ML19253C989

Forwards Verification Documents for in-house Computer Programs Sconv,Spectra,Quake & Time,In Support of Response to NRC Question 131.63

ML19253C989
Person / Time
Site:	05000584
Issue date:	11/30/1979
From:	Prieto R GIBBS & HILL, INC. (SUBS. OF DRAVO CORP.)
To:	NRC OFFICE OF STANDARDS DEVELOPMENT
References
LGH-NRC-70, NUDOCS 7912120474
Download: ML19253C989 (53)

Text

.*

Gibbs S Hill. Inc.

E NGIN E E RS DESIGNERS CONSTRUCTORS OtRECT DIAL EXTEN$lCN ai23 Teo- 5167 November 30, 1979 LGH-NRC-70 File: 5.1.4 U.S. Nuclear Regulatory Commission Standardization Branch-Washington, D.C. 20555

Subject:

GIBBSSAR (STN-50-584),

Computer Verification Documents Gentlemen:

Enclosed are the verification documents for Gibbs & Hill, Inc.,

in-house computer programs SCONV, SPECTA, QUAKE, and TIME for your review. This additional information supports our response to NRC Question 131.63.

If you have any questions or comments concerning this letter, we will be pleased to meet with you at your convenience.

Sincerely yours, GIBBS & HILL, INC.

/

/

RP:lm Robert Prieto encs. GIBBSSAR Assistant Project Manager cc: J. Conran U.S. Nuclear Regulatory Standardization Branch Commission Washington, D.C.

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0 VERIFICATION OF SCONY COMPUTER PROGRAM 022 1541

VERIFICATION OF COMPUTER PROGRAM SCONV I. PROGRAM FUNCTION SCONV uses modal analysis time history method to determine the transient response. of a linear elastic system subjected to support time history excitation.

The convolution integration is used to obtain the analytical results. The input information of SCONV program includes the support time history excitation and the free vibration characteristics of the system, such as frequencies, mode shapes, weighted damping and participation factors, obtained by program QUAKE.

Step-by-step integration by Simpson's rule :is used. The theoretical part of this program is presented in Appendix A. .

II METHOD OF VERIFICATION Continuous System Modeling Program (CSMP) , which is an IBM program in the public domain and is independent of SCONV, is used to verify GGH proprietary program SCONV.

CSMP uses the fourth order Runge-Kutta integration method with variable integration interval and the Simpon's rule for error estimate. The results from both programs compared to establish the validity of SCONV.

-1 -

III. COMPARISON OF RESULTS The dynamic analysis model of an auxiliary building of a PWR nuclear power plant shown in Figure 1 is used to verify SCONV. This mcdel, which consists of five lumped masses or 30 dynamic degrees of freedom (Table 1) , was analyzed by using both SCONV and CSMP for the same ground excitation as shown in Figure 2.

The absolute floor acceleration time history and the corresponding ficor respcnse. spectra for the mass number 1, which produced most sensitive and critical respcnse, are used for comparison. These results correspond to acceleration component in X direction due to Safe Shutdown Earthquake . in X direction for the dynamic model with upper bound soil spring.

Figure 3 and Figure 4 are. the absolute floor acceleration time histories obtained from SCONV and CSMP respectively. These time histories derived from both programs are almost identical. The equignent damping for the floor response Spectra shown in Figure 5 is 1 percent of the critical damping.

Plotted in Figure 5 are the : floor response spectra case considered. It is obvious that the difference between two floor response spectra developed from SCONV and CSMP is negligibly small.

-2 -

1541 024

T ABLE 1 DEGREE OF FREEDOM OF DYNAMIC MODEL MASS POINT DEGREE OF FREEDOM 1 Translation X 2 Translation Y 3 Translation Z 1 4 Rotation 6x 5 Rotation ey 6 Rotation Oz 7 Translation X 8 Translation Y 9 Translation Z 2 10 Rotation ex 11 Rotation ey 12 Rotation Oz 13 Translation X 14 Translation Y 15 Translation Z 3 16 Rotation ex 17 Rotation ey 18 Rotation Oz

-3 -

1541 025

4

?ABLE 1 DEGREE OF FREEDOM OF DYNAMIC MODEL (Continued)

MASS POINT DEGREE OF FREEDOM 19 Translation X 20 Translation Y 21 Translation Z 4 22 Rotation ex 23 Rotation Oy 24 Rotation Oz 25 Transla tion X 26 Translation Y 27 Translation Z 5 28 Rotation ex 29 Rotation ey 30 Rotation Oz 1541 026

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Figure 1. Dynamic Analysis Model 1541 027

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hO 2 y APPENDIX A ' d Computer Progra= (SCONV)

This program uses cenvolutien int egration to solve a general dynamic prcblem f or a, linear elastic system with the support,, time history excitation Esco where R is the scaling f actor and S ct) time dependent support a ccelera tion. FDr any linear elastic system, its governing equation f or motiens in matrix f orm

[ m] (u) + [ c ](u) + [k](u) = - [ m] (D} Rs (t) (;)

can be decoupled into a equations :

system of second crder differential

.. . 2 ..

(q ] + 2[p][w](q} + [u] ](q) = - (T) Rs (2) where (q) is in nor .al coordinate, that is:

(u) = [$] (q) and [p ], [w], (T} , (D) and [@] = the matrixes of ecefficients of critical da . ping, circular frequencies, participation factors, direc-icn vec or, and eigenvecters , re spec .ively.

The general solution of Iguatica (2) for any system is:

q =C Tn A n (t) n (t) g 2 n (t) (3) n p c) t[q +pyq nn 0 nno where C =e _ _

sin W t + q cos W t n(t)

( y n

n 0 n 2

(t q) -p 4) (t - ?)

D nn ..

A =

n (t) -

-- e Rs sin (C (; - t ) } d; (4) 0 g (t) n n

n 1541 032

d 2 2

=

(T) n k) n 1-An qnte; is and q, are the the relative initial velocity displacement of nth mode at time t and q, and displacement.

Differentiating Ege tion (2} once and twice, the relative velocity and accel.eration are obtained.

q =-Aec +D + T_n gmA

_, -3 (5) n (t) ^ n n n (t) n (t) g2 n n n (t) n (t) n ,

(- 2 22) q =. u) - # W C -2pWD +

n (t) (n n n/ n (t) n n n (t)

/2 22 ..

+ T_,,n 40 - 2% W) A +2pW3

- 7 RS g2 n '

(n n nj n (t) n n n(t) n (t)

(6)

-p 0) (t ~c) t 2.. , nn where B = 0,) RS e cos[W (t - ?) ] d -

n (t) n (t) n '(7)

O s

ot nn .

and D =e +gQq) cos e t -qe

[ (q sin u.) t ]

n (t) 0 nn0 n 0n n The solution proced ure is based upon the step-by-step method.

The tctal time duration is divided into small time int ervals er steps. Displacement and velocity frc: the previous step are used as initia1 condition for the next step. A each step, the relative displacement, velocity, and acce.'. era .icn are calculated.

The. absolute acceleration is then combined by medal superposition.

=

(W } [p] (q ] + (D) R3 (t) where h ) ,= the column . atrix o f absolute a--a ' a-= -4 cn s.

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1541 033

The numerical methed used fer integratien of Equations (4) and ( 7 ) is Si=pson's rule.

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VERIFICATION OF QUAKE COMPUTER PROGRAM 1541 035

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. . . - . . . . . . . . .-. ~. ... .. - .. ...._ _ _ _ . . . . _ . _ - . . . . . . - . . . . . . . . . _ _ _ _ _ . . .

