DYNAMICS OF STRUCTURES PDF

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DYNAMICS OF STRUCTURES Third Edition DYNAMICS OF STRUCTURES Ray W. Clough Professor of Civil Engineering University of California, Berkeley Joseph Penzien International Civil Engineering Consultants, Inc. THIRD EDITION Computers & Structures, Inc. 1995 University Ave. Berkeley, CA 94704 USA DYNA...

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DYNAMICS OF STRUCTURES Third Edition

DYNAMICS OF STRUCTURES Ray W. Clough Professor of Civil Engineering University of California, Berkeley

Joseph Penzien International Civil Engineering Consultants, Inc.

THIRD EDITION

Computers & Structures, Inc. 1995 University Ave. Berkeley, CA 94704 USA

DYNAMICS OF STRUCTURES Copyright (c) 2003 by Computers & Structures, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. Library of Congress Cataloging-in-Publication Data Clough, Ray W., (date). Dynamics of structures / Ray W. Clough, Joseph Penzien. p. cm. Includes index. 1. Structural dynamics. I. Penzien, Joseph II. Title. TA654.C6 2003

CONTENTS

Preface List of Symbols 1 1-1 1-2 1-3 1-4

1-5

1-6

Overview of Structural Dynamics Fundamental Objective of Structural Dynamics Analysis Types of Prescribed Loadings Essential Characteristics of a Dynamic Problem Methods of Discretization Lumped-Mass Procedure Generalized Displacements The Finite-Element Concept Formulation of the Equations of Motion Direct Equilibration using dŠAlembertŠs Principle Principle of Virtual Displacements Variational Approach Organization of the Text

xv xvii 1 1 2 3 4 4 5 7 9 9 10 10 11

PART I SINGLE-DEGREE-OF-FREEDOM SYSTEMS 2 2-1 2-2 2-3 2-4 2-5 2-6

Problems

Analysis of Free Vibrations Components of the Basic Dynamic System Equation of Motion of the Basic Dynamic System Influence of Gravitational Forces Influence of Support Excitation Analysis of Undamped Free Vibrations Damped Free Vibrations Critically-Damped Systems Undercritically-Damped Systems Overcritically-Damped Systems

15 15 16 17 18 20 25 26 27 32 32 v

vi

CONTENTS

3 3-1

3-2 3-3 3-4 3-5 3-6

3-7 Problems 4 4-1

4-2 4-3 Problems

Response to Harmonic Loading Undamped System Complementary Solution Particular Solution General Solution System with Viscous Damping Resonant Response Accelerometers and Displacement Meters Vibration Isolation Evaluation of Viscous-Damping Ratio Free-Vibration Decay Method Resonant Amplification Method Half-Power (Band-Width) Method Resonance Energy Loss Per Cycle Method Complex-Stiffness Damping

33 33 33 33 34 36 42 45 46 52 52 53 54 56 58 61

Response to Periodic Loading Fourier Series Expressions of Periodic Loading Trigonometric Form Exponential Form Response to the Fourier Series Loading Preview of Frequency-Domain Analysis

65 65 65 66 67 69 71

5 Response to Impulsive Loading 5-1 General Nature of Impulsive Loading 5-2 Sine-Wave Impulse 5-3 Rectangular Impulse 5-4 Triangular Impulse 5-5 Shock or Response Spectra 5-6 Approximate Analysis of Impulsive-Load Response Problems 6 6-1

6-2

Response to General Dynamic Loading: Superposition Methods Analysis Through the Time Domain Formulation of Response Integral Numerical Evaluation of Response Integral 89 Analysis Through the Frequency Domain Fourier Response Integral Discrete Fourier Transforms (DVF)

73 73 74 77 78 79 82 84 87 87 87 97 98 100

CONTENTS

6-3

Fast Fourier Transforms (FFT) Evaluation of Dynamic Response Relationship between the Time- and Frequency-Domain Transfer Functions

102 106

Response to General Dynamic Loading: Step-by-Step Methods General Concepts Piecewise Exact Method Numerical Approximation Procedures General Comments Second Central Difference Formulation Integration Methods Euler-Gauss Procedure Newmark Beta Methods Conversion to Explicit Formulation Incremental Formulation for Nonlinear Analysis Summary of the Linear Acceleration Procedure

111 111 112 116 117 120 120 121 123 124 127 132

Problems 7 7-1 7-2 7-3 7-4 7-5

7-6 7-7 Problems

vii

8 Generalized Single-Degree-of-Freedom Systems 8-1 General Comments on SDOF Systems 8-2 Generalized Properties: Assemblages of Rigid Bodies 8-3 Generalized Properties: Distributed Flexibility 8-4 Expressions for Generalized System Properties 8-5 Vibration Analysis by RayleighŠs Method 8-6 Selection of the Rayleigh Vibration Shape 8-7 Improved Rayleigh Method Problems

