Response Spectrum Method
In Seismic Analysis and Design
of
Structures
A J A Y A
K U M A R G U P T A .
Prokssor of Civil Engineering North Carolina State University
F O R E W O R D BY
WILLIAM J. H A L L
Professor and Head. Civil Engineering Universit.~ of lllinois at Urbana-
ChampaignBLACKWELL SCIENTIFIC PUBLICATIONS
BOSTON O X F O R D
Q 1990 by
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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 6 2 4 . 1 ' 7 6 2 6 ~ 2 0 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:
New
Dirdons
in Civil Engineering
S E R I E S E D C T Q ~W.
F. C H E N Purdue UniversityDedicated 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, 112.1 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, 514.1 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, 665.1 Introduction, 66
5.2 Analytical formulation, 67 5.3 Response spectra. 7 1
viii 1 C O N T E N T S
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, 1398.1 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 tech- niques in analysis and design. Although the theory of spectra has been understood for some extended period of time, it was only in the past twenty years that the approach was adopted in a major way by the profession for use in engineering practice. This development came about as a result of three major factors, namely that the theory and background of spectra was more fully understood, that the theory was relatively simple to understand and use, and because there was a need for such a simple approach by the building codes and by the advanced analysis techniques needed in the design of nuclear power plants and lifeline systems.
The author rather directly presents his interesting and informative interpreta- tions of various spectrum techniques in the topical chapters. He correctly points out that much work remains to be accomplished, which is accurate, for spectra in general only depict maxima of various effects, and in many cases, especially where nonlinear 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 considera- tions, nonlinear effects, etc. One can immediately discern the differences, directly or subtly as may be the case.
It is believed that the reader will find the interesting presentation by Dr Ajaya Gupta to be educational and informative, and hopefully such as to promote additional effort to improve even further our understanding of the theory and applications thereof.
W. J. HALL
Professor and Head, Civil Engineering University of Illinois at Urbana-Champaign
In modern earthquake engineering the response spectrum method has emerged as the most commonly used method of analysis. The primary reason for this popularity is the fact that it provides the designer with a rational and simple basis for specifying the earthquake loading. Another reason often cited is that the method is computationally economical. If a comparison is made between the computational effort required in, say, a modal superposition analysis of a multi- degree-of-freedom structure subjected to a specified ground motion history, and that in a response spectrum analysis including the evaluation of the response spectrum from the same motion history, it is not clear whether the response spectrum method would do much better. A major part of the effort, which is com- mon in both the methods, is the solution of the eigenvalue problem. In fact, if the objective is to evaluate the response of a structure subjected to a known earthquake ground motion, there should not be any question about using a standard time-domain analysis, or alternatively, an equivalent frequency- domain analysis. It is when we are designing a structure for a potential future earthquake that the response spectrum method is much more relevant.
Criticisms of the response spectrum method arise from the fact that the- temporal information is lost in the process of evaluating the spectrum. In the words of Robert Scanlan:' 'Multi-degree-of-freedom cases are thus improperly served, intermodal phasings, in particular, being unaccounted for.' Further he points out: 'The needs arising in the design of secondary responding equipment (piping, machinery, etc., on upper floors of a structure) are not adequately met by the given design response spectra. That is, the given primary shock spectra do not lead directly and simply to definition of corresponding secondary shock spectra.' Similar difficulties arise in combining the responses from three components of the earthquake.
Much progress has been made in the last decade. Lack of temporal information in the response spectrum method no longer appears to be a handicap. Rational rules are now available to combine responses from various modes, and from three components of earthquake motion. These rules account for the physics of the problem, and can be further justified in the same spirit as the design spectrum itself, as a representation of expected response values in an uncertain world. Response of secondary systems can now be evaluated using efficient modal synthesis techniques in conjunction with the response spectrum method. Alternatively, the secondary spectrum, or the instructure response 'R. H. Scanlan. On Earthquake Loadings for Structural Design, Earthquake Engineering and Sfrucfural Dynamics, Vol. 5, 1977, pp. 203-205
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 res- ponse spectrum into a 'compatible' motion history or a power spectral density function. The question of noncIassica1 damping introduced in the coupled primary-secondary system, which had not even been specifically raised 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 res- ponse 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.
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 whichf;
denotes the inertia force,f,
the spring (or the column) force andf,
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 tovary 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 getrn(ii
+
ii,)+
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.
S T R U C T U R A L D Y N A M I C S A N D RESPONSE S P E C T R U M 1 3
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 displace- ments, 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 writeFigure 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 seeFrom 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 . Thismeans,
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 S P E C T R U M / 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 oneof 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, = maxI
u,I,
and in the high frequency range S, = max1
ii,I
.
This pheno- menon 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 D Y N A M I C S 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,/kI
,,,
= ( i i , / 0 2I
,,,,
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-freedomsystems
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 threeDOF
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 byur
= [u, u2 u,]. The inertia force vector is0 i i ,
+
ii,4 = M l i j + o g ; =
['a
in2 [ i i 2 + i i g ] , (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 asIn 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, =
m 3
= m, and k,-
k,
=k,
= k. Let us denote the frequency of the ith mode of an N-DOF structure bya,
and the modal vector by
9,.
The modal vectors have the following orthogonal propertiesI
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 thatS T R U C T U R A L D Y N A M I C S A N D RESPONSE S P E C T R U M I S
Mode 1 Mode 2 Mode 3
f = o.0708 Hz f = 0 . 1 9 8 4 HZ f
-
0.2a7fi HZFig. 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 structureStructures 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 .Chapter Z/Design spectrum
2.1
IntroductionWe 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 magni- tude. 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
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 dis- placement 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 S P E C T R U M / 1 3
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: respec- tively 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 compon- ent 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 approxima- tely 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.
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 spectraThe spectral amplification factors presented in the previous section are based on the analysis of several earthquake ground motions without particular considera- tion 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.
D E S I G N 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 dif- ferences 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 compon- ents. 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 S P E C T R U M J 1 9
Soft to Medium Clay and Sand eep Cohesionless Soik (>250W Stiff Soil Conditions k 1 5 0 f t )
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 allu- vium 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
fora
site with alluvium layers underlain by rock. The values of v l a and adlv2 ratios, and of vand 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 dif- ferences 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 d o 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 S P E C T R U M / 2 1
-
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
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 con- tribution 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 = u r n / u y . 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 ) uI
G u , , and& =+
o, when ) u 12u,. After oneor 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 S P E C T R U M / 2 5
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 spec- trum 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 = p s E ,
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 D Y 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 dif- ferent 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 displace- ment spectrum is straight forward. We simply multiply the yield spectrum ordinates by the ductility factor to do so. In the resulting maximum displacementI
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 con- cluded 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].