PROGRAM VERIFICATION OF QU1KE I. PROGRAM FUNCTION Program 'QU1KE' performs the dynamic analysis of a linear elastic lumped mass system using the response spectrum approach. First, the program determines the free vibration characteristics of the lumped mass system. Then, using the ground response spectra as loading input, the program computes the seismic response of the system due-to base. excitation. The analytical formulation of the program is described in appendix 1.

The capabilities of tha program include:

(1) Extracting the eigenvalues (frequencies) and the corresponding eigenvectors (mode shapes) from the general equation of motion of the lumped mass system, (2) Computing participation factors for three orthogonal directions of earthquake motion, (3) Computing equivalent modal damping (or weighted damping, compoclte damping) according to the energy stored in each material component and in each vibraticn mode, (4) Computing spectral accelerations, absolute modal accelerations, relative modal displacements and modal forces (or internal loads) using spectrum approach, (5) Calculating modal shears and mcments of statically determinate structures, and combining them by the square rcot of the sum of squares (SRSS).

II. METHOD OF VERIFICATION The reactor building of a PWR nuclear power plant is used as a model to verify the program. Ihe dynamic model consists of two components; one represents the containment and the other represents the internal structure (see Fig. 1) . The nodal coordinates of the model are shown in Table 1. Each mass point has 6 degrees-of-freedom (3 translations and 3 rotations) ;

the entire model has a total of 54 degrees-of-freedcm.

The eff ects of soil-structure interacticn are simulated '

by soil springs attached to the f oundation mat of the reactor building.

1541 036

_1_

MRI/STARDYNE structural analysis system, which is available at control Data corporation Cybernet centers, is used to verify part of program 'QULKE'. Hand calculation is used to verify other capabilities which are exclusive to ' QUAKE'.

Program ' STAR' of STARDYNE extracts eigenvalues and eigenvectors for all dynamic degrees of freedom of a structural system. Program 'DYNRE 4' of STARDYNE analyzes the response of lumped mass system subjected to a given shock spectra input. The ground response spectra, both horizontal and vertical, are input directly into the program in the form of user furnished spectra matrices. Hand calculations are used to verify the . modal forces, and shears and bending moments.

IV. COMPARISON OF RESULTS Very gcod agreement is found between analysis results from ' QUAKE' and ' ST ARD YNE ' . It is also true for results from ' QUAKE' and hand calculations. The comparisons of results from computer runs and verification by hand calculations are presented in Tables 2 through 9 listed below. For comparison of results due to base excitation, the case .of 1/2 SSE in X-direction is selected for demonstration. However, the conclusion frem these selected comparisons holds true for the earthquakes in the other two orthogonal directions.

(a) Frequencies and weighted dampings (Table 2) ,

(b) Spectral accelerations (Table 3) ,

(c) Absolute modal accelerations for the .first and second modes (Tables 4 and 5) ,

(d) Comparison of modal combination of accelerations, based on Regulatory Guide 1.92, frcm ' QUAKE' and hand calculations for the first and second degrees-of-f reedom (Table 6) ,

(e) Relative modal displacements for the first mode (TM31e 7) ,

(f) Modal forces for the.first and second modes (Table 8) ,

(g) Comparison of modal sheex forces and base '

moments for the shield building through hand calculation for -Jie two major coupling compenents (Table 9) .

1541 037

The ecmparison of modal participation factors computed from 'QU1KE' and 'STARDYNE' can not be made dd My because the mode shapes obtained from the nr . were: normalized with respect to mass matrix while the L tter with respect to maximum modal displacement.

Note that the absolute modal accelerations are equal to the product of participation f actors, modal shapes and spectral accelerations. Since both the absolute modal accelerations and spectral accelerations were shown to be correct, it can be concluded that the participation factors are also correct.

SRSS combination of modal shear forces and base moments are computed through the same routine used in the : combinaticn of the absolute acceleration as shown in Table 6, the validity of the procedure is thus established without additional demonstration necessary.

With all the comparisons shown above, program

' QUAKE' can thus be considered a valid and reliable program.

1541 038

'Y Containment i@ "x 2'@ Z 3h Internal Structure 4@ @6l

@7 5@ ss l

eth MAT e9 1#

1 Fig. 1 Dynamic Model Of A PWR Nuclear Power Plant 1541 039

TABLE 1 NODAL COORDINATES Mass Point X (ft) Y (ft) Z (ft) 1 0. 246.6 0.

2 0. 197.0 0.

3 0. 145.0 0.

4 0. 87.75 0.

5 0.. 29.25 0.

6 9.45 89.83 4.44 7 -0.28 55.94 8.27 8 ~5.87 26.98 2.42 9 2.40 --7.75 0.38 Soil Spring 0. -18.85 0.

1541 040

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TABLE 2 COMP ARISON OF FREQUENCIES AND WEIGHTED DAMPINGS FREQUENCIES (HZ) WEIGHTED DAMPINGS MODE QU A KE STARDYNE QUAKE ST AR DYNE 1 3.289 3.2894 0.04282 0.04282 2 3.290 3.2904 0.04280 0.04280 3 6.317 6.3165 0.04320 0.04320 4 7.002 7.0019 0.04472 0.04472 5 7.462 7.4616 0.04102 0.04102 6 9.019 9.0186 0.04106 0.04106 7 9.673 9.6730 0.06347 0.06347 8 10.238 10.2380 0.04319 0.04319 9 10.252 10.2517 0.04295 0.04295 10 14.684 14.6841 0.05879 0.05879 11 15.220 15.2198 0.05055 0.05055 12 16.665 16.6648 0.05875 0.05875 13 18.524 18.5237 0.04218 0.04218 14 18.765 18.7656 0.04183' O.04183 15 19.172 19.1725 0.04177 0.04177 16 19.576 19.5761 0.04328 0.04328 17 20.119 20.1187 0.04211 0.04211 18 20.626 20.6256 0.04018 0.04018 19 20.960 20.9596 0.04094 0.04094 20 22.379 22.3791 0.04582 0.04582 21 23.356 23.3563 0.05377 0.05377 22 '23.820 23.8203 0.05050 0.05050 23 24.903 24.9027 0.04253 0.04253 24 26.171 26.1711 0.04313 0.04313 25 26.250 26.2502 0.04214 0.04214 26 27.499 27.4987 0.04779 0.04779 27 27.618 27.6179 0.04988 0.04988 28 28.317 28.3166 0.04130 0.04130 29 28.706 28.7060 0.04193 0.04193 30 28.817 28.8167 0.04487 0.04487 31 29.251 29.2507 0.05185 0.05185 32 30.477 30.4768 0.04462 0.04462 33 31.293 31.2935 0.04331 0.04331 34 33.302 33.3019 0.04178 0.04178 35 33.645 33.6448 0.~4705 0.04705 36 35.726 35.7262 0 503 0.04503 37 37.197 37.1965 0. 4029 0.04029 38 38.628 38.6280 v.94065 0.04065 39 39.501 39.53 b 0.c4048 0.04048 40 39.997 39.9969 0.06059 0.04059 41 40.155 40.1549 0.0L043 0.04043 42 41.576 41.5756 0.08211 0.04211 43 42.259 42.2596 0.Js056 0.04056 44 44.259 44.2591 0.04030 0.04030 45 49.081 49.0810 0.04034 0.04034 46 52.966 52.9656 0.04028 0.04028 47 53.047 53.0469 0.04028 0.04028 48 54.012 54.0111 0.04034 0.04034 49 56.816 56.8157 0.04010 0.04010

~'-

1541 04i

4 TABLE 2 CCMP ARISON OF FREQUENCIES AND WEIGHIED DAMPINGS FREQUENCIES (HZ) WEIGHTED DAMPINGS MODE QUAKE STARDYNE QUAKE STARDYNE 50 56.854 56.8545 0.04012 0.04012 51 59.112 59.1113 0.04011 0.04011 52 69.698 69.6981 0.04005 0.04005 53 73.706 73.7060 0.04004 0.04004 54 81.125 81.1249 0.04003 0.04003 1541 042

