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Deterministic and stochastic earthquake response analysis of the containment shell of a nuclear power plant
Nuclear Engineering and Design 72 (1982) 309-320 North-Holland Publishing Company
309
DETERMINISTIC AND STOCHASTIC EARTHQUAKE RESPONSE CONTAINMENT SHELL OF A NUCLEAR POWER PLANT
ANALYSIS OF THE
A.H. YOUSAFZAI Department of Mechanical Engineering, Shiraz University, Shiraz, Iran
Goodarz AHMADI Department of Mechanical and Industrial Engineering, Clarkson College, Potsdam, N Y 13676, USA
Received 28 June 1982 Response of the containment shell of a nuclear plant to earthquake ground motion is considered. A finite element model of the structure is developed and SAP IV structural analysis program is employed for the determination of the frequencies and the corresponding mode shapes of the structure. The response of the containment shell to several past earthquakes are analyzed and the results are discussed. Stochastic models of earthquake ground acceleration are then considered and the general expressions for the power spectra, cross correlations and the mean-square responses are derived. The root mean-square of the relative displacement responses of various nodal points of the containment shell structure subjected to stationary as well as nonstationary random support motion are evaluated. The stochastically estimated maximum displacement responses are compared with those obtained from a deterministic analysis and reasonable agreements are observed.
1. Introduction
Dynamic response analysis of structures subjected to earthquake ground motions is one of the basic requirements for their aseismic design. Such an analysis is of prime importance for the design of nuclear power plants, dams and other high rise buildings. In the early 1960s, the first designs of large scale power plants incorporating seismic analysis were being prepared. By the late 1960s, dynamic analysis of nuclear power plants structures and equipments had become a common practice for their design. For a specified base motion or any other forcing function, it is always possible to compute the time history of a given structural system using several deterministic analytical or numerical techniques as described by Clough and Penzien [1], Hunty and Rubinstein [2], Meirovitch [3], Newmark and Rosenblueth [4] and Omato [5]. At present time, due to the extensive developments in the field of digital computers software such an analysis has become a rather sophisticated art as well as technique. A n u m b e r of complex computer programs such as SAP IV [6,7], N A S T R A N [8-11] and N O N S A P [12] among others as summarized for instance in [ 12-14] are developed. Simplified multi-degrees-of-freedom models 0029-5493/82/0000-0000/$02.75
as well as finite element analysis of containment shell of nuclear power plants are discussed in [14-23], among others. In the common practice of deterministic response analysis, several past earthquake accelerograms are usually employed in order to take into account the possible variations in the frequency content of the earthquake ground motions. However, there always remains uncertainties about the nature of the future earthquakes at a site of interest due to the lack of sufficiently large n u m b e r of earthquake records for similar sites. In the recent years, statistical method of analysis is proposed as an alternative to the deterministic response analysis. In this approach the earthquake ground acceleration is modeled by a random process and the response analysis is treated as a problem in random vibrations. Several stochastic models of earthquake ground motion were proposed in the past as summarized by Clough and Penzien [1] and Ahmadi [24]. Among the most well known models are the white noise representation of Housner [25] and Bycroft [26], finite duration white noise model of Rosenblueth et al. [4,27,28], filtered white noise representation of Kanai [29] and Tajimi [30] and nonstationary models of Amin and Ang [31] and others [1,4,5]. Response of multi-degree-of-freedom systems to sta-
© 1982 N o r t h - H o l l a n d
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A.H. Yousafzat, G. Ahmadl / Deterministic and stochastic earthquake re.s7)onseanalysis
tionary excitations were considered by several investigators (see for instance [1-5], [32-35]). Recently, the mean-square response of structures to random excitation was studied by Gersch [36] and Zeman and Bogdanoff [37]. The response of single degree of freedom systems to nonstationary random excitation was investigated by Barnoski and Maurer [38,39], Bucciarelli and Kuo [40], Caughey and Stumpf [41], Corotis et al. [42,43] and Bogdanoff et al. [44]. Response of beams and plates to nonstationary random loads were studied by Ahmadi and Satter [45-47]. Response of structures to nonstationary random excitation was considered by Holman and Hart [48] and Ahmadi [49,50]. In the present study earthquake response of the containment shell of a nuclear power plant which resembles those under construction in the southern part of Iran is investigated. Both deterministic and stochastic methods of response analysis are employed. A finite element model of the shell and its foundation is developed where a single translational degree-of-freedom at each nodal point is assumed. SAP IV structural analysis program is used and the first 25 natural frequencies and the corresponding mode shapes of the structure are obtained and discussed. The response of the containment shell structure to several past earthquake ground accelerations are determined and the results are compared with those found from the response spectra of the corresponding earthquakes. In the probabilistic phase of the analysis, stochastic response of the structure to random earthquake ground motion is considered. The formal general solutions in time as well as frequency domains for a system with its damping matrix being a linear combination of the mass and stiffness matrices is discussed. Statistically stationary models of earthquake ground acceleration is considered and the general expressions for the meansquare responses of various nodal points, are derived. For small damping coefficients, a simplified approximate solution for the mean-square response is presented. Nonstationary stochastic models of earthquake ground accelerations are also employed and the response of the containment shell structure to such a nonstationary support motion is studied. General expressions for the cross correlations as well as meansquare responses are derived. Numerical examples for E1 Centro (1940) and Taft (1952) earthquakes are considered and the stochastically predicted maximum responses are compared with those obtain from the deterministic analysis and reasonable agreements are observed.
2. Deterministic method of analysis In the present study, a finite element model of the containment shell of a nuclear power plant is considered and the structural analysis program SAP IV as developed by Bathe, Wilson and Peterson [6] is employed for the numerical computations. The program uses the finite element technique and assumes the structural system to be divided into a number of discrete elements interconnected only at a finite number of nodal points. The properties of the complete structure are then found by evaluating the properties of the individual finite elements and superposing them appropriately. The corresponding nodal point equilibrium equations are of the form [M](X)+[C](x)+[K](x)=(P(t)),
(1)
where [M] is the mass matrix, [C] is the damping matrix and [KI is the stiffness matrix of the element assemblage. The vectors (x), (5:), (2) and (P(t)) are the nodal displacements, velocities, accelerations and generalized loads, respectively. When the containment shell is subjected to an earthquake ground acceleration, the generalized force is given by { P ( t ) ) = - [M](1}2g(t),
(2)
where (l} is a column matrix whose all elements are unity and £g(t) is the horizontal acceleration of earthquake ground motion. The stiffness matrices in the program are formed by direct addition of the element matrices, for example [KI=[K,]+[K2]+
... + [K,],
(3)
where [K], [Ki], and [Kn] are the stiffness matrices of the first, second and n th element, respectively. The mass matrix is established by a lumped mass matrix formulation. In SAP IV computer program, it is assumed that the damping matrix is a linear combination of the mass and stiffness matrices of the system, i.e., [ C ] = a [ M] + b[ K],
(4)
where the constants a and b are determined from the damping characteristics of the material. This types of damping matrix is usually used in the so-called method of normal modes. Alternatively, an equivalent model damping can be specified to represent the damping characteristics of the structureal material. In the deterministic dynamic response analysis of any structural system, the final objective is to find the time histories of motions of various elements of the
A.H. Yousafzai, G. Ahmadi / Deterministicand stochastic earthquake response analysis
311
system. Referring to eq. (1), given the effective load vector (P(t)), the dynamic analysis consists of finding the displacement of all the nodal points, i.e., the displacement vector (x), and the stresses in all the segments or elements. Knowing the time history of the displacement vector, it is a matter of simple calculations to find the corresponding velocity and acceleration vectors.
where (7) is the column matrix of modal displacement and (R(t)) is the column matrix of modal force which is defined as
2.1. Method of normal modes
the solution for the n th mode becomes
The following techniques are usually employed to solve eq. (1): (i) modal analysis, (ii) direct integration, (iii) complex response (transform) method. Detailed description of modal analysis and direct integration methods are provided by Clough and Penzien [1], Hunty and Rubinstein [2], and Newmark and Rosenblueth [4], among others. The application of the complex transform method to dynamic analysis of structures is described for instance by Lysmer, Uderka and Seed [51]. In the present work, the method of normal modes is employed throughout the analysis and hence a brief description of this technique is in order. When the damping matrix [C] is a linear combination of [M] and [K], the general solution of the system of eq. (1) could be found by the method of normal modes. Let [q,] be the matrix of the eigenvector satisfying the following relationships,
[~,]~[~] = [---~..J,
(5)
(R(t))=
- [~]T(1)g s.
