The uncoupling criteria for subsystem seismic analysis

Nuclear Engineering and Design 57 (1980) 245-252 © North-Holland Publishing Company

THE UNCOUPLING CRITERIA FOR SUBSYSTEM SEISMIC ANALYSIS * Chang CHEN

Gilbert/Commonwealth Companies, Reading, PA 19603, USA Received 5 September 1979

This paper discusses the uncoupling effects of a subsystem from the system based on frequency, mode shape and response variations. The two-mass system is first used to study the problem, and a closed form solution of the frequency variation is derived for the two resonant masses with different mass ratios. Since the coupled and uncoupled analyses have a different number of modes, proper selection of modes for frequency variation check is also discussed. The resonance effect of a coupled analysis is shown in the mode shapes. Thus, the closed form solution of mode shapes for two resonant masses is also derived as a function of mass ratio. Since the response variation is a function of input, the response variation of a two-mass system subjected to white noise input is discussed. Since multiple degree of freedom systems are used in the majority of cases, the results of the two-mass system are extended to the multiple degree of freedom systems by the concept of normal modes. Each normal mode can be represented by a single degree of freedom system with equivalent modal mass and equivalent modal spring. Different equations were used in the past to define the equivalent modal masses depending on the objective of the analysis. The method of defining the equivalent modal mass which takes into account the location of a subsystem, is recommended. Once the equivalent two-mass system of the multiple degree of freedom systems and subsystems is derived, the frequency, mode shape, and response variations of the multiple degree of freedom system and subsystem can be assessed. The above coupling/uncoupling analysis is applied to two different situations. The f'trst one is the building-equipment interaction usually with small mass ratios. The second one is the equipment-equipment interaction where the mass ratio can be large. Here, the equipment includes piping systems, pressure vessels, pumps, etc. The uncoupling analysis of the first case is required because the equipment information is not available during the building analysis. The uncoupling anal ysis of the second case is required due to practical need to reduce the size of the model. The recommendations of the uncoupling analysis of both cases are presented.

1. Introduction

deviation should be used as the criterion. Frequency deviation will cause response deviation if the actually recorded time history is used as input. However, when the smoothed response spectrum of USNRC Regulatory Guide 1.60 is used as input, reasonable frequency deviation should not cause appreciable response deviation. The frequency deviation of the primary system can cause the resonance peak of the floor response spectra to shift on the frequency axis. This effect can be accounted for in the floor response spectrum peak broadening process. In addition to the system frequency and input, the mode shape of the coupled system will also influence the response, as will be discussed later. The eigenvalue deviation and uncoupling criterion was discussed by Pickel [I ]. Lin and Liu [2] extended the models used in [1 ] and discussed the frequency

The US Nuclear Regulatory Commission Standard Review Plan (USNRC SRP) Section 3.7.2 specifies an acceptable criterion on the uncoupling of a subsystem from the system. This criterion is defined in terms of the mass ratio and the frequency ratio of the subsystem to the system without explaining the method of calculating the mass ratio of a multiple degree of freedom system. It does not explain either whether the criterion is based on frequency deviation or response deviation of the system or subsystem. From an engineering point of view, the response * Invited Paper K8/1 *, presented at the 5th International Conference on Structural Mechanics in Reactor Technology, Berlin (West), 13-17 August, 1979. 245

C Chen / Uncoupling criteria Jbr subsystem seismic analysis

246

deviation due to uncoupling. Hadjian [3] compared the uncoupling criteria so far published and recommended a new criterion with smooth transition zones based on frequency deviation. The uncoupling effects on the response deviation of a two degrees of freedom system subjected to white noise input was studied by Crandall and Mark [4]. The study indicates a reduced resonance response of the supported mass by coupling analysis as a function of the ratio of its mass to the supporting mass. The supporting mass resonance response is also reduced for certain mass ratios. Chen [5,6] took advantage of this phenomenon and applied it to the nuclear power plant designs. Later the results of the two degrees of freedom system subjected to white noise input were also utilized by others [2,3]. This paper discusses the two degrees of freedom system first. The closed form solution of frequencies and mode shapes are also derived. The results are extended to multiple degree of freedom system.

