ML18029A934

Speech Entitled, Complete Quadratic Combination Technique for Model Responses

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Issue date:	02/28/1985
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COMPLETE QUADRATIC COMBINATION TECHNIQUE FOR MODAL RESPONSES Presented by:

Impell Corporation February 1985

I g

r

MODAL COMBINATION RULES 0

Standard Approaches

'I ASUM SRSS Ten Percent Grouping Double Sum For Closely-Spaced Modes 0

Present Methods for Closely-Spaced Modes are Overly Conservative 0

SRSS May Be Unconservative 0

An Improved Technique is the Complete Quadratic Combination (CQC)

Method Based on Random Vibration Principles More Accurate Insignificant Additional Computational Effort

I p ir l

0

CQC METHODOLOGY R =

(~ ~

P R.

R

)

j k jk j k

where R.,

Rk

= Modal Responses for Modes J and K

j' P.k

= Correlation Coefficient Between Modes J and K (based on Random Vibration Principles)

= Combined Response.

~

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

~3G Gm I

CONCRETE FLOOR DIAPHRAGMS STEEL EXTERIOR STRUCTURE

~25.5m

~21 5m

~16 5m

~11 3 m I

jJjTl

~10 m 77 m

COMPARISON OF RESPONSES BY VARIOUS METHODS ASUM 10PC SRSS

~C C

CQC

~C C

Observations All Components 2.90 Forces and Moments 2.86 Accelerations 3.61 Displacements 2.30 2.31 2.21 3.18 1.77 1.02 1.04 1.04 0.94 1364 836 264 264 TABLE 1.

All Design Data Due to Three Directions of Input ASUM 10PC SRSS

~CC

~CC CQC

~C T-H Observations NS - Total

- In-Plane

- Out-of-Plane EW - Total

- In-Plane

- Out-of-Plane VT - Total

- In-Plane

- Out-of-Plane 3.81 2.77 1.15 2.18 1.46 0.92 4.79

';56 1.29 3.70 2.58 1.05 2.06 1.43 0.78 4.68 3.27 1.21 5.24 5.03 1.26 2.26 2.26 0.69 5.66 5.43 1.34 1.03 0.98 1.05 1.05 1.00 1.07 N/A N/A N/A 304 114 190 304 114 190 304 38 266 TABLE 2.

Forces and Moments Due to Single Earthquake Input

-4

NS

- Total In-Plane

- Out-of-Pl ane EW Total

- In-Plane

- Out-of-Pl ane VT

- Total

- In-Plane

- Out-of-Pl ane LN8 CQC 4.74 2.59 5.81 5.21 2.76 6.44 6.29 2.44 7.06 3.92 2.00

'.88 4.34 2.05 5.49 5.44 2.43 6.82 S~R QE CM T-H ~e~~

1.26 1.07 264 0.91 1.04 88 1.43 1.08 176 1.31 1.00 264 0.89 0.98 88 1.51 1.00 176 1.45 N/A 264

0. 70 N/A 44

1. 60 N/A 220 TABLE 3.

Accelerations Due to Single Earthquake Input Force and

~V~~n

~c~e~~

NS - Total

- In-Plane

- Ou,-of-Pl ane Eg - Total

- In-Plane

- Out-of-Pl ane

.09

.03

.11

.13

,06

.15

.14

.05

.16

.12

.08

.13 TABLE 4.

CgC/T-H Comparisons:

Standard Deviations

~ y

0 Typical Piping System D(~

l/5 lg V(p Cw/)

~ 5 Cg Ic~

(]pl(,)i Qsy

~

CQ

( ap/I Sg)

$/2 PIPING PROBLEM AC-03 (SONGS-1)

I

~ I

RESPONSES DUE TO SINGLE EARTHQUAKE INPUT Z-Direction Pipe Moments:

10PC T.H.

C~C Number of T.H.

