Structural Dynamics THEORY AND COMPUTATION

Transcription

1 Structural Dynamics THEORY AND COMPUTATION

2 Structural Dynamics THEORY AND COMPUTATION MARIO PAZ Professor of Civil Engineering University of Louisville Third Edition Itim1 VAN NOSTRAND REINHOLD ~ New York

3 The computer programs used in this book are available in several versions of the BASIC language. To order diskettes for ffim-compatible microcomputer formats, use the order form on the back of the book or write to MlCROTEXT, P.O. Box 351m, Louisville, KY Technical questions, corrections, and request for additional information should be directed to this address. Extreme care has been taken in preparing the programs used in this book. Extensive testing and checking have been performed to insure the accuracy and effectiveness of computer solutions. However, neither the author nor the publisher shall be held responsible or liable for any damages arising from the use of any of the programs in this book. ISBN-13: e-isbn-13: DOl: / Copyright 1991 by Van Nostrand Reinhold Library of Congress Catalog Number Softcover reprint ofthe hardcover 1st edition 1991 All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form by any means-graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems-without written permission of the publisher. Van Nostrand Reinhold 115 Fifth Avenue New York, New York Van Nostrand Reinhold International Company Limited 11 New Fetter Lane London EC4P 4EE, England Van Nostrand Reinhold 102 Dodds Street South Melbourne 3205, Victoria, Australia Nelson Canada 1120 Birchmount Road Scarborough, Ontario MIK 5G4, Canada Library of Congress Cataloging-in-Publication Data Paz, Mario. Structural dynamics: theory and computation / Mario paz.-3rd ed. p. cm. 1. Structural dynamics. I. Title. TA654.P '71-dc CIP

4 1 l I } ~ ~ 7 O ' M ~ ) m ~ ' ~ 1~ P 2 r ~ ~ '. ~ n $ 1 = ~ ~ o z,: ~ " 1 j r~ t ~ 9 ~ i? ~, : : ~ ~ ~ r ~ ~ : i.? l ~ 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

5 CONTENTS PREFACE TO THE THIRD EDITION xv PREFACE TO THE FIRST EDITION xix PART I STRUCTURES MODELED AS A SINGLE DEGREE-OF-FREEDOM SYSTEM 1 1 UNDAMPED SINGLE DEGREE-OF-FREEDOM SYSTEMS 3 l.1 Degrees of Freedom / Undamped System / Springs in Parallel or in Series / Newton's Law of Motion / 8 l.5 Free Body Diagram / 9 l.6 D'Alembert's Principle / 10 l. 7 Solution of the Differential Equation of Motion / 11 vii

6 viii Contents 1.8 Frequency and Period / A m p l i of ~ Motion u d e / 15 2 DAMPED SINGLE DEGREE-OF-FREEDOM SYSTEM Viscous Damping / Equation of Motion / Critically Damped System / Overdamped System / Underdamped System / Logarithmic Decrement / 29 3 RESPONSE OF ONE-DEGREE-OF-FREEDOM SYSTEM TO HARMONIC LOADING Undamped Harmonic Excitation / Damped Harmonic Excitation / Evaluation of Damping at Resonance / Bandwidth Method IHalf-Power) to Evaluate Damping / Response to Support Motion / Force Transmitted to the Foundation / Seismic Instruments / 56 4 RESPONSE TO GENERAL DYNAMIC LOADING Impulsive Loading and Duhamel's Integral / Numerical Evaluation of Duhamel's Integral-Undamped System / Numerical Evaluation of Duhamel's Integral-Damped System / Response by Direct Integration / Program 2-Response by Direct Integration / Program 3-Response to Impulsive Excitation / 83 5 FOURIER ANALYSIS AND RESPONSE IN THE FREQUENCY DOMAIN Fourier Analysis / Response to a Loading Represented by Fourier Series / Fourier Coefficients for Piecewise Linear Functions / Exponential Form of Fourier Series / Discrete Fourier Analysis / 101

7 Contents ix 5.6 Fast Fourier Transform / Program 4-Response in the Frequency Domain / 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 / Rayleigh's Method / Improved Rayleigh's Method / Shear Walls / NONLINEAR STRUCTURAL RESPONSE Nonlinear Single Degree-of-Freedom Model / Integration of the Nonlinear Equation of Motion / Linear Acceleration Step-by-Step Method / Elastoplastic Behavior / Algorithm for the Step-by-Step Solution for Elastoplastic Single Degree-of-Freedom System / Program 5-Response for Elastoplastic Behavior System / RESPONSE SPECTRA Construction of Response Spectrum / Response Spectrum for Support Excitation / Tripartite Response Spectra / Response Spectra for Elastic Design / Response Spectra for Inelastic Systems / Response Spectra for Inelastic Design / Program 6-Seismic Response Spectra / 192 PART II STRUCTURES MODELED AS SHEAR BUILDINGS THE MUL TlSTORY SHEAR BUILDING Stiffness Equations for the Shear Building / Flexibility Equations for the Shear Building / Relationship Between Stiffness and Flexibility Matrices / Program 7-Modeling Structures as Shear Buildings / 207

