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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.
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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
-DEG
REE-
OF-
FREE
DO
K S
TRUC
TURE
(e
M.)
<al
C1
-C
2 -
12
3.4
kN
/ •.•
ec.
San
Fer
nand
o K
exic
o C
ity
S
1l1N
lac.
d S
imu
late
d
El
C.n
cro
(1
su
pl.
) (2
0-u
lllp
le)
U1
U2
U1
U 2 U
1 U 2
Ul
U 2 U
1 U
2
(1)
1.7
6
1.0
9
1.8
9
1.1
7
0.6
1
0.3
8
1.4
3
0.8
9
1.8
4
1.1
4
(2)
1. 7
6 1
.09
1
.89
l.
17
0
.61
0
.38
l.
43
0
.89
1
.84
:.
14
(3)
1.7
6
1.0
9
1.8
9
1.1
7
0.6
1
0.3
8
1.4
3
0.8
9
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(4)
1.8
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0
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0
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l.
44
0
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(b)
C 1 -
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8 kN
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l.
69
1
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0
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0
.36
l.
16
0
.75
1
. 55
0.9
8
0- I
(2)
1.5
4
0.9
7
1.6
9
1.0
6
0.5
8
0.3
6
00
l.
16
0
.74
l.
55
0.9
8
(3)
1.5
3
0.9
5
1.6
7
1.0
3
0.5
7
0.3
6
1.1
6
0.7
3
1.5
4
0.9
6
(4)
1.6
1
0.9
8
1.6
7
1.1
1
0.6
0
0.3
2
1.1
5
0.7
7
(c)
C 1 -
24
6.'
kI/
a.l
ec .•
C
2 -
0
(1)
2.2
4
1.3
5
2.3
2
1.4
0
0.7
0
0.4
2
2.0
0
1.2
0
2.4
0
1.4
5
(2)
2.2
4
1.3
5
2.3
2
1.4
0
0.7
0
0.4
2
2.0
0
1. 2
1 2
.40
1
.45
(3)
2.1
2
l.3
1
2.2
1
1.3
6
0.6
8
0.4
2
1.9
8
1.2
2
2.3
4
1.4
5
(4)
2.2
5
l.3
0
2.3
2
1.3
9
0.7
2
0.3
8
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)
Ap
pro
xt.
ace
Cl •
• aic
a11
y D
ampe
d A
ppro
ach;
(4
) E
xAct
S
olu
tio
n
TA
iLI
2.
IlAX
IlCO
K
DE
FO
IIIA
1'IO
li o
r T
OIlE
D
IQU
Il'J
IElI
T-S
nU
c'l'
UU
SY
STE
M
(CX
. >
(we
-w
l'
C1
-C
2 -
123.
4 k
N/_
/8ec
.)
(a)
e. -
Ot
El
Cen
tro
San
Fer
MC
lClo
M
exic
o C
ity
S
!8U
lat.
4
Sl_
l.u
d
(1 •
...,
1.)
(2
0-•
...,
le)
U 1 Uz
U
U 1 U 2
U
U1
U 2 U
0
1 U 2
0 U 1
U2
U
• •
• a
• (l)
1
. 71
1.1
0
76.9
6 1
.90
1
.11
1
4.4
. 0
.61
0
.31
10
.76
1.4
4
0.9
0
53.6
2 1
.15
1
.14
70
.62
(2)
1. 7
1 1
.10
7
7."
1
.90
1
.11
1
4.1
4
0.6
2
O.l
I 5
.14
1
.44
0
.90
55
.61
1.1
5
1.1
4
70.6
5
(l)
1
. 76
1.0
9
20.0
9 1
.19
1
.17
21
.70
0.61
O
.lI
7.03
1
.43
0 .•
' 16
.17
1.9
1
1.12
n
.9'
(4)
1. 7
9 1
.05
1
2.4
3
1.1
9
1. '
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75.7
0 0
.64
0
.34
9
.44
1
.43
0
.15
1
00
.7'
(b)
e. -
5t
(1)
l. 7
6 1
.09
20
.03
1.9
0
1.1
1
21.7
3 O
. fit
0
.3'
7.04
1
.43
0
."
16
.61
1
. Il
l.ll
21
.07
~
I (2
) 1
. 76
1.0
9
20.0
3 1
.90
1
.17
n
.n
0.6
1
0.3
1
7.0
4
1.4
2
0."
16
.61
l.1
3
l.ll
21 .
07
~
(3)
1.7
6
1.0
9
20.0
3 1
.90
1
.17
21
.13
0.6
1
0.3
1
7.04
1
.42
0.
" 16
.51
1.9
2
1.1
9
22.0
5
(4)
1.1
0
1.0
5
14
.42
1
.17
1
.20
1
7.9
0
0.6
4
0.)
4
5.71
1
.44
O
.SS
15.5
3
(c)
e. -
lOt
0)
1. 7
6 1
.09
'.
97
1
.19
1
.17
1
1.1
0
0.6
1
0.1
' 1
. 511
1
.43
0
.19
1
.16
1
.1'
1.1
3
11
.69
(2)
1.