. . . . - . . . . . . . . - . . . . . . . . . . . . . . . ~ . . ~ . . . . . . . . . . . . . . . . . -

TABLE 3 SPECTRAL ACCELERATIONS (in G's)

HORIZONTAL (X AND/OR Z) VERTICAL (Y)

MODE QU1KE STAFDYNE QUAKE ST ARDYNE 1 0.189770 .190220E+00 0.180343 .178495E+00 2 0.189786 .190237E+00 0.180415 .178567E+00 3 0.172822 .172880E+00 0.173567 .172803E+00 4 0.168370 .168417E+00 0.168873 .168361E+00 5 0.171861 .171773E+00 0.172133 .171735E+00 6 0.167134 .166930E+00 0.166988 .166930E+00 7 0.135579 .135232E+00 0.135669 .135232E+00 8 0.148869 .148770E+00 0.148756 .148770E+00 9 0.148972 .148865E+00 0.148859 .148865E+00 10 0.104819 .1046 2 4E+00 0.104834 .104624E+00 11 0. 105857 .105960E+00 0.105827 .105960E+00 12 0.096080 .959297E-01 0.096092 .959297E-01 13 0.094284 .942425E-01 0.094248 .9 4 2425 E-01 14 0.093445 .934019E-01 0.093410 .934019E-01 15 0.091904 .918623E-01 0.091370 .918623E-01 16 0.089972 .899461E-01 0.089942 .899461E-01 17 0.088400 .883667E-01 0.088371 .883667E-01 18 0.087197 .871536E-01 0.087169 .871536E-01 19 0.085899 . 85 8619E-01 0.085873 . 858619 E-01 20 0.080621 .806233E-01 0.080603 806233E-01 21 0.076845 .768246E-01 0.076840 .'s682462-01 22 0.076217 .762488E-01 0.076208 .76 248 8E-01 23 0.074744 .747296E-01 0.074730 . 747296E-01 24 0.071841 .718310E-01 0.071830 .718310E-01 25 0.071773 .717603E-01 0.071762 . 7176 03E-01 26 0.068784 .687938E-01 0.068777 .687938E-01 27 0.068417 . 6 843 73E- 01 0.068412 . 6 84373E-01 28 0.067693 .676841E-01 0.067686 .676841E-01 29 0.066931 .669240E-01 0.066925 . 6 6 9240 E-01 30 0.066567 .665658E-01 0.066562 .665658E-01 31 0.065494 .654967E-01 0.065492 . 6 54 96 7E-01 32 0.063779 .637772E-01 0.063775 .637772E-01 33 0.062523 .625210E-01 0.062521 . 6 25 210 E-01 34 0.060000 .600000E-01 0.060000 .600000E-01 35 0.060000 .600000E-01 0.060000 .600000E-01 36 0.060000 .600000E-01 0.060000 .600000E-01 37 0.060000 .600000E-01 0.060000 . 6 0000 0E-01 38 0.060000 .600000E-01 0.060000 .600000E-01 39 0.060000 .600000E-01 0.060000 . 6 0000 0E-01 40 0.060000 .600000E-01 0.060000 . 6 0000 0E-01 41 0.060000 .600000E-01 0.060000 . 6 0000 0 E-01 42 0.060000 .600000E-01 0.060000 .600000E-01 43 0.060000 .600000E-01 0.060000 .6 0000 0E-01 44 0.060000 .600000E-01 0.060000 .600000E-01 45 0.060000 .600000E-01 0.060000 .600000E-01 -

46 0.060000 .600000E-01 0.060000 .600000E-01 47 0.060000 .600000E-01 0.060000 . 6 0000 0E-01 48 0.060000 .600000E-01 0.060000 .600000E-01 49 0.060000 .600000E-01 0.060000 .6 0000 0E-01 1541 D43

TABLE 3 SPECTRAL ACCELERATIONS (in G's)

HORIZONTAL (X AND/OR Z) VERTICAL Of)

MODE QUAKE STARDYNE QUAKE STARDYNE 50 0.050000 .600000E-01 0.060000 .6 0000 0 E-01 51 0.060000 .600000E-01 0.060000 .600000E-01 52 0.060000 .600000E-01 0.060000 .600000E-01 53 0.060000 .600000E-01 0.060000 .600000E-01 54 0.060000 .600000E-01 0.060000 .6 00 00 0E-01 1541 044

TABLE 4 MODAL ACCELERATION COMPARISON --Mode 1 (in G's)

DOF QU1KE ST ARDYNE 1 0.174543 E- 02 0.17452SE-2 2 -0.16006 6 E- 05 -0.160235E-5 3 0.233091E-01 0.233358E-1 4 0.786848E-04 0.787749E-4 5 -0. 24533 0 E- 07 -0. 2 4 56 42E- 7 6 -0. 589219E- 05 -0. 5 89158 E- 5 7 0.139659E-02 0.139644E-2 8 - 0.1573 66 E- 05 -0.157529E-5 9 0.186507E-01 0.186720E-1 10 -

0.7882432-04 0.789145E-4 11 -0.242828E-07 -0.243138E-7 12 - 0. 5 9 026 4 E- 05 -0.5902025-5 13 0.10190 4 E-02 0.101894E-2 14 -0.15 5626 E- 05 -0.1557 8 9 E- 5 15 0.136089E-01 0.136244E-1 16 . 0. 73 8709E- 04 0.739555E-4 17 -0. 23775 6 E- 07 -0. 2 3 80 61 E- 7 18 -0. 5 53172E- 05 -0.553114E-5 19 0.580324E-03 0.580264E-3 20 - 0.15163 5E- 05 -0.1517 95 E- 5 21 0.775012 E-02 0.775899E-2 22 0. 5945 8 5 E- 04 0.595266E-4 23 -0. 22508 5E- 07 -0.225373E-7 24 -0. 4 4 52 4 9 E- 05 0.445203E-5 25 0.186918 E-03 0.186899E-3 26 - 0.14562 8 E- 05 -0.145781E-5 27 0.249649E-02 0. 2499 3 5E- 2 28 0. 32123 6 E- 04 0.3 216 04E- 4 29 -0.205209E-07 -0.205471E-7 30 -0. 240561 E- 05 -0.240536E-5 31 0.17 504 5 E-03 0.175043E-3 32 -0. 83233 4 E- 04 -0. 8 33121E- 4 33 0. 23384 0 E- 02 0. 234108E- 2 34 0.16 9495E- 04 0.169689E-4 35 -0.188876E-05 -0.189094E--5 36 -0.148701 E- 05 -0.148716 E- 5 37 0.114839 E-03 0.114829E-3 38 - 0.131871 E- 03 -0.1320 20E- 3 39 0.159738E-02 0.159921E-2 40 0.164543 E- 04 0.164731E-4 41 -0.127531 E- 05 -0.1276 80 E- 5 42 - 0.14089 5 E- 05 -0.140907E-5 43 0.764824E-04 0. 7647 73 E- 4 44 -0. 28 9521 E- 04 -0.289927E-4 45 0.1008n1E-02 0.100956E-2 -

46 0.15232 8E- 04 0.152502E-u 47 -0.402359E-06 -0.402836E-6 48 -0.126107E- 05 -0.126109E-5 49 0.302338E-04 0.302308E-4 154i 045

TABLE 4 MODAL ACCELERATION COMPARISON - Mode 1 (in G's)

DOF OUAKE STARDYNE 50 -0.903831 E-05 -0.904547E-5 51 0.4 0426 6 E- 03 0.404729E-3 52 0.136146E-04 0.1363 01E- 4 53 -0.192120E- 07 -0.192366E-7 54 . -0.101963E-05 -0.101952E-5 1541 046 e a ,

i TABLE 5 MODAL ACCELERATION COMPARISON -Mode 2 (in G's)

