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PROGRAMMED OVERHEA

JOB NO. 94444 (l983)

REPORT NO. SFPD-C/S-83-07

DIRECT GENERATION OF PROBABXLISTIC FLOOR RESPONSE SPECTRA

DECEMBER'983

~Sf nature Date

Prepared by:K. Lilhanand

3/~c/aq

Reviewed by: grM. S. Tseng

>/ic/s+

Approved by:B. . shack

BECTTE POMER CORPORATIONSan Francisco, California

0

TABLE OF CONTENTS

SECTION PAGE

INTRODUCTION

OVERVIEW OF THEORIES

2.1 The PSDF-RS Relationship2.2 Peak Factor

2.2.1 Stationary Response2.2.2 Nonstationary Response

2.3 Evaluations and Discussions

PSDF FOR WIDE-BAND SEISMIC GROUND MOTIONS

SIMULATION OF WIDE-BAND SEISMIC GROUND MOTIONS

4.1 Stationary Ground Motion4.2 Nonstationary Ground Motion

STRUCTURAL TRANSFER FUNCTIONS

'SIMULATION OF NARROW-BAND FLOOR RESPONSES

2-1

2-32-7

2-82-13

2-16

3-1

4-1

4-14-3

5-1

6-1

6.1 Responses to Stationary Ground Motion6 ' Responses to Nonstationary Ground Motion

VALIDATIONOF THE PSDF-RS RELATIONSHIP

6-16-3

7-1

7.1 Wide-Band Seismic Ground MotionsI

7.1.1 Stationary Ground Motion7.1.2 Nonstationary Ground Motion

7.2 ,Narrow-Band Floor Responses

7.2.1 Responses to Stationary Ground Motion7.2 ' Responses to Nonstationary Ground Motion

DIRECT GENERATION OF PROBABILISTIC FLOOR

RESPONSE SPECTRA

7-1

7-17-3

7-5

7W7-7

8-1

9

10

APPLICATIONS

SUMMARY AND CONCLUSIONS

REFERENCES

9-1

10-1

ll-l

I

0

1 INTRODUCTION

The floor response spectra (FRS) is of practical importance for seismic

qualifications of equipment and piping systems supported on buildingfloors and walls in nuclear power plants. Currently, it is a common

practice to use the deterministic method to generate the FRS by performing

a time history analysis. This method, however, does not take into account

the randomness inherent in seismic ground motions. Furthermore, sinceseismic ground motions are usually prescribed in the form of the design

re'sponse spectra (DRS) such as those given by the USNRC Regulatory Guide

(R G ) 1.6O> which has been developed from statistical studies of the

actual seismic records (Ref. 1), the deterministic method requires thatseismic ground motion time histories be generated which are compatible

with the DRS. There are many time histories which may be generated

compatible with a given DRS. The effect on the floor response spectra due

to variation of different time history inputs is generally not addressed

by the deterministic method. Therefore; the deterministic method'for

generating the FRS is not methodologically rational, and generally iscostly.

An alternative to the deterministic method is the probabilistic method,

which uses the random vibration theory to determine the probabilisticfloor responses to random seismic ground motions. This method can

directly generate the FRS. for a desired level of confidence from the DRS

without having to perfo the more costly time history analysis. However,j

because the probabilistic methods that are current available forgenerating the FRS (Refs. 2 through 6) have not been throughly tested toconfirm its reliability and accuracy, additional validations are requiredbefore these methods can be implemented for applications.

The objective of this report is to present the evaluation of currentlyavailable probabilistic methods and the result of testing of theunderlying theories against the. simulation results obtained from 20

simulated time histories, in order to confirm the reliability and accuracy

of the theories for application to the generation of probabilistic floor

1-1

response spectra of nuclear power plants sub)ected to seismic ground

motion excitations.

An overview. of the theoretical background and an evaluation of the

probabilistic methods proposed by various researchers are presented inSection 2. Based on the result of the evaluation, the probabilisticmethod which ezplicitly uses the relationship between the power spectraldensity function (PSDF) and the response spectra (RS), i.e., the PSDF-RS

relationship, is selected for a detailed evaluation using simulationresults. The theoretical derivation of the PSDF-RS relationship ispresented in Section 2.1. The peak factor which is one of the key

parameter contained in the PSDF-RS relationship is presented in Section

2.2. In Section 3, the PSDF of random seismic ground motions compatible

with the USNRC R.G. 1.60 spectra, which is a typical wide-band random

process, is generated. This PSDF is used as the basis for simulation ofboth stationary and nonstationary seismic ground motions in Section 4.

Section 5 presents the structural transfer functions of a typical nuclear1

power plant structure. These transfer functions are used as thestructural filters which transfer the wide-band seismic input motions tothe narrow-band floor response output motions. Section 6 presents thesimulation of floor responses, to both the stationary and nonstationaryseismic ground motions. Comparisons of the analytical and the simulationresults for the wide-)and seismic ground motions and the narrow-band floorresponses are made in Section 7 to validate the PSDF-RS relationship givenin Section 2.1.

Based on the comparison results in Section 7, a procedure for directgeneration of probabilistic floor response spectra is described inSection 8. Section 9 demonstrates applications of the proposed procedure

described in Section 8 . The summary'and conclusions are given in Section10.

1-2

2 ~ OVERVIEW OF THEORIES

The probabilistic methods currently proposed in the literature forgenerating the FRS can be grouped according to the analytical procedure

used, into two approaches: the implicit approach, and the explicitapproach. Both approaches treat seismic ground mot1on as wide-band random

processes, and the theor1es of random vibration are used to determine the

probabilistic floor responses. It is noted that none of the proposed

methods have been adequately tested for its reliability for actualapplicat1ons.

In the implicit approach, typical of the method developed by Singh

(Ref. 2) and Atalik (Ref. 3), the PSDF-RS relationship is implicitly used

in deriving the FRS as a direct function of the prescribed seismic DRS.

The approach is shown schematically in Fig. 2.1. The derivation of thisapproach assumes that the seismic ground motion and the floor response

mot1on are stat1onary processes, that the structure possesses normal-mode

properties, and that the peak factor used for raising the root-mean-square

(rms) response to the maximum (spectral) response at a particularfrequency is the same for the wide-band seismic ground motion and for the

narrow-band floor response motion. This method also requiresapproximations to simplify the mathematical complexities involved inderiv1ng the FRS. Furthermore, because of the assumption made on the peak

factor, this method can only generate the FRS for one confidence level,i.e. non-exceedance probability, which is the same as that prescr1bed forthe DRS.

The'xplicit approach, typical of the methods developed by Vanmarcke

(Ref. 4), Grossmayer (Ref. 5), Rorno-Organista (Ref. 6), Der Kiureghian(Ref. 7), among others, uses the PSDF-RS relationsh1p explicitly ingenerating the FRS from the prescribed seismic DRS. This approach shown

schematically in Fig. 2.2, generally assumes that seismic ground motionand.structural response are stationary random processes. However, thefloor spectral response at a part1cular frequency, wh1ch can be obtainedfrom filtering the 'structural response motion through a singledegree-of-freedom (SDOF) system, can be approximately treated as either

2-1

4

stationary or nonstationary process depending on the selected peak factorused in the PSDF-RS relationship. The explicit approach requires the

following four major steps of computation:

1. Compute the PSDF of seismic ground motion from a prescribed DRS usingthe PSDF-RS relationship.

2 ~ Compute the structural transmittancy functions which is the square ofthe transfer function amplitude.

J

3 ~ Compute the structural response PSDF by multiplying the PSDF ofseismic ground motion in Step 1 with the structural transmittancyfunctions in Step 2.

4. Compute the FRS for any desired confidence levels from the structuralP

response PSDF in Step 3 using the reversed PSDF-RS relationship.

Comparing the two approaches outlined above, the explicit approach has the

following advantages over the implicit approach:

1. The FRS can be generated from the prescribed DRS for any desiredlevels of confidence as opposed to only one level of confidence

provided by the implicit approach.

2. The PSDF of seismic grourid motion compatible with the prescribed DRS

is readily available as a by-product which can be used for simulationof seismic ground motions, if needed

3 ~ The nonstationary effects, which tend to reduce the ground and floorspectral responses, can approximately be included to reduce theoverconservatism inherent in the implicit approach, especially for thelightly damped and soft {low frequency) systems.

4. The peak factors for the wide-band ground motion. and the narrow-bandstructural response which are generally different, can be prescribeddifferently.

2-2

r

5 ~ The linear structure does not require to possess normal-modeA

properties.

For actual seismic ground motions which are nonstationary random motions,

the PSDF-RS relationship which is required in the explicit approach,

generally cannot be analytically derived, thus is often obtained through

empirical means. However, by approximately treating seismic ground

motions as a random stationary process, which has been found to be

applicable when the duration of seismic ground motion is longer than the

fundamental period of structure (Ref. 8), the PSDF-RS Relationship can

approximately be derived analytically as that developed by Vanmarcke

(Ref. 4). This analytical relationship contains two critical parameters-the rms response and the peak factor. The theoretical derviation of the

analytical PSDF-RS relationship is presented in Section 2.1. The

stationary rms response can be approximated as a function of the PSDF ofseismic ground motion as discussed in Section 2.1. The theoreticalbackgrounds of the peak factor and various widely known approximated peak

factors are presented in Section 2.2.

2.1 The PSDF-RS Relationshi

The equation of motion of a SDOF system subjected to a zero mean

random seismic ground acceleration input, x(t), can be written as

y(t) + 28mny(t) + s y(t) -'x(t) (2.1)

in which y{t) is the relative displacement, the dot above y{t)indicates the derivative with respect to time, e is the circularfrequency, and 8 is the critical damping ratio.

From Eq. (2.1), the transfer function relating the groundacceleration input to the relative displacement response output can

be determined by letting

2-3

0

x(t) ei(Ot

and (2. 2)

y(t) = H (v) e

in which Hy(z) is the transfer function for y(t), and i is the

imaginary number equal to ~l. Substituting Eq. (2.2) into Eq.

(2.1), the transfer function can be obtained as follows:

Hy(m) ~ ((o„-'e + 2 ige o)) (2. 3)

For a stationary random process, it is well known from the

stationary random vibration theory (see, e.g., Ref. 9) that the

output PSDF of the relative displacement response of a linear system

can be related to the input PSDF of the stationary ground motion by

Sy (Q) [ q (0)) ] S/(6)) (2. 4)

in which Sz(o)) is the input PSDF, and I Hy(G)))2 is the transmittancyfunction which is the square of the transfer function amplitudegiven by Eq. (C.3).

With Sy(o)) given by Eq. (2.4), the mean square (ms) relativedisplacement response, a , can simply be obtained by integratingS (M) over all frequencies as follows:

Q2

Sy (Q) dQ (2 5)

Since seismic ground motion is assumed to have a zero mean, the

response of a linear system will also have a zero mean. Thus, the

ms response in Eq. (2.5) is the same as the variance of the

2-4

response, and the root-mean-square (rms) response, ay, is the same

as the standard deviation of the response (Ref. 10).

For a lightly damped system, i.e., when t)«1, the ms absolute

acceleration, a?, can be estimated from a2 by2

~2 ~4 ~2n y (2 6)

where the subscript z(t) absolute acceleration ~ z(t) + y(t).Substituting Eqs. (2.4) and (2.5) into Eq. (2 ') gives

Qr ~ ~ Q2 4

z nH (0)) I

S- (Q) dG) (2 7)

For the case of a lightly damped system sub)ected to a fairlywide-band ground motion whose PSDF is slowly varying around the

vicinity of ()I) ()()n where the transmittancy function is sharplypeaked, Eq. (2.7) can approx&ately be rewritten as

0- ~ ()()„S ((()n)~

H (()()) ) d()j)

n

+, [ S<((o) — S>((o ) ] d(() (2. 8)

The first term in Eq. (2.8) xepresents the contribution of Sz(u)around m ~ mn, the second term is the contribution of Sz(e) over

Sz(un) in the frequency range up to Qn. Since the first term inEq. (2.8) is identical to the widely known stationary ms absoluteacceleration response to white noise -ground motion having a

constant PSDF of Sz(o)n)~ i.e., 62, it can readily be obtained fromsRefs. (4 and 9) as

a,' I,"s„((u„),' H„(ro)('~

0

mm S-{<o )(2 9)

2-5

!

Substituting Eq (2,9) into Eq. (2.8) yields

Gz Aron SR{(on) + ax {+n) (2.10)

in which A (<-48)/48, and <..(+n) is the "partial" ms absoluteacceleration as given by

(+ )xS" (Gl) dQ (2.11)

Note that Eq. (2.10) gives exact value of o2 when the ground motion

,2'san ideal white noise. For this case, o,. in Eq. (2.10) reducedz

to a2 in Eq. (209).s

With oz obtained from Eq. (2 '0), the maximum absolute accelerationresponse, i.e., the spectral acceleration at frequency (on, R'z(mn),

can approximately be determined by multiplying oz with the peak

factor as follows

Rz {en) p {ton) a" (2. 12)

in which r (o) ) is the peak factor at frequency (on with confidencep nlevel (non-exceedance probability) of p. The peak factor is to be

presented in Section 2.2.

By substituting Eq. (2.10) into Eq. (2.12), the PSDF-RS

relationship for seismic ground motions can be obtained as follows

R-(~„) rp(~) I. M„S„~{e„) a„'., {e„) ) (2. 13)

2-6

Rearranging the terms in Eq. (2.13), gives the following

Sx(sn)-1

(Are„) [ R~(e„) - a~ (~) )

rp(~)

(2.14)

The PSDF-RS

essentiallyRef. 4.

relationship as given in Eq. (2.13) or (2.14) isidentical to that previously obtained by Vanmarcke in

For application to the generation of FRS as previously outlined inSection 2.1, Eq. {2.14) can be used for computing the PSDF of seismic

ground motions compatible with the prescribed DRS. The numerical

computation of Sx(m ) can be initiated at the low frequency end

where ~?(+ ) is negligible. Let Ri(+) denotes the spectraln

acceleration of the ith floor at frequency mn, Eq. {2.13) can

similarly be derived for the floor responses as follows

Ri((o„) = rp(e ) f AM Si((o ) + oi (m„) )2 (2 15)

in which Si(mn) is the PSDF of structural response at floor i, and

r'(z ) is the peak factor with confidence level of p for thep nstructural response. Accordingly, Eq (2.15) can be used forcomputing the FRS from the PSDF of structural response.

2.2 Peak Factor

The peak'factor previously defined in Eq. {2.12) is a multiplier toraise the rms response to the maximum (spectral) response. The

determination of the analytical peak factor requires the solution ofthe first-passage problem in the random vibration theory. To date,the exact solution of practical interests has not been found for theSDOF system considered herein. Consequently, ~pproximations have tobe made in deriving the analytical peak factor.

2-7

From the analytical and simulation studies of normal stationaryrandom processes in Refs. (11 and 12), it has been found that the

non-exceedance probability, p, which is the probability that the

random process remains below the threshold level, say b, follows a

decaying-exponential form, i.e.,

-QTp = p e (2.16)

in which po is the probability starting below b, + is the decay rate,and T is the duration that the process remains below b. The

parameters po and e in Eq. (2.16) are a function of the peak factor,among others ~

Once po and + are determined, the peak factor corresponding to a

specified value of p can be determined from numerically evaluating or.

inverting Eq. (2.16).

2.2.1 Stationar Res onse

In the following, some of the widely known peak factors forstationary response which. have p in the form of Eq. (2.16) are

presented.P

a. The Daven ort Peak Factor (Ref. 13). This peak factor has

been derived on the assumption that the rate of thresholdcrossing occurs independently according to the Poisson-

process. As a result of the stated assumption, the

parameters po and + in Eq. (2.16) can be written as

p = 10

G ~ v exp (-r /2)0 P

(2. 17)

2-8

in which >o is the mean rate of zero level crossing as

given by

(2. 18)

and,"i is the ith moment of the PSDF about the frequencyorigin which is given by

to S (e) dm (2.19)

i 0, 1, 2

Detailed interpretations of the first three spectralmoments in both time and frequency domain relating toreliability measures of a random motion can be found inRef. 14. It suffices to note that ~o and g2 are the

stationary ms of the process and its first time

derivative, respectively, and the center frequency, fc,which is the dominant frequency of the process is given by

Vpf a—2m

(2. 2O)

One other important parameter which can be obtained from

the spectral moments is the so-called dispersionparameter, d, a dimensionless measure of the spectralbandwidth, as given by

From Schwarz'nequality, it can be shown that 6 variesfrom 0 for a very wide-band process to 1 for a verynarrow-band process.