Observations

- Total

- In-Plane

- Out-of-Plane 1.40 1.12 1.70 1.07 1.03 1.11 40 20 20 Support Reactions:

- Total

- In-Plane

- Out-of-Plane 1.85 1.27 2.17 1.15 1.05 1.20 17 6

11 C C/T-H COMPARISONS:

STANDARD DEVIATIONS Pipe Moments Support Reactions Total In-Plane Out-of-Plane 0.19 0.20 0.18 0.33 0.20 0.38 3.000 4.316 4.204 2.500 Q 10 PC METHOD/TIME HISTORY 0 CQC/TIME HISTORY 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 Random Vibration Method) 0 This Technique is Currently Being Implemented in Program SUPERPIPE (Piping Analysis Program) 9

L

PROJECT USE CgC Method Has Been Used On Several Projects:

0 Seismic Analysis of Chemical Separation Plant (British Nuclear Fuels) 0 Seismic Analysis of.Pipe Bridge {British Nuclear Fuels) l

~ r C'

REFERENCES l.

E. L. Wilson, A. Der Kiureghian and E.

P. Bayo, "A Replacement for the SRSS Method In Seismic Analysis," Earthquake Engineering 5 Structural Dynamics, Vol. 9, 1981.

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.

EARTHQUAKE ENGlNEERlNG AND SZRVCTUaAL DYNAMICS. VOL 9, ts7-194 tl9$t)

SHORT COMMUNICATIONS A REPLACEMENT FOR THE SRSS METHOD IN SEISMIC ANALYSIS

m. L WILSON, A. DER KIUREGHIANfAND E P. BAYOUS Dr~ ofCiofl Engi>>roririp, UniorrsiiyifCafiforrila, Brrktfry,California, USA.

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 ofthis note is to present an improved technique to be used in place ofthe 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 forsystems with well-spaced natural frequencies.

Since thc CQC method only involves a small increase in numerical effort,itis recommended that the new approach be used as a replacement for the SRSS method in all response spectrum calcuhtions.

C3 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 ofthc 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 ofcomputer

'programs for the dynamic analysis of general thre<<dimensional systems.'hc problem associated with the application ofthe SRSS method can be illustrated by its application to the four-storey building shown in Figure 1. The building is symmetrical; however, the centre ofmass is located 25 inches from the geometric centre of the building. The direction ofthe applied earthquake motion, a table of natural frcquencics and the principal directions ofthe mwle shapes are illustrated in Figurc 2. ne notes the

<<los<<ness ofthc frequencies and thc complex nature ofthe mode shapes in which the fundamental mode shape FRAME Pl too Iso i ~200

%2 A3

~ C.G.

OO 8

OO d

250 55 E LEVATION PLAN TYPICAL FRAME Figure I. Simple thrxodimensional buikling example o Pcofcuor of Civil Engineering.

f Axxatant Profoscur ol'ivilEngincoring.

I Graduate Studcm.

0098-8847/81/020187-06$ 01.00 1981 by John Wiley k, Sons, Ltd.

Received 8 January 1980 Revised 24 July 1980

188 SHORT COMMUNICATIONS

~II D

FAF.OUKNCIES

~<<

Us X

I

. 2 s

8 8

7 8

9 IOII 12 (RADIANS/SEC.)

IS.869 IS. 9S I 4S.99S 44.I89 84,4 I8 77.888 78.029 I08.S2 I08.80 I72.6IS S04.80 425.00 Figure 2. Frequencies and approsimare directions ofmode 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 fivemodes are shown in Figure 3. A mode superposition solution, inwhich all 12 modes werc used, produces base shears as a function oftime. 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 ofthc 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 ofthe motion.