DOF OUAKE STARDYNE 1 0.310599E+00 0.311341E+0 2 - 0. 2144 6 5 E- 04 -0.214999E-4 3 -0.232555E-01 -0.232821E-1 4 -0.785024E-04 -0.785921E-4 5 0.518279E-06 0.519490E-6 6 -0.104850E-02 -0.105101E-2 7 0.248518E+00 0.249111E+0 8 -0.210783E-04 -0.211302E-4 9 -0.186074E-01 -0.186286E-1 10 -0.786416E-04 -0.787314E-4 11 0.512989E-06 0. 514187E- 6 12 -0.105036E-02 -0.105287E-2 13 0.181326E+00 0.181760E+0 14 -0.208448E-04 -0.208966E-4 15 -0.135767E-01 -0.135922E-1 16 -0.736966E-04 -0.737808E-4 17 0.502269E-06 0.503441E-6 18 -0.984314E-03 -0.986668E-3 19 0.103246E+00 0.103493E+0 20 -0.203102E-04 -0.203606E-4 21 -0.773068E-02 -0.7739523-2 22 -0.593087E-04 -0.593764E-4 23 0.475478E-06 0.476586E-6 24 -0.792150E-03 -0.794045E-3 25 0.332367E-01 0.333162E-1 26 -0.1950 4 9E- 04 -0.195534E-4 27 -0.248893E-02 -0.249178E-2 28 -0.320207E-04 -0.320572E-4 29 0.433460E-06 0. 4344 67E- 6 30 -0.427695E-03 -0.428718E-3 31 0.288000E-01 0.288689E-1 32 -0.228943E-02 -0. 2295 00 E- 2 33 -0.224332E-02 -0. 224577E- 2 34 -0.143306E-04 -0.143438E-4 35 0.467390E-05 0.468269E-5 36 -0. 2213 83 E-03 -0. 221913 E- 3 37 0.202740E-01 0.203225E-1 38 -0.590659E-04 -0. 5 937 05E- 4 39 -0.153119E-02 -0.153287E-2 40 -0.140853E-04 -0.140985E-4 41 0.598524E-05 0.5997923-5 42 -0.212995E-03 -0. 213505 E- 3 43 0.1320 20 E-01 0.132336E-1 44 0.11029 7E-02 0.110556E-2 45 -0.984814E-03 -0.985914E-3 46 -0.134569E-04 -0.134702E-4 47 0.299470E-05 0.300134E-5 48 -0.202454E-03 -0.202939E-3

\54\ DAl TABLE 5 MODAL ACCELERATION COMPARISON - Mode 2 (in 3's)

DOF QUAKE STARDYNE 49 0.536445E-02 0.537728E-2 50 -0.448052E-03 -0.449131E-3 51 -0.403068E-03 -0.403528E-3 52 -0.135438E-04 -0.1355 92E- 4 53 0.405787E-06 0.406732E-6 54 -0.180921E-03 -0.181354E-3 1541 048

4 e

TABLE 6 BOD &L ConBINATION COMPARISON (Check Accelerations f or First Two D.O.F. 's)

FdEQUENCY GkOUP FIkST D.O.F. SECOND D.O.{.

noos 01:1 No. sopat acc. (at caoue sun suu&eE. ___nooat Acc. Ici choue son souasE.

I 3.289 .374543E-2 -

.160066E-5 2 3.290 1 .3105998 0 9.75598-2 .2284465E-4 5.3117E-10 3 6.357 2 .329tJbE-4 1.0833E-9 .802472E-4 I.0501E-10 4 7.002 .4678848-1 .174962E-2 5 7.462 3 .2943898-J 2.2361E-3 .3062455-4 J.0985E-6 6 9.089 .J13862E-3 .458347E-4 7 9.673 4 .3327678-3 4.1881E-7 .933138E-2 8. 7 9 32 E- 5 8 10.238 .10622 5-1 .5935468-3 9 10.252 5 .9873988-1 1.1960E-2 .329356E-2 1.5108E-5 10 14.684 .3492625-2 .248648E-1 18 15.220 6 .639873E-4 1. 2650 E-5 .687655E-3 .. 5289E-4 72 16.665 7 .Ju9821E-I 8.3996E-4 .295999E-1 8.7615E-4 13 18.524 .193434E-3 .1431068-3 14 18.765 .536154E-2 .158420E-2 lb 19.172 .5815458-3 .542822E-4

[. 16 19.5)o .708787E-3 .201251E-3 f 17 20.119 8 .2820335-1 1.22848-3 .742782E-3 7.4290E-6 18 20.626 .376743E-3 . 73 4 0 6 8 E-4 19 20.960 .3019798-4 .204669E-4 20 22.379 9 .2786788-2 9.7572a-6 .108796E-3 1.2042E-4 21 23.356 .143886E-4 . 364 7 42 E-4 22 23.820 .3368238-1 .2764385-2 23 24.903 10 .725653E-3 3.1850E-3 .939570E-2 e.4880E-4 24 J6.171 .9dO660E-3 . 9847 09 E-2 25 26.250 .1693918-4 .469699E-3 26 17.499 .4057368-I .695947E-2 27 27.618 .716228E-2 .106847E-2 28 28.387 .129457E-4 .160J36E-4 29 28.706 11 .561036E-3 3.7278E-4 .491361E-3 3.5540E-4

,,, 30 28.887 .182691E-2 .504823E-3 ,

31 29.251 .433380E-2 .7748J3E-2 L3 32 30.477 .202638E-3 .440079E-5 4" 33 31.293 12 .2182778-3 4.3309E-5 .880205E-3 7.1070E-5 s

C

.p=

W .

TABLE 6 50DAL C05 BIN ATION ConP& BISON (Check Accelegations for First Two D.O.F.'s)

FSEQUENCE G kOU P FIkST D. O . F. SECOND 9.O.F.

MODE IHz) NO. 50DAL ACC. (GL GBOUP sus $O0&BE* HODAL ACC. (G) GROUP SUN SQUABE*

34 33.302 .6444888-5 .7763788-5 35 33.645 .391534E-4 .745290E-4 36 35.726 13 .391881E-J 1.9133E-7 .2954038-3 1.4265E-7 37 37.197 .1210?28-4 .6859408-4 38 38.628 .1924b48-4 .386540E-3 39 39.501 6731808-3 .162576E-3 40 39.997 .959826E-4 .609032E-4 41 40.155 14 .300691E-6 6.3810E-7 .359994E-6 3.7060E-7 42 41.576 .413844E-6 .I'42499E-6 43 42.259 .688781E-5 .356304E-6 44 44.259 15 .5090588-5 7.0430E-Il .271022E-5 1.0298E-11 45 49.081 .747342E-5 .219735E-4 46 52.966 .354846E-4 .3344768-4

, 47 53.047 16 .125383E-3 2.83398-8 .610u84F-4 9.35415-9 48 54.012 .113180E-4 .83016SE-4 49 56.816 .4596575-5 .692842E-6 50 56.854 .6410948-4 .929218E-5 51 59.112 17 .145388E-6 6.4271E-9 .536767E-4 1.2472E-8 52 69.698 .2211ubE-7 .7662615-7 L3 73.706 18 .6042368-9 5.1633E-16 .458925E-8 6.59595-15 54 81.125 19 .8674588-9 7.5248E-19 .324068E-8 1.0502E-l?

