Structural Dynamics Theory and Computation

Fou rth Edition

Structural Dynamics Theory and Computation Fourth Edition

Mario Paz Speed Scientific School University of Louisville Louisville, KY

.....

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Library of Congress Cataloging-in-Publication

Paz, Mario. Structural dynamics: theory and computation / by Mario Paz.-- 4th ed.

p. cm. Includes bibliographical references and index.

ISBN-13: 978-1-4684-0020-5 e-ISBN-13: 978-1-4684-0018-2 DOl: 10.1007/978-1-4684-0018-2 1. Structural dynamics. 1. Title.

TA654.P39 1997 624.1 '7--dc21

British Library Cataloguing in Publication Data available

Copyright <0 1997 by Chapman & Hall Fifth Printing 2003 by Kluwer Academic Publishers

Softcover reprint of the hardcover 4th edition 2003

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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 Sal rna Misri Paz

v

CONTENTS

PREFACE TO THE FOURTH EDITION / xv

PREFACE TO THE FIRST EDITION / xxi

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

UNDAMPED SINGLE-DEGREE-OF-FREEDOM SYSTEM 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 I 10 1.7 Solution of the Differential Equation of Motion I 11 1.8 Frequency and Period / 13 1.9 Amplitude of Motion I 15 1.10 Undamped Single-Degree-of-Freedom Systems Using COSMOS / 20 1.11 Summary / 22

Problems / 23

vii

viii Contents

2 DAMPED SINGLE-DEGREE-OF-FREEDOM SYSTEM 31

2.1 Viscous Damping I 31 2.2 Equation of Motion I 32 2.3 Critically Damped System I 33 2.4 Overdamped System I 34 2.5 Underdamped System I 35 2.6 Logarithmic Decrement I 37 2.7 Summary I 43

Problems I 44

3 RESPONSE OF ONE-DEGREE-OF-FREEDOM SYSTEM TO HARMONIC LOADING 47

3.1 Undamped System: Harmonic Excitation I 47 3.2 Damped System: Harmonic Excitation I 50 3.3 Evaluation of Damping at Resonance I 58 3.4 Bandwidth Method (Half-Power) to Evaluate Damping I 59 3.5 Energy Dissipated by Viscous Damping I 61 3.6 Equivalent Viscous Damping I 63 3.7 Response to Support Motion I 66 3.8 Force Transmitted to the Foundation I 76 3.9 Seismic Instruments I 79 3.10 Response of One-Degree-of-Freedom System to

Harmonic Loading Using COSMOS I 81 3.11 Summary I 88

Problems I 92

4 RESPONSE TO GENERAL DYNAMIC LOADING 96

4.1 Impulsive Loading and Duhamel's Integral I 96 4.2 Numerical Evaluation of Duhamel's Integral-Undamped

System I 105 4.3 Numerical Evaluation of Duhamel's Integral-Damped

System I 109 4.4 Response by Direct Integration I 110 4.5 Program 2-Response by Direct Integration I 116 4.6 Program 3-Response to Impulsive Excitation I 119 4.7 Response to General Dynamic Loading Using COSMOS I 124 4.8 Summary I 131

Problems I 132

5 FOURIER ANALYSIS AND RESPONSE IN THE FREQUENCY DOMAIN 139

5.1 Fourier Analysis / 139

Contents ix

5.2 Response to a Loading Represented by Fourier Series / 140 5.3 Fourier Coefficients for Piecewise Linear Functions / 143 5.4 Exponential Form of Fourier Series / 144 5.5 Discrete Fourier Analysis / 145 5.6 Fast Fourier Transform / 148 5.7 Program 4-Response in the Frequency Domain / 150 5.8 Summary / 156

Problems / 156

6 GENERALIZED COORDINATES AND RAYLEIGH'S METHOD 162

6.1 Principle of Virtual Work / 162 6.2 Generalized Single-Degree-of-Freedom System-Rigid

Body / 164 6.3 Generalized Single-Degree-of-Freedom System-Distributed

Elasticity / 167 6.4 Shear Forces and Bending Moments / 172 6.5 Generalized Equation of Motion for a Multistory Building / 177 6.6 Shape Function / 180 6.7 Rayleigh's Method / 185 6.8 Improved Rayleigh's Method / 192 6.9 Shear Walls /195 6.10 Summary / 199

Problems / 200

7 NONLINEAR STRUCTURAL RESPONSE 205

7.1 Nonlinear Single Degree-of-Freedom Model / 206 7.2 Integration of the Nonlinear Equation of Motion / 208 7.3 Constant Acceleration Method / 208 7.4 Linear Acceleration Step-by-Step Method / 211 7.5 The Newmark Beta Method / 214 7.6 Elastoplastic Behavior / 215 7.7 Algorithm for the Step-by-Step Solution for Elastoplastic