2-9

For a lightly damped system, the parameter vo in Eq.

(2.18) can be approximated by

VII

(2.22)

in which +n is the circular frequency of the SDOF system.

Substituting Eq. (2.22) into Eq. (2.20) gives

"nf a a fc 2II n (2.23)

in which fn is the fundamental frequency in cps of the

SDOF system.

With po and a given in Eq. (2.17), the mean (m) and

standard deviation (a) values of the Davenport peak

factor, i.e., rm and rg, can approximately be determined

from Eq'2.16) as follows:

r ~22nv T + Y

~22-.T(2.24)

and r

~2invoT(2. 25)

in which Y is the Euler constant (0 '77), and >o is as

given by Eq. (2.22) ~

b. The Vanmarcke Peak Factor (Ref. 15) ~ In the derivation ofthe parameters po and n in Eqi (2 '6), Vanmarcke assumes

that the successive time intervals spent below and above

the threshold crossing follow the Markov process.

Accordingly, these parameters can be written as

2-10

p = 1 — exo (-r /2)2

0 P

[ 2 — exp (-perp Jii/2) ]

exp (rp/2) — 1

(2. 26}

in which 6e ~ 61+ , 6 is defined in Eq. (2.21), and c isempirical constant equal to 0.2. It is of interest tonote that, for high threshold level (r » 1), Eq. (2.26)reduces to be essentially the same as Eq. (2.17). The

so-called "Vanmarcke-Exact" peak factor will be referredto as the peak factor obtained from numerical evaluationof p in Eq. (2.16) with po and a given by Eq. (2.26).

Substituting po and c( in Eq. (2 '6) into Eq (2.16), the

peak factor can be determined approximately as

rp = [ 22n (n [ 2-exp (-6e x( nn2n n ) ] ) ]r (2.27)

in which n ~ uoT/Sap. The approximate expression inEq. (2.27) is referred to as the Vanmarcke peak factor.

For a lightly damped system, ~o can be approximated as inEq. (2.22), and Q can be approximated as

6 ~ (48/g) (2.28)

in which 8 is the critical damping ratio of the SDOF

system.

2-11

c. The Der Kiure hian Peak Factor (Ref. 16). Der Kiureghian

obtained rm and r~ by modifying the Davenport peak factorsin Eqs. {2.24) and (2.25), respectively, to match the m

and 0 of the,Vanmarcke-Exact peak factor. Accordingly, rm

and ro can be written as

{2.29)

1.2

JSanV T 13 + (2RnveT)'

veT > 2.13.2

0.65 v T < 2.1 (2.30)

in which

(1.65a ' 0.38) u 2 0.1 C 6 < 0.690.45

Vo , . 0.69 c 8 c 1 (2.3l-a)

For a lightly damped system, uo is approximated as in Eq.

(2.22), and 6 is approximated as

(1.16B — 0 21) B > 0 01 (2. 31-b)

In this report, the limit imposed on B in Eq. {2 31-b) isextended to B ~ 5X ~

'The m and m+a values of the peak factors given above by

Davenport, Vanmarcke, and Der Kiureghian, are plottedagainst the parameter fn T, which represents theequivalent number of response cycles, for damping valuesof 0 'X, 2X and 5X in Figs. 2.3 through 2.8. These peak

2-12

0

0

factors are computed using the approximate values of ~o

and 6 given by Eqs. (2.22) and (2.28). In th'ese figures,the p levels corresponding to the m and ~ values for the

Vanmarcke-Exact peak factor are selected to be 57X and

84X, respectively. These selected p levels can be

compared with those obtained exactly from numericallyevaluated p using Eq. (2.26) with po and n of Eq. (2 .26) as

shown in Table 2.1 for the 2X damping value. It can be

seen from this table that the p level of 57X for the m

value is slightly higher than the average p value given inthis table which is about 54X, and that the p level of 84X

for the m+0 value is about the same as the average p value

given in this table. The slightly higher p level selected

for the m value has little effect on the peak factorvalue. This can also be confirmed by the fact that, inFigs. 2.3 through 2.8, the Vanmarcke-Exact peak factorwith p of 57X and 84X are compared closely with the m and

m+a values of the Der Kiureghian peak factor which were

derived from matching with the m and m+a values of theVanmarcke-Exact peak factors. Therefore, the selected p

levels of 57X and 84X are adequate for use in computing

the m and nH.G values of the Vanmarcke-Exact peak factor.

2.2.2 Nonstationar Res onse

In the following, the peak factors by Vanmarcke and by Lutes

for nonstationary response are presented. These peak factorshave assumed p in the form similar to Eq . (2 .16) except thatthe parameter u in this equation is replaced by a

time-dependent function, i.e., c(t), so that Eq. (2.16) isrewritten as

pTa(t) dt

p pe o0(2.32)

2-13

Although po in Eq. (2 .32) for this situation should, by

definition, be set at 1 since the response process builds

, up from rest, the simulation studies performed herein as

well as by Lutes indicate otherwise. Therefore, po istreated herein as an empirical parameter for nonstationry

response and should not be sub)ected to the same

interpretation as given in Eq. (2.16).

a. The Modified-Vanmarcke peak Factor (Ref 4) ~

For a non-stationary response, Vanmarcke has suggested

that p in Eq. (2.32) be approximated as in Eq. (2 .16)

except that po is equal to 1, and the damping 8 and the

duration T be replaced by the effective damping Bs and

effective duration Ts given by

Bs ~ 8 / [ 1 — exp (-28(o„T) ]

Ts T exp (-2m)

(2. 33)

in which m ~ -1 + [1-exp(-2 BmnT] /[1-exp(-BM T)]

By empirically modified po to be the same as that inEq. (2.26), the parameters po and 0 in Eq. (2.16) now

become

p = 1 — exp (-r /2)0 P

(2.34)

v~ [ 1 — exp (-6~ rpjr/2) 1

aexp (r /2) — 1

P

in which <e (48s/~) , >o is as given by Eq. (2.22),0.6

and Bs is as given by Eq. (2.33).

Another modification made on this peak factor in order toinclude the nonstationary effect, is the modification on

2-14

the rms response. This nonstationary effect can be

approximately obtained from comparing the stationary rms

response with the nonstationary rms response to white

noise ground motion. It is well known that the

nonstationary rms response to white noise'otion, on,can be written as (Refs. 4 and 9).

<n <s [ 1 — exp (-2ga)nT) ) (2.35)

in which <s is as given by Eq. (2.9). The term following0's in Eq. (2.35) is a reduction factor to account for the

nonstationary effect on the rms response.

With the above modifications, the Modified-Uanmarcke peak

factor for nonstationary processes can be computed as

follows:

1. Compute the peak factor for stationary processes for a

desired level of confidence p by numerically evaluated

Eq. (2.16) with po and a from Eq. (2.34).

2. Multiply the peak factor computed from step 1 by the

reduction factor, (1-e28~T) ~

P

b. The Lutes Peak Factor (Ref. 17). Lutes derived the

parameter po and u in Eq. (2.16) empirically based on

simulation studies of the first-passage probability ofthe absolute value of the zero-start response of a

SDOF system sub)ected to stationary white noise ground

motion. These parameters can be written as follows:

po = exp [exp (-1.195-0.316r )j) (2.36-a)

a ~ u exp (-r /2) [ 1-1.075 [ r exp (-r /2) ] )~0 P P P

2-15

in which ~o is as given by Eq. (2.22), and

w = 0. 2364 + 28. 14(2.36-b)

(48 (1-1.18)/m)

The 6 given above is about the same value as thatgiven by Eq. (2.28) when damping B is low. The Lutespeak factor can be determined for a desired level ofconfidence p by numerically evaluated Eq. (2.16) withpo and a from Eq. (2.36).

In Figs. 2.9 and 2.10, the Modified-Vanmarcke and Lutes peakfactors for nonstationary responses. with p of 57X and 84X areplotted against fnT for damping values of 5X, 2X and 5X.

Table 2.2, provides the exact p values for the m and m+cr

values of the Lutes peak factor. It is of interest to notethat the p values of !i7X and 84X selected for the Lutes peakfactor are about the same order of accuracy as that previouslyfound for the Vanmarcke-Exact peak factor for a stationaryprocess. This implies that it is adequate to use the p valuesof 57X and 84X for computing the m and aH-0 values for theLutes peak factor.

2.3 Evaluation and Discussions

'I

For a stationary process, the peak factors given in Section'.2.1 arecompared in Figs. 2 .3 through 2.8 ~ It can be seen from these figuresthat, due to the different assumptions used in obtaining the peakfactors, the Davenport peak factor generally is higher than the otherpeak factors, especially when damping is low. This is because theDavenport peak factor assumes that the rate of threshold crossingoccurs independently according to the Poisson process. Thisassumption has been shown by Cramer (Ref. 12) to be asymptoticallyexact as the threshold crossing approaches infinity. However, forthe threshold crossing level of practical interests, this assumption

2-16

0

has been found to produce result erring on the unsafe side for a

wide-band process and on the safe side for a narrow-band process.

These errors occurred because the Poisson process does not allow the

time actually spent above the threshold level for a wide-band

process, and does not take into account the dependent occurrence of

the threshold level for a narrow-band process. The peak factor by

Vanmarcke, on the other hand, includes these effects on the thresholdcrossing by assuming that the successive time intervals spent below

and above the threshold crossing follow the Markov process.Therefore, as noted before, the Davenport peak factor represents the

upper bound of the peak factor by Vanmarcke.

For a nonstationary process, the peak factors given in Section 2.2.2are compared in Figs. 2.9 and 2.10. It can be seen from these

figures that the Modified-Vanmarcke peak factor is generally lowerthan the Lutes'peak factor when fnT is less than about 10; otherwise,it is higher than the Lutes peak factor. These two peak factors,however, approach each other as damping increases from 0.5% to 5%.

The analytical peak factors discussed above, however, have not been

adequately verified for their accuracies for actual applications.Since the accuracy of the PSDF-RS relationship required in theexplici.t approach depends on the accuracy of the peak factor and therms response, it is necessary that the accuracy of these peak factorsand the rms response be confirmed against simulation results.

The accuracy of the PSDF-RS relationship for both the wide-band

seismic ground motion and the narrow-band floor response are

systematically, evaluated in Sections 3 through 7 using simulationresults. In Section 3, the PSDF compatible with the 2% damping

response spectrum from the USNRC R.G. 1.60 is generated from the ,

PSDF-RS relationship with the use of the Der Kiureghian peak factor.Using this PSDF, twenty acceleration time histories are simulated forthe wide-band stationary and nonstationary ground motions in Section4. Also presented in Section 4 are the simulated-results for

2-17

evaluating the accuracy of the PSDF-RS relationship for the vide-band

ground motions in Section. 7. The simulated narrow-band floorresponses of a typical nuclear pover plant containment structure are

obtained in Section 6 by filtering the simulated wide-band ground

motions in Section 4 with the structural response 'transfer functionspresented in Section 5. Using the simulated results obtained from

Sections 4 and 6, the accuracy of the analytical PSDF-RS relationshipis then evaluated for the wide-band seismic ground motions and thenarrow-band floor responses in Section 7.

2-18

3. PSDF FOR WIDE-BAND SEISMIC GROUND MOTIONS

For the purpose of generating the PSDF for the wide-band seismic ground

motion, the design response spectra (DRS) from the USNRC R.G. 1.60 for the

horizontal seismic ground motion with 1.0g maximum acceleration is used.

The generated PSDF compatible with the 2X damping DRS will be used laterin Section 4 for simulation of seismic ground motions.

The response spectra from the USNRC R.G. 1.60 are shown in Fig. 3.1 for0.5X, 2X, and 5X damping values. These spectra were developed from

statistical evaluations of a number of recorded seismic ground motions forthe nH-a confidence level, which was based on the 84.1X non-exceedance

probability assuming a normal distribution for the spectral values

(Ref. 1) ~ In order to examine the PSDF of these recorded motions,

forty-five of them as listed in Table 3.1 are selected. The raw data on

the response spectral values and the Fourier amplitudes of these selectedrecorded motions for the 91 frequencies listed in Table 3 2 are obtainedfrom the ground motion data tape supplied by the California Institute of

Technology (Cal Tech). Using these raw data normalized to 1.0g maximum

acceleration, the m + a level response spectra are computed and plotted as

shown in Figs. 3.2 through 3 ' for OX, 2X and 5X damping, respectively.The response spectra shown in these figures are consistent with those used

as the basis in Ref. 1 for developing the USNRC R.G. 1.60 DRS shown inFig. 3.1. Therefore„ the forty-five recorded motions selected constitutes

P

an adequate sample size for the computation of the PSDF compatible withthe USNRC R.G ~ 1.60 spectra. It should be noted that, in Ref. 1, the

normalization factors with respect to velocity and displacement are alsoused in minimizing the statistical variations of the response spectra. Inorder to show the statistical variations of the response spectra computed

in Figs. 3.2 through 3.4, the coefficient of variation (COV) which is a

dimensionless measure of the dispersion relative to the mean is computed

as follows:

COV a/m

3-1

The COV's corresponding to the the response spectra in Figs. 3.2 through

3.4 are computed using Eq. (3 ') for OX, 2X, and 5X damping values and are

...shown in Fig. 3.5. It can be seen from this figure that COV graduallydecreases as frequency increases. This trend of variation was also found

in Ref. (1) when the maximum acceleration was used as the normalizationfactor. Note that COV at .1 cps is about 1 and decreases to as low as .08

for the 5X damping curve.

In order to approximately compute the PSDF of these recorded motions,

these recorded motions are assumed to be a stationary process withduration T. Accordingly, the one-sided PSDF of these recorded motions can

be approximated as follows:

(4. 1)

where N is the total number of recorded motions equal to 45, T is the

equivalent stationary duration, and I xi(m) I is the Fourier amplitude ofthe ith recorded motion obtained from Cal Tech. The duration T used inEe. (4.1) is selected to be 15 seconds which is a typical duration ofstrong phase motion for high-intensity seismic motions. However, by

visual examination of these recorded motions in Ref. 18, their strongphase durations are found to be varying from about 3 to 20 seconds. Thus,

the duration of 15 seconds selected for T in Eq. (4.1) is merely an

approximation of the duration of an ensemble of these recorded motions,treated as a stationary process. In Fig. 3.6, the PSDF of these recorded

motions computed from Eq.(4.1) is plotted against, frequency in cps. Itcan be seen from this figure that this PSDF is fairly wide-banded coveringfrequency ranging from about .25 cps to 2.5 cps.

In order to generate the analytical PSDF compatible with the USNRC R.G.

1.60 response spectra, the PSDF-RS relationship in Eq. (2.14) with thepeak factor corresponding to the ~ level as prescribed for the spectrais used. The PSDF generated using Der Kiureghian peak factor inEq. (2.14) is shown in Fig. 3.7, and the PSDF generated using the

3-2

Modified-Vanmarcke peak factor is shown in Fig. 3.8. Three response

spectral curves corresponding to the '.5X, 2X and 5X damping are shown ineach figure. It is apparent from these figures that the PSDF's

corresponding to the different damping response spectra are different.This implies that the USNRC R.G. 1.60 response spectra for differentdamping values can not be associated with a single random process.Furthermore, it can be seen by comparing Fig. 3.7 with Fig. 3.8 that the

PSDF in Fig. 3 ' are higher than those in Fig. 3.7 especially for low

damping values and in the low frequency range up to about 2.5 cps'hisis because the peak factor used for generating the PSDF in Fig. 3.8includes the nonstationary effect which reduces the stationary ms response

for a lightly damped and soft system. Thus, in order to achieve the same

response spectral value in Eq. (2.14), the PSDF in Fig. 3.8 obtained from

using the nonstationary peak factor must be higher than that in Fig. 3.7

obtained from using the stationary peak factor. Comparing the analyticalPSDF shown in Figs. 3.7 and 3.8 with the empirical PSDF shown in Fig. 3.6,it can be seen that the empirical PSDF is generally higher than the

analytical PSDF in the low frequency range up to about 4 'cps, but is lower

in the higher frequency range. This difference can be attributed to the

approximation used in computing the empirical PSDF according to Eq. (3.2).