Y = 1.1543E-l 2.3389E-3 SSSS=p)l/2 3.3975E-l 4.8362E-2 SSSS Pkos 3.397475-1 4.83617E-2 QUAKE s

W 4

C:0 -

LD CD

• Based es the grouping method, Section 1.2.1, NBC su9ulatory Guide 1.92, February 1976

TABLE 7 RELATIVE MODEL DISPLACEMENTS (IN FT AND RADIAN)

MODE 1 DOF QUAKE STARDYNE 1 0.131459E-03 0.131558E-03 2 -0.120556E-06 -0.1207 86 E-06 3 0.175556E-02 0.1759 07E-0 2 4 0. 592626E- 05 0.593811E-05 5 -0.184774E-08 -0.185167 E- 0 9 6 -0.443779E-06 -0.444112E-06 7 0.105186E-03 0.10 5265 E-0 3 8 - 0.1185 22E- 06 -0.118747E-06 9 0.140470E-02 0.140751E-02 10 0. 593677E- 05 0.5 94864 E-0 6 11 -0.182889E-08 -0.1832 79 E-0 9 12 - 0. 4 44 56 5 E- 06 -0. 4 44 8 9 9E-0 6 13 0.767508E-04 0.768084E-04 14 - 0.117212E- 06 -0.1 17434 E-06 15 0.102497E-02 0.102702E-02 16 0.556369E-05 0.557482E-05 17 -0.179069E-08 -0.179452E-08 18 - 0. 416 629 E- 06 -0. 416 9 4 2E-0 6 19 0. 437 079 E-04 0.437407E-04 20 - 0.114206E- 06 -0.114424E-06 21 0.583711E-03 0.584879E-03 22 0.447820E-05 0.448716E-05 23 -0.169526Z-08 -0.16 988 8 E-08 24 -0.335345E-06 -0. 3 3 55 97 E- 06 25 0.140780E-04 0.140885E-0 4 26 - 0.109682E- 06 -0.109891E-06 27 0.188027E-03 0.188403E-0 3 28 0.241943E-05 0.242427E-05 29 -0.154556E-08 -0.1548 86 E-08 30 -0.1811822-06 -0.181318 E- 0 6 31 0.131838E-04 0.131949E- 0 4 32 - 0. 6 26 884E- 05 -0.628013E-05 33 0.176120E-03 0.176472E-03 34 0.127658E-05 0.127912E-0 5 35 -0.142254E-06 -0.142540 E- 0 6 36 -0.111996E-06 -0.112103E-06 37 0.864929E-05 0.865587E-06 38 -0.993203E-05 -0.995180E-05 39 0.1203093-03 0.120549E-0 3 40 0.12392 8E-05 0.124175E-05 41 -0.960515E-07 -0.962461E-07 42 - 0.106117E- 06 -0.106 217 E-06 43 0. 576 03 8 E-05 0. 5 76492E-0 5 44 - 0. 218056 E- 05 -0.218549E-05 45 0. 7 61017E-0 4 0.759498E-04 46 0.114728E- 05 0.1149575-05 47 -0.303042E-07 -0.303661E-07 48 -0.949789E-07 0.950618E-07 49 0.227710E-05 0.227882E-05 '

\54\ DS\

_1,_

TABLE 7 RELATIVE MODEL DISPLACEMENTS (IN FT AND RADIAN)

- MODE 1 DOF CUAKE STARDYNE 50 -0.680733E-06 -0. 6 81854 E- 06 51 0.304479E-04 0. 3 050 87E- 0 4 52 0.102540E-05 0.102745E-05 53 -0.144698E-08 -0.1450 07E- 0 8 54 -0.767945E-07 -0.768525E-07 1541 052

TABLE 8 MODEL FORCE COMPARISON (IN KIP AND KIP-FT)

MODE 1 MODE 2 DOF OU1KE CALCULATED QU1KE CALCULATED 1 0.971477E+01 0.971478E+1 0.172874E+04 0.172874E+4 2 -0.890901E-02 -0.890902E-2 -0.1193 68 E+0 0 -0.119368E+0 3 0.129735E603 0.129735E+3 -0.129436E+03 -0.129436E+3 4 0.471869',+03 0.471869E+3 -0.470775E+03 - 0. 470 77 5E+ 3 5 -0.268753E+00 -0.268752E+0 0.567761E+01 0.56776 E+1 6 -0.353352E+02 -0.3 533 52E+ 2 -0.628781E+04 0.628781E+4 7 0.168494E+02 0.168494E+2 0.299829E+04 0.299829E+4 8 ~0.189857E-01 -0.189857E-1 -0.254303E+00 -0.?54303E+0 9 0.225015E+03 0.225015E+3 -0.224492E+03 - 0. 7.24 4 9 2E+3 10 0.244975E+04 0.244975E+4 -0.244407E+04 -0.244407E+4 11 -0.135466E+01 -0.135466E+1 0.286182E+02 0.286182E+2 12 -0.183446E+03 -0.183446E+3 -0.326438E+05 -0.326437E+5 13 0.177696E+02 0.177695E+2 0.316188E+04 0.316187E+4 14 -0.277882E-01 -0.277882 E- 1 -0.372200E+00 -0.372199E+0 15 0.237305E+03 0.237305E+3 -0.236743E+03 -0.236743E+3 16 0.331537E+04 0.331537E+4 -0.330755E+04 -0.330755E+4 17 -0.1919 9 5E+ 01 -0.191995E+1 0.405597E+02 0.405597E+2 18 -0.248267E+03 -0.248267E+3 -0.441767E+05 -0.441766E+5 19 0.100447E+02 0.100447E+2 0.178707E+04 0.178706E+4 20 -0.2624622-01 -0. 2 624 6 2E- 1 -0.351545E+00 -0.351545E+0 21 0.13414 5E+ 03 0.134145E+3 -0.133809E+03 -0.133809E+3 22 0. 2798 62E+0 4 0.279861E+4 -0.279156E+04 -0.279156E+4 23 -0.189583E+01 -0.189583E+1 0.400484E+02 0.400484E+2 24 -0.209571E+03 -0.209571E+3 -0.372852E+05 -0.372852E+5 25 0.323532E+01 0.323532E+1 0.575288E+03 0.575287E+3 26 -0.252065E-01 -0.252064E-1 -0.337607E+00 -0.337606E+0 27 0.432113E+02 0.432112E+2 0.43 08 0SE+ 02 0.430804E+2 28 0.151200E+04 0.151200E+4 -0.150716E+04 -0.150716E+4 29 -0.172843E+01 -0.172843E+1 0.365093E+02 0.365093E+2 30 -0.113228E+03 -0.113228E+3 -0.201309E+05 -0.201309E+5 31 0.247792E+01 0.247792E+1 0.407690E+03 0.407690E+3 32 -0.117824E+01 -0.117824E+1 -0.324090E+02 -0.324089E+2 33 0.331022E+02 0.331022E+2 -0.317562E+02 -0.317562E+2 34 0.327076E+03 0.327075E+3 -0.276538E+03 -0.276538E+3 35 -0.597026E+02 - 0.5 970 27E+2 0.147739E+03 0.147739E+3 36 -0.205524E+02 -0.205524E+2 -0.305980E+04 - 0. 3 0 598 0E+ 4 37 0. 2615 8 3E+ 01 0.261582E+1 0.461805E+03 0. 4 6180 4 E+3 38 -0.225283E+01 -0.225283E+1 -0.100906E+01 - 0.10 09 0 6E+ 1 39 0.364881E+02 0.364881E+2 -0.34 9763E+02 -0.349762E+2 40 0.432182E+03 0. 43 2182E+ 3 -0.369959E+03 -0.369959E+3 41 -0.456251E+02 -0.456252E+2 0.214127E+03 0.214127E+3 42 -0.282675E+02 -0.282674E+2 -0.427327E+04 - 0. 4 273 2 7E+ 4 43 0.148130E+01 0.148130E+ 1 0.255695E+03 0.255694E+3 34 -0.596134E+00 -0.596135E+0 0.227106E+02 0.227106E+2 45 0.195307E+02 0.19530 7E+ 2 -0.190737E+02 -0.1907372+2 46 0.216074E+03 0.216075E+3 -0.190884E+03 -0.190884E+3 47 -0.127481E+02 -0.127481E+2 0.948826E+02 0.948825E+2 48 -0.238642E+02 -0.238643E+2 -0.383122E+04 -0.3 83121E+4 1541 053

