COMPLETE QUADRATIC COMBINATION
TECHNIQUE FOR MODAL RESPONSES
Presented
by:
Impell Corporation
February 1985
I
r
g
MODAL COMBINATION RULES
0
Standard Approaches
'I
ASUM
SRSS
Ten Percent
For Closely-Spaced
Grouping
Double Sum
0
Present Methods
0
SRSS
0
An Improved
May Be
for Closely-Spaced
Modes
Modes
are Overly Conservative
Unconservative
Technique
is the
Complete Quadratic Combination (CQC)
Method
Based on Random
More Accurate
Vibration Principles
Insignificant Additional Computational Effort
I
ir
p
l
0
CQC METHODOLOGY
(~ ~k
R =
j
P
jk
R.
j
R
k)
where
j'
R.,
P.k
Rk
for
Modes J and K
=
Modal Responses
=
Correlation Coefficient Between
on Random Vibration Principles)
=
Combined Response.
Modes J and K (based
'
I
~
I
0
VERIFICATION EXAMPLES
0
Complex 3-D
Structure
Chemical Separation
Plant (British Nuclear Fue'Is)
685 Nodes
161 Beam Elements
264 Dynamic DOF
Q
CONCRETE FLOOR DIAPHRAGMS
STEEL EXTERIOR STRUCTURE
~3G
Gm
I
~25.5m
I
~21
~16
5m
~11
3 m
5m
~10
jJjTl
m
77
m
COMPARISON OF RESPONSES
ASUM
~C
All
Components
Forces and Moments
Accelerations
Displacements
TABLE
NS
EW
VT
1.
All
- Total
- In-Plane
- Out-of-Plane
- Total
- In-Plane
- Out-of-Plane
- Total
- In-Plane
- Out-of-Plane
TABLE
2.
C
2.90
2.86
3.61
2.30
BY VARIOUS METHODS
10PC
SRSS
~C
CQC
Observations
1.02
1.04
1.04
0.94
2.31
2.21
3.18
1.77
Design Data Due
C
1364
836
264
264
to Three Directions of Input
ASUM
10PC
SRSS
~CC
~CC
CQC
3.81
2.18
4.79
2.77
1.46
';56
3.70
2.06
4.68
2.58
1.43
3.27
0.78
5.24
2.26
5.66
5.03
2.26
5.43
0.69
~C
T-H
Observations
1.15
0.92
1.29
1.03
0.98
1.05
304
114
190
1.05
1.05
1.00
1.07
304
N/A
N/A
N/A
304
38
266
1.21
1.26
1.34
114
190
Forces and Moments Due to Single Earthquake Input
-4
LN8
NS
EW
VT
- Total
— In-Plane
- Out-of-Pl ane
-
3.92
2.00
In-Plane
Out-of-Pl ane
5.21
2.76
6.44
Total
In-Plane
Out-of-Pl ane
6.29
2.44
7.06
TABLE
Accelerations
3.
1.07
1.04
1.08
264
88
176
4.34
2.05
5.49
1.31
0.89
1.51
1.00
0.98
1.00
264
88
176
5.44
2.43
6.82
1.45
0. 70
1. 60
N/A
N/A
N/A
264
44
220
'.88
Due
to Single Earthquake Input
~V~~n
Eg
- Total
- In-Plane
- Ou,-of-Pl ane
- Total
- In-Plane
- Out-of-Pl ane
TABLE
4.
~e~~
1.26
0.91
1.43
Force and
NS
T-H
CM
4.74
2.59
5.81
— Total
QE
S~R
CQC
~c~e~~
.03
.11
.14
.05
.16
.13
,06
.15
.12
.08
.13
.09
CgC/T-H Comparisons:
Standard Deviations
~
y
,
0
Typical Piping System
D(~
l/5
lg
V(p
(]pl
(,)i
Qsy
~
( ap/I
CQ
Cw/)
~5
Cg
Ic~
Sg)
$
/2
PIPING PROBLEM AC-03 (SONGS-1)
-6-
I
~
I
RESPONSES
DUE TO SINGLE EARTHQUAKE INPUT
Z-Direction
Number
10PC
C~C
T.H.
T.H.
1.40
1.12
1.70
1.07
1.03
1.11
40
20
20
1.85
1.27
2.17
1.15
1.05
1.20
17
6
11
of
Observations
Pipe Moments:
- Total
- In-Plane
- Out-of-Plane
Support Reactions:
- Total
- In-Plane
- Out-of-Plane
C
C/T-H COMPARISONS:
Pipe
Moments
Total
In-Plane
Out-of-Plane
0.19
0.20
0.18
STANDARD DEVIATIONS
Support
Reactions
0.33
0.20
0.38
-7-
3.000
4.316
Q
4.204
10 PC METHOD/TIME HISTORY
0 CQC/TIME HISTORY
2.500
2.000
1.500
1.000
0.900
0.500
TIME HISTORY MOMENTS
0.00
BENDING MOMENT COMPARISON
l
~
g
0
IMPELL COMPUTER CODES WITH
CQC
TECHNI UE
0
EDSGAP
- Structural Analysis
0
FLORA
- Structural Analysis (Using
0
This Technique is Currently Being Implemented in Program
SUPERPIPE (Piping Analysis Program)
9
Random
Vibration Method)
L
PROJECT USE
CgC
Method Has Been Used
On
Several Projects:
0
Seismic Analysis of Chemical Separation
(British Nuclear Fuels)
0
Seismic Analysis of .Pipe Bridge
Nuclear Fuels)
-10-
Plant
{British
l
~
r
C'
REFERENCES
l.
E. L. Wilson, A. Der Kiureghian and E. P. Bayo, "A Replacement for the
Method In Seismic Analysis," Earthquake Engineering 5 Structural
Dynamics, Vol. 9, 1981.
SRSS
2.
B. F. Maison and C. F. Neuss, "The Comparative Performance of Seismic
Response Spectrum Combination Rules in Building Analysis," Earthquake
Engineering and Structural Dynamics, Vol. 11, 1983.
-11-
EARTHQUAKE ENGlNEERlNG AND SZRVCTUaAL DYNAMICS. VOL 9, ts7-194 tl9$ t)
SHORT COMMUNICATIONS
A REPLACEMENT FOR THE SRSS METHOD IN SEISMIC ANALYSIS
L WILSON, A. DER KIUREGHIANfAND E P. BAYOUS
of Ciofl Engi»roririp, Uniorrsiiy ifCafiforrila, Brrktfry, California, USA.
m.
Dr~
SUMMARY
It is weil-known that the application of the Square-Root-of-Sumef-Squares (SRSS) method in seismic analysis for
combining modal maxima can cause significant errors. Neverthekss, this method continues to be used by the profession for
significant buildings. The purpose of this note is to present an improved technique to be used in place of the SRSS method
in seismic analysis.
A Complete Quadratic Combination (CQC) method is proposed which reduces errors in modal combination in all
«xamples studied. The CQC method degenerates into lhe SRSS method for systems with well-spaced natural frequencies.
Since thc CQC method only involves a small increase in numerical effort, it is recommended that the new approach be used
as a replacement for the SRSS method in all response spectrum calcuhtions.
INTRODUCTION
Thc SRSS method of combining modal maxima has found wide acceptance among structural engineers
«ngagcd in seismic analysis. For most twMimensional analyses, thc SRSS method appears to yield good
results when compared to time-history response calculations. Based upon the early success of thc method in
twMimcnsions, the SRSS approach is now being used for threcMimcnsional dynamic analysis without having
been verified for such structures. ln fact, the method is now an integral part of a large number of computer
'programs for the dynamic analysis of general thre«dimensional
problem associated with the application of the SRSS method can be illustrated by its application to the
four-storey building shown in Figure 1. The building is symmetrical; however, the centre of mass is located 25
inches from the geometric centre of the building. The direction of the applied earthquake motion, a table of
natural frcquencics and the principal directions of the mwle shapes are illustrated in Figurc 2. ne notes the
«los«ness of thc frequencies and thc complex nature of the mode shapes in which the fundamental mode shape
systems.'hc
C3
FRAME Pl
too
Iso
i ~200
O
O
A3
%2
8
O
O
~ C.G.
d
250
55
PLAN
E LE VATI ON
TYPICAL FRAME
Figure I. Simple thrxodimensional buikling example
o Pcofcuor
of Civil Engineering.
Axxatant Profoscur ol'ivil Engincoring.
f Graduate
Studcm.
I
0098-8847/81/020187-06$ 01.00
1981 by John Wiley k, Sons, Ltd.
Received 8 January 1980
Revised 24 July 1980
SHORT COMMUNICATIONS
188
~II
D
FAF.OUKNCIES
(RADIANS/SEC.)
~«
Us
I
IS.869
. 2
s
IS. 9S I
4S.99S
8
84,4 I 8
77.888
8
7
44.I89
IO
78.029
I08.S2
I 08.80
I72.6IS
12
425.00
8
9
II
S04.80
X
Figure 2. Frequencies and approsimare directions
of mode
shapes
has x, v, as well as torsional components. This type offiequency distribution and coupled mode shapes are very
common in asymmetrical building systems.
This structure was subjected to the Taft, 1952, earthquake. The exact maximum base shears for the four
exterior frames produced in the first five modes are shown in Figure 3. A mode superposition solution, in which
all 12 modes werc used, produces base shears as a function of time. The maximum resulting base shears in each
ofthe four frames are plotted in Figure 4(a). For this structural model and loads, these base shears represent the
'exact'esults.
Ifthese modal base shears are combined by the SRSS method, the values shown in Figure 4(b) are obtained.
