STRUCTURAL DYNAMICS Theory and Computation

Fifth Edition

STRUCTURAL DYNAMICS Theory and Computation

Fifth Edition

Mario Paz Speed Scientific School University of Louisville

Louisville, KY

William Leigh University of Central Florida

Orlando, FL .

SPRINGER SCIENCE+BUSINESS MEDIA, LLC

Library of Congress Cataloging-in-Publication Data

Paz, Mario. Structural Dynamics: Theory and Computation I by Mario Paz, William Leigh.-5th ed. p.cm. Includes bibliographical references and index. Additional material to this book can be downloaded from http://extras.springer.com

ISBN 978-1-4613-5098-9 ISBN 978-1-4615-0481-8 (eBook) DOI 10.1007/978-1-4615-0481-8 I. Structural dynamics. I. Title.

Copyright© 2004 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 2004 Softcover reprint of the bardeover 5th edition 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo­copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC.

Pennission for books published in Europe: pennissions@wkap.nl Pennissions for books published in the United States of America: pennissions@wkap.com

Printedon acid-free paper.

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CONTENTS

PREFACE TO THE FIFTH EDITION xvii

PREFACE TO THE FIRST EDITION xxi

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

1 UNDAMPED SINGLE-DEGREE-OF-FREEDOM SYSTEM 3

1.1 Degrees of Freedom 4 1.2 Undamped System 5 l.3 Springs in Parallel or in Series 7 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 12 1.8 Frequency and Period 14 1.9 Amplitude of Motion 16 1.10 Summary 22 1.11 Problems 23

2 DAMPED SINGLE-DEGREE-OF-FREEDOM SYSTEM 31

2.1 Viscous Damping 31 2.2 Equation of Motion 32 2.3 Critically Damped System 33 2.4 Overdamped System 34 2.5 Underdamped System 35 2.6 Logarithmic Decrement 37 2.7 Summary 44 2.8 Problems 45

viii Contents

3 RESPONSE OF ONE-DEGREE-OF-FREEDOM SYSTEM TO HARMONIC LOADING 49

3.1 Harmonic Excitation: Undamped System 49 3.2 Harmonic Excitation: Damped System 52 3.3 Evaluation of Damping at Resonance 60 3.4 Bandwidth Method (Half-Power) to Evaluate Damping 61 3.5 Energy Dissipated by Viscous Damping 63 3.6 Equivalent Viscous Damping 64 3.7 Response to Support Motion 67 3.8 Force Transmitted to the Foundation 75 3.9 Seismic Instruments 78 3.10 Response of One-Degree-of-Freedom System

to Harmonic Loading Using SAP2000 80 3.11 Summary 92 3.12 Analytical Problem 94 3.13 Problems 96

4 RESPONSE TO GENERAL DYNAMIC LOADING 101

4.1 Duhamel's Integral-Undamped System 101 4.2 Duhamel's Integral-Damped System 110 4.3 Response by Direct Integration 110 4.4 Solution of the Equation of Motion 112 4.5 Program 2-Response by Direct Integration 117 4.6 Program 3-Response to Impulsive Excitation 120 4.7 Response to General Dynamic Loading Using SAP2000 126 4.8 Summary 137 4.9 Analytical Problems 137 4.10 Problems 141

5 RESPONSE SPECTRA 149

5.1 Construction of Response Spectrum 149 5.2 Response Spectrum for Support Excitation 153 5.3 Tripartite Response Spectra 154 5.4 Response Spectra for Elastic Design 157 5.5 Influence of Local Soil Conditions 161 5.6 Response Spectra for Inelastic Systems 163 5.7 Response Spectra for Inelastic Design 166 5.8 Program 6-Seismic Response Spectra 171 5.9 Summary 174 5.10 Problems 174

Contents ix

6 NONLINEAR STRUCTURAL RESPONSE 179

6.1 Nonlinear Single Degree-of-Freedom Model 179 6.2 Integration of the Nonlinear Equation of Motion 181 6.3 Constant Acceleration Method 182 6.4 Linear Acceleration Step-by-Step Method 184 6.5 The Newmark Beta Method 187 6.6 Elastoplastic Behavior 188 6.7 Algorithm for the Step-by-Step Solution for Elastoplastic