Single-Degree-of-Freedom System / 217 7.8 Program 5-Response for Elastoplastic Behavioar System / 221 7.9 Nonlinear Structural Response Using COSMOS / 224 7.10 Summary / 228

Problems / 229

x Contents

8 RESPONSE SPECTRA 233

8.1 Construction of Response Spectrum I 233 8.2 Response Spectrum for Support Excitation I 237 8.3 Tripartite Response Spectra I 238 8.4 Response Spectra for Elastic Design I 241 8.5 Influence of Local Soil Conditions I 245 8.6 Response Spectra for Inelastic Systems I 247 8.7 Response Spectra for Inelastic Design I 250 8.8 Program 6-Seismic Response Spectra I 257 8.9 Response Spectra Using COSMOS I 260 8.10 Summary I 265

Problems I 266

PART II STRUCTURES MODELED AS SHEAR BUILDINGS 271

9 THE MULTISTORY SHEAR BUILDING 271

9.1 Stiffness Equations for the Shear Building I 272 9.2 P-Ll Effect on a Plane Shear Building I 275 9.3 Flexibility Equations for the Shear Building I 278 9.4 Relationship Between Stiffness and Flexibility Matrices I 280 9.5 Program 7-Modeling Structures as Shear Buildings I 281 9.6 Summary I 283

Problems I 283

10 FREE VIBRATION OF A SHEAR BUILDING 287

10.1 Natural Frequencies and Normal Modes I 287 10.2 Orthogonality Property of the Normal Modes I 294 10.3 Rayleigh's Quotient I 298 10.4 Program 8-Natural Frequencies and Normal Modes I 300 10.5 Free Vibration of a Shear Building Using COSMOS / 301 10.6 Summary I 304

Problems / 305

II FORCED MOTION OF SHEAR BUILDING 310

11.1 Modal Superposition Method I 310 11.2 Response of a Shear Building to Base Motion I 317 11.3 Program 9-Response by Modal Superposition I 324

Contents xi

11.4 Harmonic Forced Excitation / 326 11.5 Program 10-Harmonic Response / 331 11.6 Combining Maximum Values of Modal Response / 334 11.7 Forced Motion of a Shear Building Using COSMOS / 335 11.8 Summary / 346

Problems / 348

12 DAMPED MOTION OF SHEAR BUILDINGS 352

12.1 Equations for Damped Shear Building / 353 12.2 Uncoupled Damped Equations / 354 12.3 Conditions for Damping Uncoupling / 355 12.4 Program II-Absolute Damping From Damping Ratios / 362 12.5 Summary / 364

Problems / 364

13 REDUCTION OF DYNAMIC MATRICES 366

13.1 Static Condensation / 367 13.2 Static Condensation Applied to Dynamic Problems / 370 13.3 Dynamic Condensation / 380 13.4 Modified Dynamic Condensation / 387 13.5 Program 12-Reduction of the Dynamic Problem / 391 13.6 Summary / 393

Problems / 393

PART III STRUCTURES MODELED AS DISCRETE MUL TIDEGREE-OF-FREEDOM SYSTEMS 397

14 DYNAMIC ANALYSIS OF BEAMS 399

14.1 Static Properties for a Beam Segment / 400 14.2 System Stiffness Matrix / 405 14.3 Inertial Properties-Lumped Mass / 408 14.4 Inertial Properties-Consistent Mass / 410 14.5 Damping Properties / 414 14.6 External Loads / 414 14.7 Geometric Stiffness / 416 14.8 Equations of Motion / 420 14.9 Element Forces at Nodal Coordinates / 427 14.10 Program 13-Modeling Structures as Beams / 430 14.11 Dynamic Analysis of Beams Using COSMOS / 433

xii Contents

14.12 Summary I 437 Problems I 438

15 DYNAMIC ANALYSIS OF PLANE FRAMES 442

IS.1 Element Stiffness Matrix for Axial Effects I 443 IS.2 Element Mass Matrix for Axial Effects I 444 IS.3 Coordinate Transformation I 449 IS.4 Program 14-Modeling Structures as Plane Frames I 4S8 IS.S Dynamic Analysis of Frames Using COSMOS I 460 IS.6 Summary I 46S

