STRUCTURAL DYNAMICS. Theory and Computation Fifth Edition

Transcription

1 STRUCTURAL DYNAMICS Theory and Computation Fifth Edition

2 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

3 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 ISBN ISBN (ebook) DOI / 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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5 CONTENTS PREFACE TO THE FIFTH EDITION PREFACE TO THE FIRST EDITION xvii xxi PART I STRUCTURES MODELED AS A SINGLE-DEGREE-OF-FREEDOM SYSTEM 1 1 UNDAMPED SINGLE-DEGREE-OF-FREEDOM SYSTEM Degrees of Freedom Undamped System 5 l.3 Springs in Parallel or in Series Newton's Law of Motion Free Body Diagram D' Alembert's Principle Solution of the Differential Equation of Motion Frequency and Period Amplitude of Motion Summary Problems 23 2 DAMPED SINGLE-DEGREE-OF-FREEDOM SYSTEM Viscous Damping Equation of Motion Critically Damped System Overdamped System Underdamped System Logarithmic Decrement Summary Problems 45

6 viii Contents 3 RESPONSE OF ONE-DEGREE-OF-FREEDOM SYSTEM TO HARMONIC LOADING Harmonic Excitation: Undamped System Harmonic Excitation: Damped System Evaluation of Damping at Resonance Bandwidth Method (Half-Power) to Evaluate Damping Energy Dissipated by Viscous Damping Equivalent Viscous Damping Response to Support Motion Force Transmitted to the Foundation Seismic Instruments Response of One-Degree-of-Freedom System to Harmonic Loading Using SAP Summary Analytical Problem Problems 96 4 RESPONSE TO GENERAL DYNAMIC LOADING Duhamel's Integral-Undamped System Duhamel's Integral-Damped System Response by Direct Integration Solution of the Equation of Motion Program 2-Response by Direct Integration Program 3-Response to Impulsive Excitation Response to General Dynamic Loading Using SAP Summary Analytical Problems Problems RESPONSE SPECTRA Construction of Response Spectrum Response Spectrum for Support Excitation Tripartite Response Spectra Response Spectra for Elastic Design Influence of Local Soil Conditions Response Spectra for Inelastic Systems Response Spectra for Inelastic Design Program 6-Seismic Response Spectra Summary Problems 174

7 Contents 6 NONLINEAR STRUCTURAL RESPONSE Nonlinear Single Degree-of-Freedom Model Integration of the Nonlinear Equation of Motion Constant Acceleration Method Linear Acceleration Step-by-Step Method The Newmark Beta Method Elastoplastic Behavior Algorithm for the Step-by-Step Solution for Elastoplastic Single-Degree-of-Freedom System Program 5-Response for Elastoplastic Behavior Summary Problems 198 PART II STRUCTURES MODELED AS SHEAR BUILDINGS FREE VIBRATION OF A SHEAR BUILDING Stiffness Equations for the Shear Building Natural Frequencies and Normal Modes Orthogonality Property of the Normal Modes Rayleigh's Quotient Program 8-Natural Frequencies and Normal Modes Free Vibration of a Shear Building Using SAP Summary Problems FORCED MOTION OF SHEAR BUILDING Modal Superposition Method Response of a Shear Building to Base Motion Program 9-Response by Modal Superposition Harmonic Forced Excitation Program 10-Harmonic Response Forced Motion Using SAP Combining Maximum Values of Modal Response Summary Problems REDUCTION OF DYNAMIC MATRICES Static Condensation Static Condensation Applied to Dynamic Problems Dynamic Condensation Modified Dynamic Condensation Program 12-Reduction of the Dynamic Problem Summary Problems 299 ix

8 x Contents PART III FRAMED STRUCTURES MODELED AS DISCRETE MUL TI-DEGREE-OF- FREEDOM SYSTEMS DYNAMIC ANALYSIS OF BEAMS Shape Functions for a Beam Segment System Stiffuess Matrix Inertial Properties-Lumped Mass Inertial Properties-Consistent Mass Damping Properties External Loads Geometric Stiffuess Equations of Motion Element Forces at Nodal Coordinates Program 13-Modeling Structures as Beams Dynamic Analysis of Beams Using SAP Summary Problems DYNAMIC ANALYSIS OF PLANE FRAMES 353 ILl Element Stiffuess Matrix for Axial Effects Element Mass Matrix for Axial Effects Coordinate Transformation Program 14-Modeling Structures as Plane Frames Dynamic Analysis of Frames Using SAP Summary Problems DYNAMIC ANALYSIS OF GRID FRAMES Local and Global Coordinate Systems Torsional Effects Stiffuess Matrix for a Grid Element Consistent Mass Matrix for a Grid Element Lumped Mass Matrix for a Grid Element Transformation of Coordinates Program 15-Modeling Structures as Grid Frames Dynamic Analysis of Grid Frames Using SAP Summary Problems DYNAMIC ANALYSIS OFTHREE-DIMENSIONAL FRAMES Element Stiffuess Matrix Element Mass Matrix Element Damping Matrix 410