Thc sum of the absolute values of the base shears is shown in Figure 4(c). The base shears found by thc new
'Complete Quadratic Combination'(CQC) method are shown in Figure 4(d). Note that the signs of thc base
shears arc not retained in any of these approximate methods.
For this case it is clear that the SRSS method greatly underestimates the forces in the direction of the motion.
Also, the base shears in the frames normal to the motion arc overestimated by a factor of 14. It is clear that
errors of this order are not acceptable. The sum of absolute values, which is a method normally suggested for
the case where frcqucncics are close, gives a good approximation of the forces in the direction of motion but
overcstimatcs the forces in the normal frames by a factor of 25. For this example, the double sum method
required by the Nuclear Regulatory Commission produces results very close to the sum of the abolute values.
The CQC method applied to this example gives an excellent approximation to the exact results. The main
purpose of this note is to present a summary of this new technique for combining model maxima.
„
~r
s)
st.so
2+44
~ r.ss
st.se
r
~
~
4.0t
«
O.rt
Figure 3. Base shears in firsI five modes
0
r
O.ss
n. ~
'F0, ~
(b) SRSS
(rt) TIME HISTORY
I ~
(e) SUMOFABSOLUTE
(4) COO
VALUES
Figurc 4. Comparison of rnodat combination tncthods
BASIC MODAL.EQUATIONS
The dynamic equilibrium equations for a thrceMimensional structural system subjected to a ground
accckration, ugt), in the direction is written as
MV+CU+KU= M,(iieet))
0
(I)
where M, C and K are the mass, damping and stiffness matrices, respectivley. The threcMimensional relative
displacements, velocities and accelerations are indicated by U, U and V. The column vector M, contains thc
components of mass in the x4ircction and zeros in all other directions.
The mode superposition solution involves thc introduction of the transformation
U=4Y
(2)
where
is the matrix containing threeMimensional mode shapes of the system and Y is the vector of normal
co-ordinates. The introduction of this transformation and the premultiplication of equation (I) by Or yields
vMCrV+4vCQY+e
K4Y = C M,(iieet))
(3)
For proportional damping the mode shapes have the following properties:
4r Mgr
(4)
mr
$ , KQ,=
4r C4 4rWmr
(5)
(6)
in which Qr is the ith column of 4 representing the ith mode shape, m, is the ith modal mass, and (r is the
damping ratio for mode i. Due to the orthogonality properties of the mode shapes. all modal coupling terms of
theform ghfgarezcro forivsj. Thus, equation(3) reduces toa set ofuncoupled equationsin which the typical
'modal'quation is of thc form:
~i+Zrtrrr ~r+tcr )'
pr(iiJt))
where
(g)
is the participation factor for mode i.
5HORT COMMUNICATlONS
Thc evaluation of equation (7) for all modes yields the time-history solution for normal ~rdinates. The
total structural displaccmcnts, as a function of time, arc then obtained from equation (2).
DEFINITION OF RESPONSE SPECTRUM AND MAXIMUMMODAL DISPLACEMENT
Thc following equation can be solved for the response yet)
Vi+2J;i W 5+m'ri
- uP)
(9)
At thc point in time where )yet)) is maximum, the response is defined as y<,. A plot of this maximum
displacement versus thc frequency co, for each 4, is by definitio the displacement response spectrum for the
earthquake ii,(t). A plot of y<.,co, is the pseudo-velocity spectrum and a plot of y,,co~ is the pseudoaccclcration spectrum. These pseudo-velocity and acceleration spectra are of the same physical interest but are
not an essential part of a response spectrum analysis.
If thc dynamic loading on the structure is specifie in terms of the displacement spectrum, then the
maximum response of each mode is given by
~i,~a
Therefore, the maximum contribution
of mode i to the total
UI,
y,,
(10)
Pi W.~a
response
4 pr J'r,
of the structure
is
(ll)
U,,
is proportional to the
For all modes
is, by definition, positive. The maximum modal displacement
mode shape Q;, and the sign of the proportionality constant is given by the sign of the modal participation
factor. Therefore;each maximum modal displacement has a unique sign, which is given by equation (11). Also,
the maximum internal modal forces, which are consistently evaluated from the maximum modal displacements, have unique signs. These signs for the maximum modal base shears of thc'example structure are
indicated in Figure 3 by arrows.
THE COMPLETE QUADRATIC COMBINATION METHOD (CQC)
The use of random vibration theories can eliminate the previously illustrated errors which are inherent in the
absolute sum or the SRSS method. Based on this approach, several other papers have presented more realistic
The complete development of the CQC method, which is now being
methods for modal combination.
proposed as a direct replacement for thc SRSS technique, is presented by the second author in References 6 and
7. Thc CQC me'thod requires that all modal response terms be combined by the application of the following
equations:
For a typical displacement component, g:
and for a typical force component,
f„
4(Z Z Ru Ag ~g)
(12a)
fi 4KZ fuugfg)
(12b)
f„
is a typical force
where g, is a typical component of the modal displacement response v'ector, U, „and
the
modal
displacement
vector,
Note
that
this
combination
formula is
which
is produced by
component
cross-modal
all
hence,
the
reason
for
the
name
terms,
Complete
of complete quadratic form including
Quadratic Combination. It is also important to note that the cross-modal terms in equations (12) may assume
positive or negative values depending on «hether the corresponding modal responses have the same or
opposite signs. The signs of thc modal responses are, therefore, an important key to thc accuracy of thc CQC
method.
In general the cross-modal coc5cients, p,> are functions of the duration and frequency content of the
loading and of the modal frcquencics and damping ratios of the structure. Ifthe duration ofearthquakc is long
U,,
~
~
0
'IHORY COMMUNICA'nONS
0
I91
compared to thc periods of the structure, and il'he earthquake spectrum is smooth over a wide range
frequencies, then, it is possible to approximate these coeAicients
of
bye'/(g,g)(f,+rg)r
(I
r
-r ) +4),f)r(I +r )+4@, +(~)
For constant modal damping, (, this expression reduces to
r'here
cop'to,.
8P(I + r) r3I
(14)
Note that for equal damping and r = 1, p,> 1. Expressions for the cross terms, which take into account the
duration and frequency content of the loading, are given in Reference 7. Additional modal combination rules
giving the variability and the distribution of the peak response are also given in that refcrcncc. These rules are
useful in non-deterministic analysis.
Considering 5 pcr cent damping and the frequencies of the example structure, the evaluation ofequation (14)
yields the cross-correlation coeAicients given in Table I. One notes that ifthe frequencics are well-separated the
offMiagonal terms approach zero and the CQC method approaches the SRSS method.
Table
Mode
1
2
3
4
5
1.
Modal cross~rrclation cocFicicnts
1
2
3
4
5
Frcq. rad/s
1000
0998
0 998
1400
N$6
N$6
N$6
N$6
N$4
N$4
13 87
13 93
0998
0 180
1
0.186
N$6
N$6
N$4
N$6
N$6
N$4
1400
0.998
0 180
0 186
1400
43 99
44 19
54 42
COMPUTER PROGRAM IMPLEMENTATION
—
The CQC method of modal combination has been incorporated in the computer program TABS ThreeDimensional Analysis of Building Systems.'his involved the addition ofone subroutine for the evaluation of
modal correlation factors, Equation (14), and the replacement of the SRSS by the CQC method as given by
equations (12). The increase in computer execution time due to the addition of the CQC method was
insignificant (less than 0 per cent for a typical structure). Therefore, there is no justification to continue using
the potentially erroneous SRSS method. The application of the modified TABS program to several buildings
has verified the validity of the CQC method. Other examples are given in Reference 7.
1
FINAL REMARKS
bc pointed out that a method similar to the CQC method was first proposed by Rosenblueth and
Elorduy~ in 1969. Their method. which has a somewhat heuristic basis, has a more complicated cross-modal
term involving the duration ofearthquakc as well as the modal frequencies and damping values. This method
has unfortunately been neglected or misrepresented over the past several years. For example, the NRC
Regulatory Guidea recommends it for structures with closely spaced modes. however, it wrongly specifies the
cross-modal terms as being always positive. This will result in overly conservative response estimates in some
applications. Concern that this earlier method is being misunderstood. and the fact that thc CQC method is
simpler and more practical. have prompted the writing of this note.
It sho'uld also bc pointed out that the SRSS method gives good results for some structures subjected to twodircctional seismic input, cvcn when the modal frequencies are closely spaced. It can be shown that this is duc
to cancelling of the cross-modal terms corresponding to the two directions of input. This phenomenon.
howcvcr, is not generally true. For example, when thc two components of input are of different intensities, or
It should
0
~ ~
.
l92
. SHORT COMMUNICATIONS
*
when thc three-dimensional structure is highly asymmetric, the cross-modal terms would still be significant
and, therefore, the SRSS method will lead to erroneous results.
LIscd on thc prcccding numerical example and thc above discussion, it is strongly recommended that the
.usc of the SRSS method for seismic response analysis of structures be immediately discontinued. Continued
usc of thc SRSS technique may dramatically overestimate the required design forces in some structural
elements or it may significantly undcreslimate the forces in other elements. The proposed CQC method is
based on fundamental theories ofrandom vibration and consistently yields accurate results when compared to
time-history analyses.
,
REFERENCES
l.
2.
3.
L Wilson and A. HabibuUah, 'A program for threcMimcnsional static and dynamic analysis of multistory buiktings', in Strucr urof
hfrrhanirs Sofisrvrr Srrirs. Vol. Jl. University Press of Virginia, l978.
