[ajaya kumar gupta]_response_spectrum_method_in_se(book_zz.org)

Response Spectrum Method In Seismic Analysis and Design of Structures

A J A Y A K U M A R GUPTA. Prokssor of Civil Engineering North Carolina State University

F O R E W O R D BY

WILLIAM J. HALL Professor and Head. Civil Engineering Universit.~ of lllinois at Urbana - Champaign

BLACKWELL SCIENTIFIC PUBLICATIONS

BOSTON OXFORD

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Library of Congress Cataloging-in-Publication Data

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. Seismic waves. 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

: New Dirdons in Civil Engineering SERIES E D C T Q ~ W. F. CH E N Purdue University

Dedicated to my parents Dr Chhail Bihari Lal Gupta and Mrs Taravali Gupta

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-freedom systems. 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, 5 1 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. 7 1

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, 8 1

References, 87

6 Response of secondary systems, 89 6.1 Introduction, 89 6.2 Formulation of the coupled problem, 9 1 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, 1 10 6.9 Examples of instructure response spectra, 1 I4 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

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 techniques 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 effects are 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-freedom damped 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 considerations, 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

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 common 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

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 (between the 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 response 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 10 or 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 definition of the response spectrum, 1 feel 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~icing engineers. 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 response spectra. Chapter 8 deals with 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

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 as the standard. This concept is especially powerful when judging 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

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 Group charged 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 (1 98 1) 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 (1 983). 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 (1 968-7 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.

Chen of Purdue University, Editor of the series New Directions in Civil Engineering, facilitated prompt review of the manuscript.

The manuscript was produced by Engineering Publications 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 love and support. To them, 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.

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 which f; denotes the inertia force, f, the spring (or the column) force and f, denotes the damping force. The equilibrium equation for the girder is simply

Our structure is linear elastic, having the force-displacement relationship shown in Figure I . l(d). Therefore,f, = ku. The viscous damping force f, is assumed to vary linearly with relative velocity u, f, = cii, Figure l.l(e). The inertia forcef; is given by tn(u + u,). A super dot ( ') denotes the time derivative. Making the substitutions in Equation 1.1, we get

rn(ii + i i , ) + cu + ku = 0, ( 1 ..a

Equation 1.3 represents damped vibrations of the structure subjected to the -mu, force. We now use the following basic relationship of structural dynamics; k = t?ro2, and c = 2mol; which with Equation 1.3 becomes

where o is circular frequency of the structure in radians per second and [ is the damping ratio. For free response to be vibratory, ( < 1. For most structures l; is small, say < 0.1, or 10%. We note that the frequency in Hertz (Hz) or in cycles per second (cps) f = 0/2n, and that the period of vibration T = I /f = 2x10, which is in seconds.

Equation 1.4 can be solved using standard numerical techniques. As a result we can obtain the time histories of displacement, velocity and acceleration, of the spring and the damping forces. and any other related response time history. See the Appendix.

. .

Lateral Stiffness

Mass M

(a) One story model

- f .

(c) Free body diagram

C-- (-4 (b) Model subjected to

ground motion

(d) Elastic force-deformation (e) Viscous damping force- relation velocity relation

Fig. 1.1 A single-degree-of-freedom model. (Based on Chopra [ I ] . )

1.2 Response spectrum We can solve Equation 1.4 for many single-degree-of-freedom (SDOF) structures having different frequencies, each subjected to the same earthquake ground motion. For each structure we can calculate the absolute maximum value of the response of interest from the corresponding time history. In earthquake response calculations the sign of response is often not considered. For design purposes the maximum positive and negative values are assumed to have equal magnitudes, hence the absolute sign. The curve showing the maximum response versus structural frequency relationship is called the response spectrum.

STRUCTURAL DYNAMICS A N D RESPONSE SPECTRUM13

Time, sec

Fig. 1.2 Ground acceleration history of El Centro earthquake (SOOE, 1940).

For designing a structure, we are most interested in the maximum spring force f,, which can be evaluated if the maximum relative displacement u is known. A plot between maximum relative displacement and structural frequency is called the displacement response spectrum. Its ordinates are called spectral displacements, and are denoted by S,(f; (). Depending upon the context, they can also be denoted by SD(o, 0, SD( f ), SD(o), or simply by SD. Let us write

Figure 1.2 shows the ground acceleration time history of the El Centro (SOOE, 1940) earthquake. The corresponding displacement response spectrum is shown in Figure 1.3(a).

Let us consider the spring force-displacement relationship f, = ku. We have indicated earlier that if the relative displacement u is known, we can find the spring forcef,. Alternatively, if the spring force is known, we can determine the corresponding relative displacement. We can visualize this as a pseudo-static problem shown in Figure 1.4. Now let us think of-& as a pseudo-inertia force, which can be written in terms of the pseudo acceleration a as ma. The relationships, ma = f, = ku, give a = (k/m)u = 0 2 u . The absolute maximum value of a is called spectral acceleration SA. We can easily see

From Equation 1.2 we observe that when cu is small we can write m(ii + ii,) -- -ku, or the total acceleration (u + u,) 1. (-k/m)u = - 0 2 u . This means,

This pseudo-acceleration response spectrum for the El Centro earthquake is plotted in Figure 1.3(c).

Fig. 1.3 (a) Displacement response spectrum. (b) Velocity response spectra. (c) Acceleration responsc spectrum for El Centro earthquake (SOOE. 1940); damping ratio, 6 = 0.02.

STRUCTURAL DYNAMICS A N D RESPONSE SPECTRUM/S

Fig. 1.4 Pseudo-static problem.

Having defined the response spectra for relative displacement and for pseudo acceleration, we wish to define a response spectrum for velocity. It can be done in more than one way. First, let us define a spectral velocity Sv such that the kinetic energy associated with it is equal to the maximum strain energy of the spring, (1/2)mS: = ( 1 / 2 ) k ~ i . This gives

The spectral velocity Sv is really a pseudo velocity because it is not directly related to the actual velocity of the structure. This pseudo-velocity response spectrum for the El Centro earthquake is plotted in Figure 1.3(b).

We now have three spectral quantities SD, Sv and SA which have units of displacement, velocity and acceleration, respectively. Only spectral displacement SD is directly based on an actual response quantity, the maximum relative displacement. Equations 1.6 and 1.8 give their mutual relationships

Because of this relationship it is possible to read S,, Sv and S, from the same logarithmic chart shown in Figure 1.5. This chart is known as the tripartite chart because, for any frequencyJ there are three scales, one each for SD, Sv and SA.

Now consider the second way of defining a velocity spectrum. We shall denote the new quantity by S;. It is defined as the absolute maximum relative velocity

S; = max I u(t) 1. (1.10)

The relative velocity spectrum is shown in Figure 1.3(b) with the dashed lines. The two spectra in the figure are close in the intermediate frequency range; the pseudo velocity spectrum is higher in the high frequency range, and the relative velocity spectrum is higher in the low frequency range. Thus, as a rule, we cannot substitute one spectrum for the other.

For the SDOF structure, the response spectrum quantity of interest is any one of S,, S, or S,. Also, for the classically damped r n u ~ ~ ~ ~ f - f r e e d o m (MDOF) systems defined in Section 1.4, we need only one o&*drce spectra. We shall see in Chapter 5, that we also need S; for nonclassically? systems.

1.3 Characteristics of the earthquake response spectrum Let us observe Figure 1.5 again, which shows the tripartite El 1940) response spectrum, along with the maximum ground displacement, # velocity and acceleration values. It is clear that in the low frequency range S, = max I u, I, and in the high frequency range S, = max 1 ii, I . This phenomenon can be easily explained.

The low frequency range is characterized by a low value of the spring stiffness k, o = J(k /m) . As the spring stiffness becomes smaller and smaller, it progressively ceases to transmit any motion to the mass. In the limit, the total displacement of the mass tends to zero. Relative displacement of the oscillator becomes - u,, or S,, = max I u, I .

STRUCTURAL DYNAMICS A N D RESPONSE SPECTRUM17

- Maximum relative displacement can be expressed as: S, = I u I,,, - I mii,/k I,,, X (dynamic amplification factor). We know that when the oscillator (structural) frequency is sufficiently greater than the dominant frequencies of the input force (mu,), then the dynamic amplication factor = 1.

Therefore, S, = / mu,/k I ,,, = ( i i , / 02 I ,,,, or S, = 02~,, = ( ii, (

We can think of the tripartite response spectrum as 'anchored' on the two sides to the maximum ground displacement and acceleration values. In the intermediate frequency range the spectrum has amplified spectral displacement, velocity and acceleration. These observations will be useful in developing design spectra in the next chapter.

1.4 Multi-degree-of-freedom systems Figure 1.6 shows a 3-degree-of-freedom (3-DOF) structure which is a simple example of MDOF systems. The equation of motion for this structure can be derived in a manner similar to that for the SDOF structure we did earlier. For a rigorous derivation the reader is referred to books on structural dynamics [2]. Our example 3-DOF structure has three story masses, nt, , m,, m,, and three story stiffnesses, k,, k,, k,. The three DOF are associated with the lateral (horizontal) displacements of the three masses. The structure deforms under the action of earthquake ground motion, u,. The relative displacement of the structure is given by ur = [u, u2 u,] . The inertia force vector is

0 i i , + ii, 4=Mlij+og;= ['a in2 [ i i2+ i ig ] , (1 .1 1 )

0 in3 ii, + ii, where U, is a vector of ground acceleration ii,. The vector of spring forces is given

by

When damping is absent the equilibrium equation simply becomes F, + Fs = 0, which can be written as

In the above equation M is the mass matrix of the structure, K the stiffness matrix, and the vector 1 consists of unit elements. For the 3-DOF structure these matrices are explicitly defined above. For other MDOF structure these matrices

(a) Simple 3-DOF System (b) Deformed Shape

Fig. 1.6 Example of a multi-degree-of-freedom system.

can be obtained using standard procedures [2]. A more general form of the undamped equation of motion is

The vector U,, defines static structural displacements when the support undergoes a unit displacement in the direction of the earthquake. For the simple structure at hand, it is easy to see that U, becomes 1 as in Equation 1.13.

I

The mode shapes and the frequencies of the structure are obtained by solving the following eigenvalue problem

where w is a natural frequency of the structure. The solution of Equation 1.15 gives N frequencies and the corresponding mode shapes or modal vectors, where N is the number of DOF of the structure. Figure 1.7 shows the mode shapes and frequencies of 3-DOF structure when m, = m, = m3 = m, and k, - k, =

k, = k. Let us denote the frequency of the ith mode of an N-DOF structure by a, and the modal vector by 9,. The modal vectors have the following orthogonal properties

I i ( P ~ ' M @ ~ = O and @ i r ~ @ j = O f o r i # j . (1.16a)

1 The modal vectors are often 'normalized' such that

in which case, it can be shown that

STRUCTURAL DYNAMICS A N D RESPONSE SPECTRUMIS

Mode 1 Mode 2 Mode 3

f = o.0708 Hz f = 0 . 1 9 8 4 HZ f - 0.2a7fi HZ

Fig. 1.7 Unnormalized mode shapes and frequencies of a 3-DOF system, m, = m, - m , , k , = k 2 = k,.

The response of the structure is represented in terms of a linear superposition of mode shapes

where y, terms are called normal coordinates, and are functions of the time variable t. Substitution of Equation 1.17 in Equation 1.14, premultiplication by i$jT, and the application of the orthogonality conditions from Equation 1.16 gives

in which y, is called the participation factor for mode i, and is given by

Equation 1.18 is similar to Equation 1.4 for the SDOF structure for the undamped case.

It is difficult accurately to define the damping matrix for a MDOF structure. Often it is assumed that the damping matrix C has orthogonality properties similar to those of M and K, and that we can define the damping ratio for each mode just as we did for a SDOF structure

Structures that have the idealized damping matrix property given by the above equations are called classically damped. Equation 1.18 is replaced by

In the modal superposition method Equation 1.21 is solved to obtain the time histories of the normal coordinates y,, which with Equation 1.17 give the history of the relative displacement vector U, etc.

We shall now use the above concept to apply the response spectrum method to the MDOF structure. The comparison of Equations 1.4 and 1.21 shows, yi(f) = y,u(t) , when o = w, and & = 6 , . Hence, yimar = y, SD(w,. r,) = y,SD;. Thus, the maximum displacement vector in the ith mode can be written as

Given the displacement vector U,,,,, we can determine the maximum value of any response of interest. Methods of combining maximum response values from various modes, and from three components of earthquake are presented in Chapters 3 and 4, respectively.

References I . A.K. Chopra, Dwamics of Struc~ures - A Prirtier, Earthquake Engineering Research Institute, Berkley, California, I98 1 . 2. R.W. Clough and J. Penzien, D7vnarnics ofStructures, McGraw Hill, New York, 1975.

Chapter Z/Design spectrum

2.1 Introduction We have reviewed the basic concepts of dynamic structural analysis in Chapter 1. Only linear elastic behavior has been considered. The purpose was to set the stage for other topics. Indeed, theory and techniques of structural dynamics have reached a stage of advancement such that it is fair to say that any structure that can be mathematically modeled can be analyzed subject to any given transient forcing function, e.g. an earthquake ground motion. The structure may have any given linear or nonlinear constitutive properties. Large displacement effects can also be accounted for. For us the key here is the definition of the earthquake ground motion. If we know the ground motion history, we can analyze the structure and design it. But the earthquake we are talking about has not yet occurred.

In many ways the problem of specifying a future earthquake is not very different from specifying any other load for design. The actual live load on a building floor varies a great deal during its lifetime, and it is not uniformly distributed on the floor area. There are at least three idealizations involved in live load specification. First, we idealize the actual floor distribution of furniture, people and other live loads as a uniformly distributed load such that the design quantities of interest-the slab moments-have approximately the same magnitude. Second, we estimate the likely maximum magnitude of this uniformly distributed load during the lifetime of the building. Finally, we design the floor using appropriate load factors, capacity reduction factors or factors of safety. The end product is a slab, which, incidentally, has a relatively definite resistance. The art of specifying the load and the remainder of the design procedure is then a 'recipe', which more than anything else assures a resistance. Therefore, when we are specifying a load, we are really specifying the resistance or the level of resistance in a structure or a structural element.

The specification of the earthquake load consists of determining themagni- tude and intensity of the design ground shock at a given site, and of somehow converting them into the ground motion parameters. The intensity of the design earthquake is determined from the seismological and geological data concerning earthquakes and their occurrence. It is well to note that available data base is far from adequate and is the major source of uncertainty in earthquake-resistant design. It is then unrealistic in most cases to expect that we can characterize a future ground motion in any detail based on the design intensity and any other limited information available. Approximate procedures have been developed to

11

give the estimate of the peak ground acceleration associated with intensity levels. In some cases the peak ground displacements and velocity can also be approximately estimated.

Given the peak ground motion parameters-displacement, velocity and acceleration-techniques have been developed to define smooth spectrum curves, which are called design spectra. The main difference between the response spectrum presented in the previous chapter and the design spectrum we are discussing now is that the former represents the response to an actual earthquake and the latter defines the level of seismic resistance we are to design for. Just as in the example of the live load, the design spectrum idealizes the real phenomenon to fit it into a design recipe.

2.2 'Average1 elastic spectra Biot [I .2] and Housner[3] were among the first researchers who recognized the potential of the response spectrum as an earthquake-resistant design tool. Biot[l.2] developed a mechanical analyzer to evaluate experimentally the response of a single-degree-of-freedom system subjected to recorded earthquake acceleration time histories. For design purposes he suggested smoothing the response spectra obtained from actual records. The mechanical analyzer Biot used was practically undamped, although he recognized that the damplng will lower the peaks of the response spectra. Later, Housner[4] obtained design spectra by averaging and smoothing the response spectra from eight ground motion records, two each from four earthquakes, viz., El Centro (1934), El Centro (1 WO), Olympia (1949), Tehachapi (1 952). Housner's spectra for several damping values are shown in Figure 2.1. These were the first spectra used for the seismic design of structures. The spectra shown in Figure 2.1 have been scaled for 0.125 g peak ground acceleration. One could scale them to any other peak ground acceleration consistent with the design intensity of earthquakes at the site.

Newmark and coworkers[5,6] studied the response spectra of a wide variety of ground motions, ranging from simple pulses of displacement, velocity or acceleration of ground through more complex motions such as those arising from nuclear blast detonations and for a variety of earthquakes as taken from available strong motion records. They observed that the general shape of a smoothed response spectrum looks like that shown in Figure 2.2 on a logarithmic tripartite graph. As we observed in Chapter 1, in the low frequency range, the special displacement SD = maximum ground displacement d; and in the high frequency range, the spectral acceleration S, = maximum ground acceleration a. The intermediate frequency range can be divided into five regions: ( I ) an amplified velocity region in the middle which is flanked by (2) amplified displacement and (3) acceleration regions, and two regions of transition, (4) from the maximum ground displacement to the amplified spectral displacement and (5) from the amplified spectral acceleration to the maximum ground acceleration. In an earlier publication, Blume, Newmark and Corning[7] had suggested the following

DESIGN SPECTRUM/13

0.4 0.8 1.2 1.6 2.0 2.4 2.8

Period (sec)

Period (set)

Fig. 2.1 (a) Housner velocity design spectra. (b) Housner acceleration design spectra, peak ground acceleration = 0.125 g [4l.

I

Frequency, f (Log scale)

Fig. 2.2 General shape of a smoothed response spectrum.

factors for amplified spectral displacement, velocity and acceleration: respectively 1.0, 1.5, 2.0 for 5- 10% damping ratio; and 2, 3 ,4 for < 2% damping ratio.

The elastic spectra of the type we are discussing here are primarily used for critical facilities like nuclear power plants. The United States Atomic Energy Commission sponsored two comprehensive studies in the early seventies, one conducted by Mohraz, Hall and Newmarkt81 for Newmark Consulting Engineer- ing Services and the other by Blume, Sharpe and Dalal[9]. In the Blume study [9] two components of horizontal motion for sixteen earthquakes, and one component for an additional earthquake, a total of thirty-three different records were considered. In the Newmark study[8] fourteen earthquakes were considered with two components of the horizontal motion and one component of the vertical motion for each earthquake. The two studies were conducted independently and there are differences in their details. The conclusions from the two studies were quite similar, however. Therefore, we will present the results of the Newmark study[8] only.

In order to 'average' the response spectra from various ground motion records they need to be normalized using a ground motion parameter, maximum ground displacement, velocity or acceleration. It was found[8] that the normalization to the maximum ground acceleration gave a standard deviation that increased quite uniformly from high frequencies to low frequencies. Normalization to maximum ground displacement showed a standard deviation which increased from low to high frequencies. Further, the normalization to maximum ground velocity showed a nearly constant standard deviation over the whole range of frequencies. The smallest standard deviation was obtained in each region when the normaliza-

DESIGN S P E C T R U M / l S

tion was made to the particular ground motion parameter for which the response spectrum curve was most nearly parallel to the coordinate. In the Newmark study[8], the spectral ordinates were assumed to have a normal distribution; the data would also fit the log-normal distribution[9]. The design spectrum can be obtained from the maximum ground displacement, velocity and acceleration values, if the respective amplification factors are known. These amplification factors were obtained for the mean spectrum (50% probability level) and for the mean plus one standard deviation spectrum (84.1% probability level). Table 2.1 summarizes these amplification factors for the horizontal components of earthquake for four damping values. Values simiIar to those in Table 2.1 for 84.1% probability level have been adopted by a recent ASCE standard[lO]. In the Newmark study[8], it is recommended that the transition from amplified acceleration to ground acceleration begin at 6 Hz for all damping values and end at 40, 30, 20, 20 Hz, respectively for damping ratios of 0.5%, 2.0%, 5.0% and 10.0%. In the ASCE Standard[lO], this transition occurs between 9 and 33 Hz for all damping values. Corresponding to a Ig maximum ground acceleration, it was found[8] that the maximum ground displacement and velocity were approximately 36 in and 48 in sec-I, respectively, for alluvium soil, and 12 in and 28 in sec-' for rock. For both types of soiis, we have ad/u2 2: 6, the value recommended by the ASCE Standard[lO]. Figure 2.3 shows the 'Newmark spectrum' for Ig maximum ground acceleration on an alluvium soil.

The joint Newmark-Blume recornmendations[l l ] were later adopted in a modified form by the United States Atomic Energy Commission (now the US Nuclear Regulatory Commission-USNRC)[12]. Figure 2.4 shows the USNRC spectra for Ig ground acceleration. As can be seen, there are no major differences between Figures 2.3 and 2.4. The recommendations for the vertical component of

Table 2.1 Amplification factors for Newmark spectrum[8]

Damping ratio (%) Spectral Probability quantity level (%) 0.5 2.0 5.0 10.0

SD = fact0r.d. Sv = factor.^, SA = fact0r.a. d. v, a = maximum ground displacement, velocity, acceleration, respectively. adlv2 1. 6.

Frequency, Hz

Fig. 2.3 Newmark design spectra for alluvium soil; maximum ground acceleration = I g.

earthquake vary much more, and a complete discussion is beyond the scope of this book. The ASCE Standard[lOJ recommends that the design spectra for the vertical component be obtained by multiplying the corresponding spectra for the horizontal component by a factor of 2/3.

2.3 Site-dependent spectra The spectral amplification factors presented in the previous section are based on the analysis of several earthquake ground motions without particular consideration of the site conditions. The only consideration of site has been that for maximum ground motion parameters suggested by Mohraz, Hall and New- mark[8], d = 36 in and u = 48 in sec-' for alluvium site; and d = 12 in, 28 in sec-' for rock site; for I g maximum ground acceleration. One would expect that the site conditions influence the frequency content in the ground motion, and therefore, the spectral amplification factors would depend upon them too. Mohraz. Hall and Newmark indicated that the spectrum for an alluvium site is considerably different from that for a competent rock site. Since only a few accelerograms from stations on rock site were considered in their study[8], no conclusive recommendations were made.

DESIGN S P E C T R U M 1 1 7

0.1 0.2 0.5 1 2 5 10 20 50 100

Frequency. Hz

Fig. 2.4 United States Atomic Energy Commission design spectra [12].

A major problem associated with evaluating these site-dependent spectra is in the description of the site itself. One possible method is to classify the recording stations according to their shear wave velocity. For most stations the estimates of shear wave velocity are not available, nor available are details of any other soil properties at the stations. Researchers have, therefore, used the limited site information and their experience, and have subjectively classified the recording stations. As we would expect, different researchers have different classifications. Studies that have been performed do show definite trends, and in that sense they are very valuable.

Hayashi, Tsuchida and Kurata[l3] performed a study in which they averaged the normalized response spectra for sixty-one accelerograms obtained from thirty-eight earthquakes in Japan including many with peak accelerations in the relatively low range of 0.02-0.05 g. The spectra were averaged in three groups. Group A was considered to be associated with very dense sands and gravels, Group B with soils of intermediate characteristics and Group C with extremely

loose soils. As expected, they found that the soil conditions affect spectra substantially. Although the authors suggested that the spectral shapes be considered preliminary in view of the relatively low maximum accelerations associated with the earthquake records and the limited data on some of the subsoil conditions at the recording stations, their results were later found to be in substantial agreement with those obtained by Seed, Ugas and Lysmer[l4]. A limited study of the influence of local site conditions on spectral shapes for Japanese earthquakes was presented by Kuribayashi el al. [ I 51.

All the studies on the site dependence of the spectra cited above were performed for the horizontal components of the ground motions. A recent comprehensive study by Seed, Ugas and Lysmer[l4] was also performed on the horizontal components only. They considered four site conditions: rock, stiff soils less than about 150 ft deep, deep cohesionless soils with depths greater than about 250 ft, soil deposits consisting of soft to medium stiff clays with associated strata of sands or gravels. Their study was based on one hundred and four records each with maximum acceleration >0.05 g.

The mean plus one standard deviation-84.1 percentile spectra obtained by Seed, Ugas and Lysmer[l4] for the four site conditions, 5% damping ratio are shown in Figure 2.5 along with the corresponding AEC spectrum[l2]. All the spectra are normalized for 1 g maximum ground acceleration. There are wide differences in the spectral shapes for the four sites for periods greater than roughly 0.4 sec. In these period ranges, the sites on softer soils (deep cohesionless soil and soft to medium clays and sands) have much higher spectral values than those on stiffer soils (rock and stiff soil). The AEC spectrum has the best agreement with Seed, Ugas and Lysmer's spectrum on the stiff soil in the 0.5-3 sec period (0.3-2 Hz frequency) range. The agreement between the AEC spectrum and Seed, Ugas and Lysmer's spectra for all the four site conditions is generally good in 0.1-0.5 sec (2- 10 Hz frequency) range. For periods < 0.1 sec, or frequencies > 10 Hz, the AEC spectrum significantly overestimates the spectral values for the soft soils.

Another recent significant study is due to Mohraz[16]. His study included the vertical components of earthquake-in addition to the usual horizontal components. Further, he calculated and recommended spectral values for several damping ratios. He studied the effects of geological conditions on the spectra, and also on the ground motion parameters, such as peak ground acceleration, velocity and displacement. He considered the two common site conditions, alluvium and rock, and two intermediate site conditions-deposits of < 30 ft of alluvium, and deposits of approximately 30-200 ft of alluvium, both underlain by rock deposits. One reason the latter two categories were selected was that substantial earthquake records from stations located on such deposits were available. The sites labeled alluvium, were those which did not have a specified depth, and may have a depth less than or greater than 200 ft. Mohraz analyzed one hundred and six records from forty-six stations in sixteen seismic events.

DESIGN SPECTRUMJ19

Soft to Medium Clay and Sand

eep Cohesionless Soik (>250W Stiff Soil Conditions k150ft)

Regulatory Guide

0 .5 1 1.5 2 2.5 3

Period, Seconds

Fig. 2.5 Seed, Ugas and Lysmer site-dependent spectra and Atomic Energy Commission spectrum; mean plus one standard deviation (84.1 percentile), peak ground acceleration =

1 g, damping ratio = 0.05 [14].

Mohraz had three components for each earthquake. The peak accelerations for the three components are ordered from the largest to the smallest as follows: the larger horizontal a,,, the smaller horizontal a,, and the vertical a,. The mean value of the ratios aSH/aLH and a,/aLH for all the site conditions are 0.83 and 0.48, respectively. The corresponding 84.1 percentile values are 0.98 and 0.65. That means that at the 84.1 percentile level the maximum ground acceleration for the smaller horizontal component is almost equal to the maximum horizontal acceleration for the larger horizontal acceleration, and that the maximum vertical acceleration is almost 213 of the latter. This is consistent with the assumption commonly made for the design of critical facilities[lO].

Mohraz performed a detailed statistical study on the ratios of the ground motion parameters, u/a and ad/v2. When the ratios are known, we can calculate the values of u and d; the value of a is commonly given based on seismological considerations. Table 2.2 lists the mean values of the ratios v/a and ad/v2, along with the corresponding values of u and d for various site conditions for a 1 g maximum ground acceleration. For illustration here, the larger horizontal

Table 2.2 Ground motion parameters[ 161

Larger horizontal component Vertical component

Mean ratios For a- l g Mean ratios For a- l g

v l 0 adlv2 v d vla ad/u2 v d Site condition (in sec-'g-') (in sec-') (in) (in scc-'g-')

Based on Mohraz[l6]

Rock 27 6.9 27 13.0 31 7.6 31 18.9

Less than 30 ft 3 7 5.2 37 18.4 37 8.5 37 30.1 alluvium underlain by rocks

30-200 ft Alluvium 33 5.6 33 15.8 33 9.1 33 25.6 underlain by rocks

Alluvium 5 1 4.3 51 28.9 51 5.0 51 33.7

Based on Mohraz, Hall and Newmark(81

Rock 28 6 28 12 - - - - Alluvium 48 6 48 36 - - - -

component and the vertical component are included in Table 2.2. The table indicates that the v /a ratios for rock are substantially lower than those for alluvium. These ratios for the two alluvium layers underlain by rock are between those for rock and alluvium. The v /a ratios for the larger horizontal components are the same as that for the vertical component, except in the case of rock, when they are close. The ad/v2 ratios given in Table 2.2 indicate that the ratios for alluvium are smaller than those for rock and alluvium layers underlain by rocks. Since ad/u2 is a measure of the width of the spectra, the ratios indicate that the average spectrum for a rock site is flatter than for alluvium site or for a site with alluvium layers underlain by rock. The values of vla and adlv2 ratios, and of v and suggested earlier by Mohraz, Hall and Newmark(81 for the horizontal component only are also given in Table 2.2 for comparison. There are minor differences between the old and the new v/a values. A relatively greater change occurs in the value of ad/v2 for the alluvium site (6-4.3) and in the corresponding value of d (36-28.9 in). We note that Mohraz[I 61 also gives the median and 84.1 percentile values, which do show dispersion in these values. For most practical applications we believe that using the mean values of the ratios should be adequate.

Mohraz(l61 showed that his average spectrum for the rock site, 5% damping ratio compared well with the corresponding spectrum given by Seed, Ugas and

DESIGN SPECTRUM/21

- Alluvium 4 - ! \

-.- Lesa than 30 ft. Alluvium on Rock

-. ., 30-200 ft. Alluvium on Rock

,-, Rock

0 I 1 I

0 .5 1 1.5 2 2.5 3

Period, Seconds

Fig. 2.6 Mohraz average site dependent spectra; peak ground acceleration - I g, damping ration = 0.02 [16].

Lysmer[l4]. Mohraz's average spectra normalized to 1 g for the four sites are shown in Figure 2.6 for 2% damping ratio. We observe from Figure 2.6 that the acceleration amplification for the alluvium deposits extends over a larger frequency region than the amplification for other site categories. Further, the maximum spectral accelerations for the two sites with alluvium layers underlain by rock is greater than the maximum spectral acceleration for either rock site or the alluvium site. For periods ~ 0 . 2 sec (frequencies > 5 Hz), the spectral ordinates for alluvium sites are less than those for the other three site types studied. In the periods >0.5 sec (frequencies < 2 Hz), the spectral ordinates for the alluvium site are greater than those for the other three. The alluvium spectral values are approximately 2.5-3 times the rock spectral values in the 1.5-3 sec period range (frequencies, 0.3-0.7 Hz).

Mohraz [I 61 has presented comprehensive statistics of displacement, velocity and acceleration amplification factors for all the four site types, damping ratios 0-20%, and the three components of earthquake. Amplification factors are given for the mean (50% probability level) and the mean plus one standard deviation (84.1% probability level) spectral values. Amplification factors for the larger horizontal component for the alluvium site only are reproduced in Table 2.3, for three damping ratios (2%, 5% and 10%) and for 50% and 84.1% probability levels.

Table 2.3 Amplification factors for larger horizontal component for alluvium site suggested by Mohraz(l61

Damping ratio (%) Spectral Probability quantity level (%) 2.0 5.0 10.0

S,, = fact0r.d. S,, = factor-v, S, = fact0r.a. d, v , a = maximum ground displacement, velocity, acceleration, respectively; Table 2.2.

