Earthquake response analysis for a BWR nuclear power plant using recorded data

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NUCLEAR ENGINEERING AND DESIGN 20 (1972) 385-392. NORTH-HOLLAND PUBLISHING COMPANY

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E A R T H Q U A K E R E S P O N S E A N A L Y S I S F O R A BWR NUCLEAR POWER PLANT USING RECORDED DATA

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20-24September 1971

Kiyoshi MUTO Muto lnstttute o f Structural Mechantcs, Inc , Tokyo, Japan

and Kyoichi OMATSUZAWA Nuclear Power Department, The Tokyo Electric Power Company, Inc , Tokyo, Japan Recewed 27 July 1971 In the dynamic analysis of comphcated structures such as nuclear power plants, zt is necessary to consider different damping characteristics for each structural element. The authors have developed a computer program for the dynamic analys~s based on the internal viscous damping theory and have recently performed a vibration test and earthquake observations of an actual nuclear power plant. The data resulting from the test and observation were applied to the program and the dynamzc response of each part of the plant was computed. A close agreement was noted between the computed and recorded acceleration-tzme histories as well as acceleratzon-response spectra The authors conclude that their analyzmg system might be one of the most rehable methods for the design of the nuclear power plants

1. I n t r o d u c t i o n In the dynamic analysis for complicated structures such as nuclear power plants, composed of various materials or elements, it is necessary to consider different damping characteristics for each structural element based on the internal viscous damping theory [ 1 - 7 ] . In this paper, we present an analyzing system based on the internal viscous damping theory. The system was checked against a forced vibration test and earthquake observations (see fig. 1). In 1969 the authors performed a precise forced vibration test at the Fukushima Nuclear Power Plant No. 1 (BWR-type) of the Tokyo Electric Power Company, and estimated the rigidities and the damping coefficients of the structural elements. On May 26th, 1970 an earthquake [8] shook the plant, and the acceleration records at six points in the N - S direction of the reactor building were obtained. We applied the recorded acceleration-time history at the foundation mat to the computer program as an

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--~F,~';,,~'~.]

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Fig. 1. Flow diagram for dynamic earthquake response analysis.

386

K Muto and K Ornatsuzawa, Dynamtc earthquake response analysts of nuclear power plants

input earthquake and the computed acceleratlon--txme histories of the other five points were compared with the actual recorded ones. Furthermore, the response spectra from both the computed acceleration-time histories and the recorded ones were also calculated and compared. A close agreement was found between the computed and recorded acceleration-time histories as well as the response spectra. This suggests that the system of analysis described here might be one of the most reliable methods for nuclear power plants

0 LOCATIONOF SEISMOGRAPHS

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W2 ST

2. Outline of the reactor building and the earthquake observation This nuclear power plant is a boihng water type reactor (460 MWe, GE type) as shown in figs. 2 and 3. The reactor building is about 57 meters in height from the foundation mat and is composed of a reinforced concrete structure from the basement (BMT)

Fig. 2. Concentrated masses and location of seismographs.

Fig. 3. The Fukushima nuclear power plant no. 1.

K Muto and K. Omatsuzawa, Dynamic earthquake response analysis o f nuclear power plants

387

BMT

,

3F 5F

RF

Fig. 4. Recorded acceleration-time histories.

to the fifth floor (refueling floor) with a steel roof truss (ST). The foundation mat (about 42 × 42 m 2) is built on sandy mudstone and is completely separate from the other auxiliary buildings. At the center of the building there is a reactor pressure vessel (RPV) which surrounded by a concrete gamma shield wall (GSW), a bulb shaped steel primary containment vessel (PCV) and a reinforced concrete shield wall (SW) in that order. As is common in seismic devices, horizontal support is provided between the RPV and GSW: the PCV is connected to.the SW by shear lugs and to the GSW by a stabilizer. After the construction of the reactor building, in March 1970, several sets of seismographs (moving coil type) were installed. The arrangement of the seismographs is shown in fig. 2. On May 26th, 1970, a minor earthquake (whose center was located 50 kilometers offshore at a depth of 50 kilometers) hit the plant and the N - S earthquake motions were recorded at the following six points; the basement, third and fifth floors, roof, the top (GSW-t) and bottom (GSW-b) of the gamma shield wall. These records, shown in fig. 4, were digitized and used to perform the earthquake response analysis.