76
1.0
9
10
.75
1
.19
1
.17
1
1.1
4
0.6
1
1'.
3'
1. 5
2 1
.43
0.
" '.
95
1
.14
1
.13
1
2.1
2
(3)
1.7
5
1.0
1
20.0
6 1
.19
1.
17
21.1
6 0
.61
O
.l'
7.0
7
1.4
2
0.1
1
17
.00
1
.91
1
.1'
n.6
6
(4)
1.1
0
1.0
5
9.1
5
1.1
7
1.2
0
10
.'7
0
.64
0
.3.
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
) E
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
.69
l.
07
60
.50
O
. S.
0.1
6
7.80
1
.16
0
.15
)1
.05
1
. 55
0.9
' 49
.61
(2)
1.7
0
1.0
7
60
.67
1
. 70
1.0
7
60
.61
0
.58
0
.16
4
.10
1
.16
0
.15
)'
.79
I.~S
0.9
' 4
9.6
3
(1)
1. 5
4 0
."
12
.29
1
.67
1
.0)
13
.19
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
.60
0.
3Z
6.1
9
1.1
4
0.7
7
29.9
4
(b)
( -~ .
• (1
) 1
.54
0
.97
1
1.H
1
.69
1
.06
1
4.1
5
0.5
8
0.3
6
2.41
1
.15
0
.74
10
.Z1
1. 5
5 0
.91
15
.4S
(2)
1. S
4 0
\ 0
.91
1
4.0
8
1.6
9
1.0
7
14
.55
0
.58
0
.36
2
.44
1
.U
0.7
4
13. 5
9 1
.55
0
.91
1
5.9
1
I
0)
1.S
l 0
.95
12
39
1.
67
1.0
3
13.1
7 0
.51
0
.16
4
.57
l.
U
0.7
1
9.2
2
1.6
1
1.0
4
11
.61
0
(4)
1.6
1
0.9
' 1
1.7
9
1.6
7
1.1
1
14.1
1 0
.60
0
.32
2
.33
1
.U
0.7
7
12
.17
(e)
E. -
lOt
(1)
1. S
4
0.9
7
8.0
4
1.6
9
1.0
6
9."
0
.51
0
.)6
1.
)1
1
.16
o
.n
6.9
9
1. 5
4 0
.91
9
.34
(2)
1. S
4 0
.97
1
.10
1
.69
1
.06
10
.65
0.5
1
O. )
6
1. )
. 1
.16
0
.15
5
.99
1
.55
0
.91
9
.64
0)
1.5
3
0.9
5
12
.5'
1.6
7
1.0
) 1
3.2
6
0.5
7
0.1
6
4.5
8
1.1
5
0.7
6
9.2
9
1.6
5
1.0
) 1
3.6
7
(4)
1.6
1
0.9
1
7.5
9
1.6
7
1.1
1
'.4
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
"".9
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
.55
I
n)
2.1
1
1. ]
0
44
.40
(4)
2.2
4
1. )
0
26. !
~
(1)
2.2
4
1. ]
S
12 .
]7
0)
2 24
1
. )5
1S
.02
(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
• ...
,1.)
U1
U2
U
U1
U2
U
U\
U2
U
• •
• 2
.]6
1
.4]
14
7.2
0
0.7
1
O.U
1
1.6
4
2.0
4
1.2
1
97
.70
2. ]
6
1.4
3
14
7.n
0
.71
O
.U
14
.91
2
.05
1
.24
1
0l.
"
2.2
0
1.3
6
"6.0
0
0.6
1
0.4
2
14
.01
1
.K
1.2
1
40
.71
2.l
l 1
.]'
1"0
.]5
0
.72
0
.]9
1
6.0
' 2
.00
1
.19
6
7.1
4
(b)
(. -
S\
2.l
l 1
.39
26
.42
0.7
1
0.4
] ].
16
1
.99
1
.20
2
2.2
'
2.l
l 1
.)9
2
6.]
0
0.7
1
0.4
) ].
45
1
.99
1
.20
2
5.9
0
2.2
0
1.]
6
45
.51
0
.61
0
.42
1
1.9
1
1.9
6
1.2
1
41
.04
2.n
1
.]1
25
.91
0.7
2
0.]
9
].2
1
2.0
1
1.1
9
21.1
2
(e)
e. -
10
\
2.l
l 1
.40
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
.40
1
5.4
5
0.7
0
0."
2
1.1
1
1.9
9
1.2
0
13.1
)5
2.1
0
1.]
6
....
54
0
.61
0
."2
1
4.0
] 1
.96
1.
21
41
.71
2.]
2
1. 1
9 1
4.4
9
0.7
2
O.l
I 2
.11
2
.01
1
.19
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. ..
. 4
9.1
6
2.3
9
1.U
2
9.1
5
2.19
1
.44
16
.66
2.3
5
1.4
5
49
.00
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
• ...
• 1
..9
1.
17
2.1
0
0.6
1
0.)
' 0
.55
1
.4]
0."
3
.3'
1..
9
1.1
7
2.1
0
0.6
1
O.l
I 0
.54
1
.43
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
.15
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.
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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
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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.
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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