Single-Degree-of-Freedom System 190 6.8 Program 5-Response for Elastoplastic Behavior 196 6.9 Summary 198 6.10 Problems 198

PART II STRUCTURES MODELED AS SHEAR BUILDINGS 203

7 FREE VIBRATION OF A SHEAR BUILDING 205

7.1 Stiffness Equations for the Shear Building 205 7.2 Natural Frequencies and Normal Modes 209 7.3 Orthogonality Property of the Normal Modes 214 7.4 Rayleigh's Quotient 218 7.5 Program 8-Natural Frequencies and Normal Modes 220 7.6 Free Vibration of a Shear Building Using SAP2000 221 7.7 Summary 225 7.8 Problems 227

8 FORCED MOTION OF SHEAR BUILDING 231

8.1 Modal Superposition Method 231 8.2 Response of a Shear Building to Base Motion 238 8.3 Program 9-Response by Modal Superposition 244 8.4 Harmonic Forced Excitation 246 8.5 Program 10-Harmonic Response 251 8.6 Forced Motion Using SAP2000 254 8.7 Combining Maximum Values of Modal Response 265 8.8 Summary 266 8.9 Problems 267

9 REDUCTION OF DYNAMIC MATRICES 271

9.1 Static Condensation 271 9.2 Static Condensation Applied to Dynamic Problems 275 9.3 Dynamic Condensation 285 9.4 Modified Dynamic Condensation 293 9.5 Program 12-Reduction of the Dynamic Problem 296 9.6 Summary 299 9.7 Problems 299

x Contents

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

10 DYNAMIC ANALYSIS OF BEAMS 305

10.1 Shape Functions for a Beam Segment 305 10.2 System Stiffuess Matrix 311 10.3 Inertial Properties-Lumped Mass 314 10.4 Inertial Properties-Consistent Mass 315 10.5 Damping Properties 320 10.6 External Loads 320 10.7 Geometric Stiffuess 322 10.8 Equations of Motion 326 10.9 Element Forces at Nodal Coordinates 333 10.10 Program 13-Modeling Structures as Beams 336 10.11 Dynamic Analysis of Beams Using SAP2000 339 10.12 Summary 347 10.13 Problems 347

11 DYNAMIC ANALYSIS OF PLANE FRAMES 353

ILl Element Stiffuess Matrix for Axial Effects 353 11.2 Element Mass Matrix for Axial Effects 355 11.3 Coordinate Transformation 359 11.4 Program 14-Modeling Structures as Plane Frames 367 11.5 Dynamic Analysis of Frames Using SAP2000 370 11.6 Summary 376 11.7 Problems 376

12 DYNAMIC ANALYSIS OF GRID FRAMES 381

12.1 Local and Global Coordinate Systems 381 12.2 Torsional Effects 382 12.3 Stiffuess Matrix for a Grid Element 384 12.4 Consistent Mass Matrix for a Grid Element 385 12.5 Lumped Mass Matrix for a Grid Element 385 12.6 Transformation of Coordinates 386 12.7 Program 15-Modeling Structures as Grid Frames 392 12.8 Dynamic Analysis of Grid Frames Using SAP2000 395 12.9 Summary 403 12.10 Problems 403

13 DYNAMIC ANALYSIS OFTHREE-DIMENSIONAL FRAMES 407

13.1 Element Stiffuess Matrix 407 13.2 Element Mass Matrix 409 13.3 Element Damping Matrix 410

Contents xi

13.4 Transformation of Coordinates 410 13.5 Differential Equation of Motion 414 13.6 Dynamic Response 415 13.7 Program 16-Modeling Structures as Space Frames 415 13.8 Dynamic Response of Three-Dimensional Frames

UsingSAP2000 418 13.9 Summary 426 13.10 Problems 427

14 DYNAMIC ANALYSIS OF TRUSSES 429

14.1 Stiffness and Mass Matrices for the Plane Truss 429 14.2 Transformation of Coordinates 432 14.3 Program 17-Modeling Structures as Plane Trusses 438 14.4 Stiffness and Mass Matrices for Space Trusses 441 14.5 Equation of Motion for Space Trusses 443 14.6 Program 18-Modeling Structures as Space Trusses 444 14.7 Dynamic Analysis of Trusses Using SAP2000 446 14.8 Summary 459 14.9 Problems 459

