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  • 1. Response Spectrum Method In Seismic Analysis and Design of Structures AJAYA K U M A R GUPTA. Prokssor of Civil Engineering North Carolina State University FOREWORD BY WILLIAM J. HALL Professor and Head. Civil Engineering Universit.~of lllinois at Urbana -Champaign BLACKWELL SCIENTIFIC PUBLICATIONS BOSTON OXFORD LONDON EDINBURGH MELBOURNE
  • 2. Q 1990by Blackwell ScientificPublications, Inc. Editorial offices: 3 CambridgeCenter. Suite 208 Cambridge, Mas~gchusetts02142, USA " Osney Mead, Oxford OX2 OEL, England 25 John Street, London WClN 2BL, England 23 Ainslie Place, Edinburgh EH3 6M, Scotland 107 Bany Street. Carlton ' Victoria 3053, Australia I . All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storageand retrieval systems, without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review First published 1990 Set by Tima Graphics, Singapore Printed and bound at the University Press, Cambridge. England 90 91 92 93 4 3 2 1 DISTRIBUTORS USA Blackwell Scientific Publications,Inc. Published Business Services PO Box 447 Brookline Village Massachusetts 02147 (Orders: Tel: 617 524-7678) Canada Oxford University Press 70 Wynford Drive Don Mills Ontario M3C 159 (Orders: Tel: 416 441-2941) Australia Blackwell ScientificPublications (Australia)Pty Ltd 107 Barry Street Carlton, Victoria 3053 (Orders: Tek 03 347-0300) Outside North America and Australia Marston Book S e ~ c e sLtd PO Box 87 Oxford OX2 ODT (Orders: Tel: 011 44 865 791155) Library of Congress Cataloging-in-PublicationData Gupta Ajaya K. Response spectrum method in seismic analysis and design of structures/ Ajaya Kumar Gupta; foreword by W.J. Hall. P- cm. -(New directions in civil engineering) ISBN 0-86542-1 15-3 1. Earthquake engineering. 2. Structural engineering. 3. Seismicwaves. 1. Title. 11. Series. TA654,6.G87 1990 624.1'7626~20 British Library Cataloguing-in-Publication Data Gupta, Ajaya Kumar Response spectrum method in seismic analysis and design of structures, A. I. Structure. Analysis - I. Title 11. Series 624.1'7 1 ISBN 0-632-02755-X
  • 3. : New Dirdons in Civil Engineering SERIES E D C T Q ~W.F. CH EN Purdue University
  • 4. Dedicated to my parents Dr Chhail Bihari Lal Gupta and Mrs Taravali Gupta
  • 5. Contents Foreword, ix Preface, xi Acknowledgments, xv 1 Structural dynamics and response spectrum, 1 I. 1 Single-degree-of-freedom system, I 1.2 Response spectrum, 2 1.3 Characteristics of the earthquake response spectrum, 6 1.4 Multi-degree-of-freedomsystems. 7 References, 10 2 Design spectrum, 11 2.1 Introduction, I I 2.2 'Average' elastic spectra, 12 2.3 Site-dependent spectra. 16 2.4 Design spectrum for inelastic systems, 23 2.5 Comments. 27 References, 28 3 Combination of modal responses, 30 3.1 Introduction, 30 3.2 Modes with closely spaced frequencies, 31 3.3 High frequency modes-rigid response, 39 3.4 High frequency modes-residual rigid response, 45 References. 49 4 Response to multicomponents of earthquake, 51 4.1 Introduction, 51 4.2 Simultaneous variation in responses, 52 4.3 Equivalent modal responses, 55 4.4 Interaction ellipsoid, 59 4.5 Approximate method, 60 4.6 Application to design problems, 62 References, 64 5 Nonciassically damped systems, 66 5.1 Introduction, 66 5.2 Analytical formulation, 67 5.3 Response spectra. 71
  • 6. viii 1 CONTENTS 5.4 Key frequencies f L and f H , 74 5.5 Modal combination, 75 5.6 Modal combination for high frequency modes, 77 5.7 Modal combination for high frequency modes-residual rigid response, 78 5.8 Application, 81 References, 87 6 Response of secondary systems, 89 6.1 Introduction, 89 6.2 Formulation of the coupled problem, 91 6.3 Coupled modal properties, 95 6.4 Coupled response calculation, 98 6.5 Comparison of coupled response with the response from conventional IRS method, 101 6.6 An alternate formulation of the coupled response, 106 6.7 Secondary system equivalent oscillators. 108 6.8 Evaluation of instructure spectral quantities, 110 6.9 Examples of instructure response spectra, 1I4 6.10 Correlation coefficients, 1 16 6.1 1 Response examples, 1 18 References. 124 7 Decoupled primary system analysis, 125 7.1 Introduction, 125 7.2 SDOF-SDOF system, I26 7.3 MDOF-MDOF systems, 130 7.4 Application of the frequency and response ratio equations, 131 References. 138 8 Seismic response of buildings, 139 8.1 Introduction, 139 8.2 Analysis, I39 8.3 Building frequency, 144 8.4 Seismic coefficient, 144 References. 152 Appendix: Numerical evaluation of response spectrum, 153 A. I Linear elastic systems, 153 A.2 Bilinear hysteretic systems, 156 A.3 Elastoplastic systems, 158 A.4 Notes for a computational algorithm. 159 A.5 Records with nonzero initial motions, 160 References, 163 Author index, 165 Subject index, 167