For intermediate damping ratios, Mohraz[l6] suggests double logarithmic interpolation. Mohraz has given amplification factors for the other three site conditions also. For brevity, we are not reproducing that information here. Instead, in Table 2.4 are given the site design spectra coefficients, also reproduced from Mohraz[l6], which can be used to obtain the spectral bounds for the other three site conditions from those for the alluvium site. Mohraz recommends the same coefficients for the two alluvium sites underlain by rock deposits because the coefficients for these two categories do not vary significantly from each other. Since the number of available records for these two site types is not as large as that for either the rock or the alluvium deposits, the recommended coefficients are on the conservative side.

\

Table 2.4 Site design spectra coefficients[l6]

Coefficients

Site category Displacement Velocity Acceleration

Rock 0.5 0.5 1 .05

Less than 30 ft alluvium 0.75 0.75 I .20 underlain by rock

30-200 ft Alluvium 0.75 0.75 1.20 underlain by rock

Design spectrum value at the site = site coefficient X design spectrum value at the alluvium site.

DESIGN S P E C T R U M / 2 3

Frequency, Hz

Fig. 2.7 Mohraz and Newmark site dependent spectra; peak ground acceleration = 1 g. damping ratio = 0.02.

Given the maximum ground acceleration, there is sufficient information in Tables 2.2,2.3 and 2.4 to obtain 50% or 84.1% probability level design spectra for the larger horizontal component. Although the Mohraz study shows that the other horizontal component has slightly less maximum ground acceleration than does the larger horizontal component, it is a common practice to assume that the design spectra for the two horizontal components are the same. Mohraz[l6] does give much detail about the vertical component of the earthquake. However, at the end he derives the vertical design spectral ordinates as 2/3 of the ordinates of the horizontal spectrum, which is consistent with the present practice. The Mohraz horizontal spectra for 2% damping ratio, 84.1 percentile level are shown in Figure 2.7 for the three site conditions; (based on Table 2.4, the two alluvium sites underlain by rock are combined into one site type). Also shown in Figure 2.7 are Newmark's spectra[8] for alluvium and rock sites.

2.4 Design spectrum for inelastic systems The basis of applying the response spectrum method to multi-degree-of-freedom systems is the modal superposition method; see Chapter 1. Strictly speaking,

therefore, the method cannot be applied to inelastic multi-degree-of-freedom systems because the superposition is no longer valid. No such difficulty, however, exists when a single degree-of-freedom is under consideration. In that case the response spectrum simply represents the maximum value of the relative displacement-or of any other quantity of interest. The maximum value can be evaluated whether the system is linear or nonlinear.

In this section we shall present the inelastic spectrum for the single-degree-of- freedom systems. When the major response of a structure, such as a tall building, comes from the fundamental mode, then we can consider the structure to be a pseudo single-degree-of-freedom system and make use of the inelastic spectrum for evaluating the required resistance of the structural members. The inelastic spectrum is sometimes also used for calculating response in higher modes. The accuracy of such an approach is questionable. It can be justified because the contribution of higher modes is relatively small, and the error in the evaluation of higher mode response would not introduce significant error in the overall response of the structure. The question of inelastic response of multi-degree-of- freedom systems is a complex one, and it continues to be a topic of active research. A full discussion on the topic is beyond the scope of the present work. We do note that it is an important topic-the majority of buildings and many other structures are designed based on the assumption of significant inelastic response in case of a severe earthquake.

The simplest inelastic material is elastic-perfectly plastic with equal yield values in tension and compression. For a single-degree-of-freedom system, the ductility factor or ductility (p) is defined as the ratio of the maximum deformation urn to the yield deformation u,, p = urn/uy . In linear elastic analysis we assume that maximum deformation remains below u, . The member is designed such that the analytically calculated u is less than or equal to u,; or that the analytically calculated member force or stress o is less than or equal to the yield force or stress o, corresponding to u,. If the system is capable of safely undergoing inelastic deformation, it is economical to design it such that the maximum allowable deformation is achieved under the given earthquake. For a given material and structural system, the permissible ductility factor p can be judged to be known. The objective of the calculation is to evaluate u, such that u, is achieved under the given earthquake.

In Equation 1.1 ,.A + fD + -f, = 0,1; is still m(u + u,) and fD remains cu. Due to inelasticity now, S, = ku when ) u I G u, , and& = + o, when ) u 12 u,. After one or more plastic excursions these conditions are appropriately modified. The solution of the nonlinear equation is straightforward, although more involved than the solution of a linear equation; see Appendix. The response spectrum consists of the response from many single-degrees-of-freedom of systems with varying frequencies; the damping is kept constant for each response spectrum curve. Now we have another variable, the ductility ratio p. For the ductility ratio

DESIGN SPECTRUM/25

of unity, urn = u,, and we have the elastic response spectrum curve. The inelasticity is introduced when p > 1. Again, for each response spectrum curve a constant value of p is assumed. For a single-degree-of-freedom system of given frequency and damping, the solution process consists of assuming a value of u,, and integrating the nonlinear equation of motion numerically for the earthquake ground motion history which gives u,. For the assumed value of u,, the calculated urn is not likely to give the ductility factor urn/uy equal to the desired p.

For each point, therefore, several solutions have to be performed, each with a different value of u,. When the calculated urn/uy is sufficiently close to the desired p, the iteration stops.

The elastic response spectrum is based on maximum relative displacement, which is also a measure of the maximum spring force. We recall, the relationship S, = oZSD is obtained on the basis that the pseudo-inertia force given by S, is equal to the spring force given by SD. For an inelastic single-degree-of-freedom system the measure of spring force is u,. Since u, and urn are so conveniently related, urn = pu,, we can use either of the two displacements for drawing the response spectrum, as long as we know which one it is in a given case. The spectrum based on urn can be called the maximum displacement spectrum, and that based on u, , the yield spectrum.

Early studies on the inelastic response spectrum were made by Newmark and coworkers [5,6,17]. They reached the following conclusions: 1. For low frequency systems, the maximum displacement for the inelastic system (urn) is the same as for an elastic system having the same frequency. 2. For intermediate frequency systems, the total energy absorbed by the spring is the same for the inelastic system as for an elastic system having the same frequency. 3. For high frequency systems, the force in the spring is the same for the inelastic system as for an elastic system having the same frequency.

Let us denote the elastic spectral values by s E , and the corresponding maximum displacement and yield spectra values by SM and SY, respectively. The above conclusions give the following relationships:

Low frequency range, SM = s E , sY = SE/p;

P p , SY = 1

Intermediate frequency range, SM = J ( ~ P - 1) J(2p - I )

High frequency range, SM = psE, sY = sE. (2.1)

Note, in all the frequency ranges, sM = psY. These recommendations are applied to the Newmark type elastic design

spectrum in Figure 2.8. The symbols D, V, A, A, refer to the bounds of the elastic

Frequency (Log Scale)

Fig. 2.8 Newmark inelastic response spectra.

spectrum. A, represents the maximum ground acceleration. Superscripts M and Y are used to denote the corresponding maximum displacement and yield spectral values. The elastic spectrum bounds D and Vare covered by the small frequency range. The corresponding DY and V' values are obtained by dividing D and Vby p. The value ofA ' is obtained by dividing A by J(2p - 1). A: remains the same as A,. The D and Vspectral lines, and the DY, A' spectral lines intersect at the same key frequency; the key frequency at the intersection of VY and A ' is in general different from that at the intersection of V and A. Usually, we begin the transition from A ' to A, at the same frequency at which the transition from A to A, begins. Having obtained the yield spectrum, the evaluation of the maximum displacement spectrum is straight forward. We simply multiply the yield spectrum ordinates by the ductility factor to do so. In the resulting maximum displacement

I spectrum, we note D~ = D, vM = V. Riddell and Newmark11 81 performed a relatively detailed study on the topic

I

I of the inelastic spectrum. They found that the factors used to modify an elastic spectrum into an inelastic spectrum are functions of the damping ratio and of the

1 I type of the material force-deformation relationship. Overall, however, they

I confirmed the conclusions of the earlier studies[5,6,17]. They evaluated inelastic spectra for elastoplastic, bilinear and stiffness degrading systems. They concluded that the ordinates of the average inelastic spectra for the three material models did not differ significantly. They also found that the spectrum for the elastoplastic material was always slightly conservative as compared with those for the other two materials. That leaves the effect of damping values. One may use the more refined modification factors given by Riddell and Newmark[lS].

DESIGN SPECTRUM127

However, the damping independent modification factors reported here based on the earlier work[5,6,17], appear to be adequate for most practical applications.

The dependence of the inelastic spectra on the sites was studied by Elghadamsi and Mohraz[l9]. As explained above, the inelastic spectrum for a given ductility level has to be obtained iteratively. This procedure is computationally inefficient. Elghadamsi and Mohraz computed the maximum displacement spectra for a given yield displacement, u,. This eliminated the iterations. The authors found that their procedure required approximately 5-10% of the computational time needed for the ductility based inelastic response spectrum calculations. This permitted them to consider a relatively large ensemble of earthquake ground motions, fifty records on alluvium sites and twenty-six on rock sites. The maximum displacement spectra for fixed yield displacement values, in effect, represent the ductility demand curves. These curves can be easily converted into the constant ductility inelastic spectra.

Elghadamsi and Mohraz[l9] used four material models: elastoplastic, bilinear, four-parameter Nadai and a new stiffness degrading model. They found that the inelastic design spectrum computed from the elastoplastic model can be used conservatively in most cases to estimate the design spectrum from the other three models.

2.5 Comments An effort has been made in this Chapter to summarize some of the available information on design spectra in a simplified form. For many readers the information presented here may be adequate. For others, especially, those involved in the design of major facilities and those interested in pursuing the topic for research, there is a vast growing body of literature available. A philosophical and historical perspective is given by Newmark and Ha11[20] and Housner and Jennings[2 11 in two EERI monographs.

Broadly classified spectral shapes presented in this Chapter should serve a useful purpose when more detailed and precise data f0r.a site can not be obtained. It is not uncommon in many cases that the only motion parameter that is specified is the peak ground acceleration. The use of peak ground acceleration in conjunction with a standard spectral shape for a site in the vicinity of a fault can grossly overstate the response values in the frequency range of interest. It is much more desirable somehow to evaluate and use the three major motion parameters: the peak acceleration, velocity and displacement. For sites like those in Mexico City, specific knowledge about the local conditions is very important. Housner and Jennings recommend[2 I]: 'A much better method of describing the ground motion simply would be to compare it to a known accelerogram, such as recorded in Taft, California in 1952, or to a synthesized accelerogram. The decription could thus be phrased as: 1.5 times as intense as Taft 1952, with duration of strong shaking 1.2 times as long and with similar frequencies of motion.'

References 1. M.A. Biot, A Mechanical Analyzer for the Prediction of Earthquake Stresses, Bulletin of the

Seismological Societv of America, Vol. 31, 194 1, pp. 1 5 1 - 1 7 1. 2. M.A. Biot, Analytical and Experimental Methods in Engineering Seismology, Proceedings,

ASCE, Vol. 68, 1942, pp. 49-69. 3. G.W. Housner, An Investigation of the Effects of Earthquakes on Buildings, Ph.D. Thesis,

California Institute of Technology, Pasadena, California, 1941. 4. G.W. Housner, Behavior of Structures During Earthquakes, Journal of Engineering

Mechanics Division, ASCE, Vol. 85, No. EM4, 1959, pp. 109- 129. 5. N.M. Newmark and AS. Veletsos, Design Procedures for Shock Isolation Systems of

Underground Protective Structures, Vol. 111, Response Spectra of Single-Degree-of- Freedom Elastic and Inelastic Systems, Report for Air Force Weapons Laboratory, by Newmark, Hansen and Associates, RTD TDR 63-3096, June 1964.

6. A.S. Veletsos, N.M. Newmark and C.V. Chelapati, Deformation Spectra for Elastic and Elasto-Plastic Systems Subjected to Ground Shock and Earthquake Motions,. Proceedings. Third U,,'orld Conference on Earthquake Engineering, New Zealand, 1965.

7. J.A. Blume, N.M. Newmark and L.H. Corning, Design ofMuhi-Story ReinJorced Concrete Buildingsfor Earthquake Motions, Portland Cement Association, Chicago. Illinois, 1961.

8. B. Mohraz, W.J. Hall and N.M. Newmark, A Study of Vertical and Horizontal Earthquake Spectra, Nathan M. Newmark Consulting Engineering Services, Urbana, Illinois; AEC Report No. WASH-1255, 1972.

9. J.A. Blume, R.L. Sharpe and J.S. Dalal, Recommendations for Shape of Earthquake Response Spectra, John A. Blume and Associates, San Francisco, California; AEC Report, No. 1254, 1972.

10. American Society of Civil Engineering, Standard for the Seismic Analysis ofSafey Related Nuclear Structures, September 1986.

I I . N.M. Newmark, J.A. Blume and K.K. Kapur, Seismic Design Criteria for Nuclear Power Plants. Journal ofthe Power Division, ASCE, Vol. 99, 1973, pp. 287-303.

12. United States Atomic Energy Commission. Design Response Spectra for Seismic Design of Nuclear Power Plants, Regulatory Guide, No. 1.60, 1973.

13. S.H. Hayashi. H. Tsuchida and E. Kurata, Average Response Spectra for Various Subsoil Conditions. Third Joint Meeting. L!S. Japan Panel on Wind and Seismic Efects, UJNR, Tokyo, May 197 1.

14. H.B. Seed, C. Ugas and J. Lysmer, Site-Dependent Spectra for Earthquake-Resistant Design, Bulletin ofthe Seismological Society of America, Vol. 66, No. 1. February 1976, pp. 22 1-243.

15. E. Kuribayashi, T. Iwasaki, Y. lida and K. Tuji, Effects of Seismic and Subsoil Conditions on Earthquake Response Spectra, Proceedings, International Conference on Microzonation, Seattle, Washington, 1972, pp. 499-5 12.

16. B. Mohraz, A Study of Earthquake Response Spectra for Different Geological Conditions, Bulletin ofthc Seismological Socie?rl ofAmerica, Vol. 66, No. 3, June 1976, pp. 9 15-935.

17. A.S. Veletsos and N.M. Newmark, Effect of Inelastic Behavior on the Response of Simple Systems to Earthquake Motions, Proceedings, Second World Conference on Earthquake Engineering, Vol. 11, 1960.

18. R. Riddell and N.M. Newmark, Statistical Analysis of the Response of Nonlinear Systems Subjected to Earthquakes, Strrrcrural Rcsearch Series, No. 468, Department of Civil Engineering. University of Illinois at Urbana-Champaign, Urbana, Illinois. August 1974.

19. F.E. Elghadamsi and B. Mohraz, Site-Dependent Inelastic Earthquake Spectra, Technical Report, Civil and Mechanical Engineering Department, Southern Methodist University, Dallas, Texas. June 1983.

DESIGN SPECTRUM/29

20. N.M. Newrnark and W.J. Hall, Earthquake Spectra and Design (Engineering Monograph on Earthquake Criteria, Structural Design, and Strong Motion Records, M.S. Agbabian, Coordinating Editor), Earthquake Engineering Research Institute, Berkeley, California, 1982.

2 1. G. W. Housner and P.C. Jennings, Earthquake Design Criteria (Engineering Monograph on Earthquake Criteria, Structural Design, and Strong Motion Records, MS. Agbabian. Coordinating Editor,), Earthquake Engineering Research Institute, Berkeley. California, 1982.

Chapter 3/Combination of modal responses

3.1 Introduction The equation of motion for an N-degree of freedom system was presented in Chapter I, and is rewritten below

where M, Cand K are mass, damping and stiffness matrices, respectively; U is the relative displacement vector; Ub is the static displacement vector when the base of the structure displaces by unity in the direction of the earthquake; u, is the earthquake ground acceleration; and a super dot (.) represents the time derivative. The structure has N-orthogonal modal vectors $,, i = 1 - N. For the present treatment, we assume that the modal vectors have been scaled such that $, M $, = 1. Also, we recall from Chapter 1, $, K $, = of and $i C q i = 20,C,,, where o, is the circular frequency in radians per second, and ci is'the damping ratio, both for mode i. In the modal superposition method we use the following transformation (or modal superposition equation)

in which y, is called the normal coordinate. Substitution of Equation 3.2 in Equation 3.1, pre-multiplication by +,*, and use of appropriate orthogonality conditions gives

The term yi is called the model participation factor. If SD(w) represents the displacement response spectrum, and we denote the

spectral displacement by S,,, then by definition (Chapter l ) ,

Also, we can write

Equation 3.5 gives the maximum relative displacement for each mode. The superposition equation, Equation 3.2, applies only when we know the time histories of all the modal displacement vectors in all the modes. Equation 3.5, however, does not provide that information. In general, it is unlikely that the

C O M B I N A T I O N O F M O D A L R E S P O N S E S / 3 1

maximum values of U; in different modes would occur at the same time. How should we then combine these maximum modal vaiues? Given the modal displacement vector U;, we can evaluate any other response of interest in the same mode, R,. The vector Ui,,, gives Rim,,. The problem with combining various modal U;,,, stated above also applies to the response R,,,,. For brevity, we shall drop the subscript max. From now on, the term R, would denote the maximum value of the response in mode i.

It is obvious that the upper bound of the combined response is given by the absolute sum of the modal values

Goodman, Rosenblueth and Newmark[l] showed that the probable maximum value of response is approximately equal to the square root of the sum of the squares'(SRSS) of modal values

Published in 1953, the Goodman-Rosenblueth-Newmark rule, known as the SRSS rule is still used quite widely. There are circumstances, presented in subsequent sections, in which this rule must be modified. For more early research on this topic, also see Jennings and Newmark (1960)[2], Merchant and Hudson (1 962)[3], Clough (1 962)[4], and Newmark et al. (1 965)[5].

3.2 Modes with closely spaced frequencies One of the exceptions for the SRSS combination rule (Equation 3.7) arises when the responses to be combined are from modes with closely spaced frequencies. An obvious situation is when frequencies and dampings of two modes are identical. In this case the response histories of the two modes are in-phase. The maximum values in the two modes do occur simultaneously, and they should be combined algebraically. We are already denoting the maximum value of the response by R. Let us denote the response history by R(t). The combination equation in the time domain is

Let us define the standard deviation of the response as follows:

in which td is the duration of the motion. If we assume the earthquake to be a stationary ergodic process, we can write the maximum responses [6] as

R = qo, Ri = qioi, (3.10)

where q and q, are the peak factors. These peak factors are a function of the frequency, and would normally vary from mode to mode and for the combined response. However, since we are primarily interested in modal responses with close frequencies, we can make a simplifying assumption, q = q , for all values of i.

Equations 3.8 and 3.9 give

in which q, is called the modal correlation coefficient, and is defined by

Now, Equations 3.10 and 3.12 give

The alternative forms of Equation 3.13 are

In the first of Equation 3.14 we have E, = 1 for i = j; and in the second one we take into account the symmetry property, ciJ = E,~.

Equation 3.1 3 or 3.14 is popularly known as the double sum equation [7]. In the correct form of the equation all the sums are algebraic. The US Nuclear Regu- latory Commission incorrectly uses an absolute sign in .front of the double summation 181. Consider two modes with equal frequencies and damping values. Then, E , ~ = I, and Equation 3.13 gives

C U M B I N A T I O N OF MODAL RESPONSES133

As indicated earlier, the double sum rule correctly gives the combined response as the algebraic sum of the two modal response values. On the other hand, if the two modes had sufficiently separated frequencies, E , , 2: 0,and we would get

which is the SRSS rule. In general, the value of E,, varies between zero and unity. For a given earthquake ground motion the value of E, can be evaluated

numerically, in accordance with Equation 3.12. We should make two comments here. First, the double sum rule, or any combination rule for the response spectrum method, is an approximate rule. Even when the value of E,, is evaluated 'exactly' from Equation 3.12, the combined responses would be approximate. When the calculations are performed on several earthquake ground motions, the combination rule would, on the average, give a reasonably accurate estimate of the combined response. The second comment is about the response spectrum method, and is related to the first comment. If the objective were to obtain the response values for a known ground motion record, the appropriate technique would be one of the time-history integration methods, e.g. the modal superposition method. The most appropriate application of the response spectrum method is to design problems for which the future earthquake is not known. For this purpose, we not only need 'average' spectral shapes which are presented in Chapter 2, we also need representative values of the modal correlation coefficient E,,,

which on average weuld give sufficiently accurate combined response values. Rosenblueth and Elorduy [9] assumed the earthquake ground motion to be a

finite segment of white noise, and assumed the response to be damped periodic- of the form e-C"' sinoDt. Based on their work, the correlation coefficient can be written as

in which o, and o, are the circular frequencies of the two modes in radians per second;.. o,,_ .grid oD, are the corresponding damped frequencies,

"\ r'i;,, = J(l - Sf) o,/, wDJ = J(l - (j) o,; and (: and 6; are the equivalent damping rz tos wTkh account for the reduction in the response due to the finite nature of the white noise segment.

where s is the effective duration of the white noise segment. The duration s is not the total duration of the ground motion. It is not clear how exactly to evaluate it. Villaverde [ lo] obtained numerically values of s for several ground motions by exploiting its relationship with the expected value of pseudo velocities at different damping values. However, he did not suggest any method of evaluating s for a given response spectrum.

We note from Equation 3.16 that the value of effective duration has a significant contribution in the lower frequency range only. In the higher frequency range (; fi c,. For the lower frequency range, the effective duration of the earthquake may be represented by the duration of the strong motion. A measure of the effective duration of the strong motion can be obtained from the Husid plot[I 11, which is the graphical representation of the following function

I,' iii(f) dl

H ( t ) = [ iii(t) dl '

in which t , is the final value oft. By definition, 0 S H(t ) I: 1. The Husid plot for El Centro (SOOE, 1940) ground motion is shown in Figure 3.1. The function H(t) builds slowly initially because of the weak motion at the early phase of ground shaking. In the intermediate duration, the H ( t ) builds rapidly. In the final phase, very little seismic input is developed. It is clear from Figure 3.1 that the intermediate portion of the Husid plot comprises the significant strong motion contribution. For definitiveness, but arbitrarily, the first 5% and the last 5% are

A

1, = 1.67 sec 1,126.15 sec 95%

Duration

5%

t 0 10 20 30 40 50 60

Time. sec

I Fig. 3.1 Husid plot for El Centro earthquake (SOOE, 1940). (Based on Nau and Hall[l I]).

COMBINATION O F MODAL RESPONSES135

deleted from the plot. The remaining 90% is defined as the significant strong motion portion as shown in Figure 3.1. This duration for El Centro record is 24.5 sec. In using Equations 3.15,3.16 in conjunction with a design spectrum, the value of the duration s should be specified.

By substituting the expressions for r; and r' in Equation 3.15 from Equation 3.16, we obtain

To avoid the estimation of the effective duration s, Gupta and Cordero[l2] modified the above equation as follows

The coefficient c,, was evaluated numerically for ten strong ground motion records. On the basis of their study, an expression of the type given below was suggested [ 12 ]

in which r,, is the average damping value. Figure 3.2 shows a comparison of the value of E,, obtained from Equations 3.18 and 3.19, with the average of numerically obtained values from ten earthquake records[ 121. Since Equation 3.19 is based on the average of c,, values obtained from several records, it is more appropriate to use it for a broad band earthquake input.

w

f i=O. l H Z , ~ = $ = O . O I .c 9) 0 - Formula

0 Numerical

Frequency Ratio fj /fi

Fig. 3.2 Comparison of the modal correlation coefficients from the formula with the average of numerical values from ten earthquake records[12].

Using the assumption of stationarity, Singh and Chu [13] derived an equation similar to Equation 3.13 from which an expression for E,. can be derived. Assuming earthquake motion to be white noise, Der Kiureghian[l4] also obtained an expression for E~ which is given below:

The double sum equation in which Der Kiureghian's expression is used has been called the complete quadratic combination (CQC)[15]. When the two modes have equal damping values, it can be shown that the E , values obtained from the Singh-Chu equation[l3] and those from Equation 3.20 (Der Kiureghian[l4]) are about the same. Both can be significantly lower than those given by Equation 3.15 (Rosenblueth and Elorduy[9]) and 3.18 (Gupta and Cordero[l2]) within the frequency ranges of interest. As will be shown later, these differences in the 8,

values result in response variations that are not negligible. There is another important element in the expressions of the correlation

coefficient which has not been explicity recognized in the published studies so far. Equations 3.15 and 3.18 are likely to overestimate the values of E,, when the damping ratios of two modes are sufficiently different. Consider a situation when w, and wJ are large enough that the effect of the finite duration on the values of E,,

in these equations can be neglected. Equations 3.15 and 3.18 can be approximated as follows:

An approximate form of Der Kiureghian's[l4] Equation 3.20 is

Equation 3.22 includes a coefficient, J(c,l;i)/c,,,that Equation 3.21 does not. The variation in the value of the coefficient with c,/c,ratio is tabulated below.

COMBINATION OF MODAL RESPONSES/37

The coefficient is approximately unity when l,, and 6, are not very different. On the other hand it is much less than unity when 4 , and 6, are sufficiently apart. This would have serious influence on the correlation coefficients for modes with close frequencies. Consider an example, w, = o, and l,,/l,, = 0.2. Equation 3.21 would give E,, = 1.0, and Equation 3.22, E, = 0.745. Further, consider the response of a secondary oscillator in resonant modes. Assume, R, = 1.01, R, = -0.99. The first value of E,,, Equation 3.21, would give R = 0.02, the second value, Equation 3.22 would yield R = 0.7 14.

Our recent numerical experimentation using the actual earthquake motion data at North Carolina State University indicates that the correlation coefficient values and the resulting combined response values are relatively more realistic when the coefficient ,/(<,l;,)/l,,, is included than when it is not. Consequently, Equation 3.18 should be modified as follows

Equation 3.15 can also be modified in the same way. A comparison of the double sum, SRSS, and the absolute sum combination

rules was made by Mason, Neuss and Kasai [I 61. They analyzed the fifteen story steel moment resisting frame structure of the University of California Medical Center Health Sciences East Building located in San Francisco. Two building models were formulated. For both models a constant 5% modal damping was used. The first was the 'regular7 building in which the centers of stiffness and mass were coincident. The second was an irregular building with mass offset from the stiffness center of the building. The regular building did not have interaction between modes with closely spaced frequencies. Therefore, as one would expect the double sum and the SRSS rules gave comparable results, which were also very close to the time-history results for the regular building; the absolute sum rule over-estimated the response values significantly. In the irregular building, the modes in the two orthogonal directions became coupled leading to interacting modes with closely spaced frequencies. Three ground motions were used: San Fernando (Pacoima Dam, SOOE, 1971), Imperial Valley (El Centro, SOOE, 1940) and San Fernando (Orion Blvd., NOOW, 1971). The double sum calculations were performed using the modal correlation coefficient from the Rosenblueth-Elorduy equation, Equation 3.15, and from Equation 3.20, the Der Kiureghian equation. In the former, the effective duration s was taken to be 10 sec.

A statistical summary of errors is given in Table 3.1. The earthquake motion was applied in the east and west direction. The response in the north-south direction, and the rotational torque response was generated due to the eccentricity between the mass and the stiffness centers. The parallel east-west response values from the two double sum calculations are comparable; the SRSS values

Table 3.1 Error in response spectrum results with respect to time-history results[l6]. (Reprinted by permission of John Wiley & Sons Ltd)

% Error in results

Double sum SRSS ' Abs. sum

Response Rosenblueth- Der quantity Description Elorduy [9] Kiureghian (141

Parallel (E- W) response Deflection

Average error 7 6 Maximum error 19 17

Shear Average error 8 8 Maximum error 20 19

Overturning moment

Average error 6 7 25 39 Maximum error 18 18 34 120

Orthogonal (N-S) response Deflection


Shear Average error 17 24 Maximum error 31 55

Overturning moment

Average error 16 25 218 520 Maximum error 25 5 1 299 658

Torsional response Torques


have relatively higher errors; the errors from the absolute sum calculations are the highest. Similar conclusions can be made about the torsional response, except that the absolute sum values now have much higher errors. All the combination rules have the highest errors in the orthogonal north-south response. The double sum method using the Rosenblueth-Elorduy[9] modal correlation coefficient gives the best results. The results from the SRSS and the absolute sum combination rules are unacceptable. The orthogonal north-south response values from the San Fernando-Pacoima Dam excitation are shown in Figure 3.3. The

COMBINATION O F MODAL RESPONSES/39

ROOF

1 2 3 4

Story Deflection (inch)

1 I c I I

1000 2000. 3000 4000 5000

Story Shear (kips)

Story Overturning Moment ( 10' kip-inch)

Fig. 3.3 Comparison of modal combination rules: (a)story deflections, (b) story shears, (c) story overturning moments(l61. (Reprinted by permission of John Wiley & Sons Ltd.)

order of accuracy between different combination rules observed from the figure is the same as that concluded on the basis of Table 3.1.

3.3 High frequency modes-rigid response As was observed in Chapter 1 , at higher frequencies the spectral acceleration becomes equal to the maximum ground acceleration. Ideally, the highest

frequency is m, and the corresponding period is zero. Therefore, for a zero period oscillator the spectral acceleration is equal to the maximum ground acceleration, which is also called the zero period acceleration (ZPA). The minimum frequency at which the spectral acceleration becomes approximately equal to the ZPA, and remains equal to the ZPA is called the ZPA frequency or the rigid frequency, JT (Hz) or or (radians sec-'). The reason for this phenomenon-the spectral acceleration becoming equal to the ZPA, is the finite frequency content of the input motion. At oscillator frequencies sufficiently higher than the highest significant input frequency, the transient part of the response, the damped periodic response, becomes negligible; only the steady-state response remains. The steady-state response can be evaluated by a pseudo-static calculation from the equation ku = -mug or u = - i i g / 0 2 . This steady-state or pseudo-static response is also called the rigid response.

It is clear from the calculation of the rigid response that the rigid response history is in-phase with the input motion acceleration time history. It follows that the responses in all the modes having frequencies greater than the rigid frequency are in-phase with each other. In the response spectrum method, the combined response from those modes can be calculated simply by summing algebraically the responses. In the double sum equation, Equation 3.13, E, = 1, when both w, and wJ are greater than or. Note that the previous definitions of E,,, which are based on the closeness of o, and o,, do not apply here. When w, and o, are greater than or, E,, = 1, irrespective of how close or apart a, and w, are. Based on the considerations similar to those presented above, Kennedy[l7] suggested that the responses from the modes with frequencies beyond the rigid frequency be combined algebraically.

