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REPORT DOCUMENTATION 1 L It!".,. NO. PAGE NCEER-88-0020 I~ N8fnrat02 g' 0 9 lAS

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A Response Spectrum Approach For Analysis of Nonclassically April 22, 1988 Damped Structures I-

7_ 1.utfIoo1I) I. PerfoMll .. O ... nIz.tlon "1111. No.

J.N. Yang, S. Sarkani, and F. X. Long ,. ~Inc Dlpnlzlflon N.me etId _ .... ao . ......-/T .... /WI ... U,,1t .....

National Center for Earthquake Engineering Research State University of New York at Buffalo n. ~rKt(C) Of GrlnteG) No.

Red Jacket Quadrangle ICl{87-2001J Buffalo, NY 14261 ~ ECE 86-07591

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Technical Report 14.

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This research was conducted at The George Washington University and was partially supported by the National Science Foundation under Grant No. ECE 86-07591.

al. _rMt (Umlt: 20CI _I

A response spectrum approach for analysis of nonclassically damped structural systems is presented. Similar to the response spectrum procedure for analysis of classically damped systems, the only required information regarding the ground motion input is its respons\. spectrum. The procedure takes into account the effect of cross correlation between modes with closely spaced frequencies, and it is simple for practical application.

The proposed method is used to approximate the maximum response of several non-c.lassically damped structural systems. Emphasis is placed on noncJassically damped primary-secondary systems in which the effect of nonclassical damping is significant. Numerical results indicate that the maximum structural responses predicted by the proposed response spectrum approach are generally closer to the exact solutions than those obtained using an approximate classically damped procedure. The accuracy of the present approach is quite reasonable. ~\

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EARTHQUAKE ENGINEERING NONCLASSICALLY DAMPED SYSTEMS BUILDING STRUCTURES

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RESPONSE SPECTRA SEISMIC RESPONSE ANALYSIS

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NATIONAL CENTER FOR EARTHQUAKE ENGINEERING RESEARCH

State University of New York at Buffalo

A RESPONSE SPECll{UM APPROACH FOR ANALYSIS OF NONCLASSICALLY

DAMPED STRUCTURES

by

J.N. Yang, S. Sarkani and EX. Long Department of Civil, Mechanical and Environmental Engineering

The George Washington UrJversity Washington D.C. 20052

Technical Report NCEER-88-0020

April 22, 1988

This research was conducted at The George Washington University and was partially supported by the National Science Foundation under Grant No. ECE 86-<17591.

REPRODUCED BY U.S. DEPARTMENTOF COMMERCE

NATDNM. TECHNICAl. INFOA*TION SERVICE SPRINGFIELD. VA 22181

NOTICE This report was prepared by The George Was.hington University as a result of research sponsored by the National Center for Earthquake Engineering Research (NCEER) and the National Science Foundation. Neither NCEER, associates of NCEER, its sponsors, The George Washington University, nor any person acting on their behalf:

a. makes any warranty, express or implied, with respect to the use of any information, apparatus, method, or process dic;closed in this report or that such use may not infringe upon privately owned rights; or

b. assumes any liabilities of whatsoever kind with respect to the use of, or for damages resulting from the use of, any information, apparatus, method or process disclosed in this report.

l,C!.J

1111111111111 11,----

A RESPONSE SPECTRUM APPROACH FOR ANALYSIS OF NONCLASSlCALL Y DAMPED STRUCTURES

by

J.N. Yang 1 , S. Sarlcani2 and F.X. Lon~

April 22, 1988

Technical Report NCEER-88-OO20

NCEER Contract Number 87-2004

NSF Master Contract Number ECE 86-07591

1 Professo{, Dept. of Civil, Mechanical and Environmental Engineering, The George Washington University

2 Assistant Professor, Dept of Civil, Mechanical and Environmental Engineering. The George Washington University

3 Visiting Scholar, Dept. of Civil, Mechanical and Environmental Engineering. The George Washington University

NATIONAL CENTER FOR EAR1HQUAKE ENGINEERING RESEARCH State University of New Y 0It at Buffalo Red Jacket Quadrangle, Buffalo, NY 14261

PREFACE

The National Center for Earthquake Engineering Research (NCEER) is devoted to the expansion of knowledge about earthquakes, the improvement of earthquake-resistant design, and the implementation of seismic hazard mitigation procedures to minimize loss of lives and property. Initially, the emphasis is on structures and lifelines of the types that would be found in zone:; of moderate seismicity. such as the eastern and central United States.

NeEER's research is being carried out in an integrated and coordinated manner following a structured program. The current research program comprises four main areas:

• Existing and New Structures • Secondary and Protective Systems

• Lifeline Systcms • Disaster Research and Planning

This technical repon penains to the second program area and. more specifically. to secondary systems.

In eanhquake engineering research. an area of increasing concern is the perfonr.ance of M:Con­dary systems which are anchored or attached to primary structural systems. ~hny secondary systems perform vital functior,s whose failure during an earthquake could be just as catastrophic as that of the primary structure itself. The research goals in this area are to:

1. Develop greater understanding of the dynamic behavior of secondary systems in a seismic environment while realistically accounting for inherent dynamic complexities that exist in the underlying primary-secondary structural systems. These complexities include the problcm of tuning. complex attachment configuration. Donproportional damping. parametric uncertainties. large number of degrees of freedom and non­linearities in the primary struCture.

2. Develop practical criteria and procedures for the analysis and design of secondary systems.

3. Investigate methods of mitigation of potential seismic damage to secondary systems through optimizaticn or pi otection. The most direct route is to consider enhancing their performance through optimization in their dynamic characteristics. in their placement within a primary structure or in innovative design of their suppons. From the point of view of protection. base isolation of the prinw:y structure or the applica­tion of other passive or active protection devices can also be fruitful.

iii

Current research in secondary systems involves activities in all three of lhese areas. Their interaction and interrelationships with olher NCEER programs are illustrated in the accompany­ing figure.

Seotndary Systems I I Prol!am 1

Analyses and I - Structural Experiments Response

/"" - Risk and

Reliability - Seismicity

and Ground Perfonnance Optimization Motion Evaluation and Prediction and Design Prol!!m%

Criteria - ProteCtive Systems

The purpose of research documented in this report is to develop a response spectrum approach for nonclassically damped structural systems. The aim is to make the procedure simple for practical applications and similar to the response spectrum procedure commonly used for analysis of classically damped systems. Emphasis is placed on nonclassically damped primary­secondary systems in which the effect of nonclassical damping is significunt.

iv

ABSTllCT

A response spectrum approach for analysis of nonclassically damped

structural systems is presented. Similar to the response spectrum procedure

for analysis of classically damped systems, the only required information

regardi.ng the ground motion input is its response spectrum. The procedure

takes into account the effect of cross correlation between modes with closely

spaced frequencies, and it is simple for practical application.

The proposed method is used to approximated the maximum response of

several nonclassically damped st.ructural systems. Emphasis is placed on

nonclassic3lly damped primary-secondary systems in which the effect of

nonclassical damping is significant. Numerical rellultll indicate that the

maximum IItructural respon.es predicted by the proposed respon.e spectrum

approach are generally closer to the exact solutions than those obtained using

approximate classically damped procedure. The accuracy of the present

approach is quite reasonable.

v

ACENOVLEDCEHENT

This work is supported by the National Center for Earthquake Engineering

research. State University of New York at Buffalo under grant NCEER-87-2004.

vii

Preceding page blank

TABLE OF CONTENTS

SECTION TITL! PAGE

1 INTl.ODUCTION 1-1

2 BACItGaOUND . 2-1

3 CANONICAL MODAL ANALYSIS 3-1

4 EVALUATION OF COSINE SPICTaA 4-1

5 COMBINING MAXIMUM MODAL IESPONSES 5-1

6 NOHEllICJ.L EUHPL!S 6-1

7 CONCLUSIONS 7-1

8 IEFEIlDCIS . 9-1

APPENDIX CALCULATIOR OF COalELATION COEFFICIENTS 1-1

ix

Preceding page blank

LIST OF ILLUSTRATIONS

FIGUU TIlL! PAGE

1 Smooch Sine and Cosine Spectra 4-9

2a Sine and Cosine Spectra for El Centro Earthquake, e - 2t . . . . . . . . . . . . . . . . . . . . 4-10

2b Sine and Cosine Spectra for El Centro Earthquake, e - 20' . . . . . . . . . . . . . . . . . . . . 4-11

2c Sine and Co.ine Spectra for San Fernando Earthquake e - 2t . . . . . . . . . . . . . . . . . . . . . 4-12

2d Slne and Co.ine Spec era for San Fernando Earthquake, e - 20, . . . . . . . . . . . . . . .. 4-13

2e Slne and Cosine Spectra for Mexico City Earthquake, e - 2t .................... " 4-14

2f Sine and Cosine Spectra for Mexico City Earthquake, e - 20' . . . . . . . . . . .. ........ 4-15

2g Average Sine and Cosine Spectra for twenty Simulated Ground Acceleration., e - 2, . . . . . . . . . . .. 4-16

2h Averag_ Sine and Cosine Spectra for twenty Simulated Ground Accelerations, e - 20t 4-17

3 Simulated Ground Acceleration 6-7

xi

Preceding page blank

LIST OF TABLES

TABLE TITLE PAGE

1 Ma:.:1mwa Response of Two-Degree-Of-Freedoll Structure (CII.) 6-8 <a) C1 - C2 - 123.4 kN/li/sec., (b) C1 - 246.8 kN/li/

sec .• C2 - 0.0, (c) C1 - 0.0. C2 - 246.8 ~~/II/se~.