is "tuned" into ml by making kl + k2 _ k2 ml

-),'2

The equations of motion are now ((co/f) 2 - 1) ul + Uu2 = 0

The resonance effect of a small mass supported by a large mass was discussed by Biot [7]. It was described as the "whip effect". Biot's concept is extended here to derive the closed form solution of the two degrees of freedom system. Denoting the relative translocational degrees of freedom of m 1 and m2 of fig. 1 as L/1 and u2, the equations for the amplitudes of the harmonic motions are

U 1 + ( ( ~ , ) / D 2 - - 1) U 2 = 0 ,

(5)

where # = m 2 / m a is the mass ratio. The combined frequencies of the systems are obtained by substituting eq. (5) into eq. (4): (co/f) 2 - 1 = +/a v:

(6)

or

~ol,2 = f(1 + Ul/2) v2 .

(7)

The mode shapes of the system are

(8)

•

Eq. (7) indicates that the combined frequencies are close to the 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. (8) 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 the case of nonresonance, we can assume that k~ + k2 ml

m l u l w 2 = k l / d l + k 2 ( h ' 1 - H2)

(4)

and

u 2 / u 1 = +-u - v 2

2. Two degrees of freedom system

(3)

.

m2

k2 = f 2

- a--

.

(9)

m2

(1)

Eqs. (4) and (5) become and (2)

m2u2o~ 2 = k2(u2 - u l ) ,

((ff-1)

Ul +--~u2=0'a

(10)

u2 = 0

(11)

where ~ is the circular frequency. Assume that m2

--a + ,m 2

-

a

"

Substituting eq. (11) into eq. (10), we have

k2

1

-w2

ta : 0 ,

(12)

}mI

77

kl

or

r/7"

(f)4

Fig. 1. T w o d e g r e e s o f f r e e d o m s y s t e m .

-

( +1~6o)2 1 /a 1 altf! +a-~ "~0"

(13)

C. Chen/ Uncoupling criteria for subsystem seismic analysis

247

ported mass to the supporting mass is obtained by assuming that the two modes are in phase, or

Solving for (~/f)2 , /2 ~71/2

(f)2=(l+l)+I(l+l12-4(1-~,Ja/2 (14)

[(1) [(

41)1,211,2 _

The combined frequencies of the system are 1+ a -+

1+

('01,2 =

2

.

(15) The mode shapes are obtained by substituting eq. (15) into eq. (10): 1+

-+

-4

1+

us _ - a u~ p

a '

a2/J 1 .

2 (16)

Instead of eq. (3), if the "tuned" system is as defined in ref. [4], or (17)

k l / m l = k:/rn2 = f] ,

it can be shown that a system described by eq. (17) can also be described by eq. (9) with /a =(1 + p ) f ~

and

a--- 1 + p .

(18)

Thus, eqs. (15) and (16) are also the frequencies and mode shapes of the "tuned" system described by eqs. (17) and (18). When the nonresonance is defined

,Ull~

u~, 2

r\~!

'

(22)

which is the same as in ref. [8]. When there are dampings in the system, denoted by the p6rcentage of critical ~', the upper bound of the response ratio of the "tuned" system as reported in [8] is approximately [ 2 f + mv'~f~:/m~] -~ .

(23)

The two modes of the "tuned" system with small mass ratio have frequencies close to each other, and the USNRC Regulatory Guide 1.92 requires absolute addition, as is done in eq. (22). In reality, close modes seldom respond in phase because the mode shapes as shown in eq. (8) are out of phase. Refs. [9] and [10] give a more detailed description of close modes not responding in phase. The frequencies of the coupled systems described by eqs. (15), (19), and (20) are plotted in fig. 2 as 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 frequency to supporting mass frequency. During resonance 0cl = f2), the combined frequencies are close to the original frequency. Away from the resonance the first combined mode represents the motion of the supporting mass when the frequency ratio is less than one or the supporting mass frequency is less than the supported

as

k , / m , = bk2/rn2 = / ? ,

(19) I ~

it can be shown that a system described by eq. (19) can also be described by eq. (9) with f~ =/?(1 +la/b)

and

a = b +ta.