15 DYNAMIC ANALYSIS OF STRUCTURES USING THE FINITE ELEMENT METHOD 463

15.1 Plane Elasticity Problems 464 15.1.1 Triangular Plate Elementfor Plane Elasticity problems 465 15.1.2 SAP2000 for Plane Elasticity Problem 472

15.2 Plate Bending 477 15.2.1 Rectangular Elementfor Plate Bending 478 15.2.2 SAP2000for Plate Bending and Shell Problems 484

15.3 Summary 491 15.4 Problems 493

16 TIME HISTORY RESPONSE OF MULTIDEGREE-OF-FREEDOM SYSTEMS 495

16.1 Incremental Equations of Motion 495 16.2 The Wilson-BMethod 497 16.3 Algorithm for Step-by-Step Solution of a Linear System Using the

Wilson-BMethod 500 16.3.1 Initialization 500 16.3.2 For Each Time Step 500

16.4 Program 19-Response by Step Integration 505 16.5 The Newmark Beta Method 506 16.6 Elastoplastic Behavior of Framed Structures 508 16.7 Member Stiffness Matrix 508 16.8 Member Mass Matrix 511 16.9 Rotation of Plastic Hinges 513

xii Contents

16.10 Calculation of Member Ductility Ratio 514 16.11 Time-History Response of Multi degree-of-Freedom Systems Using

SAP2000 515 16.12 Summary 521 16.13 Problems 522

PART IV STRUCTURES MODELED WITH DISTRIBUTED PROPERTIES 525

17 DYNAMIC ANALYSIS OF SYSTEMS WITH DISTRIBUTED PROPERTIES 527

17.1 Flexural Vibration of Uniform Beams 527 17.2 Solution ofthe Equation of Motion in Free Vibration 529 17.3 Natural Frequencies and Mode Shapes for Uniform Beams 531

17.3.1 Both Ends Simply Supported 531 17.3.2 Both Ends Free (Free Beam) 534 17.3.3 Both Ends Fixed 535 17.3.4 One End Fixed and the other End Free

(Cantilever Beam) 537 17.3.5 One End Fixed and the other End Simply Supported 538

17.4 Orthogonality Condition Between Normal Modes 540 17.5 Forced Vibration of Beams 542 17.6 Dynamic Stresses in Beams 547 17.7 Summary 549 17.8 Problems 550

18 DISCRETIZATION OF CONTINUOUS SYSTEMS 553

18.1 Dynamic Matrix for Flexural Effects 554 18.2 Dynamic Matrix for Axial Effects 556 18.3 Dynamic Matrix for Torsional Effects 558 18.4 Beam Flexure Including Axial-Force Effect 560 18.5 Power Series Expansion of the Dynamic Matrix for Flexural

Effects 563 18.6 Power Series Expansion of the Dynamic Matrix for Axial and for

Torsional Effects 564 18.7 Power Series Expansion of the Dynamic Matrix

Including the Effects of Axial Forces 565 18.8 Summary 566

PART V SPECIAL TOPICS: Fourier Analysis, Evaluation of Absolute Damping, Generalized Coordinates 567

19 FOURIER ANALYSIS AND RESPONSE IN THE FREQUENCY DOMAIN 569

19.1 Fourier Analysis 569

19.2 Response to a Loading Represented by Fourier Series 19.3 Fourier Coefficients for Piecewise Linear Functions 19.4 Exponential Form of Fourier Series 19.5 Discrete Fourier Analysis 19.6 Fast Fourier Transform 19.7 Program 4-Response in the Frequency Domain 19.8 Summary 19.9 Problems

Contents xiii

570 573 574 575 578 580 586 586

20 EVALUATION OF ABSOLUTE DAMPING FROM MODAL DAMPING RATIOS 593

20.1 Equations for Damped Shear Building 593 20.2 Uncoupled Damped Equations 595 20.3 Conditions for Damping Uncoupling 596 20.4 Program II-Absolute Damping From Modal Damping Ratios 602 20.5 Summary 604 20.6 Problems 604