Also, the base shears in the frames normal to the motion arc overestimated by a factor of 14. Itis clear that errors ofthis order are not acceptable. The sum ofabsolute values, which is a method normally suggested for the case where frcqucncics are close, gives a good approximation ofthe forces in the direction ofmotion 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 ofthe 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

~

~ <<

0 r 4.0t O.rt Figure 3. Base shears in firsI five modes O.ss

n. ~

(rt) TIME HISTORY I ~

'F0, ~

(b) SRSS (e) SUMOFABSOLUTE (4) COO VALUES Figurc 4. Comparison of rnodat combination tncthods 0

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))

(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 ofthe system and Y is the vector ofnormal co-ordinates. The introduction ofthis transformation and the premultiplication ofequation (I) by Or yields vMCrV+4vCQY+e K4Y= C M,(iieet))

(3)

For proportional damping the mode shapes have the following properties:

4r Mgr mr (4)

$, KQ,=

(5) 4r C4 4rWmr (6) in which Qr is the ith column of4 representing the ith mode shape, m, is the ith modal mass, and (r is the damping ratio formode i. Due to the orthogonality properties ofthe 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 ofequation (7) forall 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

~i,~a Pi W.~a Therefore, the maximum contribution of mode i to the total response of the structure is UI, 4 pr J'r, (ll)

For all modes y,, is, by definition, positive. The maximum modal displacement U,, is proportional to the 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 displace-ments, have unique signs. These signs for the maximum modal base shears of thc'example structure are indicated in Figure 3 by arrows.

(10)

Thc following equation can be solved for the response yet)

Vi+2J;iW5+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 ofy<.,co, is the pseudo-velocity spectrum and a plot ofy,,co~ is the pseudo-accclcration spectrum. These pseudo-velocity and acceleration spectra are ofthe same physical interest but are not an essential part of a response spectrum analysis.

Ifthc dynamic loading on the structure is specifie in terms of the displacement spectrum, then the maximum response of each mode is given by THE COMPLETE QUADRATIC COMBINATION METHOD (CQC)

The use ofrandom 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 methods for modal combination.

The complete development of the CQC method, which is now being proposed as a direct replacement forthc 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 ofthe following equations:

For a typical displacement component, g:

4(ZZ Ru Ag~g)

(12a) and for a typical force component, f fi 4KZfuugfg)

(12b) where g, is a typical component ofthe modal displacement response v'ector, U, andfis a typical force component which is produced by the modal displacement vector, U,, Note that this combination formula is of complete quadratic form including all cross-modal terms, hence, the reason for the name Complete Quadratic Combination. Itis 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 ofthc modal responses are, therefore, an important key to thc accuracy ofthc CQC method.

In general the cross-modal coc5cients, p,> are functions of the duration and frequency content of the loading and ofthe modal frcquencics and damping ratios ofthe structure. Ifthe duration ofearthquakc is long

~

~

0

'IHORY COMMUNICA'nONS I91 0

compared to thc periods of the structure, and il'he earthquake spectrum is smooth over a wide range of frequencies, then, it is possible to approximate these coeAicients bye'/(g,g)(f,+rg)r (I-r ) +4),f)r(I+r )+4@, +(~)

r'here r

cop'to,. For constant modal damping, (, this expression reduces 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 ofthe loading, are given in Reference 7. Additional modal combination rules giving the variabilityand the distribution ofthe peak response are also given in that refcrcncc. These rules are useful in non-deterministic analysis.

Considering 5 pcr cent damping and the frequencies ofthe 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 1. Modal cross~rrclation cocFicicnts Mode 1

2 3

4 5

Frcq. rad/s 1

1000 0 998 N$6 N$6 N$4 13 87 2

0998 1400 N$6 N$6 N$4 13 93 3

N$6 N$6 1400 0.998 0 180 43 99 4

N$6 N$6 0998 1

0 186 44 19 5

N$4 N$4 0 180 0.186 1400 54 42 COMPUTER PROGRAM IMPLEMENTATION The CQC method of modal combination has been incorporated in the computer program TABS Three-Dimensional Analysis ofBuildingSystems.'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 1 per cent for a typical structure). Therefore, there is no justification to continue using the potentially erroneous SRSS method. The application ofthe modified TABS program to several buildings has verified the validity of the CQC method. Other examples are given in Reference 7.

