Seismic qualification of equipment — research needs

Nuclear Engineering and Design 59 (1980) 149-153 © North-Holland Publishing Company

SEISMIC QUALIFICATION OF EQUIPMENT - RESEARCH N E E D S *

C. CHEN and F.L. MOREADITH Gilbert~Commonwealth Engineers and Consultants, Reading, PA 19603, USA

In general, industry has followed the Institute of Electric and Electronic Engineers' (IEEE) Recommended Practices for Seismic Qualification of Class 1E Equipment for Nuclear Power Stations (IEEE Standard 344-1975). However, the U.S. Nuclear Regulatory Commission Regulatory Guide 1.100 notes exceptions to a small part of the IEEE Standards. This paper describes research needed to reconcile the differencies between the IEEE Standard and the Regulatory Guide. In addition, the paper discusses the effects of shake table mass and stiffness on the dynamic response of equipment tested, and the effect attributable to the difference between methods of attaching to the shake table and the actual in-situ attachment method.

and stiffness on the response of the equipment: ~AZhen the equipment under test is small, this effect is negligible. But when the equipment is large, heavy, and has a high center of gravity, the interaction o f the equipment with the shaker table can be severe. This kind of interaction is analyzed by a two-mass model and the research needed to obtain more meaningful results is identified.

1. Introduction

The Institute of Electric and Electronic Engineers' (IEEE) Standard 3 4 4 - 1 9 7 5 , "Recommended Practices for Seismic Qualifications o f Class 1E Equipment for Nuclear Power Stations", prepared by IEEE Working Group 2.5, has been widely followed by the nuclear industry - not only for electric and electronic equipment but also mechanical, instrumentation and control equipment. The NRC Regulatory Guide 1.100 was prepared through joint efforts the reactor vendors, consulting engineers, testing laboratories, equipment vendors, utilities, and the U.S. Nuclear Regulatory Commission (NRC). It is clear that the Standard is not completely acceptable to NRC, as indicated in the Guide (Seismic Qualification of Electric Equipment for Nuclear Power Plants). The primary concern reflected by the Guide is related to the 1.5 factor used in the static coefficient analysis. This paper first interprets the static coefficient analysis by the method o f normal modes. Recommendations are made to define research required to reconcile the difference between the two documents. One area o f concern in the testing of equipment seismic qualification is the effect of shaker table mass

2. The static coefficient analysis The static coefficient analysis is a simplified static method to replace dynamic analysis. The static coefficient, multiplied by the weight of the structure or equipment, is used as the external load in the analysis. Since it is a simplified method, a penalty must be paid by selecting a conservative static coefficient. IEEE Standard 3 4 4 - 1 9 7 5 recommends a factor of 1.5 times the peak o f applicable spectrum value as the static coefficient; the NRC Regulatory Guide takes exception to the Standard, on the ground that there is no adequate evidence to substantiate the validity of the 1.5 factor. Chen [1 ] applied the method of normal modes to prove that the factor of 1.0 instead of 1.5 is conservative enough, based on current industry practice. Proof of this recommendation is presented in the following. Defining the diagonal mass matrix and mode shape matrix as [m] and [¢], respectively, the ortho-

* Paper presented at the SMiRT-5/USNRC Panel Session "Status of Research in Structural and Mechanical Engineering for Nuclear Power Plants", Berlin (West), 14 August, 1979. 149

150

C. Chen, F.L. Moreadith /Se&mic qualification of equipment same as the pseudo-acceleration, therefore,

gonality of normal modes is expressed by [~]TIm] [q~] = [M] ,

(1)

where the superscript T indicates a transposition of the matrix, and [34] is the diagonal model mass matrix. Since an arbitrary function can be represented by the superposition of orthogonal functions (ref. [2]),

03 = [4] ( A ) ,

(2)

where {or}is an arbitrary vector assigned at all dynamic degrees of freedom, and {A} is the amplitude vector to be determined later in order to satisfy eq. (2). Assuming such expansion exists, and premultiplying both sides ofeq. (2) by [~]T[m] results in [~]W[mlQ3 = [~]T[m] [¢]{A} .

(3)

Making use of the orthogonality relationship of eq. (1) it yields [q~lT[m]{3') = [M] (A}.

(4)

When {f} is a unit vector, the amplitude vector {A} becomes the vector of participation factors under seismic-type excitation. Thus, eq. (2) becomes {1} = [q~] {,y},

(5)

where {7} is the vector of participation factors. Now, the equation of motion in normal mode coordinates are, Xi + 2~(.0i~ i + (.d2~ki = --'yi f ,

(6)

where Xi is the amplitude of ith mode, and dots indicate time derivatives, fl, co, and I7 are the percentage of critical damping, model frequency, and input acceleration, respectively. With the initial conditions as ~ki(0) = )ki(0 ) = 0 ,

(7)

the solution of eq. (6) is ([3]) as 60"2 -" --Ti F ~t~2)lJ 1/2 e - # w i ( t - r) y(7-) ~'i(t) = 602 g 6Oi(1 × sin[wi(1 -/32) 1/2 (t - 7-)] d r .