For initially motionless structures, that is (x(O)) = (J¢(O)) = O,
*/n(t) = h , ( t ) * g n ( t ),
= ["",~..],
(6)
(7) where superscript T stands for the transpose matrix, [ ~ ~0,2-...] is the diagonal matrix of the square of natural frequencies and ["~2~,~%..J is the diagonal damping matrix. It is well known that [1-5] the application of the transformation
(x) = [,~](n),
(8)
to eq. (1) gives
(qj) + 2 [ " ~#%....j(i]) + ['~'to2...] (7) = ( R ( t ) ) ,
(9)
(11)
(12)
where hn(t) is the impulsive response of the nth mode and asterisk ' *' stands for convolution integral, i.e.
,1.(t)=lffe-~.~.~t-')sin I2.(t - ~')g.(~)d~',
s2g0
(13)
where I2n = ~o.~l - ~.2.
(14)
The column (x(t)) then is given by
(x(t)) = [ q~l["~h, * .~g.....] [q~l'r(l).
(15)
If the force (P(t)) or (R(t)) persist to operate for - oc < t < + o~, taking Fourier transform of eq. (6), it
follows that ~.(~o)
= H.(io~)R.(w),
(16)
where ~.(w), R . ( ~ ) are the Fourier transforms of 7/.(t) and R.(t), respectively. Fourier transform pair is defined as
l
[,~ ]T[ M]-'[K][,~]
(10)
~/,,(t) -
f-°°e-'=%,(t)dt,
f+_fe~.
16Ol- ( w )
(17) (18)
The function H.(i~o) is the system function of the n th mode and is given by H.(i~o) = 1/(~o 2 - ~o2 + 2i~'.~o.~o).
(19)
The solution for the displacement column in the frequency domain then becomes ( ~ ( w ) ) = [ q~]["~U.(i~o)~s(to)___] [q~]T(1).
(20)
The inversion of eq. (20) gives the solution in the time domain. When the response for each mode ~/,(t) is determined from eq. (13), the displacements of various
312
A.H. Yousafzai, G. Ahmadi / Deterministic" and stochastic earthquake response analysis
nodal points can be found by use of eqs. (8) or (15). which merely represent the superposition of the various modal contributions. It is to be noted that for most types of structures the contributions of the various modes generally are greatest for the first few lowest frequencies and tend to decrease for the higher frequencies. Consequently, it is not usually necessary to include all the higher modes of vibration in the superposition process. Furthermore, the mathematical idealization of any complex structural system also tends to be less reliable in predicting the higher modes of vibration; therefore, it is usual to limit the number of modes considered in a dynamic response analysis. In the present analysis the first twenty-five modes are considered. It is clear from the foregoing discussion that the major problem in the dynamic response analysis by the mode-superposition technique is to solve the corresponding eigenvalue problem. Naturally, this is also the most time-consuming step. If the order of the matrices be large, the computer time required to find all eigenvalues and eigenvectors would also become quite enormous. Usually, an iteration routine is needed which calculates only the required frequencies and the corresponding eigenvectors with optimum efficiency. The SAP IV program which is used in the present analysis employs subspace iteration [2] technique for the solution of the eigenvalue problem.
where 6(z) is the delta function. According to Bycroft [26], So is 0.375 ftZ/s4/cps for the NS-component of the E1 Centro (1940) earthquake. The matrix of the displacement cross spectrum is defined as ..
[s(,0)]
(22)
=
S~x,(w)
When the earthquake ground acceleration 2g(t) is specified, the expression for forcing function vector is given by eq. (2) and the time histories of the motions of various nodal points could be found by the technique discussed in the previous section. Alternatively, when the earthquake ground acceleration is modeled by a random process, eq. (1) becomes a random vector differential equation and the stochastic method of analysis must be employed. There are stationary as well as nonstationary random models for the earthquake ground accelerations and the method of response analysis varies accordingly.
Sx,x,( *0)
By definition the cross spectrum is given by [1,4,5] Sx,x,(w)= lim ~ T 2 i ( ~ 0 ) ~ 7 ( w ) ,
(23)
T ~
where T is the averaging time duration. The displacement cross spectra matrix then becomes [S(w)]=
lim 2@(~(w)}(2*(w))T.
(24)
T~oo
Employing expression (20) into (24), we find [S(w)] = [n(iw)][l][H*(iw)]TS~.(w),
(25)
where matrix [l] is a square matrix whose all elements are unity. The system function matrix [H(i¢0)] is defined by [H(iw)] =
3. Stochastic method of analysis
...
[,~l[--- H.(iw)__.][,~l T,
(26)
and superscript * stands for the complex conjugate. A typical displacement cross-spectrum then is given by
=E E E p
n
l
m
× H,(iw)H*(iw)]S~.(o~),
(27)
where summations are over the number of degrees of freedom of the system. Substituting the appropriate expression for the power spectrum of ground acceleration in the righthand side of eq. (27) gives the various displacement cross-spectra of the response of the shell. The variance of the horizontal displacement of the ith nodal point then becomes
3.1. Response to stationary ground motion Stationary white noise representation of earthquake ground acceleration was introduced by Housner [25], Bycroft [26] and Rosenblueth et al. [5,27,28]. For such a model, the power spectrum and autocorrelation of the ground acceleration are given by Sz(w ) = S O,
R , . ( r ) = 2¢rS03(r ),
(21)
where angular brackets, ( ), stand for the expected value or ensemble average. The evaluation of the meansquare response as given by eq. (28) involves extensive numerical computations. However, it was shown in [40,45,49,50] that for small damping coefficients eq. (28) may be simplified considerably to a simple expression
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis for evaluation of the approximate mean-square response. From the definition (19) it is observed that for light damping the main contributions to the summation on the righthand side of eq. (28) are from the diagonal terms, m = n. Therefore, eq. (28) approximately becomes, ( ~2 . + o o Sx,(co)d ¢o
s,,,o)
For white noise model of the ground acceleration, eq. (29) reduces to
(z 4,,. ; /~.,~.~.
(30)
Similar consideration for the mean-square response of the velocity of the i th nodal point gives
.so
02 = (k2) = T
]~ (q~,,) n
( x
;
/~',~0,.
(31)
/
It is well known that the 3o level is a conservative estimate of the maximum response [1,4,6151,53]. That is, the maximum value of the displacement response of the ith nodal point is restricted to Maxlx~t < 3 - - ~
(*s,)
to have the same peak ground velocity, Amin and Ang [31] have suggested the following envelope function
eo(t/tA) e ( t ) = ~e o
for0
eo e-c(t-~a)
for t
(34)
:> l B
t A = 1.5 s, t 2 = 15 s, c =0.18 s - ' to 0.20 s - l , and e 0 = 252 gEs-3. The stationary random function w(t) then is a filtered shot noise with the filter impulsive response where
(29)
~,
313
q~,
/ff,~
,
(32)
with probability of 0.9985. This completes our discussion on the stationary response analysis.
1
h(t) =--e
twt
TM
0 sin Wdt,
(35)
with w0=10~r,
0.5<~0 < 0 ' 6 ,
~d=°~0(1--~0z"
(36)
Rosenblueth et al. [27,28] have assumed that w(t) is a stationary white noise process and e(t), is a constant equal to one for the duration of strong motion, and zero otherwise. Response of single-degree-of-freedom as well as continuous systems to nonstationary random excitations was considered by several investigators [38-47]. Response of structural systems to general nonstationary excitation was discussed in [48-50] and in [56], among others. The correlation matrix is defined by [ g ( t t , t2] = ( ( X ( t l ) ) { x ( t 2 ) ) T ) ,
(37)
the elements of which are given by
3.2. Response to nonstationary ground motion
Rx,xj( t I , t2) = ( x i ( t l ) x j ( t2) ) .
Actual earthquakes do have finite durations and hence a stationary stochastic model hardly could represent the detailed statistical properties of earthquake ground accelerations. More realistic non-stationary stochastic models of ground motions were suggested by Amin and Ang [31], Jenning Housner and Tsai [54], Shinozukz and Sato [55], Bagdanoff, Goldberg and Bernard [44], among others. In most common nonstationary models, it is assumed that the ground acceleration is given by
Employing eq. (33) in eqs. (13) and (15), the general expression for the instantaneous response becomes
Jig(t) = e( t ) w ( t),
/,(t) = Z ~ ~ n
p
(38)
f'e-~o'~o,'-,, ~n
"~0
× sin ~2,(t - ~ ) e ( , ) w ( z ) d , .
(39)
Direct substitution of eq. (39) into eq. (37), yields
Rx'%(tl't2)=~n ~p ~m~I
$2.".