9 Contents 13.4 Transformation of Coordinates Differential Equation of Motion Dynamic Response Program 16-Modeling Structures as Space Frames Dynamic Response of Three-Dimensional Frames UsingSAP Summary Problems DYNAMIC ANALYSIS OF TRUSSES Stiffness and Mass Matrices for the Plane Truss Transformation of Coordinates Program 17-Modeling Structures as Plane Trusses Stiffness and Mass Matrices for Space Trusses Equation of Motion for Space Trusses Program 18-Modeling Structures as Space Trusses Dynamic Analysis of Trusses Using SAP Summary Problems DYNAMIC ANALYSIS OF STRUCTURES USING THE FINITE ELEMENT METHOD Plane Elasticity Problems Triangular Plate Elementfor Plane Elasticity problems SAP2000 for Plane Elasticity Problem Plate Bending Rectangular Elementfor Plate Bending SAP2000for Plate Bending and Shell Problems Summary Problems TIME HISTORY RESPONSE OF MULTIDEGREE-OF-FREEDOM SYSTEMS Incremental Equations of Motion The Wilson-BMethod Algorithm for Step-by-Step Solution of a Linear System Using the Wilson-BMethod Initialization For Each Time Step Program 19-Response by Step Integration The Newmark Beta Method Elastoplastic Behavior of Framed Structures Member Stiffness Matrix Member Mass Matrix Rotation of Plastic Hinges 513 xi

10 xii Contents Calculation of Member Ductility Ratio Time-History Response of Multi degree-of-freedom Systems Using SAP Summary Problems 522 PART IV STRUCTURES MODELED WITH DISTRIBUTED PROPERTIES DYNAMIC ANALYSIS OF SYSTEMS WITH DISTRIBUTED PROPERTIES Flexural Vibration of Uniform Beams Solution ofthe Equation of Motion in Free Vibration Natural Frequencies and Mode Shapes for Uniform Beams Both Ends Simply Supported Both Ends Free (Free Beam) Both Ends Fixed One End Fixed and the other End Free (Cantilever Beam) One End Fixed and the other End Simply Supported Orthogonality Condition Between Normal Modes Forced Vibration of Beams Dynamic Stresses in Beams Summary Problems DISCRETIZATION OF CONTINUOUS SYSTEMS Dynamic Matrix for Flexural Effects Dynamic Matrix for Axial Effects Dynamic Matrix for Torsional Effects Beam Flexure Including Axial-Force Effect Power Series Expansion of the Dynamic Matrix for Flexural Effects Power Series Expansion of the Dynamic Matrix for Axial and for Torsional Effects Power Series Expansion of the Dynamic Matrix Including the Effects of Axial Forces Summary 566 PART V SPECIAL TOPICS: Fourier Analysis, Evaluation of Absolute Damping, Generalized Coordinates FOURIER ANALYSIS AND RESPONSE IN THE FREQUENCY DOMAIN Fourier Analysis 569

11 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 EVALUATION OF ABSOLUTE DAMPING FROM MODAL DAMPING RATIOS Equations for Damped Shear Building Uncoupled Damped Equations Conditions for Damping Uncoupling Program II-Absolute Damping From Modal Damping Ratios Summary Problems GENERALIZED COORDINATES AND RAYLEIGH'S METHOD Principle of Virtual Work Generalized Single-Degree-of-Freedom System-Rigid Body Generalized Single-Degree-of-Freedom System- Distributed Elasticity Shear Forces and Bending Moments Generalized Equation of Motion for a Multistory Building Shape Function Rayleigh's Method Improved Rayleigh's Method Shear Walls Summary Problems 643 PART VI RANDOM VIBRATION 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

12 xiv Contents Response to Random Excitation: Single-Degree-of-Freedom System Response to Random Excitation: Multiple-Degree-of-Freedom System Relationship Between Complex Frequency Response and Unit Impulse Response Response to Random Excitation: Two-degree-of-freedom System Response to Random Excitation: N Degree of Freedom System Summary Problems 692 PART VII EARTHQUAKE ENGINEERING UNIFORM BUILDING CODE 1997: EQUIVALENT LATERAL FORCE METHOD Earthquake Ground Motion Equivalent Lateral Force Method Earthquake-Resistant Design Methods Seismic Zone Factor Base Shear Force Distribution of Lateral Seismic Forces Story Shear Force Horizontal Torsional Moment Overturning Moment P-Delta Effect (P-A) RedundancylReliability Factor p Story Drift Limitation Diaphragm Design Forces Earthquake Load Effect Irregular Structures Summary Problems UNIFORM BUILDING CODE 1997: DYNAMIC METHOD Modal Seismic Response of Buildings Modal Equation and Participation Factor Modal Shear Force Effective Modal Weight Modal Lateral Forces Modal Displacements Modal Drift Modal Overturning Moment Modal Torsional Moment Total Design Values 737

13 Provisions ofubc-97: Dynamic Method Scaling of Results Program 24-UBC 1997 Dynamic Lateral Force Method Summary Problems Contents xv INTERNATIONAL BUILDING CODE IBC Response Spectral Acceleration: Ss, SJ Soil Modification Response Spectral Acceleration: SMS, SMl Design Response Spectral Acceleration: SDS, SOl Site Class Defmition: A, B,... F Seismic Use Group (SUG) and Occupancy Importance Factor (IE) Seismic Design Category (A, B, C, D, E and F) Design Response Spectral Curve: Sa v.s. T Determination of the Fundamental Period Minimum lateral Force Procedure [mc-2000: Section ] SimplifiedAnalysisProcedure [mc-2000: Section ] Seismic Base Shear Response Modification Factor R Vertical Distribution o/lateral Forces Equivalent Seismic Lateral Force Method: [mc-2000: Section ] Distribution o/lateral Forces Overturning Moments Horizontal Torsional Moment P-Delta Effect (P-LJ) Story Drift Redundancy/Reliability Factor Earthquake Load Effect Building Irregularities Summary 781 APPENDICES Appendix I: Answers to Problems in Selected Chapters Appendix II: Computer Programs Appendix III: Glossary Selected Bibliography Index

14 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

15 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 CD ROM containing the introductory version of SAP2000 as well as the educational set of the program developed by the author is included in this 5 th 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

16 Preface to the Fifth Edition 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 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. xix

17 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

18 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

19 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. December, 1979 MARIO PAZ