2 Maximum Responsa of Tuned Equipllent-Structure Systell (CII. ) 6-9 <a) e - Ot. (b) e - 5\. (c) e - 10\ e e a

3 MaxillUll Respon3e of Tuned Equipllent-Structure Sy. tell (CII.) 6-10 <a) e - Ot. (~) e - 5t, (c) ee - 10\

III e

4 Maximum Respons. of Tuned Equipllent-Structure Systell <CII.) 6-11 (a) e. - Ot, (b) e - S'. (c) ee - 10, e

5 KaxillUll Response of Detuned Equipllent-Structure Systell (CII.) 6-12 <a) Cl - C2 - 123.4 kNill/sec .• (b) Cl - 246.8 kN/lI/

sec .. C2 - 0.0, (c) C1 - 0.0, C2 - 246.8 kN/lI/sec.

xiii

Preceding page blank

SECTION 1

INTP..ODUCTION

In seismic analysis of multi-degree-of-freedom linear structures, modal

analysis in conjunction with the res~onse spectrum continues to be the most

widely used technique, referred to as the response spectrum approach. The

main reasons are the simplicity of the procedure and the fact that in most

design situations the input ground acceleration is specified in terms of the

displacement response spectrum (commonly referred to as the response

spectrum) .

Traditionally, the response spactrum approach requires the decoupling of

the equations of motion using the undamped modes of vibration of the system.

Then, the maximWl response in each mode can be obtained using the response

spectrum of the ground acceleration [2). Then, the maximWII value of the

response quantities are determined using a proper modal combination rule [13).

Consequently, such as approach requires that the damping matrix of the

structure is of the classical (proportional) form (i.e., the form specified by

Caughey and O'Kelly [1]). However, real structural systems may not always be

classically damped, so that the damping matrix may not be diagonalized by the

eigenvectors of the undamped systell. For these structures, referred to as

nonclassically damped structures, the classical modal analysis is not

applicable and the co~lex modal analysis procedure has been used in the

literatures.

Response spectrum approaches for nonclassically damped structures ha'.e

been suggested, recently, tn the literature [9,12,6,ll}. Based on the complex

modal analysis, the maximum response of a nonclassically damped structure had

been expressed in terms of the displacement response spectrum and the velor-tty

response spectrum of the ground acceleration [6,11]. The velocity re~ponse

1 - 1

spectrum was obtained from the displacement response spectrum using var; ous

approximations [6,5].

In this paper an alternate approach is presented in which the equations

of motion for a nonclassically damped structure are decoupled using the

canonical modal decomposition approach [14]. The decoupled equations involve

only real parameters and the maximum response of the structure is expressed in

terms of the sine spectrum, that is related to the displacement response

spectrum, as well as the cosine spectrum. The cosine spectrum is determined

in approximation from the sine spectrum. A proper modal combination rule for

sine and cosine spectra is derived and the maximum respons4r quantities are

determined taking into account the effect of cross correlation of modes with

closely spaced frequencies. Similar to the response spectrum approach for

analysis of classically damped structures, the present approach is simple fur

practical applications and only the information of the response spectrum f01:

the ground acceleration input i. needed.

The propo.ed respon.e spectrum approach is employed to approximate the

maximum reapon.e of a number of noncla •• ically damped primary-secondary

structural systells. Particular empha.is is placed on evaluating the maximum

response of primary-secondary structure. in which the effect of noncl.ssical

damping is known to be .ignificant [14]. including the tuning of the secondary

system. Numerical result. indicate that the accuracy of the proposed response

spectrum approach is quite re.sonabl. in comparison with the exact solution.

1-2

SECTION 2

BACKGROUND

The response of a linear n degree-of-freedom viscously damped structure

subjected to a ground acceleration can be obtained by solving the

following matrix equation of motion

(1)

in which ts, ~, and K denote the (nxn) mass, damping, and stiffness matrices of

the structure, respectively, and X is an n displacement vector relative to the

~oving base. The vector X is a unit vector X - [1, 1, ... ,1)'. The super dot

f.ndieates differentiation with respect to time and an underbar denotes a

vector or matrix. In Eq. (1). the argument of time, t. for X and x has been g

omitted for simplicity.

Caughey and O'Kelly (1) showed that if the damping matrix satisfies the

identity ~ H- 1 K - K H-1 ~. the matrix of the eigenvectors of the undamped

system, !. can be used to transform the equations of motion into a set of n

decoupled equations. The eigenvectors are found from the solution of the

following

2 1.01.1 ts!j - !!j - 0 (2)

2 in which Wj and!j are the j th eigenvalue and eigenvector, respectively,

where CooIj

is the jth undamped natural frequency of the structure.

The equations of motion are decoupled using the transformation X - •

l in Eq. (1), where ! - [ !1' !2' .'. I !j] is the modal matrix.

deeoupled equation of motion is given by [2)

Tne jth

(3)

in which Yj is the jth element of I, 'j - ~' ts X I .' ts. is the jth modal -j -j-j

2-1

participation factor, and Ej - !j ~ !j / 2wj is the jth modal damping ratio.

In the expressions above, a prime denotes the transpose of a matrix or vector.

The solution of Eq. (3) can be expressed in terms of the well known

Duhamel's integral

(4)

in which

1 -- t Io exp[ -Ejwj(t-r)] sin wOj(t-r) xg(r) dr (5)

is the relative displacement with respect to the ground for a SDOF oscillator

2 1/2 with frequency Wj and damping ratio Ej . Note that wDj - wj(l - Ej ) is tha

jth damped frequency. Generally, the quantity of most interest is the maximum

value of Yj(t), denoted by Yj . For small values of damping (e.g., E < 20') it

can be approximated by

(6)

in which

S (e,w) -v Jt exp[-ew(t-r)] sin w(t-r) i (r)dr o g ~

(7)

is called the spectral pseudo-velocity respon.e of the ground motion. Finally,

the maximum value of the responae vector X(t), denoted by i, can be

approxim4ted by the well known .quare-root-of-sum-of-square. (SRSS)

procedure

1/2

(8)

However, if the damping II&trix, k, is not of the cl.ssical fora, the

eigenvectors of the undamped syste. will not diagonalize the damping .. trix.

For such syste.s, the approach proposed by Foss (4) and Trai11-N •• h [10) can

be used. In this approach, Eq. (1) is converted into a 2n first order .. trix

2-2

equation

(9)

where ~ is a 2n state vector, Z -{!} and

~-[:I:l. 8- [-IS Q 1 { Q } ~j , l!- ..... Q K -Ii 1:

(10)

The eigenvalue problem of Eq. (9), I>. Al + ~I - 0, can be expressed as

follows

(11)

in which >'j and !j are the jth eigenvalue and eigenvector, respectively. If

the systell is underdamped, the eigenvalue. and ei~411vectors are n pairs of

complex conjugAte. The jth pair of eigenvalues can be written as

(12)

in which 1 - G , CoIOj - IoIj(l-(j 2)1/2, where IoIj

is different froll the jth

natural frequency of the corresponding undallped systell.

The response state vector, Z. can be expressed as a linear combination of

the eigenvectors

Z - ! ~ (13)

where ! - [ !1' !2.···. !n ] is a (2nx2n) cOlDplex 1I0dal matrix. Sub­

stituting Eq. (13) into Eq. (9) and prellu1tiplying it by the inverse of the f -1 matrix,! ,one obtaina a set of 2n decoupled equations

j - 1,2 •...• 2n (14&)

in Which Vj and Qj

are the jth e1e.ent of the ~ and Q vectors, Where

Q _ t'l {. ::.}. (14b)

Solutions of £q. (14a) together with t~e transforaation of Eq. (13) yield

2-3

the response state ve::tor ~(t) of the structural s·J'item. Since).j and !j are

complex parameters that occur in complex conjugate palrs, the final response

quantities are all. real.

Igusa and Der Kiureghian (6). and Veletsos and ventura (11) expressed

the solution of ~, Eq.(13), in terms of the Duhamel's integral and its

derivative

~ - j~l [~j hj (t) + ~j hj (t)] (15)

in which ~j - -2 Re[ !jQj).j ] and ~j- 2 Re[ fjQj]' In Eq. (1:), hj(t) is the

relative displacement of a SDOF system with frequency Wj and damping ratio

~j given by Eq. (5); whereas hj(t) is the rel.ative velocity with respect to

the ground. For classically damped systems, 13j

- 0, and as expected, the

response can be expressed in terms of hj(t) alone.

In practical design applications, the designer is provided with the

description of the ground motion in terms of a response spectrua. The

response spectrum is the plot of the maximUII pseudo-velocity, s « ,W), v

defined by Eq. (6) as a function of frequency wand damping ratio (. The

respon.e spectrUII also contains information regarding maxiCIUII relative

displacement. Sd«('w), and

they are related as follows

S « ,w)/w v

s «(,w) - w s «(,w) a v

the maximWl pseudo-acceleration, S «(,w), since a

(16a)

(16b)

Thes. thr.. quantities, i.e. , Sc:lce,w), s (e,w) and S «(,w) are plotted in v a

a slngle chart against frequency in a so-called tripartite losarlthalc plot.

It follow. fro. Eq. (15) that the determination of the maximum response

of a noncla .. lcally damped structure requires not only the maxillUll relative

displace.ent Sc1(e ,w) but also the maximum relative velodty, h(e ,w)

maxi. Ih(t)l. Note that the maxillUll relative velocity, ~(e,w), is different

2-4

from the ~ximum pseudo v~locity, S (~,w). v

Since, however. the maximUID

relative displacement Sd(~'w) is the only information available to a

designer, it is desirable to obtain the maximum velocity, h(E.w), in term

of Sd<{'w). Attempts in this regard have been made by several researchers in

the following.

Villaverde and Newmark (12) assumed that for small values of damping

ratio e. the maximum relative velocity. h({.w). can be approximated by the

maximum pseudo veloCity S (e.w). v When the ground acceleration excitation is

assumed to be a stationary white noice process. the relation between the

maxilllUll relative velocity h(E .i00i) and the maximWl relative displacement

Sd(E.w) can b. obtained in approximation. Such a relationship was assumed to

hold for general earthquake excitations to obtain h(~ ,w) from Sd<{ ,w) by

Igusa and Der Kiure~hian (6].

Gupta and Jaw (5) compared the maximum relative velocity. h(E. i00i), and

the maximum relative displacement. Sd<e.w). for several earthquake records in

different frequency ranges and concluded that for intemediate values of w

(i.e .• 1< w <10 hz.). the maximum realtive velocity h({.w} can be

approxim~ted by the maximum pseudo-velocity S (e.w). v Further. procedures are

proposed for approximating the IUxillUll relative velocity li(e .w) using the

knowledge of the IlAxillUll relative displacement Sd(e .w) for other frequency

ranges.

