analysis of Structural Continua 632

EARTHQUAKE RESPONSE AND DESIGN OF

18.1 Systems Analyzed, Design Spectrum, and Response

Quantities 641 18.2 Influence of T, and p on Response 646

18.3 Modal Contribution Factors 647

18.4 Influence of T, on Higher-Mode Response 649

18.5 Influence of p on Higher-Mode Response 652

18.6 Heightwise Variation of Higher-Mode Response 653

18.7 How Many Modes to Include 655

19.1 Allowable Ductility and Ductility Demand 660

19.2 Buildings with “Weak” or “Soft” First Story 665

20.2 Base-Isolated One-Story Buildings 686

20.3 Effectiveness of Base Isolation 691

20.4 Base-Isolated Multistory Buildings 695

20.5 Applications of Base Isolation 701

Structural Dynamics in Building Codes

Part A: Building Codes and Structural Dynamics 704

21.1 Uniform Building Code (United States), 1994 704

21.2 National Building Code of Canada, 1995 707

213 Mexico Federal District Code, 1987 711

21.4 Structural Dynamics in Building Codes 713

Part B: Evaluation of Building Codes 720

_——_!!£i Foreword a

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 Bar- bara, 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 unconcemed 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 modem 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

xix

xx Foreword The information presented in this book 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

_———_7€Hi Preface aa

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 applica- tions of this theory to earthquake analysis, response, and design 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 easily accessible to students and professional engineers because many find this subject to be difficult In order to achieve this goal, the presentation is characterized

by several features: The mathematics is kept as simple as each topic would permit Analytical procedures are summarized to emphasize the key steps and to facilitate their implementation by the reader These procedures are illustrated by over 100 worked- out examples, including many comprehensive and realistic examples where the physical interpretation of results is stressed Some 400 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

xxi

xxii Preface The preparation of this book has been inspired by several objectives:

e Relate the structural idealizations studied to the properties of real structures

e 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

e Interpret the theoretical results to understand the response of structures to various dynamic excitations, with emphasis on earthquake excitation

e Apply structural dynamics theory to conduct parametric studies that bring out sev- eral fundamental issues in the earthquake response and design of multistory build- ings

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 and design of structures, thus narrowing the gap between theory and practice

SUBJECTS COVERED

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

Part I includes eight chapters In the opening chapter the structural dynamics problem is formulated for simple elastic and inelastic structures, which can be idealized

as single-degree-of-freedom (SDF) systems, and four methods for solving the differential equation governing the motion of the structure are reviewed briefly We then study the dynamic response of linearly elastic systems (1) in free vibration (Chapter 2), (2) to harmonic and 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, a topic that is normally not included in civil engineering texts, but one that has become relevant to earthquake engineering, because energy-dissipating devices based on friction are being used in earthquake-resistant construction After presenting numerical time-stepping methods for calculating the.dynamic response of systems (Chapter 5), the earthquake response of linearly elastic systems and of inelastic systems is studied

in Chapters 6 and 7, respectively Coverage of these topics is more comprehensive than in texts presently available; included are details on the construction of response and design spectra, effects 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 of Chapter 8

Part I] includes Chapters 9 through 17 on the dynamic analysis of multi-degree-of- freedom (MDF) systems In the opening chapter of Part II the structural dynamics prob- lem is formulated for structures idealized as systems with a finite number of degrees of freedom and illustrated by numerous examples; also included is an overview of methods

Subjects Covered xxiii for solving the differential equations governing the motion of the structure Chapter 10 is coricerned with free vibration of systems with classical damping and with the numerical calculation of natural vibration frequencies and modes of the structure Also included are differences in free vibration of systems with classical damping and of systems with nonclassical damping, a topic not normally discussed in textbooks Chapter 11 addresses several issues that arise in defining the damping properties of structures, including experi- mental data—from forced vibration tests on structures and recorded motions of structures during earthquakes—that provide a basis for estimating modal damping ratios, and ana- lytical procedures to construct the damping matrix, if necessary Chapter 12 is concerned with the dynamics of linear systems, where the classical modal analysis procedure is em- phasized Part C of this chapter represents a “new” way of looking at modal analysis that facilitates understanding of how modal response contributions are influenced by the spatial distribution and the time variation of applied forces, leading to practical crite- ria on the number of modes to include in response calculation In Chapter 13, modal analysis procedures for earthquake analysis of structures are developed; both response history analysis and response spectrum analysis procedures are presented in a form that provides physical interpretation The presentation and application of modal combination rules to estimate the peak response of MDF systems directly from the earthquake re- sponse or design spectrum is more comprehensive than in textbooks presently available The procedures are illustrated by numerous examples, including coupled lateral-torsional response of unsymmetric-plan buildings and torsional response of nominally symmetric buildings