TABLE 8 MODEL FORCE COMPARISON (IN KIP AND KIP-FT) 49 0.177128E+01 0.177128E+1 0.314282E+03 0.314282E+3 50 -0.569648E+00 -0.569648E+0 -0.282389E+02 -0.282389E+2 51 0.236844E+02 0.236844E+2 -0.236142E+02 -0.236142E+3 52 0.922021E+03 0.922023E+3 -0.917229E+03 -0.917228E+3 53 -0.240378E+01 -0.240378E+1 0.507716E+02 0.507716E+2 54 -0.711516E+02 -0.711519E+2 -0.126251E+05 -0.126251E+5 1541 054

-I9-

TABLE 9 CusPakISou or SHEAk FokC ES AND BASE nosENTS (INTESM AL LOADS) sODE I Contalaamat Calculated QUAEE Elevation sodal Force Calculated Bodal shear & Base soment Besult ffti jK]P & KIP-ft) fKip & Eip-fti JKIP & KIP-ft) nass 1 246.60 Fz: 9.71478 9.71478 9.7148 sas -35.3352 - J 5. 3 3 52 -35.335 2 197.00 rz 16.84?S 9.71478 e 16.8494 = 26.56418 26.564 az -183.446 -9.71478 x 49.6-35.3352-183.446=-700.634 -700.63 3 145.00 Fx 17.7695 26.56418e17.7695=44.33368 44.334 azz -248.267 - 70 0. 6 3 4- 2 6. 56 418: 52-2 4 8. 2 67 =- 23 30. 2 384 -2330.23 4( 87.75 Fz 10.0447 44.33368e10.0447=54.37838 54.379 L aza -209.571 -2330.2384-44.33368:57.25-209.571=-5077.9126 -5077.9 O

s 5 (3) 29.25 Ps: 3.2352 5 4. 3783 s e 3. 2 352= 57. 61358 57.614 sa: -113.228 - 5077. 9126- 54. 378 3 8s 58. 5- 1 13. 228 =- 8372. 2758 -8372.3 f WVe --~ }-] 0.0 Fza 57.61358 57.64 EP (Top of nat) sa: 83 72. 2 758-5 7. 6 3 3 58: 29. 25=- 100057. 4731 -10057.0 S_

4 aY

, >=

' X r

Z LJ7

, 4=

s W

W

APPENDIX A I. Comeute r Procram (OUAXE)

This program performs the dynamic analysis of a lumped mass system. The input informacion includes mass da ta , structural stiffness, scil spring con sta nt s , structure and soil damping values and other data related to the dynamic a nalysis. model.

Program " QUAKE" has the folicwing capabilities:

(1) Extracting the eigenvalues and the corresponding eigenvectors from the following equation, l_ (I] - [6][m]l (t) =0 (1)

CG)* /

where

, [I] = the unit matrix Cd = the natural circular frequency (m) = the mass matrix (6 ) = the f1exibility matrix

($) = the column matrix of eigenvectors Equation ( 1) can be written in terms of the stiffness matrix [k] as follows:

(Ek ] - com (m]) (9) =0 (2) ,

~

In order to e xtract eigenvalues, and . eigenve ct ors ,

equation (1) is converted into the f ollowing f orm:

T (m) = (U] (U) (3)

In general, matrix [U] consists of diagonal and upper-diagonal elements only.

Substitutien- of (m] from equation (3) and (#) from

. equation (9) = (U]-1( into equation ( 2) yields: ,

                               -1        :      T         -1

((k][ 0] -

                                        &) ( U ] (U ][U )      (f)  =0                 ' ( 4)
                                     -^2-1541       056

Premultiplication of Equation '( 4) by [ U][ k ]- 1 yields the Equation: T3 [,1, [I] - [ U][ $ ][ U] ) (j) =0 ( 5) (W2 The matrix product [ U ][c3 ][U ]T is a symmetric matrix. The eigenvalue s and eigenvectors are extracted from Equation ( 5) using a subroutin e called IIGni of 2SM. This subroutine uses an algorithm known as the paccbi Diagenalization method. The mode shapes ($) as ccrresponding eige nvectors of Equation (1) are cbtained from the following equation: (@] = ( U]-1[h] ( 6) (2) Ccmputing participation f actors ]" using matrix mani;;ulation

      .            in the following manner:
                                                      -t           T (r)        = TM]       [9] (m][D}                                                      ( 7) '

d j where T - [M] = [p] (m][@] (8) and (6) = the matrix of mode shapes

                             ..T (4)     =  the transpose of mode shape matrix (r)     =  the column matrix of participatien                                       factors for j     seismic motion in the jth direction (D)    =  the column matrix governed by the seismic motion j , in jth direction
                          .[m]     =  the nass matrix The       normalization                     and       orthogonality'               con,ditions         are represented by the matrix equation                                          ( 8)
                                                              - ^.2      -

1541 057

(3) computing equivalent modal damping (or composit e modal damping) according to the energy stored in each component and in each vibration mode. Each component, such as concrete structures, steel structures and systems, and foundation materials, can have different damping prope rtie s . The effective damping in any vibration mode of the total system depends upon 'the degree of participatien of these materials in the modal response. The flexibility matrices [6 ): for each ccmponent r that have inherently different damping properties are formed. The summation of these flexibility matrices yields the total flexibility matrix of the. composite system (6 ]. where, N [b] = I C5) (9) 1 r After the free vibration characteristics are generated using flexibility ma trix (6 ], obtained according to Equation ( 9), the stif fnes s ma trix ( k ] of the entire system is cetained by the inversion of flexibility matrix as follows: [ k ] = [61- 1 (10) Then the fictitious f orce matrix ( f] is obtained as a product of the stiffness =aurix and mode shape matrix [4] as fo11cws: [f] = [k][4] (11) The mode shape matrix [9]r for each component is generated as follcus: [p] r

                          = [8]r (f)                                                          (12)

The stiffness matrix for each ccmpenent r is cbtained by inverting the correspending flexibility matrix as shown by equation, [k] = [6]-L (13)

r For foundation compon ents, that is, for each foundation spring constant, the stif fness matrix ( k]r ha s a single ialue on the diagonal.

                                                         -^3-1541       058

Matrix [ K ]r which represents the energy stored in each mode of component : is calculated from the f olicwing equation in matrix form: T [K] = [$) [k ] [p] (14} r  : r r Matrix [ 2E] which represents the total energy s:cred in the entire system is obtained as fo11cws: [ 2E] = [M ][a;z ] (1S} where T [M] = [p] [m][p] = the generali::ed mass matrix (4) = the circular natural f:equency [m] = the mass matrix [p] = the mode shape matrix T [p] = the transpo s e o f mo de shape matrix Inergy matrix corre sponding to the last cc=penent N is generated as presente?. 'in the fellcwing equaticn in matrix fcrm: r= N-1

                              -[K]      = [2I] -                 [K]                                     (16)

N  :

                                                          =1 Then           the      column matrix of weighted medal damping ratics (D}

is calculated f rom the following equation: r=N [K] (D) ( 17) (D} = [ 2E]-1 r

=1 4 s where (t} = the column matrix o'f damping ratios for component r. ,

r

                                                       - x.4 -                                                1541 059

Fraction of modal energy components is generated as follows: (MEC) = [ 2 E ]-t ( K ] (1) (18) . r r Then, as a check, total su=ma tien of these fractions is obtained, which is equal unity:

                                  ==N (MIC) = - (1)                           (19) r=1

( 4) Computing absolute =edil accelerations, relative displacements and inertia loads using spectrum approach. The matrix cf maximum =odal absolute accelerations [Id] is obtained from the follcwing equatien in matrix ferr: [W] = [p][Sa][j"] (,20) where [ p ] = the mode shape matrix [Sa] = the diagonal matrix cf maxi =um medal spectral accelerations [ p ] = the diagonal matrix of medal participatien factors The matrix of maxi =cm modal relative displacements [d] is ccmputed by

                                             .. F                                   .