8 x Contents 10 FREE VIBRATION OF A SHEAR BUILDING Natural Frequencies and Normal Modes / Orthogonality Property of the Normal Modes / Program 8-Natural Frequencies and Normal Modes / FORCED MOTION OF SHEAR BUILDINGS 230 ILl Modal Superposition Method / Response of a Shear Building to Base Motion / Program 9-Response by Modal Superposition / Harmonic Forced Excitation / 245 1l.5 Program lo-harmonic Response / Combining Maximum Values of Modal Response / DAMPED MOTION OF SHEAR BUILDINGS Equations for Damped Shear Building / Uncoupled Damped Equations / Conditions for Damping Uncoup}ing / Program ll-absolute Damping From Damping Ratios / REDUCTION OF DYNAMIC MATRICES Static Condensation / Static Condensation Applied to Dynamic Problems / Dynamic Condensation / Modified Dynamic Condensation / Program 12-Reduction of the Dynamic Problem / 296 PART III FRAMED STRUCTURES MODELED AS DISCRETE MUL TIDEGREE-OF-FREEDOM SYSTEMS DYNAMIC ANALYSIS OF BEAMS Static Properties for a Beam Segment / System Stiffness Matrix / Inertial Properties-Lumped Mass / Inertial Properties-Consistent Mass / 315

9 Contents xi 14.5 Damping Properties / External Loads / Geometric Stiffness / Equations of Motion / Element Forces at Nodal Coordinates / Program 13-Modeling Structures as Beams / 334 IS DYNAMIC ANALYSIS OF PLANE FRAMES Element Stiffness Matrix for Axial Effects / Element Mass Matrix for Axial Effects / Coordinate Transformation / Program 14-Modeling Structures as plane Frames / DYNAMIC ANALYSIS OF GRIDS Local and Global Coordinate Systems / Torsional Effects / Stiffness Matrix for a Grid Element / Consistent Mass Matrix for a Grid Element / Lumped Mass Matrix for a Grid Element / Transformation of Coordinates / Program IS-Modeling Structures as Grid Frames / THREE-DIMENSIONAL FRAMES Element Stiffness Matrix / Element Mass Matrix / Element Damping Matrix / Transformation of Coordinates / Differential Equation of Motion / Dynamic Response / Program 16-Modeling Structures as Space Frames / 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 / 409

10 xii Contents 19 TIME HISTORY RESPONSE OF MULTIDEGREE-OF-FREEDOM SYSTEMS Incremental Equations of Motion / The Wilson-O Method / Algorithm for Step-by-Step Solution of a Linear System Using the Wilson-O Method / Program 19-Response by Step Integration / Newmark Beta Method / Elastoplastic Behavior of Framed Structures / Member Stiffness Matrix / Member Mass Matrix / Rotation of Plastic Hinges / Calculation of Member Ductility Ratio / 431 PART IV STRUCTURES MODELED WITH DISTRIBUTED PROPERTIES DYNAMIC ANALYSIS OF SYSTEMS WITH DISTRIBUTED PROPERTIES Flexural Vibration of Uniform Beams / Solution of the Equation of Motion in Free Vibration / Natural Frequencies and Mode Shapes for Uniform Beams / Orthogonality Condition Between Normal Modes / Forced Vibration of Beams / Dynamic Stresses in Beams / 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 Effect of Axial Forces / 475

11 Contents xiii PART V RANDOM VIBRATION RANDOM VIBRATION Statistical Description of Random Functions / The Normal Distribution / The Rayleigh Distribution / Correlation / The Fourier Transform / Spectral Analysis / Spectral Density Function / Narrow-Band and Wide-Band Random Processes / Response to Random Excitation / 499 PART VI EARTHQUAKE ENGINEERING EQUIVALENT STATIC LATERAL FORCE METHOD: UNIFORM BUILDING CODE Earthquake Ground Motion / Equivalent Seismic Lateral Force / Distribution of the Lateral Seismic Force / Horizontal Torsional Moment / Overturning Moments and Story Shear / Story Drift / Diaphragm Forces / Program 22-UBC 85 / Design Process / EQUIVALENT STATIC LATERAL FORCE METHOD: UNIFORM BUILDING CODE Earthquake-Resistant Design Methods / Static Lateral Force Method / Distribution of Lateral Forces / Story Shear Force / Horizontal Torsional Moment / Overturning Moment / Story Drift Limitation / 553