3. Dynamic analysis 3.1. Mathematical vibration model

3.1.1. Foundation mat The mat is divided into three foundation el-_..lents, F1, F2 and F3, shown in fig. 5. These are assumed to have the same horizontal movement but independent rotahons. The rotation springs for F1, F2 and F3 are evaluated by considering the elastic effect of the soil and mat. 3.1.2. Building wall and shield wall The main resisting elements of the reactor budding are assumed to be the shield wall and the three building walls: Wl, W2 (setback wall), and W3 shown in fig. 2. The masses are concentrated at each floor level and the stiffness matrices are calculated by considering the bending and shearing deformations. The horizontal spring constants between the building walls and the shield wall are considered because the rigidity of the floor is poor compared to that of the surrounding concrete walls.

K Muto and K Omatsuzawa, Dynamtc earthquake response analysts of nuclear power plants

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3.3 3 3.1. General relations The relation between the external force vector 0"}t and the displacement vector {v} t, of the tth element, may be expressed as follows:

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(1)

in which [B] t = stiffness matrix of ith element expressed m local coordinates, deduced by a slope deflection method. The relation for all elements between {f} and {v} is given m the following form:

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3.1.3. Equipment The stiffness matrices are computed considering the bending and shearing deformations of each element. The horizontal springs of the shear lug and two stabilizers are taken into account as shown in fig. 5. The light water and core internal structures inside the RPV are assumed to have only mass effect on the building's vibration. 3.2. Physical constants The physical constants such as Young's modulus E, Poisson's ratio v and internal viscous damping coefficient r for each material are shown in table 1.

E

It/cm21

Soll Concrete Steel

45 520 2100

v

The displacement transformation matrix [A], which relates the displacement vector {V} and the velocity vector {V} m the generalized coordinates to {v} and {f } respectively, may be written as follows:

{v} = [,41 { ~ } ,

(3~

The external work in both coordinates is equal: {v}T{f} = { ~ } T { F } ,

(p} = [Xl {~} + [Cl {P}

r

[sec] 0.42 0.17 0.30

{~} -- tA] {e}.

(4)

where {F} = generalized external force vector. From eqs. (2), (3) and (4), the following equation of equilibrium may be obtained

Table 1 Physical constants.

Material

(2)

0.044 0.00064 0 00032

(5)

where [K] = [,41T [K] [A] (coupled stiffness matrix), and

389

K. Muto and K. Omatsuzawa, Dynamic earthquake response analysis of nuclear power plants

[C] = [A] T [~"][A] (coupled damping matrix).

(6)

3.3.2. Damped free vibration By substituting the inertia force -[34] {~} in {F}. of eq. (5), the dynamic equation of damped free vibration may be written as follows: [341 {V}+ [C] {1~}+ [KI {V} =

0 ,

(7)

where [34] = generalized diagonal mass matrix. The coupled damping matrix [C] is symmetric as the coupled stiffness matrix [K] is, but in general is not proportional to [K]. From eq. (7) both complex eigenvalues sX and complex mode shapes sU of sth order can be solved by the generalized eigenvalue program as follows: sX = SXRe + is)tlm,

(8)

st: = SURe + isVtm.

(9)

Furthermore the sX may be defined as follows: sX = sh s w + i

l~-sh 2 sw.

(10)

Then, the ]th point movement s~. may be expressed in the following form: s ~ =,[ sU/le -shswt c o s ( x / ~ S ~ s w t ÷ s ¢ ] )

(11)

4. Damped free vibration The natural periods, damping factors from the first to the eighth order are shown in table 2 and the complex modes are illustrated with amplitudes and phase angles m fig. 6. The characterxstlcs of each mode may be described as follows: (1) The first mode: Both the building and the mat vibrate in the same direction, but the rocking of the mat lags the displacement of the roof by phase angle of 64 ° . (2) The second mode: The amplitude of the steel truss is very dominant. The top of the steel truss leads the fifth floor by 106 ° and the mat by 154° . (3) The third mode: This mode corresponds to the first mode of the RPV. (4) The fourth mode: The horxzontal movement and rocking of the foundation mat are very pronounced. This mode may be considered the second mode of the whole vibration system. (5) (6) The fifth and sixth modes: The steel truss structure vibrates locally. The fifth mode means the torsional movement of the truss structure. On the other hand, the truss m the sixth mode moves horizontally. (7) The seventh mode: This mode is considered the third mode of the whole vibration system. (8) The eighth mode: This is considered the first mode of the GSW.

where s~ = circular frequency, s¢i = phase angle, and sh = so-called damping factor which may be calculated from eq. (8) and eq. (10). It can be found by eq. (11) that each mass of each vibration mode may vibrate with an independent phase angle. 3.3.3. Forced vibration When the foundation mat is subjected to an earthquake acceleration a, the inertia force -[M] { P +a~} may be used for eq. (7). Then, the dynamic equation of forced vibration may be expressed a,s follows:

NI

[Cl {V}+

{V} =- NI

where {~} = unit vector. Eq. (7) can be solved directly by a step-by-step integration process based upon Newmark's 3-method.