In this paper. we propose an alternate approach for approximating the

maximum resp~nse of a nonclassically damped system from the knowledge of the

response spectrum. Sd«('w). This is accomplished using the "canonical modal

analysls· formulated by the authors for the analysis of nonclassically damped

structures [141.

2-5

SECTION 3

CANONICAL MODAL ANALYSIS

The equations of motion for the state vector, Eq. (9), are rewritten as

follows:

(In

where

[ -1 -1 1 {} H & -li K -1:

~ - I I Q ; li - . 'g' (18)

The eigenvalues and eigenvectors of matrix ~ are identical to those of

Eq. (11), denoted by ~j and!j' respectively, for j - 1,2, ... , 2n, lee Eq,

(12). Further, the jth pair of eigenvectors can be expressed as

!2j-l - !j + 1 ~j

!2j - llj - 1 ~j j-l,2, ... ,n

in which llj and hj

are 2n real vector.

The (2nx2n) real matrix I constructed in the following

will transform the matrix ~ into a canonical form~, i.e.,

~ - I-I ~ I

in which 1-1 is the inverse of the I matrix and

A _ [~1 ~2 .. 0 1 o 'A

-n

where

• j - 1. 2, .... n

3-1

(19)

(20)

(21)

(22)

(23)

Let the transformation of the state vector be

Z - I ~ (24)

Substituting Eq. (24) into Eq. (17) and premu1tip1ying it by the inverse of

-1 the I matrix. I . one obtains

. v - A II + I x

- - g ( 25)

in which A is given by Eqs. (22) and (23). and

Equation (25) consists of n pairs of decoup1ed equations. Each pair of

equations represents one vibrational mode, and it is uncoupled with other

pairs. However. the two equations in each pair are coupled. The

tran~"::ormation given in Eq. (24, is referred to as the canonical

transformation [14). All the parameters in Eq. (25) are real.

The jth pair of coupled equations in Eq. (25), corresponding to the jth

vibrational mode, is given as follows:

(27a)

(27b)

in which F2j

_l and F2j are the 2j-lth and the 2jth elements of the [ vector,

respectively. Solutions of Eqs. (27) together with the transformation of Eq.

(24) yield the response state vector Z(t) of the structural system.

The advantage of the formulation given above is that the computations for

the solution. are all in the real £leld. The modal de.ccmposition approach

described above is referred to as the canonical audal decomposition.

There are a number of procedures that call be used to solve Eqa. (27) as

explained in details in Ref. 14. Let hv (t) be the impulse response function j

of the jth vibrational mode, i.e., Xg 6(t) and ~j(t) -[v2j_l,V2j)' - bv (t) j

3 -·2

where &(t) 1s the Dirac delta function. Then b (t) can be easily obtained Vj

from Eq. (27) as

-ej'-ljt cos '-IOj t) F2j -1 +

-(j'-ljt sin c.lDj t) F2j (e (e

II (t)- (2ij) Vj

t -<e-(j'-lj -E '" t ~in c.lDjt) F2j _l + (e j j cos c.lDj t) F2j

The response of the jth vibrational mode ~j<t) - [V2j _l , v2j ] 1s given

by

j - 1, 2, ... n (29)

Using Eq. (29) and the transformation of Eq. (24), the displacement vector can

be expressed as:

(30)

in which

(318)

(Jlb)

r: -{ Col (t-f)

Sj (t) - e j j dn c.lDj (t-f') i (r) dr g (llc)

r: -( Col (t-f)

Cj(t) - e j j cos "'Dj(t-f') ig(f) dr (lld)

where AjL and ~jL are the lower halve. of the &j and ~j vector., re.pectively.

For cla •• ie.lly aa-ped .tructure., the vector ~j c.n be shown to be null

and, •• expected, tbe response reducea to teras involving the slne integrals

S j (t) only. Of course, the lI&XilllWl value of integrals involving the sine

terma is nothing but the pseudo-velocity S «(,w) which can b. obtained from v

3-3

the response spectrum Sd<€'w), Eq. (16a).

For nonclassically damped syst.ems, the maximum response involves not only

sine integ"Cals. but also cosine integrals, The following

section outlines a simple procedure that can be used to evaluate the maximum

of cosine integrals from the pseudo-velocity spectrum of the ground motion,

i.e., maximum of the sine integrals.

3-4

SECTION 4

EVALUATION OF COS IN! SPECTRUM

The maximum value of Sjet) that involves sine integral denoted by 5j -

maxi. \Sj(t)\, is nothing but the pseudo-velocity, i.e., 5j - Sv(€j'wj } - Wj

Sd(€j,wjJ. The plot of Sj against the frequency, wj ' for a given damping

ratio, € j' is the pseudo-velocity spectrum, and herein referred to as the

"sine spectrum". The maximum value of Cj(t) that involves cosine integral Eq.

01), is denoted by <:j - maxi. \cjet)\. The plot of Cj against the

cre ::juency, wj ' for a given damping ratio, € j' is herein referred to as the

"cosine spectrum". Note that 5j

and <:j (or sine and cosine spectra) are

functions of Wj and €j' and because of simplicity in notation these arguments

are omitted.

The pseudo-velocity spectrUII of an earthquake record can be generated

either directly by numerical integration of Eq. (5) or using an approximate

procedure developed by Newmark, Blume and Kapur [8]. In the latter case the

pseudo-velocity spectrum (or the sine spectrum) is approximated by the

knowledge of the lIaxilllUll values of the ground acceleration, ground velocity

and ground displacellent. Such an approximate procedure is based on an

empirical study of the respon.e spectra of a large number of earthquake

records.

In this study, the pseudo-velocity spectrum or the sine spectrum will be

obtained using either one of the two approaches mentioned above, whereas the

cosine spectrum will be evaluated, in approxillAtion, froll the sine spectrum.

In order to evaluate the approxillAte relation between the sine spectrum

and the cosine spectrua, the earthquake ground acceleration i (t) is ~del.d g

expediently as a uniformly lIodulated non.tationary randoll process with zero

mean

(32)

4-1

in which a(t) is a deterministic non-negative ~odulating or envelope function

and XO(t) is a stationary random process with zero mean and a power spectral

density, ~xx(w). The stationary random process Xo(t) can be expressed 1n

terms of a summation of sine functions as follows

N xo(t) - L ~ sin(""'k t + ;k)

k-l (33a)

in which ;k's are independent random phase angles distributed uniformly in

[0,211) and ~ - [ 2;xx(wk ) w ]1/2 with ~xx(""'k) being the power spectral

density of xO(t) evaluated at frequency wk - kW. A commonly used form of the

spectral density, .xx(w), is that given by (2,7)

2 2 1+4~ (w/I.) )

g g

[ 2] 2 2 2 l-(w/w) + 4e (,,",/101 ) g g g

* (33b)

in which eg , wg ' eO' wO' and S are parameters depending on the intensity and

the characteristics of the earthquake at a particular geological location.

Various typ •• of the envelope function aCt) have been suggested in the

literature to introduce the non.tationarity of the ground acceleration into

Eq. (32). One po •• ible fora of aCt) is: aCt) - (t/tl

)2 for 0 s t s t l , aCt)

- 1 for tl s t s t 2 , and aCt) - exp [P(t-t2) ] for t > t 2 · Note that t 1 ,

t2 and ~ can b. .elected to reflect the shap. and duration of the earthquake

ground acc.leration. When aCt) - 1, the ground acceleration i. a .tationary

random proc •••. Th. stationary a •• umption i. r.a.onabl. wh.n the duration of

the .trong .haking of the .arthquake ground motion is much lonb~r than th.

natural period of the structur •.

Thus, the earthquake ground acc.l.ration can b •• xpr •••• d a. follow.

(34)

4-2

Note that for a giv~n set of ;k values (k-l, 2, ... , N). Eq. (34) represents a

sample time history of the ground acceleration x (t). Substituting Eq. (34) g

into Eq. (31) and changing tor to a new integration variable r. one obtains

N

J: o(t-<l -~ w r

Sj (t) - L ~Ksin[wk(t-r) ~ .k] e j j sin w r dr k-l

Dj (35a)

N

J:O(t-') ... in["k(t-.) + 'oj -~ w ,

Cj

(t) - L e j j cos wDj ' d, k-l

(35b)

It is mentioned that wk's for k - 1.2 ...• N are the frequency values of the

power spectral density of the random process XJ(t); whereas Wj and ej

are the

natural frequency and damping ratio of the single-degree-of-freedom

oscillator.

The maxiaull value. of expression. in Eqs. (35a) and (3Sb) have been

denoted by Sj and Cj

• r.spectively. Cj

will b. obtained in approximation .s

follows.

When Wj is very slDall, both the exponential term, and the

cosine term,

results in

Cj (t)

N

1: - r k-l

N

1: -r k-l

- x (t) g

in Eq. (3Sb) can b. approximated by unity. This

~ Q(t-r) sln[wk(t-r) + ;k] dr

d. - I: ~ Q(r) sin(wkr + ;k) x (r) dr (36) g

Therefore, the maximua value of Cj(t) for ... 11 Wj 1. the maximum ground

velocity. Le .• Cj

- maxi. lig(t) I. In other word., the cosine spectrum in

the small frequency range 1. equal to the maximum earthquake ground velocity.

4- 3

When w.l is very large, the natural period Tj of the oscillator is very

short and hence the stationary period of the earthquake is much longer than

the oscillator pe~iod Tj

. Thus, the maximum values Gj

and Sj will occur in

the stationary portion of the response. Conse~uently, the earthquake ground

acceleration can be modeled as a stationary random process, i.e., oCt) - 1,

and C.l(t) can be expressed as

N Cj(t) - L

lo.-l (37)

Using exponential relations (Euler's equation) for the sine terms and cosine

terms in Eq. (37), the integral can be evaluated as

(38)

Uhen Wj i. large compare to ~, the above expre •• ion can be approximated

by

in whlch

1 +-2 Wj

(39a)

(3gb)

• Let t b. the ti .. at which Cj(t) reaches It. maximum value, 1.e., Cj

-

* Cj(t}. Then, Eq. (39.) beeo ...