0 FINAL REMARKS It should 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, itwrongly specifies the cross-modal terms as being always positive. This willresult 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 two-dircctional 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 ofinput are ofdifferent intensities, or

~

~.

l92

. SHORT COMMUNICATIONS when thc three-dimensional structure is highly asymmetric, the cross-modal terms would still be significant and, therefore, the SRSS method willlead 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 ofstructures 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. E. L Wilson and A. HabibuUah, 'A program for threcMimcnsional static and dynamic analysis ofmultistory buiktings', in Strucr urof hfrrhanirs Sofisrvrr Srrirs. Vol. Jl. University Press of Virginia, l978.

2. 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 ofCalifornia, Berkeley, California (l975)

3. K. J, Bathr. E L Wilson and F. E Petorson.'SAP IVA Structumt Analysis Program for Response ofLinear Systems', Rrport No.

UCHrEERC-73/I l. Earthquake Engineering Roscarch Center, University of California, Bcrkeky, California (1973)

4. E. Roscnblueth and J. Etord uy, 'Responses oflinear systems to certain transient disturbances'. Proc. Fourth WldCoqf Eort hq. Engng I, San(iago, Chile. I85-l96 (I969)

5. M. V. Singh and S. LChu, 'Stochastic considerations in seismic analysis ofstructures', Earrhqu. Enf/. Srrurr. DyrL 4, 295-307 (l976)

6. A. Der Kiureghian, 'On response of structures lo stationary excitation', Rrport No. UCJJ/EERC-79/32, Earthquake Engineoring Rosoaroh Center. University ofCalifornia, Berkeley, California, (l979)

7. 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)

g. U5. Nuckar Regulatory Commission, Regulatory GuiCh l,92, Jhrlsion J (1976)

EARTHQUAKEENGINEERING AND 5TRUCTURA1 DYNAMICS,VOL II, 62~7 tl983)

THE COMPARATIVE PERFORMANCE OF SEISMIC RESPONSE SPECTRUM COMBINATION RULES IN BUILDINGANALYSIS 0

B. F. MAISON AND C. F. NEUSS J. G. Bouwkuttp, ItteBerkeley, Califorrtla, USA.

K. KASAIf Uttfoersity ofCattforrtia, 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 ofstilfness and mass are coincident resulting in uncoupled modes with well-separated periods in each component direction ofresponse; 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 ofthe sum ofthe 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.

Received 28 September J982 Revised 24 January l983 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 from'xcitation 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 ofeach 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 ofminimizing 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

'trtl Engtneer.

< Graduate Student in Civil Engineering.

0098-8847/83/050623-25$ 02.50 1983 by John Wiley & Sons, Ltd.

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 offour different modal combination rules are investigated by sample seismi analyses ofa fifteen-storey high-rise building. The four modal combination rules are: (1) the square root ofthi 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 fortlIto the earlier DSC method The study includes buildings with concentric and eccentric mass idealizations to investigate the significantx ofoneMimensional versus three-dimensional vibration response on the accuracy ofthe rules. The maximutr, building response in terms ofstorey 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 ofan actual fifteen-storey building; and (6) to make recommendations for the appropriate use ofmodal 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 briefdiscussiot regarding their formulation and application.

Square root sum ofthe sum ofthe squares (SRSS) method Form of combination rule:

where R = estimated maximum response for quantity R Rt maximum response of quantity R in mode i 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 introduced in Reference 8 for the combination rule developed by Rosenblueth er ot. It should be noted that this rule differs from the NRC double sum method.'