109 109

133 133 134 140 145 149 152 156 160

PART II MULTI-DEGREE-OF-FREEDOM SYSTEMS 9 9-1 9-2 9-3 10 10-1

Formulation of the MDOF Equations of Motion Selection of the Degrees of Freedom Dynamic-Equilibrium Condition Axial-Force Effects

169 169 171 173

Evaluation of Structural-Property Matrices Elastic Properties Flexibility Stiffness Basic Structural Concepts Finite-Element Stiffness

175 175 175 176 177 179

viii

CONTENTS

10-2

10-3 10-4

10-5

10-6 Problems 11 11-1 11-2 11-3 11-4

11-5

Mass Properties Lumped-Mass Matrix Consistent-Mass Matrix Damping Properties External Loading Static Resultants Consistent Nodal Loads Geometric Stiffness Linear Approximation Consistent Geometric Stiffness Choice of Property Formulation

184 184 185 189 189 190 190 191 191 194 196 198

Undamped Free Vibrations Analysis of Vibration Frequencies Analysis of Vibration Mode Shapes Flexibility Formulation of Vibration Analysis Influence of Axial Forces Free Vibrations Buckling Load Buckling with Harmonic Excitation Orthogonality Conditions Basic Conditions Additional Relationships Normalizing

201 201 204 208 208 208 209 210 211 211 212 214 215

Analysis of Dynamic Response Using Superposition Normal Coordinates Uncoupled Equations of Motion: Undamped Uncoupled Equations of Motion: Viscous Damping Response Analysis by Mode Displacement Superposition Viscous Damping Complex-Stiffness Damping Construction of Proportional Viscous Damping Matrices Rayleigh Damping Extended Rayleigh Damping Alternative Formulation Construction of Nonproportional Damping Matrices Response Analysis using Coupled Equations of Motion Time Domain

219 219 221 222 223 223 230 234 234 237 240 242 245 245

Problems 12 12-1 12-2 12-3 12-4

12-5

12-6

CONTENTS

12-7 12-8 12-9 Problems 13 13-1 13-2 13-3 13-4

13-5 13-6 13-7 13-8

Frequency Domain Relationship between Time- and Frequency- Domain Transfer Functions Practical Procedure for solving Coupled Equations of Motion Interpolation Procedure for Generation of Transfer Functions

14-2 14-3 14-4 14-5 14-6 14-7

14-8

246 247 251 254 256

Vibration Analysis by Matrix Iteration Preliminary Comments Fundamental Mode Analysis Proof of Convergence Analysis of Higher Modes Second-Mode Analysis Analysis of Third and Higher Modes Analysis of Highest Mode Buckling Analysis by Matrix Iteration Inverse Iteration the Preferred Procedure Inverse Iteration with Shifts Special Eigenproblem Topics Eigenproperty Expansion Symmetric Form of Dynamic Matrix Analysis of Unconstrained Structures

259 259 260 265 267 267 271 272 275 279 281 285 286 288 290 291

Selection of Dynamic Degrees of Freedom Finite-Element Degrees of Freedom One-Dimensional Elements Two- and Three-Dimensional Elements Kinematic Constraints Static Condensation Rayleigh Method in Discrete Coordinates Rayleigh-Ritz Method Subspace Iteration Reduction of Modal Truncation Errors General Comments on Coordinate Reduction Modal Contributions Static Correction Procedure Mode Acceleration Method Derived Ritz Vectors Preliminary Comments Derivation Details

293 293 294 294 295 296 298 299 304 306 306 307 311 313 314 314 316

Problems 14 14-1

ix

x

CONTENTS

Tridiagonal Equations of Motion Loss of Orthogonality Required Number of Vectors Problems 15 15-1 15-2 15-3 15-4 15-5

319 322 323 323

Analysis of MDOF Dynamic Response: Step-by-Step Methods Preliminary Comments Incremental Equations of Motion Step-by-Step Integration: Constant Average Acceleration Method Step-by-Step Integration: Linear Acceleration Method Strategies for Analysis of Coupled MDOF Systems Localized Nonlinearity Coupled Effects Treated as Pseudo-Forces

325 325 327 328 330 332 332 336

16 Variational Formulation of the Equations of Motion 16-1 Generalized Coordinates 16-2 HamiltonŠs Principle 16-3 LagrangeŠs Equations of Motion 16-4 Derivation of the General Equations of Motion for Linear Systems 16-5 Constraints and Lagrange Multipliers Problems

341 341 342 344 351 356 359

PART III DISTRIBUTED-PARAMETER SYSTEMS 17 Partial Differential Equations of Motion 17-1 Introduction 17-2 Beam Flexure: Elementary Case 17-3 Beam Flexure: Including Axial-Force Effects 17-4 Beam Flexure: Including Viscous Damping 17-5 Beam Flexure: Generalized Support Excitations 17-6 Axial Deformations: Undamped Problems