3-3

4. SIMULATION OF WIDE-BAND SEISMIC GROUND MOTIONS

Using the PSDF for the wide-band seismic ground motions generated in the

previous section, twenty acceleration time histories are simulated inSection 4 1 for the stationary seismic ground motion, and in Section .24

for the nonstationary seismic ground motion. The m and ada levels of

response spectra of these simulated ground motions are then presented. In

Section 4.1, the spectral moments of the absolute acceleration response tostationary ground motions and the simulated peak factors are 'also

presented.

4.1 Stationar Ground Motion

x(t) = —Re( g [2 jsz(~) e >] e 6m}1 i$ i~t2 7f

(4. 1)

The stationary ground motion, xs(t) is simulated using the Fast

Fourier Transform (FFT) algorithm as follows:

in which Sx (+k) is the PSDF corresponding to the 2X damping value inFig. 3.7, 4k is the phase angle with a uniform distribution in the

range from 0 to 2m, ek ~ kh a), A o) ~ 0.42 radian per second, and thetotal number of sample points, N, is 1024. Therefore, the durationof the simulated'tationary ground motion is 15 seconds digitized ata time increment of 0.0146 seconds. The maximum accelerations of the20 simulated stationary ground motions are tabulated in Table 4.1.As shown in this table, the maximum ground acceleration varies from

0.77g to 1.23g with the m value of 0.95g and the aHrr value of 1.06g,which is close to 1.0g maximum ground acceleration specified for theDRY

A typical stationary ground motion simulated using Eq. (4.1) and itsresponse spectra for ~ 5X, 2%, and 5X damping are plotted as shown inFigs. 4.1 and 4.2, respectively The response spectra are computed

for the 47 frequencies listed in Table 4.1. The m and m+0 response

spectra are plotted as shown in Figs. 4 ' through 4.5 for .5X, 2X,

4-1

Fi . 4 ~ 2and 5X damping, respectively. It can be seen from comparing g.with Fig. 4.3 through Fig. 4.5 that the m and ~ response spectra

are smoother than the response spectra generated from a single ground

motion. The statistical variations of the response spectral values

measured by the COV of Eq. (3.1) are shown in Fig. 4.6. It can be

seen that the COV of spectral value of stationary ground motions is,on the average, fairly constant at about 0 '5, and is less than the

COV in Fig. 3.5 obtained for the ensemble of 45 recorded ground

motions which varies from 1 to .1.

The properties of the PSDF of absolute acceleration response tostationaxy ground motions, which are closely related to the peak

factor as shown in Section 2, are presented in Figs. 4.7 throug h

4.9. In Fig. 4.7, the square root of the three spectral moments,

~X~, ~Ay . and ~Xp are plotted against frequency in cps for2X damping system. The spectral moments Xs in this figure are

computed numerically from Eq. (2.19) in which S(a)) is obtained fx'om

multiplying Eq (2.4) with the factor (M" + 48 +I ) ~ Note that,since ~>o is the rms absolute acceleration response the,rms ground

acceleration can be approximately obtained from Fig. 4.1 as2the ~lo value at f 23 cps which is about 120 in/sec or 0 ~ 3g.

The spectral moments in Fig. 4.1 are used for computing the center

frequency, fc, in Eq. (2.20), and the spectral dispersion parameter

in Eq. (2.21). -In Fig. 4.8, the simulated fc computed from Eq.

(2.20) is plotted as the solid line, and the analytical fc estimated

by Eq. (2.23) as the dotted line. Similarly, the simulated 6

computed from Eq.,(2.21) is plotted as the solid line in Fig. 4.9 and

.the analytical 8 estimated from Eq. (2.28) plotted as the dottedline. It can be seen from these figures that the analytical fc and 6

estimated from Eq. (2.23) and Eq. (2.28) compared on the average,quite well with the corresponding simulation results in the frequencyrange of less than 10 cps. When frequency exceeds 10 cps, theanalytical f is higher but the analytical 6 is lower than theccorresponding simulated value. This difference, however, wouldproduce small changes in the peak factor in Eq. (2.27).

4-2

Consequently, the estimated values of the analytical fc and 4 as

given by Eq. (2.23) and Eq. (2.28), respectively, can be used withsufficient accuracy for computing the analytical peak factors, which

also compare well with the simulated peak factors as will be shown inSection 7.1.1.

The simulated m and ~ peak factors for 2X damping can be computed

from Eq. (2.12) by dividing the m and mfa response spectral values inpig. 4.4 with 0 X<> in pig. 4 ~ 7. Similar computations of the peak

factor are also performed for .5X and 5X damping values. The

simulated m and ~ peak factors are plotted against frequency in cps

in Figs. 4.10 through 4.12 for the three damping values respectively.

ln Section 7.1.1, the simulation results presented in this subsectionfor the m and csc response psectra,~A~ , and the peak factor willbe used for the validation of the PSDF-RS relationship for stationarywide-band ground motions.

4.2 Nonstationar Ground Motion

The nonstationary ground motion, LAN (t) is simulated as follows:

(t) = E (t) xs (4. 2)

in which xs(t) i's the stationary ground motion as given by Eq. (4.1),and E(t) is the envelope function. The envelope functions used are

the so-called Types B and C envelope functions proposed by Housner,

Jenning and Tsai (Ref. 19). These envelope functions are shown ins

Fig. 4-13. For brevity, the nonstationary ground motion using Types

B and C envelope functions will be referred to as the Type B ground

motion and the Type C ground motion.

For the Type B ground motion, the duration used for simulation is 30

seconds digitized at a time increment of 0.0146 seconds. A typicalof this ground motion and its response spectra for .5X, 2% and 5X

4-3

damping are shown in Figs. 4.14 and 4.15, respectively. The m and

~ response spectra for these three damping values are shown inFigs. 4 .16 through 4.18. The COV of spectral values for Type B

ground motion spectra is plotted in Fig. 4.19 'y comparing Fig.4.19 with Fig. 4.6, it can be seen that the COV for the Type B ground

motion is slightly larger than the COV for stationary ground motion.

For the Type C ground motion, the duration used for simulation is 12

seconds digitized at .0117 seconds. A typical of this ground motion

and its response spectra for .5X, 2X and 51 damping are shown inFigs. 4.20 and 4.21, respectively. The m and mKf response spectrafor these three damping values are shown in Figs. 4.22 through 4.24.The COV of spectral values for Type C ground motion are plotted inFig. 4.25. By comparing Fig. 4.25 with Fig. 4.19, it can be seen

that the COV for the Type C ground motion has slightly more variationthan the Type B ground motion.

It is apparent from the above comparisons of the COV of spectralvalues for stationary ground motions with that for Type B or Type C

ground'motions that the longer is the strong phase (stationary)duration, the smaller is the amount of relative dispersion.

The m and ~ response spectra obtained from simulation for Types B

and C ground motions as shown in Figs. 4.16 through 4.18, and Figs.4.21 through 4.24, respectively, will be used for validation of the

PSDFWS relationship in Section 7.1.2 for nonstationary wide-band

ground motion.

4-4

5 ~ STRUCTURAL TRANSFER FUNCTIONS

In order to study the structural response due to seismic ground motions,

the transfer functions of the containment and its internal structure of a

pressurized water reactor (PWR) nuclear power plant are used as a typicalexample in this study. 'These structural transfer functions serves as

filters to transfer the simulated wide-band seismic ground motions inSection 4 to the output of narrow-band floor responses to be described inSection 6.

The PWR containment and its internal structures supported on a flexiblesoil foundation are shown in Fig. 5.1. The mathematical model of thesoil-structure system is also shown in Fig. 5.1 as a lumped-mass stickmodel consisting of 19 nodes and 16 beam elements, with the soil stiffnessand damping represented by springs and viscous dampers attached to the

base of structures at node 19 'he properties of the structures, and the

soil foundation are provided in Table 5.1, and Fig. 5.1, respectively.This soil-structure system has one translational dynamic degree-of-freedom

(DDOF) per node and one rotational DDOF at node 19.

Using the Bechtel in-house computer program CE933 (FASS), the transferfunctions relating the ground acceleration input to the absoluteacceleration responses are computed for node 11 (top of the containment

structure), and node 18 (top of the internal structure). Since thiscomputer program requires that the fixed-base structures be characterizedby their modal properties, the modal properties of the fixed-basestructures are computed using the Bechtel in-house computer program

CE917. The four lowest modal properties obtained from this computer runas shown in Table 5.? are then used as the input to the computer program

CE933 to compute the structural transfer functions for nodes 11 and 18

shown in Fig. 5.2.

It can be seen from Fig. 5.2 that the structural transfer functions fornodes ll and 18 exhibit typical narrow-band filtering characteristics withtwo dominant peaks. The frequencies corresponding to these two dominant

peaks are 3.4 cps and 17 cps for node ll, and 3.4 cps and 11 cps for node

18. These frequencies correspond to the system frequencies of thesoil-structure interaction system.

6. SIMULATION OF NARROW-BAND FLOOR RESPONSES

The narrow-band floor responses for nodes 11 aand 18 of the soil-structuresystem shown in Fig. 5.1 are s mu a e1 ted by filtering the vide-band seismic

ground motions in ect onS i 4 through the respective structural transferfunctions in Section ~

' oo5 'h fl r responses to stationary ground motions

and to nonstationary ground motions are separat y pel resented in Sections

6.1 and 6.2,,respectively.

6.1 Res oases to Stationar Ground Motion

nodes 11 and 18The absolute acceleration response time histories for nodes

ted to the 20of the soil-structure system shown in Pig. Sil sub)ected tostationary ground motions generated in Section 4.1 are computed usingthe computer program CE933 (FASS). This computer program has the

capability to perform the seismic soil-structure interaction analysisusing the impedance approach and frequency domain analysis method viathe Fast Fourier Transform (FFT) algorithm. The typical time historyresponses to one simulated ground motion time history input for nodes

ll and 18 are plotted in Figs. 6.1 and 6.2, respectively. The

maximum accelerations for the 20 simul'ated stationary response

motions for nodes 11 and 18 are tabulated in Table 6.1. As shown inthis table, the maximum response acceleration varies from 3.3g to4.7g with the m and m+0 values of 3.8g and 4.1g for .node 11, and from

ode 18.2.5g to 3.9g with the m and m+a values of 2.9g and 3.2g for node

The response spectra of the response motions shown in Pigs. 6.1 and

6.2 for .5X, 2X, and 5X damping are shown in Figs. 6.3 and 6.4,respectively. The m and'+0 floor response spectra for nodes 11 and

18 for all 20 time history inputs are plotted for the same threedamping values in Figs. 6.'5 through 6.7, and Figs. 6.8 through 6.10,respectively. The statistical variations of these floor response

spectral values measured by the COV computed from Eq. (3.1) are shown

in Fig. 6.11 for node ll, and in Fig. 6.12 for node 18. By comparing

Figs' .11 and 6.12 with Fig. 4 .6, it can, be seen that the COV ofspectral value of the floor response motions are: about the same order

of variation as that of the stationary ground motions in Fig. 4.6.

The PSDF's of the absolute acceleration floor response for nodes lland 18 are plotted against frequency in cps as shown in Fig. 6.13.

These PSDF's are obtained from multiplyi'ng the absolute

acceleration structural response transmittancy functions, which are

the square of the transfer function amplitudes in Fig. 5.2, with the

PSDF of the stationary ground motion in Fig. 3.7. In order to obtainthe PSDF of the absolute acceleration floor spectral response, the

PSDF of the absolute acceleration floor response in Fig. 6.13 ismultiplied by the absolute acceleration transmittancy of' SDOF

system having frequency fn and damping 8. The spectral properties ofthe PSDF of the absolute acceleration floor spectral response are

4plotted against frequency fn in cps for damping 9 of 2X in Figs. 6.1through 6.16 for node 11, and Figs. 6.17 through 6.19 for node 18.

The square root of the three spectral moments,~A~ > Aj and

shown in Figs. 6.14 and 6.17 for nodes 11 and 18, respectively, are

computed according to Eq. (2.19). It can be seen thatJX~ at 25 cps

for nodes 11 and 18, which approximately are the rms absolute

acceleration floor responses, are about 500 and 370 iw'sec , or 1.QI 3

and 0.96g, respectively. Using the three spectral moments in these

figures, the center frequency, fc, and the spectral dispersionparamarameter 6 can accordingly be computed from Eqs. (2.20) and (2.21),respectively. The computed fc and 8 are plotted in Figs. 6.15 and

6.16 respectively for node 11, and Figs. 6.18 and 6.19 respectivelyfor mode 18 'n these figures, the analytical fc and 6 computed from

Eq. (2.23) and Eq. (2.28) are also plotted as dotted lines. It can

be seen that the analytical fc compares well with the simulated fc. when frequency is less than 10 cps, and is higher than the simulated. f when frequency is greater than 10 cps. This difference betweenc

the analytical and the simulated fc is very similar to thatpreviously observed for the wide-band ground motion response inSection 4 ~ On the other hand, the comparison of the analytical 6

with the simulated 6 for the narrow-band floor response in Figs. 6.16

and 6.19 does not show as good a comparison with that for the

6-2

wide-band ground motion in Fig. 4.9. The analytical 4 is generally

lower than the simulated 4 except around the system frequencies at

3.4 and 11 cps where the analytical 6 is higher than the simulated

Thzs difference implies that the analytical peak factor in

Eq. (2.27) with the analytical 6 tends to overestimate the analytical

peak factor with the simulated 4 around the system frequencies and

underestimate the analytical peak factor with the simulated

elsewhere .- As will be shown in Section 7.2.1, the estimated fcand 8 can be used with reasonable accuracy for computing the

analytical peak fa'ctors for the narrow-band floor response, which

generally compare well with the simulated peak factors, except near

the system frequencies where slight overestimates result.

Using the m and aH.S floor response spectral values derived from the

simulation and thezms response which is equal to ~ho, the simulated

m and m+a peak factors for 2X damping can be computed for nodes 11

and 18, by respectively dividing the m and m+a floor response

spectral values in Figs. 6 ' and 6.9 with the~Ao values in Figs.6.14 and 6.17. The simulated m and nd.tx peak factors for 0.5X and 5X

damping can be computed similarly. The resulting peak factors are

shown in Figs. 6.20 through 6.22 for Node 11 and Figs. 6.23 through

6.25 for Node 18. As will be described in Section 7 ~ 2 ~ 1, these

simulated peak factors will be used for validating the analyticalPSDF-RS relation'ship presented in Section 2 for narrow-band floorresponse to stationary ground motion input.

6.2 Res onses to Nonstationar Ground Motion

The absolute acceleration response motions for nodes 11 and 18 of the

soil-structure system shown in Fig. 5.1 subjected to 20 simulated

Type B nonstationary ground motion time histories are computed usingthe computer program CE933, and representative resulting time historyresponses are plotted in Figs. 6.26 and 6.27, respectively for Nodes

11 and 18 'heir respective response spectra for .5X, 2X and 5X

6-3

damping are plotted in Figs. 6.28 and 6.29. The m and r&U floorresponse spectra for three damping values computed from all 20 Type B

time history inputs are plotted in Figs. 6.30 through 6.32 for node

ll, and Fig. 6.33 through 6.35 for node 18.

The m and ada floor response spectra presented in this section willbe used later in Section 7.2.2 for the validation of the analyticalPSDF-RS relationship for the narrow-band floor responses to Type B

nonstationary ground motion input.

The simulation of the narrow-band floor response to Type C ground

motion inputs has not been performed since, as will be explained inSection 7 '.2, the analytical PSDFWS relationship with the use ofModified-Vanmarcke peak factor and the equivalent stationary durationis found to be too conservative for the Type C ground motion response.

6-4

7. VALIDATIONOF THE PSDF-RS RELATIONSHIP

The simulation results presented in Section 4 for the wide-band seismic

ground motions and Section 6 for the narrow-band floor responses are used

in this section to validate analytical the PSDF-RS relationship presented

in Section 2.1.