  • 7. Foreword This book devoted to the Response Spectrum Method contains concise sections on a number of the major topics associated with the application of spectrum tech- niques in analysis and design. Although the theory of spectra has been understood for some extended period of time, it was only in the past twenty years that the approach was adopted in a major way by the profession for use in engineering practice. This development came about as a result of three major factors, namely that the theory and background of spectra was more fully understood, that the theory was relatively simple to understand and use, and because there was a need for such a simple approach by the building codes and by the advanced analysis techniques needed in the design of nuclear power plants and lifeline systems. The author rather directly presents his interesting and informative interpreta- tions of various spectrum techniques in the topical chapters. He correctly points out that much work remains to be accomplished, which is accurate, for spectra in general only depict maxima of various effects, and in many cases, especially where nonlinear effectsare to be treated, it is often desirable to know more about the response than just a maximum value. Research on such topics presently goes forward on such matters at a number of institutions, and in time will lead to even greater understanding of the theory, and to new approaches of application. In this connection one can cite subtle yet important differences in use and interpretation of spectra. For example, the term 'response spectrum' normally is used to refer to a plot of maximum response parameters as a function of frequency or period, for a given excitation of the base of a single-degree-of-freedomdamped oscillator, as for acceleration time history of excitation associated with a specific earthquake. On the other hand a design spectrum is a similar shaped plot selected as being representative of some set of such possible or plausible excitations for use in design; as such it is a characterization of effects that might be expected as a result of some possible range of excitation inputs, and possibly adjusted to reflect risk or uncertainty considerations, personal safety requirements, economic considera- tions, nonlinear effects,etc. One can immediately discern the differences, directly or subtly as may be the case. It is believed that the reader will find the interesting presentation by Dr Ajaya Gupta to be educational and informative, and hopefully such as to promote additional effort to improve even further our understanding of the theory and applications thereof. W. J. HALL Professor and Head, Civil Engineering University of Illinois at Urbana-Champaign
  • 8. In modern earthquake engineering the response spectrum method has emerged as the most commonly used method of analysis. The primary reason for this popularity is the fact that it provides the designer with a rational and simple basis for specifying the earthquake loading. Another reason often cited is that the method is computationally economical. If a comparison is made between the computational effort required in, say, a modal superposition analysis of a multi- degree-of-freedom structure subjected to a specified ground motion history, and that in a response spectrum analysis including the evaluation of the response spectrum from the same motion history, it is not clear whether the response spectrum method would do much better. A major part of the effort, which is com- mon in both the methods, is the solution of the eigenvalue problem. In fact, if the objective is to evaluate the response of a structure subjected to a known earthquake ground motion, there should not be any question about using a standard time-domain analysis, or alternatively, an equivalent frequency- domain analysis. It is when we are designing a structure for a potential future earthquake that the response spectrum method is much more relevant. Criticisms of the response spectrum method arise from the fact that the- temporal information is lost in the process of evaluating the spectrum. In the words of Robert Scanlan:' 'Multi-degree-of-freedom cases are thus improperly served, intermodal phasings, in particular, being unaccounted for.' Further he points out: 'The needs arising in the design of secondary responding equipment (piping, machinery, etc., on upper floors of a structure)are not adequately met by the given design response spectra. That is, the given primary shock spectra do not lead directly and simply to definition of corresponding secondary shock spectra.' Similar difficulties arise in combining the responses from three components of the earthquake. Much progress has been made in the last decade. Lack of temporal information in the response spectrum method no longer appears to be a handicap. Rational rules are now available to combine responses from various modes, and from three components of earthquake motion. These rules account for the physics of the problem, and can be further justified in the same spirit as the design spectrum itself, as a representation of expected response values in an uncertain world. Response of secondary systems can now be evaluated using efficient modal synthesis techniques in conjunction with the response spectrum method. Alternatively, the secondary spectrum, or the instructure response 'R. H. Scanlan. On Earthquake Loadings for Structural Design, Earthquake Engineering and Sfrucfural Dynamics, Vol. 5, 1977, pp. 203-205
