Structural Dynamics THEORV AND COMPUTATION

Structural Dynamics THEORY AND COMPUTATION

Structural Dynamics THEORY AND COMPUTATION

MARIO PAZ Professor of Civil Engineering

University of Louisville

Third Edition

Springer-Science+Susiness Media, S.v IGJp An International Thomson Publishing Company

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The computer programs used in this book are available in several versions of the BASIC language. To order diskettes for IBM-compatible microcomputer formats, use the order form on the back of the book or write to MICROTEXT, P.O. Box 35101, Louis­ville, KY 40232. Technical questions, corrections, and request for additional informa­tion should be directed to this address.

Extreme care has been taken in preparing the programs used in this book. Extensive testing and checking have been performed to insure the accuracy and effectiveness of computer solutions. However, neither the author nor the publisher shall be held re­sponsible or liable for any damages arising from the use of any of the programs in this book.

This edition published by Chapman & Hall One Penn Plaza New York, NY 10119

Published in Great Britain by Chapman & Hall 2-6 Boundary Row London SEI 8HN

Copyright © 1991 by Springer Science+Business Media Dordrecht Originally published by Van Nostrand Reinhold in 1991

Softcover reprint of the hardcover 3rd editioo 1991

All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or by an information storage or retrieval system, without permission in writing from the publishers.

16 15 14 13 12 11 10 9 8 7 6 5 4 3

Library of Congress Cataloging-in-Publication Data

Paz, Mario. StructuraI dynamies: theory and cornputation / Mario Paz.-3rd

ed. p. crn.

1. Structural dynamies. l. Tide. TA654.P39 1985 624.1'71-dc20 90-33615 ISBN 978-1-4684-9909-4 ISBN 978-1-4684-9907-0 (eBook) ClP DOI 10.1007/978-1-4684-9907-0

British Library Cataloguing in Publication Data available

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Honor your father and your mother, as the Lord your God has commanded you, that you may long endure and

that you may fare well... Exodus 20:12

TO THE MEMORY OF MY PARENTS

Benjamin Maman Paz Salma Misri Paz

CONTENTS

PREFACE TO THE THIRD EDITION xv

PREFACE TO THE FIRST EDITION xix

PART I STRUCTURES MODELED AS A SINGLE DEGREE-OF-FREEDOM SYSTEM 1

1 UNDAMPED SINGLE DEGREE-OF-FREEDOM SYSTEMS 3

1.1 Degrees of Freedom / 3 1.2 Undamped System / 5 1.3 Springs in Parallel or in Series / 6 1.4 Newton's Law of Motion / 8 1.5 Free Body Diagram / 9 1.6 D'Alembert's Principle / 10 1.7 Solution of the Differential Equation of Motion / 11

vii

viii Contents

1.8 Frequency and Period / 13 1.9 Amplitude of Motion / 15

2 DAMPED SINGLE DEGREE-OF-FREEDOM SYSTEM 23

2.1 Viscous Damping / 23 2.2 Equation of Motion / 24 2.3 Critically Damped System / 25 2.4 Overdamped System / 26 2.5 Underdamped System / 26 2.6 Logarithmic Decrement / 29

3 RESPONSE OF ONE-DEGREE-OF-FREEDOM SYSTEM TO HARMONIC LOADING 36

3.1 Undamped System: Harmonic Excitation / 36 3.2 Damped System: Harmonic Excitation / 38 3.3 Evaluation of Damping at Resonance / 46 3.4 Bandwidth Method (Half-Power) to Evaluate Damping / 47 3.5 Response to Support Motion / 49 3.6 Force Transmitted to the Foundation / 53 3.7 Seismic Instruments / 56

4 RESPONSE TO GENERAL DYNAMIC LOADING 63

4.1 Impulsive Loading and Duhamel's Integral / 63 4.2 Numerical Evaluation of Duhamel's Integral-Undamped

System / 70 4.3 Numerical Evaluation of Duhamel's Integral-Damped System /

74 4.4 Response by Direct Integration / 75 4.5 Program 2-Response by Direct Integration / 80 4.6 Program 3-Response to Impulsive Excitation / 83