(33)
where e(t) is a deterministic envelope function and w(t) is a stationary random function of time. In order to match the average of the earthquakes of E1 Centro (1934), (1940), Olympia (1949) and Taft (1952) adjusted
× sin I2,(t, - "1) sin g2,~(t2 - h )
× e(,,)e(,~)Rw(-i- ,2)d,,d,2,
(40)
314
A.H. Yousafzai. G. Ahmadi / Deterministic and stochastic earthquake response analysis
where Rw(? ) is the autocorrelation of w(t). The mean square displacement response of the ith nodal point is now given by
the mean-square response as given by eq. (44) becomes
~x2(t))=R~,~,(t,t)
= I~Xi,(1 / Y~.X,,(I
l
_ e - 2 , ..... )/2~',w, e - 2 ; ...... )/2~',wn
forO < t_< t o , for t > t o ,
I
Xffe-G,~,,(t
r , ) - ~ . . . . . (t
r2)
(46)
¢0 " 0
where
× sin 12,(t - rl) sin ~2m(t - r2)
Xe('r,)e('r2)Rw('C , -
1"2)dT,dT 2 .
(41) (47)
Eqs. (40) and (41) give the general expressions for the autocorrelation and the mean-square response, respectively. For given autocorrelation of the stationary part of the ground acceleration Rw(~- ) and the envelope function e(t) the integrals in eqs. (40) and (41) could be evaluated. However, the algebra in the numerical evaluation of the mean square responses and the cross correlations are quite cumbersome. It was shown in [40,45,49,50] that for the case of slightly damped system, a simple approximate expression for evaluation of the mean-square response could be obtained. For small damping coefficients ~"<< 1, the main contributions to the summation in eq. (41) are from the diagonal terms, m = n. Thus expression (41) reduces to 2
(x~,(t)) = \2~(~'--~" ftf/e-'¢°'~°~212. '}-~t"l / , oo ....-"~> × sin I 2 ( t - ~-,) sin ~2,(t - r 2 ) e ( r l ) e ( ~ ' 2 ) × R~(~', - 72)d~'0"2.
3 o bound on the maximum displacement response could also be constructed from the present nonstationary analysis.
4. Finite element model
Fig. 1 shows a typical containment shell of a nuclear power plant with its foundation slab, which resembles those under construction in the southern part of Iran. A finite element model of the structure is considered. The foundation slab of the structure is divided into 196 three-dimensional solid elements, commonly known as brick elements. Each of these elements have eight nodal points. The brick elements employed are of varying sizes. Figs. 2 and 3 show the finite element model of the foundation slab. All the points at the bottom layer of
(42)
For the special case when the ground acceleration is modeled by a finite duration white noise process (Rosenblueth model [9,10]), the autocorrelation of the stationary part of ground acceleration is given by Rw(~" ) = 2~rS03(~" ),
(43)
with So being the constant power spectrum of the white noise process. Employing eq. (43) in eq. (42), integrating over the delta function and using the approximation of [40,45,47,49,50] for small damping, the result becomes
:r-
~5 llfl
I i
(
(44)
e(t)=
1 0
for 0 < t < t o , fort>t o,
zj
[ }-
For the Rosenblueth model with
l
I
242,7 ft
/
5
R=98.4 fl
× "1
Y
(45)
Fig. l. A cross section of the containment shell with the foundation slab.
315
A.H. YousafzaL G. Ahmadi / Deterministic and stochastic earthquake response analysis 843
1
9
6
~
2
1
0
1
5
1
~
1
6
5
1
2
1
1
0
~
~
9
1
7
6
1 1
~
1 ~
3
5
2
0 9
r
I
II
~~
~
~45
0
5
",
0
6 4 7 ~
I
-
I
~~j.~689
i-
6,9r -Y 7 .
I 234 5 6 7 8 9 I0 II 121314 15 Fig. 2. Nodal point distribution pattern of the bottom layer of the foundation slab of the structure.
535 L 1 / 4 7 9 ~
-. .,. :. . .
T--
i_ ~
57z 4
9
3
332q-54 - - ~ the slab are considered to be fixed while those at the surface had a translation degree-of-freedom in the directin of x-axis. The containment shell of the plant is divided into 420 thin shell elements. Each of these elements had four nodal points. The nodes common between the brick elements and the thin shell elements transfers the motion from the foundation to the structure. The finite element model of the shell structure is shown in fig. 4. As mentioned before, all the nodal points on the bottom layer of the foundation slabs, that is, the points ranging from 1 to 225, are taken to be fixed which is a boundary condition of the problem. All the rest of the nodal points of the finite element model of the structure are assumed to have only one translational degree of freedom along the x-axis. This assumption would save
436 421[
450
III
376
I I
S4SI
/111 j 111
331 I
Nil
27i I 241i 2 26
345 ggO
IIXII I JJii IIN I J l I',.1
233
240
Fig. 3. Nodal point distribution pattern of the surface layer of the foundation slab of the structure.
275 21 247 248 249 265 I---J~29T 28 Fig. 4. Finite element mesh used for the shell.
considerable amount of computer time at the expense of reduction of the accuracy of the model. The containment shell is assumed to be made of reinforced concrete with the following properties: Weight density = 150 l b / f t 3. Modulus of elasticity = 3 × 106 psi. Modulus of rigidity = 1.5 × 106 psi. Poission's ratio = 0.18. Rayleigh damping is assumed and is supplied to the program in the form of a constant modal damping factor equal to 2% of critical.
5. Results and discussions
As the first step for the response analysis, natural frequencies and mode shapes of the structure are obtained. The present finite element model of the containment shell structure does have 445 degrees-of-freedom. Therefore, theoretically the same number of frequencies and mode shapes could be evaluated. However, evaluation of all of these natural frequencies and the corresponding mode shapes takes an enormous amount of computer time and furthermore, only the first few frequencies and the corresponding mode shapes are of practical importance in evaluation of the response by the mode superposition technique. The values of the first 25 natural frequencies are listed in table 1. It is observed that the fundamental
316
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis
Table 1 Table of natural frequencies Mode number
Natural frequency (cycles/s)
Natural period (s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
6.63 13.59 19.11 26.08 32.24 38.31 45.19 52.03 58.24 64.85 69.23 74.18 78.98 83.35 88.14 92.93 97.51 99.82 104.20 107.30 111.60 114.90 119.80 121.30 123.80
0.151 0.0735 0.0523 0.0383 0.0310 0.0261 0.0221 0.0192 0.0172 0.0154 0.0144 0.0135 0.0127 0.0120 0.0114 0.0108 0.0103 0.0100 0.0096 0.0093 0.0090 0.0087 0.0083 0.0082 0.0081
natural period of the structure is about 0.15 s which shows that the present c o n t a i n m e n t shell has the characteristics of a relatively rigid structure. Figs. 5 - 1 0 show some vertical, as well as horizontal cross sections of the
[ Fig. 5. First mode of vibration of the structure.
[
1 Fig. 6. Second mode of vibration of the structure.
d e f o r m e d structure for the first three modes of vibration. It is observed that these mode shapes bear close similarities with the corresponding modes of a simple cantilever beam.
5.1. Results of the deterministic analysis With the known natural frequencies and the corres p o n d i n g m o d e shapes, the deterministic response of the c o n t a i n m e n t shell to various earthquake ground motions could be easily calculated by the method of normal m o d e through the use of eqs. (13) and (15), as discussed in the previous sections. As the first example of the deterministic response analysis the S00E c o m p o n e n t of the E1 Centro earthquake of 1940 was considered as the input and the time
I Fig. 7. Third mode of vibration of the structure.
1
317
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis
t Fig. 8. First mode of vibration of a horizonal cross section of the structure (for nodal points 479 to 506).
/ [.
]
Fig. 11. Comparison of the estimated peak displacement response in according to stochastic theory with the exact numerical results for the S00E component of El Centro (1940) earthquake.
histories of the displacement responses of all the nodal points of the c o n t a i n m e n t structure are calculated. It was observed that the maximum relative displacement response at all the nodal points occurs at t = 4.92 s. The m a x i m u m relative displacement is found to be 0.554 cm for the top of the dome. A vertical cross section of the structure at t = 4.92 s is shown by the dotted curve in fig. 11. A similar response analysis is also carried out for the $69E c o m p o n e n t of Taft 1952 earthquake. The correFig. 9. Second mode of vibration of a horizontal cross section of the structure (for nodal points 479 to 506).
Fig. 10. Third mode of vibration of a horizontal cross section of the structure (for nodal points 479 to 506).