  • 9. spectrum can be evaluated by applying similar modal synthesis techniques to the secondary single-degree-of-freedom oscillator coupled to the primary system. These new techniques directly use the design response spectrum at the base of the primary structure as seismic input and account for the effects of mass interaction (betweenthe equipment and the structure) and of multiple support input into the secondary system. In doing so it is no longer necessary to convert the design res- ponse spectrum into a 'compatible' motion history or a power spectral density function. The question of noncIassica1 damping introduced in the coupled primary-secondary system, which had not even been specifically raised 10or 15 years ago, is now adequately addressed. This brings us to the objective of this book. It is intended to bring together in one volume the wealth of information on the response spectrum method that has been generated in recent years. Needless to say that this information has reached a critical mass suitable for a book. This book can be used as a text or as reference material for a graduate level course. Although Chapter 1 begins with the introductory information about the single-degree-of-freedom systems that leads into the definitionof the response spectrum, 1feel that most students will be more comfortable with the material in subsequent chapters if they already have had an introductory structural dynamics course. This book should also serve as a useful reference for prac~icingengineers. It should help them appreciate the analytical techniques they are already using. In many cases the book may also help them improve those techniques, especially when the improvement would lead to enhanced accuracy, often resulting in significantly lower response values. It is assumed throughout the book that we are dealing with linear systems. There are two exceptions. In Chapter 2 a brief treatment is given to inelastic res- ponse spectra. Chapter 8 dealswith conventional buildings which are customarily designed to undergo significant inelastic deformation under the worst loading conditions. Inelastic behavior has always been a part of seismic design of buildings, unintentionally in the beginning, and later with full knowledge and intention. Yet, our knowledge of the topic is relatively limited. Inelastic seismic behavior and design continue to be a topic of active research. Detailed coverage of current research on the topic goes beyond the realm of the response spectrum method, and is beyond the scope of this book. Brief treatments in Chapter 2 and Chapter 8 are intended to provide a useful link between the response spectrum method and the design of conventional buildings. It should be of particular interest to the students to see the link established and, at the same time, recognize the limitations of the link. I have emphasized deterministic modeling of the earthquake response phenomenon. For a given earthquake ground motion, the maximum response values for a single-degree-of-freedom system-which are the basis of the definition of response spectrum-are deterministic quantities. For a multi- degree-of-freedom system, therefore, the maximum response values in individual modes are also deterministic quantities. The modal combination rules are based
  • 10. partly on the physics of the problem, that is on deterministic concepts, and partly on the random vibration modeling of the phenomenon. Strictly speaking, then, these rules do not apply to responses from individual earthquakes. On the other hand, we can look upon the modal combination rules as tools for giving approximate values of the deterministic maximum response values. It is in this spirit that the response spectrum analysis results have been repeatedly compared with the corresponding time-history maxima for individual earthquakes, treating the latter asthe standard. This concept is especially powerful whenjudging two or more modal combination rules within the response spectrum method. A rule which models the physics well is likely to give results which are reasonably close to those obtained using the time-history analysis. Probabilistic concepts play an important role in the definition of the design spectrum, as they do in defining other kinds of loads too. These concepts are most useful when all the available deterministic tools have been carefully employed. One should not replace the other. Great strides have taken place in recent years in the development and application of random vibration techniques to the earthquake