Chapter 14 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 by their first few natural modes In Chapter 15 numerical time-stepping methods are presented for MDF systems riot amenable to classical modal analysis: systems with nonclassical damping or systems responding into the range of nonlinear behavior Chapter 16 is concerned with classical problems in the dynamics of distributed-mass systems; only one-dimensional systems are included In Chapter 17 two methods are presented for discretizing one-dimensional distributed-mass systems: the Rayleigh—Ritz method and the finite element method The consistent mass matrix concept is introduced, and the accuracy and convergence of the approximate natural frequencies of a cantilever beam,

determined by the finite element method, are demonstrated

Part III of the book contains four chapters concerned with earthquake response and design of multistory buildings, a subject not normally included in structural dynamics texts Several important and practical issues are addressed using analytical procedures developed in the preceding chapters In Chapter 18 the earthquake response of linearly elastic multistory buildings is presented for a wide range of two key parameters: fun- damental natural vibration period and beam-to-column stiffness ratio Based on these results, we develop an understanding of how these parameters affect the earthquake re- sponse of buildings and, in particular, the relative response contributions of the Various natural modes, leading to practical information on the number of higher modes to include

in earthquake response calculations Chapter 19 is concerned with the important subject

xxiv Preface

of earthquake response of multistory buildings deforming into their inelastic range It includes discussion of the heightwise variation of story ductility demands, large ductility demand in the first story if it is “weak” or “soft” relative to the upper stories, ductility demands for buildings designed according to the lateral force distribution of the 1994 Uniform Building Code, and how these demands compare with allowable ductility The currently active and expanding subject of base isolation is the subject of Chapter 20 Our goal is to study the dynamic behavior of buildings supported on base isolation systems with the limited objective of understanding why and under what conditions isolation is effective in reducing the earthquake-induced forces in a structure In Chap- ter 21 we present the seismic force provisions in three building codes—Uniform Building Code (United States), National Building Code of Canada, 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 dynamic analysis of buildings presented in Chapters 18 and 19

A NOTE FOR INSTRUCTORS

This book is suitable for courses at the graduate level and at the senior undergradu- ate level No previous knowledge of structural dynamics is assumed The necessary background is available through the usual courses required of civil engineering under- graduates These include:

e Static analysis of structures, including statically indeterminate structures and matrix formulation of analysis procedures (background needed primarily for Part II)

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

@ Title: Dynamics of Structures I (1 semester)

Syllabus: Chapters 1 and 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 8 to 11 of Chapter 9; Parts A and B of

Chapter 10; Parts A and B of Chapter 12; Sections 1, 2, 7, and 8 of Chapter 13;

and selected topics from Part A of Chapter 21

A Note for Instructors xxv

e Title: Dynamics of Structures II (1 semester)

Syllabus: Chapter 6 (including review of Sections 1 to 7); Chapter 7 (including

review of Sections | to 7); Sections 5 to 7 of Chapter 9; Part C of Chapter 10;

Chapter 11; Parts C and D of Chapter 12; Sections 3 to 9 of Chapter 13; and Chapters 14 to 21

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

An abbreviated version of the outline above covering two quarters can be organized

as follows:

e 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 to 5 of 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 8 to 11 of Chapter 9; Parts A and B of Chapter 10; 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 21

@ Title: Dynamics of Structures II (1 quarter)

Syllabus: Chapter 6 (including review of Sections 1 to 7); Chapter 7 (including review of Sections 1 to 7); Sections 5 to 7 of Chapter 9; Chapter 11; Parts C and