4 4

k C'

SEISMIC RESPONSE SPECfRUM COMBINATIONRULES 625 ested, but no ted complete osely spaced Imple seismic Ire root ofthe 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 in which rai = aI(Q(1-(Pi) )

2 Pi = PI+-

Stag co, = natural frequency of the ith mode P, = critical damping ratio for the ith mode S= time duration of 'white.noise'egment of earthquake excitation. For actual 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 where g v (PI PI roIre)(PI >+

PgI) I I (f-

))'+4PI PgroI gaII'+g)+4(P)'+Pg)

W'>

Absolute sum oj'odal maxima (ABS) method Form of combination rule:

(4)

(6) iefdiscussion (2) h et auld R,= Z/R

{g)

The accuracy of each of the above modal combination rules in predicting the peak time history response depends upon the characteristics ofthe earthquake record and the structure's dynamic properties. The SRSS, DSC and CQC rules are based upon the theory ofrandom vibrations. Two ofthe 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 ifthe 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 ofmodes 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). Ifthe periods are well separated, the off-diagonal cross-modal terms (i g jjof the matrix P,J become small and the DSC and CQC methods approach equivalence with the SRSS rule.

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 ofthese 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 C0 JIC 4IIL O

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DSC Fortcilatton (Ti/5 0.10)

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SElSMlC RESPONSE SPECTRUM COMBlNATlONRULES 627 3.0 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 willhave 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 ofthe 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 ofcombination 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 willyield 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 ofthe response quantity is lost; that is, the peak response may either be plus or minus. When combining the results with load cases ofknown 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 ofa 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 ofthe response quantities that are to be considered.

3.0 EXAMPLE BUILDING MODELS 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 fifteen-storey steel moment resisting frame structure of the University ofCalifornia 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

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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 offreedom (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.

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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 Analytical Experimental number model foal 1965-II 1

1 113 1 I 2

0386 043 3

0222 0.22 4

0154 Direction E-W translational (all modes)

Inegular building model The irregular building model is developed from the regular model by offsetting the centre ofmass at each storey level by 10 per cent ofthe plan dimensions ofthe 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.

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632 B. F. MAISON, C. F. NEUSS AND K. KASAl Table II. Irregular building natural periods Mode number Period (s)

Predominant direction 1

2 3

4 5

6 7

8 9

10 11 12 1 167 1 121 0773 0409 0390 0278 0238 0.225 0166 0.165 0.156 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 data Earthquake record San Fernando 1971 Pacoima Dam (SOOE component)

Imperial Valley 1940 El Ccntro (SOOE component)

San Fernando 1971 Orion Blvd.

(NOOW component)

Soil type Rock site Stiffsoil Dccp cohcsionlcss soil Epicentral distance 2 miles 5 miles 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

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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 (East-West) 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 ofthe actual peak responses. The SRSS rule significantly underestimates the parallel responses, greatly overestimates the orthogonal responses, but gives a fairly good estimate of

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% error results compiled from three earthquake excitations SRSS DSC CQC ABS Deflections Shears Overturning moments Average error

% underestimated~

Extreme errors Average error '

underestimated'xtreme errors Average error undcrestimatcd'xtreme errors 6

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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 ofthe 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 ofcross-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 cross-modal 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 ofthe 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 ofthe 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 cross-modal 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 ofthe combination methods in predicting the actual time history maxima ofstorey shear and torque for each earthquake excitation is contained in Appendix I (Tables VIIto IX). In general, the trends

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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 Vllto 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 ofaverage 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 ofinterest 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

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% error results compiled from three earthquake excitations SRSS DSC CQC ABS Parallel (E-%) response Deflections Shears Overturning moments Average error underestimated'xtreme errors Average error underestimated'xtreme errors Average error underestimated'xtreme errors 18 100

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2, 120 Orthogonal (N-S) response DeAcetions Shears Oyerturmng moments Average error underestimated'xtreme errors Average error

% underestimated~

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186, 350 217 0

133, 307 218 0

136, 299 18 100

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0 0, 67 371, 800 24 528 0

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-15, 40 9

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-27,18

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'Number of underestimated responses expressed as a percentage 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 oflarge 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 ofthe sum ofthe squares (SRSS) method; (2) the double sum

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SEISMIC RESPONSE SPECTRUM COMBINATIONRULES 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 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 ofthe 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 ofthe 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 ifthe strong motion duration parameter S is set to infinity,it is possible to select a value ofS 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 ofthe 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 cross-modal 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 ofS 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.