In Kennedy's method, in effect, there are two groups of modes. those with frequencies less than or and those having frequencies greater than or. The responses from the second group are summed algebraically. This combined response from the second group and the modal responses from the first group are combined using the double sum equation, Equation 3.13. This procedure is an improvement over the procedures which did not recognize the presence of the rigid modes. As will be shown later, however, the boundary between the rigid and the nonrigid part is too abrupt in Kennedy's method, which needs further consideration. To solve the problem, various forms of relative acceleration response spectra based procedures have been proposed by Lindley and Yow [ 181, Hadjian [ 191, and by Singh and Mehta[20]. Lindley and Yow[l8] perform a static analysis using the ZPA and the usual modal analysis using a relative acceleration response spectrum, ordinates of which are square root of the difference of the squares of the regular spectral acceleration and the ZPA. Hadjian[l9] does something similar except that he obtains the relative acceleration spectrum by directly subtracting ZPA from the regular spectral accelerations. Kennedy1171 has pointed out that this procedure leads to an inconsistency in the method and suggests a modification which would make Hadjian's method very similar to that


of Lindley and Yow[l8]. Singh and Mehta[20] formulate the problem using the modal acceleration approach and suggest making use of relative velocity and relative acceleration spectra. The Singh-Mehta method is theoretically rigorous, and gives accurate results. The availability of the relative velocity and relative acceleration spectra may be a problem. Among the methods proposed by Kennedy [ 171, Lindley and Yow [ 181 and Hadjian [ 191, the Lindley-Yow method appears to be most rational and is likely to give most accurate results for a structure having frequencies in the neighborhood of or greater than the rigid frequency. The method is likely to run into trouble for modes having frequencies significantly lower than the rigid frequency. Even for frequencies immediately below the rigid frequency, the method of calculating the relative spectral acceleration is somewhat arbitrary.

We will now present a method developed by Gupta and coworkers[12,2 1-25]. It has been pointed out above that the responses from the modes having frequencies greater than the rigid frequency are perfectly correlated with the input acceleration history. What happens at a frequency immediately below the rigid frequency? Let us call the correlation between a modal response and the input acceleration as the rigid response coefficient, because at frequencies equal to or higher than the rigid frequency the response is rigid and the correlation is unity. Figure 3.4 shows the variation of rigid response coefficient for the San Fer- nando earthquake (Hollywood Storage, EW, 197 1). The rigid response coefficient becomes almost unity at about 20 Hz, which is much less than the rigid frequency, which is about 30 Hz. Below 20 Hz the coefficient shows a gradual diminishing trend, and becomes zero at about 2.5 Hz. This means that even the modes having frequencies below the rigid frequency have a 'rigid content.' This is the natural consequence of gradually changing proportions of the contributions of the transient response (damped periodic response) and the steady state response (rigid response).

Based on the above discussion, we can divide a modal response. R,, at a frequency w, c or, into two parts: the rigid part, R,', and the damped periodic part, RP. Denoting the rigid response coefficient by a,, the rigid part is defined by

It is assumed that the rigid part and the damped periodic part are statistically independent

R: = ( ~ j ) ~ + (R,?)', R,? = J(l - a,?) R, . (3.25)

The statement of Equations 3.24 and 3.25 immediately leads to an appropriate combination rule. Since the rigid parts are all perfectly correlated they are summed algebraically:

The damped periodic pans are combined using the standard double sum equation:

f2

I 1 / > I

Finally, the total response becomes

1 0,

0.8-

C C

.g

Ii 06-

1 04- K

S 0 .- a 0 2 -

0

The rigid response coefficient can be numerically evaluated if the earthquake time history is known. As indicated before, that is not usually the case when the response spectrum method is used. As shown in Figure 3.4, the numerically calculated rigid response coefficient can be idealized by a straight line on the semi-log graph. The idealized equation is given by

I T I 1

-

-

-

-

- I I l l

in which J; is the modal frequency in Hz, J; = oi/(21c). The key frequencies .fl and f * can be expressed as

I 0-' lo0 f' 10' 10'

Frequency, HZ

Fig. 3.4 Variation of rigid response coefficient with frequency, San Fernando earthquake (Hollywood Storage, EW, 1971)[23].

C O M B I N A T I O N O F M O D A L RESPONSES143

j - 1 = SA,,, , Hz; f = ( f ' + 2f ')/3, Hz. 2~ Svrnax

Equations 3.24-3.28 constitute a complete modal combination procedure, including cases when modes have closely spaced frequencies, and the cases when there are modes with a rigid response content, (ai > 0). For the sake of notation brevity, we can reframe these equations into a modified double sum equation:

in which

B,, = a,., + ,/[(I - a f ) ( l - a;)] E,,.

Equations 3.31 and 3.32 include the effec :t of rigid resp lonse in the modified correlation coefficient T,,. A comparison of the average numerical values of 8, obtained from ten earthquake records and the values calculated from Equation 3.32 is shown in Figure 3.5. The agreement between the numerical values and Equation 3.32 is good. Singh and Mehta[20] have also proposed an expression for 8, which includes the effect of rigid response.

The time history and the response spectrum analyses were performed on five 3-degree-of-freedom systems, each subjected to three actual earthquake ground motion records, and to three instructure calculated motions. The five buildings had fundamental frequencies from 2-64 Hz. The building model and the unnormalized mode shapes (which are same for all the five buildings) are shown in Figure 3.6. In the response spectrum method, the modal responses were combined using four methods: SRSS, Kennedy [l7], Hadjian [I 91, and Gupta and Chen[21]. The combined response spectrum values were compared with the corresponding maximum values from the time-history method. The following responses were considered: story displacements, story inertia forces, story shears and story moments. It was found that the story displacements were dominated by the fundamental mode and their response spectrum values were insensitive to the method of combination. Therefore, displacements were eliminated from the comparison. A statistical summary of the errors in the four methods of modal combination is given in Table 3.2. The mean error in all the methods is relatively small. The most important error parameter for consideration is the standard deviation. Relative to the value for Gupta's method, the standard deviation for Hadjian's method is 2.4, that for Kennedy's method is 2.9, and for the SRSS method is 4.6. A comparison of the time-history response results with those from the response spectrum method using Gupta's method of modal combination is shown in Figure 3.7 for the San Fernando (Hollywood Storage, EW, 1971) earthquake. The dotted lines in Figure 3.7 show the response spectrum results only when they are not superimposed by the time-history results shown by the solid lines.

l i d o Hz

C - Formula

o Nwnerlcal 0 0 1 - Formula

o Numerical

Frequency Ratio fjlfl Frequency Ratio fj Ifi

Fig. 3.5 Comparison of modal correlation coefficients including the effect of rigid response [ 121.

Building Model Mode Shapes

Fig. 3.6 Building model and unnormalized mode shapes (same for all five building)[21].

Table 3.2 Statistical summary of percent errors[21]

Method

Descri~tion SRSS Kennedv Hadiian G u ~ t a

Maximum absolute error 4 1 3 7 38 2 1 Mean error - 1.3 - 1.5 4.1 -0.8 Standard deviation 16.9 10.7 8.8 3.7 Relative standard deviation 4.6 2.9 2.4 1 .O

ASCE Standard1261 has adopted Gupta's method of modal combination with a simplification. The standard uses a modal combination equation like Equation 3.32. Rather than varying the rigid response coefficient a from 0 to 1 between f' andf2 in accordance with Equation 3.29, the standard assumes a sudden

COMBINATION O F MODAL RESPONSES145

Fig. 3.7 Comparison of maximum building response from time-history analysis and from response spectrum method using Gupta's modal combination method for San Fernando earthquake (Hollywood Storage, EW, 1971)[21].

change in a from 0 to 1 at a frequency approximately midway between f' and f 2,

viz., at f '/2. Accordingly, E,, is given by the ei, equations for the damped periodic part, unless both L a n d 4 are equal to or greater than f '/2. In the latter case, the standard recommends E,, = 1, irrespective of relative magnitude of frequencies. The standard permits use of the detailed procedure explained above.

3.4 High frequency modes-residual rigid response In many practical applications, the structure model has a large number of degrees-of-freedom. The structure has as many number of modes as the number of degrees-of-freedom. Often the significant response of the structure can be obtained from the first few modes; the response contribution of the higher modes is negligible in those cases, and it can be neglected. There are situations where it is not certain how many modes to include in the analysis. For example, in the nuclear power plant piping systems, some parts get much of their response from very high modes because of nonuniform distribution of stiffnesses. If those higher modes were ignored, unacceptably high error would be introduced in the calculated response values. There is clearly a need for techniques for including

I the higher mode effects without having to perform calculations for all the modes. b !

In one type of existing techniques[21-25,27-291, the inertia effect of modes having frequencies greater than the rigid frequency is lumped into a 'missing mass' term which yields the 'residual rigid response.' In another type[20], the analysis is performed by the mode acceleration method. This latter method requires the use of relative acceleration and relative velocity spectra which are not readily available. We shall present the residual rigid response approach[24].

We recall the vector Ub introduced in Equation 3.1. The following is an exact linear transformation of this vector:

Let us premultiply Equation 3,33 by +; T ~ . Equation 3.3, and the orthogonality condition: +; M +j = 0 for i # j, and = 1 for i = j, give

Equation 3.1 with Equation 3.34 becomes

From Equations 3.2 and 3.35 we infer

Equation 3.36 gives the response in the ith mode of vibration. Let us assume that the number of degrees-of-freedom is N, which is also the

number of modes of the structure. Further, let us assume that there are n modes having frequencies less than the rigid frequency, f '. We denote the response in these n modes by U', and the response in the remaining modes by Uo

The summations in Equation 3.37 are in the time-domain. Equations 35-37 give

As we pointed out earlier, the response of the structure in modes having frequencies greater than the rigid frequency is pseudo-static, i.e. U , and uo terms in Equation 3.38 can be ignored, or


The displacement vector U, gives the residual rigid response. In the modal superposition method, the response history in modes having frequencies up to the rigid frequency is obtained by the usual algorithms. The response in all the remaining modes is given by U,, in Equation 3.39. Although the vector Uo is time- dependent, we need not solve the simultaneous equation at each time step. Since the time dependence is introduced by the term ii,, which in the present case is a scalar, we solve the simultaneous equation once with - M U,,, on the right hand side, and multiply the solution by ii, at each time step. The complete displacement vector is given by Equation 3.37. The above procedure is not only economical compared with doing calculations for all the modes, but in most cases it will also be more accurate. It is usually difficult to evaluate the frequencies and the mode shapes of the higher modes very accurately.

In the response spectrum method, the responses in modes having frequencies

Fig. 3.8 Example piping problem[24].

Table 3.3 Some important forces and moments near the supports[24]

Percent error in method Direct

Element Node integration number number Value result I I1

Mean error 4.2 -42.6 Error standard deviation 11.2 47.5 RMS error 12.0 64.1

, F - axial force, V - shear force, M - moment, subscript X, Y, Z refer to the corresponding global axes. Forces in kips, moments in kips ft. Method I - including residual rigid response. Method I1 - without residual rigid response.

I up to the rigid frequency are calculated by Equation 3.5 as before. For the , /

. residual rigid response, Equation 3.39 is replaced by

i 4 K Uoma, = M UbO (ZPA). (3.40)

I

$ , The residual value of any response, Ro, is then calculated from U,,, . The modal

1: I I combination procedure described in the previous section, Equations 3.24-3.28,

is applied with one change. Equation 3.28 is modified as follows:

This method of residual rigid response calculation was applied to a piping system shown in Figure 3.8 by Gupta and Jaw[24]. The piping was subjected to Taft

(N2 1 E, 1952) ground motion which has rigid frequency, f r - 20 Hz. There were nine modes having frequencies <20 Hz, n = 9. We are presenting here some of the results from Reference[24], comparing the time-history results with the response spectrum results with or without residual rigid response. Table 3.3 gives the significant element forces and moments in the vicinity of the supports obtained from the time-history analysis along with the errors in the response spectrum calculations. Ignoring the residual response gives error as high as - 98.6%, which means the calculated value is 1.4% of what it should be, clearly an unacceptable situation. Excluding the residual rigid response introduces a mean error of -43% as opposed to mean error of only 4% when the residual rigid response is included. This shows a definite bias in the response results towards under-estimation when the residual rigid response is not included. The exclusion of residual rigid response also gives much larger error standard deviation and RMS error than when it is included.

References 1. L.E. Goodman, E. Rosenblueth and N.M. Newmark. Aseismic Design of Elastic

Structures Founded on Firm Ground. Proceedings, ASCE, November 1953, pp. 349-1, 349-27.

2. R.L. Jennings and N.M. Newmark, Elastic Response of Multi-Story Shear-Beam-Type Structures Subjected to Strong Ground Motion, Proceedings, Second World Conference on Earthquake Engineering, Vol. 11, Tokyo, 1960.

3. H.D. Merchant and D.E. Hudson, Mode Superposition in Multi-Degree-of-Freedom Systems Using Earthquake Response Spectrum Data. Bulletin of the Seismological Society ofAmerica, Vol. 52, No. 2, 1962, pp. 405-416.

4. R.W. Clough, Earthquake Analysis by Response Spectrum Superposition, Bulletin of the Seismological Society of America, Vol. 52, No. 3, 1 962, pp. 647-680.

5. N.M. Newmark, W.H. Walker. A.S. Veletsos and R.J. Mosborg, Design Procedures for Shock Isolation Systems of Underground Protective Structures, Report for Alr Force Weapons Laboratory, by Newmark, Hansen and Associates, RTD TDR 63-3096, December 1965.

6. A.G. Davenport, Note on the Distribution of the Largest Value of a Random Function with Application to Gust Loading, Proceedings. lnstltution of Civd Engrneers, Vol. 28. 1963, pp. 187- 196.

7. A.K. Singh, S.L. Chu and S. Singh, Influence of Closely Spaced Modes in Response. Spectrum Method of Analysis, Proceedings, Speciality Conference on Structural Design of Nuclear Power Plant Facilities, ASCE, Chicago, Illinois, 1973.

8. United States Nuclear Regulatory Commission, Design Response Spectra for Nuclear Power Plants, Nuciear Regulatory Guide, No. 1.60, Washington, DC, 1975.

9. E. Rosenblueth and J. Elorduy, Response of Linear Systems in Certain Transient Disturbances, Proceedings, Fourth World Conference on Earthquake Engineering, Santiago, Chile, 1969, A-1, pp. 185-196.

10. R. Villaverde, On Rosenblueth's Rule to Combine the Modes of Systems with Closely Spaced Natural Frequencies, Bulletin of the Seismological Society of America, Vol. 74, No. 1, February 1984, pp. 325-338.

I I. J.M. Nau and W.J. Hall, An Evaluation of Scaling Methods for Earthquake Response Spectra, Structural Research Series, No. 499, Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, May 1982.

I 12. A.K. Gupta and K. Cordero, Combination of Modal Responses, Transaction. Srxth I Internatronal Confirence on Structural Mechanrcs in Reactor Technology, Paper No. K7115, 1 Paris, August 198 1.

r 13. M.P. Singh and S.L. Chu, Stochastic Considerations in Seismic Analysis of Structures,

i Earthquake Engineering and Structural Dynamics, Vol. 4, 1976, pp. 295-307.

I 14. A. Der Kiureghian, A Response Spectrum Method for Random Vibrations, Report No. UCBIEERC-8O//I5, Earthquake Engineering Research Center, University of California, Berkeley, California, 1980.

15. E.L. Wilson, A. Der Kiureghian and E.P. Bayo, A Replacement for the SRSS Method in Seismic Analysis, Short Communrcation. Earthquake Engineering and Structural Dyna- mics, Vol. 9, 198 1, pp. 187-1 94.

16. B.F. Maison, C.F. Neuss and K. Kasai, The Comparative Performance of Seismic Response Spectrum Combination Rules in Building Analysis, Earthquake Engineering and Structural Dynamrcs, Vol. 11, 1983, pp. 623-647.

17. R.P. Kennedy, Recommendations for Changes and Additions to Standard Review Plans and Regulatory Guides Dealing with Seismic Design Requirements for Structures, Report prepared for Lawrence Livermore Laboratory, Published in NUREGICR-I 161, June 1979.

18. D.W. Lindley and J.R. Yow, Modal Response Summation for Seismic Qualification, Second ASCE Specralitj~ Confirence on Civil Engineering and Nuclear Power, Vol. VI, Paper 8-2. Knoxville, TN, September 1980.

19. A.H. Hadjian, Seismic Response of Structures by Response Spectrum Method, Nuclear Engineer~ng and Design, Vol. 66, No. 2, August 198 1, pp. 179-201.

20. h . ~ . Singh and K.B. Mehta, Seismic Design Response by an Alternative SRSS Rule, Earthquake Engrneering and Structural Dynamics, Vol. 11, 1983, pp. 77 1-783.

21. A.K. Gupta and D.C. Chen, Combination of Modal Responses: A Follow Up, Report, Department of Civil Engineering, North Carolina State University, Raleigh, November 1982.

22. A.K. Gupta and D.C. Chen, A Simple Method of Combining Modal Responses, Transactions, Sewnth International Conference on Structural Mechanics in Reactor Technology, Paper No. K3/ 10, Chicago, August 1983.

23. A.K. Gupta and D.C. Chen, Comparison of Modal Combination Methods, Nuclear Engineenng and Design, Vol. 78, March 1984, pp. 53-68.

24. A.K. Gupta and J.W. Jaw, Modal Combination in Response Spectrum Analysis of Piping Systems, in Seismic Eflects in PVP Components, PVP. Vol. 88, ASME, 1984, pp. 1-12; presented at Pressure Vessel and Piping Conference, San Antonio, Texas, June 1984.

25. A.K. Gupta, Modal Combination in Response Spectrum Method, Proceedings, Eighth World Conference on Earthquake Engineering, San Francisco, 1984.

26. American Society of Civil Engineers, Standard for the Sersmic Analysis of Safety-Related Nuclear Structures, September 1986.

27. K.M. Vashi, Computation of Seismic Response from Higher Frequency Modes, Journal of Pressure Vessel Technology, ASME, Vol. 103, February 198 1, pp. 16- 19.

28. G.H. Powell, Missing Mass Correction in Modal Analysis of Piping Systems, Transactions, Fifih International Conference on Structural Mechanics in Reactor Technology, Paper No. K1013, 1979.

29. A.J. Salmonte, Consideration on the Residual Contribution in Modal Analysis, Earthquake Engineering and Structural Dynamics, Vol. 10, 1982, pp. 295-304.

Chapter 41Response to multicomponents of earthquake

4.1 Introduction The earthquake motion at any point can be resolved into three orthogonal components, two horizontal and one vertical. Penzien and Watabe[l] have shown that two horizontal components, which are approximately radial and tangential with respect to the epicenter, are uncorrelated. Any other orientation of horizontal axes leads to partially correlated horizontal components. This would normally be the case because the buildings or the structures are, in general, unlikely to be placed along the radial and tangential directions. The vertical component of the earthquake motion always has some correlation with the horizontal components. Because of the wave motion, a building is also subjected to three rotational components[2-71. These rotational components are mutually correlated, and have a strong correlation with the translational components. The structures supported on multiple supports, such as bridges can be considered to be subjected to multiple 'components.' These components would have variable degrees of correlation.

Let us denote the earthquake component by the capital letter subscripts, and modes by the small letter subscripts. For example, R,, represents the response spectrum value of a response in the ith mode of vibration due to Ith earthquake component. Following a procedure similar to that used for the derivation of Equation 3.13, we can write the following equation for component and modal response equation [8]:

in which EIiJj represents the correlation between the response in mode i due to I th component, and the response in mode j due to J t h component. For derivation of an equation like Equation 4.1 see also Amin and Ang[9]. Ghafory-Ashtiany and Singh[lO] have performed a stochastic study on the combination of responses from six components of earthquake (three horizontal and three rotational). Making worst-case assumptions, they have shown that the seismic response of the building can be influenced significantly depending upon its orientation with respect to the epicenter of the earthquake.

If the correlation coefficient EIi4is known, its application is straightforward. On the other hand if the knowledge about the correlation is uncertain, it may be desirable to make simplifying assumptions. Assuming that the two horizontal

components have equal intensities, Rosenbiueth and Contreras[l 1] have concluded that one can take zero correlation between motions along any two orthogonal horizontal directions. Usually the intensity of the tangential component is slightly smaller than that of the radial component. However, a given site may expect to receive ground shocks from more that one source. Therefore, it is reasonable to assume that the two horizontal components have equal intensities for design purposes; and consequently, that the two components are uncorrelated irrespective of the orientation of the horizontal axes. The vertical component is considered to be of a lesser significance than the two horizontal components are. Therefore, we may justify ignoring the correlation between the vertical component and the two horizontal components. Finally, for most buildings and structures the rotational components make only a minor contribution to the structural response. Thus, for many practical situations, it would suffice to consider the effect of two equal horizontal components and a vertical component, and to assume that the three components are statistically uncorrelated. We can write

in which ci, includes the effect of modes with close frequencies and that of the rigid response at higher frequencies. In Chapter 3, we denote the corresponding correlation coefficient by BO; we are dropping the bar in Equation 4.2 here for brevity without any loss of meaning. Equation 4.2 can be rewritten as

Equations 4.1-4.3 yield the maximum probable value of any response. Often the design of a structural element is based upon more than one response, for example, a column subjected to axial force P and bending moment M, or a metal element subjected to stresses ox, o,, o,, T,,, T,,, T ~ . It is unlikely that the maximum values of these responses would occur simultaneously. In most conventional design procedures, it is implicitly assumed that the maximum responses do occur simultaneously. This assumption introduces error on the safe side which may be significant. Various aspects of this problem are discussed in subsequent sections.

4.2 Simultaneous variation in responses The following development is based on Gupta and Chu[I 2,131. In the response spectrum method of analysis, the maximum value of the response in mode i due to I th earthquake component is known, denoted here by R,,. At a given instant of time, the response can be expressed as the weighted algebraic sum of responses in various modes due to all the components of earthquake.

RESPONSE TO MULTICOMPONENTS OF EARTHQUAKE/53

The value of R ( t ) given by Equation 4.4 is bounded by the maximum value of R given by Equation 4.3, which we shall denote here by R,,. Therefore,

Before we proceed, let us consider the nature of Equation 4.3. It can be viewed as the definition of the length R of a vector Rli in' a 3N-Riemannian space [ 141 having a metric tensor E,,, where N is the number of modes. Now consider the instance of time when a particular response U reaches its maximum value Urnax. Equation 4.4 gives

In Equation 4.6, u,, is a unit vector in the Riemannian space. Equations 4.3 and 4.6 give

Let us consider another arbitrary unit vector v,,

At the instant, Kli has a value which gives U,,, we will have

Since, v, is an arbitrary unit vector, without any loss of generality we can write

Equations 4.8-4.10 give

JCls 1. l i j

Let us now substitute Equation 4.10 into Equation 4.6, which gives

C 1 1 1 E,, uIi v I = 1. (4.12) l i j

In the Riemannian space, the cosine of the angle between unit vectors uli and uIJ is given by

COS (u, V) = 1 E;, MI; U1,. l i j

Equations 4.1 2 and 4.13 yield

According to Equation 4.14,l CI is either equal to or greater than unity. On the other hand, Equation 4.1 I gives ( Cl equal to or less than unity. The only common value is ] C I = 1. Hence, Equation 4.12 becomes

which is possible when the unit vectors ull and v, are parallel. If we assume that the positive direction of both the vectors is same, then we conclude

Equation 4.16 with Equation 4.10 gives

Equation 4.17 gives the value of K,,, at the instant when U,,, is attained. We shall now proceed to find a relationship for K,, which will be applicable to any arbitrary response. The coefficient E,, is an element of the N X N matrix e, where N is the number of modes. The matrix e is positive-definite by definition, see Chapter 3. Let us define the inverse of the matrix E by e-. The solution of

Equation 4.17 gives uIj = 16, K,, which with Equation 4.6 yields i

In Equation 4.1 8, KIl (or K,,) values are unknown. Any set of values of Kit, which satisfy Equation 4.18, will yield the maximum value of a response Rr

where r is the response number. The value of other responses at the time when Rr is maximum, can be obtained by another equation similar to Equation 4.19,

R E S P O N S E T O M U L T I C O M P O N E N T S O F EARTHQUAKE155

where s # r, the K,, values are the same as those in Equation 4.19. Thus, all the R'and I? values occurring simultaneously are evaluated. In practical applications, one may not be able to identify the response Rr, which is maximum for a given set of K, satisfying Equation 4.18. However, that is not necessary. In practice, Equations 4.18 and 4.19 can be used as follows. Evaluate sufficient sets of K , values which satisfy Equation 4.18. Then for each set of K, values evaluate Rr from Equation 4.19, for all responses R' of interest. For example, in case of a beam column, we have two responses of interest: axial force P and bending moment M. Then, R ' = P, R * = M. Each set of Rr values (example, P, M) gives the simultaneously occurring responses which should be considered in the design.

4.3 Equivalent modal responses [l2,15] Equations 4.18 and 4.19 form the basis of evaluating the simultaneous variation in responses. However, the application of these equations is tedious. Consider, for example, we have three components of earthquake, and for each component we have twenty modal responses. Then each set of K , consists of 3 X 20 = 60 values. To have sufficient sets of K,, which satisfy Equation 4.18 to cover all the possibilities may be all but impossible. It is shown in this section that a large number of modal responses for all the components of earthquake can be replaced by a very few numbers of 'equivalent modal responses.'

Consider a design problem in which only a small number, M, of the response values Rr, r = 1 -M, contribute. For instance, for the beam column problem referred to in the previous section, M = 2, R ' = P, and R' = M. There are many design problems with M = 2, 3, etc. Let the design criterion be specified as

The design function @ in Equation 4.20 is a linear function of the responses Rr, provided the influence coefficients A' are constants. It is not unusual for a design function to be a nonlinear function of the response values, in which case the influence coefficients, A', are functions of the response values also. For the purpose of Equation 4.20, however, A' is assumed to be locally constant. This assumption is justifiable in view of the fact that the response analysis which gives the Rr values itself is linear. For the ith mode, Ith component of earthquake, we have

Equations 4.3 and 4.20 give the maximum response value of the design function as

$2 = C 7 1 1 &A~ASR;R; , r s l i j

It is obvious from Equation 4.22 that for design purposes, it is sufficient to know G" values, and it is not necessary to know R;, values. At least, it is not necessary to work with R;, values.

We shall now introduce the 'equivalent modal responses' denoted here by a:. The subscript in a: denotes the equivalent mode number. The equivalent modes are defined such that

Consider M responses Rr, r = 1 - M. For these M responses, we can specify M unique vectors a:, i = 1 - M. Each vector has M elements, r = 1 - M. Hence, we need to define M~ unknown elements of a:. The response matrix G is symmetric, Grs = G". Therefore, there are only M(M+ 1)/2 equations to define

values, Equation 4.23. That means the other M(M- 1)/2 values can be defined arbitrarily. A suggested set of these arbitrary values is

For example, when M = 2, Equation 4.23 gives

According to Equation 4.24, K \ = 0. Hence, E ( = JG", iF: = GI2/W1, and = 4[GZ2 - ( W : ) 2 ] . The equivalent modal vectors for M = 6 case are given in

Table 4.1. We note from Equations 4.3, 4.22 and 4.23 that the maximum value of a

response is given by

The subscripts i, j in the triple summation in Equation 4.25 refer to the actual mode numbers, and the single summation refers to the equivalent mode numbers. Whereas, the modal response values R, constitute a 3N-Reimannian space with E,, as the metric tensor, Equation 4.25 indicates that a: values are part

RESPONSE TO MULTICOMpONENTS OF EARTHQUAKE157

of an M-Cartesian space whose metric tensor is given by an identity matrix. Since the Cartesian space is a special form of Reimannian space, we can also apply the conclusions of the previous section here.

Equations 4.18 and 4.19 are now replaced by

For illustration[l6], consider a beam-column problem with the following calculated G array.

The equivalent modes are

According to Equation 4.26, the variation in P and M is represented by

It is clear that the above equations represent an ellipse, which we shall call 'interaction ellipse.' In general, when M > 2, Equation 4.26 represents an ellipsoid in an M-space. R, , E2 are like direction cosines. For design purposes, several sets of i?, , RZ values which satisfy the equation R: + K: = I should be used, as shown in Table 4.2. The calculated values of P and M are plotted in Figure 4.1, the points are joined to form a polygon. Also shown in Figure 4.1 are the exact ellipse, and a rectangle which will be obtained when we assume that the maximum values of + P and -+ M occur simultaneously.

Table 4.2 Calculation seismic force and moment

No. K , P M No. K , P M

RESPONSE TO MULTICOMPONENTS OF EARTHQUAKE159

Interaction t 1 2 . 0

r---- ,

Rectangle for I

Conventional Method

Fig. 4.1 Interaction ellipse for the column problem[l6].

4.4 lnteraction ellipsoid [l2,17] As shown in the previous section, Equation 4.26 parametrically represents an ellipsoid in M-space, which we call the interaction ellipsoid. We shall derive a nonparametric equation of the ellipsoid here. We shall use matrix notations in the derivation. Let us denote the responses R' by the vector R; the equivalent modal responses R; by the square matrix K, in which each column represents one equivalent modal vector: and R, values are contained in the vector K. Equation 4.26 is rewritten as

Eliminating the parameters K we get

From Equation 4.23

Denoting C-' by H, Equations 4.28 and 4.29 give

Equation 4.30 is the desired equation of the interaction ellipsoid. This derivation is based on Gupta and Chu[l2]. An alternative derivation is given by Gupta and Singh [ 1 71.

4.5 Approximate method Equations 4.26 or 4.30 represent the simultaneous variation in the values of various responses due to earthquake loading. Several points must be calculated on the interaction ellipsoid defined by these equations, to represent the surface adequately for design purposes. An approximate method based on Gupta[l8] is presented here. In this method the interaction surface is replaced by a few discrete points. The points can be joined to form a convex polyhedron in the M-response space that completely inscribes the interaction ellipsoid.

Consider any equivalent response 1,. Let us assume for the time being that the equivalent modal values of the response are arranged in a descending order, - a, r R, r . . . r RM. The maximum response is given by Equation 4.25,

R' = 1 R : . It is proposed to represent approximate response by I

in which C, values are constant coefficients, and are so defined that R,,,,,,,,, 2 R. This condition can be satisfied with minimum conservatism by taking

Maximum relative error in the response is ( F C:)'I2 - I . The values of e l a n d

the corresponding errors are given in Table 4.3. Rosenblueth and Contreras[l I ] give values of C, when the maximum errors on the safe and unsafe side are equal.

Many problems of practical interest have M = 2 or 3. As can be seen from Table 4.3, the maximum error on the conservative side will be only 8% or 13% in the two cases, respectively. The highest possible error is 30%. These are maximum possible errors. In most cases the errors will be somewhere between zero and the maximum value. It is shown by Gupta, Fang and Chu[19] that the

RESPONSE TO MULTICOMPONENTS OF EARTHQUAKE161

Table 4.3 Values of C,

Equivalent mode Maximum error number c, (%)

maximum relative error in the conventional method in which the maximum values of all the responses are assumed to occur simultaneously, is ./M - 1. For M = 2, 3 and 6 the error values are 41%, 73% and 145%, respectively.