D of Chapter 12; Sections 3 to 9 of Chapter 13; and Chapters 18 to 21

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

e Title: Structural Dynamics and Earthquake Engineering

Syllabus: Chapter 1; Sections 1 and 2 of Chapter 2; Sections 1 to 4 of Chapter 3; Sections 1 to 5 of Chapter 4; Chapters 6 and 7; selected topics from Chapter 8; Sections 1 to 4 and 8 to 11 of Chapter 9; Parts A and B of Chapter 10; Part A of

Chapter 11; Parts A and B of Chapter 12; Sections 1, 2, 7, and 8 of Chapter 13;

and Part A of Chapter 21

As every instructor knows, solving problems is essential for students who are learning structural dynamics For this purpose the first 17 chapters include 233 problems Chapters 18 through 21 do not include problems, for two reasons: (1) no new dynamic analysis procedures are introduced in these chapters; (2) this material does not lend itself to short, meaningful problems However, the reader will find it instructive to

xxvi Preface work through the examples presented in Chapters 18 to 21 and to reproduce some of the results Most of the problems can be solved with an electronic hand calculator and a sufficient quantity of patience and perseverance; a computer is most helpful, of course The computer is essential for solving some of the problems, and these have been identified In solving these problems, it is assumed that the student will have access to computer programs such as MATLAB or CAL A solutions manual is available

In my lectures at Berkeley I use transparencies of figures in this book Instructors wishing to utilize these visual aids can make transparencies from the enlarged versions

of the figures available from the publisher

A NOTE FOR PROFESSIONAL ENGINEERS

Many professional engineers have encouraged me to prepare a book more comprehensive 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 for students,

but these features should not deter the professional from using the book because, to the extent possible, its philosophy and style are akin to those of the monograph

For professional engineers interested in earthquake analysis, response, and design

of structures, I suggest the following reading path through the book: Chapters 1 and 2; Parts A and B of Chapter 3; Chapters 6 to 9; Parts A and B of Chapter 10; Part A of

Chapter 11; and Chapters 13, 18, 19, 20, and 21

YOUR COMMENTS ARE INVITED

Since this is a new book, I request that instructors, students, and professional engineers write to me if they have questions, suggestions for improvements or clarifications, or if they identify errors I thank you in advance for taking the time and interest to do so

Anil K Chopra

_— Acknowledgments ia

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

e Dr Rakesh K Goel, a partner from beginning to end, assisted in numerous ways and played an important role His most significant contribution was to develop and execute the computer software necessary to generate the numerical results and create the over 450 figures

e Professor Gregory L Fenves read the first draft, discussed it with me weekly, and

provided substantive suggestions for improvement

e Six reviewers—Professors Luis Esteva, William J Hall, George W Housner, Don-

ald E, Hudson, Rafael Riddell, and C C Tung—examined a final draft They provided encouragement as well as perceptive suggestions for improvement

e Professor W K Tso reviewed Chapter 21 and advised on interpretation of the National Building Code of Canada

@ The late Professor Emilio Rosenblueth provided much valuable advice and back- ground on the building code for the Mexico Federal District

e Several students, present and former, assisted in preparing solutions for the worked-

out examples and end-of-chapter problems: Juan Chavez, Juan Carlos De la Llera,

Rakesh K Goel, and Tsung-Li Tai Han-Chen Tan did the word processing and graphics for the Solutions Manual

e Julie Reynolds and Eric Eisman did the word processing of the text in TEX Mr Eis- man also assisted in the preliminary editing of some of the material

e Ms Katherine Frohmberg helped in selecting and collecting several photographs

xxvii

xxviii Acknowledgments

e Professor Joseph Penzien assumed my duties as Associate Editor of Earthquake Engineering and Structural Dynamics from June 1993 until August 1994 while I was working on the book

I also wish to express my deep appreciation to Professors Ray W Clough, Jr., Joseph Penzien, Emilio Rosenblueth, and A S Veletsos for the influence they have had on

my professional growth In the early 1960s, Professors Clough, Penzien, and Rosenblueth exposed me to their enlightened views and their superbly organized courses on structural dynamics and earthquake engineering Subsequently, Professor Veletsos, through his research, 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