Fig. 12. Comparison of the estimated peak displacement response in according to stochastic theory with the exact numerical results for the $69E component of Taft (1952) earthquake
318
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis
sponding peak displacement response which occurs at t = 4.58 s is shown by the dotted curve in fig. 12. The peak response for the top of the dome is found to be 0.319 cm for Taft 1952 earthquake. From the dotted curves in figs. 11 and 12, it is observed that the first mode of vibration is quite dominant at the peak response. The dominancy of the first mode was also observed to hold throughout the time history of the response. From the available response spectra curves of El Centro (1940) and Taft (1952) earthquakes, for an oscillator with natural period of 0.15 s and damping coefficient of 2%, the corresponding maximum relative displacement responses are found to be about 0.6 cm and 0.32 cm, respectively. These values are quite close to the maximum deflections of the top of the dome found in the present deterministic analysis. These observations are also compatible with dominancy of the first mode in the response analysis of the present structure. The calculations of the responses of the containment structure to E1 Centro (1934) and Olympia (1949) earthquake are also carried out and the results are discussed in [56].
\ °3°I 0 0
5
IO 15 20 TIME (seconds)
\
/
25
50
Fig. 13. Time development of the 3o~ for the top of the dora for El Centro (1940) earthquake.
o75/ g
I
0145I-
0
,~ --
5
I0 15 20 TIME (secondsl
25
~50
Fig. 14. Time development of the 3ax for the top of the dom for Taft (1952) earthquake.
5.2. Results of the stationary analysis With the natural frequencies and the modal vectors found from the free vibration analysis of the containment shell structure, the variances of the relative displacements of various nodal points could be evaluated from eq. (31) for the stationary white noise model of the earthquake ground motion. Three times standard deviation provides an upper bound on the maximum response as is noted by eq. (32). Employing the available estimates for the power spectrum So of the El Centro (1940) and Taft (1952) earthquakes, the maximum displacement responses of the various nodal points of the containment shell are evaluated by use of the bound provided by eq. (32) and the results are plotted by solid curves in figs. II and 12. It is observed that the estimated peak responses as found from the stationary stochastic analysis do indeed form reasonable bounds on the actual maximum responses found from the deterministic analysis for those earthquakes. Similar calculations for the El Centro 1934 and Olympia 1949 earthquakes are also carried out and the results are discussed in [56].
5. 3. Results of the nonstationary analysis With known natural frequencies and the corresponding mode shapes of the containment structure, the time
development of the root mean square responses to various nonstationary models of earthquakes could be evaluated from eqs. (44) and (46). For the Rosenblueth finite duration white noise model of the ground acceleraton, such analysis have been carried out for the various nodal points of the containment shell with appropriate values of SO for the principle horizontal components of El Centro (194) and Taft (1952) earthquakes. The corresponding variations of 30x for the top of the dom are shown in figs. 13 and 14. The time duration of both earthquakes are taken to be 25 s and integrations are carried out for 30 s. It is observed that in both cases the values of root mean square displacement responses reach to their stationary levels after about 7 s. In order to compare the estimated 3o levels with the peak responses, the maximum of 3o(t) over the time duration must be considered. In the case of El Centro (1940) and Taft (1952) earthquakes with their time durations of 25 s, the peaks of o(t) are clearly their stationary values found previously. Therefore, the solid curves in figs. 11 and 12 also correspond to the estimated maximum responses according to nonstationary analysis. Similar analysis for the response of the containment shell to E1 Centro (1934) and Olympia (1949) earthquakes are also carried out and the results are discussed in [56].
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis
6. Conclusions and further remarks In the present investigation the response of the cont a i n m e n t shell of a nuclear power plant to deterministic as well as r a n d o m e a r t h q u a k e support m o t i o n is studied. A finite element model of the c o n t a i n m e n t shell is constructed a n d the m e t h o d of modal analysis is employed. The response of the shell to E1 C e n t r o (1940) and Taft (1952) earthquakes are obtained and the results for the peak relative displacement responses are shown to b e compatible with those o b t a i n e d from the corresponding response spectra curves of those earthquakes. Stationary a n d n o n s t a t i o n a r y stochastic models of earthq u a k e ground m o t i o n are considered and the general expressions for the m e a n square responses of various n o d a l points are derived a n d discussed. It is shown that the 3o levels of the relative displacement responses provide reasonable b o u n d s on the corresponding maxim u m responses. In b o t h deterministic as well as stochastic response analysis of this relatively rigid structure, it is observed that the first m o d e of vibration is quite dominant.
Acknowledgements The authors would like to t h a n k Professor N. Mostaghel, Professor H. Z o h o o r a n d Professor H. Seyyedian a n d Shiraz University for m a n y helpful discussions. The earlier stages of this work was carried out at the D e p a r t m e n t of Mechanical Engineering of Shiraz University a n d is supported in part by the Atomic Energy Organization of Iran.
References [1] R.W. Clough and J. Penzien, Dynamics of Structures (McGraw Hill Book Company, 1975). [2] W.C. Hunty and M.F. Rubinstein, Dynamics of Structures (Prentic-Hall, Inc., 1964). [3] I. Merirovitch, Analytical Methods in Vibration (Macmillan Book Company, 1967). [4] N.M. Newmark and E. Rosenblueth, Fundamentals of Earthquake Engineering (Prentice-Hall, Inc., 1971). [5] S. Omato, Introduction to Earthquake Engineering (University of Tokyo Press, 1973). [6] K.J. Bathe, E.L. Wilson and F.E. Peterson, SAP IV, a structural analysis program for static and dynamic response of linear systems, Report No. EERC 73-11, College of Engineering University of California, Berkeley (1973). [7] K.J. Bathe, E.L. Wilson and F.E. Peterson, SAP I V - - a
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structural analysis program for static and dynamic analysis of linear structural systems, Nucl. Engrg. Des. 25 (1973) 257-274. [8] R.H. MacNeal (ed.), NASTRAN Theoretical Manual, NASA Sp-221 (01) (April, 1972). [9] C.W. McCormick (ed.), NASTRAN User's Manual, NASA Sp-222 (01) (June, 1972). [10] C.W. Hennrich (ed.), NASTRAN Programmer's Manual, NASA Sp-223 (01) (Sept., 1972). [11] R.H. MacNeal, Some organization aspects of NASTRAN, Nucl. Engrg. Des. 29 (1974) 254-265. [12] K.J. Bathe and E.L. Wilson, NONSAP--a nonlinear structural analysis program, Nucl. Engrg. Des. 29 (1974) 266-293. [13] T. Belytschko, A survey of numerical methods and computer programs for dynamic structural analysis, Nucl. Engrg. Des. 37 (1976) 23-24. [14] V. Svalbonas, Transient dynamic and inelastic analysis of shells of revolution--a survey of programs, Nucl. Engrg. Des. 37 (1976) 73-93. [15] C.W. Lin, Seismic response of nuclear reactor containment vessels, Nucl. Engrg. Des. 9 (1969) 234-238. [16] D.C. Jeng, Transient seismic analysis of HTCR nuclear power plants, Nucl. Engrg. Des. 21 (1972) 358-367. [17] J.K. Aldersebaes and K. Wiedner, Containment structure for Trojan nuclear power plant, J. Power Division ASCE P02 (1971) 351-366. [18] K. Muto, Earthquake response analysis for a BWR nuclear power plant using recorded data, Nucl. Engrg. Des. 20 (1972) 385-392. [19] G.E. Howard, Some investigations on nonlinar dynamic response of nuclear power systems to ground motions, Nucl. Engrg. Des. 25 (1973) 42-50. [20] N.M. Newmark, Earthquake analysis of reactor structures, Nucl. Engrg. Des. 20 (1972) 303-322. [21] T. Ichiki, T. Matsumoto and Gunyasu, A seismic analysis of nuclear power plant components subjected to multi-excitations of earthquakes, Trans. 4th Internat. Conf. on SMiRT, Aug. 15-19, 1977, San Francisco, Calif., Paper K 6/12. [22] I.W. Yu, A finite element method for seismic response analysis, Trans. 4th Intenat. Conf. on SMiRT, Aug. 15-19, 1977, San Francisco, Calif., Paper K 3/6. [23] T. Takemori, Y. Ogiwara, T. Kawakatsu, Y. Abe and K. Kitade, Comparative aseismic response study of differrent analytical models of nuclear power plants, Trans. 4th Internat. Conf. on SMiRT, Aug. 15-19, 1977, San Francisco, Calif., Paper K 2/2. [24] G. Ahmadi, Generation of artificial time histories compatible with given response spectra--a review, SM Archives 4 (1979) 207-239. [25] G.W. Housner, Properties of strong motion earthquakes, Bull. Seismol. Soc. Am 45 (1955) 197-218. [26] G.N. Bycroft, White noise representation of earthquakes, J. Engrg. Mech. Div. ASCE 86 EM2 (1960) 1-16. [27] E. Rosenblueth and J. Bustamante, Distribution of structural response to earthquakes, J. Engrg. Mech. Div. ASCE 88, EM3 (1962) 75-106.