12 xiv Contents 24.8 P-Delta Effects (p-a) / Diaphragm Design Force / Program 23 UBC-1988: Equivalent Static Lateral Force Method / DYNAMIC METHOD: UNIFORM BULDING CODE Modal Seismic Response of Buildings / Total Design Values / Provision of UBC-88: Dynamic Method / Scaling Results / Program 24 UBC-1988: Dynamic Lateral Force Method / 583 APPENDIX I: ANSWERS TO PROBLEMS IN PART I 593 APPENDIX II: COMPUTER PROGRAMS 601 APPENDIX III: ORGANIZATIONS AND THEIR ACRONYMS 605 GLOSSARY 609 SELECTED BIBLIOGRAPHY 617 INDEX 621

13 Preface to the Third Edition The basic structure of the two previous editions is maintained in this third edition, although numerous revisions and additions have been introduced. Three new chapters on earthquake-resistant design of buildings have been incorporated into Part VI of the book. The computer programs formerly written in FORTRAN for execution on mainframe computers have been expanded and rewritten in BASIC for implementation on microcomputers. Two independent packages of computer programs are used throughout the book: A structural dynamics package of 20 interactive computer programs and an earthquake-resistant design package of 10 interactive programs. The structural dynamics package includes programs to determine the response of a single oscillator in the time domain, or in the frequency domain using the FFT (Fast Fourier Transform). It also includes a program to determine the response of an inelastic system with elastoplastic behavior, and another program for the development of seismic response spectral charts. A set of seven computer programs is included for use in modeling structures as two-dimensional and three-dimensional frames and trusses. Finally, other programs, incorporating modal superposition or a step-by-step time history xv

14 xvi Preface to the Third Edition solution, are provided for calculation of the responses to forces or motions exciting the structure. The new chapters in earthquake-resistant design of buildings describe the provisions of both the 1985 and 1988 versions of the UBC (Uniform Building Code) for the static lateral force method and for the dynamic lateral force method. Other revisions of the book include the presentation of the Newmark beta method to obtain the time history response of dynamic systems, and the direct integration method in which the response is found assuming that the excitation function is linear for a specified time interval. A modification of the dynamic condensation method, which has been developed recently by the author for the reduction of eigenproblems, is presented in Chapter 13. The proposed modification substantially reduces the numerical operation required in the implementation of the dynamic condensation method. The subjects in this new edition are organized in six parts. Part I deals with structures modeled as single degree-of-freedom systems. It introduces basic concepts and presents important methods for the solution of such dynamic systems. Part II introduces important concepts and methodology for multidegree-of-freedom systems through the use of structures modeled as shear buildings. Part III describes methods for the dynamic analysis of framed structures modeled as discrete systems with many degrees of freedom. Part IV presents the mathematical solution for some simple structures modeled as systems with distributed properties, thus having an infinite number of degrees of freedom. Part V, which contains one chapter, introduces the reader to the fascinating topic of random vibration. Finally, Part VI presents the important current topic of earthquake engineering with applications to earthquakeresistant design of buildings following the provisions of the Uniform Building Codes of 1985 and Several considerations-different sizes of diskettes (3 1/2 inch or 51/ 4 inch), different versions of the programs (compiled or not compiled), and the anticipation of future updating-contributed to a final decision not to include in the book the program diskettes. Diskettes with the computer programs, in several versions, are available directly from the author. A convenient form to order the selected version of the. programs is provided in the back of the book. The author believes that, just as knowledge of a combination of figures and turns is needed to open a safe, a combination of knowledge on the subjects of mathematics, theory of structures, and computer programming is needed today for successful professional practice in engineering. To provide the reader with such a combination of knowledge has been the primary objective of this book. The reader may wish to inform the author on the extent to which this objective has been fulfilled. Many students, colleagues, and practicing professionals have suggested improvements, identified typographical errors, and recommended additional top-

15 Preface to the Third Edition xvii ics for inclusion. All these suggestions were carefully considered and have been included in this third edition whenever possible. During the preparation of this third edition, I became indebted to many people to whom I wish to express my appreciation. I am grateful to John D. Hooper and to Robert D. Anderson, consulting engineers on the west coast, who most diligently reviewed the three new chapters in earthquake engineering. Their commentary and suggestions were most useful in making the presentation of the material in these chapters closer to present engineering practice. I am thankful to my colleague Dr. Joseph Hagerty for carefully reading and editing the manuscript of the new chapters. I am also grateful to Dean Leo Jenkins of Speed Scientific School and to Dr. Manuel Schwartz, of the Department of Physics, who reviewed the manuscript at the early stages of its preparation. A special acknowledgment of gratitude is extended to my friend Dr. Edwin A. Tuttle, Professor of Education, who most kindly spent many hours checking my English grammar. My thanks also go to Miss Debbie Gordon for her competent typing of the manuscript. To those people whom I recognized in the prefaces to the first and second editions for their help, I again express my wholehearted appreciation. Finally, I give due recognition to my wife, Jean, who with infinite patience and dedication helped me in editing the whole manuscript. Any remaining errors are mine. MARIO PAZ

16 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 a c ~ o m p in l ia spractical h e d 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 xix

17 xx Preface to the First Edition 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