This book has been influenced by my own research experience in collaboration with my students Since 1969, several organizations have supported my research in earthquake engineering, including the National Science Foundation, U.S Army Corps

of Engineers, and California Strong Motion Instrumentation Program I am especially grateful to the National Science Foundation, in particular Dr S C Liu, for sustained support

This book was prepared during a year of sabbatical leave, a privilege for which I

am grateful to the University of California at Berkeley

Anil K Chopra

DYNAMICS OF STRUCTURES

<< PART I

Single-Degree-of-Freedom

Systems

serail

a |

Equations of Motion, Problem

Statement, and Solution Methods

PREVIEW

In this opening chapter, the structural dynamics problem is formulated for simple struc- tures that can be idealized as a system with a lumped mass and a massless supporting structure Linearly elastic structures as well as inelastic structures subjected to applied dynamic force or earthquake-induced ground motion are considered Then four methods for solving the differential equation governing the motion of the structure are reviewed briefly The chapter ends with an overview of how our study of the dynamic response

of single-degree-of-freedom systems is organized in the chapters to follow

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 understanding the vibration of these structures when subjected to a lateral (or horizontal) force 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 & 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 very stiff 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

3

4 Equations of Motion, Problem Statement, and Solution Methods Chap 1

a .s“

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

damaged by the earthquake of July 29, 1967 The Magnitude 6.5 event, which was

centered about 15 miles from the hotel, overstrained the steel pipe columns (Courtesy

of G W Housner.)

mass m equal to the mass of the roof shown and its lateral stiffness & is equal to the sum

of the stiffnesses of individual pipe columns A similar idealization, shown in Fig 1.1.3b,

is appropriate 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 is small 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 # denotes the ve-

locity of the mass and # 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 dis- placed 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 equilib- rium position As shown in Fig 1.1.3c, the same maximum displacement occurs oscil-

Sec 1.1 Simple Structures

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

K V Steinbrugge Collection, courtesy

of the Earthquake Engineering Research

Center, University of California at

Berkeley.)

lation after oscillation; these oscillations continue forever and these idealized systems

would never come to rest This is unrealistic, of course Intuition suggests that if the roof of the pergola or the top of the water tank were pulled laterally by a rope and the rope were suddenly cut, the structure would oscillate with ever-decreasing amplitude

mO) u uO),

Massless tower

(c)

Figure 1.1.3 (a) Idealized pergola; (b) idealized water tank; (c) free vibration due to

initial displacement

6 Equations of Motion, Problem Statement, and Solution Methods Chap 1 and eventually come to rest Such experiments were performed on laboratory models

of one-story frames and measured records of their free vibration response are presented

in Fig 1.1.4 As expected, the motion of these model structures decays with time,

Figure 1.1.4 (a) Photograph of aluminum

and plexiglass model frames mounted on a

small shaking table used for classroom

demonstration at the University of

California at Berkeley (courtesy of T

Merport); (b) free vibration record of

aluminum model, (c) free vibration record

Sec 1.2 Single-Degree-of-Freedom System 7 with the decay being more rapid for the plexiglass model relative to the aluminum frame

The process by which vibration steadily diminishes in amplitude is called damping

In damping the kinetic energy and strain energy of the vibrating system are dissipated by various mechanisms that we shall mention later For the moment, we simply recognize that an energy-dissipating mechanism should be included in the structural idealization in order to incorporate the feature of decaying motion observed during free vibration tests

of a structure The most commonly used damping element is the viscous damper, in part because it is the simplest to dea] with mathematically In Chapters 2 and 3 we introduce other energy-dissipating mechanisms

1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEM

The system considered is shown schematically in Fig 1.2.1 It consists of a mass m concentrated at the roof level, a massless frame that provides stiffness to the system, and a viscous damper (also known as a dashpot) that dissipates vibrational energy of the system The beam and columns are assumed to be inextensible axially

This system may be considered as an idealization of a one-story structure Each

structural member (beam, column, wall, etc.) of the actual structure contributes to

the inertial (mass), elastic (stiffness or flexibility), and energy dissipation (damping) properties of the structure In the idealized system, however, each of these properties is concentrated in three separate, pure components: mass component, stiffness component, and damping component