5 FOURIER ANALYSIS AND RESPONSE IN THE FREQUENCY DOMAIN 95

5.1 Fourier Analysis / 95 5.2 Response to a Loading Represented by Fourier Series / 96 5.3 Fourier Coefficients for Piecewise Linear Functions / 98 5.4 Exponential Form of Fourier Series / 100 5.5 Discrete Fourier Analysis / 101

Contents ix

5.6 Fast Fourier Transform I 104

5.7 Program 4-Response in the Frequency Domain I 106

6 GENERALIZED COORDINATES AND RAYLEIGH'S METHOD 116

6.1 Principle of Virtual Work I 116 6.2 Generalized Single Degree-of-Freedom System-Rigid Body I 118 6.3 Generalized Single Degree-of-Freedom System-Distributed

Elasticity I 121 6.4 Rayleigh's Method I 129

6.5 Improved Rayleigh's Method I 136

6.6 Shear Walls I 139

7 NONLINEAR STRUCTURAL RESPONSE 149

7.1 Nonlinear Single Degree-of-Freedom Model I 150

7.2 Integration of the Nonlinear Equation of Motion I 152 7.3 Linear Acceleration Step-by-Step Method I 153 7.4 Elastoplastic Behavior I 156 7.5 Algorithm for the Step-by-Step Solution for Elastoplastic Single

Degree-of-Freedom System I 158 7.6 Program 5-Response for Elastoplastic Behavior System I 162

8 RESPONSE SPECTRA 170

8.1 Construction of Response Spectrum I 170

8.2 Response Spectrum for Support Excitation I 174 8.3 Tripartite Response Spectra I 175 8.4 Response Spectra for Elastic Design I 178

8.5 Response Spectra for Inelastic Systems I 182 8.6 Response Spectra for Inelastic Design I 186 8.7 Program 6-Seismic Response Spectra I 192

PART II STRUCTURES MODELED AS SHEAR BUILDINGS 199

9 THE MULTISTORY SHEAR BUILDING 201

9.1 Stiffness Equations for the Shear Building I 202

9.2 Flexibility Equations for the Shear Building I 205 9.3 Relationship Between Stiffness and Flexibility Matrices I 206

9.4 Program 7-Modeling Structures as Shear Buildings I 207

x Contents

10 FREE VIBRATION OF A SHEAR BUILDING 213

10.1 Natural Frequencies and Normal Modes / 213 10.2 Orthogonality Property of the Normal Modes / 220 10.3 Program 8-Natural Frequencies and Normal Modes / 223

11 FORCED MOTION OF SHEAR BUILDINGS 230

11.1 Modal Superposition Method / 230 11.2 Response of a Shear Building to Base Motion / 236 11.3 Program 9-Response by Modal Superposition / 243 11.4 Harmonic Forced Excitation / 245 1l.5 Program lO-Harmonic Response / 250 11.6 Combining Maximum Values of Modal Response / 253

12 DAMPED MOTION OF SHEAR BUILDINGS 259

12.1 Equations for Damped Shear Building / 260 12.2 Uncoupled Damped Equations / 261 12.3 Conditions for Damping Uncoupling / 262 12.4 Program ll-Absolute Damping From Damping Ratios / 269

13 REDUCTION OF DYNAMIC MATRICES 273

13.1 Static Condensation / 274 13.2 Static Condensation Applied to Dynamic Problems / 277 13.3 Dynamic Condensation / 287 13.4 Modified Dynamic Condensation / 293 13.5 Program 12-Reduction of the Dynamic Problem / 296

PART III FRAMED STRUCTURES MODELED AS DISCRETE MUL TIDEGREE-OF-FREEDOM SYSTEMS 303

14 DYNAMIC ANALYSIS OF BEAMS 305

14.1 Static Properties for a Beam Segment / 306 14.2 System Stiffness Matrix / 311 14.3 Inertial Properties-Lumped Mass / 314 14.4 Inertial Properties-Consistent Mass / 315

14.5 Damping Properties / 319 14.6 External Loads / 320 14.7 Geometric Stiffness / 322 14.8 Equations of Motion / 325 14.9 Element Forces at Nodal Coordinates / 332 14.10 Program 13-Modeling Structures as Beams / 334