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[28] E. Rosenblueth, Probabilistic design to resist earthquakes, J. Engrg. Mech. Div. ASCE 40, EMS (1964) 189-220. [29] K. Kanai, Semi-empirical formula for the seismic characteristics of the ground, Univ. Tokyo Bull. Earthquake Res. Int. 35 (1957) 309-325. [30] H. Tajimi, A statistical method of determining the maximum response of a building structure during an earthquake, Proc. 2nd World Conf. Earthquake Engrg. Tokyo and Kyoto, Vol. II (1960) 781-798. [31] M. Amin and A.H.S. Ang, Nonstationary stochastic model of earthquake motions, J. Engrg. Mech. Div. ASCE 94, EM2 (1968) 559-583. [32] S.H. Crandall and W.D. Mark, Random Vibration in Mechanical Systems (Academic Press, 1963). [33] H. Parkus, Random Processes in Mechanical Systems (UDINE Springer-Verlag, 1969). [34] J. Robson, C.J. Dodds, D.B. Macvean and V.R. Paling, Random Vibrations (UDINE Springer-Verlag, 1971). [35] Y.K. Lin, Probabilistic Theory of Structural Dynamics (McGraw-Hill, 1967). [36] W. Gersch, Mean-square responses in structural systems, J. Acoustical Soc. Am. 48 (1970) 403-413. [37] J.L. Zeman and J.L. Bogdanoff, A comment on complex structural response to random vibrations, AIAA 7 (1969) 1225-1231. [38] R.L. Barnoski and J.R. Maurer, Mean square response of simple mechanical system to nonstationary random excitation, J. Appl. Mech. Trans. ASME 36 (1969) 221-227. [39] R.L. Barnoski and J.R. Maurer, Transient characteristics of simple systems to modulated random noise, J. Appl. Mech. Trans. ASME 40 (1973) 73-77. [40] L.L. Bucciarelli, Jr. and C. Kuo, Mean square response of a second order system to nonstationary random excitation, J. Appl. Mech. Trans. ASME 37 (1970) 612-616. [41] T.K. Caughey and H.J. Stumpf, Transient response of a dynamic system under random excitation, J. Appl. Mech. Trans. ASME 28 (1961) 563-566. [42] R.B. Corotis, E.H. Vanmarcke and C.A. Cornell, First passage of nonstationary random processes, J. Engrg. Mech. Div. ASCE 98 EM2 (1972) 401-414. [43] R. Corotis and T.A. Marshall, Oscillator response to mod-
[44]
[45]
[46]
[47]
[48]
[49]
[50] [51]
[52] [53] [54]
[55]
ulated random excitation, J. Engrg. Mech. Div. ASCE 103 (1977) 501-513. J.L. Bogdanoff, J.E. Goldberg and M.C. Bernard, Response of a simple structure to a random earthquake-type disturbance, Bull. Seismol. Soc. Am. 54 (1961) 292-310. G. Ahmadi and M.A. Satter, Mean-square response of beams to nonstationary random excitation, A1AA 13 (1975) 1097- 1100. M.A. Satter and G. Ahmadi, Mean-square response of a pipe carrying flowing fluid to nonstationary random excitation, J. Sound Vibration 51 (1977) 577-581. G. Ahmadi and M.A. Satter, Response of plate to nonstationary random load, J. Acoustical Soc. Am. 65 (1979) 926-930. R. Holman and G.C. Hart, Nonstationary response of structural systems, J. Engrg. Mech. Div. ASCE 100, EM2 (1974) 415-431. G. Ahmadi, Stochastic earthquake response of multi-degree-of-freedom systems, Shiraz University. Dept. Mech. Engrg. Research Report (1979). G. Ahmadi, Earthquake response of linear continuous systems, Nucl. Engrg. Des. 50 (1978) 327-345. J. Lysmer, T. Udaka and H.B. Seed, Report No. EERC 74-4, Earthquake Engineering Research Center, University of California, Berkeley. R.L. Wiegel, Earthquake Engineering and Structural Dynamics (Prentice-Hall, 1970). M.P. Singh, Generation of seismic floor spectra, J. Engrg. Mech. Div. ASCE 101 (1975) 593-607. P.C. Jenning, G.W. Housner and N.C. Tsai, Simulated earthquake motions, Rept. Earthquake Engrg. Res. Lab. Calif. Inst. Technol. (April, 1968). M. Shinozuka and Y. Sato, Simulation of nonstationary random process, J. Engrg. Mech. Div. ASCE 93 (1967) 11-40.
[56] A.H. Yousefzai and G. Ahmadi, Stochastic and deterministic response of structures: Application to containment structure of nuclear power plants, Shiraz University, Dept. of Mechanical Engineering Research Report 80-2 (July, 1980).
309
DETERMINISTIC AND STOCHASTIC EARTHQUAKE RESPONSE CONTAINMENT SHELL OF A NUCLEAR POWER PLANT
ANALYSIS OF THE
A.H. YOUSAFZAI Department of Mechanical Engineering, Shiraz University, Shiraz, Iran
Goodarz AHMADI Department of Mechanical and Industrial Engineering, Clarkson College, Potsdam, N Y 13676, USA
Received 28 June 1982 Response of the containment shell of a nuclear plant to earthquake ground motion is considered. A finite element model of the structure is developed and SAP IV structural analysis program is employed for the determination of the frequencies and the corresponding mode shapes of the structure. The response of the containment shell to several past earthquakes are analyzed and the results are discussed. Stochastic models of earthquake ground acceleration are then considered and the general expressions for the power spectra, cross correlations and the mean-square responses are derived. The root mean-square of the relative displacement responses of various nodal points of the containment shell structure subjected to stationary as well as nonstationary random support motion are evaluated. The stochastically estimated maximum displacement responses are compared with those obtained from a deterministic analysis and reasonable agreements are observed.
1. Introduction
Dynamic response analysis of structures subjected to earthquake ground motions is one of the basic requirements for their aseismic design. Such an analysis is of prime importance for the design of nuclear power plants, dams and other high rise buildings. In the early 1960s, the first designs of large scale power plants incorporating seismic analysis were being prepared. By the late 1960s, dynamic analysis of nuclear power plants structures and equipments had become a common practice for their design. For a specified base motion or any other forcing function, it is always possible to compute the time history of a given structural system using several deterministic analytical or numerical techniques as described by Clough and Penzien [1], Hunty and Rubinstein [2], Meirovitch [3], Newmark and Rosenblueth [4] and Omato [5]. At present time, due to the extensive developments in the field of digital computers software such an analysis has become a rather sophisticated art as well as technique. A n u m b e r of complex computer programs such as SAP IV [6,7], N A S T R A N [8-11] and N O N S A P [12] among others as summarized for instance in [ 12-14] are developed. Simplified multi-degrees-of-freedom models 0029-5493/82/0000-0000/$02.75
as well as finite element analysis of containment shell of nuclear power plants are discussed in [14-23], among others. In the common practice of deterministic response analysis, several past earthquake accelerograms are usually employed in order to take into account the possible variations in the frequency content of the earthquake ground motions. However, there always remains uncertainties about the nature of the future earthquakes at a site of interest due to the lack of sufficiently large n u m b e r of earthquake records for similar sites. In the recent years, statistical method of analysis is proposed as an alternative to the deterministic response analysis. In this approach the earthquake ground acceleration is modeled by a random process and the response analysis is treated as a problem in random vibrations. Several stochastic models of earthquake ground motion were proposed in the past as summarized by Clough and Penzien [1] and Ahmadi [24]. Among the most well known models are the white noise representation of Housner [25] and Bycroft [26], finite duration white noise model of Rosenblueth et al. [4,27,28], filtered white noise representation of Kanai [29] and Tajimi [30] and nonstationary models of Amin and Ang [31] and others [1,4,5]. Response of multi-degree-of-freedom systems to sta-
© 1982 N o r t h - H o l l a n d
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A.H. Yousafzat, G. Ahmadl / Deterministic and stochastic earthquake re.s7)onseanalysis
tionary excitations were considered by several investigators (see for instance [1-5], [32-35]). Recently, the mean-square response of structures to random excitation was studied by Gersch [36] and Zeman and Bogdanoff [37]. The response of single degree of freedom systems to nonstationary random excitation was investigated by Barnoski and Maurer [38,39], Bucciarelli and Kuo [40], Caughey and Stumpf [41], Corotis et al. [42,43] and Bogdanoff et al. [44]. Response of beams and plates to nonstationary random loads were studied by Ahmadi and Satter [45-47]. Response of structures to nonstationary random excitation was considered by Holman and Hart [48] and Ahmadi [49,50]. In the present study earthquake response of the containment shell of a nuclear power plant which resembles those under construction in the southern part of Iran is investigated. Both deterministic and stochastic methods of response analysis are employed. A finite element model of the shell and its foundation is developed where a single translational degree-of-freedom at each nodal point is assumed. SAP IV structural analysis program is used and the first 25 natural frequencies and the corresponding mode shapes of the structure are obtained and discussed. The response of the containment shell structure to several past earthquake ground accelerations are determined and the results are compared with those found from the response spectra of the corresponding earthquakes. In the probabilistic phase of the analysis, stochastic response of the structure to random earthquake ground motion is considered. The formal general solutions in time as well as frequency domains for a system with its damping matrix being a linear combination of the mass and stiffness matrices is discussed. Statistically stationary models of earthquake ground acceleration is considered and the general expressions for the meansquare responses of various nodal points, are derived. For small damping coefficients, a simplified approximate solution for the mean-square response is presented. Nonstationary stochastic models of earthquake ground accelerations are also employed and the response of the containment shell structure to such a nonstationary support motion is studied. General expressions for the cross correlations as well as meansquare responses are derived. Numerical examples for E1 Centro (1940) and Taft (1952) earthquakes are considered and the stochastically predicted maximum responses are compared with those obtain from the deterministic analysis and reasonable agreements are observed.