365 365 366 368 369 370 373 375

18 Analysis of Undamped Free Vibrations 18-1 Beam Flexure: Elementary Case 18-2 Beam Flexure: Including Axial-Force Effects 18-3 Beam Flexure: With Distributed Elastic Support 18-4 Beam Flexure: Orthogonality of Vibration Mode Shapes 18-5 Free Vibrations in Axial Deformation 18-6 Orthogonality of Axial Vibration Modes Problems

377 377 386 388 389 391 392 394

CONTENTS

19 19-1 19-2 19-3 19-4 19-5

Analysis of Dynamic Response Normal Coordinates Uncoupled Flexural Equations of Motion: Undamped Case Uncoupled Flexural Equations of Motion: Damped Case Uncoupled Axial Equations of Motion: Undamped Case Wave-Propagation Analysis Basic Axial-Wave-Propagation Equation Consideration of Boundary Conditions Discontinuity in Bar Properties

Problems

xi 397 397 400 403 407 411 411 415 418 423

PART IV RANDOM VIBRATIONS 20 Probability Theory 20-1 Single Random Variable 20-2 Important Averages of a Single Random Variable 20-3 One-Dimensional Random Walk 20-4 Two Random Variables 20-5 Important Averages of Two Random Variables 20-6 Scatter Diagram and Correlation of Two Random Variables 20-7 Principal Axes of Joint Probability Density Function 20-8 Rayleigh Probability Density Function 20-9 m Random Variables 20-10 Linear Transformations of Normally Distributed Random Variables Problems 21 21-1 21-2 21-3 21-4 21-5 21-6 21-7 21-8 21-9 21-10 21-11 21-12 21-13

Random Processes Definition Stationary and Ergodic Processes Autocorrelation Function for Stationary Processes Power Spectral Density Function for Stationary Processes Relationship Between Power Spectral Density and Autocorrelation Functions Power Spectral Density and Autocorrelation Functions for Derivatives of Processes Superposition of Stationary Processes Stationary Gaussian Processes: One Independent Variable Stationary White Noise Probability Distribution for Maxima Probability Distribution for Extreme Values Nonstationary Gaussian Processes Stationary Gaussian Process: Two or More Independent Variables

427 427 432 434 442 451 455 458 461 463 465 466 471 471 473 478 484 485 488 490 492 498 501 506 510 511

xii

CONTENTS

Problems 22 22-1 22-2 22-3 22-4 22-5 22-6 Problems 23 23-1 23-2 23-3 23-4 23-5

512 Stochastic Response of Linear SDOF Systems Transfer Functions Relationship between Input and Output Autocorrelation Functions Relationship between Input and Output Power Spectral Density Functions Response Characteristics for Narrowband Systems Nonstationary Mean Square Response Resulting from Zero Initial Conditions Fatigue Predictions for Narrowband Systems

517 517 518

Stochastic Response of Linear MDOF Systems Time-Domain Response for Linear Systems using Normal Modes Frequency-Domain Response for Linear Systems using Normal Modes Normal Mode Forcing Function due to Discrete Loadings Normal Mode Forcing Function due to Distributed Loadings Frequency-Domain Response for Linear Systems having FrequencyDependent Parameters and/or Coupled Normal Modes

539 539 541 543 547

Problems

522 524 528 532 535

548 550

PART V EARTHQUAKE ENGINEERING 24 24-1 24-2 24-3 24-4 24-5 24-6 24-7

Seismological Background Introductory Note Seismicity Earthquake Faults and Waves Structure of the Earth Plate Tectonics Elastic-Rebound Theory of Earthquakes Measures of Earthquake Size

555 555 556 558 559 563 567 571

25 25-1 25-2 25-3

Free-Field Surface Ground Motions Fourier and Response Spectra Factors influencing Response Spectra Design Response Spectra Dual Strategy of Seismic Design Peak Ground Accelerations Response Spectrum Shapes Uniform-Hazard Site-Specific Response Spectra Two Horizontal Components of Motion

575 575 581 586 586 587 590 595 597

CONTENTS

25-4

Design Accelerograms Response Spectrum Compatible Accelerograms Principal Axes of Motion Spatially Correlated Motions

xiii 597 598 603 607

26 26-1 26-2

Deterministic Earthquake Response: Systems on Rigid Foundations613 Types of Earthquake Excitation 613 Response to Rigid-Soil Excitations 615 Lumped SDOF Elastic Systems, Translational Excitation 615 Generalized-Coordinate SDOF Elastic Systems, Translational Excitation 617 Lumped MDOF Elastic Systems, Translational Excitation 623 Comparison with ATC-3 Recommended Code Provisions 638 Distributed-Parameter Elastic Systems, Translational Excitation 640 Lumped MDOF Elastic Systems, Rotational Excitation 642 Lumped MDOF Elastic Systems, Multiple Excitation 644 Lumped SDOF Elastic-Plastic Systems, Translational Excitation 647 26-3 Combining Maximum Modal Responses 650 Mean Square Response of a Single Mode 650 Covariance of Response Produced by Two Modes 652 SRSS and CQC Combination of Modal Responses 653 Combining Two-Component-Excitation Responses 657 Problems 662 27 27-1