[d] = (W]'L ( 21) u)2 s , where 1

                                     = the diagenal matrix cf eigenvalues, if the flexibility g          matrix is used in the equaticn for the extraction of 4

eigenvalues and eigenvectors. The matrix cf maximum medal inertia forces ( 71 is -the prod,uct of mass matrix [m] and maximum modal acceleration matrix ;,W ]. tr] = tm]tUI (22)

                                                         -A5-1541     060

The maximum modal inertia forces [F] are cc=bined by the square rect of the sum cf the squares (in accordance with NFC Regulatory Guide 1.92), 'cy absolute sum and by algebraic su=. (5) calculation of modal shears and mcments of statically

       . determinate structures, and ccmbining them by the square rect of the sum of the squares of medal values, by absolute st=1 and by algebraic sum.

The matrix of =odal s hears and bending =cments (Q] is ccmputed f cm: T T [Q) = [?] [J) (23) where T (F] = the transpose of th e matrix of inertia forces and inertia m'ement (J] = summation = atrix 1541 061 8

                                                - A.6 -

4*- se eh. . mm. @ 6w+Me equeM VERIFICATION OF SPECTRA COMPUTER PROGRAM

                                                              \SA\ Dt2
       . .    .- . . . - .       - - - - .              .. ..  --..----e- -

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VERIFICATION OF COMPUTER PROGRAM SPECTRA I. PROGRAM FUNCTION This is a two part computer program which stores both the . horizontal and vertical ground (acceleration) design response spectra, as given by NRC Regulatory Guide 1.60, in the fcrm of polynomial functions. The program can also compute the spectral value for a given pair of frequency and damping value. Basically, the program divides into two parts. The first part of the program is to find, for each cont'rol frequency, a polynomial expression for the spectral amplification curve which is a function of damping values as determined from Regulatory Guide .1.60. This is done by using the least- square fitting method described in References 1 and 2. A total of six curves, with three each for horizontal and vertical directions, are being fitted in such a manner. The polynomial expressicn so established are then built into the second part of the program which calculates spectral accelerations between the control frequencies for a given damping value in acccrdance with linear variations with frequencies in the log field. Thus, for a given pair of frequency and damping the spectral acceleration can be computed from one.cf the polyncmial functions or from interpolaticn of the spectral values so computed. 0

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II. METHOD OF VERIFICATION Six polynomial functions are established to fit the six sets of the NRC design response spectral acceleration values at six control points: the: horizontal spectra at 0.25, 2.5 and 9 Hz, and the vertical spectra at 0.25, 3.5 and 9 Hz. Using these functions, the program can compute and interpolate in log-field the : spectral value for a given frequency and damping value. These program calculated values are then compared with the values obtained from Regulatory Guide 1.60. III. COMPARISON OF RESULTS The spectral accelerations for various frequency and damping pairs calculated by the program are listed in Tables 1 and 2. The corresponding spectral values according to NRC Regulatory Guide 1.60 are also listed for comparison. Based on the closeness of the results, the program ' SPECTRA' can be considered satisfactory.

References:

1. Kelly, L.G. , Handbook of Numerical Methods and loolication, Addison --Wesley.

2. Library of Mathematical Subprograms, Reference Manual, control Data company.

1541 064

TABLE 1 HORIZONTAL GROUND (ACCELERATION) DESIGN RESPONSE SPECTRA Percent Program Percent of Calcu- - of Program Critical Freq. NRC lated Critical.Freq. NRC Calculated Damoing (cos) Values values Damoing (cps) Values Values 0.25 0.74 0.73 0.25 0.43 0.43

1. 0 2.59 2.58 1.0 1.31 1.31 0.5 2.5 5.95 5.93 7.0 2.5 2.72 2.72 5.0 5.39 5.37 5.0 2.47 2.46 9.0 4.96 4.94 9.0 2.27 2.26 20.0 1.85 1.85 20.0 1.37 1.37 33.0 1.00 1.00 33.0 1.00 1.00 0.25 0.65 0.66 0.25 0.41 0.41

1. 0 2.25 2.28 1.0 1.24 1.24 2.5 5.10 5.18 2.5 2.55 2.54 1.0 5.0 4.62 4.69 8.0 5.0 2.31 2.30 9.0 4.25 4.31 9.0 2.13 2.12 20.0 1.75 1.76 20.0 1.34 1.34 33.0 1.00 1.00 33.0 1.00 1.00 0.25 0.58 0.58 0.25 0.39 0.39 1.0 1.89 1.92 1.0 1.13 1.13 2.0 2.5 4.25 4.25 10.0 2.5 2.28 2.28 5.0 3.85 3.85 5.0 2.07 2.07 9.0 3.54 3.54 9.0 1.90 1.91 20.0 1.63 1.63 20.0 1.28 1.28 33.0 1.00 1.00 33.0 1.00 1.00 0.25 0.53 0.53

1. 0 1.72 1.71 2.5 3.75 3.73 3.0 5.0 3.41 3.39 9.0 3.15 3.12

20. 0 1.56 1.55 33.0 1.00 1.00 0.25 0.47 0.47

1. 0 1.47 1.48 5.0 2.5 3.13 3.14 5.0 2.84 2.85 -

9.0 2.61 2.62 065 20.0 1.45 1.45 1541 33.0 1.00 1.00

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TABLE 2 VERTICAL GRCUND (ACCELERATION) DESIGN RESPONSE SPECTRA Percent Program Percent of Calcu - of Program Critical Freq. NRC lated Critical Freq. NRC Calculated Damping (cos) Values Values Damoing (cps) Values Values 0.25 0.49 0.49 0.25 0.29 0.29 1.0 1.77 1.77 1.0 0.91 0.91 0.5 3. 5 5.67 5.65 7.0 3.5 2.59 2.58 5.0 5.39 5.37 5.0 2.46 2.46 9.0 4.96 4.94 9.0 2.27 2.26 20.0 1.85 1.85 20.0 1.37 1.37 33.0 1.00 1.00 33.0 1.00 1.00 0.25 0.43 0.44 0.25 0.28 0.28 1.0 1.54 1.57 1.0 0.87 0.86 3.5 4.86 4.94 3.5 2.43 2.42 1.0 5.0 4.62 4.69 8.0 5.0 2.31 2.30 9.0 4.25 4.31 9.0 2.13 2.12 20.0 1.75 1.76 20.0 1.34 1.34 33.0 1.00 1.00 33.0 1.00 1.00 0.25 0.38 0.38 0.25 0.26 0.26 1.0 1.33 1.32 1.0 0.79 0.79 2.0 3.5 4.05 4.05 3.5 2.17 2.18 5.0 3.85 3.85 10.0 5.0 2.06 2.07 9.0 3.54 3.54 9.0 1.90 1.91 20.0 1.63 1.63 20.0 1.28 1.28 33.0 1.00 1.00 33.0 1.00 1.00 0.25 0.35 0.35 1.0 1.20 1.19 3.5 3.60 3.56 3.0 5.0 3.42 3.39 9.0 3.15 3.12 20.0 1.56 1.55 33.0 1.00 1.00 0.25 0.31 0.32 1.0 1.02 1.03 5.0 3. 5 2.98 2.99 . 5.0 2.83 2.85 9.0 2.61 2.62 15 066 20.0 1.45 1.45 33.0 1.00 1.00 _4