E L Wilson. J. P. Hollings and H. H. Dovcy, 'ThroeMimensional analysis of building systems'extended version), Rcport No.
t/CH EERC-75r I3, Earthquake Engineering Research Center, University of California, Berkeley, California (l975)
A Structumt Analysis Program for Response of Linear Systems', Rrport No.
K. J, Bathr. E L Wilson and F. E Petorson.'SAP IV—
E.
UCHrEERC-73/I l. Earthquake Engineering Roscarch Center, University of California, Bcrkeky, California (1973)
of linear systems to certain transient disturbances'. Proc. Fourth Wld Coqf Eort hq. Engng I,
San(iago, Chile. I85-l96 (I969)
M. V. Singh and S. L Chu, 'Stochastic considerations in seismic analysis of structures', Earrhqu. Enf/. Srrurr. DyrL 4, 295-307 (l 976)
A. Der Kiureghian, 'On response of structures lo stationary excitation', Rrport No. UCJJ/EERC-79/32, Earthquake Engineoring
Rosoaroh Center. University of California, Berkeley, California, (l979)
A. Dcr Kiureghian.'A response spectrum method for random vibrations', Rcport Na t/CJJ/EERC49/l5, Earthquake Engineering
Rrscarch Center. University of California, Berkeky, California, (l980)
U5. Nuckar Regulatory Commission, Regulatory GuiCh l,92, Jhrlsion J (1976)
4. E. Roscnblueth and J. Etord uy, 'Responses
5.
6.
7.
g.
EARTHQUAKE ENGINEERING AND 5TRUCTURA1 DYNAMICS, VOL I I, 62~7 tl983)
THE COMPARATIVE PERFORMANCE OF SEISMIC RESPONSE
SPECTRUM COMBINATION RULES IN BUILDING ANALYSIS
0
B. F. MAISON AND C. F. NEUSS
J. G. Bouwkuttp, Itte„Berkeley, Califorrtla, USA.
K. KASAIf
Uttfoersity of Cattforrtia, Berkeley, California, USA.
SUMMARY
The peak dynamic responses of two mathematical models of a fifteen-storey steel moment resisting frame building
subjected to three earthquake excitations are computed by the response spectrum and time history methods. The mode)s
examined are: a 'regular'ui1ding in which the centres of stilfness and mass are coincident resulting in uncoupled modes
with well-separated periods in each component direction of response; and an 'irregular'uilding with the mass otfset from
the stiffness centre of the building causing coupled modes with the trans1ational modes having closely spaced periods.
Four response spectrum modal combination rules are discussed and are used to predict the peak responses: (I) the square"
root of the sum of the squares (SRSS) method; (2) the double sum combination (DSC) method; (3) the complete quadratic
combination (CQC) method; and (4) the abso1ute sum (ABS) method. The response spectrum results are compared to the
corresponding peak time history values to evaluate the accuracy of the different combination rules. The DSC and the
CQC methods provide good peak response estimates for both the regular and irregu1ar building models. The SRSS
method provides good peak response estimates for the regular building, but yields significant errors in the irregular
building response estimates. The poor accuracy in the irregular building results is attributable to the effects of coupled
modes with closely spaced periods. It is concluded that the DSC and CQC methods produce response estimates of
equivalent accuracy. Both methods are recommended for general use. In addition to the DSC and CQC rules, the SRSS
method is recommended for systems where coupled modes with closely spaced periods do not dominate the response.
INTRODUCTION
The response spectrum method is a widely used procedure for performing elastic dynamic seismic analysis.
The response spectrum, by definition, represents the sct of the maximum acceleration, velocity or
displacement responses of a family of single-degrcc-of-freedom (SDOF) damped oscillators, resulting
by a specific earthquake ground motion..The application of response spectrum analysis
procedures to structures which cannot be adequately described as SDOF systems requires modal analysis
Iques to transform the coupled multi-degree-of-freedom equations of motion to a set of uncoupled
pns in normal co-ordinates. This transformation allows the response of each mode to be evaluated as a
~t". system. The response spectrum can be used to predict the individual modal response maxima, but
lacks modal time phasing information. Therefore, the relative times at which each peak modal response
occurs are unknown. To estimate the total peak response, techniques which combine the individual
maximum modal responses are required. Numerous response spectrum modal combination rules have been
proposed with the intent of minimizing the total peak response prediction errors when compared to the time
history analysis values. The most common rule is the square root of the sum of the squares (SRSS) method,
which is recommended for use in the nuclear power,'ffshore oil and building industries. However, it is
generally recognized that the SRSS method can be a poor estimator of peak responses when applied to
from'xcitation
'trtl Engtneer.
<
Graduate Student in Civil Engineering.
0098-8847/83/050623-25$ 02.50
1983 by John Wiley & Sons, Ltd.
Received 28 September J982
Revised 24 January l983
r
l
~
624
B. F. MAISON, C. F. NEUSS AND K. KASAI
systems with closely spaced natural periods. For these cases, various other rules have been suggested, but nt
single method has gained wide acceptance although a candidate may be the recently presented complete
quadratic combination method.4 This method accounts for the infiuence of modes with closely spacet
periods using the principles of random vibration theory, and is relatively easy to use.
In this paper, the performance of four different modal combination rules are investigated by sample seismi
analyses of a fifteen-storey high-rise building. The four modal combination rules are: (1) the square root of thi
sum of the squares (SRSS) method; (2) the double sum combination (DSC) method (3) the complett
quadratic combination (CQC) method; and (4) the absolute sum (ABS) method. The SRSS and ABS method:
are well known whereas the CQC method is a recent development similar in fortlI to the earlier DSC method
The study includes buildings with concentric and eccentric mass idealizations to investigate the significantx
of oneMimensional versus three-dimensional vibration response on the accuracy of the rules. The maximutr,
building response in terms of storey deflections, shears, overturning moments and torques is computed by th<
response spectrum method using each rule and is compared with the time history results for three different
single component translational earthquake records. The objectives are: (I) to present the formulations for the
different modal combination rules and summarize the assumptions used in their development; (2) to illustrate
situations where the SRSS rule leads to significant errors in peak response prediction; (3) to highlight the
lesser known DSC and CQC rules and contrast these methods by comparison of their formulations with
respect to the system's dynamic characteristics and by inspection of numerical results; (4) to explain the
physical significance of modal cross-correlation effects which are accounted for in the DSC and CQC
methods; (5) to present representative peak response prediction error magnitudes relative to time history
response values for different characteristic earthquakes based on example analyses of an actual fifteen-storey
building; and (6) to make recommendations for the appropriate use of modal combination rules in the seismic
analysis of building systems.
RESPONSE SPECTRUM MODAL COMBINATION RULES
In this section, the equation forms of the modal combination rules are presented along with a brief discussiot
regarding their formulation and application.
Square root sum
of the
sum
of the
squares (SRSS) method
Form of combination rule:
where
R
„= estimated
maximum response for quantity R
R in mode i
Rt —maximum response of quantity
n = number of modes considered.
Double sum combination~ (DSC) method
Form of combination rule
7
R
„=
g g
RIPttRt
(2)
where
'pparently
the name 'double
sum'as
be noted that this rule differs from the
introduced in Reference 8 for the combination rule developed by Rosenblueth er ot. It should
NRC double sum method.'
4
4
k
C'
SEISMIC RESPONSE SPECfRUM COMBINATION RULES
ested, but no
ted complete
osely spaced
Imple seismic
Ire root of the
the complete
EBS methods
DSC method.
e significance
he maximum
puted by the
hree different
ations for the
2) to illustrate
highlight the
ulations with
o explain the
IC and CQC
time history
fifteen-storey
in the seismic
625
in which
rai
= aI(Q(1-(Pi)
Pi
(4)
)
2
= PI+Stag
= natural frequency of the ith mode
= critical damping ratio for the ith mode
S= time duration of 'white. noise'egment of earthquake excitation. For actual
co,
P,
earthquake
records, this may be represented by the strong motion segment characterized by extremely
irregular accelerations of roughly equal intensity.
'
Complete quadratic combination (CQC) method
Form of combination rule:
=
Z ZRIPIIR
(6)
where
I I
gaII'+g)+4(P)'+Pg)
g v (PI PI roI re)(PI >+ PgI)
(f- ))'+4PI
Absolute sum
oj'odal
PgroI
W'>
maxima (ABS) method
Form of combination rule:
R,„= Z/R
ief discussion
h et
(2)
{g)
The accuracy of each of the above modal combination rules in predicting the peak time history response
depends upon the characteristics of the earthquake record and the structure's dynamic properties. The SRSS,
DSC and CQC rules are based upon the theory of random vibrations. Two of the major assumptions used in
the development of these rules are: (I) the excitation is a sample of a wide frequency band (covering the
structure's natural frequencies) stationary Gaussian random process; and (2) the vibration responses of the
structure's normal modes are also stationary. In general, these assumptions are reasonably accurate if the
earthquake has a time segment with extreme irregular accelerations of roughly equal intensity which is
several times longer than the fundamental period of the'tructure.'he simple form of the SRSS rule as
compared to the DSC and CQC rules is a consequence of the additional assumption that the modal
vibrations are statistically independent; that is, the vibration of any mode is not correlated to that of any
other mode. In systems with closely spaced periods, the SRSS rule may be a poor estimator of the actual
maximum response.
By introduction of a modal cross-correlation coe5cient matrix P,;, the DSC and
CQC rules account for the mutual reinforcement and/or cancellation of modes with closely spaced periods.