2. Deterministic method of analysis In the present study, a finite element model of the containment shell of a nuclear power plant is considered and the structural analysis program SAP IV as developed by Bathe, Wilson and Peterson [6] is employed for the numerical computations. The program uses the finite element technique and assumes the structural system to be divided into a number of discrete elements interconnected only at a finite number of nodal points. The properties of the complete structure are then found by evaluating the properties of the individual finite elements and superposing them appropriately. The corresponding nodal point equilibrium equations are of the form [M](X)+[C](x)+[K](x)=(P(t)),
(1)
where [M] is the mass matrix, [C] is the damping matrix and [KI is the stiffness matrix of the element assemblage. The vectors (x), (5:), (2) and (P(t)) are the nodal displacements, velocities, accelerations and generalized loads, respectively. When the containment shell is subjected to an earthquake ground acceleration, the generalized force is given by { P ( t ) ) = - [M](1}2g(t),
(2)
where (l} is a column matrix whose all elements are unity and £g(t) is the horizontal acceleration of earthquake ground motion. The stiffness matrices in the program are formed by direct addition of the element matrices, for example [KI=[K,]+[K2]+
... + [K,],
(3)
where [K], [Ki], and [Kn] are the stiffness matrices of the first, second and n th element, respectively. The mass matrix is established by a lumped mass matrix formulation. In SAP IV computer program, it is assumed that the damping matrix is a linear combination of the mass and stiffness matrices of the system, i.e., [ C ] = a [ M] + b[ K],
(4)
where the constants a and b are determined from the damping characteristics of the material. This types of damping matrix is usually used in the so-called method of normal modes. Alternatively, an equivalent model damping can be specified to represent the damping characteristics of the structureal material. In the deterministic dynamic response analysis of any structural system, the final objective is to find the time histories of motions of various elements of the
A.H. Yousafzai, G. Ahmadi / Deterministicand stochastic earthquake response analysis
311
system. Referring to eq. (1), given the effective load vector (P(t)), the dynamic analysis consists of finding the displacement of all the nodal points, i.e., the displacement vector (x), and the stresses in all the segments or elements. Knowing the time history of the displacement vector, it is a matter of simple calculations to find the corresponding velocity and acceleration vectors.
where (7) is the column matrix of modal displacement and (R(t)) is the column matrix of modal force which is defined as
2.1. Method of normal modes
the solution for the n th mode becomes
The following techniques are usually employed to solve eq. (1): (i) modal analysis, (ii) direct integration, (iii) complex response (transform) method. Detailed description of modal analysis and direct integration methods are provided by Clough and Penzien [1], Hunty and Rubinstein [2], and Newmark and Rosenblueth [4], among others. The application of the complex transform method to dynamic analysis of structures is described for instance by Lysmer, Uderka and Seed [51]. In the present work, the method of normal modes is employed throughout the analysis and hence a brief description of this technique is in order. When the damping matrix [C] is a linear combination of [M] and [K], the general solution of the system of eq. (1) could be found by the method of normal modes. Let [q,] be the matrix of the eigenvector satisfying the following relationships,
[~,]~[~] = [---~..J,
(5)
(R(t))=
- [~]T(1)g s.
For initially motionless structures, that is (x(O)) = (J¢(O)) = O,
*/n(t) = h , ( t ) * g n ( t ),
= ["",~..],
(6)
(7) where superscript T stands for the transpose matrix, [ ~ ~0,2-...] is the diagonal matrix of the square of natural frequencies and ["~2~,~%..J is the diagonal damping matrix. It is well known that [1-5] the application of the transformation
(x) = [,~](n),
(8)
to eq. (1) gives
(qj) + 2 [ " ~#%....j(i]) + ['~'to2...] (7) = ( R ( t ) ) ,
(9)
(11)
(12)
where hn(t) is the impulsive response of the nth mode and asterisk ' *' stands for convolution integral, i.e.
,1.(t)=lffe-~.~.~t-')sin I2.(t - ~')g.(~)d~',
s2g0
(13)
where I2n = ~o.~l - ~.2.
(14)
The column (x(t)) then is given by
(x(t)) = [ q~l["~h, * .~g.....] [q~l'r(l).
(15)
If the force (P(t)) or (R(t)) persist to operate for - oc < t < + o~, taking Fourier transform of eq. (6), it
follows that ~.(~o)
= H.(io~)R.(w),
(16)
where ~.(w), R . ( ~ ) are the Fourier transforms of 7/.(t) and R.(t), respectively. Fourier transform pair is defined as
l
[,~ ]T[ M]-'[K][,~]
(10)
~/,,(t) -
f-°°e-'=%,(t)dt,
f+_fe~.
16Ol- ( w )
(17) (18)
The function H.(i~o) is the system function of the n th mode and is given by H.(i~o) = 1/(~o 2 - ~o2 + 2i~'.~o.~o).
(19)
The solution for the displacement column in the frequency domain then becomes ( ~ ( w ) ) = [ q~]["~U.(i~o)~s(to)___] [q~]T(1).
(20)
The inversion of eq. (20) gives the solution in the time domain. When the response for each mode ~/,(t) is determined from eq. (13), the displacements of various
312
A.H. Yousafzai, G. Ahmadi / Deterministic" and stochastic earthquake response analysis
nodal points can be found by use of eqs. (8) or (15). which merely represent the superposition of the various modal contributions. It is to be noted that for most types of structures the contributions of the various modes generally are greatest for the first few lowest frequencies and tend to decrease for the higher frequencies. Consequently, it is not usually necessary to include all the higher modes of vibration in the superposition process. Furthermore, the mathematical idealization of any complex structural system also tends to be less reliable in predicting the higher modes of vibration; therefore, it is usual to limit the number of modes considered in a dynamic response analysis. In the present analysis the first twenty-five modes are considered. It is clear from the foregoing discussion that the major problem in the dynamic response analysis by the mode-superposition technique is to solve the corresponding eigenvalue problem. Naturally, this is also the most time-consuming step. If the order of the matrices be large, the computer time required to find all eigenvalues and eigenvectors would also become quite enormous. Usually, an iteration routine is needed which calculates only the required frequencies and the corresponding eigenvectors with optimum efficiency. The SAP IV program which is used in the present analysis employs subspace iteration [2] technique for the solution of the eigenvalue problem.
where 6(z) is the delta function. According to Bycroft [26], So is 0.375 ftZ/s4/cps for the NS-component of the E1 Centro (1940) earthquake. The matrix of the displacement cross spectrum is defined as ..
[s(,0)]
(22)
=
S~x,(w)
When the earthquake ground acceleration 2g(t) is specified, the expression for forcing function vector is given by eq. (2) and the time histories of the motions of various nodal points could be found by the technique discussed in the previous section. Alternatively, when the earthquake ground acceleration is modeled by a random process, eq. (1) becomes a random vector differential equation and the stochastic method of analysis must be employed. There are stationary as well as nonstationary random models for the earthquake ground accelerations and the method of response analysis varies accordingly.
Sx,x,( *0)
By definition the cross spectrum is given by [1,4,5] Sx,x,(w)= lim ~ T 2 i ( ~ 0 ) ~ 7 ( w ) ,
(23)
T ~
where T is the averaging time duration. The displacement cross spectra matrix then becomes [S(w)]=
lim 2@(~(w)}(2*(w))T.
(24)
T~oo
Employing expression (20) into (24), we find [S(w)] = [n(iw)][l][H*(iw)]TS~.(w),
(25)
where matrix [l] is a square matrix whose all elements are unity. The system function matrix [H(i¢0)] is defined by [H(iw)] =
3. Stochastic method of analysis
...