4-4

For ~j ~ 0, Eq. (40) can be further simplified to

Cj - 12 [~j"'j Xg(t*) + Xg(t*) ] "'j

(40)

(41a)

since the summation of sin ~k or cos ~k for k-1,2, ... ,N is close to zero.

Eq. (41a) represents the maximum value of Cj(t) when "'j is large. Note

that Ej

is generally much smaller than unity. EXaDlination of Eq. (41a)

reveals that ~ "'j 1I D2k extremely lAI&A, Cj ~ ~ approximated AI ~

inversely proportional !2 ~~.

Cn the other hand, when "'j is extremely large, Ej"'j Xg(t) is the dominant

term and Cj reduces to

(41b)

In a similar manner using Eq. (35a) with aCt) - 1, the following

approximate expression for the sine spectrum Sj(t) for "'j very large can be

obtained as

(42)

If t denotes the time at which Sj(t) reaches its maximum value, i.e.,

" Sj - Sj(t), then Eq. (42) can be simplified as

.. Sj - Xg(t)/"'j - maxi. IXg(t)/"'jl (43)

provide that (j ~ O. Equation (43) indicates that in the high frequency

region (i.e., Wj very large), the relative pseudo-spectral acceleration ",j Sj'

is constant and equal to maximum ground acceleration i (t) as expected. g

A comparison of Eqs. (41b) and (43) indicat.s that when "'j is extremely

large and (j .. 0, the cosine spectrum is related to the sine spectrum as

follows

4-5

(44)

In the intermediate frequency range for Wj' closed form expressions for

Sj and Cj are not tractable.

ground acceleration X (t) is g

However, for a comparison of 5j

and Cj

, the

approximated by stationary white noise with

zero mean and a power spectral density SO' Then, and are

st.:ltionary random process with zero mean and their mean square values (or

variance) can be obtained as follows (see Appendix).

(45a)

(4Sb)

It follows from above equations that for small damping ej

the variances

of Sj(t) and Cj(t) are identical. Thus, the maximum values of Cj(t) and Sj(t)

are approximately equal, i .•.• Cj

= Sj' This, indeed, has been verified by

comparing the exact Cj

and Sj values for several earthquake records for

intermediate values of wj

.

Noted that in Dlost situations the =esponse spectrulll used for design

purposes is approximated by a series of straight lines in the tripartite

logarithmic paper and it i. referred to as a smooth spectrum herein. These

straight linea correspond to regions of amplification of the ground

acceleration. velocity and displacement. There are also guidelines available

that can be used to approxilll4te a smooth spectrum based on only the knowledge

of the maxilllUla ground acceleration, velocity and dlaplace .. ent [8 J .

Therefore the cosine spectrum will be obtained in approxilll4tion froll the

available .lIOoth alne spectrum using the result. derived above. Tht. will

lead to a smooth co.ine spectnmt, and the procedures are described in the

following.

Exact sina and cosine apectra for .everal earthquakes have been

constructed and all sine spectra have been smoothlad out with straight lines.

4-6

A comparison betw •. en the smooth sine spectrum with the respective cosine

specttum indicates that at low frequency values of "'j' the cosine spectrum i&

constant and is equal to maximum value of ground velocity, Eq. (36), up to the

frequency value, ~ . at which it intersects the sine spectrum as shown in Fig. , 1 by segment 1. This point of intersection, ~ , determines the beginning of

a

the intermediate region in which the two spectra are almost identical. The

upper frequency, ~, at which the two spectra are almost identical can be well

approximated by the lar~est frequency at which the sine spectrum is equal to a

constant amplification of the ground acceleration. This frequency, wb ' can

easily be determined from the smooth sine spectrum, see Fig. 1. Beyond ~,

the cosine spectrum can be well approximated by a straight line on the

log-log paper with a slope of -1. see Eq.(41a), as shown in Fig. 1 by the

segment (3).

When Wj is extremely large, the cosine spectrum can be approximated by

multiplying the sine spectrum by the damping ratio {j as shown in Eq. (44).

Thus. in the extremely large frequency region, say Wj > 100 cps, a straight

line parallel to the sin. spectrum but its ordinate being equal to {j Sj can

be drawn as shown in Fig. 1 by the segment (4). The intersection of segments

(3) and (4), denoted by we' is the beginning of the region where Cj

= ejsj,

Eq. (44).

Thus, the cC"sin. spectrum is completely define if the maximum value of

the ground velocity is known. Note that the value. of the maximum ground

acceleration and ground displacement can b. obtained from the sin •• pectrum.

Unfortunately, the value of the maximWl ground velocity cannot be directly

extracted from the slne spectrum. Based on emplrical study of a large number

of earthquakes, the average values for the maximum ground velocity was

expre.sAd in teras of the .. ,ilium ground acceleration and the site condltion,

(i .•. , aol1 type), In Ref.B. For rock, the aaximua value. of ground velocity

4-7

and ground displacement are 36 in/sec. and l~ in., respectively, for every 1 g

in/sec. 2 of maximum ground acceleration. For alluvium, the maximum values of

ground velocity and displacement are 48 in/sec. and 36 in, respectively, for

2 every 1 g in/sec of maximum ground ar.:celeration. Therefore, by comparing the

above values with the maximum values of ground acceleration and displacement

obtdined from the sine :;pectrum, one can determine the soil types and the

approximate maximum value of the ground velocity.

Figures 2a through 2h presents the exact and smooth sine spectra for

several earthquake ground accelerations. Also shown in these figures a~e the

approximate smooth cosine spectrum obtained from the knowledge af the

smooth sine spectrum. Fa~ comparison purposes the exact cosine spectrum which

was obtained using a O\'JIlerical itegration of Eq. 35b is also presented.

4-8

,a 3

to ... .. U III ~ til

GI d ...t III

8

~ III d

...t til

.,;: U o

3<fl ~ ---- --- - ---- - ---- ----- - ---- -.-

4-9

PICOU 2a

10· ,-ry"..," J " v~ :J ...,

-= ::; 10

3 ~ oJ U ~ Q. .."

<II C

.."

10- 1

Exact

10-.1 Smooth

1( •. ~ 10- 1 10 101

103

lOa u GI oJ> ..... 5 --! 10 ~ .. u ~ '" ~ ... .... 0 U

1

10- 1 Exact

Appr.Jximate

10-a 10-2 10·' 10 102

Frequency, Cpt

Sine and Cosine Spectra for E1 Centro Earthquake, E - 2'

4-10

FlCUU 2b

103 ---------------

102

':.t "'::- 10 :: ~

• --~ .r. it

/". 1(; 1 - Exact

Smooth

10"'2 10-2 10- 1 10 102

103

~ f;! 102 ~ ,=.

• c:: ... ~ -T.

10

l! ... III

8 1

Exact

Approxilllate

Frequency. c p8

Sine and Co.in. Spectra for 11 Centro Earthquake, ( - 20t

4-11

FlGuai 2c

102 I\jY ....

,.. :.;

10 " "II .. :.;

~ .. .. '" 41 CI. III .. c:

10' , ... :1\ - Exact

Smooth

102

10-2 10-' 10 102

103

10~

-tJ . ____ .... _<0_ U .. .... • u 10 --• " .. .. l III

I ..4 • 8

10" Exact

"pproxill.ate

S1ne and Co.La. Spectra for San Fernando Earthquake ~ - 2'

4-12

FIGUU 2ei

10l

OJ :. 11 -... e 10 '" -~ .. ... './ III Q.

'J'J

III C ...

en

10' Exact

Smooth 10.2

10.2 10·' 10 102

10'

... 102 u

GI • .... • u -~ 10 .. .. u X-CII

III c: , .. • 8

10" Exact

4pprol i.a te

10-2

10.2 10- 1 10 102

Frequency, cp.

S1ne anei C081ne Spec era for San Fernanclo Earehquak •• ( - 20,

4-13

FIaval 2.

102 ,1'\

'" .. :II .... 10 • y ~

~ ... .. y ~ a.

<II

" C .... '" 10. 1

162 I 10-2 10. 1 10 102

101

102

-y • • ..... • ~ 10 ! .. ... I en

• 1 a ... e 8

10"

"pproxi.ate

10'2 10.2 10- 1 10 102

Frequency, cpe

Sin. and Co.lna Spectra for Mexico City Earthquake, E - 2t

4-14

FlGOU 2f

.J

'" " --ii u

B ~ .. ~

" 1I ... :r. OJ C .. I/)

102

10

10·'

10.2

10=·~2~--~'~0~-'~--~-----1-0----~'02

103 r-------------------------~

Sine and COline 3pectra for Kexico City Earthquake, E - 20,

4-15

FIGOU 2,

.... !I .;.

~ .. ~

~ ;; ;, c

'"

... ~ • -• ... -! .. .. t

CI'I

II = ... • a

102

~---- --/ .... , " / \

'0 I

\ /'

\l - 1 I

10 Exact

Smooth

10-2

10-2 10- , 1 10 102

10

1

ElI.act

A.pprox imate

Frequ.ncy. cpa

Avera,e Sine and COline Spectra for Twenty Si.ulated Ground Acceleratlon., ( - 2'

4-16

~ --~ ~ .. .. '" ;, c. ;/l

" = ;/l

u -II • -• u

I .. ... 1 :II

= -• d

IIGUU 2h

102 I

._-_.

/--~,

10 /""\

~ " "

- 1 10

Exact

102

Smooth

10-2 10- 1 10 102

10

1

-. 10 E'(act

~pprOXlmate

102

10-2 10- 1 1 10 10 2

Frequency. cpa

Avera.e S1na and Co.lne Sp.ctra for Tven~y Slmulated Ground Aceeierationa, e - 20.