21 GENERALIZED COORDINATES AND RAYLEIGH'S METHOD 607

21.1 Principle of Virtual Work 607 21.2 Generalized Single-Degree-of-Freedom System-Rigid Body 609 21.3 Generalized Single-Degree-of-Freedom System-

Distributed Elasticity 612 21.4 Shear Forces and Bending Moments 617 21.5 Generalized Equation of Motion for a Multistory Building 621 21.6 Shape Function 624 21.7 Rayleigh's Method 628 21.8 Improved Rayleigh's Method 636 21.9 Shear Walls 639 21.10 Summary 642 21.11 Problems 643

PART VI RANDOM VIBRATION 649

22 RANDOM VIBRATION

22.1 Statistical Description of Random Functions 22.2 Probability Density Function 22.3 The Normal Distribution 22.4 The Rayleigh Distribution 22.5 Correlation 22.6 The Fourier Transform 22.7 Spectral Analysis 22.8 Spectral Density Function 22.9 Narrow-Band and Wide-Band Random processes

651

652 654 656 657 659 663 665 669 671

xiv Contents

22.10 Response to Random Excitation: Single-Degree-of-Freedom System 675

22.11 Response to Random Excitation: Multiple-Degree-of-Freedom System 681

22.11.1 Relationship Between Complex Frequency Response and Unit Impulse Response 681

22.11.2 Response to Random Excitation: Two-degree-of-freedom System 683

22.11.3 Response to Random Excitation: N Degree of Freedom System 688

22.12 Summary 691 22.13 Problems 692

PART VII EARTHQUAKE ENGINEERING 697

23 UNIFORM BUILDING CODE 1997: EQUIVALENT LATERAL FORCE METHOD 699

23.1 Earthquake Ground Motion 700 23.2 Equivalent Lateral Force Method 703 23.3 Earthquake-Resistant Design Methods 703 23.4 Seismic Zone Factor 703 23.5 Base Shear Force 704 23.6 Distribution of Lateral Seismic Forces 711 23.7 Story Shear Force 711 23.8 Horizontal Torsional Moment 712 23.9 Overturning Moment 713 23.10 P-Delta Effect (P-A) 713 23.11 RedundancylReliability Factor p 715 23.12 Story Drift Limitation 715 23.13 Diaphragm Design Forces 716 23.14 Earthquake Load Effect 717 23.15 Irregular Structures 717 23.16 Summary 726 23.17 Problems 726

24 UNIFORM BUILDING CODE 1997: DYNAMIC METHOD 731

24.1 Modal Seismic Response of Buildings 731 24.1.1 Modal Equation and Participation Factor 732 24.1.2 Modal Shear Force 733 24.1.3 Effective Modal Weight 734 24.1.4 Modal Lateral Forces 735 24.1.5 Modal Displacements 735 24.1.6 Modal Drift 736 24.1.7 Modal Overturning Moment 736 24.1.8 Modal Torsional Moment 737

24.2 Total Design Values 737

24.3 24.4 24.5 24.6 24.7

Contents xv

Provisions ofUBC-97: Dynamic Method Scaling of Results Program 24-UBC 1997 Dynamic Lateral Force Method Summary Problems

738 740 750 754 755

25 INTERNATIONAL BUILDING CODE IBC-2000 757

25.1 Response Spectral Acceleration: Ss, SJ 757 25.2 Soil Modification Response Spectral Acceleration: SMS, SMl 758 25.3 Design Response Spectral Acceleration: SDS, SOl 759 25.4 Site Class Defmition: A, B, ..... F 760 25.5 Seismic Use Group (SUG) and Occupancy Importance Factor (IE)760 25.6 Seismic Design Category (A, B, C, D, E and F) 761 25.7 Design Response Spectral Curve: Sa v.s. T 763 25.8 Determination of the Fundamental Period 766 25.9 Minimum lateral Force Procedure

[mC-2000: Section 1616.4.1] 767 25.10 SimplifiedAnalysisProcedure [mC-2000: Section 1617.5] 768