When there are M responses of interest, Rr, r = 1 - M, the condition previously imposed, W ; ;r: i?; r . . . r &, cannot be explicity satisfied for all values of r. Therefore, we replace Cl in Equation 4.31 by a variable Dl, where

R p x l m , = D l , Dl = Permutations (t C,). I

Equation 4.33 will give 2 M ~ ! sets of R~pp,,,m, values. Let us apply this procedure to the beam-column problem of Section 1.2. The values of D, and the corresponding P, M points are given in Table 4.4. These points are plotted in Figure 4.2, to give the polyhedron enveloping the interaction ellipse.

When M is relatively large, the number of permutations in Equation 4.33 will become very large. For example, for M = 6, 2M M ! = 46 080. Although such

Table 4.4 Calculation of seismic force and moment by approximate method

No. D, D, P M

Polygon for approximate method t "SO

Rectangle for conventional method

Fio. 4.2 Interaction ellipse and the approximate polyhedron for the column problem[l6].

a large number of points can be handled without much difficulty in a modem computer, it may be desirable to make a further approximation to reduce the number of points. It can be done by setting

C, = 1.0, C, = 0.41 for i > 1. (4.34)

Equations 4.33 and 4.34 will give M 2M points. For M = 6 again, the number of points = 384, a reduction by a factor of 120. In general, the reduction is by a factor of (M - 1 )!.

4.6 Application to design problems One can calculate several points on the interaction ellipsoid using Equation 4.26, or approximately, using Equation 4.33 or 4.34. These points represent the

" seismic response values. The structure is subjected to other static loads which

RESPONSE TO MULTICOMPONENTS OF E A R T H Q U A K E / 6 3

3

r Conventional

Fig. 4.3 Interaction diagrams for combined static and seismic loads[l6].

must be resisted simultaneously with the seismic loads. Thus, for design purposes,

R;,I R L c + Rkismic . (4 .35 )

Equation 4.35 amounts to shifting the center of ellipsoid to the point R&,,,. For a safe design, all the possible RW,, points should be within the resisting capacity of the structural element under consideration.

The design problem of a reinforced concrete beam-column is illustrated in Figure 4.3. It is shown that the interaction ellipse representing the seismic loading with origin shifted for the static loading is completely inscribed by the capacity

interaction diagram of the beamcolumn section. Also shown in the figure is the rectangle given by the conventional method in which the maximum seismic Pand M values are assumed to occur simultaneously. The conventional method would have required a stronger, and therefore, less economical section.

Methods similar to those presented here have been applied to the design of building cross-sections, base slabs and reinforced concrete columns, and to the analysis of base slabs with local uplift by Gupta and Chu[12]. The design of steel beam-columns have been studied by Gupta, Fang and Chu[l9]. Gupta has applied these methods to a reinforced concrete nuclear pressure vessel, shear- walls[l6] and piping[l6,20] systems.

References 1. J. Penzien and M. Watabe, Simulation of 3-Dimensional Earthquake Ground Motion,

Bulletin of International Institute of Seismology and Earthquake Engineering, Vol. 12, 1974, pp. 103-1 15.

2. J.L. Bogdanoff, J.E. Goldberg and A.J. Schiff, The Effect of Ground Transmission Time on the Response of Long Structures, Bulletin o f the Seismological Society ofAmerica, Vol. 55, No. 3, June 1965, pp. 627-640.

3. N.M. Newmark, Torsion in Symmetric Buildings. Proceedings. Fourth U'orld Conference on Earthquake Engineering, Vol. 2, Paper A-3, Santiago, Chile, 1968, pp. 19-32.

4. E. Rosenbueth, The Six Components of Earthquake, Proceedings, Twelfih Regional Conference on Planning and Design o f Tall Buildings, Sydney, Australia, 1973, pp. 63-81.

5. N.D. Nathan and J.R. MacKenzie, Rotational Components of Earthquake Motion, Canadian Journal ofCivil Engineering, Vol. 2, 1975, pp. 430-436.

6. E. Rosenblueth, Tall Building Under Five-Components Earthquake, Journal of the Structural Division, ASCE, Vol. 102, No. 2, February 1976, pp. 453-459.

7. W.K. Tso and T.I. Hsu, Torisonal Spectrum for Earthquake Motion, JournalofEarthquake Engineering and Structural Dynamics, Vol. 6, 1972, pp. 375-382.

8. A.K. Gupta, Multicomponent Seismic Design, Proceedings, Seventh U'orld Conference on Earthquake Engineering, Istanbul, Turkey, September 1980.

9. M. Amin and A.H.S. An& Nonstationary Stochastic Model of Earthquake Motion, Journal o f Engineering Mechanics Division, ASCE, Vol. 94, No. EM2, April 1968.

10. M. Ghafory-Ashtiany and M.P. Singh, Seismic Response for Multicomponent Earth- quakes, Technical Report, NSF Grant No. CEE-82 14070, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, April 1984.

I I . E. Rosenblueth and H. Contreras, Approximate Design for Multicomponent Earthquakes, Journal of Engineering Mechanics Division, ASCE, Vol. 103 No. EMS, 1977, pp. 88 1-893.

12. A.K. Gupta and S.L. Chu, A Unified Approach to Designing Structures for Three Components of Earthquake, Proceedings, International Symposium on Earthquake Struc- tural Engineering, St. Louis, Missouri, August 1976, pp. 58 1-596.

13. A.K. Gupta and S.L. Chu, Probable Simultaneous Response by the Response Spectrum Method of Analysis, Nuclear Engineering and Design, Vol. 44, 1977. pp. 93-95.

14. J.L. Synge and A. Schild, Tensor Calculus, University of Toronto Press, Toronto, 1949. 15. A.K. Gupta and S.L. Chu, Equivalent Modal Response Method for Seismic Design of

Structures, Nuclear Engineering and Design, Vol. 44, 1977, pp. 87-9 1. 16. A.K. Gupta, Design of Nuclear Power Plant Structures Subjected to Three Earthquake

Components, Proceedings. ASCE Speciality Conjerence on Civil Engineering and Nuclear Power, Knoxville, Tennessee, September 1980.

RESPONSE TO MULTICOMPONENTS O F EARTHQUAKE/^^

17. A.K. Gupta and M.P. Singh, Design of Column Sections Subjected to Three Components of Earthquake, Nuclear Engineering and Design, Vol. 41, 1977, pp. 129-1 33.

18. A.K. Gupta, Approximate Design for Three Earthquake Components, Journal of Engin- eering Mechanics Division, ASCE, Vol. 104, No. EM6, December 1978, pp. 1453- 1456.

19. A.K. Gupta, S.J. Fang and S.L. Chu, A Rational and Economical Seismic Design of Beam- - Columns in Steel Frames, Transactions. Fourth International Conference on Structural Mechanics in Reactor Technology, Paper No. K9/7, San Francisco, 1977.

20. A.K. Gupta, Rational and Economic Multicomponent Seismic Design of Piping Systems, Journal of Pressure Vessel Technology, ASME, Vol. 100, November 1978.

Chapter 5/Nonclassically damped systems

5.1 Introduction Equations of motion of the classically damped multi-degree-of-freedom (MDOF) systems can be transformed into a set of independent modal equations using the real-valued eigenvectors and eigenvalues of the undamped systems as was done in Chapter 1. However, in many real systems the modal equations are coupled by the nonclassical damping maxtrix[l]. In many cases, nonclassically damped systems can be approximated by a classically damped system without a significant loss of accuracy. On the other hand, there are important practical situations when the nonclassical nature of the damping matrix cannot be ignored. Such is the case when a structure is made up of materials with different damping characteristics in different parts. For example, a combined analytical model of a soil-structure system is nonclassically damped. Another example is a coupled structure-equipment (primary-secondary) system.

Whether damped classically or nonclassically, one can always evaluate the response of a MDOF system using the direct time-history integration analysis. In classically damped systems, the advantage of modal analysis is that often the response can be represented by a few modes only. For nonclassically damped systems, one may still use the undamped mode shapes to obtain the transformed coupled modal equations as was done by Clough and MojtahediI2J. Because the resulting equations are coupled, the complete transformed system must be integrated simultaneously, as in the direct integration method. The degree of efficiency thus achieved is much less than when the equations are uncoupled. On the other hand, this procedure can be used to reduce the number of degrees-of- freedom by considering only a few of the undamped modes.

Frequencies and mode shapes of a nonclassically damped system are complex and can be calculated using Foss' method[3]. The complex mode shapes and frequencies can be used to obtain first order uncoupled modal differential equations[4]. Itoh[5] solved these equations in conjunction with the FFT (fast Fourier transform) procedure.

SinghI61, using the random vibration approach, developed a response spectrum method for nonclassically damped systems. Implicit in the method is the dependence of response on two spectra, one based on the maximum relative displacement, and another one based on the maximum relative velocity. Singh[6] assumed that the relative velocity spectrum is the same as the relative displacement spectrum (when expressed in same units) in the low and intermediate frequency ranges, that the relative velocity spectrum has zero ordinates in

66

NONCLASSICALLY D A M P E D SYSTEMS/67

the higher frequency range, and that there is a linear transition on the logarithmic scale between the frequencies the relative velocity spectrum diminishes from relative displacement equivalence to zero. As will be shown later, these assumptions are not entirely true. Although these assumptions may introduce errors in certain special cases, in most cases, the method should yield satisfactory results. Singh [6] also presented a comprehensive modal response combination procedure. The main criticism of Singh's method is that it is relatively tedious.

Villaverde and Newmark[7] performed a deterministic formulation starting with the complex frequencies and mode shapes. For each complex mode shape and its conjugate, they explicitly showed that the response can be represented in two parts, one based on the relative displacement spectrum and another one based on the relative velocity spectrum. They assumed that the two spectra are equivalent when expressed in the same units. This assumption does not hold true in the high frequency range.

Igusa and Der Kiureghian[8], and later Gupta and Jaw[9], and Veletsos and Ventura[lO] showed that the displacement vector of a nonclassically damped MDOF system may be expressed as a linear combination of the displacements and velocities of equivalent single-degree-of-freedom (SDOF) systems. Igusa and Der Kiureghian[8] treated the earthquake as white noise and, thus, were able to simplify the final modal response combination equation greatly. Gupta and Jaw[l 11 extended their earlier formulation[9] to the response spectrum method. They also proposed a method for estimating the relative velocity spectrum that would be needed for the analysis.

5.2 Analytical formulation Let us consider the following equation of motion for an N-DOF system

where M, Cand Kdenote mass, damping and stiffness matrices, respectively; Uis the relative displacement vector; Ub is a displacement vector obtained by statically displacing the support by unity in the direction of the input motion; 14, is the ground (or support) displacement; the super dot (.) represents a derivative with respect to the time variable. In the Foss approach[3], Equation 5.1 is cast into a 2N-dimensional matrix equation as follows:

where

The free vibration equivalent of Equation 5.2 yields N complex eigenvectors and eigenvalues, along with their conjugates. These eigenvectors satisfy various

orthogonality conditions[4]. The complex eigenvectors can be written in terms of N- dimensional vectors as

when hi is the complex eigenvalue for mode i . The response is given by

in which zi and 2, can be called complex normal coordinates. The bar (-) in Equa- tion 5.5 and elsewhere denotes the complex conjugate. Equation 5.5 with standard operations yields the following uncoupled equations.

in which

where

Similar equations can be written for the conjugate F,. Given the time history of jig, Equation 5.6 can be solved to give the histories

of the complex normal coordinates z, and Z,[1 21. In turn, we can calculate the history of the displacement vector U from Equation 5.5. Whereas the individual terms on the right-hand side of Equation 5.5 are complex, the summation of the conjugate pairs yield real U.

Now we proceed to develop an alternate modal superposition formulation [9].Consider a kth element of the vector U in the mode i, u,,. From Equation 5.5

Let us write various parameters in Equations 5.6 and 5.9 in terms of their real and imaginary parts.

F, = g + hi, yr, = a + pi. (5.10)

We have omitted subscripts on the right-hand side of Equation 5.10 for brevity. The expression for h, is written in this particular form in view of the similar expression of a SDOF system; w, is the circular frequency, (, is the damping ratio, and o,, is the corresponding damped circular frequency. Equations 5.9 and 5.10 yield


For a given mode, 6 and q are functions of the time variable only. The parameters (a, p) define the mode shape and differ among the various DOF as they should. In a classically damped system, the variation in (a, p) can be represented in terms of a real vector, which, in fact, represents the mode shape. In a nonclassically damped system, on the other hand, both a, P vary independently. Thus, each DOF has its own relative phase. In a classically damped system, we can write one differential equation in terms of the real normal coordinate and real coefficients. In a nonclassically damped system, if we wish to deal with real values, we must write a differential equation for each DOF.

Substituting Equation 5.10 into Equation 5.6 and separating real and imaginary parts, and then solving for 6 and q, we get

i j + 2 c ,oi 5 + of 6 - -(<,wig - wDih) ii, - gu,,

Equations 5. l 1 and 5.12 yield

u,, + 2 <,w,ii,, + o: u,, = -r[ugcos 0 - i&sine/o,], (5.13)

in which

In a numerical experimentation with 2-DOF systems, it was found that 0 can have practically any value from 0 to 2 ~ . For a classically damped system, 0 = 0.

An equation similar to Equation 5.13 is used in the conventional approach for the classically damped system. However, in those cases, the iib term does not appear on the right-hand side of the equation. Consider the following standard normal equation

In terms of the solution of Equation 5.1 5, the solution of Equation 5.13 can be written as

u,, = r (x, cose - i, sin 8/o,). (5.16)

Equation 5.16 can be used to obtain the history of the displacement u,,. It can be written as

u : ' . = ~ ~ x , , y(,=rcos8,

= , , tyy' = r sin 8/61,. (5.17)

We would like to write Equation 5.17 in matrix notations. We note from Equations 5.10

Substitution of Equations 5.18 into Equations 5.14 gives

Equations 5.19 can be rewritten in matrix notations now.

Equations 5.5, 5.16 and 5.17 become

The advantage of Equations 5.21 is that most present modal superposition programs can be adapted to these equations with relatively small change, as opposed to adapting them to Equation 5.5. Instead of the usual complex mode shape, in this case, we are dealing with real mode shapes y? and y f . We are using the conventional modal integration routine, from which we also obtain i, in addition to the usual x,. It was shown by Gupta and Jaw[9] that the above modal superposition method gives response values identical to those obtained by the direct integration of equation of motion, Equation 5.1.

As in the conventional response spectrum method, we define the spectral displacement by

s", = max I x , ( f ) 1. (5.22)

Here, we also define a spectral velocity:

Sb, = max I i , ( t ) I. (5.23)

The nature of these spectral properties is discussed in more detail in the next section. Equations 5.19, 5.22 and 5.23 give

d Uimax = y : ~ & , UP,,, = yPS& (5.24)

N O N C L A S S I C A L L Y D A M P E D S Y S T E M S 1 7 1

For each complex mode i and its conjugate, Equation 5.24 gives two response displacement vectors. These vectors, in turn, yield two modal values of any response, R: and R,", in which the suffix 'max' has been dropped for brevity. In a classically damped case the vector yrP becomes null, so the response Ry is zero. In those cases the problem reduces to combining the maximum modal responses from various modes, R:'. Now, we have two maximum responses from each mode. The method of combining these responses is presented subsequently.

5.3 Response spectra Two spectral quantities were defined in the previous section, a spectral displacement S& and a spectral velocity S&. In these notations, the superscripts d and v (small letters) indicate that the two are based on the maximum relative displacement and velocity, respectively. The subscripts D and V (capital letters) indicate the unit of the spectral value, viz, displacement (D), velocity (V ), and we can also include acceleration (A). We recall the conventional D-V-A relationship (Chapter 1):

in which all the quantities are based on the maximum relative displacement. We can write a similar relationship for the other spectral value:

where, now all the quantities are based on the maximum relative velocity. To distinguish the quantities in Equations 5.25 and 5.26 we shall call all the sJquan- tities in Equation 5.25 the displacement spectrum values, and the S"quantities in Equation 5.26, the velocity spectrum values. Again, in both cases the units are defined by the subscripts D, V or A.

For a given earthquake motion history, the procedure for obtaining the velocity response spectrum is straightforward. For design purposes, the displacement spectrum is specified, the velocity spectrum is not. Theoretically, it should be possible to specify the velocity spectrum also. We shall investigate here the characteristics of the velocity spectrum and its relationship with the displacement spectrum. The objective is to estimate the velocity spectrum from a given displacement spectrum.

Figure 5.1 shows the displacement and the velocity spectra for El Centro (SOOE, 1940) record. A 2-sec initial pulse was added to correct for the initial ground motion values[13,14]. Both spectra are in acceleration(g) units. As such we are dealing with the ~ d , and S: values. We observe from Figure 5.1 that the two spectra are approximately equal in the intermediate frequency range. The displacement spectrum is higher in the higher frequency range, and the velocity spectrum is higher in the lower frequency range. Similar observations have been made by others in the past; see for example Nau and Ha11[14].-

Frequency. Hz

Fig. 5.1 Displacement and velocity spectra for El Centro Earthquake (SOOE, 1940); damping ratio = 0.05 [ l 11.

As is well known, the conventional spectrum' reaches a constant S: in the higher frequency range. This s;, which is often called the zero period acceleration (ZPA), is equal to the maximum ground acceleration. The reason for this phenomenon is that in the higher frequency range the structure acts as a rigid body, and there is practically no dynamic amplification. In roughly the same frequency range Sd, is constant, S: is varying linearly with the frequency (on the log-log chart). The same logic by which Sd, is equal to ii,,,,, S i can be shown to be equal to ugma,/m, which explains the linear variation in S: on the log-log chart. In the conventional spectrum, in the low frequency range SL = ugmar or ~ d , = m2 ugmax, which is represented by a straight line on the chart. Similarly, Sl = w icgmax, which is the other line. Thus, in the low frequency range, we can write

and in the high frequency range

where, the definitions of wL and oH are obvious:

NONCLASSICALLY DAMPED SYSTEMS/73

. . . maa ,,)L = - L'g mar (,," = - . .

Ug max Ug max

Also

wL OH f = - (Hz), f = - (HZ).

2a 2a

If we know f and f ", we can accurately determine S: from S; in the high and low frequency ranges. Further, we know that in the intermediate frequency range, S;- ~ d , . This, for most part, completes the estimation of S; . The exception is a band of frequencies between 'intermediate' and 'high' frequencies. This is the band of frequencies in which the displacement spectrum transitions from almost totally non-rigid (damped periodic) to almost totally rigid, see Chapter 3. This transition is characterized by a change of correlation between the response history and the input ground acceleration history from 0 to 1. This correlation coefficient is called the rigid response coefficient, to be denoted here by a d . ~ h e coefficient ad varies from 0 to 1 between two frequencies f' and f *. These frequencies are estimated from the following equations

W ' d

S A maa f l = % , W 1 = - , f Z = ( f ' + 2f ')/3, (5.30) S L a a

where f 'is the minimum frequency at which S: z ii,,,,, and is commonly known as the rigid frequency.

Now, for f s f ', Equation 5.27 holds; for/" r f 2 f L, S" = sd, and for f 2 f' Equation 5.28 holds. In the range f z r f 2 f ', we need to account for the transition from the damped periodic to rigid response. As in Chapter 3, we separate Sd into two parts, the damped periodic and the rigid as follows:

Then, we apply a relationship similar to Equation 5.28 to the rigid part:

Further,

The above relationships were applied to estimate S from sd for twelve earthquake records[l 11. A comparison of the estimated and the actual velocity spectra for El Centro (SOOE, 1940) earthquake is shown in Figure 5.2.

As for the displacement spectrum, let us denote the rigid response coefficient for the velocity spectrum by a".

Frequency, Hz

Fig. 5.2 Comparison of actual and estimated velocity spectra for El Centro earthquake (SOOE, 1940); damping ratio = 0.05 [ I I].

Svr = av Su, .Srb = J[ 1 - Sv,

Equations 5.3 1-5.34 yield

We observe the following from Equation 5.35.

At f = f', aV = ad = 0,

a t f = f 2 , a v = a d = 1,and

at f = f H , a V = ad.

5.4 Key frequencies f and f To estimate Si from Sd, we need to know two key frequencies f and f ". For actual ground motions these frequencies vary depending upon the frequency content and distribution. As such, one would not expect an exact expression for these frequencies. Indeed, if an actual ground motion is known one need not estimate these frequencies or the velocity response spectrum. Both the displacement and the velocity spectra can be determined directly from the ground motion record. For design purposes, the spectra represent an averaged phenomenon.

NONCLASSICALLY DAMPED SYSTEMS175

Therefore, we are interested in the average representative values off andf which can be used in a design environment to estimate the velocity spectrum.

From a given displacement spectrum certain key frequencies can be defined, viz.,

and the rigid frequency f ', defined earlier. For twelve earthquake records, these frequencies (f ', f and f ') along with the two key frequencies of interest ( f L and f H, are listed in Table 5.1. We observe that f0 and f are of approximately similar magnitude and that f O < f ' < f H c f '. This observation is important in that it defines the neighborhood of the desired key frequencies f Land f H. Also given in Table 5.1 are the ratios f L/fO, f H/f I and f H/fr. There is quite a bit of scatter in these ratios among various earthquake records. Such a scatter is not unexpected. The standard smooth shape of the design spectrum is obtained from averaging much scattered values. It is reasonable to use the average values of these ratios to estimate f and f frequencies. Therefore

The two estimates of f H are comparable in most cases. The equation in terms of f' may be preferable, since the frequency f can be evaluated directly from Equation 5.36. A certain degree of judgment may be required in defining f r in a non-smooth spectrum.

5.5 Modal combination It was shown earlier that for each complex mode i and its conjugate, any response has two maximum values R! and R;. Let us denote the time history of these values by Rf(t) and RY(t). Thus,

The time history of the combined response is given by

Assuming that the earthquake motion is stationary and ergodic, the standard deviation can be expressed as (Chapter 3)

Table 5.1 Key frequencies for various ground motions[l I]

Frequencies (Hz) Ratios

No. Earthquake record

San Fernando, Pacoima 0.366 Dam, S16E (1971)

Parkfield, Cholame- 0.633 Shandon No. 2, N65E ( 1966)

Bear Valley (CA), Melendy 0.848 Ranch, N29W (1972)

Coyote Lake (CA), Gilroy 0.669 Array No. 6. 230' (1979)

Imperial Valley, Bonds 0.795 Corner, 230' (1 979)

Imperial Valley, El Centro, 0.297 SOOE ( 1940)

Kern County. Taft-Lincoln 0.165 School Tunnel, S69E ( 1 952)

Andreanof lsland (Alaska), 0.202 Adak U.S. Naval Station, West (1971)

Kilauea, Hawaii National 1.1 7 Park, Namakani Paio Camp, S30W ( 1 973)

Managua (Nicaragua), Esso 0.8 17 Refinery, South (1 972)

Bucarest (Rumania), 0.450 Building Research Inst., SN (1977)

Off Central Chile Coast, 0.124 Univ. of Chile, Santiago Engineering Building, N IOW, (1971)

Averages 1.02 4.34 0.451

NONCLASSICALLY D A M P E D SYSTEMS177

where td is the duration of response, op is the standard deviation for RP(t), etc. and

in which a;' is the standard deviation for x,, and o:'for f,. The maximum value is given by a peak factor times the standard deviation. If

we assume that the peak factor is the same for all quantities, and that the maximum values are the same as the maximum of the time-history values, we can write the desired modal combination equation

In the classically damped systems R," = 0, and Equation 5.42 degenerates into the standard double sum equation. Various authors have given expressions for the correlation coefficient E;; see Chapter 3. For frequencies less than the key frequency, based on Igusa and Der Kiureghian[8], we observe E; - ~ , j l = E,,.

Further, we can write

As is well known, the correlation coefficients E: and ~ , l j in Equation 5.42 are unity (or nearly unity) when i==j, or when o , = w, and 6, = 6,. These coefficients diminish rapidly as the ratio o,/o, departs from unity. On the other hand, the correlation coefficient p,, is zero when i = j or when o, = o,. This coefficient first increases in magnitude to a maximum value as o,/o,departs from unity, but it soon diminishes too by virtue of the diminishing E,, in the expression.

5.6 Modal combination for high frequency modes As in Chapter 3 for the classically damped systems, the modal combination procedure must be modified in the high frequency range for the nonclassically

damped system also. The high frequency modifications would start at the key frequency f' . We note that for the frequencies > f ' , the rigid response coefficients ad and a", both are nonzero and are less than or equal to unity. The value of ad can be obtained from Chapter 3, that of a" is given by Equation 5.35.

Following Chapter 3, we can separate the damped periodic and the rigid modal response parts.

R: = a: R: RP = ~ [ l - (af)']~:,

R; = a: R:, R,"~ = J[I - (a:)'] R:. (5.44)

The rigid parts are in-phase and can be combined algebraically

It can be assumed that the two rigid parts are uncorrelated. Therefore,

The damped periodic parts can be combined in accordance with Equation 5.42.

Finally, the total combined response is

The above equations can be condensed into one equation with modified correlation coefficients.

F; = J{[1 - (a:)'] [ l - (a;)']} E; + a: a;,

5.7 Modal combination for high frequency modes-residual rigid response The inertial vector Q from Equations 5.2 and 5.3 can be written as

Let us perform a linear transformation,

in which the coefficients r, are complex and yet to be determined. The summation from i = 1 - 2 N implies that all the conjugate parts are included. We now substitute Equation 5.52 into Equation 5.51, and premultiply the resulting equation by vj7. Making use of the orthogonality condition and of Equations 5.7 and 5.8 we get

a, F, = a, r,, r, = F,. (5.53)

Hence,'


In view of Equation 5.54, Equation 5.1 or 5.2 can be expressed as

The eigenvector Fi v;/& should satisfy the free vibration equation. Therefore,

which, with Equation 5.56, gives

Writing the conjugation pairs explicity, we have

Making substitutions from Equation 5.10 and 5.20 we get


We have denoted the response vector in the ith complex conjugate pair of modes as Ui in Equation 5.21. From Equation 5.59 we infer

The structure has N-DOF and N pairs of conjugate modes. Let us assume that there are n pairs of modes that either have frequencies less than the rigid frequencies fr or have significant values of U:. The higher of the two values of n given by these conditions should be used. We denote the combined response in the modes beyond those n pairs by U,. The total response vector is given by

Equations 5.1, 5.60 and 5.6 1 give

1 The response of the structure in the modes having frequencies greater than the 1 rigid frequency is pseudo-static. Therefore, u0 and & vector terms in Equation

5.62 can be ignored and we get


The above formulation allows us to perform the complex eigenvalue analysis only for the first n pair of modes. The term K-' M Ubin Equation 5.63 represents

n

only one static analysis. The other term y:/o: is known from the modal i- l

analysis. The history of the residual response U, is evaluated by multiplying the vector

by ii, for each time step. In the response spectrum method the term -u, is replaced by the ZPA

The vector U, gives the value of any residual rigid response R,. The modal combination equations, Equations 5.45-5.48 are appropriately modified. For example, Equation 5.45 becomes

5.8 Application The response spectrum method was applied to nine primary-secondary systems of the type shown in Figure 5.3 [I 11. The story stiffness and mass of the primary system were kept constant, and those of the secondary system were varied, Table 5.2, to obtain different coupled systems. The first eight systems were selected such that the uncoupled analysis of the primary system would introduce approximately 10% error in the fundamental frequency. The ninth system was selected to make the fundamental frequencies of the primary and secondary systems identical, the tuned case.

The uncoupled frequencies and mode shapes of the primary and secondary systems are shown in Figure 5.4. The damping ratio of the primary system is 796, and of the secondary system is 2%. A method of obtaining the coupled nonclassical damping matrix is described in Chapter 6. The coupled frequencies and damping ratios for all the nine cases under consideration are given in Table 5.3. The coupled modes which have damping ratios close to 7% are dominated by the primary system modes, those having damping ratios close to 2% have predominant secondary system modes, and those which have damping ratios in between have significant participation from both the primary and secondary system modes. The coupled modal vectors y%nd y " are shown in Figures 5.5 and 5.6.

Primary system Secondary system Coupled system

Each story

Mass rn= 1 kip s2/in

Stiffness k= 5000 kipstin

Mass (mo and stiffness (ko) Node number rn

are varied to obtain a range @ Element number n

of r, and r, values

Fig. 5.3 Examples o f primary, secondary, and coupled systems[l4].

All nine systems were subjected to the twelve earthquake motions listed in Table 5.1. Two methods of analysis were used, the direct time-history integration using the Newmark's f3 method and the proposed response spectrum approach. The direct time-history integration results were treated as standard and the response spectrum results were compared against them.

Details of Case 2 subjected to El Centro ground motion are given in Tables 5.4 and 5.5. It is observed that the response spectrum method does yield satisfactory results. In all, for nine systems, twelve time histories each, there were one

Table 5.2 Description o f secondary system [9, 15)

Frequencies (Hz)

Case (kips in-') (kips s2 in-') Mode 1 Mode 2


Mode 1 f = 2.713Hz

Mode 4 f = 16.85Hz

(b)

Mode 2 f = 7.980Hz

Mode 5 f = 19.93Hz

Mode 3 f = 12.79Hz

Mode 6 f = 2 1.86Hz

F i g 5 4 Unnonnalized mode shapes and frequencies of (a) the uncoupled primary system, (b) the uncoupled secondary system[l I].

Table 5.3 Coupled frequencies and damping ratios[l I ]

Frequency (Hz)/damping ratio (%) for coupled mode

Case I 2 3 4 5 6 7 8

hundred and eight direct time history, response spectrum analyses pairs. In most cases the errors were sufficiently small. The highest error in displacements was 41% and that in element forces 45%. In a comparative study performed by Maison, Neuss and Kasai in Reference[l6] of Chapter 3, the maximum error was found to be 67% when double sum combination was used for a classically damped building. In other modal combination methods (SRSS, absolute sum) the error

Table 5.4 Comparison of nodal displacements (in), Case 2, El Centro (SOOE, 1940)[1 I]

Node no. Direct integration Response spectrum % error


Mode 6 Mode 6 f = 12.8Hz. 1~7.0% f = 16.9Hz. {=?.o%

0.038 0.038 -0.009 -0.009

0.05f'08~.0 [::::: l : l ~ . o ~ ~ ~ ~ ~ -0.00 1

0.0 19

-0.074 -0.03 1

0.065 0.0 17

Mode 7 Mode 8 f = l9.9Hz, {=7.O% f = 2 1.9Hz. {=7.0%

Fig. 5.5 The modal vectors [ I I].

I I

Mode 1 (lo-') Mode 2 (lo-') f = 2.5Hz,(=5.7% f = 4.2~z.(=3.2%

- Mode 3 (10-3 Mode 4 (lo-')

f =6.5Hz,(=2.2% f =8.2~z.(=6.7%

-0265

-0.49 1

-0.559

Mode 5 (109 Mode 6 (lo-') f = 12.8Hz. r=7.0% f = ~~.QHz,(=?.o%

-0.438

-0.002 9.002 -0.759 -0.759

-1.720 0.622

-0293

ode 7 ( 1 ~ ' ) Mode 8 (109 f =l9.9 & (=7.0% f =219 &{=7.0%

Fig. 5.6 The coyu modal vectors [ I I].