The number of independent displacements required to define the displaced posi- tions of all the masses relative to their original position is called the number of degrees of freedom (DOFs) for dynamic analysis More DOFs are typically necessary to define the stiffness properties of a structure compared to the DOFs necessary for dynamic analysis Consider the one-story frame of Fig 1.2.1, constrained to move only in the direction

of the excitation The static analysis problem has to be formulated with three DOFs— lateral displacement and two joint rotations—to determine the lateral stiffness of the frame (see Section 1.3) In contrast, the structure has only one DOF—lateral

Figure 1.2.1 Single-degree-of-freedom system: (a) applied force p(t); (b) earthquake-

induced ground motion.

8 Equations of Motion, Problem Statement, and Solution Methods Chap 1 displacement—for dynamic analysis if it is idealized with mass concentrated at one location, typically the roof level Thus we call this a single-degree-of-freedom (SDF)

system

Two types of dynamic excitation will be considered: (1) external force p(t)

in the lateral direction (Fig 1.2.1a), and (2) earthquake-induced ground motion u,(¢) (Fig 1.2.1b) In both cases u denotes the relative displacement between the mass and the base of the structure

1.3 FORCE-DISPLACEMENT RELATION

Consider the system shown in Fig 1.3.la with no dynamic excitation subjected to an externally applied static force fy along the DOF u as shown The internal force re- sisting the displacement u is equal and opposite to the external force fs (Fig 1.3.1b)

It is desired to determine the relationship between the force fs and the relative dis- placement u associated with deformations in the structure This force—displacement relation would be linear at small deformations but would become nonlinear at larger deformations (Fig 1.3.1c); both nonlinear and linear relations are considered (Fig 1.3.1¢ and d)

Sec 1.3 Force—Displacement Relation 9

To determine the relationship between fs and uw is a standard problem in static structural analysis, and we assume that the reader is familiar with such analyses Thus the presentation here is brief and limited to those aspects that are essential

1.3.1 Linearly Elastic Systems

For a linear system the relationship between the lateral force fs and resulting deformation

u is linear, that is,

fs = ku (1.3.1)

where k is the lateral stiffness of the system; its units are force/length Implicit in

Eg (1.3.1) is the assumption that the linear fs—u relationship determined for small deformations of the structure is also valid for larger deformations Because the resisting force is a single-valued function of u, the system is elastic; hence we use the term linearly elastic system

Consider the frame of Fig 1.3.2a with bay width L, height A, elastic modulus

E, and second moment of the cross-sectional area (or moment of inertia)! about the axis of bending = J, and /, for the beam and columns, respectively; the columns are clamped (or fixed) at the base The lateral stiffness of the frame can readily be determined for the two extreme cases: If the beam is rigid [i.e., flexural rigidity EJ, = oc (Fig 1.3.2b)],

10 Equations of Motion, Problem Statement, and Solution Methods Chap 1 Observe that for the two extreme values of beam stiffness, the lateral stiffness of the frame is independent of L, the beam length or bay width

The lateral stiffness of the frame with an intermediate, realistic stiffness of the beam can be calculated by standard procedures of static structural analysis The stiffness matrix of the frame is formulated with respect to three DOFs: the lateral displacement u and the rotations of the two beam-column joints (Fig 1.3.2a) By static condensation or elimination of the rotational DOFs, the lateral force—displacement relation of Eq (1.3.1)

is determined Applying this procedure to a frame with L = 2h and Ef, = EJ,, its lateral stiffness is obtained (see Example 1.1):

.— Ele

7 hề The lateral stiffness of the frame can be computed similarly for any values of J, and

I, using the frame stiffness coefficients developed in Appendix 1 If shear deformations

in elements are neglected, the result can be written in the form

— 24EI,12ø+l

where p = [,/4I, is the beam-to-column stiffness ratio [see Eq (18.1.1)] For p = 0, ©, and i> Eq (1.3.5) reduces to the results of Eqs (1.3.3), (1.3.2), and (1.3.4), respectively The lateral stiffness is plotted as a function of p in Fig 1.3.3; it increases by a factor of

4 as p increases from zero to infinity