25.10.1 Seismic Base Shear 768 25.10.2 Response Modification Factor R 768 25.10.3 Vertical Distribution o/Lateral Forces 769

25.11 Equivalent Seismic Lateral Force Method: [mC-2000: Section 1617.4] 769 25.11.1 Distribution o/Lateral Forces 771 25.11.2 Overturning Moments 771 25.11.3 Horizontal Torsional Moment 772 25.11.4 P-Delta Effect (P-LJ) 772 25.11.5 Story Drift 773

25.12 Redundancy/Reliability Factor 774 25.13 Earthquake Load Effect 775 25.14 Building Irregularities 775 25.15 Summary 781

APPENDICES

Appendix I: Answers to Problems in Selected Chapters

Appendix II: Computer Programs

Appendix III: Glossary

Selected Bibliography

Index

783

785

793

795

803

807

PREFACE TO THE FIFTH EDITION

The basic structure of the four previous editions is maintained in this fifth edition, although numerous revisions and additions have been introduced. The three chapters on Earthquake Engineering have been rewritten to present the most recent versions of the Uniform Building Code (UBC-97) and of the new International Building Code (IBC-2000) as in the fourth edition. A new chapter to serve as an introduction for the dynamic analysis of structures using the Finite Element Method has been incorporated in Part III, Structures Modeled as Discrete Multidegree-of-Freedom Systems. The chapter on Random Vibration has been extended to include the response of structures modeled as a multidegree-of-freedom system, subjected to several random forces or to a random motion at the base of the structure. The concept of damping is discussed more thoroughly, including the evaluation of equivalent viscous damping. The constant acceleration method to determine the response of nonlinear dynamic systems is presented in addition to the linear acceleration method presented in past editions. Chapter 5 on Response Spectra now includes the development of seismic response spectra with consideration of local soil conditions at the site of the structure. The secondary effect resulting from the lateral displacements of the building, commonly known as the P-L1 effect, is explicitly considered through the calculation of the geometric stitfuess matrix. Finally, a much larger number of solved illustrative examples using the educational computer programs developed by the author or using the professional program SAP2000 have been incorporated in various chapters of the book.

The use of the professional computer program SAP2000 for the analysis and

xviii Preface to the Fifth Edition

solution of structural dynamics problems is introduced in this new edition. This program was selected from among the various professional programs available because of its capability in solving complex problems in structures as well as its wide use in professional practice by structural engineers. SAP2000 includes routines for the analysis and design of structures with linear or nonlinear behavior subjected to static or dynamics loads; (material non-linearity or large displacements non-linearities) and may be used most efficiently in the microcomputer. The larger versions of SAP2000 have the capability for the analysis of structures modeled with virtually any large number of nodes. This new fifth edition of the book uses, almost exclusively, the introductory version of SAP2000 which has a capability limited to 25 nodes or 25 elements. A CD­ROM containing the introductory version of SAP2000 as well as the educational set of the program developed by the author is included in this 5th edition of Structural Dynamics: Theory and Computation.

The set of educational programs in Structural Dynamics includes programs to determine the response in the time domain or in the frequency domain using the FFT (Fast Fourier Transform) of structures modeled as a single oscillator. Also included is 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 modeling structures as two or-three-dimensional frames and trusses. Finally, other programs, incorporating modal superposition or a step-by-step time-history solution, are provided for calculation of the responses to forces or motions exciting the structure. This fifth edition also includes a program to determine the response of single or multiple-degree-of-freedom systems subjected to random excitations.