7.1 Wide-Band Seismic Ground Motions

The validity and accuracy of the analytical PSDF-RS relationship isevaluated in Section 7.1.1 for the stationary ground motion, and

Section 7.1.2 for the nonstationary ground motion. In Section 7.1.1,the accuracies of the analytical rms response and peak factorcontained in the PSDF-RS relationship are also explicitly evaluated

for the stationary ground motion. In Section 7.1.2, the validity ofthe PSDF-RS relationship for the nonstationary ground motion isevaluated using the equivalent stationary duration to represent the

stationary portion of Types B and C groundmotions'.1.1

Stationar Ground Motion

In order to evaluate the accuracy of the analytical rms

responses given in Eg. (2.10), these analytical rms responses

are compared with the simulated rms responses obtained inSection 4, as shown in Fig. 7.1 for .5$ , 2$ and 5$ damping.

The PSDF compatible with the 2g damping DRS is used inEq.'2.10) for Sz(z) in computing, the analytical rms

response. It can be seen from these comparisons that the

analytical rms response generally compares well with thesimulated rms response. Thus, the accuracy resulting from

using the analytical rms response given in Eq. (2.10) topredict the rms response to the stationary wide-band ground

motion is confirmed.

In order to evaluate the accuracy of the analytical peak

factors shown in Figs'.3 through 2.10, these analytical

peak factors with duration T of 15 seconds are compared with

the simulated m and m+0 peak factors shown in Figs. 4.10

through 4.12 for .5g, 2$ and 5$ damping, respectively. Eased

on these comparisons, the analytical peak factor that compares

best with the simulated peak factor, is the Modified-Vanmarcke

peak factor. The comparisons of the Modified-Vanmarcke peak

factor with the simulated peak factor at the m and m+0 levelscan also be seen in Figs. 7.2 through 7.4 for .5$ , 2$ and 5$

damping, respectively. It is apparent from these figures thatthe Modified-Vanmarcke peak factor compares closely with the

simulated peak factors.

The comparisons of the analytical peak factors in Figs. 2.3through 2.8 with the simulated peak factors in Figs. 4.10

through 4.12 show that the Davenport and Der Kiureghian peak

factors overestimate the simulated m and mw peak factorsespecially vhen the frequency and damping are low. As

mentioned in Section 2-5, this overestimation can be

attributed to the inaccuracy of the independent thresholdcrossings assumption made for the Davenport peak factor, and

the neglect of the nonstationarity of the response by these

two peak factors eventhough the input ground motion isstationary.

By comparing Figs. 2.9 and 2.10 with Figs. 4.10 through 4.12,it can be seen that the Lutes peak factor slightlyunderestimates the simulated m and m+0 peak factors vhen

frequency erceeds 1 cps, and,slightly overes'timates thesimulated m and m+a peak factors when frequency is less than 1

cps. This difference may be due to the use of ideal whitenoise instead of the more realistic vide-band seismic ground

motion for the derivation of the Lutes empirical peak factor.In summary, among the analytical peak factors surveyed inSection 2.2, the Modified-Vanmarcke peak factor gives the

V ~

most accurate prediction as compared to the simulated peak

7-2

factor for the case of stationary wide-band ground motioninput.

In order to evaluate the validity and accuracy of the PSDF-RS

relationship for the stationary wide-band ground motion, the m

and m+g response spectra generated from the analytical PSDF-RS

relationship are compared with the simulated m and m+0

response spectra in Figs ~ 7 5 through 7 7 respectively for.5$ , 2$ and 5$ damping. The analytical m and m+0 response

spectra, shown as dotted lines in these figures, are computed

from Eq. (2.13) using the analytical rms response and the

Modified-Vanmarcke peak factor. The simulated m and m+0

response spectra, shown as solid lines in these figures, are

obtained from Section 4.1, Figs. 4.3 through 4.5. As can be

seen in Figs. 7.5 through 7.7, the analytical' and m+0

response spectra compare closely with the simulated m and m+a

response spectra. Thus, the validity and accuracy of the

PSDF-RS relationship in Eq. (2.13) with the use of the

analytical rms response and the Modified-Vanmarcke peak

factor, is confirmed for the stationary wide-band ground

a otion input"

7.1.2 Nonstationar Ground Motion

I

Since the PSDF-RS relationship derived in Section 2.1 is based

on the assumption of stationary ground motion input, it isnecessary that, for this relationship to be applicable toTypes B and C nonstationary ground motion inputs, the ground

motion duration T be characterized by an equivalent stationarymotion duration, Te. Various methods of determining Te fornonstationary ground motions have been proposed in the

literatures (e.g., Refs. 20 and 21). The method selected here

for determining Te is similar to that proposed by Trifunac and

Brady in Ref. 20. The equivalent stationary duration, Te, 'is

defined as follows:

T T 9p T p5

T 90 and T 05 are the times respectively corresponding

to 90$ and 5$ of the total energy of the ground motion

represented by the integral Z(t) as follows:

(7. 2)

Note that the limits on input energy used by Trifunac and Brady

in Ref. 22 are 5C and 95$ . These limits have been modified to

5$ and 90$ because it has been found that Te withoutmodification would produce too conservative predictions on the

response spectra. Using Eq ~ (7.1), the average values of Te

are determined to be 14 and 5 seconds, from the totaldurations of 30 and 12 seconds used for the Types B and C

ground motion simulation, respectively.

For Type B ground motions, the validity and accuracy of the

PSDF-RS relationship is evaluated by comparing the analyticalm and m+c response spectra with the simulated m and m+g

response spectra, as shown in Fige. 7.8 through 7.10 for .5$ ,

2g and 5$ , respectively. The analytical m and m+0 response

spectra plotted as dotted lines in these figures are computed

from using Te 14 seconds, the analytical rms response, and

the Modified-Vanmarcke peak factor in Eq. (2.13) ~ The

simulated m and m+g response spectra plotted as solid lines inthese figures are obtained from Section 4.2, Figs. 4.16

through 4.18. As can be seen from Figs. 7.8 through 7.10, the

analytical m and m+0 response spectra compare closely with the

simulated m and m+a response spectra, respectively. Thus, the

validity and accuracy of the PSDF-RS relationship inEq. (2.13) with the use of Te, the analytical rms response,

and the Modified-Vanmarcke peak factor, is confirmed for Type

B nonstationary ground motions.

7-4

For Type C ground motions, the validity and accuracy of the

PSDF-RS relationship is evaluated by comparing the analyticalm and m+a response spectra with the simulated m and m+g

response spectra, as shown in Figs. 7.11 through 7.13

respectively for .5$ , 2$ , and 5g. damping. The analytical m

and m~ response spectra, plotted in these figures as dottedlines, are computed from using Te 5 seconds, the analyticalrms response, and the Modified-Vanmarcke peak factor inEq. (2.13). The simulated m and m+0 response spectra plottedas solid lines in these figures are obtained from Section 4.2,Figs. 4.22 through 4.24- As can be seen from Figs. 7.11

through 7.13, the analytical m and m+v response spectragenerally overestimate the corresponding simulated m and mat

response spectra, especially in the low to medium frequencyrange.

This is because the Type C ground motion has relatively shortstrong intensity portion to allow response to reach

stationary. Thus, the PSDF-RS relationship in Eq.(2.13) withthe use of Te, the analytical 'rms response, and the

Modified-Vanmarcke peak factor result in conservativeestimates on the response spectra of Type C ground motion.

Based on,the above findings, it can be concluded that thePSDF-RS relationship in Eq. (2.13) can be applied withsufficient accuracy. for the nonstationary wide-band ground

motion whose transient characteristics are similar to the

Type .B ground motion with an equivalent stationary duration as

de'fined by Eq. (7.1) to be about 14 seconds.

7.2 Narrow-Band Floor Res onse

The validity and accuracy of the analytical PSDF-RS relationship isevaluated in Section 7-2.1 for the narrow-band floor responses tostationary ground motion, and in Section 7.2.2 for the narrow-band

7-5

floor response~ to nonstationary ground motion. The accuracies ofthe analy'tical rms response and 'the peak factor contained in thisrelationship are also explicitly evaluated in Section 7.2.1. Using

the same equivalent stationary duration determined in Section 7.1.2for the Type B ground motion, the validity and accuracy of the

PSDF-RS relationship is evaluated for narrow-band floor responses to

Type B nonstationary ground motion. The validity and accuracy of the

analytical PSDF-RS relation for narrow-band floor responses to the

Type C ground motion will not be evaluated, since it has been found inSection 7.1.2 that it results in too conservative response estimatesfor the Type C ground motion response.

7.2.1 Res onses to Stationa Notion

En order to evaluate the accuracy of'the analytical rms

response given in Eq. (2.10) for this case, the analytical rms

floor spectral responses for .5$ , 2g, and 5$ damping are

compared with the corresponding simulated rms floor spectralresponses, as shown in Figs. 7.14.a through 7.14.c for node

ll, and Figs. 7.15.a through 7.15.c for node 18. The

analytical rms floor spectral responses for nodes ll and 18

are computed from using Eq. (2.10) with Sx(+) equal to theirrespective PSDF shown in Fig. 6.15. The simulated rms floorspectral responses for nodes ll and 18 are obtained from

1

Section 6, e.g., the simulated rms floor spectral responses

for 2$ damping are shorn as J 1o in Figs. 6.14 and 6 ~ 17 fornodes ll and 18, respectively. As can be seen from thecomparisons in Pigs. 7.14 and 7.15, the analytical rms floorspectral response overestimates the peaks and underestimatesthe valleys of the simulated rms floor spectral response.

Thus, the analytical rms response given in Eq - (2.1$ ) cannotbe accurately used for predicting the narrow-band rms floorspectral response to stationary ground motions. Therefore,

~ the simulated rms floor spectral response should be used

instead of the analytical rms response for a more accurateprediction of the floor response spectra.

7-6

In order to evaluate the accuracy of the analytical peak

factors shown in Figs. 2.3 through 2.10, these analytical peak

factors are compared with their respective simulated peak

factors in Figs. 6.20 through 6.22 for node 11, and Figs. 6.23

through 6.25 for node 18. The results from these comparisons

are that the Modified-Vanmarcke peak factor can be used withreasonable accuracy for pre'dieting the narrow-band floorresponse to stationary ground motion input, except near thesystem frequencies where slight overestimates occur.

Due to the inaccuracy of the analytical rms response, the

analytical PSDF-RS relation in Eq. (2.15) cannot be expected

to be accurate for predicting the narrow-band floor response

spectra. To improve the accuracy, Eq. (2.12) with thesimulated rms response in Figs. 7.14 and 7.15 and the

Modified-Vanmarcke peak factor, can be used for generating thefloor response spectra. The analytical m and mKf floorresponse spectra as generated are compared respectively withthe simulated m and ~ floor response spectra in Figs. 7.22

through 7.24 for node ll, and Figs. 7 '5 through 7.27 for node

18. The analytical and simulated m and ~ floor response

spectra are plotted in these figures as dotted and solidlines, respectively. As can be seen from Figs. 7.22 through

7.27, the analytical m and mar floor response spectra generallycompare reasonably well with the simulated m and ~ floorresponse spectra. Thus, these comparison results confirm thatthe analytical floor response spectra of the narrow-band floorresponse motion to the stationary wide-band ground motioninput, can be computed with reasonable accuracy from

Eq. (2.12) with the use of the simulated rms floor spectralresponse, i.e. the rms floor spectral response given byJX~and the Modified-Vanmarcke peak factor, except near the system

frequencies where slight overestimates occur.

7-7

7.2.2 Res onse to Nonstationar Ground Motion

The equivalent stationary duration of 14 seconds as previouslydetermined in Section 7.1.2 is used in generating the

analytical m and aHe floor response spectra of the

narrow-band floor responses to the Type B ground motion. As

in Section 7.2.1, the analytical m and m+< floor response

spectra are computed from using Eq. (2.L?) with the simulated

rms response and the Modified-Vanmarcke peak factor. These

computed analytical m and aHa floor response spectra are

compared with the simulated m and m+a floor response spectrain Figs. 7 .28 thr'ough 7 .30 for node ll, and Figs. 7 .31 through

7 .33 for node 18 . The simulated and analytical response

spectra are plotted in these figures as solid and dottedlines, respectivley. The simulated m and aHa afloor response

I

spectra shown in these figures are obtained from Section 6.2,Figs. 6.30 through 6.32 for node 11, and Figs. 6.33 through6.35 for node 18. As can be seen from Figs. 7.28 through 7.33,the analytical m and m+0 floor response spectra generallycompare reasonably well with the simulated m and m+0 floorresponse spectra. Thus, these comparison results confirm thatthe analytical floor response spectra of the narrow-band floorresponse motion to the Type B ground motion input, can be

computed with reasonable accuracy from Eq. (2.12) with the use

of the simulated rms floor sPeotral resPoosa gives by ~ioP

and the Modified-Vanmarcke peak factor, except .near the systemfrequencies where slight overestimates occur.

7-8

8 ~ DIRECT GENERATION OF PROBABILISTIC FLOOR RESPONSE SPECTRA

Based on the evaluation results in Section 7, the probabilistic floorresponse spectra (FRS) with any desired confidence levels can be generated

directly with reasonable accuracy from the prescribed design ground

response spectrum (DRS) as follows, see Fig. 8.1:

1. Compute the ground motion PSDF from the prescribed DRS using the

PSDF-RS relationship in Eq. (2.14) with the Modified-Vanmarcke peak

factor given in Section 2.2.1-b. The approximate values of 8 inEq. (2.28) and u in Eq. (2.22) are used for computing the Modified-Vanmarcke peak factor.

2. Compute the structural response transmittancy function which is the

square of the amplitude of the structural response transfer function.

. 3 ~ Compute the structural response PSDF by multiplying the ground PSDF inStep 1 with the structural response transmittancy function in Step 2.

4. Compute the floor spectral response PSDF by multiplying the structuralresponse PSDF in Step 3 with the transmittancy function of a SDOF

system with specified spectral frequency and the desired spectraldamping ratio.

5. Compute the rms f2ooi spectral response by taking the square root ofthe area under the floor spectral response PSDF in Step 4.

6. Compute the. floor spectral response value by multiplying the rms floorspectral response in Step 5 with the Modified-Vanmarcke peak factor inStep l.

7. Generate the floor response spectrum for the desired spectral damping

ratio by repeating the computations from Step 4 to Step 6 usingdifferent spectral frequencies for the SDOF system in Step 4.

I

8-1

.~

The probabilistic FRS generated from the above procedure generally compare

reasonably well with those obtained from simulations as shown in Section

7, although the floor spectral values at the peaks corresponding to the

system frequencies are found to be slightly overestimated relative to the

simulated results. These slight overestimates on the floor spectralresponse values at the peaks can be attributed to the slight overestimates

on the Modified-Vanmarcke peak factor values at the system frequencies

resulting from using the approximate value of 6 in the above Step 6. As

shown in Section 6.2, the approximate value of 6 for the narrow-band floorresponse tends to overestimate the simulated (exact) value of 4. near the

system frequencies ~ and underestimate the exact value of 4 elsewhere. As

a result, the Modified-Vanmarcke peak factor computed from using the

approximate value of 5 for the narrow-band floor response tends to be

higher than that computed from using the exact value of 5 near the system

frequencies, and lower than that computed from using the exact value ofelsewhere. Therefore, in order to improve the accuracy of the

probabilistic FRS at the system frequencies relative to the simulated

results, the exact value of d in Eq. (2.21) may be used, instead of the

approximate value of 6 in Eq. (2.28), for computing theModified-Yanmarcke peak factor for the narrow-band floor response in the

above Step 6. This, in effect, would lower the floor response spectralvalues near the peaks and raise the floor response spectral values,elsewhere, producing more accurate estimates near the peaks and slightoverestimates elsewhere. The slight overestimates on the floor response

Is ectral values in the high frequency range, however, can be reduced toP

improve the accuracy relative to the simulated results, by using the exactvalue of uo in Eq. (2.18) instead of the approximate value of vo in Eq.

(2.22) for computing the Modified-Vanmarcke peak factor for thenarrow-band floor response in the above Step 6. As shown in Section 6.1,the simulated center frequency computed from using the exact value of vo

generally compares closely with the analytical center frequency computed

from using the approximate value of vo, except in the high frequencyregion where the simulated frequency is smaller than the analytical centerfrequency. In fact, as frequency increases, the simulated centerfrequency approaches the center frequency of the floor response motion,

8-2

whereas the

result, the

value of vo

approximate

analytical center frequency approaches a large value. As a

Modified-Vanmarcke peak factor computed from using the exact

is always smaller than that computed from using the

value of vo in the high frequency range.