[,~l[--- H.(iw)__.][,~l T,
(26)
and superscript * stands for the complex conjugate. A typical displacement cross-spectrum then is given by
=E E E p
n
l
m
× H,(iw)H*(iw)]S~.(o~),
(27)
where summations are over the number of degrees of freedom of the system. Substituting the appropriate expression for the power spectrum of ground acceleration in the righthand side of eq. (27) gives the various displacement cross-spectra of the response of the shell. The variance of the horizontal displacement of the ith nodal point then becomes
3.1. Response to stationary ground motion Stationary white noise representation of earthquake ground acceleration was introduced by Housner [25], Bycroft [26] and Rosenblueth et al. [5,27,28]. For such a model, the power spectrum and autocorrelation of the ground acceleration are given by Sz(w ) = S O,
R , . ( r ) = 2¢rS03(r ),
(21)
where angular brackets, ( ), stand for the expected value or ensemble average. The evaluation of the meansquare response as given by eq. (28) involves extensive numerical computations. However, it was shown in [40,45,49,50] that for small damping coefficients eq. (28) may be simplified considerably to a simple expression
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis for evaluation of the approximate mean-square response. From the definition (19) it is observed that for light damping the main contributions to the summation on the righthand side of eq. (28) are from the diagonal terms, m = n. Therefore, eq. (28) approximately becomes, ( ~2 . + o o Sx,(co)d ¢o
s,,,o)
For white noise model of the ground acceleration, eq. (29) reduces to
(z 4,,. ; /~.,~.~.
(30)
Similar consideration for the mean-square response of the velocity of the i th nodal point gives
.so
02 = (k2) = T
]~ (q~,,) n
( x
;
/~',~0,.
(31)
/
It is well known that the 3o level is a conservative estimate of the maximum response [1,4,6151,53]. That is, the maximum value of the displacement response of the ith nodal point is restricted to Maxlx~t < 3 - - ~
(*s,)
to have the same peak ground velocity, Amin and Ang [31] have suggested the following envelope function
eo(t/tA) e ( t ) = ~e o
for0
eo e-c(t-~a)
for t
(34)
:> l B
t A = 1.5 s, t 2 = 15 s, c =0.18 s - ' to 0.20 s - l , and e 0 = 252 gEs-3. The stationary random function w(t) then is a filtered shot noise with the filter impulsive response where
(29)
~,
313
q~,
/ff,~
,
(32)
with probability of 0.9985. This completes our discussion on the stationary response analysis.
1
h(t) =--e
twt
TM
0 sin Wdt,
(35)
with w0=10~r,
0.5<~0 < 0 ' 6 ,
~d=°~0(1--~0z"
(36)
Rosenblueth et al. [27,28] have assumed that w(t) is a stationary white noise process and e(t), is a constant equal to one for the duration of strong motion, and zero otherwise. Response of single-degree-of-freedom as well as continuous systems to nonstationary random excitations was considered by several investigators [38-47]. Response of structural systems to general nonstationary excitation was discussed in [48-50] and in [56], among others. The correlation matrix is defined by [ g ( t t , t2] = ( ( X ( t l ) ) { x ( t 2 ) ) T ) ,
(37)
the elements of which are given by
3.2. Response to nonstationary ground motion
Rx,xj( t I , t2) = ( x i ( t l ) x j ( t2) ) .
Actual earthquakes do have finite durations and hence a stationary stochastic model hardly could represent the detailed statistical properties of earthquake ground accelerations. More realistic non-stationary stochastic models of ground motions were suggested by Amin and Ang [31], Jenning Housner and Tsai [54], Shinozukz and Sato [55], Bagdanoff, Goldberg and Bernard [44], among others. In most common nonstationary models, it is assumed that the ground acceleration is given by
Employing eq. (33) in eqs. (13) and (15), the general expression for the instantaneous response becomes
Jig(t) = e( t ) w ( t),
/,(t) = Z ~ ~ n
p
(38)
f'e-~o'~o,'-,, ~n
"~0
× sin ~2,(t - ~ ) e ( , ) w ( z ) d , .
(39)
Direct substitution of eq. (39) into eq. (37), yields
Rx'%(tl't2)=~n ~p ~m~I
$2.".
(33)
where e(t) is a deterministic envelope function and w(t) is a stationary random function of time. In order to match the average of the earthquakes of E1 Centro (1934), (1940), Olympia (1949) and Taft (1952) adjusted
× sin I2,(t, - "1) sin g2,~(t2 - h )
× e(,,)e(,~)Rw(-i- ,2)d,,d,2,
(40)
314
A.H. Yousafzai. G. Ahmadi / Deterministic and stochastic earthquake response analysis
where Rw(? ) is the autocorrelation of w(t). The mean square displacement response of the ith nodal point is now given by
the mean-square response as given by eq. (44) becomes
~x2(t))=R~,~,(t,t)
= I~Xi,(1 / Y~.X,,(I
l
_ e - 2 , ..... )/2~',w, e - 2 ; ...... )/2~',wn
forO < t_< t o , for t > t o ,
I
Xffe-G,~,,(t
r , ) - ~ . . . . . (t
r2)
(46)
¢0 " 0
where
× sin 12,(t - rl) sin ~2m(t - r2)
Xe('r,)e('r2)Rw('C , -
1"2)dT,dT 2 .
(41) (47)
Eqs. (40) and (41) give the general expressions for the autocorrelation and the mean-square response, respectively. For given autocorrelation of the stationary part of the ground acceleration Rw(~- ) and the envelope function e(t) the integrals in eqs. (40) and (41) could be evaluated. However, the algebra in the numerical evaluation of the mean square responses and the cross correlations are quite cumbersome. It was shown in [40,45,49,50] that for the case of slightly damped system, a simple approximate expression for evaluation of the mean-square response could be obtained. For small damping coefficients ~"<< 1, the main contributions to the summation in eq. (41) are from the diagonal terms, m = n. Thus expression (41) reduces to 2
(x~,(t)) = \2~(~'--~" ftf/e-'¢°'~°~212. '}-~t"l / , oo ....-"~> × sin I 2 ( t - ~-,) sin ~2,(t - r 2 ) e ( r l ) e ( ~ ' 2 ) × R~(~', - 72)d~'0"2.
3 o bound on the maximum displacement response could also be constructed from the present nonstationary analysis.
4. Finite element model
Fig. 1 shows a typical containment shell of a nuclear power plant with its foundation slab, which resembles those under construction in the southern part of Iran. A finite element model of the structure is considered. The foundation slab of the structure is divided into 196 three-dimensional solid elements, commonly known as brick elements. Each of these elements have eight nodal points. The brick elements employed are of varying sizes. Figs. 2 and 3 show the finite element model of the foundation slab. All the points at the bottom layer of
(42)
For the special case when the ground acceleration is modeled by a finite duration white noise process (Rosenblueth model [9,10]), the autocorrelation of the stationary part of ground acceleration is given by Rw(~" ) = 2~rS03(~" ),
(43)
with So being the constant power spectrum of the white noise process. Employing eq. (43) in eq. (42), integrating over the delta function and using the approximation of [40,45,47,49,50] for small damping, the result becomes
:r-
~5 llfl
I i
(
(44)
e(t)=
1 0
for 0 < t < t o , fort>t o,
zj
[ }-
For the Rosenblueth model with
l
I
242,7 ft
/
5
R=98.4 fl
× "1
Y
(45)
Fig. l. A cross section of the containment shell with the foundation slab.
315
A.H. YousafzaL G. Ahmadi / Deterministic and stochastic earthquake response analysis 843
1
9
6
~
2
1
0
1
5
1
~
1
6
5
1
2
1
1
0
~
~
9
1
7
6
1 1
~
1 ~
3
5
2
0 9
r
I
II
~~
~
~45
0
5
",
0
6 4 7 ~
I
-
I
~~j.~689
i-
6,9r -Y 7 .
I 234 5 6 7 8 9 I0 II 121314 15 Fig. 2. Nodal point distribution pattern of the bottom layer of the foundation slab of the structure.
535 L 1 / 4 7 9 ~
-. .,. :. . .
T--
i_ ~
57z 4
9
3
332q-54 - - ~ the slab are considered to be fixed while those at the surface had a translation degree-of-freedom in the directin of x-axis. The containment shell of the plant is divided into 420 thin shell elements. Each of these elements had four nodal points. The nodes common between the brick elements and the thin shell elements transfers the motion from the foundation to the structure. The finite element model of the shell structure is shown in fig. 4. As mentioned before, all the nodal points on the bottom layer of the foundation slabs, that is, the points ranging from 1 to 225, are taken to be fixed which is a boundary condition of the problem. All the rest of the nodal points of the finite element model of the structure are assumed to have only one translational degree of freedom along the x-axis. This assumption would save
436 421[
450
III
376
I I
S4SI
/111 j 111
331 I
Nil
27i I 241i 2 26
345 ggO
IIXII I JJii IIN I J l I',.1
233
240
Fig. 3. Nodal point distribution pattern of the surface layer of the foundation slab of the structure.