Problems I 466

16 DYNAMIC ANALYSIS OF GRIDS 469

16.1 Local and Global Coordinate Systems I 470 16.2 Torsional Effects I 471 16.3 Stiffness Matrix for a Grid Element I 472 16.4 Consistent Mass Matrix for a Grid Element I 473 16.5 Lumped Mass Matrix for a Grid Element I 473 16.6 Transformation of Coordinates I 474 16.7 Program IS-Modeling Structures as Grid Frames / 480 16.8 Dynamic Analysis of Grids Using COSMOS I 483 16.9 Summary I 487

Problems I 488

17 THREE-DIMENSIONAL FRAMES 491

17.1 Element Stiffness Matrix I 491 17.2 Element Mass Matrix I 493 17.3 Element Damping Matrix I 494 17.4 Transformation of Coordinates / 494 17.S Differential Equation of Motion I S03 17.6 Dynamic Response I 504 17.7 Program 16-Modeling Structures as Space Frames / S04 17.8 Dynamic Response of Three-Dimensional Frames Using

COSMOS / S07 17.9 Summary I SIO

Problems I SIO

18 DYNAMIC ANALYSIS OF TRUSSES 511

18.1 Stiffness and Mass Matrices for the Plane Truss I S12 18.2 Transformation of Coordinates I S14

Contents xiii

18.3 Program 17-Modeling Structures as Plane Trusses I 520 18.4 Stiffness and Mass Matrices for Space Trusses / 522 18.5 Equation of Motion for Space Trusses / 525 18.6 Program 18-Modeling Structures as Space Trusses / 526 18.7 Dynamic Analysis of Trusses Using COSMOS / 528 18.8 Summary / 536

Problems I 536

19 DYNAMIC ANALYSIS OF STRUCTURES USING THE FINITE ELEMENT METHOD 538

19.1 Plane Elasticity Problems / 539 19.1.1 Triangular Plate Element for Plane Elasticity

Problems / 540 19.1.2 Library of Plane Elasticity Elements

(2D Elements) / 552 19.2 Plate Bending I 555

19.2.1 Rectangular Finite Element for Plate Bending / 556 19.2.2 COSMOS Library of Plate and Shell Elements / 565

19.3 Summary / 573 Problems / 575

20 TIME HISTORY RESPONSE OF MULTIDEGREE-OF-FREEDOM SYSTEMS 577

20.1 Incremental Equations of Motion / 578 20.2 The Wilson- (J Method I 579 20.3 Algorithm for Step-by-Step Solution of a Linear System Using

the Wilson- (J Method I 582 20.3.1 Initialization / 582 20.3.2 For Each Time Step / 582

20.4 Program 19-Response by Step Integration / 587 20.5 Newmark Beta Method / 588 20.6 Elastoplastic Behavior of Framed Structures / 589 20.7 Member Stiffness Matrix / 590 20.8 Member Mass Matrix / 593 20.9 Rotation of Plastic Hinges / 595 20.10 Calculation of Member Ductility Ratio / 596 20.11 Time-History Response of Multidegree-of-Freedom Systems

Using COSMOS / 597 20.12 Summary I 602

Problems / 604

xiv Contents

PART IV STRUCTURES MODELED WITH DISTRIBUTED PROPERTIES 607

21 DYNAMIC ANALYSIS OF SYSTEMS WITH DISTRIBUTED PROPERTIES 609

21.1 Flexural Vibration of Uniform Beams I 610 21.2 Solution of the Equation of Motion in Free Vibration I 611 21.3 Natural Frequencies and Mode Shapes for Uniform Beams I 613

21.3.1 Both Ends Simply Supported I 613 21.3.2 Both Ends Free (Free Beam) I 617 21.3.3 Both Ends Fixed I 618 21.3.4 One End Fixed and the other End Free (Cantilever

Beam) I 620 21.3.5 One End Fixed and the other End Simply

Supported I 622 21.4 Orthogonality Condition Between Normal Modes I 622 21.5 Forced Vibration of Beams I 624 21.6 Dynamic Stresses in Beams I 630 21.7 Summary I 632

Problems I 633

22 DISCRETIZATION OF CONTINUOUS SYSTEMS 635

22.1 Dynamic Matrix for Flexural Effects I 636 22.2 Dynamic Matrix for Axial Effects I 638 22.3 Dynamic Matrix for Torsional Effects I 641 22.4 Beam Flexure Including Axial-Force Effect I 642 22.5 Power Series Expansion of the Dynamic Matrix for Flexural

Effects I 646 22.6 Power Series Expansion of the Dynamic Matrix for Axial and

for Torsional Effects I 648 22.7 Power Series Expansion of the Dynamic Matrix Including the