4-17

SECTION S

COMBINING MAXIMUM MODAL aESPONSES

We are now in the position to obtain the approximate maximum structural

response from the sine and cosine spectra. Recall that the vector of

structural response ~ is given by Eq. (30), i.e.,

The earthquake ground acceleration is modeled as a random process with zero

mean. Further, it is assumed that the stationary segment of the ground

acceleration is long compared with the period of the structure, so that the

structural response can be considered as a stationary random process with zero

mean. Thus, Sj (t) and Cj (t) are all stationary random processfls with zero

mean. The stationary variance of the structural response can be obtained from

Eq. (46) as follows

n n - L L [ri rj P(Si,Sj) DS as + 2rj ~i p(Sj'C i ) DS DC

i-l j-l i j j i

1n which a Sj

are the

+ ~i ~j p(Ci,Cj ) aCiaCj

]

standard deviations of Sj(t)

(47)

and

respectively, and p(:Ji,Sj) - E[Si(t) Sj(t)] I aSiaSj

coefficiant of Si(t, and Sj(t). Similar definition holds

co.ffic~ants p(Ci,Cj ) and P(Si'Cj ),

is the correlation

for the correlation

For a stationary random process with zero mean, such a. Sj(t) and Cj(t),

the mean of the maxiJaulll value may b. expre .. ed in approximation by a peak

5-1

factor 1 multiplied by the standard deviation. Assuming that the peak factor

-, for all stationary processes Cj(t) and Sj(t). for j - 1.2 •.... n. as well as

the response vector process ~(t) is identical. i.e .• Sj- -'C7Sj

' Cj - .., C7Cj

for j - 1.2, .... n and ~ - maxi. 1~(t)1 - -, ~~. one obtains from Eq. (47) the

square of the maximum response as follows

I~I ~ ~ I [ri rj P(Si,Sj) 5i 5j + 2r i ~j P(Si,Cj) 5i Cj i-l j-l

(48)

2 in which the jth element of the vector I~I given above is the square of the

maximum value of the relative displacement to the moving bllse of the j th

floor.

For classically damped system. ~j - 0 and Eq. (48) redu~es to the

Complete Quadratic Combination (CQC) methed (13). Since the sine and cosine

spectra. Sj and Cj

, have been estimated previously, the remaining step is to

evaluate the correlation coefficients in approximation. Der Kiureghian [ 3 )

computed the correlation coefficients P(Si,Sj) for the sine spectrum using

the filtered white noise as the input excitation and compared with the

corresp<'nding results when the input excitation is white noise. His

correlation coefficientll P(Si' Sj) under filtered

white noise input can very well be approxim3ted by that due to a white noise

conclusion is that the

input whf'n the damping is not very large (e.g. E<20t). Hence. the correlation

coefficiAnts P(Si,Sj) obtained using the white noise excitation can be used

for the computation of the maximum response.

Here, we suggellt the use of the white noise excitation as input to

compute the correlation coefficients, and

appearing in Eq. (48). The resulta are given in the following (detailed

5- 2

derivations are presented in the Appendix)

(49a)

(49b)

(49c)

where

(50a)

(SOb)

- eiwi ... e'j Wj (SOc)

2 2 2 - b ij ~Oi WOj (SOd)

5-3

SECTION 6

NUMERICAL EXAMPLES

In order to demonstrate the accuracy of the response spectrum approach

developed in this paper, the maximum response of several nonc1assically damped

structural systems subjected to several earthquake ground excitations will b.

com:idered. Based on a detailed study conducted in Ref. 14, the effect of

nonclassical damping is significant for certain primary-secondary syst~ms

suO cted to earthquake excitations; particularly if the equipment is lhl- j-

an lt is tuned to a frequency of the primary structure. Therefore, emphasis

is placed on approximating the maximum response of several nonclassica11y

damp~d primary-secondary systems.

All the example problems are subjected to the 1940 E1 Centro, 1971 San

Fernando, 1985 Mexico City earthquakes, and simulated nonstationary ground

accelerations described in Eqs. (33) and (34). The parameters that describe

the envelope functlon, aCt), and the spe~tral density, ~ii(w), of the

earthquake modal are: tl - 3 sec., t2 - 13 sec., ~ - 0.26, Wg - 3.0 Hz.,eg -

2 -4 2 3 0.65, Wo - 0.5 Hz., eO - 0.71 and S - 74.7 x 10 m Isec. jrad. Wlth these

parameters a simulated ground acceleration is shown in Fig. 3.

The maximWl structural responses (1. e. story diap'.acements and story

deformatlons) are obtalned uslng the following approaches.

1. The wilson-' dlrect time hiatory lntegration method. The maximWD

response thus obtained i. exact, referred to as the exact solution.

2. The re.ponse spectrum approach proposed. Recall that this approach

requires the knowledge of the sine spectrum (pseudo-velocity response

spectrum) and co.ine spectrum of the ground motion. The eosine

spectrum may be generated from the time history of the ground

acceleration (if available) or approximated from the .ine spectrum using tbe

guideUnes discu .. ed in section IV. For the example problems studied the

6-1

maximum response is obtained using both t.he excat and approximate cosine

spectra.

3. The approximate classically damped approach. In this approach, the

second order equations of motion are decoupled using egenvect:Jrs of the

undamped system by disregarding the off-diagonal terms of the .' ~ ~ matrix,

~here • is the (nxn) modal matrix of the undamped system. Then the maximum

response is obtained from the sine spectrum using the SRSS procedure.

The above solutions are obtained for a particular earthquake, such as El

Centro or simulated sample eartllquake. Likewise, The maximWII response of all

the example problems were also computed using the average sine and cosine

spectra for twenty simulated ground motions having the power spectral density

and envelope function described previously. Ho~ever for this situation one

cannot obtain the exac -: maximum values of the response quanti ties. The

maximum structural response obtained using various approaches will be compared

to demonstrate the validity of each approach.

The first example problem consists of a two-degree-of-freedom shear beam

type structure. This structure is classically damped if Cl/kl - C2/k2' The

mass and stiffness of each story unit are: ml - m2 - m - 30 tons and kl - k2

- k - 19,379 kN/m. The natural frequencies of the structure are 2.S Hz and

6.5 Hz, respectively.

Let values of Cl and C2

be equal to 123.4 kN/m/sec. so that the structure

is classically damped with first modal damping ratio of S,. Now the

distribution of the damping is varied so the structure becomes nonclassically

damped. Two nonclassically damped structures are considered. Firat all the

damping Jf the structure is placed in the first story unit; with the results

CI - 246.8 kN/m/sec. and C2 - 0.0. Next all the damping of the structure is

placed in the second story unit, leading to the results C1 - 0.0 and C2 -

246.8 kNIll/sec. Tables I-a through l-c pre.ents the DSxillWII f~Ory

6-2

deformations (Ul

,U2

) of the structure with these three different damping

distributions. An examination of the Table indicates that, as expected, the

effect of nonclassical dampi.ng on this type of structural system is not

si.gnificant and that either of the response spectrum procedures (i.e.

classical or nonclassical damping) predicts maximum structural responses that

are in close agreement with the axact solutions.

Next consider the classically damped two-story structure of example 1 in

which a light single-degree-of-freedom equipment is mounted on the top floor.

Such a primary structure is classically damped if Clikl - C2/k2 where Cl - C2

- 123.4 kN/m.sec. and the combined equipment-structure system is classically

C2/k2 - Celke where the subscript e referes to the

equipment. In Ref. 14 it was shown that the effect of nonclassical damping

for this equipment-structure system is significant if (i) the eqUipment is

tuned to a frequency of the primary structure, (11) the equipment mass is

light compared to the tuned modal ~.ss of the primary structure, and (iil) the

equipment damping ratio is smaller than the damping ratio ~ that results in ec

a classically damped equipment-structure system.

Let the equip.ent frequency, w , be tuned to the fundamental frequency of e

the primary structure, 1.e. we - 2.5 Hz. and the mass ratio (equip.ent rna ..

over the fir.t modal ma •• of the primary structure which is 30 tonal be equal

-4 to 10 . For this equip.ent-structure .ystem the value of ~ i. equal to 5t. ec

Tables 2-a through 2-c pre.ents the maximum response of the equipment

structure .yste. for 3 different damping ratios of the equip.ent i.e. ee - Ot,

St. and lOt.

Examination of the re.ult. in Table 2-a, which correspond to E - Ot, e

indicate. that the displacement of the equipment relative to the attachment

point using the re.pon.e spectrum approach developed in this paper is only 6t

higher than the exact solution for the El Centro earthquake. 12t higher for

6-3

the San Fernando earthquake. l4t higher for the Mexico City earthquake. and

3lt higher for the simulated earthquake. Whereas using the approximate

classically damped approach, the response is 73t lower than the exact solution

for the [1 Centro earthquake. 71t lower for the San Fernando earthquake, 25'

lower for the Mexico City earthquake. and 60t lower for the simulated

earthquake. A comparison of the results obtained using the approximate cosine

spectra with those obtained using the exact cosine spectra for various

earthquakes considered except the Mexico City indicates excellent agreement.

For Mexico City earthquake the maximum equipment response using the

approxiaate cosine spectra is 46t lower than that using the exact cosine

spectra. ~ .xp.ctod, the effect of nonclassical damping on the response of

the primary structure is inSignificant and the maximum response. obtained

using differ.nt approaches are in close agreement. Results obtained using the

response spectrum of a single grcund acc.leration reco~d and those obtain.d

using the av.rage response spectrum of twenty records exhibit similar trends.

Table 2-b pres.nts the maximum response when the equipment- structure

system is classically damped (i.e. ~ -~ -5t). e ec Of course for this situation

the nonclassically damp.d response spectrum approach reduces to classically

damped response spectrum prodecure and these results are in good agr •••• nt

with the exact solutions.

Table 2-c pr.s.nts the results when the equipment damping ratio is lOt.

EXAllination of the equip •• nt displace.ent relative to the attachment point

indicat.s that the equip.ent response using the response spectrum approach

developed is again in good agr.e.ent with the exact solutions. Th. aa1tillUll

deformation of the equipment is within 2lt of the exact solution for all four

earthquake •. However, the equipment respon •• is 250t higher than the exact

solution when the approxiaate classically damped approach !.. used. The

results obtained using approxlaate eosine spectra are in reaarkable agr.ement

6-4

with those using the exact cosine spectra for all earthquakes. Again. for the

simulated earthquakes the results obtained using one and twenty sample records

exhibit similar trends.