In particular, the important quality of retaining the signs when combining the cross-modal components
(allowing cancellation) can be most significant. Elements of the matrix P,> can assume values ranging from
zero to one (where zero represents no modal cross-correlation) depending primarily upon the relative
proximity of the natural periods (Figure 1). If the periods are well separated, the off-diagonal cross-modal
terms (i g jj of the matrix P,J become small and the DSC and CQC methods approach equivalence with the
SRSS rule.
auld
Both the DSC and CQC modal cross-correlation coefficient matrices are functions of the modal
frequencies and damping ratios. In addition, the DSC formulation includes a parameter for the strong
motion duration. To contrast the two methods, the effects of these parameters on the modal cross-correlation
coefficient relating two modes are presented in Figure 1. For both the DSC and the CQC methods, modal
626
B. F. MAISON, C. F. NEUSS AND K. KASAI
1.0
5X
Critical
Damping
(all
aodes)
rI
C
Ie
C
0
DSC
Formulation
(Ti/5
0.20)
DSC
Fortcilatton
(Ti/5
0.10)
Ct}C
Foriautatton
DSC
Formulation (Ti/5
JI
~
C
'
4I
I
L
O
rIII
~
0.0;
5
)
O
I
IJ
0.0
1.0
1.5
2.0
Ratio
2.5
3.0
(or u3/ I)
Tt/T3
1.0
Iox Critical Damping
(all
codes)
~I
~«
V
0
I
C
O
r'
~J
~
I
DSC
Foiswlat ton
(Ti/5
~
0.20)
DSC
Forculat ton
(Ti/5
~
0.10)
CQC
Forcutat ton
DSC
Forcutation (Ti/5
V
I
O
IJ
Vl
Vl
~
0.0;
5 ~ ~ )
CI
'll
~
rs
0.0
1.0
1.5
2.0
R~tio Tt/T~
'.i
(or
2.5
3.0
j/ I)
(b)
J'a+
Figure I. Comparison of DSC and CQC modal cross. correlation coefficients: (a) 5 per cent critical damping: (b) IO per cent critical
damping
k ..",.:"
k,
*
t
~
!~':''.'.''
'
~,
~i
~
~
~
q
~,'
~.„r
I~
r
a:,
~'
'~rJ'
'...
',.'~;';, ll,. '-t:~
p
S
~
-
r
"
.
mr I 4«.
Jr-.«Y
~
.
'
''m
0
e'
SElSMlC RESPONSE SPECTRUM COMBlNATlON RULES
3.0
627
cross-correlation coefficients increase as adjacent modal periods approach the same value, and as the modal
damping increases. In addition, for the DSC method, as the ratio of the natural period to the earthquake
strong motion duration (ratio Tr/S) increases, the modal cross-correlation coefficients increase. Therefore, for
a given period ratio Tt/Tj, modes with the longest periods will have the largest cross-modal effects. When the
DSC strong motion duration is se! to infinity, the DSC and CQC methods become virtually identical.
As a guide to the approximate natural vibration period range in which random vibration theory based
rules (i.e. SRSS, DSC, CQC) are most appropriate, it has been suggested'hat structures having their most
significant natural periods in the range bounded by the intersections of the a and v lines and the v and d lines
that are used in the construction" of a tripartite logarithmic response spectrum earthquake plot are best
suited for these types of combination rules (where a, v and d are the peak ground acceleration, velocity and
displacement, respectively). For earthquake records associated with firm ground sites and moderate
distances from the focus (El Centro 1940 record type), the corresponding period range is from about 0 5 s to
4 s. An example where a combination rule not based on random vibration theory would be more effective is in
the analysis of very short period (very stiff) structures where the spectral accelerations approach the peak
ground acce1eration. For this case, an algebraic sum of the modal responses will yield the best accuracy in a
response spectrum analysis. This approach is equivalent to a static analysis using the peak ground
acceleration times the structure's mass to develop external forces. In the analysis of high-rise buildings, the
modes contributing significantly to the response generally have periods greater than 05s; therefore, the
algebraic combination rule is not considered in this study. However, it should be noted that situations can
arise where other special rules are more appropriate.
The ABS rule is an upper bound estimate of the response. It assumes that all modes reach their maxima
with the same sign at the same instant in time. In general, this method results in response estimates that are
very conservative and is usually not used for design purposes. It is presented in this study because it is of
interest to compare the other combination methods against the upper bound values for the response.
In the application of the four aforementioned combination rules, several properties regarding the peak
response quantity estimations should be noted. First, the sign of the response quantity is lost; that is, the peak
response may either be plus or minus. When combining the results with load cases of known signed responses
{e.g. static gravity load cases) judgement must be exercised to formulate the appropriate loadings for design
purposes. Secondly, a collection of response quantities produces an estimated maximum response envelope.
When considering an envelope of maximum response quantities, it.should be recognized that they do not
necessarily occur at the same time, consequently if additional response parameters are generated from
combinations of these envelope values, inconsistencies are introduced. For example, the use of a storey
inertia force envelope to calculate cumulative storey shears results in values larger than the combined modal
storey shears. In addition, the use of a storey displacement envelope to calculate storey drifts results in values
smaller than the combined modal drifts. Regarding design applications, the former case may be considered
conservative, whereas the latter case is unconservative. Thus, to arrive at the best estimates of the peak
response values, modal combinations should be perforined separately for each of the response quantities that
are to be considered.
EXAMPLE BUILDING MODELS
3.0
In order to produce representative results that may be expected in actual design situations, a model of an
existing modern high-rise building has been formulated for use in the comparative analyses. The fifteenstorey steel moment resisting frame structure of the University of California Medical Center Health Sciences
East Building(located in San Francisco) is used as an example building for this study (Figure 2). The building
is 195 ft in height and is square in plan with an outside dimension of 115 ft 3 in. The columns are located near
the periphery along frame lines 10ft 10in from the building perimeter (twelve vertical column lines with no
interior columns). Four moment resisting frames are located in each of the North-South and East-West
directions. Two building models are formulated. The first is a 'regular'uilding in which the centres of
stiffness and mass are coincident. The second is an 'irregular'uilding with mass offset from the stiKness
J
L
B. F. MAISON, C. F. NEUSS AND K. KASAI
628
Interior
Frame
Exterior View
ROOF
~
ice+
FLOOR 2
7
'I
n»
7~
Figure 2. Example building
j4
~
~
"7
centre of the building. The building's actual dynamic characteristics (natural periods and mode shapes) have
been determined from an experimental vibration study'nd a detailed analytical model of the building has
been developed which represents the small amplitude behaviour with good accuracy.
A modified version of the ETABS's program is used for the analytical study. Floor diaphragms are
idealized as being rigid in their own plane, allowing each floor level to be represented with three mass degrees
of freedom (two lateral translational and one torsional). Both building models are assumed to be fixed at the
ground level (floor 2) and to have 5 per cent of critical damping in all modes.
~~
\
.h
%P '>>
*
Wi;- '.,
Regular building model
A characteristic of this model, due to symmetry, is that it has uncoupled translational and torsional modes;
that is, each mode responds in a purely translational or torsional sense. This implies that for translational
earthquake ground motion input along either of the building's main axes only those translational modes
7
"
w'. *..
I
r
~
i
r
629
SHSMIC RESPONSE SPECTRUM COMBINATION RULES
Ro
F
P
'0
I
Bl
I
Er''4
15
I
Bl
—
I
BII
13
I
Bl
12
Br
r
Analyst cal
Model
//
I
6
/8
/
/8
IB
I
P
IB
I
I
s8
Experlmntal
Value
I
I
P'
-1.0
1.0
RDDE
1.0
I
IODE 2
RO
F
Br
15
8
8
13
12
r
I
I
8
8
8
3e shapes) have
he building has
8
}
8
iaphragms are
8
e mass degrees
r
\
be fixed at the
o
I
/
r/
/
/
rsional modes;
r translational
modes
atio
-1. 0
1.0
HDDE
3
Figure 3. Regular building mode shapes
'I
'I
k
~
630
B. F. MAISON, C. F. NHJSS AND K. KASAI
parallel to the ground motion are excited. The orthogonal translational and torsional modes have zero
participation factors. The structural response may be described as one-dimensional since the floor motions
have only a single displacement component. The actual building was experimentally tested at various stages
during the construction sequence. The East-West dynamic characteristics of the completed building are
presented in Table I and Figure 3 for comparison with the analytical model properties. Comparing the
analytical with experimental periods from Table I, the first and third periods agree within 2 per cent and the
second period is within 11 per cent. Note that the periods are well separated. In Figure 3, it can be seen that
the analytical mode shapes fit the experimental results favourably.
Table I. Regular building natural periods
Period (s)
Mode
number
Analytical Experimental
model
1
0386
0222
0154
1965-II
Direction
E-W translational
1 I
043
113
1
2
3
4
foal
(all modes)
0.22
Inegular building model
The irregular building model is developed from the regular model by offsetting the centre of mass at each
storey level by 10 per cent of the plan dimensions of the exterior frame lines as shown in Figure 4. This results
in coupled natural modes; that is, each individual mode contains both translational and torsional floor
displacement components whereby earthquake excitation from any direction will cause three-dimensional
response with all the modes participating. Representative mode shapes are shown schematically in Figure 5,
and the natural periods are contained in Table II. Modes containing predominantly translational
components have closely spaced periods.