N O N C L A S S I C A L L Y D A M P E D SYSTEMS187

Table 5.5 Comparison of element forces (kips), Case 2, El Centro (SOOE, 1 940) [ 1 1 ]

Element no. Direct integration Response spectrum % error

was much higher. The combined error statistics for all the cases analyzed are summarized in Table 5.6. The mean error is in the range of 1%, and the standard deviation of the percent error is in the order of ten. This shows that the response spectrum method for the nonclassically damped system is reasonably accurate. The degree of accuracy is of the same order as is usually expected in the response spectrum analysis of classically damped systems.

Table 5.6 Summary of percent error statistics[l I]

Descri~tion Mean Standard deviation

Displacements 1.33 9.65 Forces 0.48 10.31 Displacements and forces 0.88 10.1

References I . G.B. Warburton and S.R. Soni, Errors in Response Calculations of Non-Classically

Damped Structures, Earthquake Engineering and Strzictural Dynamics, Vol. 5. 1977. pp. 365-376.

2. R.W. Clough and S. Mojtahedi, Earthquake Response Analysis Considering Non- Proportional Damping, Earthquake Enginrering and Structural Dynamics, Vol. 4 , 1976, pp. 489-496.

3. K.A. Foss, Coordinates Which Uncouple the Equations of Motion of Damped Linear Dynamic Systems, Journal ofApplied Mechanics, Vol. 25, September 1958, pp. 361-364.

4. W.C. Hurty and M.F. Rubenstein, Dynamics of Structure, Prentice Hall, Clifton. New Jersey, 1964.

5. T. Itoh, Damped Vibration Mode Superposition Method for Dynamic Response Analysis, Earthquake Engineering and Structural Dynamics, Vol. 2, 1973, pp. 47-57.

6. M.P. Singh, Seismic Response by SRSS for Nonproportional Damping, Journal of the Engineering Mechanics Diuision, ASCE, Vol. 106, No. EM6, December 1980, pp. 1405-1419.

7. R. Villaverde and N.M. Newmark, Seismic Response of Light Attachments to Buildings, Structural Research Series, No. 469, University of Illinois at Urbana-Champaign, February 1980, Chapter 5.

8. T. lgusa and A. Der Kiureghian, Dynamic Analysis of Multiply Tuned and Arbitrarily Supported Secondary Systems, Report No. UCB/EERC-83/07, University of California, Berkeley, July 1983.

9. A.K. Gupta and J.W. Jaw, Seismic Response of Nonclassically Damped Systems, Nuclear Engineering and Design, Vol. 91. January 1986, pp. 153-1 59.

10. AS. Veletsos and C.E. Ventura, Modal Analysis of Non-Classically Damped Linear Systems, Earthquake Engineering and Structural Dj~namics, Vol. 14, 1986, pp. 2 17-243.

I I. A.K. Gupta and J.W. Jaw, Response Spectrum Method for Nonclassically Damped Systems, Nuclear Engineering and Design, Vol. 91, January 1986, pp. 16 1 - 169.

12. M.P. Singh, M. Ghatory-Ashtiany, Modal Time History Analysis of Nonclassically Damped Structures for Seismic Motion, Earthquake Engineeringand Structural Dynamics, Vol. 14, No. 1, January - February 1986, pp. 1 13-146.

13. D.A. Pecknold and R. Riddell, Effect of Initial Base Motion on Response Spectra, Journal of the Engineering Mechanics Division, ASCE, Vol. 105, No. EM6. December 1979, pp. 1057-1 060.

14. J.M. Nau and W.J. Hall, An Evaluation of Scaling Methods for Earthquake Response Spectra, Structural Research Series, No. 499, University of Illinois at Urbana-Champaign, May 1982.

15. A.K. Gupta and J.M. Tembulkar, Dynamic Decoupling of Multiply Connected MDOF Secondary Systems, Nuclear Engineering and Design, Vol. 81, 1984, pp. 375-383.

Chapter 6JResponse of secondary systems

6.1 Introduction In major industrial buildings, such as nuclear power plants, as with other common constructions such as high rise buildings, it is impractical to perform a coupled dynamic analysis of the primary system (building) and the secondary system (HVAC, piping, equipment, etc.) using the conventional analytical tools. The stiffness and inertia properties of the two systems may be quite dissimilar, which is likely to cause numerical problems in a coupled analysis. For other practical reasons also, it is customary to perform seismic analysis of the two systems separately. The effect of decoupling on the primary system is presented in Chapter 7. We shall discuss in this Chapter the problems associated with the decoupled response of secondary systems, and the techniques by which the accurate response of a secondary system can be calculated.

A popular method of calculating the response of secondary systems is by using the floor response spectrum, or more accurately, the instructure response spectrum (IRS). In the conventional IRS method, interaction between the primary and secondary systems is ignored, which may have significant effect in the resonant frequency range. For the multiply supported secondary systems, i t is customary to use a common IRS input, which is obtained by enveloping the IRS at various connecting degrees of freedom (DOF). The effect of relative motions between supports is incorporated by performing a separate static analysis. This procedure may lead to considerable over-estimation of the seismic response of the secondary system.

Often the earthquake input to the primary system is defined in terms of a design response spectrum, and the ground motion time history is not known. In the conventional procedure, a response spectrum compatible ground motion history is created and is used for generating the IRS. The main criticism of the procedure is that it is non-unique. The problem can be overcome in part by using several different ground motion histories compatible with the same input response spectrum. The time-history solutions are uneconomical to start with, and the use of several time histories further adds to the cost. In view of all the problems associated with the conventional method, the last two decades have been an ongoing search for a direct method. Penzien and Chopra[l] were among the first (1 965) to explore the topic. They were followed by Biggs and Roesset[2] (1 97O), and Kapur and Shao[3] ( 1 973).

These early efforts were semi-empirical and heuristic, and were not found to be generally acceptable. The time-history procedure mentioned above continued

to be the commonly used method, as it still is today. Other rational alternatives, however, have been presented, and some are being used. Perhaps the first among these alternatives was a stochastic method developed by Singh[4]. The modal properties of the primary system are used to obtain the power spectral density function at any connecting degree of freedom directly from the input response spectrum. The method assumes the ground motion to be stationary. This assumption leads to over-estimation of response in the lower frequency range. Singh[S] later suggested corrective measures to rectify some of the problems. The stochastic method is computationally efficient. Further, the method avoids the use of time history for analysis of the primary system. In recent years, therefore, the method has been gaining in acceptability. Singh's method, as originally developed[4,5], has some of the same problems as the conventional IRS method.

Peters, Schmitz and Wagner's[6] presented a method of determining the IRS by evaluating the mode shapes of the coupled primary and secondary system. As is usual, the secondary system was assumed to be a massless SDOF oscillator. In this case, it is assumed that the frequencies and mode shapes of the primary system do not change. One simply includes the appropriate modal displacement terms related to the secondary mass. Plus, a new modal vector is added for the extra DOF. Like most other methods, this method has problems in cases of tuned and nearly tuned secondary systems.

Peters, Schmitz and Wagner's[6] evaluation of the mode shapes of the coupled system based on the uncoupled mode shapes may be considered to be a turning point in the evaluation of the coupled system response calculation. It opened the way for consideration of more complex secondary systems. Sackman and Kelly have made much contribution in the area of secondary system response and have many publications, see for example Reference[7]. They considered the secondary system with nonzero mass and introduced a rationally derived expression for mass ratio, r,,,, between the secondary and primary system masses. They, for the first time, rationally tackled the problem of tuned and nearly-tuned secondary systems and established the role of the mass ratio in this context.

Ruzicka and Robinson[8] have 'studied the tuned systems in detail and propose three different approximate methods. However, they conclude, 'knowledge of the Fourier amplitude spectrum is essential if the response of a tuned secondary system is to be estimated accurately.' We have since learned that one can find the response of a secondary system for tuned and untuned systems without the knowledge of the Fourier spectrum. Villaverde and Newmark[9] have developed approximate methods for evaluating secondary system response.

Sackman, Der Kiureghian and Nour-Omid [ 101 used a perturbation technique to develop modal properties of the coupled system. As in Reference[7], the secondary system was still a SDOF system. They also accounted for changes in the frequencies of the primary system and for their effect on the mode shapes.

RESPONSE O F SECONDARY SYSTEMS/91

They used orthogonality conditions to improve mode shape of the modes with close frequencies, a situation that arises when the secondary system is tuned or nearly-tuned. Unlike reference[7], however, this solution is not extended to a deterministic evaluation of the IRS. Instead. in a companion paper Der Kiureghian, Sackman and Nour-Omid[l I] used the mode shapes and frequencies to evaluate response to a stochastic input. The new stochastic method is an improvement on the old method[4,5] in that it accounts for interaction between equipment and structure, correlation between closely spaced modes, etc. How- ever, other problems inherent in the stochastic method remain. Unlike Singh [4,5], Der Kiureghian, Sackman and Nour-Omid [ 1 1 ] assumed the earthquake to be a white noise. Hernried and Sackman[l2] used the perturbation technique to develop mode shapes of a coupled MDOF primary and MDOF secondary system.

Gupta[l3], and Gupta and Jaw[l4] developed an approximate method for evaluating the complex eigenvalues and eigenvectors of nonclassically damped primary-secondary systems. The method was applied to evaluate the coupled response of the secondary system [13,15,16]. An improved IRS method was also developed which accounts for the interaction effects and the correlation between responses from various support motions [13,17]. All these methods[ 1 3,15- 171 used the response spectrum at the base of the building as input without converting it into a compatible time history, or to a power spectral density function. Igusa and Der Kiureghian(l81 have proposed a perturbation method for evaluating the complex eigenvalues and eigenvectors of nonclassically damped primary-secondary systems. As in Reference[l 11, the emphasis is on the response to stochastic input. Asfura and Der Kiureghian[l9] have applied a similar technique to develop an IRS method. The present treatment is based on References [I 31-[ l 71.

6.2 Formulation of the coupled problem The free vibration equation of the coupled system is

when M, Cand K are the mass. damping and stiffness matrices, respectively, and U is the displacement vector. It is assumed that the uncoupled primary and secondary systems are classically damped. Let us denote the ith mode shape of the uncoupled primary system by and the ath mode shape of the uncoupled secondary system by $,,. The mode shapes are normalized such that $riT M, $ci = 1 and $,' M, $, = 1. The subscript p denotes a primary system property, and the subscript s a secondary system property. The subscript i and other lower case letters denote the primary system modes, and the subscript a and other Greek letters denote the secondary system modes.

In terms of the uncoupled mode shapes we can write

Substituting Equation 6.2 in Equation 6.1 and premultiplying by a T , we get

The elements of and are defined below.

= 0, a # P, (6.4)

where o , and 6 , are the circular frequency and the damping ratio, respectively, for the ith uncoupled mode of the primary system; w, and l,, are the corresponding values for the ath secondary system mode. In Equation 6.4 and later in this Chapter, subscript c denotes the primary DOF which are connected with the secondary system; and subscript s, as before, denotes the secondary DOF. The matrices K: and C& are the stiffness and damping contributions of the secondary syFtems.

To evaluate the term with K, in Equation 6.4, let us define the following matrices[20]

The matrix U, contains one secondary system vector for each connecting DOF. Each such vector represents the static deformation shape of the secondary system when the corresponding connecting DOF undergoes a unit displacement. y, is a

RESPONSE OF SECONDARY SYSTEMS193

row of participation factors for the secondary system, one element for each connecting DOF. Following the derivation for U, in Chapter 3, we can write

Equations 6.5 and 6.6. give

T Kc= --CK,O.Y~. Lh= -@:Yea a

Therefore,

Ria = -of, f)dT ymT .

For SDOF primary and secondary system y, $,

(6.8)

degenerates into the square root of the mass ratio J(m,/m,). It has been shown in Reference[19] that the product $dry,Ty, $ei can be viewed as the ratio of the kinetic energies of the secondary and primary systems. An energy-mass ratio is, therefore, defined as

4a = $cT yeaT yea $ei, r,h'2 = +idT year = ym h i - (6.9)

Equations 6.8 and 6.9 give 2 1/2 Ria = -o,,rIa . (6.10)

To evaluate KA, let us subject the secondary system to a rigid body motion, which gives

Static condensation of Us, yields

If all the DOF in U, can be specified arbitrarily, we can write

which, with Equation 6.7, gives

Equations 6.9 and 6.1 1 yield

In the derivation above, U, can be defined arbitrarilxwhen the secondary system does not offer any static constraint to the primary system. There may be cases, when it is not so. The effect of constraint is to increase the magnitude of the coupled frequencies. Therefore, we adopt the following definition.

This essentially completes the definition of K. All the expressions written so far are exact within the modeling assumptions of the primary and secondary systems. The definitions of $cT CS, $cj and $,.T C , $, are not so direct. We know that for a SDOF system c / k = 2[/w. For the expressions which are already defined for and in Equation 6.4, the same type of equality holds. It is, therefore, reasonable to assume that the same type of relationships would hold for the yet undefined terms. Based an Equations 6.1 1 and 6.13, we write

In Equation 6.14 we have omitted the expression which would be equivalent to ~ o i ~ , of Equation 6.13.

If the coupled complex eigenvalue is h, Equation 6.3 becomes

where K* = R + 31 C + h2[l]. Various elements of K* are defined below

The coupled eigenvalue problem is defined in a similar manner in Refer- ence[l8], which makes it possible to compare the two. In our notations, we shall define the coefficient matrix of Reference[18] as K*'~', IDK standing for the authors Igusa and Der Kiureghian. Various terms of K " ~ ~ are defined below.

R E S P O N S E O F S E C O N D A R Y SYSTEMS/95

We can 2crive the elements of K * ' ~ ~ , in Equation 6.18, from those of K', in Equation 6.17, by omitting terms which are of the order of r,,, and retaining the terms which are of the order r,!Lz. Further, Equation 6.18 also ignores the AW;, term. If *,=' is of the same order as c,, and c,, then the assumption[18] to omit r,, terms is justifiable. Therefore, K * ' ~ ~ can be viewed as a simplified form of K' when r:' is of the same order as c,, and <,, and when the secondary system does not offer significant static constraint represented by mi,,.

Dynzmic coupling between primary and secondary systems is caused because of the t e r n r,,, the energy-mass ratio in Equations 6.17 and 6.18. In particular, when r , = 0. the terms K: = 0, thus, dynamically uncoupling the two systems. The t e n lo:, is independent of the energy-mass ratio, and accounts for the static constraint only. It is like introducing a massless spring between the DOF i and j.

6.3 Coupled modal properties The fret \-ibration equation defined in Equation 6.16 represents an exact modal synthesis equation for a coupled system consisting of individually classically dampeC jrimary and secondary systems, irrespective of the mass ratios. In fact, to compare the accuracy of the approximate method in Reference[l4], exact complex eigenvalue analysis was performed using Equation 6.16. An approximate itextive scheme is presented here[l3,14] which is suitable for moderately light equipment attached to a structure.

We need to calculate both the eigenvalues and eigenvectors. If we know the eigenvalue somehow. i t is relatively straightforward to evaluate the eigenvector, and vice versa. As a rule, if we use an approximate eigenvector in evaluating the eigenvalue. the error in eigenvalue is relatively less (of higher order). We shall use this rule to establish our eigenvalues. Consider the coupled eigenvector corre- spondine to the ith uncoupled primary system mode. In Equation 6.16, take x,, =

1, and as an approximation assume x,, = 0, i # j. From Equations 6.16 and 6.17 we can w i t e

Since x,, = 1, and we have assumed x,, = 0, i # j, we get

We also have

which with Equation 6.20 gives

Equations 6.1 7 and 6.22 yield

Equation 6.23 forms the basis of evaluating the complex coupled eigenvalues corresponding to primary system modes.

Equation 6.23 is solved iteratively. This last equation is a quadratic in 3,. if the last expression is known. This last expression, however, includes the unknown h, terms. Therefore, we assume a trial value fork,, evaluate the expression and then solve for k,. The new value of A, is used as the next trial value and so on, until a convergence is reached. This algorithm yields accurate eigenvalues[l4].

Now that the eigenvalue is known, the eigenvector can be improved. We already have x,, = 1 and x,, = 0 for i # j, and the xsa terms can be calculated from Equation 6.20. We found that the results are uniformly improved after one iteration [ I 41. Using the previous values of the x, terms, we now calculate all x, terms from a variation of Equation 6.21.

In Equation 6.24 we have omitted the off-diagonal terms, except the K$ which is likely to have a relatively significant contribution. Next, we calculate the improved x,. Equation 6.19 gives

RESPONSE OF SECONDARY SYSTEMS/97

When the accurate eigenvalues evaluated above are used, it was found that the one iteration solution also gives very accurate eigenvectors[l4]. These eigenvectors are in transformed coordinates. The eigenvectors in the original coordinates can be obtained from Equation 6.2.

The procedure for evaluating the coupled eigenvalues and eigenvectors corresponding to the uncoupled secondary system modes is similar. For ath secondary mode, we take x, = 1 and assume xsB = 0 for a # P. Equations 6.16 and 6.1 7 give

Ignoring the off-diagonal terms, we can simply write

Also, from Equations 6.16 and 6.17

which with Equation 6.28 gives

As before Equation 6.30 is solved iteratively to obtain &. Now we shall develop the one iteration scheme for the improved eigenvectors.

We can write an equation, similar to Equation 6.29, for all the P terms

Next we calculate the improved xpi from Equation 6.27:

The eigenvectors calculated above are also transformed into the original coordinate using Equation 6.2.

When the secondary system constraint terms ~ ~ w , , , are small or zero and the square roots of energy mass ratio terms ry2 are of the order of the damping ratios, Equations 6.23 and 6.30 can be solved to give closed-form values obtained by Igusa and Der Kiureghian[18]. In detuned modes the eigenvalues of the coupled system are approximately equal to the corresponding uncoupled eigenvalues, h, = - C,plwp, + i wpl J(l -C,;,), and ha = -emom + i wsa J(l - c:a). In transformed coordinates, the eigenvectors are given by Equation 6.20 and Equation 6.28, respectively. A primary mode i and the secondary mode a are considered tuned when

in which e represents the allowable relative error in the response of the tuned modes. The term PI, is called the tuning parameter. The coupled eigenvalues in this case are

Corresponding eigenvectors can be calculated from Equations 6.20 and 6.25, and from Equations ,6.28 and 6.31, respectively, using the values of A, and A,, calculated from Equation 6.34. Equation 6.34 along with Equations 6.20, 6.25, 6.28, and 6.31 can also be used to calculate the eigenvalues and eigenvectors in the untuned modes.

6.4 Coupled response calculation We shall now use the complex eigenvalues and eigenvectors evaluated in the previous section to calculate the response of the coupled primary-secondary system. The complex value 1 , gives the coupled frequency w, and the damping ratio (Chapter 5). Let us denote the complex eigenvector in transformed


coordinates by vj. According to Equation 6.2, the corresponding eigenvector in the original coordinates, ijii, can be determined from

We shall repeat here some of the steps of Chapter 5 for the calculation of response of nonclassically damped systems. Each complex mode shape gives two response vectors.

U; = Y;S&, UP = W , ~ P S & , (6.36)

where s:, and SL, are the spectral displacements for the ith modal frequency from the displacement and the velocity spectra of the input motion for the primary system, and

The term Xi in the above equation represents the conjugate of A,, and F, is given

by

where Ub is the static displacement vector of the coupled system when the primary system support displaces by unity in the direction of the earthquake, and

a, = 2A, v i T M y i + y i T C y i . (6.39)

Equation 6.38 can be written as

in which Ub and U, represent the primary and secondary DOF, respectively, in the vector Ub. Also, Equation 6.40 can be written as

The motive behind the above rearrangement of the expressions is to write them in forms which can be readily obtained. For example, the damping matrix C is not explicitly defined, the transformed matrix is, see Equation 6.4.

Let us re-examine Equation 6.36. The vectors v! and w , y : represent the displacement responses in mode i, when the spectral displacements from displacement and velocity spectra both are unity. These vectors may be considered to be normalized real vectors representing the complex modal vector

yr,. Unlike the complex vector vi , the real vectors y! and oivP are unique. Therefore, we shall first use these vectors to evaluate the accuracy of the approximate scheme.

The same nine coupled primary-secondary systems, which were presented in Chapter 5, were analyzed in reference[14]. The coupled modal frequencies, damping ratios, and the vectors \y:, o,yrP were evaluated for nine modes for each of the nine systems using the exact method[2 11, the Gupta-Jaw method[14], and the Igusa-Der Kiureghian (IDK) method [18]. The results for Case 2 are typical of all the cases and are presented in Tables 6.1 and 6.2. The exact frequencies, and damping ratios for the eight coupled modes along with the percent error introduced by the Gupta-Jaw 'method and the IDK method[l8] are given in Table 6.1. The exact modal vectors v!, and o , yp are shown in Chapter 5. The differences between the modal vectors from the present algorithm and the exact vectors were so small that they could not be shown graphically. In Table 6.2, only the mean and standard deviation of the percent errors of the modal vectors are given. The vectors y(. and o,\y; have the same unit, and as indicated earlier, they have a 90" phase difference. Often w,\y; vector is smaller in magnitude. To avoid over- emphasizing error in a small quantity, the percent error for the DOF was evaluated against the vector sum of the elements of the two vectors.

Finally, the mean and the standard deviation of percent error in frequency, damping ratio and the modal displacement in all the modes for all the nine cases are summarized in Table 6.3. At the bottom of Table 6.3 are the cumulative statistics of all the cases. It is clear that the mean errors are generally small, showing a lack of a consistent bias. The standard deviations of the percent error in frequency, damping ratio and the modal displacement, on the other hand, show a more significant spread between the two approximate methods. In all cases the Gupta-Jaw method yields better results than the IDK method. It is so, as stated

Table 6.1 Comparison of coupled frequencies. damping ratios, Case 2 [ I 41

Frequency, (Hz) Damping ratio (%)

, Percent error Percent error

Mode Exact Gupta-Jaw IDK Exact Gupta-Jaw IDK


Table 6.2 Percent errors in modai displacements, Case 2 [ I 41

Percent error in modal displacement

Mean Standard deviation

Mode Gupta-Jaw IDK Gupta-Jaw IDK

I ' I

earlier, because the problems presented here do not satisfy the assumption of the I ,

lightness of the secondary system, and the implied assumption of the lack of constraint offered by the secondary system made in Reference[lS]. It was found that the difference between results from the Gupta-Jaw algorithm and those based on IDK narrowed as the mass of the secondary system diminished[l4].

6.5 Comparison of coupled response with the response from conventional IRS method

A comparison of the coupled response from the present method with the response from the conventional IRS method has been made in Reference[l6]. The same nine primary-secondary systems, considered in Chapter 5 and in the previous section, were analyzed. Since we are considering the response values of the secondary system only, the node and element numbers of the secondary system were numbered separately as shown in Figure 6.1.

A11 nine coupled systems were first analyzed for the El Centro (SOOE, 1940) ground motion using the time-history method. The results of this time-history method were used as the exact reference values for comparison purposes. The response of the secondary system for all nine cases was then calculated by the present method, and also by the conventional IRS method. The response values obtained from the present method were almost the same as those obtained from the coupled response spectrum analysis using exact mode shapes. In the conventional IRS method, a companion static analysis is performed in which the I maximum support motions obtained from the uncoupled primary system analysis are applied out-of-phase to obtain the worst possible member forces (or I stresses). The same technique was used in Reference[l6] in calculating the member forces by the conventional method. The particular scheme, however, would not give the maximum possible displacement in the secondary system.

Table 6.3 Statistics of percent errors[l4]

% error in frequency . % error in damping ratio- % error in modal displacement

Mean o Mean o Mean CJ

Case Gupta-Jaw IDK Gupta-Jaw IDK Gupta-Jaw IDK Gupta-Jaw IDK Gupta-Jaw IDK Gupta-Jaw IDK

All cases 0.01 -0.23 0.06 2.84 -0.26 0.06 0.51 12.77 0.17 0.68 1.26 10.72

Node Number n @ Element Number e

Fig. 6.1 Example of a secondary systemjlS].

Further, in the present problem, the two nodes supporting the secondary system have a high degree of correlation. Therefore. the two support displacements were applied in-phase for calculating the 'static' component of the secondary system displacements by the conventional method.

A comparison of nodal displacements and spring forces from the present method and those from the conventional IRS method is made in Tables 6.4 and 6.5. On an average, the present method under-estimates the nodal displacement by 3%, the conventional method over-estimates them by 58%. The standard deviations of the percent error in displacement from the present and the conventional methods are 6.9 and 44.1, respectively. The present and the conventional methods have an average error in spring forces of - 11.2 and 1 16.1%, respectively. The standard deviations of the percent error in spring forces from the two methods is 9.9 (present) and 46.9 (conventional). We note that the error values in the present method are in the order of errors commonly I

introduced in the response spectrum method and are well within the acceptable range. The average errors of 58% in displacement and of 1 16.1% in spring forces from the conventional method are rather high. It shows a consistent bias of significant over-estimation of the response values. A relatively large value of standard deviations of the percent errors also indicates a considerable dispersion i in the response values from the conventional method. In comparison, the standard deviations from the present method are much smaller.

In modal displacement calculations from the conventional method, the effect of the static displacement component is significant in only the first two cases. In

Table 6.4 Comparison of nodal displac&nents from the present method and the conventional floor response spectrum method[l5]

Conventional IRS method

Present program Combined

Time-history Earthquake Static Case Node displacement (in) , Displacement % error displacement (in) displacement (in) Displacement (in) % error

Mean - 3.0 58.0 Standard deviation 6.9 44.1

Table 6.5 Comparison of spr~ng Ibrccs t'rorn the prcscnl nmhod and rhc conventional floor response spectrum rnelhod[lS\

Conventional 1RS method Present program Combined force

Time-history Earthquake force Static force Case Element spring force Spring force (kips) % error (kips) (kips) (kips) % error

(kios)


Case 1, the static component helps reduce the error. In Case 2, it slightly increases the over-estimations. A similar conclusion can also be reached for the spring forces for elements 1 and 3. The floor response spectrum method gives zero spring force in element 2. The spring force in element 2 is zero in the first secondary system mode, which is symmetric. Since the earthquake motion is input in- phase at the two supports, the response in all the elements in the secondary system mode, which is antisymmetric, is zero. As such, the consideration of the static component of the spring force in element 2 is quite important. In most cases, however, the static component of the spring force in element 2 is much greater than the corresponding value from the coupled time-history analysis.

In the tuned system, Case 9, the error in displacements is the highest of all the cases. The same is true of the spring forces in elements 1 and 3. The spring force in element 2 is not affected by tuning because all the response is coming from the static analysis. It is significant to note, however, that the conventional method over-estimates the response in all cases by a considerable margin, not only in the tuned case.

The 7% damping value for the primary system and 2% damping value for the secondary system are on the high side of the commonly used damping values in the nuclear power plant design. We expect that at lower damping values the conventional method would introduce much larger errors in the response of the secondary system. On the other hand, the response errors from the conventional method will be lower for relatively lighter secondary systems. The errors in the tuned case are likely to be more significant as compared to those in other cases for lighter secondary systems and for lower damping values.

6.6 An alternate formulation of fhe coupled response We shall develop an alternate formulation of the coupled response of the secondary system with a view towards developing an instructure response spectrum (IRS) method. The equation of motion of the coupled system is

Equation 6.42 is identical to Equation 6.1, except for the right-hand side. On the right-hand side Ub is a displacement vector obtained by statically displacing the support by unity in the direction of the input motion, and u, is the ground displacement. We will continue to use the notation of Section 6.2, in which subscript p denotes a primary system property, subscript s a secondary system property, and subscript c is a subset of p which is used to denote the primary system DOF connected to the secondary system. From Equation 6.42, the equation of motion for the secondary system can be written as

Let us define a relative displacement vector of the secondary system &, as follows:


where us, is defined in Equation 6.5. Substituting Equation 6.44 in Equation 6.43 we get

It is customary to ignore the damping terms of the type on the right-hand side of Equation 6.45; further we can write UbS = Us, Ubc. Equation 6.45 becomes

The vector { + U,,, ii,) represents the total acceleration at the connecting DOF. We can write in terms of the normalized mode shapes of the uncoupled

secondary system

The secondary system normal coordinate defined here, x',, is different from that defined in Equation 6.2, x,,. The latter is used to represent the displacement vector of the secondary system relative to the primary system support displace-

ment, Us = $,xsa. Equations 6.46 and 6.47, and the use of orthogonality a

conditions give

in which r, is a row of participation factors, one participation factor for each connecting DOF. Let us denote elements i;. and { 8, + Ubc U s } for a given DOF c by y,, and u, + u, u,, respectively. The relative displacement response of a SDOF oscillator, circular frequency w,, and damping ratio c,, subjected to acceleration history u, + uk ii, is denoted by LC,. The equation of motion of the oscillator is


Equations 6.44, 6.47 and 6.50 give

Secondary system response values can be written as linear functions of the elements of Uc and Us. For a secondary system response R we can write:

where A, and B,, are known constant coefficients. When the secondary system does not offer any static constraint to the primary system the displacement vector of the connecting DOF, Uc, will merely introduce rigid body motion in the secondary system, and it will not cause any stresses. Therefore, in those cases, the coefficient A, will be zero for stress related response values. On the other hand, when the secondary system does offer static constraint, the stresses in the secondary system generated by the vector U, can be significant, as is shown in Section 6.1 1.

Equations 6.5 1 and 6.52 are the alternate equations for the coupled response of the secondary system. We note here the definition of the instructure spectral displacement.

Sea = I .vca 1 max 9

where Y,,(t) is given by Equation 6.49.

6.7 Secondary system equivalent oscillators In the conventional method, J,, values, which give the IRS, are evaluated from the time history of ii, + u, ii,, obtained from a decoupled solution of Equation 6.49. Such a procedure does not account for interaction between primary and secondary systems. Consequently, Equations 6.51 and 6.52 would not yield the coupled response. This problem can be remedied ify',,, itself, is evaluated using a coupled system.

Response in each mode of the secondary system can be viewed as the response of a SDOF oscillator attached to an appropriate primary system DOF. In doing so, it is assumed that various secondary system modes do not influence the interaction between any one secondary system mode (the oscillator) and the primary system. This assumption would allow the use of formulation in Sections 6.2-6.4, considering one secondary system mode at a time. A secondary system mode is characterized by the modal frequency o,, modal damping ratio c,,. and the energy mass ratio r,,. It is this latter parameter r,, which accounts for the interaction between the secondary and primary systems. For the interaction-less calculations, as is done in the conventional method, the implicit assumption is

RESPONSE O F SECONDARY SYSTEMS/lOS

r,, - 0. Since we are going to deal with one secondary system mode (oscillator) at a time, let us drop the subscript a, and denote the three oscillator parameters by us, c , and r,.