15 DYNAMIC ANALYSIS OF PLANE FRAMES 343

15.1 Element Stiffness Matrix for Axial Effects / 344 15.2 Element Mass Matrix for Axial Effects / 345 15.3 Coordinate Transformation / 350

Contents xi

15.4 Program 14-Modeling Structures as Plane Frames / 357

16 DYNAMIC ANALYSIS OF GRIDS 364

16.1 Local and Global Coordinate Systems / 365 16.2 Torsional Effects / 366 16.3 Stiffness Matrix for a Grid Element / 367 16.4 Consistent Mass Matrix for a Grid Element / 368 16.5 Lumped Mass Matrix for a Grid Element / 368 16.6 Transformation of Coordinates / 369 16.7 Program IS-Modeling Structures as Grid Frames / 374

17 THREE-DIMENSIONAL FRAMES 380

17.1 Element Stiffness Matrix / 380 17.2 Element Mass Matrix / 382 17.3 Element Damping Matrix / 383 17.4 Transformation of Coordinates / 383 17.5 Differential Equation of Motion / 390 17.6 Dynamic Response / 390 17.7 Program 16-Modeling Structures as Space Frames / 391

18 DYNAMIC ANALYSIS OF TRUSSES 395

18.1 Stiffness and Mass Matrices for the Plane Truss / 396 18.2 Transformation of Coordinates / 398 18.3 Program 17-Modeling Structures as Plane Trusses / 403 18.4 Stiffness and Mass Matrices for Space Trusses / 406 18.5 Equation of Motion for Space Trusses / 409 18.6 Program 18-Modeling Structures as Space Trusses / 409

xii Contents

19 TIME HISTORY RESPONSE OF MUL TIDEGREE-OF-FREEDOM SYSTEMS 413

19.1 Incremental Equations of Motion / 414 19.2 The Wilson-a Method / 415 19.3 Algorithm for Step-by-Step Solution of a Linear System Using

the Wilson-a Method / 418 19.4 Program 19-Response by Step Integration / 422 19.5 Newmark Beta Method / 423 19.6 Elastoplastic Behavior of Framed Structures / 425 19.7 Member Stiffness Matrix / 425 19.8 Member Mass Matrix / 428 19.9 Rotation of Plastic Hinges / 430 19.10 Calculation of Member Ductility Ratio / 431

PART IV STRUCTURES MODELED WITH DISTRIBUTED PROPERTIES 435

20 DYNAMIC ANALYSIS OF SYSTEMS WITH DISTRIBUTED PROPERTIES 437

20.1 Flexural Vibration of Uniform Beams / 438 20.2 Solution of the Equation of Motion in Free Vibration / 439 20.3 Natural Frequencies and Mode Shapes for Uniform Beams / 441 20.4 Orthogonality Condition Between Normal Modes / 450 20.5 Forced Vibration of Beams / 452 20.6 Dynamic Stresses in Beams / 457

21 DISCRETIZATION OF CONTINUOUS SYSTEMS 462

2l.1 Dynamic Matrix for Flexural Effects / 463 21.2 Dynamic Matrix for Axial Effects / 465 2l.3 Dynamic Matrix for Torsional Effects / 467 2l.4 Beam Flexure Including Axial-Force Effect / 469 2l.5 Power Series Expansion of the Dynamic Matrix for Flexural

Effects / 472 2l.6 Power Series Expansion of the Dynamic Matrix for Axial and for

Torsional Effects / 474 2l. 7 Power Series Expansion of the Dynamic Matrix Including the

Effect of Axial Forces / 475

PART V RANDOM VIBRATION 477

22 RANDOM VillRA TION 479

22.1 Statistical Description of Random Functions / 480 22.2 The Normal Distribution / 482 22.3 The Rayleigh Distribution / 485 22.4 Correlation / 486 22.5 The Fourier Transform / 488 22.6 Spectral Analysis / 489 22.7 Spectral Density Function / 493