4 - . ._. . . . . - . . . . . . .. _ ._. . . . . VERIFICATION OF TIME COMPUTER PROGRAM 1541 067

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VERIFICATION OF COMPUTER PROGHA5 TIME I. PROGRAM FUNCTION The program TIME is a routine used to develop floor response spectra for various equipment damping values' from the floor time histories of a building under either ground excitation or force vibration. The program can also develop floor response spectra for soft floor condition by assuming the floor being a single degree of freedom system itself. The program solves the uncoupled equations of motion using the me chod of Laplace transforms. Exact solutions are obtained for these second order differential equations by assuming the forcing f unction (i. e . , floor time histor y) to be linear between each time step. The analytical formulation of the problem is given in Reference 1. II. METHOD OF VERIPICATION The analysis results of a BWa nuclear power plant are chosen for the verifica tion of this program. The dynamic model shown in Figure 1 is associated with a cracked concrete condition, unflooded case and upper bound soil springs. The model was subjected to a Safe Shutdown Earthquake in I-direction. The floor response spectra in I-direction at the top node (node 47) of reactor vessel for 2 and 3 percent equipment damping are obtained by using two different approaches. First , the floor response spectra are developed by TIME, then they are again developed by DINHE5 of STARDYNE which is an established and well known prog ram in public domain developed by sechanics Research Incorporated (MRI). The results obtained by the latter approach, therefore, is used as the benchmark for verification of the program TIME. III. COMP ARISON OF RESULTS The floor response spectra obtained for node 47 of the reactor vessel from programs TI5E and DYNHES are shown in Table 1 for 2 and 3 percent equipment damping values. As can be seen from the comparison, the results from both programs agree exactly for periods above 0.10 ; however, the discrepencies, which range from 1 to 6 percent, are observed for the spectral values below that period. It is to be noted that the spectral values obtained from TI5E are exact as mentioned previously, while the results obtained from DYNHES, though exact theoretically, are amenable to round-off approximation due to the integration technique used, particularly in

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the higher frequency range. For the spectral values which are obtained by TIME and differ from those obtained by DYNRES over 2 percent, are further verified by the program DYNRE1 of STARDYNE. The results from TIME and DYNRE1 are exactly che same as shown in Table

1. It can, therefore, be concluded that the program TIME is correct and dependable.

References:

1. D. Vanccuvering, DYNRE1 Technical Description, Theoretical Manual of MRI/STARDYNE, pp. C-400-C-480.

1541 069

TABLE 1 COMPARISCN OF FLOCR RESPONSE SPECTRA Period Spectral Acceleration (c) (sec) 2% equipment damoina 31 equipment damping TIME DYNRES DYNRE1 TIME DYNRES DYNRE1 0.0100 0.7669 0.7777 0.7670 0.7812 0.0143 0.7560 0.7756 0.7560 0.7564 0.7752 0.7564 0.0200 0.7668 0.7765 0.7668 0.7766 0.0222 0.7701 0.7796 0.7701 0.7796 0.0250 0.7755 0.7842 0.7752 0.7840 - 0.0263 0.7766 0.7859 0.7766 0.7858 0.0278 0.7783 0.7872 0.7785 0.7874 - 0.0303 0.7865 0.7945 0.7862 0.7944 0.0333 0.7932 0.8058 0.7928 0.8052 0.0364 0.7976 0.8110 0.7976 0.7975 0.8110 0.7975 0.0385 0.8057 0.8184 0.8048 0.8178 0.0400 0.8108 0.8230 0.8108 0.8238 0.0435 0.9272 0.9330 0.9089 0.9105 0.0455 0.9847 0.9847 0.9577 0.9577 0.0476 0.9560 0.9678 0.9316 0.9544 0.0500 1.0170 1.0631 1.0170 1.C009 1.0270 1.0009 0.0526 1.2583 1.2583 1.1383 1.1883 0.0556 1.3943 1.3943 1.3021 1.3021 0.0588 1.5135 1.5135 1.3488 1.3488 0.0606 1.5037 1.5903 1.5037 1.3556 1.4380 1.3556 0.0625 1.5162 1.5163 1.3862 1.3862 0.0667 1.3885 1.4437 1.3885 1.1978 1.2497 1.1979 0.0f90 1.4080 1.4080 1.3037 1.3114 0.0714 1.5321 1.5569 1.5321 1.4084 1.4291 1.4084 1541 070

Period Spectral acceleration (a) (sec) 2% equicment damoino 31 equipment damping TIME DYNRES DYNRE1 TIME DYNRES DYNRE1 0.0741 1.5607 1.5695 1.4546 1.4546 0.0769 1.4678 1.5567 1.4678 1.3889 1.4192 1.3889 0.0800 1.5216 1.5216 1.4334 1.4334 0.0833 2.1793 2.1793 1.8582 1.8608 0.0870 1.9997 2.0243 1.8192 1.8337 0.0909 1.8859 1.8859 1.5661 1.5858 0.0952 1.7714 1.8091 1.7714 1.6160 1.6338 1.6160 0.1000 1.8304 1.8304 1.7602 1.7602 0.1042 2.1360 2.1360 1.9823 1.9823 0.1075 2.3906 2.3906 2.1645 2.1645 0.1111 2.4647 2.4647 2.1469 2.1469 0.1176 1.9994 1.9994 1.7935 1.7935 0.1250 2.57u2 2.5742 2.2998 2.2998 0.1290 3.2954 3.2954 2.7776 2.7776 0.1333 3.7542 3.7542 2.9873 -2.9873 0.1379 3.5311 3.5311 2.8838 2.8838 0.1429 3.2579 3.2579 2.9290 2.9290

0. 1481 4.8062 4.8063 4.0953 4.0953 0.1515 5.6867 5.6867 4.6958 4.6958 0.1538 5.6070 5.6070 4.7719 4.7720 0.1613 6.5735 6.5734 5.7952 5.7951 0.1667 8.7033 8.7034 6.6873 6.6873 0.1695 6.8596 6.8596 '

5.8582 5.8582 0.1818 4.0481 4.0481 3.5149 3.5149 0.1905 4.1172 4.1172 3.4990 3.4990 1541 071

a Period Spectral Acceleration (a) (Sec) 2% equipment damping 3% equioment damoing TIME DYNRES TIME DYNRE5 0.2000 5.7318 5.7318 4.6577 4.6577 0.2083 5.0152 5.0151 4.0309 4.0309 0.2174 3.1199 3.1199 2.7205 2.7206 0.2222 2.6946 2.6947 2.4514 2.4514 0.2273 2.4151 2.4151 2.2294 2.2294 0.2381 2.4698 2.4698 2.1699 2.1699 0.2500 1.5553 1.5553 1.4919 1.4919 0.2632 1.6646 1.6646 1.4922 1.4922 0.2778 2.4781 2.4780 2.0004 2.0004 0.2941 2.0811 2.0811 1.8997 1.8997 0.3125 2.2204 2.2204 1.8821 1.8821 0.3333 1.7806 1.7805 1.4784 1.4784 ' 0.3704 1.6777 1.6777 1.4821 1.4821 0.4000 1.7672 1.7672 1.5651 1.5652 0.4444 1.2379 1.2379 1.1917 1.1917 0.5000 0. 9.557 0.9557 0.9143 0.9144 0.5714 0.7112 0.7112 0.6834 0.6834 0.6667 0.6126 0.6126 0.5609 0.5609 1.0000 0.3689 0.3690 0.3262 0.3262 _S 1541 072

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