In summary, the accuracies of the probabilistic FRS at the systemI

frequencies as well as in the high frequency region relative to the

simulated results may be improved by the use of the exact values of 0 and u

for determining the Modified-Vanmarcke peak factor for the narrow-band

floor response in the above Step 6. In the following section, the

application of the proposed procedure of generating the probabilistic FRS

discussed above is demonstrated for a typical nuclear power plantsubjected to seismic ground motions. The effects on the probabilistic FRS

due to the use of the exact values of 5 and vo will also be discussed.

The spectra generated with the procedure proposed herein carry the

following implied assumptions:

1. The structural system is linear.

2. The prescribed DRS for the seismic ground motion is for a low damping

value and a known confidence level.

3. The seismic ground motion is a normal wide-band random process, and

its nonstationara'.ty can be characterized by an equivalent stationaryprocess.

4. Similarly, the structural response to the seismic ground motion can be

characterized by an equivalent stationary random process.

For practical applications, assumptions 1 and 2 can generally be met by

the design requirements. However, in order to satisfy assumptions 3 and

4, the seismic ground motion, which is a nonstationary process, should

have a sufficiently long equivalent stationary duration Te as defined in, Eq. 7.1, as compared to the lowest period Tl of the structural system.

8»3

0

The results in Section 7 indicate that, when the ratio of Te/Tl is greaterthan about 40, the probabilistic response spectra can be generated from

the above procedure with reasonable accuracy. However, when Te/Tl is lessthan about 40, the probabilistic response spectra as generated from the

above procedure can be expected to be conservative.

8-4

9. APPLICATIONS

Applications of the proposed procedure for generating the probabilisticFRS directly from the prescribed DRS in Section 8 are demonstrated in thissection. The probabilistic FRS of a typical nuclear power plant structuresubjected seismic ground motions prescxibed in the form of DRS are

generated "using the proposed procedure, and presented in details. Also,

presented are the comparisons between the probabilistic FRS and the

deterministic FRS generated from using a single time history compatible

with the prescribed DRS, in order to provide some insights into the

relative merits of the proposed procedure and the deterministic method

currently used in practice. The analysis model used for generating the

FRS is essentially the same as the soil-structure model of a typicalnuclear power plant shown in Fig. 5.1, except that the structures are now

fixed at the base. The material and sectional properties of the

fired-base structures are provided in Table 5.1. A constant damping value

of 2$ is assigned for the five lowest modes used in the analyses.

The 2$ damping response spectrum from the USNRC R.G. 1.60 shown as a

dashed line in Fig. 9.1 is used to prescribe the DRS at mu level (i.e.,p ~ 84$ ) for seismic ground motions. Also shown as a solid line in thisfigure is the 2$ damping response spectrum of the synthetic time historyshown in Fig. 9.2. This synthetic time history is used as input forgenerating the deterginistic FRS. The equivalent duration, Te, of thissynthetic time history used as input for generating the probabilistic FRS

is determined from Eq. (7.1) to be about 12 seconds.

Using the proposed procedure in Section 8, the 2$ damping FRS at m+0 levelfor nodes ll and 18 are generated. The results obtained from varioussteps in the proposed procedure are plotted as shown in Fig. 9.3 through

Fig. 9.16 'ig. 9.3 shows the Modified-Vanmarcke peak factor for seismic

ground motions, computed from using the approximate values of 6 and vo ~

The PSDF of ground motions computed according to Step 1 of the proposed

procedure is shown in Fig. 9.4. The structural response transferfunctions for nodes 11 and 18 required in Step 2 of the proposed procedure

9-1

are shown in Figs. 9.5 and 9.6, respectively. The PSDP's of the floorresponse motions at nodes ll and 18 computed according to Step 3 of the

proposed procedure are shown in Figs. 9.7 and 9.8, respectively. The

spectral dispersion parameter 5 for the PSDP's of the floor response

motions at nodes ll and 18 are plotted in Figs. 9.9 and 9.10,

respectively. In each of these figures, the approximate value of 6

computed from Eq. (2.21) and the exact value of 4 computed numericallyfrom Eq. (2.28) are plotted as dashed and solid linea,

respectively.'imilarly,

the approximate and exact values of center frequency fccomputed from Eqs. (2.20) and (2.25), respectively, which equal to vo/2,are plotted in Fig. 9.11 for node ll, and Fig. 9.12 for node 18. The

Modified-Vanmarcke peak factor for floor spectral response computed from

using the approximate and exact values of 6 and 'oo are plottedrespectively as dashed and solid lines in Fig. 9.13 for node ll, and Fig.9.14 for node 18. Following Steps 4 through 6 of=the proposed

procedure, the probabilistic FRS at m+a level computed from using the

approximate values and the exact values of 6 and vo are generated and

shown as dotted lines in Figs. 9.15 and 9.17 for node ll, and Pigs. 9.16

and 9.18 for node 18, respectively.

The deterministic FRS generated from performing a time history analysis ofthe fixed-base structures subjected to the synthetic time history shown inFig. 9.2 are also shown as solid lines in Figs. 9.15 and 9.17 for node ll,and Figs.9.16 and 9.18 for node 18.

It can be seen from Figs. 9.15 and 9.16 that the probabilistic FRS

computed from using the approximate values of 4 and vo compare reasonably

well wi'th the deterministic PRS, except at the spectral peaks and in the

high frequency region where the probabilistic FRS are higher than the

deterministic FRS.

As discussed in Section 8, the accuracy of the floor response spectralvalues at the peaks and in the high frequency region can be improved by

using the exact values of 6 and vo for computing the Modified-Vanmarcke

peak factors in Step 6 of the proposed procedure. The effects of usingA

9-2

the exact values rather than the approximate values of 6 and ~o on the

Modified-Vanmarcke peak factor can be seen in Figs. 9.13 and 9.14 fornodes 11= and 18, respectively. The comparisons in these figures clearlyshow that the Modified-Vanmarcke peak factor using the exact values of

e i hand vo is generally lower near the system frequencies and in the higfrequency region, but is slightly higher elsewhere than that using the

approximate values of 8 and vo. This same trend applies to the FRS as

can be seen from comparing Figs. 9.15 and 9.16 with Figs. 9.17 and 9.18,respectively.

The comparison of the probabilistic FRS generated from using the exact

values of d and vo with the deterministic FRS as shown in Figs. 9.17 and

9.18, shows that the probabilistic FRS remain slightly higher near theH

system frequencies and in the high frequency region than those of the

deterministic FRS. These differences can be attributed to the fact thatthe probabilistic FRS are generated based on treating an ensemble ofseismic ground motions as an equivalent normal stationary random process,

whereas the deterministic FRS are generated from using a single time~ history analysis. Furthermore, these differences could also be

contributed by the fact that the nonstationarity of seismic ground motions

is treated approximately as an equivalent stationary process in the

generation of the probabilistic FRS.

Since the comparisons in Figs. 9.17 and 9.18 show that the probabilistict

FRS at m+a level generated from the prescribed MS at the same confidence

level generally are consistently slightly more conservative than thedeterministic FRS currently used in practice, and since the cost ofgenerating the probabilistic FRS is much less than that by thedeterministic. method, the proposed approach in Section 8 can effectivelybe used to'generate the FRS for use in a preliminary or generic seismic

qualification of equipment and piping in a nuclear power plant.

9-3

10. SUMMARY AND CONCLUSIONS

The procedure for a direct generation of the probabilistic FRS from the

prescribed DRS for seismic ground motions has been developed and- presented in Section 8. This procedure, which can be used to generate

the FRS for any desired level of confidence, makes explicit use of the

PSDF-RS relationship in Section 2.1, which contains two key parameters-the rms response and the peak factor. This relationship has been

systematically evaluated using simulation results for the wide-band

grounround motions and for the narrow-band floor response motions. Various

widely-known peak factors were also included in the evaluation.

Twenty time histories were simulated in Section.4 for each of the three

types of the wide-band ground motions, namely„ the stationary, Types B

and C nonstationary ground motions, considered in the evaluations. These

simulated time histories were generated from the PSDF compatible with theI

2$ damping response spectrum from the USNRC R.G. 1.60. The PSDF s

corresponding to the different damping of the USNRC R. G. 1.60 response

spectra were found to be different; therefore, they cannot be associated

with a single random process. The narrow-band floor response motions

considered in the evaluation were simulated in Section 6 for a typicalnuclear power plant structure subjected to both the wide-band stationaryand Type B nonstationary ground motions.

Eased on the evaluation results of the PSDF-RS relationship in Section 7,t

the major findings can be summarized as follows:

l. For the wide-band stationary ground motion, the response spectra at m

and m~ levels generated from the PSDF-RS relationship, with the use

of the Modified-Vanmarcke peak factor computed from using the

approximate values of 6 and vo, are sufficiently accurate as compared

with those obtained from simulations.

2. For the wide-hand Type B nonstationary ground motion, the findings on

the va3.idity and accuracy of PSDF-RS relationship were similar tothose found for the wide-band stationary ground motion. An equivalent

10-1

stationary duration of this type of ground motions computed from

Eq. (7.1) and used in the PSDF-RS relationship,was about 14 seconds.

For the wide-band, Type C nonstationary ground motion, the m and ne'er

response spectra generated from the PSDF-RS relationship were found

to be too conservative, especially when damping is low, as compared

with the simulation results. An equivalent stationary duration ofthis ground motion computed from Eq. (7.1) and used in the PSDF-RS

relationship, was about 5 seconds.

4. For the narrow-band floor response motions to the wide-band

stationary ground motion input, the PSDF-RS relationship as used forthe vide-band motions can not be used with sufficient accuracy as

compared with the simulation results, due to the inaccuracy of the.

rms response contained in the PSDF-RS relationship. lt was, however,

found that the accuracy can be improved with the use of the rms floorspectral response computed numerically from the floor spectralresponse PSDF. Vith this improvement, the generated floor response

spectra at the m and m+0'evels were found to be reasonably accurate,with slight overestimates at the spectral peaks, as compared to the

simulation results.

5. For the narrow-band floor response motions to the wide-band Type B

nonstationary ground motion input, the findings on the validity and

accuracy of the PSDF-RS relationship are similar to those found forthe narrow-band floor response motions to the vide-band stationaryground motion input.

Based on the above findings, it can be concluded that the procedure

presented in Section 8 can be used for generating .the probabilistic FRS

directly from the prescribed DRS with reasonable accuracy as compared tothe simulation results, except for the values near the floor response

spectral peaks where slight overestimates result. The accuracy of the

floor response spectral values at the peaks and in the high-frequencyregion can, however, be improved by using the exact values of 5 and vocomputed numerically from the floor spectral response PSDF instead of

10-2

using the approximate values of 6 and 'uo for computing the Nodified-Vanmarcke peak factor for the floor spectral response. This was

demonstrated in the application in Section 9. Thus it may be concluded- that, in order to improve the accuracy'f the floor response spectral

values at the peaks and in the high frequency region generated from the

'rocedure in Section 8, different peak factors should be used for the

wide-band ground motions and for the narrow-band floor response motions.

The compariso'ns between the probabilistic FRS and the deterministic FRS

generated from using the conventional time histor'y analysis demonstrated

in Section 9, show that the probabilistic FRS generally are slightly more

conservative than the deterministic FRS, and that the use of the exact

values of 4 and uo gives spectral values at the peaks and in the highfrequency region closer to those of the deterministic FRS than those

generated from using the approximate values of 6 and oo.

The generation of probabilistic FRS using the proposed procedure is found

to be very cost-effective as compared with the use of the deterministicmethod. For the example demonstrated in Section 9, the cost ofprobabilistic FRS is about one fourth of the cost of the deterministicFRS. The cost effectiveness is expected to be even larger for system

with many more degrees-of-freedom. Because of its cost effectivenesss,this procedure can effectively be used for generating FRS for a

preliminary or gen'eric design and qualification purposes. Furthert

refinements of the proposed procedure for generating the probabilisticFRS may be possible. However, such refinements must be baaed upon

further studies on the floor responses to actual seismic ground motions

which are inherently nonstationary.

ll. REFERENCES

1. Newmark, N. M., Blume, J. A., and K.K. Kanpur, "Seismic Design

Spectra for Nuclear Power Plants," ASCE Structural Engineering

Meeting,'an Francisco, April, 1973.

2. Singh, M. P., "Seismic Design Input for Secondary Systems," ASCE,

ST2, February, 1980.

3. Atalik, T.S., "Instructure Spectra from Ground Spectra," Journal ofEarthquake Engineering and Structural Dynamics, to be published in1984

'.

Vanmarcke, E. H., "Structural Response to Earthquakes" in Seismic

Risk and Engineerin Decisions, Lomnitz, C. and E. Rosenblueth,

Editors, Elsevier Publishing Co., Amsterdam, 1976.

5 ~ Grossmayer, R., "A Response-Spectrum Based Probabilistic Design

Method," the Sixth European Conference on Earthquake Engineering,Yugoslavia, September, 1978.

6. Rorno-Organista, M. P., "Soil-Structure Interaction in a Random

Seismic Environment," Ph.D. Dissertation, U.C., Berkeley, January,

1977.

7. Der Kiureghian, A., Sackman, J. L., and B. Nour-Omid, "Dynamic

Analysis of Light Equipment in Structures: Response to StochasticInput," ASCE, EMl, February, 1983.

8. Newmark, N. M., and Rosenblueth, E., Fundamentals of Earth uake

Engineering, Prentice Hall, Inc., 'N. J., 1971.

9.'in, Y. K., Probabilistic Theor of Structural D amies, McGraw-Hill

Book Co., N. Y., 1967.

10. Feller, V., An Introduction to Probability Theory and ItsN. Y. 1968.

11. Crandall,.S. H., "First-Crossing Probabilities of the LinearOscillator," Journal of Sound and Vibration, London, Vol. 12, No.',1970

'2.

Cramer, H., "On the Intersections Between the Trajectories of a

Normal Stationary Stochastic Process and a High Level," Arkiv. Mat.,Vol. 6, 1966.

13. Davenport, A. G., "Note on the Distribution of the Largest Value ofa Random Function with Application to Gust Loading," Proc. Inst.Civ. Eng., Vol. 28, 1964.

14. Vanmarcke, E. H., "Properties of Spectral Moments with Applicationsto Random Vibration," ASCE, EM2, Vo. 98, April, 1972.

15. Vanmarcke, E. H., "On the Distribution of the First-Passage Time forNormal Stationary Random Processes," Jour. Appl. Mech., Vol. 42,March, 1975.

It16. Der Kiureghian, A., "Structural Response to Stationary Excitation,I

ASCE, EM6, December, 1980.

17. Lutes, L.D., Chen, Y.-T. T., and Tzuang, S.H., "First-PassageApproximations for Simple Oscillators," ASCE, EM, December, 1980.

~ I18. "Strong-Notion Earthquake Accelerograms Digitized and Plotted Data,

Vol. II - Corrected Accelerograms and Integrated Ground Velocity and

Displacement Curves, Cal Tech Reports, EERL 71-50, 72-50> 72-51~

72-52~ 74-50'4-53.

11-2

19. Jennings, P. C., Housner, G. W., and Tsai; N. C "Simulated

Earthquake Motions," Earthquake Engineering Research Laboratory, Cal

Tech, April, 1968.

20. Trifunac, M. D., and Brady, A. G., "A Study on the Duration ofStrong Earthquake Ground Motion," Bulletin of the SeismologicalSociety of America, June, 1975-

21. Vanmarcke, E. H., and Lai, S.-S. P., "Strong-Motion Duration ofEarthquakes," MET Publication Ho. R77-16, Dept. of CivilEngineering, July, 1977.