275 21 247 248 249 265 I---J~29T 28 Fig. 4. Finite element mesh used for the shell.
considerable amount of computer time at the expense of reduction of the accuracy of the model. The containment shell is assumed to be made of reinforced concrete with the following properties: Weight density = 150 l b / f t 3. Modulus of elasticity = 3 × 106 psi. Modulus of rigidity = 1.5 × 106 psi. Poission's ratio = 0.18. Rayleigh damping is assumed and is supplied to the program in the form of a constant modal damping factor equal to 2% of critical.
5. Results and discussions
As the first step for the response analysis, natural frequencies and mode shapes of the structure are obtained. The present finite element model of the containment shell structure does have 445 degrees-of-freedom. Therefore, theoretically the same number of frequencies and mode shapes could be evaluated. However, evaluation of all of these natural frequencies and the corresponding mode shapes takes an enormous amount of computer time and furthermore, only the first few frequencies and the corresponding mode shapes are of practical importance in evaluation of the response by the mode superposition technique. The values of the first 25 natural frequencies are listed in table 1. It is observed that the fundamental
316
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis
Table 1 Table of natural frequencies Mode number
Natural frequency (cycles/s)
Natural period (s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
6.63 13.59 19.11 26.08 32.24 38.31 45.19 52.03 58.24 64.85 69.23 74.18 78.98 83.35 88.14 92.93 97.51 99.82 104.20 107.30 111.60 114.90 119.80 121.30 123.80
0.151 0.0735 0.0523 0.0383 0.0310 0.0261 0.0221 0.0192 0.0172 0.0154 0.0144 0.0135 0.0127 0.0120 0.0114 0.0108 0.0103 0.0100 0.0096 0.0093 0.0090 0.0087 0.0083 0.0082 0.0081
natural period of the structure is about 0.15 s which shows that the present c o n t a i n m e n t shell has the characteristics of a relatively rigid structure. Figs. 5 - 1 0 show some vertical, as well as horizontal cross sections of the
[ Fig. 5. First mode of vibration of the structure.
[
1 Fig. 6. Second mode of vibration of the structure.
d e f o r m e d structure for the first three modes of vibration. It is observed that these mode shapes bear close similarities with the corresponding modes of a simple cantilever beam.
5.1. Results of the deterministic analysis With the known natural frequencies and the corres p o n d i n g m o d e shapes, the deterministic response of the c o n t a i n m e n t shell to various earthquake ground motions could be easily calculated by the method of normal m o d e through the use of eqs. (13) and (15), as discussed in the previous sections. As the first example of the deterministic response analysis the S00E c o m p o n e n t of the E1 Centro earthquake of 1940 was considered as the input and the time
I Fig. 7. Third mode of vibration of the structure.
1
317
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis
t Fig. 8. First mode of vibration of a horizonal cross section of the structure (for nodal points 479 to 506).
/ [.
]
Fig. 11. Comparison of the estimated peak displacement response in according to stochastic theory with the exact numerical results for the S00E component of El Centro (1940) earthquake.
histories of the displacement responses of all the nodal points of the c o n t a i n m e n t structure are calculated. It was observed that the maximum relative displacement response at all the nodal points occurs at t = 4.92 s. The m a x i m u m relative displacement is found to be 0.554 cm for the top of the dome. A vertical cross section of the structure at t = 4.92 s is shown by the dotted curve in fig. 11. A similar response analysis is also carried out for the $69E c o m p o n e n t of Taft 1952 earthquake. The correFig. 9. Second mode of vibration of a horizontal cross section of the structure (for nodal points 479 to 506).
Fig. 10. Third mode of vibration of a horizontal cross section of the structure (for nodal points 479 to 506).
Fig. 12. Comparison of the estimated peak displacement response in according to stochastic theory with the exact numerical results for the $69E component of Taft (1952) earthquake
318
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis
sponding peak displacement response which occurs at t = 4.58 s is shown by the dotted curve in fig. 12. The peak response for the top of the dome is found to be 0.319 cm for Taft 1952 earthquake. From the dotted curves in figs. 11 and 12, it is observed that the first mode of vibration is quite dominant at the peak response. The dominancy of the first mode was also observed to hold throughout the time history of the response. From the available response spectra curves of El Centro (1940) and Taft (1952) earthquakes, for an oscillator with natural period of 0.15 s and damping coefficient of 2%, the corresponding maximum relative displacement responses are found to be about 0.6 cm and 0.32 cm, respectively. These values are quite close to the maximum deflections of the top of the dome found in the present deterministic analysis. These observations are also compatible with dominancy of the first mode in the response analysis of the present structure. The calculations of the responses of the containment structure to E1 Centro (1934) and Olympia (1949) earthquake are also carried out and the results are discussed in [56].
\ °3°I 0 0
5
IO 15 20 TIME (seconds)
\
/
25
50
Fig. 13. Time development of the 3o~ for the top of the dora for El Centro (1940) earthquake.
o75/ g
I
0145I-
0
,~ --
5
I0 15 20 TIME (secondsl
25
~50
Fig. 14. Time development of the 3ax for the top of the dom for Taft (1952) earthquake.
5.2. Results of the stationary analysis With the natural frequencies and the modal vectors found from the free vibration analysis of the containment shell structure, the variances of the relative displacements of various nodal points could be evaluated from eq. (31) for the stationary white noise model of the earthquake ground motion. Three times standard deviation provides an upper bound on the maximum response as is noted by eq. (32). Employing the available estimates for the power spectrum So of the El Centro (1940) and Taft (1952) earthquakes, the maximum displacement responses of the various nodal points of the containment shell are evaluated by use of the bound provided by eq. (32) and the results are plotted by solid curves in figs. II and 12. It is observed that the estimated peak responses as found from the stationary stochastic analysis do indeed form reasonable bounds on the actual maximum responses found from the deterministic analysis for those earthquakes. Similar calculations for the El Centro 1934 and Olympia 1949 earthquakes are also carried out and the results are discussed in [56].
5. 3. Results of the nonstationary analysis With known natural frequencies and the corresponding mode shapes of the containment structure, the time
development of the root mean square responses to various nonstationary models of earthquakes could be evaluated from eqs. (44) and (46). For the Rosenblueth finite duration white noise model of the ground acceleraton, such analysis have been carried out for the various nodal points of the containment shell with appropriate values of SO for the principle horizontal components of El Centro (194) and Taft (1952) earthquakes. The corresponding variations of 30x for the top of the dom are shown in figs. 13 and 14. The time duration of both earthquakes are taken to be 25 s and integrations are carried out for 30 s. It is observed that in both cases the values of root mean square displacement responses reach to their stationary levels after about 7 s. In order to compare the estimated 3o levels with the peak responses, the maximum of 3o(t) over the time duration must be considered. In the case of El Centro (1940) and Taft (1952) earthquakes with their time durations of 25 s, the peaks of o(t) are clearly their stationary values found previously. Therefore, the solid curves in figs. 11 and 12 also correspond to the estimated maximum responses according to nonstationary analysis. Similar analysis for the response of the containment shell to E1 Centro (1934) and Olympia (1949) earthquakes are also carried out and the results are discussed in [56].
A.H. Yousafzai, G. Ahmadi / Deterministic and stochastic earthquake response analysis
6. Conclusions and further remarks In the present investigation the response of the cont a i n m e n t shell of a nuclear power plant to deterministic as well as r a n d o m e a r t h q u a k e support m o t i o n is studied. A finite element model of the c o n t a i n m e n t shell is constructed a n d the m e t h o d of modal analysis is employed. The response of the shell to E1 C e n t r o (1940) and Taft (1952) earthquakes are obtained and the results for the peak relative displacement responses are shown to b e compatible with those o b t a i n e d from the corresponding response spectra curves of those earthquakes. Stationary a n d n o n s t a t i o n a r y stochastic models of earthq u a k e ground m o t i o n are considered and the general expressions for the m e a n square responses of various n o d a l points are derived a n d discussed. It is shown that the 3o levels of the relative displacement responses provide reasonable b o u n d s on the corresponding maxim u m responses. In b o t h deterministic as well as stochastic response analysis of this relatively rigid structure, it is observed that the first m o d e of vibration is quite dominant.
Acknowledgements The authors would like to t h a n k Professor N. Mostaghel, Professor H. Z o h o o r a n d Professor H. Seyyedian a n d Shiraz University for m a n y helpful discussions. The earlier stages of this work was carried out at the D e p a r t m e n t of Mechanical Engineering of Shiraz University a n d is supported in part by the Atomic Energy Organization of Iran.
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