Contents xiii

22.8 Narrow-Band and Wide-Band Random Processes / 496 22.9 Response to Random Excitation / 499

PART VI EARTHQUAKE ENGINEERING 509

23 EQUIVALENT STATIC LATERAL FORCE METHOD: UNIFORM BUILDING CODE 1985 511

23.1 Earthquake Ground Motion / 512 23.2 Equivalent Seismic Lateral Force / 513 23.3 Distribution of the Lateral Seismic Force / 521 23.4 Horizontal Torsional Moment / 523 23.5 Overturning Moments and Story Shear / 524 23.6 Story Drift / 524 23.7 Diaphragm Forces / 524 23.8 Program 22-UBC 85 / 530 23.9 Design Process / 536

24 EQUIV ALENT STATIC LATERAL FORCE METHOD: UNIFORM BUILDING CODE-1988 545

24.1 Earthquake-Resistant Design Methods / 546 24.2 Static Lateral Force Method / 546 24.3 Distribution of Lateral Forces / 550 24.4 Story Shear Force / 552 24.5 Horizontal Torsional Moment / 552 24.6 Overturning Moment / 552 24.7 Story Drift Limitation / 553

xiv Contents

24.8 P-Delta Effects (P-d) / 553 24.9 Diaphragm Design Force / 555 24.10 Program 23 UBC-1988: Equivalent Static Lateral Force

Method / 561

25 DYNAMIC METHOD: UNIFORM BULDING CODE-1988 568

25.1 Modal Seismic Response of Buildings / 568 25.2 Total Design Values / 574 25.3 Provision of UBC-88: Dynamic Method / 576 25.4 Scaling Results / 577 25.5 Program 24 UBC-1988: Dynamic Lateral Force Method / 583

APPENDIX I: ANSWERS TO PROBLEMS IN PART I 593

APPENDIX II: COMPUTER PROGRAMS 601

APPENDIX III: ORGANIZATIONS AND THEIR ACRONYMS 605

GLOSSARY 609

SELECTED BIBLIOGRAPHY 617

INDEX 621

Preface to the Third Edition

The basic structure of the two previous editions is maintained in this third edition, although numerous revisions and additions have been introduced. Three new chapters on earthquake-resistant design of buildings have been incor­porated into Part VI of the book. The computer programs formerly written in FORTRAN for execution on mainframe computers have been expanded and rewritten in BASIC for implementation on microcomputers. Two indepen­dent packages of computer programs are used throughout the book: A struc­tural dynamics package of 20 interactive computer programs and an earth­quake-resistant design package of 10 interactive programs.

The structural dynamics package includes programs to determine the re­sponse of a single oscillator in the time domain, or in the frequency domain using the FFT (Fast Fourier Transform). It also includes a program to deter­mine the response of an inelastic system with elastoplastic behavior, and an­other program for the development of seismic response spectral charts. A set of seven computer programs is included for use in modeling structures as two-dimensional and three-dimensional frames and trusses. Finally, other programs, incorporating modal superposition or a step-by-step time history

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xvi Preface to the Third Edition

solution, are provided for calculation of the responses to forces or motions exciting the structure.

The new chapters in earthquake-resistant design of buildings describe the provisions of both the 1985 and 1988 versions of the UBC (Uniform Building Code) for the static lateral force method and for the dynamic lateral force method. Other revisions of the book include the presentation of the New­mark beta method to obtain the time history response of dynamic systems, and the direct integration method in which the response is found assuming that the excitation function is linear for a specified time interval. A modifi­cation of the dynamic condensation method, which has been developed re­cently by the author for the reduction of eigenproblems, is presented in Chap­ter 13. The proposed modification substantially reduces the numerical operation required in the implementation of the dynamic condensation method.

The subjects in this new edition are organized in six parts. Part I deals with structures modeled as single degree-of-freedom systems. It introduces basic concepts and presents important methods for the solution of such dynamic systems. Part II introduces important concepts and methodology for multi­degree-of-freedom systems through the use of structures modeled as shear buildings. Part III describes methods for the dynamic analysis of framed struc­tures modeled as discrete systems with many degrees of freedom. Part IV pre­sents the mathematical solution for some simple structures modeled as sys­tems with distributed properties, thus having an infinite number of degrees of freedom. Part V, which contains one chapter, introduces the reader to the fascinating topic of random vibration. Finally, Part VI presents the important current topic of earthquake engineering with applications to earthquake­resistant design of buildings following the provisions of the Uniform Building Codes of 1985 and 1988.