Effect of Axial Forces I 649 22.8 Summary I 650

PART V RANDOM VIBRATION 651

23 RANDOM VIBRATION 653

23.1 Statistical Description of Random Functions I 654 23.2 Probability Density Function I 657 23.3 The Normal Distribution I 659 23.4 The Rayleigh Distribution I 660

Contents xv

23.5 Correlation / 662 23.6 The Fourier Transfonn / 666 23.7 Spectral Analysis / 668 23.8 Spectral Density Function / 672 23.9 Narrow-Band and Wide-Band Random Processes / 675 23.10 Response to Random Excitation: Single-Degree-of-Freedom

System / 679 23.11 Response to Random Excitation: Multiple-Degree-of-Freedom

System / 685 23.12 Random Vibration Using COSMOS / 696 23.13 Summary / 700

PART VI EARTHQUAKE ENGINEERING 705

24 UNIFORM BUILDING CODE 1994: EQUIVALENT STATIC LATERAL FORCE METHOD 707

24.1 Earthquake Ground Motion / 708 24.2 Equivalent Seismic Lateral Force / 712 24.3 Earthquake-Resistant Design Methods / 712 24.4 Static Lateral Force Method / 713 24.5 Distribution of Lateral Forces / 718 24.6 Story Shear Force / 718 24.7 Horizontal Torsional Moment / 719 24.8 Overturning Moment / 720 24.9 Story Drift Limitation / 720 24.10 P-Delta Effect (P-.J) / 721 24.11 Diaphragm Design Force / 723 24.12 Program 23 UBC-94 Equivalent Static Lateral Force

Method / 732 24.13 Simplified Three Dimensional Earthquake Resistant Design of

Buildings / 739 24.13.1 Modeling the Building / 739 24.13.2 Transfonnation of Stiffness Coefficients / 740 24.13.3 Center of Rigidity / 742 24.13.4 Story Eccentricity / 743 24.13.5 Rotational Stiffness / 744 24.13.6 Fundamental Period / 745 24.13.7 Seismic Factors / 745 24.13.8 Base Shear Force / 746 24.13.9 Equivalent Lateral Seismic Forces / 746 24.13.10 Overturning Moments / 747

xvi Contents

24.13.10 Story Shear Force / 747 24.13.12 Torsional Moments / 747 24.13.13 Story Drift and Lateral Displacements / 748 24.13.14 Forces and Moments on Structural Elements / 749 24.13.15 Computer Program / 750

24.14 Equivalent Static Lateral Froce Method Using COSMOS / 756 24.15 Summary / 761

25 UNIFORM BUILDING CODE 1994: DYNAMIC METHOD 766

25.1 Modal Seismic Response of Buildings / 766 25.1.1 Modal Equation and Participation Factor / 767 25.1.2 Modal Shear Force / 768 25.1.3 Effective Modal Weight / 770 25.1.4 Modal Lateral Forces / 771 25.1.5 Modal Displacements / 771 25.1.6 Modal Drift / 772 25.1.7 Modal Overturning Moment / 772 25.1.8 Modal Torsional Moment / 772

25.2 Total Design Values / 773 25.3 Provisions of UBC-94: Dynamic Method / 774 25.4 Scaling of Results / 776 25.5 Program 24-UBC 1994 Dynamic Lateral Force Method / 783 25.6 Summary / 787

Problems / 788

APPENDICES / 789

Appendix I: Answers to Problems in Part I / 791

Appendix II: Computer Programs / 801

Appendix III: Organization and their Acronyms / 804

Glossary / 807

Selected Bibliography / 815

Index / 819

Diskette Order Form / 825

Preface to the Fourth Edition

The basic structure of the three previous editions is maintained in this fourth edition, although numerous revisions and additions have been introduced. 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 multidegree-of-freedom systems, SUbjected to several random forces or to a random motion at the base of the structure. The concept of damping including the evaluation of equivalent viscous damping is thoroughly discussed. 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 8, 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-LJ. effect, is explicitly considered through the calculation of the geometric stiffness matrix. Finally, a greater number of illustrative examples have been incorporated in the various chapters of the book using the educational computer programs developed by the author or the professional program COSMOS.

xvii

xviii Preface to the Fourth Edition

The use of COSMOS for the analysis and solution of structural dynamics problems is introduced in this new edition. The COSMOS program was selected from among the various professional programs available because it has the capability of solving complex problems in structures, as well as in other engin­eering fields such as Heat Transfer, Fluid Flow, and Electromagnetic Phenom­ena. COSMOS includes routines for Structural Analysis, Static, or Dynamics with linear or nonlinear behavior (material nonlinearity or large displacements), and can be used most efficiently in the microcomputer. The larger version of COSMOS has the capacity for the analysis of structures modeled up to 64,000 nodes. This fourth edition uses an introductory version that has a capability limited to 50 nodes or 50 elements. This version is included in the supplement, STRUCTURAL DYNAMICS USING COSMOS 1.