response problems. Important contributions have been made to the response spectrum method using the random vibration concepts. This book has not covered those techniques and concepts for most part. My interest in the response spectrum method has been the primary motiva- tion for writing this book. This interest has been sustained through many years of research on related topics in collaboration with coworkers and students. Such personal involvement in the topic has its advantages and disadvantages in writing a book. The advantages are obvious. The main disadvantage is that I may not be able to do full justice in presenting the works of other researchers. To that end, I shall welcome criticism and suggestions from the readers, which I hope will improve the future editions of this book. A. K. GUPTA
  • 11. Acknowledgments My interest in the response spectrum method started during my years at Sargent and Lundy in Chicago (1971-76). My division head, Shih-Lung (Peter) Chu, asked me to work on the combination of responses from three components of an earthquake. A former graduate student colleague from the University of Illinois at Urbana-Champaign, Mahendra P. Singh (now at Virginia Polytechnic Institute and State University) was also a coworker at Sargent and Lundy and was among those who willingly shared their knowledge. During my association with Illinois Institute of Technology (1976-80), I joined the American Society of Civil Engineers (ASCE)Working Groupcharged with preparing a Standard for Seismic Analysis of Safety Related Nuclear Structures. Robert P. Kennedy, who chaired the effort, encouraged me to become involved in the combination of modal responses. Another colleague in the group, Asadour H. Hadjian from Bechtel, Los Angeles actively participated in the resolution of the topic. I came to North Carolina State University in 1980 and have had a series of students who have participated in the efforts related to the response spectrum method. Karola Cordero and Don-Chi Chen worked on the modal combination methods. The ASCE Working Group was deliberating on developing the criterion for decoupled analysis of primary systems (1981)when I became interested in the topic along with another former student Jawahar M. Tembulkar. The decoupling study serendipitously led me and Jing-Wen Jaw into the coupled response of secondary systems (1983). Jerome L. Sachman and Armen Der Kiureghian were very helpful in keeping us informed about the related developments at the University of California at Berkeley. Min-Der Hwang and Tae-Yang Yoon are present graduate students who have helped in this project in many ways. Ted B. Belytschko, of Northwestern University and an editor of Nuclear Engineering and Design,has been responsible for the publication of many of our papers. He also reviewed early outlines of the present work, suggesting valuable improvements. William J. Hall of the University of Illinois; Robert H. Scanlan of the Johns Hopkins University; Bijan Mohraz of Southern Methodist University and formerly my graduate advisor at the University of Illinois (1968-7 I);Takeru Igusa of Northwestern University; and Vernon P. Matzen, James M. Nau, Arturo E. Schultz and C. C. (David) Tung, my colleagues at North Carolina State University, have read all or part of the manuscript and offered valuable comments. It has been a pleasure to work with Blackwell Scientific Publications, in particular with Navin Sullivan, Edward Wates and Emmie Williamson. W. F.
  • 12. Chen of Purdue University, Editor of the series New Directions in Civil Engineering, facilitated prompt review of the manuscript. The manuscript was produced by EngineeringPublications at North Carolina State University under the direction of Martha K. Brinson, who was assisted by Sue Ellis and Kraig Spruill in word processing and by Mark Ransom and his coworkers in preparing illustrations. My talented and beautiful daughters Aparna Mini and Suvarna (Sona) gave me their unconditional loveand support. Tothem, to everyone named above and to the many other coworkers and students who have assisted me on various occasions, I acknowledge a deep sense of gratitude.
  • 13. Chapter l/Structural dynamics and response spectrum 1.1 Single-degree-of-freedom system Figure I. l(a) shows an ideal one story structure model. It has a rigid girder with lumped mass m which is supported on two massless columns with a combined lateral stiffness equal to k. The energy loss is modeled by a viscous damper, also shown in the figure. This structure has only one degree of freedom, the lateral displacement of the girder. Under the action of the earthquake ground motion, u,, the structure deforms, Figure l.l(b). The relative displacement of the girder with respect to the ground is u. The total displacement of the girder is u-(- u,) = u +u,. Figure I. l(c) shows the free body diagram of the girder, in
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