9 ~ 4N
NORTH
z
z z
Center of ttass
I
I
O\
H
0
Center of Sttffness
93 ~ 7H
Figure 4. Irregular building plan view
r--
I
I
I
I
I
I
I
I
L
I
I
I
I
I
I
I
I
I
as ~ .937
I
L
J
at
~
ar
bs
I
1.0
-.677
I
L
S.OS x 10
r"
J
~
aT
bo ~
bT ~
I
.636
L
I.ti x 10 i
1.0
1.0
bs + 90. ~ x
I&
1
I
I
I
"..,"Ppgj/4»
I
I
I
I
I
r
I
I
L
17th Floor
I
1
J
17th Floor
I
I
L
J
1
I
I
ltth Floor
T
I
I
I
I
1
I
I
I
I
I
I
I
I
ax ~,.733
as ~ s,a66
ae ~ -3.79 x 10
I
I
I
J
1
a.
as
ao ~
.9t9
~
aT ~
aX ~ .i80
&T ~ .730
x 10
.6$ 3
-.698
ao ~ 68.1 x 10
i
i
r
I
7th Floor
~
rc'I 4'S
's,rt'. Iri
(
I
I
I
I
I
I
I
I
I
I
I
7th Floor
I
I
I
I
I
I
J
F
bx ~ . 337
bT ~
ao
710
-1.78 x 10
I
I
I
1
I
I
I
I
I
7th Floor
1
L
I
I
I
I
I
I
I
I
I
I
at
~
.t16
aT ~ .378
L
ae ~
.i]S
J
x 10-i
IRST IRXE
SECOND hDDE
Ftgure 5. Irregular buddmg mode shapes
ax ~
ar
.tlS
-.790
bo ~ 37.1 x 10
i
b
B. F. MAISON, C. F. NEUSS AND K. KASAl
632
Table II. Irregular building natural periods
Mode
number
Period
Predominant direction
(s)
1
1
167
2
1
121
3
0773
0409
0390
0278
0238
4
5
6
7
8
0.225
9
0166
10
11
0.165
0.156
12
0124
First E-W mode
First N-S mode
First torsional mode
Second E-W
Second N-S
Second torsional
Third E-W
Third N-S
Fourth E-W
Third torsional
Fourth N-S
Fifth E-W
EXAMPLE EARTHQUAKES
Each building model is analysed using three single component translational earthquake records input
parallel to the East-West building axis. Data for the selected earthquake records'~ are as in Table III.
As shown in Figure 6, the acceleration time histories have different characteristics. These records are
chosen to be representative of the different earthquake excitation types that may be encountered. For this
study, the earthquake records are scaled to 02g peak ground acceleration and their response spectra are
shown in Figure 7.
Table
III. Earthquake
Earthquake
record
San Fernando 1971 Pacoima Dam
component)
Imperial Valley 1940 El Ccntro
(SOOE component)
San Fernando 1971 Orion Blvd.
(NOOW component)
data
Soil
type
Epicentral
distance
Rock site
2 miles
Stiff soil
5 miles
(SOOE
Dccp cohcsionlcss
soil
10 miles
By inspection of the earthquake records (Figure 6), it is not obvious what value of the DSC earthquake
duration parameter S may be optimal for each earthquake. In this study, S is taken as 10s for each
earthquake, although longer values may be more appropriate, especially for the El Centro and Orion records.
REGULAR BUILDING ANALYSIS RESULTS
The storey deflection, shear and overturning moment maximum envelope values resulting from the Pacoima
Dam excitation are shown in Figure 8. Note that the estimated maximum values using the SRSS, DSC and
CQC methods are virtually identical. This is expected because the structural periods are well separated
(Table I), which produces very small DSC and CQC modal cross-correlation coefficients (Figurc I), justifying
the SRSS assumption of statistically independent modal responses for this model. A comparison with the
time history result indicates that the SRSS, DSC and CQC methods all give good estimates of the peak
response. However, the ABS combination appreciably overestimates the actual response at most storey
levels, illustrating the important influence of the individual modal maximum response time phasing to the
total response.
SEISMIC RESPONSE SPECIRUM COMBINh'DON RULES
633
1 ~ 20
D. ~ D
0 40
~
'R
D
a
lal
II
0 ~ 00
.~0
Ql
lt
El
a
.00
1.20
O,DD
$
.00
1$ .0
1C F 00
210E
0"
2C.OC
2$
20.DO
2$ ,00
20.00
2S.C
~
tsces
.(8) PACDIt1A DAN 1971
=ords input
able III.
records are
d. For this
spectra are
0 ~0
0 F 20
0 ~ OD
a
~
~
2D
J
EP
a
,10
0.00
S.OD
10.00
1$ .00
TlnE tsEet
(b) EL CENTRD
1940
0. ~ 0
arthquake
for each
~
0 ~ 20
n records.
'R
O
0.00
~4
a
a
4l
lal
Pacoima
DSC and
separated
:
justifying
with the
the peak
storey
tg to the
st
0
II)II! ilI III
.20
Ll
a
D,DO
S.OD
10
~
1$ ~ DC
DC
110$
I SEC I
(C) DRIDN 1971
Figure 6. Example earthquake excitations
~
~
C3
634
B. F. MAISON, C. F. NEUSS AND K. KASAI
San Fernando
1971
Orion Blvd.
rI
I
0.6
t
I
t
ltapertal Valley 1940
El Centro
.1
CO
I
4l
V
'IA
Pacoitaa Data
\
t
C
0
a
u~
San Fernando 1971
I
I
t
0.4
I
't
I~
02
\
\
t
I
\
la
'L
rr
r
CP
4I
CL
IA
0.0
2.0
1.0
Period
(Seconds)
Figure 7. Example earthquake response spectra
For each earthquake analysis, the accuracy of each combination rule in predicting peak storey shears is
presented in Appendix I (Table Vl) in terms of a percentage error relative to the time history response (a
negative value implies underestimation). Regarding the response in the building's lower storeys, the SRSS,
DSC and CQC rules tend to overestimate the actual deflections and shears (by as much as 15 and 14 per cent
for deAection and shear, respectively), whereas the overturning moments are estimated within 5 per cent.
Upper storey deflections are accurately predicted (within 5 per cent). However, shears and overturning
moments are both overestimated (Pacoima Dam) and underestimated (El Centro) at several levels by more
than 10 per cent. ABS combination is conservative, being 30 per cent greater than time history response for
lower storey deAections, upper and lower storey shears, and upper storey overturning moments.
In Table IV, the average error (calculated from absolute error values), the percentage of the response
predictions over the height of the building that are underestimated when compared with total number of
response predictions (error bias), and the error'xtremes for each combination rule considering all three
earthquake inputs are shown. For reasons discussed previously, the SRSS, DSC and CQC rules yield similar
results with average errors in the response ranging from 6 to 8 per cent. The error bias indicates generally
conservative deflection (only about 20 per cent of the responses are underestimated) and storey shear (only
about 30 per cent underestimated), but slightly unconservative overturning moment predictions (about 60
per cent underestimated). The ABS rule overestimates response at all levels with average errors of 24, 41 and
33 per cent for deAection, shear and overturning moment, respectively.
IRREGULAR BUILDING ANALYSIS RESULTS
Plots of storey deflection, shear, overturning moment and torque envelopes from the Pacoima Dam
excitation are shown in Figures 9, 10 and 11. These quantities are categorized as response parallel (EastWest) to the earthquake excitation direction, orthogonal (North-South), and torsional about a vertical axis.
For the parallel, orthogonal and torsional response quantities, the DSC and CQC rules yield similar values
which provide the best estimates of the actual peak responses. The SRSS rule significantly underestimates the
parallel responses, greatly overestimates the orthogonal responses, but gives a fairly good estimate of
'I
I
<l
~~
rs -'" O
!
ew
so
ui
O
~79
so
Q.OVV
A
g
A O
ro
ro
0cn
~ Eo
O Q
V
ro
\ O
t
g ~ o
ro
O g
ro
jg
Cs
s
OOHB
A 00
CA
un
Ttw History
SX55,05C,CQC
SS
13
12
//
I
II
I/
/
/I
/I
/I
/
/
C
x
c)
\
0
x
l
l
/;I
l
I
I
5
0
X
l000
Story Deflection
(inch)
3000
3000
StOry Sheer
(b)
l000
(klpS)
3
5000
e
Story Overturning Htnnrnt ( ~ )05 tip-!noh)
(c)
t
shears: (c) storey overturning momenls
Pigure 8. Regular building peak response envelopes from Ihe Pacoima Dam excitation: (a) storey del)ec(ious: (b) storey
s
C
r
g
C~
~
4
0
636
B. F. MAISON> C. F. NHJSS AND K. KASAI
Table IV. Regular building compiled error results
% error results compiled from three
earthquake excitations
Deflections
Shears
Average error
% underestimated~
Extreme errors
'
Average error
underestimated'xtreme
Overturning
moments
SRSS
DSC
CQC
ABS
6
22
6
22
6
24
20
-3, 12
7
8
8
33
31
-17,
Average error
%
errors
6
56
~Number of underestimated
rcsponscs
-18,
expressed
.-3,
29
errors
undcrestimatcd'xtreme
-3, 15
15
15
-19,
-18,
6
6
62
60
-19,
as a percentage
14
13
-19,
0
15
2, 54
41
0
14
10, 91
33
0
14
1,
92
of the total number of response
prcdicuons.
torsional responses. The upper bound ABS response values arc extrcme overestimates for the orthogonal and
torsional response quantities.
The irregular building model study illustrates the importance of accounting for the correlation between
coupled modes with closely spaced periods when combining modal responses in response spectrum analysis.