Next we ~xamine the response of the equipment mounted on top of the two

nonclassically damped structures of example 1. Again the equipment is tuned

to the fundamental frequency of the primary structure and three different

damping ratios for the equipment are considered (i.e. ~e - 0 •• 5' and 1o,).

Tables 3-a through 3-c present the maximum story dp.formation for the

equipment-structure system in which all the dampings of the primary structure

are placed in the lowp.r story unit. The corresponding results for the case in

which all the dampings of the primary structure are placed in the second story

unit are presented in Tables 4-a through 4-c.

It is observed from these tables that the maximum eqUipment deformation

based on the response spectrum approach developed in this paper is within 44'

of the exact solutions. However, based on the approximata classically damped

procedure the equipment responses deviate up to 490t of the exact solutions.

On the other hand, the effect of nonclassical damping on the primary structure

response is insignifant. The re.ult. obtained using the approximate cosine

spectra are very close to thos4 obtained from exact cosine spectra.

Suppose the equipment is detuned and its frequency is chosen to be the

average of the first tvo natural frequencies of the primary structure.

Similar to the tuned equipment-structure system, three different damping

diatributiona for the primary .tructure are considered and the equipment is

undamped, i.a., ~-o. a

Table. 5-a through 5-c pre.ent the respon.e of the

datuned equipment-structure .y.tem. It 1s ob.ervad from these tabla. that the

accuracy for tha propo.ed responae apectrwa approach, is well within 40',

wherea. the accuraey for the approxi_te cla .. 1caUy damped procedure h

within 711 of the exact .olution. Fur the more , the re.ults obtainad using

6-5

exact and approximate cosine spectra are very close. Finally. the response

due to one simulated earthquake or twenty samples of simulated earthquakes

exhibits similar trends.

6-6

__ --------~--------_.o M

it') c: 0

N ~ u

en • '" ~

Q/ .-I Q/ U U

0 U < ~ N ~ c: ::I 0

~ '" C)

'0

~ III u

U') II .-4

~ ::s ... • ~ (I)

0 C¥) .... a u ... fit

~ N M I I

6-7

TAB

LE 1

: K

AX

IKU

H R

ESPO

NSE

OF

TWO

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REE-

OF-

FREE

DO

K S

TRUC

TURE

(e

M.)

<al

C1

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2 -

12

3.4

kN

/ •.•

ec.

San

Fer

nand

o K

exic

o C

ity

S

1l1N

lac.

d S

imu

late

d

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cro

(1

su

pl.

) (2

0-u

lllp

le)

U1

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U1

U 2 U

1 U 2

Ul

U 2 U

1 U

2

(1)

1.7

6

1.0

9

1.8

9

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(2)

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(3)

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(b)

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16

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(2)

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0

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55

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(3)

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3

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3

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(4)

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(2)

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1

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(3)

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2

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2

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4

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5

(4)

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5

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0

2.3

2

1.3

9

0.7

2

0.3

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2.0

1

1.1

9

(1)

Ex.c~

CO

line

Spe

ctru

m;

(2)

App

rox1

aate

C

oain

e S

pect

rum

;

(3)

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pro

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ace

Cl •

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a11

y D

ampe

d A

ppro

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(4

) E

xAct

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2.

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DE

FO

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1'IO

li o

r T

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SY

STE

M

(CX

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(we

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l'

C1

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2 -

123.

4 k

N/_

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.)

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e. -

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tro

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S

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lat.

4

Sl_

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d

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le)

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U

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U

U1

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0

1 U 2

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U2

U

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1

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1.1

0

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6 1

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1

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0

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10

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1.4

4

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2 1

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70

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(2)

1. 7

1 1

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7

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1

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1

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1

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4

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55

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1.9

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21 .

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~

(3)

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1

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0.

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2

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5

(4)

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1

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1

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4

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1

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O

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15.5

3

(c)

e. -

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1. 7

6 1

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97

1

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1

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1

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0

0.6

1

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0

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1

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1

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1.1

3

11

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(2)

1.

76

1.0

9

10

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1

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1

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3'

1. 5

2 1

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0.

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95

1

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1

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1

2.1

2

(3)

1.7

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6 1

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17

21.1

6 0

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7.0

7

1.4

2

0.1

1

17

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1

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1

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6

(4)

1.1

0

1.0

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9.1

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7

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0

10

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0

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0

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2.0

0

1.4

4

0.1

6

9.6

6

(1)

Ex

act

Co

sin

. S

pectn

a;

(2)

Ap

pro

Kla

at.

Co

aln

. S

pec

tru

a;

0)

Ap

pro

d_

te C

lusi

c:a

l1y

D..

.,.d

Ap

pro

ach

; (4

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xact

so

lutl

on

TABL

E 3.

M

AllK

UN

D

ErO

aKA

TIO

M o

r TU

NID

IQ

UIP

K!M

T-ST

RU

CTU

lZ

SYST

EM

(eM

.)

(we

-w

1'

C 1 -

246.

8 k

N/_

/ •• c

.,

C 2 -

0.0

) (a

) (.

-O

t

El

Can

tro

S

.n

F.r

nan

do

ll

ed

co

Cl~y

Sl_

lat.

d

Sl .

.. la

ted

(l .... 1.)

(2

0-..

..,h

)

U1

Uz

U

U1

Uz

U

U1

U2

U

U1

Uz

U

U1

Uz

U

e e

• •

a

(1)

1.5

5

0.9

8

56

.25

1

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l.

07

60

.50

O

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0.1

6

7.80

1

.16

0

.15

)1

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1

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0.9

' 49

.61

(2)

1.7

0

1.0

7

60

.67

1

. 70

1.0

7

60

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0

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0

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4

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1

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0

.15

)'

.79

I.~S

0.9

' 4

9.6

3

(1)

1. 5

4 0

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12

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1

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1

.0)

13

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0

.57

0

.36

4

.57

l.

U

0.1

3

9.2

0

1.73

1

.01

17

.71

(4)

1.6

1

0.9

' S

4.17

1

.67

1.

41

57.9

7 0

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0.

3Z

6.1

9

1.1

4

0.7

7

29.9

4

(b)

( -~ .

• (1

) 1

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0

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1

1.H

1

.69

1

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1

4.1

5

0.5

8

0.3

6

2.41

1

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0

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10

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1. 5

5 0

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15

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(2)

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4 0

\ 0

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1

4.0

8

1.6

9

1.0

7

14

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0

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0

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2

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1

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0.7

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9 1

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0

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39

1.

67

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3

13.1

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0

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4

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1

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2

1.6

1

1.0

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11

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(4)

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1.7

9

1.6

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1.1

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0

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12

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(e)

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12

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1.0

) 1

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6

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6

4.5

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5

0.7

6

9.2

9

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5

1.0

) 1

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7

(4)

1.6

1

0.9

1

7.5

9

1.6

7

1.1

1

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1

0.6

0

0.1

2

1.7

1

1.1

5

0.7

7

7.69

(1)

luct

Co

lin

e Spect~;

(2)

Ap

pro

.leat.

C

O.l

na Spect~;

(1)

Ap

pro

.lu

c.

C1 ...

leall

·' D

-..

4 A

pp

ro.c

h;

(4)

Ex

ect

So

luti

on

TABL

E It

.

El

Cen

tro

UI

U2

U •

(I)

2.2

1

1. ]

1

1]2

.91

(2)

2.2

9

1.]

1

Ill.

S)

(J)

2.1

1

1. ]

0

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6

(4)

2.2

4

1.]

0

12

2.1

0

(I)

2.24

1

. )5

21

. 79

0'1

(2)

2.2

4

1. )

S

26

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n)

2.1

1

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0

44

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(4)

2.2

4

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0

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~

(1)

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4

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2 24

1

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1S

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(J)

2.1

1

1.]

0

4".

12

(4 )

2.2

5

1. ]

0

1'.

1"

MAX

IMUM

DEF

ORM

ATIO

N OF

TU

NED

EQU

IPM

ENT-

STRU

CTU

RE S

YST

EM

(CM

.)

(we

-w

I'

C1

-0

.0,

C2

-2

46

.8 k

N/_

/s.c

.)

(a)

e. -

0\

San

F.r

nand

o ll

ed

co

Cit

y

Sl .. la

tH

(1

• ...

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U1

U2

U

U1

U2

U

U\

U2

U

• •

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.]6

1

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14

7.2

0

0.7

1

O.U

1

1.6

4

2.0

4

1.2

1

97

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2. ]

6

1.4

3

14

7.n

0

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O

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14

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2

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1

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1

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2.2

0

1.3

6

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0

0.6

1

0.4

2

14

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1

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1.2

1

40

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2.l

l 1

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1"0

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0

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0

.]9

1

6.0

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1

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6

7.1

4

(b)

(. -

S\

2.l

l 1

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26

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0.7

1

0.4

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16

1

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1

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2

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2.l

l 1

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2

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0

0.7

1

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45

1

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1

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2

5.9

0

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0

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6

45

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0

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0

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1

1.9

1

1.9

6

1.2

1

41

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1

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25

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0.7

2

0.]

9

].2

1

2.0

1

1.1

9

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2

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10

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l 1

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1

5.2

4

0.7

0

0.4

2

2.1

0

1.9

9

1.2

0

11.9

6

2.n

1

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1

5.4

5

0.7

0

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2

1.1

1

1.9

9

1.2

0

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2.1

0

1.]

6

....

54

0

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0

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1

4.0

] 1

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1.

21

41

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2.]

2

1. 1

9 1

4.4

9

0.7

2

O.l

I 2

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2

.01

1

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12

.79

(1)

txa~'

co

.in

. Spect~;

(2) Appro~l .. te

Co

.ln

e S

pee

tru

.;

(3) Appro~l .. t.

Cla

•• le

all

, D

a.pe

d A

ppro

ach;

(4

) E

xact

S

olu

tio

n

Sla

alat

.1t

(20

-• ..

,le)

°1

U2

U •

2.4

5

1.4

. 1

2'.

09

2.4

5

1 ..

..

12

1.1

)

2.0

0

1.1

9

67

.80

2.]

9

1.4

S

16.4

9

2.1

9

1.10

4 2

9.0

0

2.1

4

1. ..