(8)

Representing the integral by rli(t) results in )~i(t) = Tir/i(t)/co 2. For systems with small damping, the maximum absolute acceleration response is approximately the

(9)

(~i + Ti17)max = (~ki)max60/2 = Ti(r/i)max •

(10)

It would be conservative to assume: ( I ) Maximum modal responses occur at the same time, and (2) the maximum modal responses (rl/)max are replaced by the maximum of the maximums, r/max . Using these assumptions, the superposition of normal modes, and making use of eq. (10) becomes ([q~] (~() + (1) 17)max = {1}r/max •

(11)

Making another conservative assumption that r/max is equal to the maximum of the response spectrum calculated from a recorded time history (Sa)max, eq. (11 ) becomes ([4q {X} + {1} 17) = {1}(Sa)ma x .

(12)

Eq. (12) proves that the peak of the applicable design spectrum in the unit o f g is a very conservative static coefficient. The only remotely possible spoiler of this conservative derivation is at the time when [0l {')'r/(t)) > [41 {"/)r/max •

(13)

Statistical study can be performed to prove that this possibility is of sufficiently low probability to be ignored. In a statistical study, the static analysis can be performed with the peak of the NRC Regulatory Guide 1.60 spectrum as the static coefficient. Then timehistory analyses can be performed with the set of time-histories that were used to derive Regulatory Guide 1.60 as input. The absolute maximum responses can be used to calculate means and standard deviations. These results can then be compared to the static analysis responses, using the mean-plus-one standard deviation. If the static analysis results are greater, the point is proved. It is expected that this will be the case for simple and multiple-span type structures and equipment. The 1.5 factor used in the past was derived in previous studies [4,5,6], and utilized the response spectrum method with SRSS combination. Ref. [7] indicated that SRSS combination can overestimate the responses of close modes.

C. Chen, F.L. Moreadith /Seismic qualification of equipment

Further, the peak of the spectrum value was used as the response of every mode. This is a highly conservative and unrealistic assumption. It is believed that the more realistic static coefficient of 1.0 can be proven when the above suggested statistical study is carried out.

When the equipment being testes is heavy and has a high center of gravity, it will interact with the shaker table. Tl~is interaction interferes with measurement of the equipment's natural frequency, and causes inaccurate measurement of the testing response spectrum (TRS) because the rocking motion of the table interferes with the translational input of the table to the equipment. This interaction effect can be studied theoretically by a two-mass model [8]. Designating the relative translational degrees of freedom o f m 1 and m2 of fig. 1 as Ul and u2, the equations for the amplitudes of the harmonic motions are

(14)

and m=u260 2 = k2(u 2 - Ul) ,

(15)

where co is the circular frequency. Assume that mz is in resonance with m 1 ('tuned') by making kl + k 2 _ k2 _ ] 2 , ml

Using the equations of motion, ((60//0 2 - 1) ul + #u2 = 0 , and Ul + ((60/f)2 _ 1)Uz = 0 ,

(18)

(60/I) 2 - 1 = -+x/if,

(19)

or 601,2 =fx/1 +X/ft.

(20)

The mode shapes of the system are u2/ul =

-+X/q-~

(21)

•

Eq. (20) indicates that the combined frequencies are close to original frequency when the mass ratio is small and are different from the original frequency when the mass ratio is large. The mode shapes shown in eq. (21) indicate that the ratio of the relative response of the supported mass to that of the supporting mass is proportional to the square root of the inverse of the mass ratio. In case of non-resonance, it is assumed that kl + k 2 _ a k 2 = f 2 . ml

(22)

m2

Eqs. (17) and (18) become 2

1)u,+_u2o=0

(16)

m2

(17)

where/l = m 2 / m l is the mass ratio. The combined frequencies of the systems are obtained by substituting eq. (I 8) into eq. (1 7):

3. Interaction of shaker table with equipment

mlu160:2 = k l t t 1 + k2(u 1 - u2) ,

151

(23) (24)

u da + ( (60/f)z _ 1/a) us = O .

Substituting eq. (24) into eq. (23); m2 -

k2

11{{7;-'t

(25)

i

or

(60/f)4 _ (1 + 1/a)(60/f) 2 + 1/a - la/a 2 = 0 .

Iit

Im I

Solving for (60/f)2,

kl

(60/f)2=½(1 + I / a ) + ~

Ill

1

I/0 J +

-4

-

(26)

j) P

. (27)

The combined frequencies of the system are

Fig. 1. Two degrees of freedom system. 601,2 =

-

\a

a2]_]

f (28)

152

C Chen, F.L. Moreadith /Seismic qualification of equipment

The mode shapes are obtained'by substituting eq. (28) into eq. (23): U2 _

ui

-a

/2

X 1-1+~)+-1/~+~)2-4(~-a~-)2 -

• (29)

Instead of eq. (l 6), if the 'tuned' system is k i / m 1 : k2/m2 =/-2 .