Since the SRSS method neglects all cross-modal contributions, the significance of this effect may be assessed
by comparison of the DSC and CQC results with the SRSS computed responses in Figures 9 and 10. For the
parallel response quantities, the DSC and CQC rules estimate the cross-modal reinforcement effectively to
account for more than 20 per cent of the deflection, shear and overturning response. On the other hand, the
striking feature of cross-modal cancellation is apparent in the orthogonal response quantity estimations by a
reduction greater than 60 per cent in the DSC and CQC responses compared to the SRSS results (Figure 10).
However, the cross-modal contributions are not significant for the torsional response since the SRSS method
yields similar results to the DSC and CQC methods which estimate the actual response well (Figure.l 1).
The mechanism by which the DSC and the CQC methods provide the appropriate amounts of crossmodal reinforcement or cancellation may be explained by consideration of the individual modal responses.
For the irregular building studied herein, the parallel and orthogonal responses are dominated by the first
two modes, that is, their modal contributions to overall response are much larger than any of the higher
modal contributions. Because the first two modes have closely spaced periods, it may be expected that each
reaches its maximum or minimum response at nearly the same time. The modal cross-correlation matrix
accounts for this effect by scaling the cross-modal contributions. As discussed previously, this scaling is
primarily dependent upon the relative spacing of the natural periods. Whether the modes are nearly in-phase
(cross-modal reinforcement) or nearly 180'ut-of-phase (cross-modal cancellation) is dependent upon the
relative signs of the modal responses. These are set by the modal participation factors which include the
earthquake directionality information. The parallel response modal components of the first two modes have
the same sign, indicating cross-modal reinforcement, The orthogonal response modal components are of
opposite signs, thus cross-modal cancellation results. This behaviour can be visualized by inspection of the
mode shapes in Figure 5. Ifthe first two modes oscillate such that the X components are nearly in-phase, it is
apparent that the Ycomponents must be nearly 180'ut-of-phase. Regarding the torsional response, the first
and third modes have the largest modal components; however, their natural periods are well separated,
which implies small cross-modal contributions. This explains the relatively small differences between the
DSC, CQC and SRSS results. Also, by inspection of Figure 5, ifthe first and third modes oscillate with the X -.
components nearly in-phase, the torsional components will be nearly 180'ut-of-phase, thus some crossmodal cancellation results and this is reflected in the storey torque envelope plot by the reduced DSC and
CQC values compared to SRSS results.
The accuracy of the combination methods in predicting the actual time history maxima of storey shear and
torque for each earthquake excitation is contained in Appendix I (Tables VII to IX). In general, the trends
0
A
<a
ha
rnA 5'A g
tis
tss
CD
Pt
~
e
O
cn
~
O
ha
A
A
CD
cn
C~ 5'9
cis
g
C O
A
~
sn
As~ g g
Y
Cn 0
CD
C
O I
A
as O. C"ta
~
O
—.
O.
'
O s
't3
g
CD
O
.
A C C. C'o A e,
A C
~ ~ pa 00 ~ to
$Q
MMsn
.D.CD A g CD
O
A A A in
v
O
~
O
'A O
as
O
O iri
A
~
CD
O
O
uia
So
CD
Q
CL
e
rI
'
p
R
-
~
I
I
I
<art
~
0SC
l
I
I
1
/
32
/
/I
/
I
j
tine History
I
I
I
I
I
I
I
't
/
//
I~r
/
\
I1
~
t
1'
.Cr
tt
\
I
I
I
I
/
I
3
l000
3
story Deflection
(Inch)
3000
e 000
3000
Story Sheer
(hips)
5000
I
t
3
Story Orerturnlnp Honent
t t00 ttp-Inch)
Figure 9. Irregular building peak response cnvelo pes from the Pacoima Dam excitation (East-West response) (a) storey dcllcctions: (b) storey shears; (c) storey overturning moments
'
';C
ss)
,J
I(
r
~
IC
p
I
I
I
I
I
1
I
I
jj
I
I
I
1
I
j
I
/
I
I
I
I
I I'
I
/
j
j,C
I
i
I
I
\
1
I
(I
I
1
I
j
1
1
I
\
I
1
\
1
I
\
1
\
I
I
1
\
I
I
C
1
\
\
I
I
\
1
'1
\
t
I
1
1
Iy
Ir
C
l
1
1
I
I
'( >
I
I
'C'
osc
I
I
I
12
SRSS
I
I
I
I
I
I
13
floe Hfstory
I
I
Story Oaf lectfon
(3)
3
(Inch)
I
1000
3000
f000
3000
Story Shear
(klps)
5000
I
3
3
story orertornfns Iaonent (
0
10
kfp fnch)
(b)
Figure lo. Irregular building peak response enve(opes from Ihe Pacoima Dam excitation (North-South response): (a) storey defleclions; (b) storey shears; (c) storey overturning
moments
.
e
639
SHSMIC RESPONSE SPECTRUM COMBINA'IION RULES
r
~
Tine Hit tory
e
0SC
'I
I
~
tl
11
1
Story Torque
d
~
I
10S
d
10
td
hip-inch)
Figure I I. Irregular building peak storey torque envelope
illustrated in the Pacoima Dam response plots discussed previously are the same for the El Centro and Orion
results; that is, the DSC and CQC methods yield similar results th'at are the best estimates of the actual peak
response. The SRSS method consistently underestimates the parallel response, greatly overestimates the
orthogonal response and reasonably predicts the torsional response. For the parallel response quantities, the
SC and CQC methods agree closely and generally do not show error trends consistent for all earthquakes
garding response overestimation or underestimation; that is, at a given floor level, the responses may be
either overestimated or underestimated, depending upon the earthquake excitation. Regarding the
orthogonal response predictions, the DSC and CQC methods have definite error trends whereby the DSC
underestimates and the CQC overestimates the actual peak responses. As shown in Figure 1, the DSC
method closely approximates the CQC method as the strong motion time duration parameter S approaches
infinity. This implies that the DSC method with an S value longer than 10s as used in this study would
achieve superior orthogonal response predictions as compared to the CQC method. The DSC results from
the Pacoima Dam excitation have the smallest average errors (Tables Vll to IX) suggesting that S equal to
10s may be a better value for Pacoima Dam excitation than for either the El Centro or the Orion Blvd.
records. The orthogonal responses as computed by the DSC and CQC methods have larger error
percentages than the corresponding parallel responses.
The combined results for all three earthquakes in terms of average error, error bias and error extremes are
presented in Table V. The parallel response predictions using the DSC and CQC methods yield the smallest
average errors (6 to 8 per cent). The SRSS method produces consistent underestimations with average errors
of 18 to 25 per cent and the ABS method has overestimations of 27 fo 49 per cent. Regarding orthogonal
response quantity predictions, the DSC and CQC methods have average errors of 16 to 18 per cent and 24 to
32 per cent, respectively. The DSC underestimates whereas the CQC overestimates the actual peak response.
The SRSS and ABS rules produce extreme overestimations with average errors of217 to 251 per cent and 491
to 528 per cent, respectively. For storey torques, the SRSS, DSC and CQC methods yield predictions with
similar average errors (7 to 13 per cent). The ABS rule considerably overestimates the actual peak torques
with an average error of 137 per cent.
It is of interest to note that the peak response quantity estimations by the DSC and CQC methods involve
more numerical calculations than those as computed by the SRSS method; yet, for this study, the diferences
~
v
n
Cr
~ ~
in 'rPoorr cc
~
V
B. F. MhlSON, C. F. NEUSS AND K. KASAI
Table V. Irrcgular building compiled error results
% error results compiled from three
earthquake excitations
SRSS
DSC
CQC
ABS
7
22
6
27
33
Parallel (E-%) response
Deflections
Average error
%
18
100
Shears
errors
Average error
%
Overturning
moments
-8
-26,
underestimated'xtreme
22
100
underestimated'xtreme
errors
Average error
%
errors
-7, 17
36
47
-18, 20
25
100
6
7
56
76
-18,
12
-19,
-18,
0
2,67
49
8
-35, -10
-34, -11
underestimated'xtreme
-5, 19
8.
0
1?
12, 122
39
0
11
2, 120
Orthogonal (N-S) response
DeAcetions
Average error
%
errors
Average error
% underestimated~
Extreme errors
Average error
%
underestimated'xtreme
Shears
Oyerturmng
moments
undercstimatcd'xtrcme
251
18
32
0
100
0
186, 350
-33,0
217
17
100
0
133, 307
218
0
136, 299
errors
-31,
-3
16
100
-25,
-6
0, 67
24
0
1, 55
25
0
491
0
371, 800
528
0
418, 661
520
0
5, 51
366, 658
7
58
137
Torsional response
Torques
Average error
%
errors
13
29
underestimated'xtreme
'Number of underestimated
responses
-15, 40
expressed
9
89
-27,18
as a percentage
-26,21
0
71,288
of the total number of response
predictions.
in execution times between analyses using the diferent methods are negligible. This is because relatively few
response quantities are calculated. For the solution of large numbers of response quantities involving many
modes, the DSC and CQC methods may require significantly more computational effort than the SRSS
method. For these cases, the DSC and CQC double summation calculations t'equations (2) and (6)] can be
truncated when the value of P> becomes small, therefore avoiding unnecessary computations. 'u
SUMMARY AND CONCLUSIONS
The peak dynamic responses of two mathematical models of a fifteen-storey steel moment resisting frame
building subjected to three earthquake cxcitations are computed by the response spectrum and time history
methods. The models examined are: a 'regular'uilding in which the centres of stiffness and mass are
coincident resulting in uncoupled modes with.,well-separated periods in each component direction of
response; and an 'irrcgular'uilding with the mass offset from the stiffness centre of the building causing
coupled modes with the translational modes having closely spaced periods. The building response quantities
examined are the storey dcflcctions, shears, overturning moments and torques. These quantities are
categorized as parallel and orthogonal response with respect to the earthquake direction and as torsional
response about a vertical axis. Four response spectrum modal combination rules are discussed and are used
to predict the peak responses: (1) the square root of the sum of the squares (SRSS) method; (2) the double sum
r
SEISMIC RESPONSE SPECTRUM COMBINATION RULES
combination (DSC) method; (3) the complete quadratic combination (CQC) method; and (4) the absolute
sum (ABS) method. The response spectrum analysis results are compared to the corresponding peak time
history analysis values to evaluate the accuracy of the different combination rules,—
For the regular building, the SRSS, DSC and CQC methods yield virtually identical peak response
predictions, and agree well with the time history values. The calculated peak response values have average
errors in the range of 6 to 8 per cent (see Table IV) with respect to time history analysis. Depending upon the
earthquake, the modal combination rules may overestimate or underestimate the peak response at a given
floor level over the height of the building. The ABS method substantially overestimates the actual response at
most storey levels.