27-2

27-3

28 28-1

Deterministic Earthquake Response: Including Soil-Structure Interaction Soil-Structure Interaction by Direct Analysis Kinematic Interaction for Translational Excitation; the Tau Effect Direct Inclusion of a Bounded Soil Layer Substructure Analysis of SSI Response Lumped SDOF Systems on Rigid Mat Foundation General MDOF System with Multiple Support Excitation Generation of Boundary Impedances Response of Underground Structures Free-Field Ground Motions due to Propagating Plane Waves Racking Deformations of Cross Sections Overall Axial and Flexural Deformations Influence of Transverse Joints on Axial Deformations

669 669 670 673 674 674 679 689 704 704 705 706 709

Stochastic Structural Response Modeling of Strong Ground Motions

711 711

xiv

CONTENTS

28-2

28-3

28-4 28-5

Stochastic Response of Linear Systems SDOF Systems MDOF Systems Extreme-Value Response of Nonlinear Systems SDOF Systems MDOF Systems Design Considerations Allowable Ductility Demand Versus Ductility Capacity

711 711 712 713 713 723 726 729

Index

731

PREFACE Since the first edition of this book was published in 1975, major advances have been made in the subject "Dynamics Of Structures." While it would be impossible to give a comprehensive treatment of all such changes in this second edition, those considered to be of most practical significance are included. The general organization of text material remains unchanged from the first edition. It progresses logically from a treatment of single-degree-of-freedom systems to multi-degree-of-freedom discrete-parameter systems and then on to infinite-degreeof-freedom continuous systems. The concept of force equilibrium, which forms the basis of static analysis of structures, is retained so that the experienced engineer can easily make the transition to performing a dynamic analysis. It is essential therefore that the student of structural dynamics have a solid background in the theories of statics of structures, including matrix methods, and it is assumed that the readers of this text have such preparation. The theoretical treatment in Parts I, II, and III is deterministic in nature because it makes use of dynamic loadings which are fully prescribed eventhough they may be highly irregular and transient with respect to time. The treatment of random vibrations in Part IV is however stochastic (random) in form since the loadings considered can be characterized only in a statistical manner. An understanding of basic probability theory is therefore an essential prerequisite to the study of this subject. Before proceeding with this study, it is recommended that the student take a full course on probability theory; however, if this has not been done, the brief treatment of probability concepts given in Chapter 20 can serve as minimum preparation. The solution of a typical structural dynamics problem is considerably more complicated than its static counterpart due to the addition of inertia and damping to the elastic resistance forces and due to the time dependency of all force quantities. For most practical situations, the solution usually is possible only through the use of a high-speed digital computer, which has become the standard tool of the structural dynamicist. However, most of the problems in the text, which are intended to teach the fundamentals of dynamics, are quite simple in form allowing their solutions to be obtained using a hand calculator. Nevertheless, the student of dynamics of structures should have previously studied computer coding techniques and the associated analytical procedures. Such background will permit an early transition from solving dynamics problems by hand calculator to solving them on a PC computer using programs specially developed for this purpose. The program CAL-91, developed by Professor E. L. Wilson of the University of California, Berkeley, is such a program which has been used very effectively in teaching even the first course in Dynamics Of Structures. Instructors using this book are encouraged to implement such PC computer solutions into their courses so that more realistic problems can be considered. xv

xvi

PREFACE

A large number of example problems have been solved in the text to assist the reader in understanding the subject material. To fully master the analytical techniques, it is essential that the student solve many of the homework problems presented at the ends of chapters. They should be assigned sparingly however because dynamicresponse analyses are notoriously time consuming. The authors have found that from one to four problems may constitute an adequate weekly assignment, depending on the subject matter and type of solution required. On this basis, the book includes many more problems than can be assigned during a one-year sequence of courses on structural dynamics. The subject matter of this text can serve as the basis of a series of graduate-level courses. The first course could cover the material in Part I and a portion of that in Part II. The full extent of this coverage would depend, of course, upon whether the course is of quarter or semester duration. If of quarter duration, the material coverage in Parts I and II is sufficient to provide the basis of a sequence of two quarter courses and some material from Part III also could be included in the second course. It is now generally expected that nearly all Masters-Degree students in structural engineering should have had at least the basic first-course in dynamics of structures and it is recommended that the advanced (fourth-year level) under...