Several considerations-different sizes of diskettes (3 1/2 inch or 51/ 4 inch), different versions of the programs (compiled or not compiled), and the antic­ipation of future updating-contributed to a final decision not to include in the book the program diskettes. Diskettes with the computer programs, in several versions, are available directly from the author. A convenient form to order the selected version of the programs is provided in the back of the book.

The author believes that, just as knowledge of a combination of figures and turns is needed to open a safe, a combination of knowledge on the subjects of mathematics, theory of structures, and computer programming is needed today for successful professional practice in engineering. To provide the reader with such a combination of knowledge has been the primary objective of this book. The reader may wish to inform the author on the extent to which this objective has been fulfilled.

Many students, colleagues, and practicing professionals have suggested im­provements, identified typographical errors, and recommended additional top-

Preface to the Third Edition xvii

ics for inclusion. All these suggestions were carefully considered and have been included in this third edition whenever possible.

During the preparation of this third edition, I became indebted to many people to whom I wish to express my appreciation. I am grateful to John D. Hooper and to Robert D. Anderson, consulting engineers on the west coast, who most diligently reviewed the three new chapters in earthquake engi­neering. Their commentary and suggestions were most useful in making the presentation of the material in these chapters closer to present engineering practice. I am thankful to my colleague Dr. Joseph Hagerty for carefully read­ing and editing the manuscript of the new chapters. I am also grateful to Dean Leo Jenkins of Speed Scientific School and to Dr. Manuel Schwartz, of the Department of Physics, who reviewed the manuscript at the early stages of its preparation.

A special acknowledgment of gratitude is extended to my friend Dr. Edwin A. Tuttle, Professor of Education, who most kindly spent many hours check­ing my English grammar. My thanks also go to Miss Debbie Gordon for her competent typing of the manuscript.

To those people whom I recognized in the prefaces to the first and second editions for their help, I again express my wholehearted appreciation. Finally, I give due recognition to my wife, Jean, who with infinite patience and ded­ication helped me in editing the whole manuscript. Any remaining errors are mine.

MARIO PAZ

Preface to the First Edition

Natural phenomena and human activities impose forces of time-dependent variability on structures as simple as a concrete beam or a steel pile, or as complex as a multistory building or a nuclear power plant constructed from different materials. Analysis and design of such structures subjected to dy­namic loads involve consideration of time-dependent inertial forces. The re­sistance to displacement exhibited by a structure may include forces which are functions of the displacement and the velocity. As a consequence, the governing equations of motion of the dynamic system are generally nonlinear partial differential equations which are extremely difficult to solve in math­ematical terms. Nevertheless, recent developments in the field of structural dynamics enable such analysis and design to be accomplished in a practical and efficient manner. This work is facilitated through the use of simplifying assumptions and mathematical models, and of matrix methods and modem computational techniques.

In the process of teaching courses on the subject of structural dynamics, the author came to the realization that there was a definite need for a text which would be suitable for the advanced undergraduate or the beginning graduate engineering student being introduced to this subject. The author is

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familiar with the existence of several excellent texts of an advanced nature but generally these texts are, in his view, beyond the expected comprehension of the student. Consequently, it was his principal aim in writing this book to incorporate modem methods of analysis and techniques adaptable to com­puter programming in a manner as clear and easy as the subject permits. He felt that computer programs should be included in the book in order to assist the student in the application of modem methods associated with computer usage. In addition, the author hopes that this text will serve the practicing engineer for purposes of self-study and as a reference source.

In writing this text, the author also had in mind the use of the book as a possible source for research topics in structural dynamics for students work­ing toward an advanced degree in engineering who are required to write a thesis. At Speed Scientific School, University of Louisville, most engineering students complete a fifth year of study with a thesis requirement leading to a Master in Engineering degree. The author'S experience as a thesis advisor leads him to believe that this book may well serve the students in their search and selection of topics in subjects currently under investigation in structural dynamics.

Should the text fulfill the expectations of the author in some measure, par­ticularly the elucidation of this subject, he will then feel rewarded for his efforts in the preparation and development of the material in this book.

MARIO PAZ December, 1979