(30)

It can be shown that a system described by eq. (30) can also be described by eq. (22) with

a=l +/l.

f2=(l+tx)f21,

(31)

Thus, eqs. (28) and (29) are also the frequencies and mode shapes of the 'tuned' system described by eqs. (30) and (31). When the non-resonance is defined as (32)

kl - b k-~-2=f21 , ml m2

it can be shown that a system described by eq. (32) can also be described by eq. (22) with f2 :f12 (I + b ) ,

a=b+li.

(33)

Therefore, eqs. (28) and (29) are also the frequencies and mode shapes of the 'non-resonant' system described by eqs. (32) and (33). The frequencies of the coupled systems described by eqs. (28), (32), and (33) are plotted in fig. 2 as

" - - T - - S

I

a function of mass ratio. The abscissa is the frequency ratio of supporting mass to supported mass, and the ordinate is the ratio of combined frequencies to supporting mass frequency. During resonance (./'1 = f2), the combined frequencies are close to original frequencies. Away from resonance, the first combined mode represents the motion of the supporting mass when frequency ratio is less than one or supporting mass frequency is less than supported mass frequency; the second combined mode represents the motion of the supporting mass when frequency ratio is larger tan one, or supporting mass frequency is larger than supported mass frequency. Thus, when frequency variation is used as the criterion for uncoupling, the combined frequency of the proper mode should be used for comparison. In other words, the fundamental frequency of the coupled system is not always the proper one for comparison. The mode-shape ratios of the coupled system described by eqs. (29), (32) and (33) are plotted in fig. 3 as a function of mass ratio. The abscissa is the frequency ratio, and the ordinate is the mode-shape ratio of supported mass to supporting mass. During resonance, both modes contribute similar motion to either supporting mass or supported mass. Away from resonance, the first combined mode shape contributes to the motion of supporting mass when frequency ratio is less than one; the second combined modeshape contributes to the motion of supported mass when frequency ratio is larger than one. The mode-shapes of systems with mass ratio of 0.01 and frequency ratios of 0.1, 1.0, and I0.0 are

7O

I I I I L

2 ~

~o

2 nd MODE

2 11

kl

Lk2

, k,_bk2 L,-b{2..... / /

k k ~ 2 f2_

k2 b

30

50

-

I

I - m I-

m2 -

=0.01

,,,1o7, j,=

2

.t2

= ~I:D ~-2=DI2

.<<>0 '~/ u2/u l lO

#=01

-3050

2rid M O D ~ I

~:001 #~:0001

=05 0

OI

I_

I MODE ] I

I I I I I I0 05 fb=fl/f 2

50

Fig. 2. Combined frequencies vs. frequency ratios.

I00

7O

OI

I

I 05

LO ~b:fl/f 2

I

[ I I II 50 I00

Fig. 3. Mode shape ratio vs. frequency ratios.

C Chen, F.L. Moreadith /Seismic qualification o f equipment

plotted in fig. 4. It indicates that m I and m 2 are uncoupled for frequency ratios of 0.1 and 10.0. The testing laboratories can extend the above derivation and figs. 2 and 3 to evaluate influence of the interaction on the equipment frequency measurement and equipment responses.

4. Conclusions IEEE Standard 344-1975 recommends a factor of 1.5 in the static coefficient analysis. NRC Regulatory Guide 1.100 does not accept this factor, for lack of evidence. A statistical study is proposed to prove that a factor of 1.0 is sufficient for static coefficient method. The interaction effects of heavy equipment and shaker table is studied by a two-mass model. The combined frequencies and mode-shapes are as plotted in figs. 2 and 3. Testing laboratories are encouraged to extend the result and to evaluate the influence of the interaction on the equipment frequency measurement and equipment response.

153

References [ 1] C. Chen, ASME Energy Technology Conference, Houston, TX, September 1977, Paper no. 77-PVP-59. [2] F.B. Hildebrand, Advanced Calculus for Applications (Prentice Hall, Englewood Cliffs, NJ, 1964). [3] W.C. Hurry and M.R. Rubinstein, Dynamics of Structures (Prentice Hall, Englewood Cliffs, NJ, 1964). [4] J.D. Stevenson and W.S. Lapay, ASME Paper no. 74-NE-9, presented at Pressure Vessels and Piping Conference, Miami, 1974. [5] J.M. Gwinnand N.A. Goldstein, ASME Paper no. 74-NE-6, presented at Pressure Vessels and Piping Conference, Miami 1974. [6] E.N. Liao and R.S. Marda, On static loads method for seismic design of nuclear power plant facilities, Earthq. Engrg. Conf., University of Michigan, 1975. [7] J. Penzien, Earthquake response of irregularity shaped buildings, 4th World Conf. Earthq. Engrg., Chile, 1969, Vol. II. [8] C. Chen, SMiRT-5, Berlin (West), August 1979, Paper no. K8/l*.