~"

' 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

~ '

W O 8

cn 00 CA OO n

n n a

Pacoima Dam Table Vl. Regular building storey shear results El Centro Orion Blvd.

tg

'Cs, Level (I)

RI'16 15 14 13 12II 10 9

8 7

6 5

4 3

Average 1,803 2,053 2.269 2,453 2.6()9 2.738 2,843 2,928 2,991 3,029 3,044 416 414 4.14

rror Time history results V

Time (kips)

Is)

(2)

(3) 559 848 968 848 1.223 476 1.540 476 416 Response spectrum (/)

Error i>s time history Time history results Response spectrum (%)

Error us time history Time history results Response spectrum (%)

Error es time history 0

-2 I

80 621 6

4 5

82 1,136 15 13 14 82 1,552 10 9

10 65 1,891 8

7 7

52 2,172 4

3 4

47 2,347 2

I 2

36 2,426 I

I I

24 2,441 I

I I

16 2,495 3

3 3

25 2,666 5

5 5

30 2,777 7

7 7

36 2,840 8

9 9

46 2,893 10 11 10 54 2,945 IO 11 11 57 2,967 6

6 6

49 354 354 352 3.52 350 I

I 350 354 304 304 302 3.02 300 300

-17

-19 17

-18 15

-17 15 16 14 14

-10 10

-4

-4 4

4 9

9 8

8 8

9 10 II 12 13 12 13 12 13 11 12

-18 51

-17 44'16 32

-15 24 14 17 10 22

-4 25 4

26 9

25 8

29 9

29 Il 35 12 45 13 52 13 55 12 34 636 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 11 84 11 84 13 12 11 88 11 88 1248 7

5 6

3 2

3 I

-I 0

-I

-2 2

-2

-3

-3

-3

-3

-3

-4

-4

-4

-4

-4

-4

-2

-3

-2 0

0 0

3 3

3 7

7 7

10 11 11 13 14 13.

12 13 13 5

5 5

91 76 57 45 35 33 25 15 10 19 24 32 44 53 55 41 V

Time V

Time SRSS DSC CQC ABS (kips)

(s)

SRSS DSC CQC ABS (kips)

(s)

SRSS DSC CQC ABS (4)

(5)

(6)

(7)

(8)

(9)

(IO)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

OO 5

5O 4))h I'ig I

i h.r fF $i

'4 t

~

.S ps:

• 's

'Ap pV S,', +s(S

~ p(,

'ihk,44Rsh ~~s

~ ~

"~

sp Q s

ssp(,

'v Pacoima Dam Table VII.Irregular building storey shear results, East-West response El Centro Orion Blvd.

Time history results Response spectrum (%)

Time history Response spectrum (%)

Time history Response spectrum (%)

iss Error i)s. time history results Error us. time history results Error i)s time history V

Time V

Time V

Time X

Level

[kips)

(s)

SRSS DSC CQC ABS (kips)

(s)

SRSS DSC CQC ABS (kips)

(s)

SRSS DSC CQC ABS (I)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

RF l6 IS 14 13 12II IO 9

8 7

6 5

4 3

Average 446 835 1,183 1,474 1,720 1,976 2,188 2,359 2,498 2,617 2.705 2,772 2,828 2.863 2,880 "r'rror 4 80