11-5

TABLE 2.1 THE VANthARCKE-EXACT PEAK FACTOR AT m AND m + g LEVELS

FOR 2X DAMPING

rm "m+a p for rm p for rm+a

1

1

1

1

1

1

1

)

2223'2

3

F 000h ~ 5)062 ~ 0 0075550< F 055

10 '0012 F 00013 ~ 50015 ~ QCD16 'i"01 S ~ 00019 ~ SiOG

21 ~ OCO"2 ~ 5CO2h ~ GCC27000030 '0033 F 00036e."00

~ ".0030 '3

~ 3 0 h

hs ~ 5GG229 F 505she 0006ih F 503SaeDGC75 F 55575 ~ 30382 '0090 '0397 ~ 500i"5 ~ . 0)12 ~ 53520 ~ 055'27550035 'O"bu ~ "i"e

as ~ Coo80 ~ CCC"5 ~ CCDloe'0525 F 005v7 ~ 5CG75 ~ "iGCC ~ GDC30 ~ 55075 F 000

~ 23585 01 e'53h io C 0 ~ S397o0 0~ 2holo01 . ~ 5319ooo . ~ Sh06ooo

~ 1727ool5186ho. 1

~ 1971o01~ 2 "she01 ~ 3912ooo. 1'33.C1 .see. 00~ 21CSoo 1 ~ 5703e GO

~ 2251ool ~ 5613eOC~ 2350551 „ ~ 5532e 00~ 234hool ~ 5h59eoo~ 23&22501 ~ 5392oOC~ Ch -0501 ~ 5331oDO~ 2v53ool ~ 5275500~ 22383ool .5223ooC~ 2812551 ~ 5175eDO~ 2538ool 55130e00~ 2586ool ~ 50h9ooo~ 2a28531 ~ h977o00~ 2665eCl ~ halhooo52 701e 01~ 2732oo 1

~ 2 Ialeol~ 2 I."".o01 ~ o 715e GO~ 22I230e 01 ~ 46 )6o Co~ 2857e" 1 ~ h65hooo~ 2897ool ~ hsh3oCD~ 29 "~ 01 ~ hh89 DC."«f: e C 1 ~ h h52o 00~ 29 79o 01 ~ 400102o 0 0~ 301%o01 ~ v367eoo933429eol ~ V322eCG~ 3573ool ~ '2281eoo~ 3ocveol ~ 42vsoCD~ 31235 01 ~ h211e00~ 3103>o" 1 ~ h1832o 00~ 3166ool eh 152ooo~ 3189o51 ~ v126oOC~ 322leol eho'I9eoo~ 3253501 ~ 40375 000328'eVl ehCOCooo533.Bool~ 3332501~ 355".e901~ 3385 01 53873505.3hll""1 .38voeoo

vvao31~ 3h7I3501~ 3515o G1

~ 6312o00~ 6170500~ 6G35eoo ~ Bh13ooD

~ Sh coop~ 2575ool ~ 530ho00

~ Bh2hooo~ 27135 Ol ~ 52 looo~ 2766ool ~ 5290ooo ~ Sh29ooo.2813ool ~ 52al DG ~ 9h32ooo

~ 285ho01 e5292ooo, ~ Sh35o50~ Sh38ooo~ 28905 0 1 ~ 5295e 00~ II hh Oooo~ 29235 0 1 ~ 5298 00

~ 2953o 01 55301oo G

~ 29Soo C 1 ~ 530ho00~ 8hh 2ooo58hhhooo

~ 35C6o 01 ~ 530 7oo0 ~ 9hh6ooo~ 9 hh7eoo~ 3029e Ol ~ 5310e00.

~ 3051501 ~ 5313OOC ~ Shh CO50,3".91 01,5320 00 ~ II451 00~ 312'1 532'". ~ 0453 00

. ~ 3158ool . ~ 5331 00 . „~ Sh55+00.„~ 3186501 ~ 533aooo ~ Sh56ooo.h856o00

~ h805ooo~ v758o00

~ 321 35 01 7534 le oo ~ ~ 8 ISAooo.~ '237o 01 ~ 53h 6o 0 0~ ~"ceo 01 ~ 535 0oo 0~ 329Co Gl e5356e50~ 3318o 01 ~ 536 2o 0 0~ 3351 o G 1 ~ 36 9e 0 0~ 3381 Ol ~ 5375 00~ 3h Ole 01 ~ 537 9oo0

~ 9h59ooo~ Bh6oooo~ Sh62ooo~ 9h63ooo~ Sh6hoOC~ Bh66ooo~ Sh67ooo~ 8468o50~ 3421ool ~ 5383o50

~ !hSCoOl .5389oOG e Sh69o 00~ 3h77ool ~ 539heo0 ~ Bh70ooo~ 350 le C 1 ~ 5394 5 G 0 ~ 8 h 7 lo 00~ 3523501 ~ sh03o00 59h72ooo~ 35hhool ~ sh07oC0 ~ Sh73o05

~ 8h7ho00~ Bh755 50

~ 3563o 01 ~ 5hl 1 o 00~ 3581ool ~ svlsooo

~ Sh75ooo~ 8v77eoo~ 9 h7 Boo 0

~ 3598o 01 ~ 5h18o 0 0~ 36295 01 ~ 542h o GC

~ 3657 01 ~ sh2c 00~ 2Ih7co50~ 682oCl ~ sh33500

~ 39237500~ 3937oof1~ 3916o CO

~ Bhooooo~ 3705 5 01 ~ 5 h3 7o 00~ 37265 Ol ~ Shh leoo ~ BVBlooo~ 37h6551 ~ svv235CQ ~ 8h8]oOO~ 3772501 ~ 5449oCO ~ 8hSJ.ooo

~ Sh83oDD~ 3797e 01 ~ ~ 5 v535 D 0~ 38olooo 538265 01 ~ 5h58o GC ~ Bhoho00~ 37c7 00 ~ 3852oCl ~ 5hf.2 00 Sh85ooo

3123 00~5TA~VH 00 44II2 03

TABLE Z.Z THE LUTES PEAK FACTOR AT m AND m + g LEVELS FOR

ZX DAMPING

rmrna

p for r p for r

3 ~ ODD4 ~ 5006 ~ 0007 ~ 5009 ~ 000

10 ~ 50012 ~ 00013 ~ 50015 ~ 00016 ~ 50018 F 00019 ~ 50921 ~ DCG22 '0024 F 90027 F 00030 F 00033 F 00036 F 90039 '0042 ~ OOG45 ~ Goo49 '0054 F 00060 F 00966 F 0007D ~ SOG75 F 00082 '0090 '9097 F 500

105 ~ 000112 F 500129 ~ GCO127 ~ 500135 F 900150 F 900165 ~ GDO]80 ~ 000195 ~ 9002]3 ~ OGG225eDGD247 ~ 509279 '99300 F 90033O.OOO375 F 000

~ l]S4+D1~ 1425+01~ 1589+ 01e1712ool~ 1S]1+01~ 1894+01~ 1964eol~ 2026+01~ 2080+91e2]24oo]~ 2173+01~ 2213+01~ 2250+01~ 2285+01~ 2317e0]~ 2374+91~ 2425+01~ 247oo01~ 2511+01~ 2549+0]~ 2583+01~ 2615+01~ 2658o9]~ 2697o9]~ 2744+ 01e2785+01~ 2814+01~ 284oool~ 28SD+01~ 2917+Gl~ 2950+Cl~ 2980+01~ 3008+01'3933oo]

~ 3957eo]~ 3089eolAS]2]eo]~ 3157eo]~ 319G+Gl~ 3219e0]~ 324 7+0] .~ 327Qe G].33 Fb. 0]~ 3337eo 1

~ 3374eol~ 3497e91~ 3451+01

~ 5405+ 00 ~ 2857ool ~ 5454+00~ 5363+ 0 0, ~ 291 0+ 01 ~ 5435+00~ 5322eoo ~ 2957e0] e'5419ooo.528]+DO .299a.O] .54O6+OO~ 5241+00 ~ 3035+01 e5396+00~ 5203+00 . ~ 3069ool e'5387ooo~ 5166+ 00 ~ 3100+01 *e5380ooo~ 5131+00 ~ 3128 01 ~ 5374+00~ 5081+ 00 ~ 3166+01 ~ 536 7ooo~ 5034+00 ~ 3200+ 01 ~ 5362+00~ l975o 0 0 ~ 3241+ 01 e'535 7e 00~ 4922e00, ~ 3277+01 ~ 5354+00~ 4884+00 ~ 3302+01 e5353+00~ 4849+00 ~ 3325+0] ~ 5352ooo~ 4794o00 ~ 3360+01 e5352+00~ 4744o 00 ~ 339]o 01 ~ 5353+00~ 4698+00 ~ 3419e 01 ~ 5354+00~ 4 655+ 0 0 ~ 34 4 5 o 01 ~ 5356+0 0~ 4616+00 ~ 3469+01 e5357+00~ 4580+ 00 ~ 349]o 01 e5 359+0 0~ 4546ooo ~ 3512+ 01 ~ 536]ooo~ 4514+ 00 ~ 353]o 01 ~ 5364+ 0 0~ 4456+00 ~ 3566+01 ~ 536Aooo~ 440h+00~ 435Seoo~ 4317ooo

~ 3597+01~ 3625e 01~ 365]e 01

~ 5372eoo~ 5376e00~ 5380+00

~ 4279ooo ~ 3675eG] ~ 5 S IeCO~ 4244+ 00 ~ 3696+ 01 e5388e C 0~ 4197+00 ~ 3726e 01 ~ 5393+00~ 4155eoo~ 41 G5+ 00~ 4062+00~ 4005ooo

~ 3753+01~ 3785e 01~ 3813eol~ 3851+ 01

~ 5398eoo~ 5403+00~ 5408eooc54]5ooo

~ 4'913+00 ~ 1675e Ol ~ S334eoo~ 5233+00 . ~ 1949ool ~ 5522eoo.~ 5383+00 ~ 2127+01 ~ 5589e 00~ 5461ooo ~ 2259+01 e5608+00~ 5502+ 00 ~ 2362+ 01 ~ 5604+ 0 0~ 5520+ 00 ~ 2446 ~ 01 ~ 5592+00~ 5526eoD e2517+01 ~ 5575+00~ 5522+ 0 0 ~ 2578+ 0 1 ~ 555 7+ 0 0~ 5513ooo ~ 2631+01 e'5539+00~ S499+00 ~ 2679+01 ~ 5522+00~ 5483+ 00 ~ 2721o 01 ~ 5506+0 0~ 5465+ 00 ~ 276 G+ 0] ~ 5491+0 0~ 5446+00 ~ 2795o Dl ~ 5478+00~ 5426+00 ~ 2827+01 e5466+00

~ 8391+00~ 8450+00~ 8441+ 00~ 8425+00~ 8411+00~ 840]ooo~ 8395+00~ 839Cooo~ 8388ooo~ 8387+00~ 8386+00~ 8387+00~ 8387+00~ 8389+00~ 8390+00~ S393+ 00~ 8396+00~ Shooe 00~ 8403ooo~ 8407+00~ 841C+00~ 8412+00~ 8416e 00~ 8420ooo~ S424+00~ 8428eoo~ 8431+00~ 8433e 00~ S436+00~ 8439+00~ 8 442ooo~ 8444ooo~ 84 46ooo~ 9448+00~ 8450+00~ 845]ooo~ S454ooo~ 8 456+ 00~ 9458eoo~ 8460+ 00~ 8462+00~ 8463+00edhbS.OO~ 8466ooo~ 846Aeoo~ 847oooo~ 84 72o 00

0

Earthouake Year Recordinn Station Component

TABLE 3.1 EARTH(UAKE ACCELEROGRAHS CONSIDERED

Peak GroundAcceleration,in q Units'l Centro ( Imperial Valley) 1940 El Centro SOOE

S90W.35.21

Ferndale (NW California)

Kern Co.

1951 Ferndale

1952 Taft

S44'~1

N46M

N21ES69E

.'104

.112

,16.18

Kern Co.,

Kern Co.

1952

1952

Hol lywood StorageBasement

Hollywood StoragePE Lot

SOOM

N90M

SOOW

N90E

. 055

.044

.059

.042

Eureka

Eureka (Ferndale)

El. Centro (El Alamo)

San Francisco

Hollister~ ~

El Centro (Borrego Htn)

1954 Eureka

1954 Ferndale

1956 El Centro

1957 Golden Gate Park

1961 Hol 1 i ster

1968 El Centro

Nl 1 W

N79E

N44EN46W

SOOM

S90W

N10ESBOE

S011IN89M

SOOM .

S90W

.17

.26

.159

.201

. 033

.051'08

.10

.065

.179

.130

. 057

El Centro (Lower Califorhiaf 1934 El Centro S0011S90M

.16

.18

Hel ena

Olympia (W. Washington)

Olympia (Puget Sound)

Parkfield

1935 Helena

'949 Olympia

1965 Olympia

1966 Cholame-ShandonNo. 2

SOOM

S901I

N04W

N86E'04E

S86! I

N6SE

.15

.15

.16

.28

.14

.20

.49

0

TABLE,3.1 EARTHQUAKE ACCELEROGRAM CONSIDERED (CONT'D}

Earth uake

Parkfield

Parkfield 1966 Temblor A'2 N65>lS25M

Year Recording Station Comoonent

1966 Cho1 arne-Shandon N051lNo. 5 N85E

Peak GroundAcceleration,in o Units

.35

.43

.27

.35

San Fernando 1971 Pacoima Dam S16ES74W

1.1711,076

San Fernando

San Fernando

San Fernando

San Fernando

1971 V. N, Holiday Inn NOON

(8244 Orion) S90ll

1971 Castaic ORR N2lEN69lg

1971 Bank of California NllE05250 Ventura Bl) N79llbasement

1971 Universal-Sheraton NOOE

(3838 Lankershim) S90Mbasement

.25

.13

.32

.27

.22,15

,17.15

TABLE 3.2 LIST OF 91'FREQUENCIES IN CPS

.06670

. 11800~ 20800~ 3 1300~ 55600

'1.000001.670002.630004 '50007 '9000

13.3000022.70000

,.07140. '12500. 21700.33300.seeoo .

1.050001.820002.780005.000008.33000

14.3000023.80000

.07690

. 13300

.22700

.35700

.625001. 1 10002.000002.940005.260009.09000

15.4000025.00000

.08330

. 14300

.23800

.38500

.667001. 180002.080003. 120005.56000

10.0000016.70000

.09090

. 15400

.25000

. 4 1700

. 714001.250002. 170003.330005.88000

10.5000018.20000

. 10000

. 16700

.26300

.45500

.769001.330002.270003.570006.25000

11. 1000020.00000

. 10500

. 18200

.27800'.50000.83300

1.430002.380003.850006.67000

1 1. 8000020.80000

. 11100

.20000

.29400

.52600~ 90900

1.540002.500004. 17000'7. 14000

12 F 5000021. 70000

TABLE 3.3 MAXIMUMACCELERATION OF SIMULATED STATIONARY GROUND MATRONS

Ground Motion No.Maximum Acceleration,

in Unit

6

10

12

13

14

15

16

17

18

19

20

1. 09

0. 83

0. 86

l. 13

0. 77

1. 04

0 '41.06

0. 86

0. 91

0. 83

0.830.86

0.89

0. 92

0. 92

0.98

1.23

0 '20.88

TABLE 4.1 LIST OF 47 FRE(UENCIES IN CPS

.200+00

. 100+01

. 200+01

. 400+01

. 750+01

. 140+02

.300+00

. 110+01

.220+01,44Q+Qt.800+01. 150+02

.400+00

.120tot

. 240+01

.470+01

.850+01

.165+02

.500+00

. 130+01

.260+01,

.500+01

.900+01,

. 180+02

.600+00

. 140+01

. 280+01

.55040t

. 100+02

.200+02

.700+00

. 150+01

. 300+01

. 600+01

. 1 to+02

.220+02

.800+00

. t60~01

.330iot

.650iot

.t20i02

.250+02

.900<00

. t80+Qt

.360<01

.700+01

.130>02

TABLE 5.1PROPERTIES OF THE STRUCTURAL MODELS OF

THE CONTAINMENT BUILDING AND INTERNALS

(*Concrete Modulus E = 6.9 x 10 ksf, G=2.7 x 10 ksf)5'

Joint Pro erties Member Pro ertiesMass m.