In developing an equivalent SDOF system for the present analysis, no equivalence for the ~ o i , , term exists; see Equations 6.13 and 6.17. To evaluate ~ o i , , , we need to know the properties of both the primary and secondary systems at the time the IRS are being developed. The main advantage of an IRS method is that the spectra for the secondary system input can be generated without any knowledge of the secondary system. To retain this advantage, therefore, let us set do;, = 0. Clearly, if the static constraint offered by the secondary system significantly affects the coupled response of the primary system, the IRS method (present or conventional) cannot be used. On the other hand, we have discussed earlier the effect of static constraint on the coupled response of the secondary system, which can be quite important and the present method is capable of accounting for it. It is shown later that the static constraint can introduce significant stresses in the secondary system even when it does not significantly affect the coupled primary system response.

Let us now proceed to define the secondary system oscillator, whose frequency and damping ratios are o , and c , , respectively. Let us assume that the oscillator has a mass m,. For the normalized 'mode shape,' $, = 11 Jm,, u, = 1 , y, = Jm, . Equation 6.9 gives

Equation 6.54 presents a problem. We intend to develop the IRS for specified mass ratios, just as the conventional IRS are developed for specified damping ratios. For a given oscillator mass, Equation 6.54 would give a different mass ratio with respect to each primary system mode. The solution of this dilemma is as follows. The purpose of defining the r,or m,is to take into account interaction between the primary and secondary systems. Interaction will be most significant with the primary system mode. I, whose frequency o,, is closest to the oscillator frequency o,. Therefore, let us assume that the specified mass ratio is between oscillator and the primary system mode I. This assumption in conjunction with Equation 6.54 give the oscillator mass

Values of r, for other primary system modes can be calculated from Equation 6.54. In the conventional IRS method, r, = 0, for each IRS curve cs is held constant and w, is varied. In the present case, r, # 0. For each IRS curve, we shall hold r, and 5, constant, and then vary w,. For each new set of r, and c,, a new IRS will be obtained. It is obvious that in the course of obtaining a set of several IRS curves, the computer program will have to solve a large number of these coupled

problems. There is, therefore, a need to develop an efficient computational algorithm.

Let us denote the coupled eigenvalue by 1, where 1 = -c w + ioD, w , = w J(l - c2); in which w and ( are the appropriate coupled frequency and damping ratio, respectively. Equations for the eigenvalue and eigenvector here have been adopted from Section 6.3. Equations derived in Section 6.3 show that there will be some change in all the eigenvalues of the primary system due to coupling. However, the changes are usually small, except in the eigenvalue of the mode 'I' whose frequency is very close to the secondary system frequency. Therefore, we can write

We ignore the interaction between the secondary system and all the primary system modes other than the Ith mode when evaluating the eigenvalues hi and A,. Modifying the appropriate Equations 6.23 and 6.30 we get

+ r, (of + 2 0 , C,, 31,) A: = 0. (6.57)

Equation 6.57 is quartic and can be solved exactly. The solution would yield two pairs of conjugate eigenvalues.

For each uncoupled primary system mode i, excluding i = I, there is a coupled eigenvector which can be expressed in terms of the transformed coordinates X:

and

r,"' (of + 2 0 , c1 hi) xs -

wf + 20 ,c sh i + h;

There are two more eigenvectors corresponding to the Ith primary system mode and the SDOF oscillator, one for each pair of A, obtained from Equation 6.57.

rj"' (of + 2wS cl X I ) x, =

o;, + r, o: 4- 2(w,<, + rjos<,) XI + Ah:

for all values of j.

6.8 Evaluation of instructure spectral quantities The coupled eigenvalue and eigenvector information developed above can be used to obtain the instructure spectral quantities and other related information. We are interested in the displacement of the connecting DOF u, and that of the

j : I / I SDOF oscillator us. In the coupled mode corresponding to the ith uncoupled i primary mode, the complex modal values of u, and us (w, and yrs, respectively) can 1 I

be calculated based on Equations 6.2 and 6.58 i :I

RESPONSE OF SECONDARY SYSTEMS/111 I

Similarly, in the coupled mode corresponding to the uncoupled oscillator mode, we can write, based on Equations 6.2 and 6.59,

i I

I ! '

L I I

The summation in Equation 6.61 is on all the primary system modes. In the response spectrum method for nonclassically damped systems, Chapter 5, each complex mode shape and its conjugate give two response vectors

and

where s:, and S:, are the spectral displacement and velocity for the ith coupled modal frequency from the displacement and the velocity spectra of the input motion for the primary system; S& and S;, are the corresponding values for the coupled modal frequency of the oscillator; x, and Xs are the complex conjugate of 1 , and I , , respectively. The terms Fl and F, are defined below.

a, = 2ki (1 + x f ) + 2 ~ , , ~ < ~ , + 2 ~ ~ 6 ~ (r:l2 - x,)~,

The spectral displacement S, is the relative displacement of the oscillator with respect to the connecting DOF.

Sc = us - uC. (6.64)

The values of spectral displacements corresponding to various values of uc and us in Equation 6.62 can be evaluated based on Equation 6.64, and denoted by

I' s:, S:,, s fs, S:$. To simplify the notations, we shall use the subscript i to include all the coupled modes, henceforth. Thus, all the instructure spectral displace-

I ments can be denoted by Sf, and S:,.

It can be seen from Equations 6.58-6.63 that when m, = 0, expressions of the form 0/0 are obtained. This problem can be easily remedied by either carrying on many of the algebraic substitutions, and in the process, eliminating all the problematic m, (or, r,, rJ) terms; or by normalizing the x, and xj values in Equations 6.58 and 6.59 by dividing by rfI2 and r;l2, respectively, and by making corresponding changes in Equations 6.60-6.63.

The modal combination equation for a nonclassically damped system is given by

in which R represents the maximum probable combined response, R: is the response in the ith mode from the displacement spectrum, and R: is that due to the velocity spectrum. The terms E:, E: and p,) are various correlation coefficients. Equation 6.65 is the same as Equation 5.40. In order to compress the notations, we define E$, in which the superscript a can be either d or u, and b can also be d or v , independently. Further, sf = s t , s r = ~f = -plJ, and

= - plI. Equation 6.65 can be rewritten as

Based on Equations 6.52 and 6.53 we have

The term S;,, (a = d or v) represents the maximum value of F,, for an oscillator with frequency o,, and damping ratio (,, in the ith coupled mode. Previously it was denoted by S:', and S:, without the subscript a, because at that time only one oscillator was under consideration. Equations 6.66 and 6.67 give


Let us define

( U C ) ~ = 1 1 1 &$ ' : J 9

r j a b

t w2 - 1 1 1 &I XI. s:Ja. i j a b

and


in which

Equation 6.70 constitutes the modal combination equation for the IRS method. The desired spectral values along with various correlation coefficients are defined in Equation 6.69.

In an IRS approach the properties of the secondary system (o,, C,,, r,) are not known in advance. In the conventional method, r, is assumed to be zero, and IRS curves are drawn for several values of C,, by varying o,. In the proposed method r, # 0, the spectral curves are drawn for several sets of (r,, C,,) values by varying o,. Another variable is the connecting DOF. Separate IRS are evaluated for each connecting DOF. Basically, for each point of an IRS curve, a coupled analysis of the type described earlier is to be performed. The data from the coupled analysis are saved to go back and evaluate the correlation coefficients, see Equation 6.69.

Each of the multitudes of the coupled analyses described above would give a set of u t values. Which set should we use in Equation 6.69, and elsewhere? We feel that the u: values from an uncoupled primary system analysis are a reasonable approximation. Note that the primary system, itself, is classically damped; therefore, the uncoupled analysis would give u:, = 0.

6.9 Examples of instructure response spectra The same six-story building used earlier (Chapter 5, and Sections 6.4 and 6.5), as the primary system in Reference [17] was used to illustrate the present IRS method. This building was subjected to twelve different earthquake motions, and in each case, total acceleration time histories at various floor levels were evaluated, assuming a 7% damping ratio for the building. These time histories were used to obtain instructure response spectra for those floors, using an oscillator (secondary system) damping of 2%. These IRS correspond to a mass ratio of zero. The IRS for nonzero mass ratio were not calculated using the time- history analysis because that would require a great number of coupled time- history analyses. The IRS for the same twelve earthquakes were obtained using the present method directly from the displacement and velocity spectra of these earthquakes. Note, in the proposed method we do not need to know the earthquake time history. We need only the earthquake displacement and velocity spectra. If the velocity spectrum is not known, it can be estimated from the displacement spectrum, see Chapter 5.

Figure 6.2 shows a comparison of the time-history generated IRS with those generated using the proposed method, the El Centro (Imperial Valley, SOOE, 1940) earthquake. Clearly, the agreement between the spectra from the present method and those from the time-history analyses is very good. Similar agreement between the two sets of spectra was observed for the other eleven records in Reference [I 71.


Fig. 8.2 Instructure response specirum (IRS). (a) fint floor, @) third floor, (c) rdp floor, El Centro earthquake (SOOE, 1940); primary system damping 7%; secondary system damping 2%. mass ratio 0[17].

Damping Values

Frequency, Hz

Fig. 6.3 Instructure response spectra (IRS). top floor, El Centro earthquake (SOOE, 1940); mass ratio 0[17].

The effect of damping ratios on IRS is illustrated in Figure 6.3. Both IRS in Figure 6.3 are at the top floor of the building for the El Centro earthquake. In one case the primary and secondary system damping ratios are 7% and 2%, respectively; and in the other case 1% and 0.5%, respectively. As would be expected the spectral peaks are much higher for the case with lower damping values than those for the case with higher values.

The effect of mass ratio on IRS is shown in Figure 6.4. Three IRS with mass ratios of 0,O.O 1 and O.'1 are compared. All of the three spectra are at the top of the building and for the building damping ratio of 1% and the oscillator damping ratio of 0.5%. The higher mass ratios result in lower spectral peaks, as one would expect.

6.10 Correlation coefficients The present IRS method requires evaluation of three sets of correlation coefficients, which are defined in Equation 6.69. When the secondary system does not offer any static constraint, the support motions do not stress the secondary system, and the corresponding correlation coefficients need not be evaluated. Theoretically, the procedure of evaluating the correlation coefficients is straightforward. However, it involves handling of a large amount of data at the time of the evaluation of the correlation coefficients, and then transmittal of voluminous data to the user of the proposed method. An approximate algorithm

I for the evaluation of these coefficients is presented here.


Mass Ratio - 0

Frequency. Hz

Fig. 6.4 Instructure response spectra (IRS), top floor, El Centro earthquake (SOOE, 1940); primary system damping 1%; secondary system damping 0.5% [17].

The correlation between the displacements at various connecting DOF can be evaluated from Equation 6.69, and it does not require a great deal of storage for transmittal. We shall concentrate here on E , , , ~ ~ ~ , which is the correlation in the response of the oscillator at the connecting DOF cl in ath frequency with that of the oscillator at the connecting DOF c2 in the Pth frequency; and on E, , ,~~ , which is the correlation between the displacement at the connecting DOF cl, and the response of the oscillator at connecting DOF c2 in Pth frequency.

Based on Equation 6.29, we can write

in which E* represents the correlation between the responses of two oscillators having frequencies and damping ratios of (w,,, c,) and (mSp, Cs8), assuming that the response is damped periodic; a,, is a rigid response coefficient identifying the steady-state content in the oscillator response. In Equation 6.72 both the oscillators are subjected to the same motion at the connecting DOF c. The new expression for E , , , ~ ~ , for the general case of two different DOF should, of course, give the same correlation value for the special case when the two connecting DOF are the same. We observed from the numerical data that the effect of the correlation between the two different connecting DOF can be approximately represented by the value of E,,,,, at two extreme ends of the oscillator frequency, when o,, is very low (- 0.01 Hz), and when w, is very large (- 100 Hz). At the

low frequency end, the oscillator response is primarily damped periodic and the response correlation is denoted by E:,,~. At the high frequency end, the oscillator response is rigid, and the support correlation is denoted by E:,,,. Both E : ~ ~ ~ and E:,,~ will become unity, when cl = c2. The proposed equation is

Clearly, Equation 6.73 gives Equation 6.72 when c l = c2 = c.

For low frequencies

and for high frequencies 'l

In developing a'n expression for E , , ,~~ , we note that at higher frequency, oSg, we have E~~~~~ = EclcZBB = E:,,~. We propose that at lower frequencies the correlation between the motion at cl and the oscillator response at c2 will diminish in proportion to the rigid response coefficient aCzg.

Several sets of correlation coefficients given by Equations 6.73 and 6.74 were compared with those obtained numerically from Equation 69 in Reference [17] and a reasonably good agreement was shown. A few of these are shown here in Figure 6.5.

6.1 1 Response examples The same primary-secondary systems used in Chapter 5, and in Sections 6.4 and 6.5, were used in Reference [17] to illustrate the response values from the present IRS method. See Figure 6.1 for the node and the element numbers. Results from the analysis of all nine systems for the El Centro (SOOE, 1940) ground motion using the time-history method, conventional floor response spectrum method and the coupled response spectrum method are given in Section 6.5, Tables 6.4 and 6.5. We will compare those results here with the results from the present IRS method. Tables 6.6 and 6.7 give a comparison of nodal displacements and element spring forces, respectively, from the present IRS method and the conventional floor response spectrum method with those from the time-history method. Percent errors in the results from the two methods are also shown in the same tables, using the time-history results as the reference or standard values. It is clear that the results from the present IRS method are much closer to the time- history results than those from the conventional floor response spectrum method.

We note here that the proposed IRS method is in fact an approximation of the coupled response spectrum method. The response spectrum method is a well- established method of seismic analysis. It can be looked upon as a tool to evaluate an average seismic response for design purposes. Normally, a good correspon-

RESPONSE OF S E C O N D A R Y SYSTEMS/119 I

I I

Proposed o

Numerical - C1-3. C2-6. We-1 Hz

1 10 100

Frequency. HZ

-1 I I I 1 10 100

Frequency. Hz

Proposed o

Nwnerkal - C1-J. C2-6, y-10 Hz

Frequency. Hz

-1 1 I I I

1 10 100

Freauency. Hz

g

Fig. 6.5 Comparison of correlation coefficients from the numerical procedure and the approximate equation, El Centro earthquake (SOOE, 1940); primary system damping 7%; secondary system damping 2%; mass ratio 0[17].

- - 111 - t 0' 0

dence between the time-history results and the response spectrum results exists. A total agreement between the two results is neither intended nor required. We believe the orders of mean percent errors and the standard deviations of the percent errors for the proposed IRS method in Tables 6.6 and 6.7 are well within the acceptable margins.

A better measure of the accuracy of the present IRS method is given in Table 6.8 in which the results from the present IRS method are compared with those from the coupled response spectrum method. The errors in response values, especially in the spring forces obtained from the proposed method are very low.

There may be an impression[l9] that when the correlations between responses from various support motions are accounted for, we need not consider

Proposed o ,

Numerical -

Table 6.6 Comparison of nodal displacements from the present method and the conventional floor response spectrum method[l7]

Present IRS Conventional floor method response spectrum method

Time-history Case Node displacement (in) Displacement (in) % error Displacement (in) % error


the effect of support displacements. It is not always true, as was indicated by Equations 6.5 1 and 6.52 and the related discussion. When the secondary system applies static constraint on the primary system, it also develops stresses due to support displacements. This particular point is illustrated in Table 6.9. The effect of the support displacements is particularly noticeable on the spring forces in element 2. Clearly, the effect of support displacement should be considered along with the response values from the IRS.

References 1. J. Penzien and A.K. Chopra, Earthquake Response of Appendages on a Multistory

Building, Proceedings. 3rd World Conkrence on Earthquake Engineering, Vol. XI, New Zealand, 1965.

2. J.M. Biggs and J.M. Roesset, Seismic Analysis of Equipment Mounted on a Massive Structure, in Seismic Design for Nuclear Power Plants, Ed. by R.J. Hansen, MIT Press, Cambridge, Massachusetts, 1970.

RESPONSE OF SECONDARY S Y S T E M S I ~ ~ ~

Table 6.7 Comparison of spring forces from the present IRS method and the conventional floor response spectrum method[l7]

Present IRS Conventional floor method response spectrum method

Time-history spring force Spring force Spring force

Case Element (kips) (kips) % error (kips) % error


Table 6.8 Comparison of nodal displacements and spring forces from the present IRS method and the coupled response spectrum method[l7]

Present IRS method Present IRS method Coupled response Coupled response spectrum method spectrum method displacement Displacement spring forces Spring force

Case Node (in) (in) % error Element (kips) (kips) % error

I I 1.20 1.13 -6 1 148 147 - I 2 1.3 1 1.24 -5 2 48.8 46.1 - 6

3 9 1.3 86.3 - 5 2 1 1.39 !.37 -1 I 134 132 - I

2 1.5 1 1.48 -2 2 25.8 25.9 0 3 106 100 - 6

3 I 1.79 1.84 3 I 124 124 0 2 1.89 1.95 3 2 15.3 16.0 5

3 1 1 1 108 - 3 4 I 2.20 2.32 5 I 1 16 117 I

2 2.30 2.44 6 2 10.4 11.2 8 3 109 106 - 3

5 1 2.56 2.75 7 I 105 106 I 2 2.64 2.86 8 2 7.9 1 8.68 10

3 100 98.5 - 2 6 I 4.16 3.39 - 19 I 173 165 - 5

2 4.27 3.39 - 2 1 2 10.4 10.2 - 2 3 166 167 1

7 I 4.25 3.52 - 17 I 2 13 204 - 4 2 4.36 3.52 - 19 2 16.5 16.2 - 2

3 204 206 I 8 I 4.94 4.02 - 19 I 362 343 - 5

2 5.07 4.02 -21 2 38.4 35.2 - 8 3 344 343 0

9 I 3.1 1 2.52 - 19 I 87.1 83.9 - 4 2 3.17 2.50 -21 2 4.94 5.61 14

3 85.1 86.5 2


Table 6.9 Response components from the present IRS method[l7]

Displacement (in) Spring force (kips)

Instructure support Instructure Support Case Node resoonse soectrum motion Combined Element reswnse s~ectrum motion Combined

K.K. Kapur and L.C. Shao, Generation of Seismic Floor Response Spectra for Equipment Design, Speciality Conkrence on Structural Design ofNuclear Power Plant Facilities, ASCE, Chicago, Illinois, 1973. M.P. Singh, Generation of Seismic Floor Spectra, Journal of Engineering Mechanics Division, ASCE, Vol. 101, No. EM5, Proceedings Paper 1 165 1, October 1975, pp. 593-607. M.P. Singh, Seismic Design Input for Secondary Structures, Journal of the Structural Division, ASCE, Vol. 106, No. ST2, Proceedings Paper 15207, February 1980, pp. 505-5 17. KIA. Peters, D. Schmitz and U. Wagner, Determination of Floor Response Spectra on the Basis of the Response Spectrum Method, Nuclear Engineering and Design, Vol. 44, 1977, pp. 255-262. J.L. Sackman and J.M. Kelly, Equipment Response Spectra for Nuclear Power Plants, Nuclear Engineering and Design, Vol. 57, 1980, pp. 277-294. G.C. Ruzicka and A.R. Robinson, Dynamic Response of Tuned Secondary Systems, Structural Research Series, No. 485, Department of Civil Engineering, University of Illinois, Urbana, 1980. R. Villaverde and N.M. Newmark, Seismic Response of Light Attachments to Buildings, Strucrural Research Series, No. 469, Department of Civil Engineering, University of Illinois, Urbana (I 980). J.L. Sackman, A. Der Kiureghian and B. Nour-Omid, Dynamic Analysis of Light Equipment in Structures: Modal Properties of the Combined System, Journal of Engin- eering Mechanics, ASCE, Vol. 109, February 1983, pp. 73-89. A. Der Kiureghian; J.L. Sackman and B. Nour-Omid, Dynamic Analysis of Light Equipment in Structures: Response to Stochastic Input, Journal ofEngineering Mechanics, ASCE, Vol. 109, February 1983. A.G. Hernried and J.L. Sackman, Response of Equipment in Structures Subjected to Transient Excitation, Report No. UBC/SEM, University of California. Berkeley, 1982. A.K. Gupta. Seismic Response of Multiply Connetted MDOF Primary and Secondary ~ ~ s t e h s . Nuclear Engineering and Design, Vol. 81, September 1984, pp. 385-394. A.K. Gupta and J.W. Jaw, Complex Modal Properties of Coupled Moderately Light Equipment-Structure Systems, Nuclear Engineering and Design, Vol. 91, January 1986, pp. 153-1 59. A.K. Gupta and J.W. Jaw, Coupled Response Spectrums Analysis of Secondary Systems Using Uncoupled Modal Properties, Nuclear Engineering andDesign, Vol. 92, March 1986, pp. 61-68. A.K. Gupta and J. W. Jaw, CREST, A Computer Program for Coupled Response Spectrum Analysis of Secondary Systems, User's Manual, Department of Civil Engineering, North Carolina State University, Raleigh, June 1985.' A.K. Gupta and J.W.'Jaw, A New Instructure Response Spectrum (IRS) Method for Multiply Connected Secondary Systems with Coupling Effects, Nuclear Engineering and Design, Vol. 96, September 1986, pp. 63-80. T. Igusa and A. Der Kiureghian, Dynamic Response of Multiply Supported Secondary Systems, Journal of Engineering Mechanics, ASCE, Vol. 111, No. 1, January 1985, pp. 20-4 1. A. Asfura and A. Der Kiureghian, Floor Response Spectrum Method for Seismic Analysis of Multiply Supported Secondary Systems, Earthquake Engineering and Structural Dvnamics. Vol. 14. 1986. pp. 245-265. A.K. Gupta and J.M. Tembulkar, Dynamic Decoupling of Multiply Connected MDOF Secondary Systems, Nuclear Engineering and Design, Vol. 81. September 1984, pp. 375-383. IMSL, Inc., International Mathematics and Statistics Library, 1979.

Chapter 7JDecoupled primary system analysis

7.1 Introduction We have pointed out in Chapter 6 that for various practical reasons it is customary to perform the analyses of the primary system (building) and the secondary system (HVAC, piping, equipment, etc.) separately, or to assume that the two systems are uncoupled. It is shown there that the uncoupled analysis of the secondary system introduces considerable error in the seismic response on the conservative side. A method of performing the analysis of the coupled system is presented in Chapter 6, which is based upon using an approximate technique to evaluate the coupled mode shapes and frequencies. In fact the same method can be employed to give the coupled response of both the primary and secondary systems. There is another practical problem, however. in chronological order, the design of primary system precedes the design of the secondary system. At the time the primary system is being designed, only tentative information, if any, about the secondary system is available. Therefore, the uncoupled analysis of primary system is a fact which cannot easily be altered. There is a need to have approximate criteria which can be used to evaluate the effect of decoupling on the primary system response.

A decoupled analysis of the primary system can be rationally justified if the decoupling results in a relatively insignificant error in the response calculation. A necessary condition for small error in response would be a small change in the frequencies of the primary system. In practice, this has also been taken to be the sufficient condition.

Lin and Liu[l], United States Nuclear Regulatory Commission[2] and RDT Standard[3] came out with decoupling criteria about the same time (1974-75). They considered single-degree-of-freedom (SDOF) primary and secondary systems. Curves were presented between the secondary-primary system frequency ratio, rf, and mass ratio, r,, to designate regions in which decoupling could or could not be permitted. The three sets of curves are similar in shape. But in certain ranges of rf, r, values, they may lead to widely different results. These curves are characterized by abrupt discontinuities, lack sufficient justification, and to some they portray the dynamics as very arbitrary. In developing their relationships between r, and r , Hadjian and Ellison[4] actually formulated the frequency of the coupled SDOF-SDOF systems, thus leading to a much more rational approach. When the primary system, or both the primary and secondary systems have multi degrees of freedom (MDOF), approximate heuristic methods have been used to evaluate the equivalent mass ratios, which can then be used in

conjunction with the criterion developed for SDOF-SDOF systems. Gupta and Tembulkar[5,6] showed that it is not sufficient to limit the changes in frequency. They developed approximate relationships to predict the change in response for the SDOF or MDOF primary system connected to a SDOF or MDOF secondary system. Approximate procedures similar to that used in Chapter 6 were employed. The following treatment is primarily based on Gupta and Tembul- kar[5,6].

7.2 SDOF-SDOF system Consider the system shown in Figure 7.1. The primary system has a stiffness k,, mass m, and the frequency o, = J(k,/m,). The corresponding values for the

. secondary system are k,, ms and o,. The free vibration of this system is represented by the following equation:

As shown in Chapter 6, primary and secondary systems with unequal damping values give rise to nonclassically damped systems. The effect of the nonclassical nature of damping is likely to be much less significant on the primary system than on the secondary system. Therefore we use the undamped free vibration equation.

If the coupled frequencies are represented by R , and we write RF =

Rlw,, r, = ws/op, r, = mS/mp, the characteristic equation can be written in the following form:

Equation 7.2 will give two positive values of RF, and hence R, the frequency of the coupled system. If decoupling has to be achieved, one of those frequencies should be close to o,, the corresponding R F close to 1, say R: = 1 + E, where E is small. The other mode is not likely to contribute significantly to the response. Substituting ( I + E) for R: in Equation 7.2, we get

Equation 7.3 represents relationship between r, and r, for any change in frequency. For example, for a 10% change in frequency, R , = ( 1 k 0.1 ) = 1.1,

Fig. 7.1 An undamped SDOF-SDOF primary-secondary system.

D E C O U P L E D P R I M A R Y SYSTEM A N A L Y S I S I ~ ~ ~

Mass Ratio r,

Fig. 7.2 Frequency and mass ratio (r, versus r,) curves for 5%, 10% and 15% changes in the natural frequency [ 5 ] .

0.9, R: = 1.21, 0.81, E = 0.21, -0.19. Note, a positive E means that the frequency of the coupled system is greater than that of the uncoupled system.

The rf versus4r,,, curves for 5%, 10% and 15% changes in the value of o, are shown in Figure 7.2. Roughly, E is positive for values of rf < I, and negative for rf > 1. The region on the left side of any curve will assume an error less than that used for the particular curve, As &*d expe&, to limit the error, r, should be small and rf away from unity. The obtained by Hadjian and Ellison[

The uncoupled SDOF p r i m a j system equal to the spectral displacement coupled system is ,I. 1

1 c I {:} = {" - t 2 m s } = rnSwi 1 !vi 1 :

' ! Using the standard MDOF formulation, the relative displacement of the primary mass in the coupled system can be shown to be

If we assume that the spectral acceleration SD does not change significantly between the frequency of the uncoupled primary system and the coupled system, the response ratio RR becomes

The 2DOF coupled system considered here will have two modes, and hence two values of RF for any set of r,, r, values. It can be shown that for both these modes, the response ratio, R R , is always less than unity, i-e., the response of the primary mass in the coupled analysis in any one of the two modes is always less than that in the uncoupled analysis. Figure 7.3 shows r, versus r, curves for lo%,

20% and 30% reduction in the primary system response (RR = 0.9,0.8,0.7) in the mode for which RR is closest to unity. For each reduction value there are two curves. The 'acceptable' domain (which would limit the reduction) is below the bottom curve and above the top curve for any given reduction.

Mass Ratio r,,,

Fig. 7.3 Frequency and mass ratio (r,,, versus r,) curves for 10%, 20% and 30% changes in response values[5].

Let us make a few observations concerning the frequency curves of Figure 7.2 and the response curves of Figure 7.3. If the mass ratio is sufficiently small, one can limit the change in frequency even in a tuned system, r, = 1. Apparently, one cannot control the change in response at rf = 1. All the curves are asymptotic to the r, = 1 line. This particular aspect will be discussed in more detail later. In the r, > I region, the response ratio is not adversely affected by a large secondary system mass, even by what could be considered a ridiculously high mass. However, such a system could not be decoupled because of the restriction on the change in frequency. Further, the change in frequency which would accompany such large secondary masses, would invalidate the assumption that the spectral displacement S, did not change. In the r, < 1 region, on the other hand, the secondary system becomes more and more isolated as its frequency decreases, hence the changes in frequency and the response both remain small even if its mass is large.

To investigate what would happen to the response if we uncoupled the secondary system based on an assumed 10% change in the frequency, RF = 0.9 or 1.1, we picked up a number of r,, rm points from the corresponding curve in Figure 7.2. These rf, r,,, values are summarized in Table 7.1. For each r,, rm point we have two modal frequencies of the coupled system or two RF values, viz. RF,

Table 7.1 Coupled versus uncoupled response ratios[5]

Combined R,

Case r, rm RFI R n R R I RRZ Sum SRSS

and RF2. RF, = 0.9 is accompanied by a higher R,, and RF2 = 1.1 by a lower RFl . Note, RFl RF2 = r ~ . Ratios of responses of the primary mass in each mode of the coupled systems with that of the uncoupled primary mass are also given in the table. Normally, we consider only one of the two modes for which RR is greater and ignore the other. (Indeed, if the r,,, and rf values are selected on this basis, as they are in Figure 7.3, the discarded RR would be small.) The two values of RR are combined algebraically and by the SRSS (square root of the sum of the square) rule. The RR, , RR2 values and their two combinations are also plotted in Figure 7.4. The algebraic sum is practically unity, which should be expected, since it essentially means that the inertia is preserved in the two modes. It also means, that in tuned or nearly tuned cases, where the algebraic sum can be justified, uncoupling may be permitted. However, as one goes farther from the tuned case, the SRSS would be a more appropriate combination rule. Case 8 has an SRSS value of 0.78 and Case 13 has 0.72. This would indicate that the uncoupled system can over-estimate the response by up to approximately 3896, i.e., when the decoupling is based on frequency change only. The present estimate of error will be further influenced by possibly different S,values between two coupled modes, which are assumed to be the same here.

7.3 MDOF-MDOF systems The approximate equations of Chapter 6 can also be used here. Since we are dealing with the undamped systems, the equations of Chapter 6 are simplified. Equation 6.23 becomes:

in which R, denotes the frequency of the coupled system corresponding to the ith uncoupled mode, o,, is the frequency of the ith uncoupled primary system mode, w,, is the frequency of ath uncoupled secondary system mode, and r,,, is the energy-mass ratio between the ath secondary system mode and the ith primary system mode defined in Chapter 6. Equation 6.23 also has a ~ o i , , term which accounts for the static constraint offered by the secondary system. It is assumed here that such constraint is negligible. Denoting RF; = R,/wp,, r/,, = oSa/op,, Equation 7.7 can be rewritten as[6]:

Equation 7.8 can be solved iteratively for the desired RFi value. The right-hand side of Equation 7.8 is likely to be dominated by a few terms for which (r;, - R:~) values are relatively small. To avoid iterations, one may use the equation with only one term for which (ria - R:,) is the smallest, leading to what is called the

DECOUPLED PRIMARY SYSTEM ANALYSIS/131

I I I * 1 .o 2.0 3 0

Frequency Ratio, rc

Fig. 7.4 Variation in response ratios and their combinations with the frequency ratio[5].

single mode approximation or the SMA. The summation sign in Equation 7.8 is dropped. The modified or the SMA equation is

R:, - ( 1 + r;. + rEla r/,J R:, + r;,. = 0. (7 .9)

The advantage of using Equation 7.9 is that it allows the use of r~ verses r, curves, see Figure 7.2, for the SDOF-SDOF systems.