.12 4 80 13 4 78

!6 4 78 17 4

6

-19

-22

-24 4 16

-24 4 18

-23 4 18

~ -22 4 18

-20 4 16

-18 4 16 17 4.16 16 4 14 16 19 13 12 II 10 8

7 7

5 5

4 2

0 0

-2 0

-2 I

I 3

0 5

3 8

5 10 7

11 8

12 9

6 5

122 580 103 1,032 81 1,407 70 1,742 57 1,968 48 2,086 38 2.233 28 2,336 19 2,419 29 2,463 35 2,490 41 2,554 50 2,580 59 2,609 63 2,633 56 356 356 354 352 352 584 584 586 586 586 344 3W 3W 300 3.00

-35

-18

-34 15

-34 14

-34

-14

-32 11

-28

-5

-26

-2

-23 2

-20 5

17 IO 14 14

-13 16

-11 19 10 20 IO 20 23 12

-19 16 IS

-16 7

-4 0

3 7II 13 16 17 17 12 71 61 46 38 31 37 37 34 29 40 46 51 61 73 7?

49 636 1316 1,772

%38 13 16 2,639 13 12 2,981 3,263 3,493 3,669 3,789 3,855 3,876 3,869 3,854 3,846 13 12

-21 0

-24

~ -3

-27

-6

-28

-7

-29

-8

-29

-9

-29

-8

-28

-7

-27

-5

-25

-2

-22 2

19 5

-17 9

IS 12

-14 13 24 6

I 94

-5 73

-7 53

-9 44 10 33

-10 29

-10 24

-9 17

-7 12

-4 21 I

27 3

35 6

46 9

56 10 61 7

42

?;

vh

r

, t

~

g

~

tlr it,

,i u

~7i"'I'6 15 14 12II 10 9

8 7

6 5

4 3

Average 144 244 335 407 436 439 424 413 421 430 467 511 549 578 590

".r'rror 846 844 898 898 898 840 838 610 612 558 566 566 564 564 564 Time history results V

Time I.cvcl 1k tps)

(s)

(2)

(3)

Pacoima Dam

Response

spectrum (%)

Error't c time history SRSS DSC CQC ABS (4)

(5)

(6)

(7) 133

-22 155

-20 155

-24 156

-28 174'26 201

-23 238 16 273 IO 291

-7 307

-3 296

-4 279

-6 266

-8 256

-9 252

-9 229 14 I

510 6

526 4

486 2

457 8

461 17 505 30 551 43 565 49 544 55 617 51 612 46 600 42 609 38 621 37 625 29 533 Time history results V

Time (kips)

(s)

(8)

(9) 116 5 34 194 406 268 4 02 344 4.02 394 400 421 400 451 7 26 498 7.28 551 730 600 730 639 7 32 675 674 710 674 754 442 778 440 El Centro Response spectrum (%)

Error trr time history SRSS DSC CQC ASS

( IO)

( I I)

(12)

(13) 171

-9 17 639 199

-8 24 661 200 14 21 601 189

-20 16 532 194

-21 17 496 211 19 22 519 221

-18 26 518 215

-20 23 471 205

-24 19 418 196

-26 15 427 192

-27 14 421 188

-27 12 424 182

-28 11 440 171

-29 7

445 165

-31 4

443 193 21 17 497 Time history results V

Time (kips)

(s)

(14)

(15) 153 10 52 vis 394 1052 443 1544 502 1504 561 1508 620 14 56 674 717 751 779 800 813 818 1456 Table VIII.Irrcgular building storey shear results, North-South response Orion Blvd.

Response spectrum (%)

Error vs time history I

SRSS DSC CQC ABS (16)

(17)

(18)

(19) 173 13 14 612 184 16 16 595 204 15 21 594 241

'-9 34 633 254

-9 37 613 8

250 14 34 591 5

244 17 31 550 237

-21 28 494 232

-22 26 449 231

-23 25 478 R

232

-21 26 489 234

-20 27 506 236 18 29 538 237 17 30 568 238 16 30 580 228 17 27 553

j i sr ) )I

)L(

's

~ (t

~

v,'2 i

'R g<.,

vvnsvsrqvv snss 7Bsv~s s kseennvvlveskos(ISYLLAA s4 nw ssnw Pacoima Dam Table IX. Irregular building storey torque results El Centro Orion Blvd.