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9.1

6

2.3

9

1.U

2

9.1

5

2.19

1

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16

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2.3

5

1.4

5

49

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TAB

L!

S.

El

Cen

tro

U 1 U 2

U •

(1)

1.7

6

1.0

9

2.42

(2)

1. 7

6 1

.09

2.

42

(l)

1

. 76

1.0

9

1.4

2

(4)

1.7

6

1.1

3

2.3

'

(1)

1. 5

4 0

.97

2.

51

0\

I (2

) 1

.54

0

.97

2.

51

N

()

1. 5

] o

.n

1.2

6

(4)

1.6

1

0.9

8

2.6

3

(1)

2.2

4

1.3

5

2.61

(2)

2.2

4

1.3

5

2.61

(3)

2.13

1

.31

1

.16

(4)

2.25

1

.30

2

.63

MAX

IMUM

DE

FORM

ATIO

N 01

TUN

ED

EQU

IPM

ENT-

STRU

CTU

RE

SYST

ER

(CR

.)

(we

-(w

1 +

w2)/

2,

~.

-0

.)

(a)

C 1 -

C 2 -

12

3.4

kN

/a/a

ec.

Sen

Fern

ando

N

exic

o C

ity

S

l ...

late

d

(l

....

. 1.)

U 1

U 2 U

U

l U 2

U

U.

U 2 U

e

• ...

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..9

1.

17

2.1

0

0.6

1

0.)

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1

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0."

3

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1..

9

1.1

7

2.1

0

0.6

1

O.l

I 0

.54

1

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0

."

3.3

'

1.1

9

1.1

7

1."

0

.61

D

.,.

0.4

7

1.4

3

0.1

9

1.4

0

1.'

7

1.2

0

2.4

' 0

.64

0

.34

0

.40

1

.44

0

.16

).

54

(b)

C 1 -

2c

-2

46

.8 k

N/_

/sec

.,

C 2 -

0.0

1.6

9

1.0

6

2.13

O

.SI

0.3

6

0.54

1

.16

0

.75

).

61

1.6

9

1.0

6

2.14

0

.51

1

.39

0

.54

1

.16

0

.75

3.

61

1.6

7

1.0

3

1.3

2

0.57

0

.36

0

.44

1

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0

.73

1

.09

1.6

7

1.11

2.

69

\.1.6

0 0

.32

0

.39

L

iS

0.7

7

3.13

(c)

C 1 -

0.0

, C 2

-2

46

.8 k

N/_

/aec

.

2.32

1

.40

2

.30

0

.70

0.

42

0.6

0

2.0

0

1.2

0

3.51

2.32

1

.40

2.

30

0.7

0

0.42

0

.60

2

.00

1

.21

3.

51

2.21

1.

36

2.03

0

.61

0.

42

0.5

5

1.9

1

1.22

2.

71

2.32

1.

39

2.71

0

.72

0

.31

0

.43

2

.01

1

.19

3.

65

(1)

Ex

act

Co

.ln

. S

p.c

tru

.;

(2)

App

roxl

.. t.

Co

.ln

a S

pactr

u.;

(1)

Ap

pro

xl8

ata

Cla

•• lc

all

y D

..,a

d A

ppro

ach;

(4

) E

xac

t S

olu

tlo

n

Sl .

.. la

ted

(2

0-a

....

la)

U 1 U

2 U

• 1

.14

1

.14

2.

60

1.1

4

1.13

2.

60

1.1

4

1.13

1

.51

1.5

5

0.9

1

2.69

1.5

5

0.9

' 2

.70

1.5

4

0.9

6

1.27

2.40

1

.45

2.

11

2.4

0

1.4

5

2.1

1

2.34

1

.45

2.

42

SECTION 7

CONCLUSIONS

A response spectrum approach for the seismic analysis of non- classically

damped structural systems has been presented. Similar to the r~sponse

spectrum approach for analysis of classically damped structures, the only

information required for the gt.-ound motion input is the response spectrum.

The maximum re'lponse of the nonclassically damped structure is expressed in

terms of the sine spectrum and the cosine spectrum. The stne spectrum is

directly related to the response spectrum of the ground acceleration, whereas

the cosine spectrum is obtained in approximation from tne response spectrum as

well. The formulation takes into account the effect of cross correlation of

modes with closely spaced frequencies.

The proposed approach has been applied to approximate the maximum

response of several nonclassically damped structural systems subjected to

several earthquake ground accelerations. Particular emphasis is placed on

evaluating the maximum respon~e of structural systems in which the effect of

nonclassical damping is known to be significant.

Numerical results are compared with the exact solutions obtained by

numerically integrating the equations of motion. It is shown that the

accuracy of the propo •• d approach i. quite rea.onabl •. Al.o pre.ented are

maximum re.pon •• quanti tie. obtained using the approximate classically damped

~?lution.. Numerical re.ult. indicate that the accuracy of proposed approach

is generally better than that of the approximate cla •• ically damped solutions.

7 -1

SECTION 8

REFERENCES

1. Caughey, T.H., and O'1<elly, E.J., "Classical Normal Modes in Damped Linear Dynamic Systems," Journal 2f Applied Mecbanics, AS ME 32, 583-588, 1985.

2. Clough, R.W., and Penzien, J., "pynamics 2f Structures," McGraw Hill, New York, 1975.

3. Der Kiureghian, A., "A response Spectrum Approach for Random Vibrations," Report No. UCB/EERC-80/15, Earthquake Engineering Research Center, University of California, Berkeley, California, 1980.

4. Foss, 1<.A., "Coordinates Which Uncouples the Equations of Motion of Damped Linear Dynamic Systems," Journal 2f Applied Mechanics, ASME 25, 361-364, 1958.

5. Gupta, A.K. and Jaw, J.Y., "Response Spectrum Method for Non­classically Damped Systems," Nuclear Endneerine AWl pes lin , Vol. 91, pp.161-l69.

6. 19u1a, T. and Der Kiureghian, A., "DynamiC Analysis of Multiply Tuned and Arbitrarily SupporteJ Secondary Systems," Report No. UCB/EERC-83/07, University of California, Berkeley, July 1983.

7. Kanai, K., "Semi-Empirical Formula for Seismic Characterization of The Ground," Bulletin of Earthquake Research Institute, University of Tokyo, JL~an, Vol. 35, June 1967.

8. Newmark, N.M., BIUllle, J.A., and Kapur, K.K.," Seismic Design Spectra for Nuclear Power Plants," Journal 2f ~ ~ Diyi.ion, ASeE, Vol, 99, 1973, pp. 287-303.

9. Singh, M.P., "Seilmic Responle by SRSS for Proportional Damping," Journal 2f thA Endneerinl Mechanics Diyision, ASCE, Vol. 106, 1980, pp. 1405-1419.

10. Traill-Nalh, R.W., "Modal Methods in the Dynamics of Systems with Noncla.sica1 Damping," Earthquake Enlineerini ADd Structural Qynaaics, Vol. 9, 153-169, 1983.

11. Velet.o., '; .. 5., and Ventura, C.E., "Mocla1 Analysis of Non­c1a .. ic&Uy Damped Linear Syste •• , " Earthquake IncIneerinl and Structural Dyn .. ic., Vol. 14, 217-243, 1986.

12. Villaverde, R. and Newmark, N.M., "Sets.ic Responn of Light Attachments to Buildings," SRS No. 469, Univ. of Illinois, Feb. 1980.

8-1

13. Wilson, E.L., Oer Kiureghian, A. and Bayo, E.P.," A Repla~ement for the SRSS Method in Seismic Analysis," Earthquake En&ineerin& ADQ ~~ Dynamics, Vol. 9, 187-194, 1981.

14. YanF, J.N., Sarkani, S. and Long, F.X., "Modal Analysis of Non­classicially Damped Structural Systems Using Canonical Tran.­formation,- National Center For Earthquake Engin.ering R •••• rch, Technical Report No. NCEER-87-0019, September 1987, SUNY, Buffalo, N.Y.

8-2

APPENDIX

CALCULATION OF CORRELATION COEFFICIENTS

In this Appendix, the various correlation coefficients needed to compute

the approximate maximum response will be evaluated. These correlation

coefficients are defined as

(1. la)

(1. lb)

(I.le)

in which

-EO: r: (1. 2)

Similary E(CiCj ) is given by Eq. (1.2) with sin c.lOi"l and sin c.lDj '2 being

replaced by cos c.lDi"l and cos c.lOj "2' respectively. and E(SiCj) is also given

by Eq. (1-2) with sin c.lDj "2 being replaced by cos c.lOi"2'

With the assumption that the eqrthquake ground acceleration Xg(t) is a

stationary white noise with zero lIean and a power spectral density SO' Eq.

(1.2) can be obtained a. follows.

1-1

- [ ,.... R···· (1' -1' ) o JO xx 1 2

(1. 3a)

Similarly, the expressions for E(C1

Cj

) and E(SlCj) can be obtained as

follows

[

(i wi + (, wJ «(1 wi + (j Wj )2 + (wD1 - wDj )2

+

(1. 3b)

and

1-2

(1. 3c)

The mean square values are obtained by setting indicies i and j equal to

each other in Eqs. (1.3a) and (1.3b) as follows

2

E(Si2

) 2 "SO(l-e i )

- u - 2e i "'i S1 (I.48)

2

E(C i2) 2 "SO(l+e i )

- u -2(1 "'1 C

i (1.4b)

Substituting Eqs. (1.3) and (1.4) into Eqs. (1.1). one obtains the three

correlation coefficients in the following

(1. Sa)

(1. 5b)

(1. 5c)

and

aij - (e1ejW1Wj)

1/2 (1.6a)

b ij 2 2

- 1.11 + "'j + 2(1e j"'iWj (1. 6b)

c 1j -, (11.11 + Ej Wj (1. 6c)

dij

2 - 4c.ID1

2 2 - b ij "'Dj (I.6d)

1-3

NATIONAL CENTER FOR EARTHQUAKE ENGINEERING RESEARCH LIST OF PUBUSHED TECHNICAL REPORTS

The NatioIW Cella em Eartt.quUe Enaineerin& Reaean:h (NCEER) publishes technical reports on a variety of subjects related 10 earthquake enaineerin& written by aulhon funded through NCEER. 1'heae reports are available from both NCEER's Publicaliona Depar1ment and the NationAl Technicallnfonnation Service (NTIS). Requesta for reports should be directed to the Publicaliona Depar1men1, National Callei' for Eanhqualte fn&ineerin& Reselrch, State University of New York at Buffalo, Red Jacket QuIdnn&Ie, Buffalo, New York 14261. Reports Clll also be requested Ihrou&h NTIS, S28S Port Royal Road, Springfield., Vir,inia 22161. NTIS acceuion numbers are shown in pII1iIlIheais, if available.