If the ith normalized modal vector of the uncoupled primary system is &, the participation factor is y, and the spectral displacement is S,,, the response

displacement vector is given by y, $,SD,. It is assumed that in the coupled system, the primary system mode shape remains approximately the same as that in the uncoupled system, except that it is no longer normalized. Say $ j represents the primary part of the normalized coupled system mode shape. The $ j = A $,., where A is a scalar. Let the corresponding participation factor be y:, and assume that S,, remains the same because the change in frequency is small. The primary system displacement response in the coupled system becomes y; $hSDi. The response ratio is given by

The participation factors are

where U,, and Ub6 are the displacement vectors of the primary and secondary system, respectively, when the primary system support displaces by unity in the direction of the earthquake; $,i and @ii are the secondary system displacement vectors corresponding to the primary system vectors $,. and +;;. To evaluate the scalar A we note,

Equation 6.20 yields

in which $,, is the ath uncoupled secondary system mode, $,i is part of the ith uncoupled primary system mode containing the connecting degrees of freedom only, and y, is a row of participation factors for each connecting degree of freedom. Equations 7.10-7.13 give

D E C O U P L E D P R I M A R Y SYSTEM ANALYSIS1133

in which r,,, is the inertia mass ratio[6]

The numerator and the denomintor of Equation 7.15 represent inertia forces for the secondary and primary system, respectively. Further, for a SDOF-SDOF system, Equation 7.15 becomes the square root of the familiar mass ratio, J(rn,/rn,); hence the terminology.

Equation 7.14 gives the desired response ratio. As for the frequency ratio, we can use the single mode approximation (SMA) for the response ratio also. Unlike the frequency ratio, however, the SMA does not allow the use of the SDOF- SDOF rfversus r,,, curves for the response change, Figure 7.3, due to the presence of the inertia mass ratio, r,i,, term in Equation 7.14.

7.4 Application of the frequency and response ratio equations The frequency ratio and the response ratio equations were applied to the primary-secondary systems presented in Chapter 5, by Gupta and Temulkar[6]. In Chapter 5, only one coupled configuration was used. On the other hand, in Reference[6], three different configurations were used to investigate the applicability of the frequency and response ratio equations to different circumstances. The three coupled systems configurations are shown in Figure 7.5 and are designated Models 1,2 and 3, respectively.

The secondary system has only one independent rigid body mode, but it has two connecting DOF. Therefore, we have an over-constrained system. As stated

Model 1 Model 2 Model 3

Fig. 7.5 Coupled primary-secondary system models[6].

earlier, the present formulation ignores the effect of this constraint, thus, in most cases slightly under-estimates the coupled frequency.

The effect of coupling on the fundamental mode of the primary system using the coupled Model 1 is investigated first. The secondary system mass and stiffness are so varied that a range of r,,, and rJ values are obtained but RF for the fundamental mode stays approximately 0.9 or 1.1. That amounts to a change of 10% in the frequency, which we consider a reasonable nontrivial change for illustration purposes and to be of a practical value. (A much smaller change would be trivial, and a much larger change may be beyond the range of applicability of the algorithm). Eight such cases are reported in Table 7.2 in which RF values are given. Both the iterative and the SMA solutions are in good agreement with the exact solution.

The response ratios for all the eight cases are given in Table 7.3. Since we assume that the mode shapes of the primary system do not change, one response ratio is calculated for each mode. In reality, however, there is some change in mode shapes. Therefore, we have an exact response ratio for each story in Table 7.3. Both the iterative and the SMA values of RR in Table 7.3 are in good agreement with the corresponding six values in the table.

All the six modes of the primary system were investigated for the coupled system Model 1 of Figure 7.4, case 3 of Tables 7.2 and 7.3. It was found that for higher modes, the changes in frequencies and responses were very small, RF and RR - 1. Therefore, in Tables 7.4 and 7.5 the RF and RR values for the first three modes only are reported. The present algorithms are able to predict the changes accurately to within 1%.

Next, the effect of coupling on the fundamental mode of case 3 for all the three coupled models in Figure 7.4 is investigated. The RF and the RR values are

Table 7.2 R, values for the coupled Model 1, fundamental model61

Secondary mode , RF

Present Exact

Case r, r r Iterative SMA

D E C O U P L E D P R I M A R Y SYSTEM ANALYSIS/135

Table 7.4 RF values for the coupled Model 1, Case 3 [6]

Secondary mode RF

I 2 Present Exact

Case rE rr '-E rr Iterative SMA

Table 7.5 Response ratios for the coupled Model I , Case 3[6]

Response ratio R,

Present Exact Primary Secondary mode mode rf r~ ri Iterative SMA Story l Story 2 Story 3 Story 4 Story 5 Story 6

O r i l n 222

o r i m 9 9 a! 0 0 0

o r i m S S a! 0 0 0

o'a- d Q I W d w m 0 0 -

" 8 8 86,:

0 d - 8 c r ; - 9 9 d o 0

- ( U M

reported in Tables 7.6 and 7.7. Two effects are clear. First, there is a clear difference in the R,and R,values for the three models. Second, the present algorithm can predict this difference. The agreement between the RF values from the proposed algorithm and the exact values is excellent. For response ratio, the agreement is reasonable, quite good for the top story, not so good for the bottom story.

References I . C.W. Lin and T.H. Liu, A Discussion of Coupling and Resonant Effects for Integrated Sys-

tems Consisting of Subsystems, Proceedings, Extreme Load Conditions and Limit Analysis Procedures for Srrucfural Reactor Safiguards and Containmenf Structures, Paper U3-3, Berlin, September 1975.

2. United States Nuclear Regulatory Commission, Standard Review Plan, Section 3.7.2, Seismic Sysfern Analysis, June 1975.

3. RDT Standard F9-2T. Seismic Requirements for Design of Nuclear Power Plants and Test Facilities, January 1974.

4. A.H. Hadjian and B. Ellison, Decoupling of Secondary Systems for Seismic Analysis, ASME-PVP Conference. Reprint No. 84-PVP-59, San Antonio, Texas, June 1984.

5. A.K. Gupta and J.M. Tembulkar, Dynamic Decoupling of Secondary Systems, Nuclear Engineering and Design, Vol. 81. September 1984. pp. 359-373.

6. A.K. Gupta and J.M. Tembulkar. Dynamic Decoupling of Multiply Connected MDOF Secondary Systems, Nuclear Engineermg and Design, Vol. 81, September 1984, pp. 375-383.

Chapter $/Seismic response of buildings

8.1 Introduction The majority of buildings are analyzed and designed in accordance with the building codes. An introduction to the historical development of the US and international building codes is given by Berg[l]. Typically, the earthquake loading is defined in terms of an equivalent lateral force, and a static analysis of the building is performed. In recent years, building codes have adopted more and more features of the formal dynamic structural analysis, while retaining their original formats. Perhaps the most popular and relatively rigorous building code presently in use in the profession is the Uniform Building Code (UBC)[2]. The seismic design and analysis requirements of the UBC are based on the procedures developed by the Structural Engineers Association of California (SEAOC)[3]. The developments of the UBC and the SEAOC requirements have been gradual and evolutionary in nature. In a somewhat different approach, the Applied Technology Council, a research and development organ of the SEAOC, under- took to produce a model seismic code development in 1974 under the sponsorship of the National Science Foundation. The effort was coordinated by the National Bureau of Standards. Nearly a hundred scientists and engineers contributed to the project report, 'Tentative Provisions for the Development of Seismic Regulations for Buildings', popularly known as ATC3-06, or simply as ATC3[4]. The report is being reviewed by engineers and building officials to evaluate its utility, cost and effectiveness in leading to earthquake resistant structures.

We will show in this chapter that the modern building codes, UBC, ATC3, and others, use seismic response calculation procedures which are closely related to the response spectrum method.

8.2 Analysis Consider the ith mode of vibration, with circular frequency a, and modal vector $,.. Let us assume that the vector is normalized, $ i T ~ $; = 1, where M is the mass matrix. Maximum displa&ment of the building in this mode is given in Chapter 1.

where y, = (pTM U,,, the modal participation factor, and S,, is the spectral acceleration corresponding to the frequency o,. As defined before, U, is the displacement vector of the building, when the base of the building is displaced statically by unity in the direction of the earthquake. For buildings with only a horizontal degree-of-freedom in the direction of the earthquake, U,, - 1, where 1 is a vector with every element equal to unity. The pseudo-static force which will cause the displacement given by Equation 8.1 is given by

According to Equation 1.15, K gi = of M $;. Hence,

Equation 8.3 gives the pseudo-static force, which, when applied to the building, will yield the displacement vector given by Equation 8.1. The elements of mass matrix M have mass unit, $i yiSA, terms have acceleration units. As such, we can also view the Fi force vector as a pseudo-inertia force vector. It is common in the modeling of simple buildings to lump the masses associated with a story at the corresponding floor level. The mass maxtrix M is defined as a diagonal matrix; the rth diagonal being rn: the mass of the rth story. The pseudo-inertia force of the rth story becomes

F : = mr@:~,SA,, w4) in which $: is the rth element of the vector $;. The corresponding base shear is

If V, is known, based on Equations 8.4 and 8.5, the story force can be'etatyated , . =

from

To define the base shear, Ifi, from Equation 8.5, we need to evaluate the participation factor y,, which can be written as

SEISMIC RESPONSE OF BUILDINGS/ 141


When the vector +i is not normalized, +: M gi # 1, we can easily show that Equation 8.8 becomes

Equation 8.6 does not change, whether the vector is normalized or not. Equations 8.6 and 8.9 can also be written in terms of story weights w r = mrg,

Let us define an effective modal weight Wi, such that

which is a modified form of Equation 8.1 1. It can be shown that the sum of all the

modal weights, W,, is equal to the total weight of the building W. 1

Equations 8.10 and 8.12 constitute the basis of evaluating the modal base shear and the modal story forces in ATC3 in its modal analysis procedure. The responses from several modes are combined by the square root of the sum of the square (SRSS) method (see Chapter 3). The expression SAl/g is termed the modal seismic design coefficient. We shall discuss this coefficient in greater detail later.

We shall now present the 'equivalent lateral force procedure'. In the design of conventional buildings, much of the seismic force comes from the fundamental mode. For buildings with relatively small periods (TG 0.5 sec,/ 3 2 Hz), the fun-

Concentrated Forc

(a) Linear

Fig. 8.1 Distribution of story forces.

'.. \, *

'-.\ \

Parabolic )$.. C -

Distribution v \ - ';\., L \-

Linear Distribution 1 ',

(b) Parabolic

damental mode shape can be approximated as a straight line, 9; cc kt, where hr is the story height about the base of the building, Figure 8.l(a). For tall buildings, which are relatively flexible and have higher periods ( T 3 2.5 sec, f 6 0.4 Hz), the fundamental mode lies approximately between a straight line and a parabola with a vertex at the base. In the high period buildings, the influence of higher modes of vibration can be significant. To account for both effects-the nonlinear shape of the mode and the higher modes, ATC3 recommends a parabolic shape of the fundamental mode, t$: a (hr)', Figure 8.l(b). As is shown in the figure, the curved mode shape transfers greater force to the top stories. In general, ATC3 recommends, 9: a (hr)", where n is an exponent whose values vary from 1 to 2 between 0.5 3 T 2 2.5. In ATC3's equivalent lateral force procedure, Equations 8.10 and 8.12 become

Note in Equation 8.13, U7, is assumed to be equal to the total weight U'. In the UBC, the effect of a higher period of the fundamental mode shape and

the effect of higher modes is accounted for by a concentrated force F, applied at the top story.

The fundamental mode shape is assumed to remain linear. But only V - F, is applied to various stories. The UBC version of Equation 8.13 is

SEISMIC RESPONSE O F BUILDINGS/143 / I

F r = w r h r SA

( V - F , ) , V = W - . 9

The overall effect of the ATC3 and UBC equations should be comparable. 1 The story shear is calculated by summing all the story forces above a t~

I particular story. The structural elements within a particular story receive shear , due to another source, the torsional moment which is generated due to eccentricity between the centers of rigidity and mass. ATC3 recommends using

F

an accidental eccentricity of 5% of the building plan dimension perpendicular to the direction, which is in addition to the known eccentricity, which may exist due to designed architectural and structural features. ATC3 requires analysis for two orthogonal horizontal components of earthquake separately. The effects of the two components are combined before design. In the UBC procedure, on the other hand, the design for the two horizontal components is done independently-the effects of the two components are never combined. In the UBC procedure, the accidental eccentricity is assumed equal to 5% of the higher of the plan dimensions, irrespective of the direction of the earthquake under consideration. Further, the known designed and the accidental eccentricities are not combined in the UBC. The torsional moment due to the greater of the two is considered.

In the present UBC, the overturning moment is evaluated statically from the calculated story forces. In the earlier versions, a reduction for out-of-phase multiple modes was permitted. This reduction was eliminated from UBC after SEAOC dropped it in 1970, at least in part because of bad results observed in the Caracas, Venezuela earthquake of 1967. In that earthquake, several tall buildings designed in compliance with seismic building codes similar to the UBC had exterior column damage attributed to overturning moment. The exterior column damage can be attributed to another effect not included in the UBC, as we mentioned earlier, viz., the combined effect of the two horizontal components of earthquake. This particular effect is accounted for by ATC3. Perhaps because of this mitigating effect, ATC3 does allow some reduction in overturning moments calculated statically from the story forces, to account for out-of-phase multiple modes. ATC3 does not allow any reduction for the top ten stories. Between the top ten and twenty stories a reduction factor which varies linearly between 1.0 and 0.8 is used. At the building foundations (inverted pendulum structures excluded), an overturning moment reduction factor of 0.75 is recommended by ATC3, in recognition of the observation that actual foundation overturnings or the lack of them in past earthquakes indicate that the statically calculated moments represent a significant over-estimation. To avoid overturning, ATC3 requires that the resultant of the seismic forces and vertical loads at the I

foundation-soil interface remains within the middle one-half of the base of the building. I

8.3 Building frequency In case the building configuration and the member sizes have already been worked out, the mode shapes and frequencies (periods) can be evaluated easily using the available analysis programs. In fact, the complete building response can be calculated using the response spectrum method, or using the equivalent ATC3 method. At the preliminary design stage, or in case of simple buildings when the designer chooses to perform manual calculations, an accurate estimate of the building frequency can be obtained using Rayleigh's method.

Let $ represent the assumed fundamental mode shape. The maximum strain energy associated with the mode shape is $TK$, and the maximum kinetic energy is 02 $T M $, when o is the circular frequency of the fundamental mode under consideration. Equating the maximum potential and kinetic energies, we get

Let F denote the force vector which would give the $ displacement vector, F = K $, Equation 8.16 becomes

or in terms of story masses, forces and displacements

To apply Equations 8.17 and 8.18, we assume a force vector F which approximately represents the distribution of the pseudo-inertia force in the building in the fundamental mode. A static analysis is performed to evaluate $ from F. The value of o given by these equations is very accurate and is not very sensitive to variations in the assumed vector F. Both ATC3 and UBC recommend approximate expressions for the building frequency, which can be used at preliminary stages of design.

8.4 Seismic coefficient The modal seismic design coefficient, S,,/gused in Section 8.2 is a nondimensionalized representation of the spectral acceleration. As we will show here, the coefficients used in ATC3 are directly evaluated from a Newmark type design spectrum, see Chapter 2. The coefficient used in the UBC has its roots in the

SEISMIC RESPONSE O F BUILDINGS1145

historic evolution of the pseudo-static design. Nevertheless, in its present form, the UBC seismic coefficient yields seismic forces which are comparable to those given by ATC3.

Figure 8.2 schematically shows a design spectrum on logarithmic scales. We shall first concentrate on the amplified acceleration region, BC, which has a constant spectral acceleration S,; and the amplified velocity region, CD, which has a constant spectral velocity So. The ATC3 coefficients are nominally based on a 5% damping ratio. ATC3 defines two parameters, the equivalent peak acceleration (EPA), and the equivalent peak velocity (EPV). The two parameters are obtained by dividing the respective S, and S, values by a constant factor of 2.5:

s a EPA = - S" EPV = -

2.5 ' 2.5 '

The EPA and the EPV parameters are not directly related to the corresponding maximum ground acceleration (a) and velocity (u) values. The two sets of values are of the same order of magnitude. In many cases, a rough equivalence between EPA and a, and between EPV and v can be seen, see Chapter 2. ATC3 further defines two nondimensionalized coefficients A, and A, based on EPA and EPV. The acceleration coefficient A, is numerically equal to the value of EPA in g units:

Period, T (Log Scale)

Fig. 8.2 A schematic diagram of elastic design response spectrum.

The velocity related acceleration coefficient, A,, is obtained by first establishing a relationship between the spectral velocity and spectral acceleration. It was observed that a 0.4 g of EPA is accompanied by 12 in sec-' of EPA on firm ground. Therefore, the acceleration related to EPV is (0.4 g) EPV/ 12 =

(EPVl30) g, when EPV is in units of in sec-'.The coefficient A, is then defined as

EPV A,=-.

30

ATC3 has provided maps giving the values of the coefficients A, and A, in the United States. These maps are based on an approximate probability of nonexceedance of 90% over a fifty year period. The values of A, and A, are assumed to vary between 0.05 and 0 .4 .

Given A, and A,, the spectral acceleration in the regions CD and DE of Figure 8.2 can be defined based on Equations 8.19-8.2 1.

In Equation 8.22, the value of spectral acceleration, S,, is inversely proportional to the period 7'. The period of a building goes up with the number of stories in it. Because of a number of reasons associated with the structural behavior of long- period buildings, ATC3 decided that the ordinates of the design spectra do not decrease as rapidly with T. Equation 8.22 was modified as follows:

The reasons for designing long-period buildings more conservatively include the following: (1) tall buildings have a higher number of the degrees of freedom, which increases the likelihood that the ductility requirements are concentrated in a few stories of the building; ( 2 ) the number of potential modes of failure increases with the number of degrees of freedom; (3) instability of a building is more of a problem for a taller building.

In the buildings with very long period ( T > 4 sec), we know that the spectral displacement becomes constant, see region DE in Figure 8.2. For this region, the spectral acceleration is inversely proportional to T 2 . In view of the conservatism intended for tall buildings, T 2 is replaced by T ~ ' ) .

3Avg s* = - T4/3 , for T 2 4 sec.

At T = 4 sec, Equations 8.22 and 8.23 give the same values of S,. ATC3 has defined three types of soil profiles. The following description is

based on Berg's[l] abbreviated version:

SEISMIC RESPONSE O F BUILDINGS1147

Soil type S,-Rock or shale stiff soil over rock; Soil type S,-Deep stiff soil over rock; Soil type S3-Soft soil.

As we have seen in Chapter 2, softer soils have a greater spectral amplification in the intermediate and higher period range, see region CDE in Figure 8.2. In the , amplified acceleration region, BC, the soft soil shows a smaller amplification,

I I

particularly when the ground shaking is strong. ATC3 has accommodated this observation by introducing a factor S, and other changes in Equations 8.23 and 8.24.

2.0 A, g, for S, soil when A, 2 0.3;

In the BC region of Figure 8.2, Equation 8.25 uses a reduction factor of 0.8 for S3 soil when A, is large (3 0.3). The soil coefficient S modifies the spectrum in the longer period region. The value of S for the three soil conditions is given as S = 1.0 for S, , S = 1.2 for St, and S = 1.5 for Sj.

Equation 8.25 gives an elastic design spectrum. As we discussed in Chapter 2, a majority of conventional buildings are designed to undergo a significant inelastic deformation when subjected to a strong motion earthquake. In effect, the building absorbs the earthquake energy inelastically by a relatively larger deformation and smaller strength than it would if it had to absorb the energy elastically. The consequence of this design philosophy is a reduced strength requirement, but a much greater emphasis on details which would make the inelastic deformation possible without a building collapse. The ratio of the maximum inelastic deformation in the structure to the yield deformation was defined as the ductility ratio, p, in Chapter 2. A method of obtaining an inelastic response spectrum from the corresponding elastic response spectrum was also presented. In Figure 8.2, the spectral ordinates in the CDE region are divided by the ductility ratio, p, and in the BC region the ordinates are divided by J(2p - 1). ATC3 replaces both p and J(2p - 1) factors by a single response modification factor, R. For calculating the inelastic deformations, another deflection amplification factor C,, is used, which is similar to R, but not the same. Table 8.1 gives a summary of the R-factors based on Berg[l]. Equation 8.25 becomes

1.2A,Sg 2.5Aag For T a 4 sec,SA = 6

R T = I 3 R ' 2.0 A, g

4 , for S, soil when A, 2 0.3; R

Table 8.1 Response modification factor, R. (Based on Berg[l])

Structural system Range of R

Bearing wall system (type 1) From 1 - 114 for unreinforced masonry walls to 6-112 for light framed walls with shear panels

Building frame system (type 2) From 1- 112 for unreinforced masonry shear walls to 7 for light framed walls with shear panels

Moment-resisting frame system (type 3) From 2 for ordinary moment frames of reinforced concrete to 8 for special moment frames of steel

Dual system (type 4)

Inverted pendulum structure

From 6 for braced frames to 8 for reinforced concrete shear walls

From 1-114 for ordinary moment frames of steel to 2-1/2 for special moment frames of either steel or reinforced concrete

3A,Sg For T 2 4 sec, S, = --- R ~ 4 / 3 '

ATC3 gives almost identical spectral equations for the equival'ent lateral load procedure, which is a single mode approximation, and for the modal analysis procedure.There are two exceptions. For the equivalent lateral load procedure, the extra equation for the longer period buildings ( T a 4 sec) is not allowed to assure sufficient conservatism in the approximate method. A reduced spectral value for tall buildings may also be viewed as an incentive to perform a modal analysis. The second exception is a reduction in spectral value in the short period range, AB in Figure 8.2, in the modal analysis procedure in modes other than the fundamental mode. The ATC3 equation for this range may be written as

A0 9 SA = - (0.8 +. 4.0 0, for S, soil when T < 0.3 sec.

R

When T is very small, Equation 8.27 gives SA -- 0.8 A, glR. The numerator of 0.8 A, g is roughly the maximum ground acceleration. The spectral value is further reduced by the response modification factor, R. As we pointed out in Chapter 2, the ductility does not reduce the required resistance in the high frequency (short period) range. Equation 8.27 is inconsistent with that observation. On the other hand, in most buildings, the response in modes other than the fundamental mode may not contribute significantly. Therefore, Equation 8.27 is unlikely to affect the design response values much.

* 111 ?j:i jj j

According to the UBC, the spectral acceleration can be written as I,:

in which %is a zone coefficient, I a n importance factor, Ka structural coefficient, i:

S a site-structure resonance factor, and C a seismic coefficient. The coefficient C 1 is specified as j:

*I

The UBC has five zones. the highest being seismic Zone 4 and the lowest Zone 0. The latter has no seismic design requirement. For Zones 4- 1, the values of Z are 1, 3/4, 3/8 and 3/16, respectively. There are three categories based on the importance of the building. For essential facilities the importance factor, I, is 1.5, for large occupancy buildings it is 1.25, and for all other buildings, 1 .O. The building coefficient, K, depends on the type of structure, ranging in value from 213 to 4 /3 for buildings and on up to 2.5 for elevated tanks. The K-factor, in effect, lowers the margin of reserve strength required for structural systems that have performed well in past earthquakes and raises the margin for systems that have performed poorly. The site-structure resonance factor is a function of the ratio of the building period to the site period and varies between I and 1.5. If the site period is not available, a factor of 1.5 must be used. However, the value of the product CSneed not exceed 0.14. The UBC is intended to be used with the working stress design method. The code allows a 113 increase in the allowable stresses. Conversely, the seismic loads can be multiplied by a factor of 314.

As we pointed out earlier, the Uniform Building Code is an evolutionary code, whereas, ATC3 was created from scratch based on current knowledge. Of course, the current knowledge which went into ATC3 includes the experience of the profession with the UBC and other codes. It will be interesting to compare the design values from the two codes. Let us consider the highest seismic zone (A, = A, = 0.4, Z = I ) , site type S, without soil-structure resonance ( S = I for ATC3 and UBC), and a moment resisting steel frame ( R = 8, K = 2/3). The ATC3 equations give

The ATC3 response value is intended for ultimate load design. For a steel frame, the capacity reduction factor is 0.9. T o obtain the nominal ultimate value we should divide the above I.'/ Wvalue by 0.9. T o compare the ATC3 value with the UBC value we need to convert the ATC3 value into an equivalent working stress value, which can be done by dividing the nominal ultimate value by a factor of safety of 1.7. Completing both the divisions, we get

'I

The UBC equations along with the 3/4 factor give

The V/ W values from ATC3 and UBC are compared in Figure 8.3, We observe that the ATC3 and UBC curves in Figure 8.3 are comparable.

Let us consider another example similar to the first one, except that we now have the site type S, in ATC3, and have the maximum soil-structure resonance for the UBC. Both give S = 1.5. Following steps similar to those above, we get

The two V/W values are plotted in Figure 8.4. Again, the two codes have comparable values.

Period, eec

Fig. 8.3 Comparison of ATC3 and UBC seismic forces, A, = A, = 0.4, Z - 1 , site type S,, S = I.

SEISMIC RESPONSE O F BUILDINGS/lSl

Period. sec

8

0 $ 8 - * 5 - E Q - Q 4 - F @ g 3 - U.

Fig. 8.4 Comparison of ATC3 and UBC seismic forces, A, = A, = 0.4, Z = I , site type S,, S - 1.5.

- UBC

AT C3

From the maximum seismic intensity zones to the lowest, the ATC3 A, and A,

values go down from 0.4-0.05, by a factor of 8; the UBC Z values go down from 1-3/ 16, by a factor of 5.3 only. At a given location, the actual design values will depend upon the respective maps. It is believed, however, that for low seismic areas, the UBC requirement is relatively conservative. One reason may be that the framers of the UBC seismic requirements, or those of its parent-the SEAOC, paid much greater attention to the high seismic western United States. The ATC3 committees, on the other hand, appear to have spent a great deal of effort in sorting out the seismicity of the eastern United States and have accounted for such factors as the relative infrequency of earthquakes in many regions. We may also add that ATC3 has considered the lower attenuation of seismic waves in the eastern United States. Therefore, certain regions, which may have been excluded from the seismic requirements in the UBC, are now included in the ATC3 maps. A detailed discussion on this topic is beyond the scope of this book. Interested readers should examine the respective seismic maps and other supporting documents.

The range of the R factor, which accounts for ductility in the ATC3 procedure, is 1.25-8. This means that an inverted pendulum having the same period as a moment resisting steel frame will be designed for a seismic force which is 6.4 times that for the latter according to ATC3. The corresponding term in the UBC is the K-factor, which varies from 2.5-0.67, giving the inverted pendulum force

3.7 times the steel frame force. Assuming that the forces for the steel frame are comparable between UBC and ATC3, the ATC3 calculated forces for the low ductility structures are likely to be relatively conservative compared to those calculated from the UBC.

References 1 . G.V. Berg, Seiswtic Design Codes and Procedures, Earthquake Engineering Research

Institute, Berkeley, California, 1983. 2. International Conference of Building Oficials, Un$orm Building Code, Whittier, Califor-

nia, 1982. 3. Structural Engineers Association of California, Recommended Lateral Forces and Corn-

mentary, San Francisco, California, 1980. 4. Applied Technology Council, Tentative Provisions for the Development of Seismic

Regulations for Buildings, ATC3-06, National Bureau of Standards. Washington, DC, 1978.

Appendix/Numerical evaluation of response spectrum'

A.l Linear elastic systems As explained in Chapter 1, the elastic response spectrum is obtained by integrating the following equation of motion for a single-degree-of-freedom system:

in which all the variables are defined in Chapter 1, except the ground acceleration history, which is now denoted by a( t ) . A number of efficient methods are available for integrating Equation A.I. An exact technique was developed by W.D. Iwan in an unpublished study at the California Institute of Technology and was later reported by Nigam and Jennings[2]. In this method, Equation A.l is solved analytically within each successive time step assuming the ground acceleration varies linearly between designated points.

where the ground acceleration, a( t ) , has been replaced by its piecewise-linear approximation. The solutions for the relative displacement, u , and velocity, u, are

U ( t ) = e - F ~ ( ~ - ~ i ) [C, cos oD ( t - t i ) + C2 sin wD ( t - t , ) ]

I Aa, -A-

21; Aai ai ( t - t i ) + - - - - ,

o2 At, o3 At, o2

1 Aa, - (C, oD + 1;o C2) sin oD ( t - t i )] - - - .

o' At,

In these expressions oD is the damped circular natural frequency, o, = o J( I - 1; ' ) , and C, and C2 are constants. These constants are evaluated by defining

u ( f = t i ) = U , , u( t = t i ) = u , .

Thus, C, and C2 are

a 21; Aa, c, = U i + -2 - -- 1; 1 -2C2 Aa, u , + ~ o u , + - a , + - - . (AS) o2 o3 At, ' 0 o2 At,

*Based on Nau and Hall [I].

The relative displacement and velocity at the end of the time step, ui+! and ir,+, , may be determined by substituting Equation A S into Equation A.3 and setting t = ti+,. The resulting recursion formulae for u,+, and u,+, may be conveniently expressed in matrix form as,

[ u , + l } = A([, o, At,) + B(5, 0, At,) (01 } , ui+ I a;+ I

where

bll b12 A([, o, At,) = ['I' B(4, o , At,) = [ ] . a21 a22 b21 b22

The elements of matrices A and B are functions of c , w and At, and are given by Nigam and Jennings[2]. After simplifying elements b2, and b2*, the coefficients ofA and Bare

-<mAr, a , , = e cos wD AI, + 1;

4 1 - C 2 ) e-50ati

a I2 = - sin o, At,, OD

o a2] = - e-w'l sin OD ~ t , ,

J(1 - c2)

-gwAr, a22 = e cos oD All - c

J(1 - C 2 )

and

I I 25 6 = e-cwA'~ [ (e + 5 ) o~ + ( 25 + $) cos oD - - ,

W ' A ~ , W OD o3 At, o3 Ati 0

' sin o D A t i 21; 12 - +-

02 At, W D 03 A ti

If the record is digitized at equal time intervals, the coefficients of A and B are constant for a given frequency. Hence, given the initial conditions for the single-degree- of-freedom system, usually u(0) = u(0) - 0, response computations proceed rapidly by applying the recursion relationships defined by Equation A.6. Monitoring the response quantities as computation proceeds enables the determination of the maximum relative displacement, i.e. the spectral displacement. The calculations are repeated for a family of frequencies for each selected damping value. Thereby an entire set of elastic response spectra is developed for the given earthquake record.