The book is organized in six parts. Part I deals with structures modeled as single-degree-of-freedom systems. It introduces basic concepts in Structural Dynamics and presents important methods for the solution of such dynamic systems. Part II introduces important concepts and methodology for multi-degree-of-freedom systems through the use of structures modeled as shear buildings. Part III describes in detail the Matrix Structural Analysis for modeling skeletal type of structures (beams, frames, and trusses) as discrete systems in preparation for dynamic analysis. Part III also includes a chapter to serve as an introduction to the Finite Element Method (F.E.M.) for modeling continuous structures such as plates for dynamic analysis. Part IV presents the mathematical solution for some simple structures, such as beams, modeled as systems with distributed properties, thus having an infinite number of degrees of freedom. Part V on Special Topics presents: an introduction to the magnificent Fourier Method and the use of the Fast Fourier Transform; an extension of the modeling complex structures as one degree-of-freedom systems through the use of Generalized Coordinates and of Rayleigh Method; and methods to evaluate absolute damping in structures from estimated modal damping coefficients. Part VI, which contains one chapter, introduces the reader to the complex but fascinating topic of Random Vibrations for the analysis of single degree of freedom systems, as well as for the analysis of structures modeled as multi-degree of freedom systems. Finally, Part VII presents the important current topic of Earthquake Engineering with applications to earthquake-resistant design of buildings

Preface to the Fifth Edition xix

following the provisions of the Uniform Building Code and of the new International Building Code in use in The United States of America.

The author believes that a combination of knowledge on the subjects of applied mathematics, theory of structures, and computer programs is needed today for successful professional practice of engineering, just as knowledge of a combination of numbers and turns is needed to open a safe. 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 topics for inclusion. In this new edition all these suggestions were carefully considered and have been included in this fifth edition whenever possible.

During the preparation of this fifth edition, I became indebted to many people to whom I wish to express my appreciation: First of all I am most grateful to many of my students who helped me through their inquisitive discussions in class to improve and clarify my presentation of the various topics in this book. It is now with great pain that I wish to recognize posthumously the preliminary work done by my student Elaine Fonseca, who prepared changes to some drawings from the Fourth Edition. Her tragic death was most unfortunately a great loss of a most promising engineering student. She will be sorely missed by her family, friends, fellow students and this instructor. I wish also to recognize and thank my graduate students, Xiaobing Cui and Zhiyong Zhao, for their diligent collaboration and expert use of scanning equipment to retrieve text and figures from the previous edition of this textbook. I am most grateful to my former colleague; Dr. Michael A. Cassaro, who diligently checked the chapter on the Finite Element Method and to Dr. Julius Wong, of the Department of Mechanical Engineering, whose comments and discussions helped me to refme my exposition. I am also grateful to my friend Dr. Farzad Naeim who has collaborated with me on the chapter Seismic Response Spectra in the "International Handbook of Earthquake Engineering: Codes, Programs and Examples" of which I am the editor. I have incorporated some of the material from the Handbook in updating the chapter on Response Spectra. I also wish to acknowledge Dr. Luis E. Suarez from the University of Puerto Rico in Mayaguez, who provided me with copies of his work in Random Vibration and a copy of his class notes on the Finite Element Method.

I also like to take this opportunity to thank my colleague, Dr Joseph Hagerty for his past help of many years ago, in the 1970s, at the time when I was just playing with the plan of writing a textbook in Structural Dynamics, without my knowledge, he approached a publishing company a initiated a contract in my name for the publication of the first edition of this textbook in 1980.

A special acknowledgement of gratitude is extended to my friend Dr. Assraf Habibullah, president of Computers and Structures Inc., who most kindly authorized me to include in this volume the introductory version of SAP2000. In addition, Dr.

xx Preface to the Fifth Edition

Habibulla provided me with the full version of SAP 2000 so I could solve problems beyond the capability of the introductory version. I am also most grateful to two other computer scientists of that company, Drs. Syed Hasnain and Bob Morris who most patiently tutored me and clarified many of the intricacies in the use of SAP2000. The senior author is certainly very grateful to the co-author, Dr. William Leigh for his contribution in reviewing and editing this volume, especially those sections which used the computer programs. To those people whom I recognized in the prefaces to the previous editions for their help, I again express my wholehearted appreciation.

Finally, I thank my wife, Annis, who most diligently helped me with great proficiency in the final preparation of this new edition to be camera ready for publication. Her dedication to the work as well as her continuous support and encouragement is deeply appreciated. In recognition of her indispensable help, this new edition is duly dedicated to her.

Mario paz September 2003

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 defmite need for a text which would be suitable for the advanced undergraduate or the beginning graduate engineering student

xxii Preface to the First Edition

being introduced to this subject. The author is 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