For the irregular building, the parallel, orthogonal and torsional response quantity predictions by the
DSC and CQC methods have similar values which agree well with the time history values. The DSC and
CQC methods both predict nearly identical values for parallel and torsional response having average errors
ranging from 6 to 8 per cent and 7 to 9 per cent, respectively (see Table V, Parallel and Torsional response).
Both may overestimate or underestimate the parallel and torsional response values at a given floor level over
the height of the building, depending upon the earthquake. The orthogonal response quantities as computed
by the CQC method consistently overestimate the actual peak responses with average errors ranging from 24
to 32 per cent (see Table V, Orthgonal response). Those computed by the DSC method consistently
underestimate the peak responses with average errors ranging from 16 to 18 per cent (see Table V,
Orthogonal response). The magnitude of the DSC orthogonal estimation errors is influenced by the strong
motion time duration parameter S. Because the DSC method closely approximates the CQC method if the
strong motion duration parameter S is set to infinity, it is possible to select a value of S that achieves superior
DSC orthogonal response predictions. The parallel, orthogonal and torsional response predictions as
computed by the SRSS method have average errors 18 to 25 per cent, 217 to 251 per cent and 13 per cent (see
Table V), respectively. The parallel responses are consistently underestimated and the orthogonal responses
are consistently overestimated for all earthquakes. The torsional responses at different floor levels over the
height of the building, are either overestimated or underestimated depending upon the earthquake. The poor
accuracy in the parallel and orthogonal response predictions is attributable to the efl'ects of coupled
translational modes with closely spaced periods. The ABS method substantially overestimates the actual
response at most storey levels, especially the orthogonal response.
Based on the results of this study, the following conclusions are made.
(1) The DSC and the CQC methods provide good peak response estimates, for both regular and irregular
building models irrespective of modal coupling. Both methods are recommended. In addition to the DSC.
and CQC rules, the SRSS method gives accurate response predictions for regular buildings and is
recommended for systems where coupled modes with closely spaced periods do not dominate the response.
(2) For irregular buildings with coupled modes having closely spaced periods, the SRSS rule generally
underestimates response in the direction parallel to the earthquake ground motion by neglecting crossmodal reinforcement and generally overestimates response in the direction orthogonal to the earthquake
ground motion by neglecting cross-.modal cancellation. Because of these error tendencies, the SRSS rule is
not recommended for use in these situations.
(3) The use of a modal cross-correlation matrix is an effective procedure for combining the results from
coupled modes with closely spaced periods. Because the DSC modal crossworrelation matrix formulation
'ncludes the strong motion duration (parameter S), the DSC method has the potential to provide better peak
response estimates than the CQC method. However, the selection of S is somewhat arbitrary when utilizing
design or actual earthquake spectra. Furthermore, for the irregular building studied, only the DSC
orthogonal response predictions could be significantly ilnproved over those by the CQC method by
adjustments of S. The difference betweeen the DSC and CQC orthogonal response estimates may not be
significant in design applications since the magnitude of the orthogonal responses is considerably less than
the corresponding parallel values that may ultimately control the design. For design applications that
involve independent designs in both building principal directions,ts the orthogonal responses probably
would not govern. For these reasons, the DSC and CQC methods may be considered to yield results of
equivalent accuracy.
1SC
relatively few
volving many
Ian the SRSS
Id (6)] can be
8. 10
sisting frame
'ime history
nd mass are
direction of
ding causing
se quantities
Iantities are
as torsional
nd are used
double sum
~"
'
a&
.C
B. F. MAISON, C. F. NEUSS AND K. KASAI
(4) When interpreting response spectrum results utilizing the DSC, the CQC and the SRSS (for regular
type buildings) methods, it is important to recognize that underestimation of the actual peak response is
possible. In situations where the possibility of underestimation is unacceptable, the use of the ABS
combination method or time history analysis should be considered.
ACKNOWLEDGEMENTS
The study reported herein is part of a National Science Foundation sponsored project involving the
evaluation of computer modelling procedures for the earthquake response of multistorey buildings. The
authors gratefully acknowledge the support under Grant PFR-7926734 which made this study possible.
APPENDIX I
Regular and irregular building peak response prediction errors from the inditndual earthquake excitations
~
'
8cn
W O
00
CA
OO
n
n n
Table Vl. Regular building storey shear results
a
Time history
Response spectrum
Error
results
Level
V
(kips)
(I)
(2)
RI'16
15
14
13
12
6
5
2,991
4
3,029
3,044
10
9
8
7
3
Average
tg
1.223
1.540
1,803
2,053
2.269
2,453
2.6()9
2.738
2,843
2,928
II
'Cs,
559
968
'„'rror
i>s
848
848
476
476
416
416
414
4.14
(/)
Time history
results
time history
Response spectrum (%)
Error
time history
us
Time history
results
ABS
(s)
SRSS
DSC
CQC
ABS
V
(kips)
Time
CQC
V
(kips)
Time
DSC
(5)
(6)
(7)
(8)
(9)
(IO)
(11)
(12)
(13)
(14)
(15)
0
-2
—I
80
6
4
5
10
8
9
14
10
-19
-18
-17
-18
-17
13
354
354
352
-17
—17
15
621
1,136
1,552
1,891
3.52
7
7
350
4
2
3
I
4
—10
—14
—10
I
I
I
I
I
I
3
5
7
3
7
3
5
7
8
10
9
9
10
IO
11
11
11
54
57
2,172
2,347
2,426
2,441
2,495
2,666
2,777
2,840
2,893
2,945
2,967
6
6
6
49
Time
Is)
(3)
Orion Blvd.
El Centro
Pacoima Dam
SRSS
(4)
5
2
82
82
65
52
47
36
24
16
25
30
36
46
I
I
350
354
304
304
302
3.02
300
300
—15
—15
—14
-10
-4
—16
—14
-15
-4
-4
4
4
4
9
8
9
8
8
9
9
8
9
10
12
12
12
11
II
Il
13
13
13
12
12
12
13
13
51
44'16
32
24
17
22
25
26
25
29
29
35
45
52
55
34
636
(s)
11
84
1,215
1,725
2,165
2,542
2,863
3,159
3,415
3,610
3,742
3,816
3,844
3,846
3,839
3,890
Response spectrum (%)
Error
es
SRSS
(16)
DSC
CQC
ABS
(17)
(18)
(19)
7
3
5
6
91
2
3
76
57
45
35
-I
-I
-2
-2 -3
I
11
84
13 12
11
88
time history
0
—2
-3
-4
-4
-2
-3
-4
-4
-3
-3
-3
-4
-4
-2
0
0
0
3
3
7
3
7
33
25
15
10
19
10
11
11
24
32
44
88
13
1248
12
14
13
13.
13
53
55
5
5
5
41
11
7
4))h
I
'ig
I
O
O
5
5
O
i
h.r
fF $
i
'4
t
~
'Ap
.S
ps:
S,', +s(S
pV
* 's
~
p(,
'ihk,44Rsh
~~s
~
"~
~
sp Q
s
ssp(,
'v
Table VII. Irregular building storey shear results, East-West response
El Centro
Pacoima Dam
Time history
Error
V
Time
Level
[kips)
(s)
(I)
(2)
(3)
RF
446
l6
835
IS
1,183
1,474
1,720
1,976
14
13
12
II
IO
9
8
7
6
5
4
3
Average
2,188
2,359
2,498
2,617
2.705
2,772
2,828
2.863
2,880
"r„'rror
SRSS
(4)
—.12
4 80
4 80
4 78
4 78
4 6
4 16
4 18
4 18
4 18
4 16
4 16
4.16
4 14
Time history
Response spectrum (%)
results
~
—13
—!6
—17
-19
-22
-24
-24
-23
-22
-20
-18
—17
—16
—16
19
i)s.
time history
DSC
CQC
(5)
(6)
.
ABS
(7)
V
Time
(kips)
(8)
(s)
122
580
103
81
7
12
10
7
5
5
4
2
0
57
48
38
28
1,032
1,407
1,742
1,968
13
II
8
0
0
-2
-2
I
—I
3
0
5
11
3
5
7
8
12
9
6
5
8
10
70
19
29
35
41
50
59
63
56
2,086
2.233
2,336
2,419
2,463
2,490
2,554
2,580
2,609
2,633
Response spectrum (%)
Error
results
(9)
356
356
354
352
352
584
584
586
586
586
344
3W
3W
300
3.00
Orion Blvd.
time history
us.