The sets of educational programs in Structural Dynamics and Earthquake Engineering that accompanied the third edition have now been extended and updated. These sets include programs to determine the response in the time or frequency domain using the FFf (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 a program for the development of seismic response spectral charts. A set of seven computer programs is included for modeling structures as two-dimensional and three­dimensional frames and trusses. 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. In addition, in this fourth edition, a new program is provided to determine the response of single­or multidegree-of-freedom systems subjected to random excitations. The com­puter programs for earthquake-resistant design have been updated using the latest published seismic codes.

The book is organized into six parts. Part I deals with structures modeled as single-degree-of-freedom systems. It introduces basic concepts and presents methods for the solution of such dynamic systems. Part II introduces concepts and methodology for solving multidegree-of-freedom systems through the use of structures modeled as shear buildings. Part III describes methods for the dynamic analysis of skeletal structures (beams, frames, and trusses) and of continuous structures such as plates and shells 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 introduces the reader to the fascinating topic of random vibrations, which is now extended to multidegree-of-freedom systems. Finally, Part VI presents the current topic of earthquake engineering with applications for the design of earthquake-resistant

J A convenient form to order this supplement is provided in the back of the book.

Preface to the Fourth Edition xix

buildings following the provisions of the Unifonn Building Code in use in the United States. There is a detailed presentation of the seismic analysis of buildings modeled as three-dimensional structures with two independent hori­zontal motions and one rotational motion about a vertical axis for each story of the building. A computer program for the implementation of this simplified method for seismic analysis of buildings is included in the set of educational programs.

Scientific knowledge may be presented from a general all-encompassing theory from which particular or simple situations are obtained by introducing restricting conditions. Alternatively, the presentation may begin by considering particular or simple situations that are progressively extended. The author has adopted this latter approach in which the presentation begins with particular or simple cases that are extended to more general and complex situations. Furthennore, the author believes that a combination of knowledge of applied mathematics, theory of structures, and the use of computer programs is needed today for the successful professional practice of engineering. To provide the reader with such a combination of knowledge has been the primary objective of this book. The reader is encouraged to infonn the author on the extent to which this objective has been fulfilled.

Many of my students, colleagues, and practicing professionals have sugges­ted improvements, identified typographical errors, and recommended additional topics for inclusion. All these suggestions were carefully considered and have been included in this fourth edition whenever possible.

I was fortunate to have received valuable assistance and insight from many individuals to whom I wish to express my appreciation. I am grateful to Jeffrey S. Janover, a consulting engineer from New Jersey, who shared his expertise in the implementation of professional computer programs for the solution of complex engineering problems. I appreciate the discussions and comments offered by my colleagues Drs. Michael A. Cassaro and Julius Wong who helped me in refining my exposition. I am also grateful to my friend Dr. Farzad Naeim who has collaborated with me on Seismic Response Spectra in the International Handbook of Earthquake Engineering: Codes, Programs and Examples (Paz, 1994) 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 vibrations and of his class notes on the Finite Element Method.

It is with great satisfaction that I acknowledge the help received from four of my fonner students: Christopher Biles, who carefully studied and com­mented on Chapter 23, Random Vibrations, as he worked on his Masters' thesis on that subject; Mahomet Sharif for providing me with actual cases of random vibration problems selected from his professional practice; Zair Hillal, who made skillful use of the computer in preparing some of the new figures

xx Preface to the Fourth Edition

in the book; and Cleryl Hoskins who most carefully checked the solution of the problems for some chapters of the book.

A special acknowledgement of gratitude is extended to Dr. Edwin A. Tuttle, emeritus professor of education, who provided many suggestions that helped to improve the clarity of my presentation. I also wish to express my sincere gratitude to my friend Jack Bension for his professional help in editing the revised sections of the book. My thanks also go to Ms. Debbie Jones for her competent typing skills in the revisions.

LO those people whom I recognized in the prefaces to the previous editions for their help, I again express my wholehearted appreciation. To my wife Jean a special thanks for carefully checking the structure of the book and for most graciously allowing me time to prepare this new edition, particularly during several "working vacations." As with the third edition, this volume is dedicated to the everlasting memory of my parents.

MARIO PAZ

March, 1997

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 familiar with the existence of several excellent texts of an advanced nature but gen-

xxi

xxii Preface to the First Edition

erally 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 modern 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