Table 2 Natural period and damping factors. Period [sec] sT =

1 2 3 4 5 6 7 8

2~

StON/-1-~ 2

0.25 0.18 0.089 0.077 0.051 0.048 0.050 0.045

Frequency [ sec- 1]

Damping factor

sf = l/sT

sh

4.0 5.5 11.2 13.0 19.6 20.8 20.0 22.2

33.7 8.4 1.6 70 4 24 2.9 35.9 4.3

396

K Muto and K Omatsuzawa,Dynamic earthquakeresponse analysis of nuclear power plants

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5. Comparison between computed and recorded responses 5. I. Comparison o f acceleration-time histories

The digitized acceleration values t~ at the foundation mat are used as an input earthquake to the computer program of earthquake .response analysis. The.displacement {V} , velocity { V} and acceleration" {V} of each mass are comRl.lted by the method mentioned in section 3.3.3. The V at the recorded points are compared with the recorded acceleration-time histories. Both the computed and recorded acceleration-time histories at the third and fifth floor, and the top and bottom of the GSW are shown in fig. 7. The computed and recorded maximum accelerations are illustrated in fig. 8.

K. Muto and K. Omatsuzawa, Dynamic earthquake response analysts of nuclear power plants

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Fig. 9. Comparison b e t w e e n c o m p u t e d and recorded acceleration response spectra.

5.2. Comparison of acceleration response spectra The response spectra from both the computed and the xecorded time histories are calculated and the response spectra at the third and fifth floor and at the top and bottom of GSW are shown in fig. 9. In these figures, a fairly dose correspondence between the response spectra can be found by comparison. The first peak of each point almost agrees with the

Fig. 8. Comparison of m a x i m u m acceleration.

first period (0.25 sec) at the reactor building. On the other hand the second peak of each point is based on the dominant period (0.11 sec) of the input earthquake.

6. S m m ~ A close agreement was found between the computed and recorded acceleration-time histories as well as acceleration response spectra. Obviously, this suggests that the vibration model provides a fairly accurate simulation of the nuclear reactor building for earthquake-resistant characteristics. The authors would especially like to emphasize that further study on the internal viscous damping coefficient r of each material or element is of the greatest importance. Finally, we hope that our system of analysis will be endoresed by all engineers in this field and help in the improvement of all earthquake.resistant designs of nuclear power plants in the future.

392

K Muto and K Omatsuzawa, Dynamic earthquake response analysts of nuclear power plants

References [1 ] K. Muto, The Damping Vibration (Japanese), Journal of A 1J., m July (1929) [2] K Muto and others, Earthquake Resistant Design of Nuclear Power Plant, National Conference on Bridge and Structural Engineering in Tokyo (1968). [3] K Muto, Earthquake Response Analysis of Nuclear Power Plant, J Atomic Energy Society of Japan (1969) [4] K Muto and others, Modal Analysis for Equation of Motion with Various Different Dampmgs (Japanese), Leaflet of the Muto Institute of Structural Mechanics (1970) [5] Japan Electric Association, Studies on the Vibrational Characteristics of a Nuclear Power Plant Model (Japanese),

Report (1968) [6] K Muto and others, Studies on Aselsmlc Design of Nuclear Power Plant (No. 5) - A Forced Vibration Test and Simulation Analysis of a Nuclear Power Plant Model based on Equation of Motion with Various Different Dampmgs - (Japanese), Leaflet of the Muto Institute of Structural Mechamcs ( 1970). [7] K. Muto and others, Studies on Aselsmlc Design of Nuclear Power Plant (No 6) - Elgen Value Analysis of a Nuclear Power Plant Model based on Equation of Motion with Various Different Dampmgs - (Japanese), Laeflet of the Muto Institute of Structural Mechamcs (1970) [8] K Omatsuzawa, Earthquake Observations at the Fukushlma Nuclear Power Plant No. 1 (Japanese), J. Thermal Power 22 (1971).