I

I

I

I

'~

Pl,2 -

2ml

The upper bound of the response ratio of the sup-

I

I

I

I I I

nd MODE

-

f2~' = re't" k, b m28°12 ~ .,2

~5 I.O

0

(21)

I

2.0

]sl N

/

+ml

I

q: 2

+ ~

I I I

3.0

(20)

Therefore, eqs. (15) and (16) are also the frequencies and mode shapes of the "nonresonant" system described by eqs. (19) and (20). For the "tuned" system described by eq. (3), and with mode shapes described by eq. (8), the participation factors due to support acceleration input are

I

i

0.!

i

l

l

0.5

l

l l i

1.0 '/-ff:f,/f2

i

i

i

l

i

5.0

Fig. 2. Combined frequencies vs. frequency ratios.

l l I

10.0

C Chen/ Uncoupling criteria for subsystem seismic analysis

248

70

::oo S07

I I I I I I II kJ m2 ,2 : kl =bk_2.. . . . k~m,~ 'l m~ m R ° 1 2

50

30

'

'

'I

....

P

~/.~

'

'

'I

....

-0.00

--"

p=O001

. . . . . .

u2/uI

~.

-lo

.=Ol

-5030

2nI M O D / I ~

70

I00

0.1

/Ii

0.5

oooo,

?x 10

I

I

1

Jb:ll/f 2

1

5.0

t

il

LO

mass frequency; the second combined mode represents the motion of the supporting mass when the frequency ratio is larger than one or the supporting mass frequency is larger than the supported mass frequency. Thus, when one intends using the frequency variation as the criterion for uncoupling, the combined frequency of 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. (16), (19), and (20) 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

J

mI

lml

-01 9o1~ . -]

q~m I

i.o

)ml

i

i

I !

:,,, ¢ lstMode

2ndMode

fl/f2=O.1

Hill

H,

2ndMode f)/f2=lO.O

Ist Mode 2ndMode IstMode

f)/f2=l.O

Fig. 4. Mode shapes of system with mass ratio o f 0.01.

-

IO.O

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

I.II

;

j&.o,,o

,

/

~1.0.05 (Q,.IO)

/ o.I

"-~

(o,.,o) +

,I

....

I

I.O fz / f)

,

I

t t I,,ld

Fig. 5. M e a n square a c c e l e r a t i o n o f m 2 as a f u n c t i o n

IO

of

system parametersf2/f I and u = m2/ml for ~'1 = t2 = 0.05 when excitation is ideal white noise acceleration foundation. ([4], copyright: Academic Press.)

motion to either supporting mass or supported mass. Away from resonance, the first combined mode shape contributes to the motion of the supporting mass when the frequency ratio is less than one; the second combined mode shape contributes to the motion of the supported mass when the frequency ratio is larger than one. The mode shapes of the systems with mass ratio of 0.01 and frequency ratios of 0.1, 1.0, and 10.0 are plotted in fig. 4. It indicates that m 1 and rn2 are uncoupled for frequency ratios of 0.1 and 10.0. If one intends using frequency deviation as the uncoupling criterion, fig. 2 can be used directly provided that the proper mode is selected for comparison. However, the response variation should be used as the criterion instead of the frequency deviation. Response variation is a much more complicated subject. From the point of view of normal mode method, response is a function of mode shapes,

C. Chen / Uncoupling criteria for subsystem seismic analysis '

'

'

[

'

'

'

~l

f # :ooo

I.O

N

'

'

'

249

I

'

'

'

'

I

,

, , ,

~x

~- o.5 tan

0

,

I

,

0.I

,

,

,

I

t, :005

(O,:lO)

~2 , 0 0 5

( Q 2 : I0)

,

J.O

,

,

1(3.0

f2/fl Fig. 6. Mean square acceleration of ml as a function of system parametersf2/f 1 and p = m2/ml for ~'1 = ~'2 = 0.05 when excitation is ideal white noise acceleration foundation. ([4], copyright: Academic Press.)