Time history results Response spectrum (%)

Error i)s time history Time history Response spectrum (%)

Time history Response spectrum (%)

results Error i)s time history results Error i)s time history 85 932 161 1252 40 18 21 288 o

35 15 19 259 28 10 14 219 21 5

10 180 16 I

5 140 5

12

-2 2

133 12

-3 2

131 14 I

3 124 14 I

3 103

?'4

- I 4

102 13

-2 3

100 12

-3 2

107 11

-4 I

118 10

-5 0

126 10

-6

- I 130 17 5

6 150

-5 227

-5 214

-2 201 2

I&?

3 161 0

156

-2 155

-2 142

-2

~ 114

-3 109

-3 108

-2 121

- I 142 0

159 I

167 2

158

-27

-26

-25

-23

-23

-20

-20

-17 19 16

-16

-12 I I

-7

-14 IO 16 12 15 12

-15

-11

-15

-11 14

-10

-14

-10 13

-9 17 14 117 3 58 205 3 58 266 3 56 311 2.70 359 272 388 272 404 3 10 ees 524 3 10 548 30&

567 579 587 590 3.08 15

-8 IS

-8 15

-5 18

-3 19

-2 16

-5 14

-7 15

-7 14

-7 12

-9 11

-8 13

-7 15

-6 17

-5 18

-5 IS 6

141 135 125 114 90 99 114 97 76 73 71 79 94 106 112 102

-15

-14 12

-9

-9

-4 2

-2

-3

-4 4

-3

-2 0

6 482 16 73 15 130 14 168 l3 194 12 216 I I 243 10 267 9

284 8

301 7

325 6

344 5

357 4

365 3

370 2

371 Average:error 230 296 361 418 462 482 418 416 416 418 422 424 424 426 426 491 1252 526 1248 558 589 620 647 667 675 1248 T

( x IOs Time T

Time T

Time Level k.in).

(s)

SRSS DSC CQC ABS (k.in)

(s)

SRSS DSC CQC ABS (k-in)

(s)

SRSS DSC CQC ABS

( I)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(I5)

(16)

(17)

(18)

(19) en Rt "V'e I

I>> I J

SEISMIC RESPONSE SPECTRUM COMBINATIONRULES REFERENCES

1. 'Combining modal responses and spatial components in seismic response analysis',

U.S. Nuclear Regulatory Commission, Regulatory Guide 1.92, Recision I, 1976.

2. 'Recommended practice for planning, designing and constructing lixed oifshore platforms', American-Petroleum institute, API RP 2A, 13th edn. 1982, p. 22.

3. "fentative provisions for the development ofseismic regulations for buildings', Applied Technology Council, US. Iqatianal Bureau ofStandards Special Publication $ 10, 197&, p. 63.

4. 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}.

S. E. Roscnblueth and J. Elorduy, 'Response oflinear systems to certain transient disturbances', Proc. 4th world conf. earthquake eng.

Santiago, Chile I (1969).

6. LE Goodman, E. Roscnblucth and N. M. Ncwmark, 'A seismic design ofIirmlyfounded elastic structures', Trans. hSCE 120, 782-802 (1955).

7. N. M. Newmark and E Roscnblucth, Fundamentals ofEarthquake Engineering, Prentice Hall, Englewood Cliffs,NJ1971, pp. 309-313.

8. A.K.Singh, S. L. Chu and S. Singh.'Influence ofclosely 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 ofDisaster 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$.

14. H. B. Seed, C. Ugas and J. Lysmer, 'Sile-dependent spectra for earthquake-resistant design', Rcport Na. VCB/EERC 74-I2, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1974.

IS. 'Uniform building code, Int. conf. build. oJPeiais, Whittier, California (1979).

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1 I

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