DSC
CQC
(11)
(12)
-35
-34
-34
-34
-32
-28
-26
-23
-20
—17
—14
-13
-11
—10
—IO
-18
—15
—14
-14
—11
-19
—16
—IS
-16
-13
23
12
12
SRSS
(10)
-5
-2
-7
-4
ABS
(13)
%38
31
2,639
2,981
40
19
16
17
17
-21
38
7
20
20
1316
1,772
3
II
(15)
46
5
IO
13
(14)
SRSS
(16)
636
0
14
Time
46
51
61
73
7?
49
3,263
3,493
3,669
3,789
3,855
3,876
3,869
3,854
3,846
(s)
13 16
13 12
13 12
i)s
iss
time history
X
V
(kips)
71
61
2
Response spectrum (%)
Error
results
37
37
34
29
16
Time history
-24
-27
-28
-29
-29
-29
-28
-27
-25
-22
—19
-17
—IS
-14
24
vh
DSC
CQC
ABS
(17)
(18)
(19)
.
~
0
—I
-3
-6
-7
-8
-9
-8
-7
-5
-2
—10
-10
-10
2
5
9
3
-5
-7
-9
-9
-7
-4
—I
6
9
94
73
53
44
33
29
24
17
12
21
27
35
46
56
12
13
10
61
6
7
42
?;
r
,
''t
~
~
g
tlr
Table VIII. Irrcgular building storey shear results, North-South response
Time history
results
V
I.cvcl
7i"'I'6
15
14
,i
~
(3)
144
846
844
898
898
898
840
838
610
612
558
566
566
564
564
564
10
439
424
413
9
421
8
7
430
467
6
511
5
549
578
590
12
u
(s)
(2)
244
335
407
436
II
4
3
Average ".r„'rror
Response spectrum (%)
Error't c time history
SRSS
(4)
133
155
155
156
238
273
291
307
296
279
266
256
252
229
results
Response spectrum (%)
Error trr time history
V
Time
CQC
ABS
(kips)
(s)
SRSS
(5)
(6)
(7)
(8)
(9)
( IO)
-22
-20
-24
-28
I
510
526
486
457
116
194
5 34
171
6
406
199
4 02
4.02
200
174'26
201
Time history
DSC
Time
1k t ps)
-23
—16
—IO
-7
-3
-4
-6
-8
-9
-9
14
Orion Blvd.
El Centro
Pacoima Dam
it,
4
2
8
17
30
43
49
55
51
46
42
38
37
29
.
461
505
551
565
544
617
612
600
609
621
625
533
268
344
394
421
451
498
551
600
639
675
710
754
778
400
400
7 26
7.28
730
730
7 32
674
674
442
440
189
194
211
221
215
205
196
192
188
182
171
Time history
ASS
V
(kips)
Time
CQC
I)
(12)
(13)
(14)
(15)
-9
-8
17
639
153
661
601
vis
10 52
24
22
26
532
496
519
518
394
443
502
1052
1544
1504
1508
23
471
19
15
14
12
418
427
620
674
717
421
751
424
440
445
443
497
779
800
813
818
DSC
(I
—14
-20
-21
—19
-18
-20
-24
-26
-27
-27
-28
-29
21
16
17
11
7
165
-31
4
193
21
17
Response spectrum (%)
Error
results
(s)
SRSS
(16)
173
184
204
561
14 56
241
254
250
244
237
232
231
1456
232
234
236
237
238
228
vs time
history
DSC
CQC
ABS
(17)
(18)
(19)
14
16
612
595
594
633
613
—13
—16
—15
'-9
-9
—14
—17
-21
-22
-23
-21
-20
—18
—17
—16
17
21
34
37
34
31
28
26
25
26
27
29
30
30
27
591
550
494
449
478
489
506
538
568
580
553
I
8
5
R
vvnsvsrqvv snss
ji
sr )
)I
(t
~
)L(
'R
v,'2
's
~
7Bsv~s s
g<.,
i
s4
kseennvvlveskos(ISYLLAA
nw ssnw
Table IX. Irregular building storey torque results
El Centro
Pacoima Dam
Time history
Response spectrum (%)
Error
results
i)s
T
( x IOs
T
Time
CQC
ABS
(k.in)
(s)
(5)
(6)
(7)
(8)
-5
-5
-2
227
214
117
2
13
15
17
18
-8
-8
-5
-3
-2
-5
-7
-7
-7
-9
-8
-7
-6
-5
-5
IS
6
k.in).
(s)
( I)
(2)
(3)
SRSS
(4)
16
73
130
168
194
216
243
267
284
482
15
l3
12
II
10
9
8
7
301
3
325
344
357
365
370
2
371
6
5
4
Average:„error
IS
482
418
416
416
418
422
424
15
18
19
16
14
15
14
12
11
424
426
426
i)s
Time history
time history
T
Response spectrum (%)
Error
results
DSC
CQC
ABS
(k-in)
(s)
(9)
(11)
(12)
(13)
(14)
(I 5)
SRSS
(16)
201
205
266
3 58
3 58
3 56
-15
-14
—12
I&?
311
2.70
-26
-23
-20
-17
3
161
0
156
155
142
114
109
108
359
388
404
272
272
-27
-25
-23
-20
—19
-16
—I I
-14
—16
—15
-15
-15
—14
-14
—13
-9
106
112
17
14
102
~
121
I
142
0
I
159
167
2
158
ees
524
548
567
579
587
590
3 10
3 10
30&
3.08
-9
-9
-4
2
-2
-3
-4
4
-3
-2
0
6
—16
-12
-7
—IO
—12
—12
-11
-11
-10
-10
141
85
932
40
135
125
114
161
1252
230
296
35
28
90
361
99
418
462
114
97
76
73
71
79
94
491
526
558
589
620
647
667
675
i)s
time history
Time
SRSS
(10)
-2
-2
-2
-3
-3
-2
-
Response spectrum (%)
Error
results
DSC
Time
Level
15
14
Time history
time history
Orion Blvd.
1252
1248
21
16
12
12
14
14
CQC
ABS
(17)
(18)
(19)
18
15
10
5
21
288
259
219
19
14
10
5
2
I
-2
-3
2
—I
—I
-I
180
140
133
131
3
3
124
4
103
10
10
-2
-3
-4
-5
-6
-I
102
100
107
118
126
130
17
5
6
150
13
12
11
1248
DSC
en
Rt "V'e
3
2
I
0
I
o
5
?'4
I» I
J
SEISMIC RESPONSE SPECTRUM COMBINATION RULES
REFERENCES
'Combining modal responses
and spatial components in seismic response
6.
analysis', U.S. Nuclear Regulatory Commission,
Regulatory Guide 1.92, Recision I, 1976.
'Recommended practice for planning, designing and constructing lixed oifshore platforms', American-Petroleum institute, API RP
2A, 13th edn. 1982, p. 22.
"fentative provisions for the development of seismic regulations for buildings', Applied Technology Council, US. Iqatianal Bureau
of Standards Special Publication $ 10, 197&, p. 63.
E L Wilson, A. Der Kiureghian and E. P. Bayo, 'A replacement for the SRSS method in seismic analysis,'arthquake eng. struct.
dyn. 9, 187-194 (1981}.
E. Roscnblueth and J. Elorduy, 'Response of linear systems to certain transient disturbances', Proc. 4th world conf. earthquake eng.
Santiago, Chile I (1969).
L E Goodman, E. Roscnblucth and N. M. Ncwmark, 'A seismic design of Iirmlyfounded elastic structures', Trans. hSCE 120, 782-
7.
N. M. Newmark and
1.
2.
3.
4.
S.
802 (1955).
E Roscnblucth,
Fundamentals
ofEarthquake Engineering, Prentice Hall, Englewood Cliffs, NJ„1971, pp. 309-
313.
A.K. Singh, S. L. Chu and S. Singh.'Influence of closely spaced modes in response spectrum method ofanalysis', Proc. specialty cunf.
struct. des. nuri. plan>facilities, Chicago 2 (1973).
9. A. Der Kiureghian, 'On response of structures to stationary cxcitations', Report No. UCB/EERC-79/32, Earthquake Engineering
Research Center, University of California, Berkeley, CA, 1979.
10. A. Der Kiureghian.'A response spectrum method for random vibrations', Rcport hfo. UCB/EERC40/IS, Earthquake En'gineering
Research Center, University of California, Berkeley, CA, 1980.
11. N. M. Newmark and W.J. Hall, 'Procedures and criteria for earthquake resistant design,'uilding Practices of Disaster M(tigation,
Building Science Series 46, Department of Commcrce, U.S. Bureau of Standards, 1973.
12. D. Rea, J. G. Bouwkamp and R. W. Clough, 'Thc dynamic behavior of steel frame and truss buildings', AuterJeaii Iron and Steel
Institute. Buiietin Iqo. 9 (1968).
13. E. L Wilson, J. P. Hollings and H. H. Dovey, 'Three dimensional analysis of building systems (extended version)', Report Ão.
UCB!EERC 73-13, Earthquake Engineering Research Ccntcr, University of California, Berkeley, CA, 197$ .
design', Rcport Na. VCB/EERC 74-I2,
14. H. B. Seed, C. Ugas and J. Lysmer, 'Sile-dependent spectra for earthquake-resistant
Earthquake Engineering Research Center, University of California, Berkeley, CA, 1974.
IS. 'Uniform building code, Int. conf. build. oJPeiais, Whittier, California (1979).
8.
>ra,a
I >~
1
I
l'