participation factors, and modal equation time-history responses. To obtain rigorous results, one has to use an ensemble of recorded time histories as input and perform repeated calculations to obtain mean values and standard deviations of responses. Another approach is to assume some statistical properties of the input function and then calculate the statistical properties of the responses by the random vibration theory. Fortunately, the latter approach was performed in [4] by assuming the input to be white noise. The results were obtained by solving simultaneous differential equations instead of the normal mode method. The mean square acceleration responses o f supported and supporting masses are as shown in figs. 5 and 6. The case/a = 0.0 is equivalent to the uncoupled analysis, and case tz :/: 0.0 are for coupled analyses. As shown in the figures, the uncoupled analyses always produce higher acceleration

responses than the coupled analysis. The reduction of acceleration response of the supporting mass is the same as in mechine design, i.e. the supported mass is acting as a vibration absorber.

3. Multiple degree of freedom system The results of the two degrees of freedom system are applicable to multiple degree of freedom systems, as shown in fig. 7 by the method of normal mode. The supporting structure, for example, can be represented by five normal modes. The first question one would ask is how can one calculate the modal mass of each normal mode of the supporting structure taking into account the location of the supported mass. Different methods have been used in the past to calculate the modal mass. One of them is to define

C. Chen / Uncoupling criteria for subsystem seismic analysis

250 m1 m2

m3 J m0 m4 m5 IIII1

Fig. 7. Multiple degrees of freedom system.

it as M=

( ~ miui) 2

(24)

~m,u~ This definition is independent of the amplitude of mode shapes. The other way of defining it can be

M = ~ miu?,

(25)

which is a function of the amplitude of the mode shape. The most straightforward way of defining modal mass is to equate the kinetic energy of the normal mode and that of the equivalent system. Using fig. 7 as an example, the supported mass m0 is supported at ma. The equivalent system will be located at ma, and the equation of conservation of kinetic energy is

~ mi 02 = Mo~ ,

4. Application to nuclear power plant design

(26)

where oi is the velocity at m i. The velocity due to a velocity spectrum input is related to the mode shape by ~

miui

ot -

SvOe)

tti,

(27)

mi u2

In nuclear power plant design it is not practical to include all the equipment in the coupled building model. Instead, a general practice is to include the weight of the equipment in the building model. The interaction effect can be described as kl m i + m2

where Sv(f) is the velocity spectrum value at corresponding frequency. Substituting eq. (27) into eq. (26) we have

M = (~a miu~/u]).

unity. Physically, this means that the modal mass for the fundamental mode of a cantilever beam-type structure is smaller when the supporting point is at the top than it is at lower points. Extending the conclusion to higher modes, the modal mass is smallest when the supporting point is at the point with the largest mode shape value of that mode. When the results of the two degrees of freedom system are applied to the normal modes of multiple degree of freedom systems represented by eq. (25), it is assumed that the change of mode shapes of the multiple degree of freedom system due to interaction with the supported system is negligible. This is a reasonable assumption with the mass ratios encountered in nuclear power plant design. When the small supported mass is "tuned" into one of the normal modes of the supporting structure, the coupled system will have modes with frequencies close to each other. When the root sum square method is used to combine the modal responses due to response spectrum input, the small supported mass acceleration response is overestimated in relation to the time history analysis [11,12]. The cause of this overestimate is explained in [9,10]. In spite of the above, the USNRC Regulatory Guide 1.92 still requires close modes to be combined by the absolute sum method.

(28)

It becomes apparent that eq. (28) is identical to eq. (25) when the mode shape is normalized such that the mode shape value at the supporting point is

k2 - c - - = f~a.

(29)

m2

This is the same as eq. (19) with the substitution of f~l =(1 + tt)f~

and

b =c(1 +/1).

(30)

Thus, figs. 2 and 3 are also applicable in this case. There is numerous equipment installed in a building. We can simulate the situation by using the simple case of two supported masses and a supporting mass, as shown in fig. 8. The equations of harmonic

C Chen / Uncoupling criteria for subsystem seismic analysis

251

conclusions made before are also applicable in this case. The equipment and piping interaction is similar to the equipment-building interaction except that the mass ratio is larger and the multiple support inputs also complicate the issue. Nevertheless, the two degrees of freedom system results are still applicable when the modal mass is properly defined.

kl

Fig. 8. Two masses supported by a common mass.