No. ~(ki s)I. x10

(ki -ft )

Location Are Sh Aye Moment o

Joint No. (ft ) (ft ) (ft )

19 20000

1 4600

2 4200

21. 1

9 '

8.5

19 to-.l 1400 700

1 to 2

2to33 to 4

2.8

7 4610

8 3020

9 2470

10 2120

11 190

12 2800

13 2510

14 6290

15 3760

16 8540

17 1220

18 820

4 to 5

05 to 6

T9.4 A 6 to7I5.9 N 7 to 8 990M37E8to9N

1 7 T 9 to 10

500

0.1 10 to ll2.4

1.9

5.0

6.6

12. 6

0.8

0.1

19 to 12 2000 1320

12 to 13 2560 1560N

13 to 14 2210 1460E

14 to 15 1960 730

15 to 16 1740 600L

16 to 17 780 360

17 to 18 190 70

1.9

0.8

0.2

1.2

1.2

1.3

0.9

0.2

0.004

TABLE 5.2 FIXED-BASE MODAL PROPERTIES OF THE

CONTAINMENT BUILDING AND INTERNALS

Mode No. Fre uenc CPS

5.26*

11.95-

16.24¹

17.48¹

Partici atin Factor

30.1

13.5

-13.4

-20.4

Dam in

Notes: * - Frequencies of the fixed-base containment structure

¹ - Frequencies of the fixed-base internal system

TABLE 6.1 MAXIMUM RESPONSE ACCELERATIONS

TO STATIONARY GROUND MOTIONS

ResponseMotion No.

Maximum Acceleration inNode No. 11 Node No. 18

1

2

3

4

5

6

7

8

9

10

ll12

13

14

15

16

17

18

19

20

3. 93

3.68

3.33

3. 79

3.603.89

3.68

3.92

4.02

4.25

4.06

3.52

3.96

3. 41

4. 05

3.67

3.56

3.63

3. 97

4.68

2. 92

2.51

2.81

2.69

3. 24

2. 93

2.97

2.79

3.47

3.18

2.73

2.76

2.76

3.25

2.62

3.94

2. 93

2.69

2.61

2.72

0

ftu. 2.l Srheoattc btatran of lnpllclt Approach fla ~ 2 ~ 2 Scbeaatle btaaran ol KÃPllclt APProach

Davenport

Vaenarcte-ExactDer Klureghlan

DavenportVanoarcte-Exact-Der Klureghlan

Daopfnq ~ 21

~ ~rr

lt Vanaarcte

rc«'4rx ~

4 «

Vatvxarcte

Danp t n'g ~ . 5Xpeeping ~ 2X

~ a

sa )~ E C 5 ~0'FHEOufr>C f~f

~v pl 60'ttEOUErtCY«f

f 'l ~ 2 ~ 2 Analytical Peat factors ilth tlon.fxcrcdance l'robablllty of 5$ fig 2 ~ Analytical peat factors wtth tron.frere<lance prohatiitity of 5>K

Davenport

Vanuarcke-ExactDer Kfureghlan

Davenport

Vanuarcke-Exact

Der Klureghlan

III IIL«1»(Ia.

u

Varviiarcke

crx~ ~I

n'»~ ~0

Vannarcke

Daupfng ~ St u Daxip log ~ .SE

V0'AEOUENCVuf

F lg. 2 ~ 5 Analytical Peak Factors ufth Non.Exceedance Probablllty of 51S

FREOUErrCTef

Flg. 2.6 Analytical peak Factors ufth Mon.Exceedance probabflfty of 8as

DavenportVaiinarcke.Exact

Der Klureghlan

Davenport- Vanuarcke-ExactDer Kfureghfan

II3

~ ~l

rl 3»

Ik

Vannarcke

Darplng ~ 2s

ILVI~ Iic

IX u«» «

Vanuarcke

Danp lng ~ St

l ig, 2.1 aiialrtr<~ I

3 3 ihip 3 3 rvr3~3»f fiEU»lrrCv«f

Peak farpns I»3th fran.lrcricarice Pro liahllltv of hat

'71 p'. ~

0 'I pi 'V 0'HEOUEUCV II1

fl . 2.8 Analytical f'eak factors ulth non-Exceedance Profor fillyof 8ft9 ~ ~

Oasping ~ SX

Os«Ping «St

5

Cx

xxcc

xc an.«x ~

t

.SX

Hodt fled-Yannarcke

Lutes

Hodlf ted-Yatwarcke

———Luies

V 0'FREOVKNCf«Z

Fi9 ~ z.9 nnalytica) peak factors utth Non-Exceedance probability of Syz

kx

FREOllENCy«7

Fig > lO Analytical Peak Factors with lion-Exceedance Probability of Stt

CV

Danpf nq ~ . 5Z

ltI <~ 1 V

J V

~ 4 ~< ~ 4

r ~'I4;II4V<.

CJ

V

5$

4

1.0 gbco

D 4<Q a

CC

4: J<

C:

CJ~

VID

<J

<4V<

4<<4

Oanp ing ~ 05

10 ~ 5 L~v U' I0'AEOUEtlCTICPSI

f ig. ). I 'ltu IISIRII. II.G. I.t»t Vc Slv ns I Sl ctra

0'FREOUEltCT ICPSI

Fig. 3.2 The s and n<o Response Spectra of 45 Recorded Selsnlc Ground itotlons

~ J

~4

Co

D4'IJ

WCCt. 4

KIc

4

~ .vi D

V'<oDanplng v "T

CJ

~ ~~~ZV<

CX 4r <V

IL

4-u<r

IJVl

Danp I ng ~ 5T

'

L)

0 I I '~"E I I T)tj ~ ~ ~~0 $~: j-CIj1

I nl Out UC I IEPSI

I t<I. 3,1 Ihu n an I ~« I R< sou<<so 5I» ctra of 45 peto«l<'d <'Isntc rrnuud ts<t«<ns

0 4 < 'i~« I 4144g'AEOUTIICIICPSI

Fig. 3.4 The n and I<In Response Spectra of 45 Recorded 5«tsntc O<nuud tt<tinnS

VI~U0

4IJ»Iu

~ 4C »»

4

Vl

W

4

2S

Oaeplng ~ OX UtD I»

UI~»

Z I»

IC

LIvla.ul

»I4

0'FREOUEIICf ICPSI

fig. J.o COf of kespunte Spectra ot 46 hecorded Seismic Grtw

0'FRFOUENCf ICPSI

Ol

Flq. 3.6 PSOF of 4S Recorded Selsalc Ground tiotfons

»I

III

»I

~ ~

Oanp lnq ~ SS

2X

~»~ 1IC

VII44

elu

D ulI ~

D ln~K

Sf

Oautlitui » 6

?.

I~~c'I ~

Ik

L r»

IC~ IVDUIuVlI

ul

I" r~ I' .D'HEOUCtICI ICPSI

. I. 1 I",';f Gill r.ltelt frur tte I.".ltec R.', I.fo I'espunse Spectra Using tt» per Klurethian I'eak Facturfig

FAEOUENCT ICPSI

fl'i. 3.tt PSUf rwnerated Iron the Usttec R.G. I.60 Responte SI Intra Using Ihe I ~IIIIIId thin» trite feat laslnl.

'

Oauplng n .SS

~'

O'Z~

I. ~

s

cu~ ~

C

~ ILC

~ s

u

LCcu c

C

ccrrn

cu

IJn~nr

~C

~ ~

' ~~ ~ C ~ C ~ ~ ~ ~ ~ ~ ~ ~ ~ LC ~%. ~ CL~

Iirt ln StlacO

to

0 c CC0'

4EOUEIICt ICPSI—

fig. h.l A typical Stationary Ground Notion fig. h.2 Response Spectra of the Stationary Ground llotlon In fig. h.t

Ck

LLCCC 0'

~ Cr n

ru:LC

~n.a>

~A

I <0

Oanplng n,SS

Cu

W»CCW

XCKr u

r~L

0~cir

VC

nao

Oarpln9 n ES

C ~

lli' > E <> S>hu'HI,OUtlIC'I ICPSI

f Cg. ~ .3 Sl~latt-I keaiuncce Sin CICa Of Stattanary rrOund IaltlunS

f4EOUEIICI ICPSI

f19 ~ h ~ h Slacllated Rrcnonse SpeCtra of Stationary Ground ISction,

0

0'

Cl~bne

ZCCgt.IC

LJnVl

C

~>

V

~ 4 o

Danpt ng ~ SS

25

Doap lng ~ ~ SS

0PNEOUENCT ICPSI

FIIEOUENCf ICPSI0'ig

~ 4 6 Slee>lated Rcspnnte Spectre Of Sthtlonerr Ground Hotionsfig. 4.6 COY of Slauleted Response Spectre Of Stettonery Ground Notlon5

~0Ul

~5VI

neap t ng ~ 2!

~IACLLI u

EP u7

8»~I

CL'J

~0

Anelytlcot g ~5loula ted

04>>ping ~ 25

0'HEOUENC I ICPSI

0>

I >9. 4,2 ',;c st>el H n N>ts of Reslc>nse Isut tons In stationery Ground Hot lnns

II 0'0'IIEOUENCfICPSI

fig. 4.8 Spectral Center frequencies of Response Hot>on5 tn Stat>nn>ry G>nund It>tines

~-

L j

11

ty(sec.) t3tsec.)

15 30 .099

18 . 268 I,~

I.~

E<t)

O, ~

I

'IP~ v

~C

I.O

0t3

I '

~ . ~ ~. ~ ILO Ioa Io.~~ stat III SICOIO

~O.O

fig. 4.I3 Envelopes for Types 8 Ond C Ground flattensr

fig. 4.14 8 Typical Type 8 Ground IIotton

u

~O

04np lng ~ . SS

~O

u ~

~ ~ ~

~ I

Ir~ ISraOI r ~

W

OI0 0Q rr

ZIa uIrOr Or

ICrv

hr~ .OI~

C1

~O

W

aoo

Donp Ing ~ . 51

I' i 3 . ! r3T33~g3rTT333vf fitOUI IICI ICI'SI

I 19 4, IS f33 .IOOIsr rrlvlIra uf tin [yla 0 I33uund A 3tlun In Flg. 4.I4

'O 0f AtOUEIICT ICPSI

f Ig r lG Sf~lot d POSIO3 ISe Si+ctrJ of ly

tt t o

paopfng ~ 21

co

hhtt3~h

Za rKr hh

IC

LJhu4~n

Nh

ato

Panp Ing ~ ST

h0u

ct

u0'AEOUCNCTICPSI

Ff9. d.ll Sfnulated Response Spectra of Type 8 Ground Notions

0'IICOUENCTICPSI

Fi9. e.lB Sinulated Response Spectra of Type 8 riround Ilotlons

t ~

t. ~

ptnp lng ~ . ST P n ~

cfI

~'J

SX

~ n ~

~t. ~

ph 'Tii 0FNCOVtNCl ICPSI

fig. l.tg (OT of Sttnllated ResonnSe SI<Ctra Of lyly 8 Ground Ifutfonl

~ . ~, ~ ~ , ~ ~ . ~ '«4 ~ ~ r. ~ ~ ~ Iu h I ~ . ~

Ilic Iu sICOOO

I lg. O.To A fyplcal Type C Ground llotlon

Oaaplng ~ SS

LS-bLl~u a

a~a aIK4- r4

I+I4elr

L1

III 4ea a~ I

aLO uIIe rgIe« IrLI4uI ~

auI

~ 4 o

Oanplng 4 .St

'n

0FIIEUUEIIcf lcPSI

fig. 4.2l Response Spectra of the type C Ground llotlon ln fig~ 4 ~ 20

d'FIIEOUEIlcf ICPSI

fig. 4.22 Siuulated Response Spectra of lype C Ornund Notions

~4

Ll ~rl a~ ~

a~4 aIL

K a.I

Ic'

~ ~441 rl

aW

u le

Oaap Ih5 ~

«buler ~Q a« lQ aIeKI: a4

Ll ~III

O

Iaa

1 ra

Paap Ing ~ SZ

I ~

II 0''Ht.OUtUC1 ICVSI

I I'I. 4 ~ 2J Slaulated VI Slnrlte Slactra ul lyla C IIruund HutlunS

0 ~ 'I~

0'UEOUEIICIICVSI

F lg. 4.24 Slnulalrd Pesswnte Sluvl tra ol lylv C rco I 4 III I II I i

~

'anpi' . SX

u u . 0'REOUEl(CT ICPSl

fly. 4.2S COY of Slnulated l(oslenSe Spectra of type C Ground Muttons

El.207'l.

198.5'0El 184 ''

Contain@:entStructure

El. 165. 3'

El. 143. 8'

El. 123. 8 '

El. 103. 8'

El. 83. 8'

El 63. 8'

El. 43. 8,'r

El. 23.5> 1

El. 0base

c~ k

Internals ~

62'

II

IIII

181IIII

17'

16

1514

1312

3 50

El. 93

El.61'l.

49

El. 33.5'l.

22'l-

13'l.

8~

kx = 5.17

c = 9.67X

k = 1.7'7

c, = 8.14

x 10 kip/ft HI

x 10 kip-sec/4

x 10 kip-ft10

x 10 kip-sec- ft7

R=65'9cx k

X

Fig. 5.1 Soil-Structure System of the PNR Containment and Its Internal System

O

CU

OOCOM

UJ OO ODa:I- III

O

CC OOO

C>III

I

ZoDOU ~tEUJuu)OZO~AJI

Node ll

OOCO

Node l

OO

CIOO

0I

FREOUENCY tCPS)

Fig. 5.2 Structural Response Transfer Functions

~ ~

~ . ~

La

I.~

a. ~

IIr

a I ~

l(tiI

bCK

uuC

I.~ ~

~ O ~~ ~ ~, ~ «O ~ ~ ~ . ~ I~ . ~ nn ~ I«O ~O. ~

Iloc III stcaoo

LO.O. ~ «O ~ . ~ ~. ~ ~O. ~ I~ . ~ I«~ ~ O, ~

tint Ir SICaoO

fig. 6.1 A fyptcal Response itntlon at 'lode 11 to the Stationary Ground isationfig. 6.2 A Typical Response Notion at Node 18 to Ihe Stationary Ground Notion

O'

O

Oanp ing ~ . St

Oaop ing ~ . Sz

Qco~ «

Oa V

'g rs

n~ OVuiVIg

I~O

V

SC

0'I V TrsfRCOUI HCT ICPSI

I Ig. I,.1 fl,r I'Isii se st<cti ~ of tt . I'".lance ltutlon ot ta Ie Il in Iig, 6.1

O~ u Ot O' K1'T| O

fHI.OUCIICS ICPSI

fig. 6.a floor Response spectra of the R»IIIonsr l4tiun at,ttud» 18 In fig. 6.2

~i

aJna

ZECg ~

IC

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~ia

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aV

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"3'0 0'FREOUENCT ICPSI

0'RGOUENC1 ICPS)

lip. 6.5 Simulated floor Response Spectra of Response IK<tlons at Node ll to Statioiary Ground!stlons Fig. 6.6 SiswIatcd Floor Response Spectra of Response Notions at Rale ll to Stationary Ground Notions

~i

L t>4

<.'i a

VoaECzZ a<Lv~4<avlCi

aDanp log ~ SX

<o

ri a

<o aacraIK

AwVic

V

Danplng - .5%

fl<IOUFNC'I ICI'SIOi

CIO'I V V

fIltOUf NC1 ICPSI

Ii<1. 6.6 sieutatrd floor Kvsponse spectra of Resin<use ls)tip<<5 J( N <do 18 4< slallnnary Ground NotionsI il. b.f 5«a<l ~ I< I I 1<v<i K< s<«nsv clast ~ r ui pespnnse is<I lens ~ I,'avle ll to slat<<wiry Gro«nd It<lions

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~ ~ o

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~I

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0' 0' 0'u0 I

FNEOUC'NCT ICPSl fftCOUENCT (CPSS

Fig. 6.9 sse sla.ed Floor Response spectra of Response Hotions at ttode 18 to stationary Ground Hot tons Fig. 6.10 Sinu)aled Floor Response spectra of Response Notions at ttode 18 to stationary Ground Notions

U~t

Oaap1ng ~ . St

6 ~n

~n~I~

St

4—i--I~-rrlil~r~~rllr~~rrrrrr

fntoutnct ters>

: ~ Sp stra at tlude 11 ivr Stationary fruund Notions1 lg, 6.11 (Ny ~ 1 Irvlate~l

flu�>r

prstuvlse ptc fa ~

v lOfntontNCT (CPSI

fig. 6.lf CUV of Siaulated Floor Response Sprctra at thvtr 18 fur Stationary Ground Nutinns

«<

~~ b~ <

~<

Mode ll

Mode I8~ <

«<

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Danp log ~ 2'!