NCEER·87-OOO1

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"First-Year Proarun in Resecch. Education and Technology Transfer," 3/5/87, (pBS8-13427S/AS).

"Experimenral EvalUltion of InsWlWleOIII Optimal A1&orithrru for Structural Control," by R.C. Lin. T.T. Soon, and A.M. Reinhom. 4/20187. (PB88-134341:AS).

"Experimentation U.in, the Eanhquake Sirlulation FlCilitiel at University at Buffalo," by A.M. Reinhom and R.L. Ketter, to be published.

'The System Ch.acteriJtics and Performance of a Shakin, Table," by 1.S. Hwana. K.C. Chm, IIld G.C. Lee, 611/87, (PB88-134259IAS).

"A finite E1cment Formulation for Nonlinear Vilcopiutic Material Usina a Q Model." by O. Gyebi IIld G. Duaupl&. 1112/87, (pB88-2137641AS).

"Symbolic MlIlipulation Proanm (SMP) - A1&ebnic Coclea for Two and Three Dimensional Finite Element Fonnulations," by X. Lee and G. DuSUJlll, 1119187, (pB88-219S22/AS).

'1nstmtaneous Optimal COJUroI Law. for Tall Buildings Under Seismic Excitations," by J.N. Yang, A.. Akbcpour and P. Ghllemm&&huni, 6/10187, (pB88-134333/AS).

'IDARC: Inelastic Damage Analysis of Reinforced Concrete-Frame Shear· Wall Structures," by Y.1. P .. k, AM. Reinhorn md S.K. Kunnath., 7(20/87, (PB88-13432S/AS).

"Liquefaction Potential for New York State: A PreiiminlrY Report on Sites in MmhatllD and Buffalo," by M. Budhu. V. Vijayakum .. , R.F. Giele and L. Baum&JU, 8/31/87, (PB88-1637041AS).

·Vertical and Torsiona1 Vibnlion of Foundalicna in Inhomopneoua Media," by A.S. Veleuoa and K.W. DoIlOn, 6/1187, (PB88-134291/AS).

"Seilmic Probabilialic Risk AIMumeftt IIId SeiImic Milam S~ for Nuclear Power Plants," by Howard H.M. Hw ..... 6115187. (PB88-134267/AS).

"P1r'mIeII'K: SIUdieI of Frequmcy RetpOIIII of Sec:oadary Syltellll Under Orcnmd-Acceleration R1tcitlliana," by Y. Yonan Y.K, Un. 6110117, (PB88-134309/AS),

"Frequency ReIpOIIIe of Secondary S)'ItemI Under Seiamic Excitations," by I.A. HoLuna, J. eai IIld Y.K. Lin, 7/31/87, (PB88-134317/AS).

"ModeIlin& Elrthqulke Oround Motions in Seismically AA:tive RepOlll Uaina Panmetric Time Series Melhodl,· O.W. ElliIInd A.S. CIkmIk. 8/2S/87, (PB88-13428~IAS).

"Detection IIId ANeument of Seiamic Structura1 Damaae," ';)y E. DiPuquale and 10..5. Cakmak, 8t2S/87, (PB88.1637121AS).

"Pipeline ExperimenI at Pukf'leld, California,· by J. laenber& and E. Richardson, 9/lS/87, (PB88.16372OJAS).

"DipaI Simulationa of Seilmic Ground MOIion,· by M. ShinoDlka" O. DeocIatia mel T. Handa, 8/31/87, (PBll-1SS197/AS),

A-I

NCEER-87-OO18

NCEER-87-OO19

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"PrKcic;U Con.iderlliona fur SII'UCtUr.J ConIrol: System Uncertainly. System Time Delay .00 Trunc. tionof Small !"'OIQII," J. Yana .... A. Akb.rpour.IIIUI/81. (PB88-16373I11AS).

"Modal AnaIysia of NoncJMSica1ly Damped Structural Syslanl Ulin, Cmonic.J Transformation." by J.N. Yq. S. Sabni IIId FJ(. Lon&. 9/27/81. (pB88-1818SI/AS).

"A NonItIIionIry Solulion in Random Vibrllion Theory," by J.R. Red-Horse Ind P.D. SpiIlOl, Il!3/8'T (PB88·I 637461AS).

"Horizontal ImpcdInces for R8dially lnhomoacneoua Vileoelulic Soil Layers," by A.S. Vele1lOl II\d K.W. DoIlOD, 1001SI87. (PB88·ISOSS9/AS).

"Seismic Dunap As_t of Reinfon;ed Conc:retc Members," by Y.S. Chung. C. Meyer ard M. Shinozuk .. 10JI)181. (PB88·150867JAS).

"Active SIrUct1Ir. COIlIroI in Civil En&ineerin.," by T.T. Soong, 11/11/81, (PB88-18777S/AS).

"Vertic. IIId Torsional Impedlncs for R-m.ly Inhomogeneous Viscoel_tic Soil Layers." by K.W. DoIlOn ard A.S. VeIetma, 12187. (PB88-1877861AS).

"Proceedinp from !be Sympoaium on Seismic: Hazuds. Ground Motions. Soil·LiqueflCtiOll II\d ~ PrlCtige in Euum Nonh AmerU:a. OclOber 20-22, 1987, edited by K.H. JICOb. 12187, (PB88-18811S/AS).

"Report em !be Whittier-NaroW&, C.ifomia, E.uthquake of October I, 1987," by 1. Pmtelic mel A. Reinham, 11181. (PB88·1877S2lAS).

"Deaip of a Module Prolflllll for T..-ieIIl NonlineIr AnalYlis of lar&e 3·D Buildinl Struc:ture&, • by S. Sriv_lav met I.F. Abel, 12J30/87, (PB88·1879501AS).

"Second·Yeu- Pmanm in Research. Education mel Technology Tr_fer," 3/8/88, (pB88.219480/AS).

"Wcrbhop 01\ Seismic: Computer Analysis and Desilll Wirh Inren.clive Graphics," by I.F. Abel md C.H. Conley, 1118/88. (PB88-187760/AS).

"Optimal ConlrOl of NonlineIr Sauc;auea," J.N. Yml. FX. I..oni mil D. Wona. 1122188. (PB88·2137721AS).

"Subllnlc1urina Techniques in !be Time Domain fOl' Primuy-Seconduy Structur. SYltems," by G. D. Mmolia IIId G. Juhn, 2110188. (PB88-2137801AS).

"llenlive Seismic: Analysis of Prinwy-Serondcy Systems,· by A. Sin&haJ, L.D. lAitei .00 P. SpiIlOl, 2/23188. (PB88-l13798/AS).

"Stoch.tic: Finite Eiemall EaJ*Uion for Rmlom Modia," P. D. Spmo& IIIId R. G"-, 3/14/88, (PB88-2138061AS).

"Combinina SII'UCtUnl Optimization IIIId SIrUctUral ConlrOl," F. Y. Chcna IIId C. P. Plllteliciel. tn0/88. (PB88-213814/AS).

"SeiImic: ~ ~ of Code-Deaiped SII1&:IUna," H.H-M. Hw .... 1. Jaw and H. Shall, 3f20111l. (PB88-219423/AS).

"Re1imi lily AnIlysii of CocIe-Desiped S~ Under Natur. Hazmia." H.H-M. Hwq, H. Ulhiba

and M. Shinozuka. 2Il9188.

A-2

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"Seismic frl&ility Analysis of Shear Will SaucIlRS," J-W Jaw IIld H.H-M. Hwang, 4/30,'e8.

"8ue Isolalion of a Multi-Story Buildina Under a HlIlTIonic Ground Motion . A Comparison of Performances of Various SyllemJ," F-G Fill, G. Ahmldi md I.G. TadJbakhsh, to be publimed.

"Seismic Floor Respmue Speen for I Combined System by Green's Fun;tions," F.M. Luelle, L.A. Bergman and P.D. Spanos, !i/1/88.

"A New Solution Technique for Randomly Excited Hysteretic Structures," G.Q. Cai and Y.K. Lin, !i/16l88.

"A Study of Rldillion DlIlIpini and Soil-StrucllR lntcnction Effects in the Centrifuge," K. Weissman. supervised by 1.H. Prevosl, !i/24/88, to be published.

"Puamel.er Idcntificillion md Implementation of a Kinematic Plasticity Model for Frictional Soils," 1.H. PreVOIt and D.V. Griffiths, 10 be published.

'"I'wo- .00 Three-Dimensional Dynamic Finite Element Analyses of the 1..011& Valley Dam," D.V. Griffiths md 1.H. Prevost, to be published.

"Damqe Aueument of Reinfon:ed C<Jngete Structure. in Eastern United States," A.M. Reinhorn, M.l. Seidel, S.K. Kunnalh and Y.I. P,rk, 6/1!i!88.

"Dynamic Compliance of Vertically Load Snip FDlDldations in Multilayered Viscoelastic Soils," S. Ahmad and A.S.M. I.ail, 6/17/88.

"An Experimental Study of Seismic SauclUrll Response Willi Added Viscoelastic Dunpet'S," R.C. Lin, Z. Uana, T.T. Soon& IIId R.H. Zhana, 6130/88, to be published.

"Experimerltal lnv_liBation of Primuy - Secondcy System lntenction." G.D. Manolis, G. Juhn lnd A.M. Reinhom. 5(27/88, to be published.

"A Response Spectrum Approach For Analysis of Nonc:lusically Dunped Sauctures," J.N. Yang, S. S.-kani and F.x. Lon" 4(22188.

A-3