5. Conclusions motion are mlul60 2 = k~ua + k2(ul

m2u260 2 = k2(u 2 m3u360 2

=

k3(u 3

- -

- -

- u2) + k3(Ul - u3) ,

(31)

Ul) ,

(32)

Ul).

(33)

For a "tuned" system: kl + k2 + k3_ k2 ml

k3

m2 -1

(34)

rna

(35)

u l + lalu2 + l~2u3 = O ,

t/l+

-- I

ul+

-1

)

/../2=0,

(36)

u3=0,

(37)

where ]21 "=

m2/ml

and

/a2 = m 3 / m l

•

Solving eqs. (35), (36), and (37) simultaneously, we have (60/f)2 _ 1 = +-6ul +/a2) 1/2 ,

(38)

and the frequencies are 601,3 = (1 + ~ x +/a2)1/2) 1/2 f4 •

(39)

The mode shapes of the system are

{u3, u2, u~} =

The above discussion can be concluded as follows. (a) If one intends to use frequency deviation as the basis of the uncoupling criterion, fig. 2 can be used directly once the acceptable deviation is established. However, the proper combined mode which is not always the fundamental mode should be used for comparison. (b) If one intends using the mode shapes as the basis of the uncoupling criterion, figs. 3, 4 can be used directly. (c) From an engineering point of view, the response deviation should be used as the uncoupling criterion. Using white noise as input, the uncoupling analysis always produces a more conservative acceleration response, as shown in figs. 5, 6. (d) The results of the two degrees of freedom system can be applied to multiple degree of freedom systems shown in fig. 7 when the modal mass is as defined in eq. (25) and with the mode shape normalized to unity at the supporting point. (e) When the supported mass is rigidly attached to the supporting mass, the results are still applicable, as shown in eqs. (29), (30). (f) When several masses are supported by one common mass, the "tuned" system frequencies and mode shapes of eqs. (39), (40) are similar to the results of the two degrees of freedom system. (g) The results are also applicable to equipment and piping to equipment and piping interaction provided that proper modal masses are used.

(-+(~ + u2) -~/2, +-0a, + ~2) -~/2, 1}. (40)

The frequencies and mode shapes of eqs. (39) and (40) are for first and third mode. The second mode, with 602 = f , , is a trivial mode. Eqs. (39) and (40) are similar to eqs. (7) and (8). Thus, the general

References [1] T.W. Pickel, Jr., SMiRT-1, Paper K1/2*, Berlin, Sept. 1971; also Nucl. Eng. Des. 20 (1972) 323. [2] C.W. Lin and T.H. Liu, Proc. Extreme Load Conditions and Limit Analysis Procedures for Structural

252

[3] [4] [5] [6]

C Chen / Uncoupling criteria for subsystem seismic analysis Reactor Safeguards and Containment Structures, Paper No. U3/3, Berlin, 1975. A.H. Hadjian, Proc. 6th World Conference of Earthquake Engineering, India, 1977. S.H. Crandall and W.D. Mark, Random Vibration in Mechanical Systems, (Academic Press, 1963). C. Chen, Proc. SMiRT-2, Berlin, 1973. C. Chen, Proc. ASCE Specialty Conference on Structural Design of Nuclear Power Plant Facilities, Chicago, 1973; also Nucl. Eng. Des. 30 (1974) 100.

[7] M.A. Biot, Trans. ASCE, Paper no. 2183 (1943). [8] N.M. Newmark, SMiRT-1, Berlin, 1971; also Nucl. Eng. Des. 20 (1972) 303. [9] C. Chen, Letter to USNRC (March 1975). [ 10] C. Chert, ASME Energy Technology Conference, Houston, Sept. 1977, Paper No. 77-PVP-59. [11] J. Penzien and A.K. Chopra, Proc. 3rd World Earthquake Engineering Conference, New Zealand, 1965. [12] J. Penzien, Proc. 4 th Earthquake Engineering Conference, Chile, 1969.