D~

<I

FACQUCIICF ICPSI

fig. 6.13 pSOf of ttesponse Hotlons to Stationary Ground Motions

0 ~uT a 7 i l IFFAEOUCNCS ICPSI

Fig. 6.ld Spectral Moments of Response Hotlons at Mode ll to Stationary Ground Motions

Slnulated

ubu~<S <<

CC

Anafytfcalz

Slnulated

Analytical

Ku

ler« Qa«<ping ~ 2

~0Danp ing ~ 2t

t<1 ~i<TITrr \ rj iTTI I~T<

I AfOut UC I ICPSI

O0' M~ft'~v 1 ET'y

FAtoutllcf ICPSI

t Ig, I< ls sIN<tlat Ir Irr I ~ I i«wi »s nl kusl <nse a II s al 'a<de ll to stationary r<round Is tlons I<i I~ ~ I Ill.I+rut««pu<~ I< r «I kitlts<~ <l< Is<i I < ~'I <1il'l I ~ Ial l»wry I<nun l ts<ll<u<$

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eV

C7

EC~i

Analytical

6 lou lated

Oaup tug ~ 21

Oauplng ~ 2T

FIIEOUEHC'I ICPSI0'F

0'IIEOUEHCTICPSI

fig. 6.lf Spectral Hnnents of Responle Itotlons at Node 18 to Stationary Ground Ibtlons fig. 6.18 Spectral Center Frequencies of Response Iiottons at Node 18 to Stationary Ground .'lotions

Slnula ted

Analytical XIC

LJcc

C~sl ~4

Had

Oanp lng ~ . 5

Oanp ing ~ 2T

TlllT( !~1f NEOutHC I ICPSI

6.18 Spurt ~ al toilers«wi I'ara ~ Irr 6 oi pr!ponce ltut>uns *t Huile 18 to Stationary Iirounl Notions

00' 0

FAEOUEIICT ICI'Sl

I lg. 6iZU Sinuiat«Peat faltnrS Ius flour Praliiunr ut H>dr II ln Stat>nn)rr INnunl ltutlnns

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LIC

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s

O

0 ~

fAEOUEIICT ICPSI

fig. 6.21 Slnulated Peak factors for floor Responses at Ilode ll to Stationary Grolmd lotions

V Ol

0'REOUEIICYICPSI

fig. 6.22 Slnulated Peak factors for floor Responses at Ilode 11 to Stationary Ground Hotlons

a:

LJIZ

Ir 0~Ua.

Ia a 4

Io

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Ie lnW»l0

aao

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Oanp lng ~ . 5S

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I lg. 6.2j SI» Ilatel I'i»I lastuls Ilr I lour vl slenses at Ilode III to stationary rlround Illtluns

C)

0 v 0 V 0'fAEOUENCY ICPSI

fig. 6.2a Sinulalld viva lartuls Iur I ts»ir 4"Iv»lsl's»t I>»lu III tu Statlnnlly Gruu»J 'title»ls

If

I

I

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St

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0 40'REOUKNCTICPSI

f>g. (.89 Iluor Pesponse»uttra uf tl» Response Itotlon at Mode 18 In fig. 6.2I

D4 0'

FRCOVCI>CI ICPSI

fig. 6.3) sinulatcd Floor Response spect>a of Response Itotions at 'fode ll to Type 8 I'round Motions

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0'RCOutt>C'IICPSI

I >9, L. Il «i~ >fat»f I four »' ms ~ fust>J «>I I'«".I«r>. IS t>lr>5 4> tt>rfr II Iu ly>4 t F>round Motions I >g. 6.)z si~lai«4I Flour I>«'sf>«>use sf»tl>'J uf kesf>on>e Is>i>«>ns Jt t>«>d«'l lu Ivl» 8 > >uu»d ts>tions

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r)

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nl v

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b

D~ v

VC ~KK

1V4.

4 ~VI ~

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0 0''AEOUENCt ICPSI

I lg. 6.33 lnul ~ teJ floor kespvnse spectra of aesponse lotions at Node 18 to type 8 Ground Ifvtfons

C1 0''0'REOVENCIICPSI

3Fig. 6.3l Slnulated floor kesponse 5pectra ol kespunse Ilotions 4\»de l8 tn fype 8 Ground I4tfons

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4

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nlv

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sO cciTrjj~ a'HI 01lt IIC I ICI'SI

Ilg. 6,I5 Snatlatru 1 luvr krstrnnv 5141.tra ul kv'sinlnse l4tlons at NuJe IO tn fyle 8 Orovvl 14tlons

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I.

0

4

L

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———Analytical

D 'bW

Z~I~

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Z Ie

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0-Vlf

V1

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n>o

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5 lnula ted—Analytical

DE

FREOUENCT ICPSI

fig. 1.5 Conparlson of the Analytical with the Sinulated Response Spectraetre of 5't&tlonary Ground Notions

0''0'FIIEOUENCTICPSI

fig. 7.6 Conparlson of the Analytical with the Sinulated Response Spectra of Stationary Ground Notions

D

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ca~aI4

V

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5 lnula ted

——— Analytical

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Sinu la ted

Analytical

D

I ig

cf

I NEDUENcv scpsi

1.1 Cunparison o tl» a 1 IcaI - An I t I ith the Sinulated pwsirIIne bfwstra nf Stat lIuIaI y ( ~ <un~i ItotiInIs

0flllOULNCS ICPSI

O' 3~ +

F lg. 1.8 Conparlson of the Analyllcal with Ihe Slnulated ResfuuI~e Spectra uf lype 8 GIuund:4IIIuis

0

0

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IO

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Analytical

5 ioui a ted

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CL

0~ 0'

FIIEOUENCT ICPSI

fig. 1.13 Cocparison of the Analytical «tth the slnulated Response Spectra of fype C Orovnd Iiotlons

br

fig.

p ~ II p< 0FREOuthCT ICPSI

7,14 ~ 4 C<ns<artsnn Of thc Analytical « 1th the Stnutatcd Ir<IS flou< Sl'CCtl' 1 Kcslu<nse at ttudc 11 loStast<mary Gr<wnd ls<t lnns

4M

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rr~I

C

b~ 4r

4

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Analytical

- Sinulated

IV ba<~I 4C

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nt 4'anping~ SS

Analytical

Slnutated

'4

Mr

tf v 0' 1 fibI « I Ql<i CC I <C<'Sl

na rt I.<l I<lth thr 5<nutat< d NS 1 lOur rostral Kr lr<r C at th„ie 11

tn lal louafy rr<ound IQ tl<n5

p ~

fig. f.la.c

II 0' ~rr<T rFIIEO<lt NCT ICPSI

E<uS<aris<m of thc Analytic~ 1 4<th the Sirwtated ISls f lnur '< et<a l KI slnmse at <lode 11

tn Stat lnnrry l>rO<md I<otinns

'h

Oanp Ing ~ . ST

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Sinu la ted~'lnulated

0~ u 0'S

FREOUEHCT ICPSI

fig, I.lJ-a A Conparlson of the Analytical sslth the SI4utated RHS Floor Spectral Response at Node 18

to Statlo«ary Ground Hotions

0''FREOUEHCT ICPSI

fig. I.IS.I Cueetearltnn OF the AnalytlCal Vlth the Sluulated NHS flOOr SpeCtral ReSpOnae at IIOde 18

to Stdliondry Ground Hotions

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sc

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v

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I ~ I Ji ISslsr/ GI ~ I I I led e

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01

FREOIII IICT ICPSI

tise Asedirtirdl vsth ihs'suesldts'll I' 'Ji Ide Iles s Ius' Ieeees'e'Sienese ILIIsus \ OI Ilvie I I

(suuul Set sunS

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ll 1\6

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Danp ing ~ SX

S inu Ia ted

—Analytical

0''4EOUEHCY ICPSI

fig. 1.11 Co«parison of the Analytical «lth the Sinulatcd Peat Factors for floor Response HotlonS atIavte ll to Stationary Ground Hotions

0'C0''HEOUEHCVICPSI

fig. F.la conparlson of the Analytical «Ith the sinulated peat factors for floor Response Hotlons atRode ll to Stationary Ground Ilotions

CCIEJcc

CC a~Un,

neo

CC

LICXll 'taYCC aVC an.

had

DC«ping ~ . StSinula ted

lAnalytical Danp lng ~ CX

Slnulated

Analytical

~»~| 0'Infouthcs ICI SI

fli. 1 Iy Icvt ac Cola Of II» Acwlytccal vctn tne Sin«I ~ ted Prat Iactors for floor kesf»»cse Hotions atIC«hr lh tu Statlullary rirlWnd IsitsunS

0 l V 0'0''IIEOUEIICfICPSI

fig. I~ 10 Contlarlson of the Analytical «iih Ihe sinu'lated Peas lactocs Inr I luor krslense Hotions atI!c»Ie IU tu Slatcc»cary crccuc»I II»tin ls

0

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———Analytical

l 0

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b0

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HO 0'FRCOllfllCT ICPSI

Fig. ).RI ccnsurison o( t >e Ana( tica\»Ith the sinulated peak Factors for Floor Response Hot(ons at'k»te 18 to Slat(a>.ary Or>a>nd Il>tin»s

Fig. 1.22 Coaparison of the Analytical uith the Slnulated floor Response Spectra of Response Not(ons at(rode 11 to Stationary Ground Notions

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au>b

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Sir»>la(gal

Analytical

ol 0>

FRcov((res (cps) fR(OllERC I ICPSl

u(th the shn>lated Floor Resi>ons» Spectra o( I'eslurnse Isotions at(lotionsI > I 1.21 C>»cu> is> n > I lr». Analyt>cal >>cth t»e S>r>ulated I lour krsponle Spectra ol krsi>nnle lk>lions at fig. l.tl Coaiurison Of lhe A»alytltal

h>J> II tu stat>r»u>y o»»und Is>tlclns 'lode II to Stationary f>round

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——- Analytical

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Analytical

0 ~ 4 40''AEOUEIICTICPSI

f fg. 2.25 c<y<partson of the fy<alytfcal «Ith the sleulated Floor Response Spectra of Response Hotlons atNode 13 to Stationary Ground Notions

0'FAEOUENC1 ICPSI

Fl<f. 7.26 Coeparlson of the Analytical «1th the Sfeulated Floor Response Spectra of Response Notions atRode 18 to Stationary Ground Hotlons

e

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O4~rrr ~

sya.

'b

VIOaep in<2 ~ SS

4 I I4

Analytical

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Z 4<cZr

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M

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Oanp i' . 5S5 ieul a I~I

———Analytical

00'

IIEO<ll<<C1 ICPSI0 ~ rrr<T<t< I ~ rllT t<

I l<f OUEI<C1 ICPSI

fle 1.2g I<nq Iri;On of Ihr A<~I Ii<41 4<th 1<4< Sluutatef floor kes14<nSe Slectr I Of keslu<nte Hotlnns at

Nni<'8 Io SI ~ Iloo4<y riy<w<vI IS<I<on'I

fly. 7.28 Co«turison nf tho Analytical <lith the Sin<liat<el I lour krs<44<'<'in'LI<a of kvslwn<s< I'Lit<oui .<I

k<4k. 11 Io lyi<u 8 Gro Inl Ik<t <onS

C1~&

Z u4 ~r'r

orLJUSn

4 la

Oanping ~ 2S

\Sinulated

———Analytical

tuLI

U PO~ v

a o,

A

LI

u1 ~

~ au

0anp Ing ~ 55 Inul a ted

—Analytical

4 4 0'fFRE'0UCMCT ICPSI

fig. 1.2g conparlson of the Analytical lrit.l the Sinulated floor Response Spectlsl of Response Hotlons

at Node ll to lype 8 riround Notions

40'IICOUEHC'IICPSI

fig. 1.30 Conpartson of the Analytical Pith the Slnulated floor Response Spectra of ResponseNotions at Node ll to Type 8 Ground Hotlons

~5 ~0

E7 ~V

Z ua ~r

o

Vl

/z<

4 >0

0anp ing ~ . SSSinulated

Analytical

U ~O

Zr ear4

u1»

w 0anping ~ 2S

Stnulated

Analytical

~ 1

0 ' V 0' fiT UfF ni out IIC I IfPSI

fig. 1.1l to surlson oi tip adalytlcal vitn the sinulatrd floor Reslense Spectate of ResponseHulioos at Issle IU to lyp. 8 Ground Hutions

01F4COUIIICT ICPSI

fig. 1.32 Conparison of the Analytical with the Sluulatrd floor Aspnnse Spectra of Response lbtions atlinda III to lype 8 Ground HutiOnS

~0

s

00

n~0

2 rK

a~LI

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Oanpsn5 ~ SC

5 larval ar&

Analytfcal

0 II d D'

IIEOIICIICS ICI'5)

fly 3.33 Ccnoarfcen Of II+ nnalytleal ~ith tne Slnulated ReSPOnSe SPCCtra Of ReSPOnte IIOtlenS atIIodc IC to lyIe 8 Ground lsotions

PSDF SDOF PSDF

PSDF

Fig. 8.1 Schematic Diagram of the Approach Adopted in the Computer Program PROSPEC

GAOUNO AESPONSE SPECTAUN

4 Ilrll ~cuuu Inc ~,454

I4.4 ~

iclN:c Lc

I~ ~

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~

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4~ I

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~ I~ I I. ~ I~ . ~

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4 I~ . IS ~lint ISCCI74,

fi9. 9.l Cunpurtson ol the Celculetcd Spcctna of the Synthetic Tb»lllstorr ln f Iq. 9.2 ulth the Gcsl9n Stu ctrm tron theUS'llC R.l;. l.69.

y thetlc Rcceleretfon line History

GAOUHO PEAK PACTOA

~50 -:IAGAOUNO PONEA SPECTAAL OENSlfy

E

5

5

ML'

kuulnc ~ CISrkovCNCI Crs

ful. 9.) lhc Itudifled YII455rttc Pent fICtor for Gro»5d ISItlonwith Ihe Ilse Of thc'PPrOIfuIte YJlues Of 0 end Vo

fl9. 9.0 PSllf of the Ground Notion

TRANSfER fUNC'TIOH RT NRSS POINT II TRAHSFER FUNCTION AT NR55 POINT IO

~ ~

~ 0

I0

8

UR

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OlIICOvCKI Cte

IKwcncl cts

f19. 9.S Structural Transfer function at!lode ll f19. 9.6 Structural 'transfer function ~ t Rode 18

FLOOR I'ONER SPECTRAL OEHSITT AT NASS FOIHT II fLOOR FOHER SPECTRAL DENSITT RT NASS POINT IO

b

u b

'b

g b

pl~ ICIWlucl Cts

o'acetstKI Cts

f I9. 9.1 PSOF of floor Response Stetson at Rode ll f19. 9.8 psDf of floor Response station at Rode 18

o'

EnactEaact

Approaluate ~ Approllnate

a

Ounplny ~ 2C

Oaup ln9 ~ 2$

0 u 0' 0'IIEOUCIICT ICPSI

FI9. 9.9 CoNurlsun letueen Ihe Caact and AIqroainale Values of d for Floor Response IIotton at blade ll

ca

0 u 0' 0'ffIIEOUENC1 ICPSI

f19. 9.10 Couparlson betueen the Coact and Approslwle Values of d for floor Pasponse IIotlon at bode lb

o4 ~C rM

Erect

+ Approslna'te

VlCEJ w

rExact

~ ~Approalnate

Oaupln9 ~ 2Z

Oaupln9 ~ 2C

IffREOUENCY ICPSI

Appro Iuate Values of Fc for floor Response slots

rcs

FIIEOUEIICS ICPSI

V lw s or I Ior Floor ResIonse Isation ~ t Ilnde 18C+%«ISOn 8"Inc~ the Caaet and Appruainate V~ W S O

FLOOR RESPONSE SPECTBUN 4T NASS POINT IIr itrtx .aio0+vlxc ~ .aao

FLOOII llE5PONSE SPECTIIUH Al H455 POINT IO

> xtrxx axooavixa .eao

~I

ic Probab ll

CUB

<ns

Ox

rxcoocncx crs

i xd. 9.ll fonparison Betxeen the Orternlnlstlc fRS and the Probablllstlc FRS Generated fron Usln9the lxaCt faint Of the 'Mlfled-Yannareke Peak faetar fOr ROde 'll

~ xtovtlc'I crs

fl9. 9.18 Conparlson Betneen the Oeternlntstlc fRS and the Probablllstlc fRS Generated fron Usin9

the fxact Valve of the itodlfled-Yannarcke Peak factor for node 18

(

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