The procedure described above can, of course, be applied to accelerograms digitized at unequal time intervals. However, the evaluation of matrices A and B at

N U M E R I C A L E V A L U A T I O N OF R E S P O N S E S P E C T R U M / 155

each step of integration, i.e. for each Ati, increases the computation time considerably. Experience has shown that this increase in computation time may be 100% or more. To maintain computational efficiency for records digitized at unequal time intervals, Nigam and Jennings[Z] recommend an approximate method involving time coordinate rounding. However, with the development of uniform processing and correction procedures, records are routinely digitized at equal time steps of 0.0 1 or 0.02 sec. Hence, it is unnecessary, insofar as the discussion here is concerned, to consider the treatment of records digitized at unequal time intervals.

The time step used in the response computations is selected as the smaller of the digitized interval of the earthquake accelerogram or some fraction of the period of free vibration, for example TjlO. For systems whose natural period governs the selection of Ati, i.e. for high frequencies, At, must be chosen so that an integral number of time steps comprises the digitized interval of the accelerogram. This restriction on At, preserves uniform time intervals and guarantees that response quantities will be computed at times corresponding to those of the given earthquake record. For example, suppose that the response of a system with T = 0.12 sec is to be determined. In addition, assume that the earthquake accelerogram is digitized at intervals of 0.02 sec. If the time step is not to exceed, say, T/10 or the digitized interval, At, must be selected as 0.01 sec, providing two time steps between successive digitized values of acceleration.

Aside from the uncertainties associated with the recording and processing of the accelerogram itself, errors in spectral calculations result from approximations employed in the numerical integration technique used for response computation. In this sense, the method described herein is exact. However, error is introduced by discretization. That is, the true maximum displacement or velocity, i.e. the spectral quantities, will not, in general, occur at one of the discrete times at which computations are made. The maximum error results when the true maximum falls midway between two consecutive time points, as depicted in Figure A.1. If the

Fig. A.l The true maximum and the computed values.

response within the time step is approximated by a sinusoid of frequency equal to the natural frequency of the single-degree-of-freedom system[2], the maximum error is

maximum error, % = (A. 1 0)

The true spectral quantities are greater than those computed at the discrete time points. By appropriately selecting the time step, however, the maximum error in the spectral ordinates may be controlled. For example, the expression above gives 4.9% error for At, = TIIO, 1.2% for T/20, and 0.3% for T/40. Thus, a time step corresponding to At, = TI20 is generally adequate.

A.2 Bilinear hysteretic systems[3] The bilinear hysteretic load-deformation model is shown in Figure A.2. In this figure, u, represents the initial yield level; u, and u,, are the current positive and negative yield levels; s, the current set remaining after an excursion of yielding; k, the initial elastic and unloading stiffness; and a, the ratio of the strain-hardening stiffness to the elastic stiffness. Initially, of course, s = 0, u, = u,, and u,, = -u,. Note that kinematic hardening for the bilinear system is shown, in which the current positive and negative yield levels are separated by a region of linearly elastic deformation of magnitude 224,.

Fig. A.2 Bilinear hysteretic load-deformation model.

N U M E R I C A L E V A L U A T I O N O F RESPONSE S P E C T R U M / 1 5 7

consider first the linear elastic response which follows unloading. For this case, the equation of motion for ti s t S ti+, is

Aa, ii + 26o1i + w2(u - s) = -ai - - (t - r , ) ,

A ti (A. 1 1 )

where all symbols are as previously defined. This equation may be more conveniently expressed as

Aii; ii + 2@u+ 02u = -a ' , - - ( [ - t , ) ,

At, (A. 1 2)

where

a',= a;- 02s, = a,,, - w2s. (A. 13)

The notation Aa', in Equation A. 12 is used for convenience since AZ, % hai. The solution for Equation A. 12 is given by Equation A.6 with the substitution of a', and a',+, for a, and ai+ t ,

(A. 14)

in which the coefficients of matrices A and B are defined by Equations A.8 and A.9. The set s required in Equation A.13 is computed a t the instant of unloading.

Following an excursion of positive yielding, the set is given by s = ( 1 - a) (uunl - u,); following an excursion of negative yielding, s = (1 - a) (u,,, + u,). In these equations, u,,, is the relative displacement computed at the instant of unloading. At the same time, the current yield levels are updated. For example, following a positive yield excursion, uyp = uunl and uyn = u,,, - 2uy.

Now consider excursions of loading beyond the current yield levels for the bilinear system. With reference to Figure A.2, the equation of motion for relative displacements greater than the current positive yield level uyp is

Aa, ii + 26wu + 02(u, - s) + am2 (u - u,) = -a, - - (t - t,). (A. 1 5 )

At,

This differential equation applies for u > u,, until unloading is detected, when the product ui X u,, < 0. Simplifying Equation A. 15 gives

A& ii + 2~ ,o z l i + o:u = -a', - -(t - ti),

At;

in which

and

Z i i 3 a, + wZuy( l -a ) , = a,,, + o2uY(1 - a).

(A. 16)

(A. 17)

(A. 18)

Note that C,, and o,, equivalent properties associated with the strain-hardening branch of the force-deformation model, are defined only for a > 0. For an extursion of negative yielding, for u < u,,, Equation A.16 applies with the modification,

a',= a , - w2u,(1 - a ) , Zi+, = a,,, - o 2 u Y ( l - a ) . (A. 19)

The character of the solution of Equation A.16 may be underdamped (c2 c l), critically damped (C,, = I), or overdamped (4, > 1). However, for the majority of bilinear systems of practical interest, the response is underdamped. For example, when C, = 0.05 and a = 0.02, 0.05 and 0.10, the largest value of c, is 0.051 J0.02 or 0.35. Thus, the solution as expressed by Equation A. 14 holds with the substitution of c2 and o2 for C, and o in the elements of A and B given in Equation A.8 and A.9.

A.3 Elastoplastic systems [3] The discussion regarding the linear elastic portions of the response for the bilinear system also applies to the elastoplastic system. For yielding excursions, however, the equation of motion for the elastoplastic system is

AZi ii + 2cwu = - a', - - ( I - ti)

A ti

where ii, and a',,, are computed with a = 0, in accordance with either Equation A. 18 for positive yielding or Equation A.19 for negative yielding. The solution for Equation A.20 may also be expressed by Equation A. 14 in which the elements of matrices A and B are:

-2Cma1, a2, = 0, a,, = e (A.21)

For the special case of no viscous damping ( 6 = O), the coefficients of A and B are:

N U M E R I C A L E V A L U A T I O N O F R E S P O N S E S P E C T R U M / l 5 9

The coefficients in Equations A.23 and A.24 may be obtained from those in Equation A.21 and A.22 by taking the limit as < approaches zero.

A.4 Notes for a computational algorithm[3] To maintain satisfactory accuracy in the response computations for the bilinear hysteretic and elastoplastic systems, the points at which the character of the solution changes-at yielding and unloading-must be detected reasonably precisely. This may be accomplished conveniently as follows. Before response computations begin, matrices A and B are evaluated and stored for the time interval At, and for one or several fractional time steps. The fractional time steps may be selected, for example, as At,/lO, At,/100, and At,/1000. Note that two sets of matrices A and B corresponding to the linear elastic and strain-hardening branches of the load- deformation model are required. When yielding or unloading is detected within a time step At,, the first (largest) fractional time step and corresponding A and B are used to locate the time subinterval during which yielding or unloading occurs. Once this subinterval is determined, the second fractional time step is employed further to refine the subinterval during which yielding or unloading takes place. The foregoing scheme is repeated until the smallest fractional time step is used or until the response quantities at yielding or unloading are determined to within some prescribed accuracy. It is important to note that the fractional time intervals are used progressively, as described above, to refine the previously determined time subinterval during which a change in response behavior is detected. Because the computations in Equation A. 14 are solely arithmetic and the required matrices A and B have been computed beforehand and stored, the method of fractional time stepping to detect yielding and unloading is efficient.

For the computation of inelastic spectra, the basic time step At, = TI 10 and three fractional time steps, At,/ 10, At,/ 100 and At,/ 1000, may be used. Experience with undamped elastoplastic systems, however, has shown that satisfactory accuracy is generally obtained using At, = TI10 and one fractional time step. At,/10. For this choice, response maxima differed from those using the three fractional time step scheme by about 0.2%[1]. The computation times using three fractional time steps ranged from 3-8% greater than those using one fractional time step; hence, economy is not significantly compromised when several fractional time steps are used.

One additional point should be mentioned regarding the calculation of the coefficients of matrices A and B. That is, caution must be exercised in the evaluation of Equations A.8, A.9, A.21 and A.22 to avoid roundoff or truncation errors. For sufficiently small oAt,, loss of accuracy may result when differences are taken between two values which are very nearly equal, as for example 1 and e-2S"A". One remedy, of course, is to use double (or higher) precision computer arithmetic to compute those coefficients prone to roundoff error. How small oAt,must be before roundoff becomes troublesome depends, of course, on the number of significant digits available for computation. However, no matter how many digits are used, a value of oAt, may be chosen so that roundoff errors result.

Perhaps a better method of eliminating the truncation errors is to evaluate the coefficients by first expanding the analytical expressions in power series form. In this

manner, lower order terms vanish identically. Hence, roundoff is avoided since the first remaining terms are of like order. The coefficients in which difficulties arise are those given in Equations A.9 and A.22, and coefficient aI2 in Equation A.21. Experience has shown that those in Equations A.21 and A.22 are particularly troublesome for small values of @A(,. On the CDC Cyber 175, in which 14 significant figures are available in the single precision mode, roundoff errors are evident in Equation A.22 for CoAf, less than about 0.06. Expanding coefficient a12in Equation 6.2 1 and those coefficients in Equation A.22 gives

Thus, when CoAt, is < 0.06, the expressions in Equations A.25 and A.26 are used to evaluate the elastoplastic coefficient a,, and those coefficients for matrix B. Including terms in each series up to the eighth order provides results accurate to about twelve significant figures.

A.5 Records with nonzero initial motions The Caltech accelerogram processing procedures provide estimates of the ground motions at the instant at which the instrument is triggered and recording begins. These initial motions may be expressed as a(0) = ao, v(0) = vo, and d(0) - 4, where a(t), u(t) and d(r) are, respectively, the ground acceleration, velocity and displacement. The time coordinate 2, of course, is measured from the instant at which recording commences. A difficulty arises when response computations are made for systems subjected to base excitation with nonzero initial conditions. Namely, the initial conditions for the single-degree-of-freedom oscillator are not known. To clarify this point, consider the initial conditions for the relative displacement and velocity given by

where x(t) and x(t) are the absolute displacement and velocity of the mass, respectively. It is apparent that the absolute motions, x(0) and x(0), depend upon the ground motions not recorded, i.e. those before the instrument is triggered. Hence, the x(0) and x(0) are unknown, and u(0) and u(0) are unknown.

NUMERICAL EVALUATION OF RESPONSE SPECTRUM/161

In spite of the foregoing problem, at-rest initial conditions are commonly assumed. However, an inconsistency arises when considering very flexible systems, i.e. for w -- 0. With u(0) = u(0) = 0 , Equation A.27 gives x(0) = do and x(0) = vo. For the infinitely flexible system, these initial conditions are obviously incorrect since the mass of the system must remain motionless for all time. Hence, the proper initial conditions for the very low frequency systems result from x( t ) = x(t) = 0 , from which u(0) = -do and u(0) = -vo. However, for very high frequency systems, i.e. for w -, co there is no relative motion between the mass and the ground, and the initial conditions are precisely u(0) = u(0) - 0 . In view of these limiting cases, it is clear that one set of initial conditions does not apply for all frequencies. Accordingly, one early approach for treating records with nonzero initial motions was to change initial conditions for the oscillator at some intermediate frequency.

Pecknold and Riddell[4] have proposed a successful method of treating the problems encountered in response computations from records with nonzero initial motions. In this method, a short acceleration pulse is added at the beginning of the earthquake record. For this prefixed pulse, let r i, 6 and d denote, respectively, the pulse acceleration, velocity and displacement. Also, assume that the pulse acts from 0 a t s H, or - H a t S 0. The prefixed acceleration pulse consists superposition

of three influence functions which were derived by minimizing

the constraints

bH d(t)dT = vo and

The prefixed pulse is piecewise linear so that conventional integration methods yield the velocity u, and displacement do at the end of the pulse. The ordinates of the prefixed acceleration pulse are given by

where ci, = a(iAt). The pulse is divided into N ( N 2 3) intervals such that At = HIN. The influence functions g,( i ) , g2(i) and g3(i) are cubic polynomials in the discrete variable i and are given by

in which j = 1,2,3 and i = 1,2, . . ., N. The coefficients bj, c, and a', depend upon the number of intervals Nand are given by

in which C = ( N ~ - 1) (N' - 4). The undamped spectra computed from Melendy Ranch record (N20W, 1982) with

and without a 2 sec prefix pulse are shown in Figure A.3. It is clear from this figure that the correct asymptotic behavior at low frequencies is achieved only for the spectrum computed from the record with the prefixed pulse. That is, at low frequencies, the spectral displacement approaches the peak ground displacement, d,, in this case 1.28 in. In fact, it can be shown that the low frequency asymptote for spectra computed from records with nonzero initial motions corresponds to a constant pseudovelocity equal in magnitude to the initial ground velocity, o,. This behavior is clearly evident in Figure A.3 for the spectrum computed from the Melendy Ranch record with no prefixed pulse, for which ( u, / = 1.17 in sec-'. In addition, note in Figure A.3 that the significant differences between the spectra extend up to a frequency of about 0.5 Hz. Above about 2 Hz, the spectra are identical, consistent with the previous discussion regarding the initial conditions for high frequency systems. Pecknold and Riddell(41 estimate that the frequency below which spectral

Frequency, Hz

Fig. A.3 Undamped spectra for Melendy Ranch record (N20W, 1982) with and without prefix pulse.

NUMERICAL EVALUATION O F RESPONSE SPECTRUM/163

ordinates may be in error is j; = o,/(2xdP), which for the Melendy Ranch record is 0.15 Hz. It is evident from Figure A.3 that the spectral errors may extend to a frequency several times the value given by the expression above.

References I. J.M. Nau and W.J. Hall, An Evaluation of Scaling Methods for Earthquake Response

Spectra, Structural Research Series, No. 499, Department of Civil Engineering. University of Illinois, Urbana-Champaign, May 1982.

2. N.C. Nigam and P.C. Jennings, Calculation of Response Spectra from Strong Motion Earthquake Records, Bulletin of the Seismological Society ofAmerica, Vol. 59, No. 2, April 1969, pp. 909-922.

3. J.M. Nau, Computations of Inelastic Response Spectra, Journal of Engineering Mechanics, ASCE, Vol. 109, No. 1, February 1983, pp. 279-288.

4. D.A. Pecknold and R. Riddell, Effect of Initial Base Motion on Response Spectra, Journal of Engineering Mechanics Division, ASCE, Vol. 104, No. EM2, April 1978, pp. 485-491.

Author index

American Society of Civil Engineers (ASCE) 14, 19, 28, 45, 50

Amin, M. 5 1, 64 Ang, A. H. S. 5 1, 64 Applied Technology Council ( A X ) 1 39, 152

ATC3, ATC3-06 139, 141 -1 52 Asfura, A. 9 1, 120, 124

Bayo, E. P. 36, 50 Berg, G. V. 139. 147, 152 Biggs, J. M. 89, 124 Biot. M. A. 12, 28 BIume and associates 14 Blume, J. A. 12, 14, 15, 28 Bogdanoff, J. L. 51, 64

Chelapati, C. V. 12. 25, 27, 28 Chen, D. C. 41, 43, 45, 50 Chopra, A. K. 2, 89, 124 Chu, S. L. 32, 36, 50, 52, 55, 59, 60, 64, 65 Clough, R. W. 7, 10, 31, 50, 66, 87 Contreras, H. 52, 60, 64 Cordero, K. 35, 36, 41, 50 Corning, L. H. 12, 28

Dalal. J. S. 14. 28 Davenport. A. G. 32. 50 Der Kiureghian, A. 36, 37, 38, 50, 67, 77,

88, 90, 91. 94, 98, 100, 120, 125

Elghadamsi. F. E. 27, 28 Ellison. B. 125. 127, 139 Elorduy. J. 33, 36, 37, 38, 50

Fang, S. J. 60, 64, 65 Foss, K. A. 66. 67, 87

Ghafory-Ashtiany, M. 51, 64, 88 Goldberg. J. E. 51. 64 Goodman, L E. 3 1, 48 Gupta, A. K. 12, 35, 36, 41, 43, 46, 49, 50,

51, 52, 55, 58, 59, 60, 64, 65, 67, 68, 70,

Hadjian, A. H. 40, 43, 50, 125, 127, 139 Hall, W.J. 11, 14, 16,20,23,27,28,29,34,

71, 88, 153, 159, 163 Hayashi, S. H. 17. 28 Hernried, A. G. 91, 124 Housner, G. W. 12, 27, 29 Hudson, D. E. 3 I, 49 Hurty, W. C. 66, 67, 68, 87

Igusa, T. 67, 77, 88, 91, 94, 98, 100, 124 lida. Y. 18, 28 International Mathematics and Statistics

Library 100, 124 Itoh, T. 66, 87 [wan, W. D. 154 Iwasaki, T. 18. 28

Jaw, J. W. 41, 46, 48, 50, 67, 68, 70, 73, 76, 81, 82. 83, 84, 85, 86, 87, 88. 91, 95, 97, 101. 102, 104, 105, 118, 119, 120

Jennings, P. C. 27, 29. 154, 164 Jennings, R. L. 3 1, 49

Kapur, K. K. 15, 28, 89, 124 Kasai, K. 37, 50, 84 Kelly, J. M. 90, 124 Kennedy, R. P. x. 40. 43, 50 Kurata. E. 17, 28 Kuribayashi, E. 18, 28

Lin, C. W. 126, 137 Lindley, D. W. 40, 50 Liu, T. H. 125, I 38 Lysmer, J. 18, 21, 28

Maison, B. F. 37, 50, 84 Mehta 40, 43, 46, 50

Merchant. H. D. 3 1, 49 M0hraz.B. 14, 16, 18,20,21,23,27,28 Mojtahedi, S. 66, 87 Mosberg. R. J. 3 1. 50

Nathan, N. D. 51, 64 Nau. J. M. 34. 50. 71, 88, 153, 159, 163 Neuss. C. F. 37, 50, 84 Newmark Consulting Engineering Services, 14 Newmark,N. M. 12. 14. IS. 16, 20.23, 25,

26, 27, 28, 29, 31, 50, 51, 64,67, 88, 90, 124

Nigam, N. C. 153, 163 North Carolina State University 37 Nour-Omid. B. 90, 91. 124

Pecknold, D. A. 71, 88. 161, 162, 163 Penzien, J. 7, 10, 51, 64, 89, 124 Peters. K. A. 90. 124 Powell. G. H. 46, 50

RDT 125, 135 Ridde1.R. 26. 28, 71. 88, 161, 162, 163 Robinson, A. R. 90, 124 Roesset, J. M. 89, 124 Rosenblueth, E. 31, 33, 37, 38, 49, 51, 52,

60. 64 Rubenstein, M. F. 66, 68, 87 Ruzicka, G. C. 90, 124

Sackman, J. L 90, 91, 124 Salmonte, A. J. 46, 50 Scanlan, R. H. xi Schiff, A. J. 51, 64 Schild, A. 53, 64 Schmitz, D. 90, 124 Seed, H. B. 18, 20, 28 Shao, L. C. 89, 124 Sharpe, R. L 14, 28

Singh, A. K. 32. 50 Singh, M. P. 36, 40, 43, 46, 50, 51, 59, 60,

64, 65, 66. 67, 90, 91, 123 Singh, S. 32, SO Soni, S. R. 66, 87 Structural Engineers Association of California (SEAOC) 139, 143, 151, 152

Synge, J. L 53. 64

Tembulkar, J. M. 82, 88, 92, 124, 126, 127, 131, 133, 139

Tw, W. K. 51, 64 Tsuchida, H. 17, 28 Tuji, K. 18, 28

Ugasa, C. 18, 20, 28 Uniform Building Codc (UBC) 139, 142,

143. 149, 151, 152, 153 US Atomic Energy Commission

(USAEC) 14, 15, 18, 28 see also US' Nuclear Regulatory Commission

US Nuclear Regulatory Commission (USNRC) 15, 28, 32. 50 see also US Atomic Energy Commission

Vashi, K. M. 46, 50 Veletsos, A. S. 12, 25, 27, 28, 31, 50, 67, 88 Ventura, C. E. 67. 88 Villaverde, R 34, 50, 67. 88, 90, 124

Wagner 90, 124 Walker, W. H.' 31, 50 Warburton, G. 9. 66, 87 Watabe, M. 51, 64 Wilson, E L, 36, 50

Yow, J. R., 40, 50

Subject index

Amplification factors 14 Mohraz (site-dependent) 2 1, 22 Newmark Spectrum 14, 15, 23

Buildings 139 allowable stresses 149 base shear 140 diagonal mass matrix 140 equivalent lateral force procedure 141 frequency 144 long period 146 mass matrix 139 modal base shear 141 modal story force 142 modal vector 140 modal weight 141 pseudo-inertia force vector 140 pseudo-static force 140 seismic coefficient 14 1. 144 spectral acceleration 140 story pseudo-inertia force. 140 ultimate load design 149 working stress design method 149

Cartesian space 58 metric tensor 58

Combination of modal responses 30 absolute sum method . 31, 38 complete quadratic combination (CQC) 36 double sum method 32, 38 high frequency modes 40. 43, 45, 77 modes with closely spaced frequencies 3 1,

52 nonclassically damped systems 75 square root of sum of squares (SRSS)

method 31, 38, 43 Compatible motion history xii Correlation coefficient

instructure response spectrum method 114, 1 I6 approximate algorithm 1 16

modal 32 modal (nonclassically damped systems) 77 multiple earthquake components 52

Coupled response of secondary systems 89, 95, 98, 101

complex eigenvalue problem 94, 95 iterative scheme 95 modal properties 95 modal synthesis 95 perturbation technique 91 static constraint 94, 95, 1 16, 124

Damped frequency 33 vibration 1

Damping classical 6, 10, 66 force 1 matrix 9. 30, 67, 91 modal (nonclassical) 68

nonclassical 6, 66 ratio 1. 30 viscous 1

Decoupled analysis of primary systems 125 energy-mass ratio 130 frequency ratio (MDOF) 1 30 frequency ratio (SDOF) 126 multi-degreesf-freedom (MDOF)

systems 130 response ratio (MDOF) 133 response ratio (SDOF) 128 r,- r, curves 125, 127. 128

change in frequency 127 change in response 128

single-degree-of -freedom systems 126 single mode approximation (SMA) 13 1

Design spectrum 1 1, 12 elastic systems 12 horizontal component 15, 23 inelastic systems 23

bilinear 26. 27 buildings 45, 147 ductility demand curves 27 effect of damping 26 elastic-perfectly plastic 24, 26, 27 four-parameter Nadai 27 maximum displacement 25 stiffness degrading 26. 27 yield 25

site-dependent 16, 1 7 vertical component 15, 23

Deterministic methods xii

168 / INDEX

Direct time-history integration method 66 Newmark's method 82

Ductility 24 factor 24

Earthquake motion records Adak 76 Bonds Corner 76 Bucarest 76 Caracas 145 Coyote Lake 76 El Centro (1 934) 12 El Centro (1940) 3, 34, 35, 37, 71, 73, 84,

101. 114, 120 Kilauea 76 Managua 76 Mclcndy Ranch 76, 162 Mexico City 27 Olympia (1949) 12 Pacoima Dam 38. 76 Parkheld 76 San Fernando 37, 41, 42, 43 Santiago 76 Taft 27. 49, 76 Tehachapi 12

Earthquake motions broad band 35 effectivc durirtion 34 horizontal components 14, 5 1 Husid plot 34 multicomponents 51 normalized 14 prefix pulse 71, 162 records with nonzero initial motion 161 rotational components 51 stationary ergodic process 32, 36, 75 stochastic process 9 1 vertical components 14, 51 white noise 33, 34, 91

Eigenvalue problem xi, 8 complex eigenvalue 67 complex eigenvector 66 orthogonality conditions 68

Equivalent lateral force procedure for buildings 141

ATC3 139, 142.143,144,145,146.148. 149, 150. 152

accidental eccentricity 143 building frequency 144

approximate expressions 144 Rayleigh's method 144

center of mass 143 ccntcr of rigidity 143 concentrated force (top story, UBC) 142 eccentricity 143 fundamental mode shape 142 horizontal components 143 overturning moment 143

reduction for out-of-phase multiple modcs 143

seismic cocfficicnt 141

torsional moment 143 UBC 139, 142. 143, 149, 150, 152

Fast Fourier transform 66 Floor response spectrum see Instructure

response spectrum Fourier amplitude spectrum 90 Frequency 1

building 145 circular 1, 30 complex 66 damped 33 key .f f 43. 73 ,

key .PY' 73, 74 key S' 75 natural 8 nonclassical systems 69 ratio 126 rigid 40. i3ZPA 40

Frequency-domain analysis vii

High frequency modes 40, 45 missing mass 46 nonclassically damped systems 77 residual rigid response 46 rigid response 40

Higher mode effects 40. 45 nonclassically damped systems 77

Inertia force 1 pseudo 3 vector 7

Instructure response spectrum 89. 106, 110 correlation coefficients 1 14, 1 16 effect of damping ratio 116 effect of mass ratio 116 energy mass ratio 108 equivalent oscillator 108 interaction with primary system 89 resonant frequency range-in 89 response examples 120 static constraint 108, 109 tuned secondary system 90

Interaction ellipsoid (ellipse) 58. 59

Mass interaction xii. 89, 95. 108 matrix 7, 30. 67, 91, 139 ratio 90, 125

energy 93. 95, 108, 130 inertia 133

Metric tensor Cartesian space 58 Riemannian space 53

Modal correlation coefficient 32. 52 Der Kiureghian 36

Gupta and Cordero 35 nonclassically damped system 77 Rosenblueth and Elorduy 33 with rigid response 43, 52

Modal superposition method xi, 10, 30 nonclassically damped system 68, 69

Modal vectors 8 buildings 140 complex 66 normalized 8. 139 orthogonal properties 8, 30 uncoupled primary and secondary

systems 91, 132 Mode acceleration method 46 Mode shapes see Modal vectors Multi-degreesf-freedom (MDOF) systems 7

classically damped 6, 10, 30, 66 coupled analysis of primary and secondary

systems 89 damping matrix 9, 30, 67, 91 decoupled analysis of primary systems 125

I30 direct time-history integration method 66 equation of motion 30, 67 free vibration equation 91 inertia force vector 7 mass matrix 7, 30, 67, 9 1, 139 modal superposition method vii, 10, 30 nonclassically damped 6. 66 pseudo single-degreesf-freedom system 24 spring force vector 7 stiffness matrix 7, 30, 67, 91

Nonclassically damped systems- 66 complex eigenvalue problem 67 correlation coeficients 77 high frequency modes 77 modal damping 68 modal frequency 68 modal superposition method 68 residual rigid response 78 response spectra 7 1

Normal coordinates 9, 30 Numerical evaluation of response

spectrum 153 bilinear hysteretic systems 156 computational algorithm 159 elastoplastic systems 158 error of discretization 155 fractional time step 159 linear elastic systems 153 maximum error 156 Nigam-jennings (Iwan. Cal Tech)

method 153 prefixed pulse 1 6 1 record with nonzero initial motion 160 time step 155

Participation factor 9, 30 secondary system 93

Peak factor 32, 77 ,

Period of vibration 1 Power spectral density function xii Primary system 89, 125

decouplcd (uncoupled) analysis 125 Probabilistic methods ix

Random vibration methods xiii Response

buildings 139 damped periodic 33, 40, 41 equivalent modal 55 multicomponents of earthquake 5 1 multi-degree-of-freedom (MDOF) system 9,

30 - - nonclassically damped system 70 pseudo-static 40, 46 ratio, uncoupled and coupled primary

systems 128 residual rigid 45-48 rigid 40, 41 secondary systems 89 single-degree-of-freedom (SDOF) system 2 steady state 40, 41 three components of earthquake 5 1 transient 40. 41

Response spectrum 2, 12 acceleration (pseudo) 5 acceleration (relative) 40. 4 1, 46 bilinear hysteretic system 156 characteristics of 6 D-V-A relationship 5, 7 I displacement (relative) 3. 66, 71, 99 elastoplastic systems 158 inelastic xii. 147 linear elastic systems 153 numerical evaluation 153 tripartite S velocity (pseudo) 5 velocity (relative) 5, 41, 46. 66. 71, 99

Response spectrum method xi. 10 nonclassically damped systems 70

Riemannian space 53 metric tensor 53

R~gid response 40. 4 1 coefficient 41, 73 coefficient (velocity spectrum) 73 residual 45-48 residual (nonclassically damped system) 78

Secondary systems xi, xv. 89, 125 coupled response xv, 89 detuned 98 equivalent oscillator 108 floor response spectrum 89 instructure response spectrum 89, 106

alternate formulation 1, 6

170/ INDEX

interaction with primary system 89, 91, 108

relative support motions 89 statlc constraint 94. 95, 108, 109, 124 static displacement cornponcnt (effect

of) 103 tuned 90, 98 tuning parameter. 98

Seismic coefficient 141. 144, 147, 148 acceleration coefficient 145 deflection modification factor 147 ductility ratio 147 equivalent peak acceleration 144 equivalent peak velocity 145 importance factor 149 ~nelastic deformation 147 response modification factor 147 site-structure resonance factory 149 soil coefficient 147 soil profiles 146 structural coefficient 149 \.elocit! related acceleration coefficient 146 zone coeffic~ent 149

Single-degree-of-freedom (SDOF) system I damped vibration* 1 daniping force 1 damping ratio I '

decoupled malysis of,primary system 126 equatlon of motion 1, 153 equilibrium equation 1, 153 frequency 1 frequency ratio 126

inertia force I period I pseudo-static problem 3 response ratio 128 spring force I

Spectral acceleration 3, 14, 140 zero pcriod acceleration (ZPA) 40, 72

Spectral displacement 3, 12, 70. 7 1, 99. 1 1 1 Spectral velocity 12, 99

pseudo 5 relative 5, 70, 7 1, 1 1 1

Spring force I vector 7

Stiffness matrix 7, 30. 67, 91 Stochastic process 91

white noise 91

Three components of earthquake 5 approximate method 60 design problems 63 equivalent modal responses 55 interaction diagrams

capacity 62, 63 conventional 63 ellipsoid (ellipse) 58. 59

simultaneous variation in response 52 Time-domain analysis xi Total acceleration 3

Zero period acceleration (ZPA) 40, 72