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Research on FSI effect and simplified method of PCS water tank of nuclear island building under earthquake
Progress in Nuclear Energy 100 (2017) 48e59
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Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene
Research on FSI effect and simplified method of PCS water tank of nuclear island building under earthquake Chenning Song a, Xiaojun Li a, b, *, Guoliang Zhou c, Chao Wei c a
The College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China Institute of Geophysics, China Earthquake Administration, Beijing 100081, China c Nuclear and Radiation Safety Center, Ministry of Environmental Protection of the People's Republic of China, Beijing, 100082, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 November 2016 Received in revised form 17 April 2017 Accepted 25 May 2017
Water tank is one of the most important components of passive containment cooling system (PCS) in advanced nuclear power plant. Moreover, the dynamic characteristics of shield building and auxiliary building might be changed because of fluid-structure interaction (FSI) of the PCS. Therefore, FSI between cooling water and the shield building should be considered in the dynamic analysis of the nuclear island building. To study the FSI effect and propose an effective and simplified method, three numerical models were established and analyzed under three-dimensional seismic loadings. Acceleration time histories of reference points in the shield building and auxiliary building were got and floor response spectra were calculated. The results show that FSI effect could not be regarded to decrease the seismic response simply but it should be considered in structure design and analysis. A simplified model is proposed in this study and proved to have a good agreement with the FSI model in the curves of floor response spectra. The conclusions could provide reference for further study in the field of nuclear island building. © 2017 Published by Elsevier Ltd.
Keywords: Water tank Nuclear island building Fluid-structure interaction (FSI) Floor response spectrum
1. Introduction The nuclear island building studied in this paper is one of the third-generation nuclear power plants (Schulz, 2006). One of the most obvious characteristics is the passive containment cooling system (PCS). Shield building, steel containment vessel and auxiliary building are the three important components of nuclear island building. Shield building is used to protect the steel containment vessel and internal structure. Auxiliary building is used to protect and shield the mechanical equipment and electrical equipment outside of the safety containment. The PCS and the numerical model of nuclear island building are shown in Fig. 1. Because of FSI between cooling water and shield building, the dynamic characteristics of nuclear island building would be changed. In recent years, the research around the FSI of nuclear power plant has been done to evaluate the security of structure. Lo Frano and Forasassi (2009, 2011, 2012, 2015) did some research on liquid metal nuclear reactor considering the FSI effect subjected to seismic loads. Choi et al. (2013) designed a 1/12 scale model of the
* Corresponding author. The College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.pnucene.2017.05.025 0149-1970/© 2017 Published by Elsevier Ltd.
System-Integrated Modular Advanced Reactor (SMART) and established the three-dimensional finite element model based on ANSYS. The numerical results considering FSI effect are well matched with the test. Leonardo et al. (2007) established finite element model of AP1000 nuclear island and analyzed the seismic response of the structure at generic and rock sites. Liu et al. (2016) designed a 1/20 scale concrete containment vessel (CCV) and did the shaking table tests of CCV for CPR1000 nuclear power plant. It resulted that the prototype PCCV for CPR nuclear power plant has sufficient seismic safety margin. Lee et al. (2013) did some thermodynamic and seismic analysis for AP1000 structure considering water levels and the location of air intake. Moreover, some optimizing opinions were proposed. In the field of FSI effect of AP1000 nuclear power plant, some advanced numerical methods were used. Liu (2014) used the Coupled Eulerian Lagrange (CEL) technique to analyze the FSI of the PCS water tank and the seismic response of the shield building based on ABAQUS. Zhao et al. (2014a,b) used the Arbitrary Lagrange Eulerian (ALE) algorithm to simulate the FSI, fluid sloshing and oscillation of water tank under earthquake. The influence of water tank and the air intake were systematically studied. The research shows that water levels and the location of air intake could change the dynamic characteristics of shield building. In addition, some
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
Acronyms PCS FSI SSI SMART CCV PCCV CEL ALE SPH PBFE CWT CCWT ATH PGA
Passive Containment Cooling System Fluid-Structure Interaction Soil-Structure Interaction System-Integrated Modular Advanced Reactor Concrete Containment Vessel Prestressed Concrete Containment Vessel Coupled Eulerian Lagrange Arbitrary Lagrange Eulerian Smoothed Particle Hydrodynamics Potential-Based Fluid Elements Cylindrical Water Tank Circular Cylindrical Water Tank Acceleration Time History Peak Ground Acceleration
optimized design advices were proposed. On the basis, Xu et al. (2016) used the smoothed particle hydrodynamics (SPH) and finite element method (FEM) coupling method to simulate the FSI between cooling water and shield building of AP1000. The research shows that the water tank can decrease the natural frequency and response of the shield building. But the auxiliary building was not considered in these studies. The auxiliary building is another important component of nuclear island building and there is also some important equipment in it, so it is necessary to do some study on auxiliary building. The existing research mainly paid attention to the shield building, but there was little study on auxiliary building and the FSI effect of the whole nuclear island building has not been studied systematically. Zhao et al. (2015) established the numerical model and did some research on the natural frequency of nuclear island building considering FSI effect. The frequency characteristics of structure considering various water levels and the elevation of air intake were got from analysis. However, this study was not involving the seismic response. In the field of simplified method, researchers have proposed
49
some models to analyze the dynamic response of liquid containers in the past decades and these models are almost based on Housner model. Housner (1957, 1963) developed an approximate method for regular containers on the basis of some assumption and the core idea was that the water could be split into impulsive mass and convective mass. This method was based on results from experimental as well as analytical method. Haroun (1980) developed the model by considering the flexibility of the storage. Aimed at the influence of high modes of sloshing, Manos and Clough (1983) did shaking table tests to study seismic response of the storage model. The previous researches show that the water tank has an influence on the seismic response of shield building. But the auxiliary building was not considered in these studies. In addition, the simplified model for sloshing liquid has not been used to analyze the seismic response of nuclear island building. In this study, one simplified method is proposed based on Housner model. Then, numerical models of shield building and auxiliary building are established to analyze the seismic response of structure. The FSI effect of nuclear island building is investigated. By comparing the FSI model and simplified model, the simplified method is proved reasonably and could be used for further study of structure design and analysis of nuclear power plant.
2. Analysis methods The formulation of potential-based fluid elements (PBFE) and simplified model of water are reviewed in this section.
2.1. Potential e based fluid elements The equations of the f u formulation in this study are briefly introduced in this part. As the special fluid element in ADINA (ADINA R&D 2010), the PBFE can be used for frequency and timedomain analyses of structure (Wei et al., 2013; Bouaanani and Lu, 2009). There are two basic parameters in the formulation: velocity potential f and displacement u. The velocity potential f satisfies the wave equation:
Fig. 1. The passive containment cooling system and profile of numerical model.
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2
1 v2 f V f¼ 2 Cw vt 2 2
(1)
2
where V is the Laplace differential operator, t is the time variable, Cw is the wave velocity in fluid. And Cw is given by
Cw ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi kw =rw
(2)
where rw is water density, kw is the bulk modulus. Based on the standard theories, the variational form of Eq. (1) can be got as
rw 2 Cw
Z Vw
v2 f dfdV þ rw vt 2
Z Sw
vu ndfdS þ rw vt
Z Vf$VdfdV ¼ 0
(3)
Vw
where Vw is the volume of water, Sw is water boundary where normal velocity is prescribed, n is unit normal on Sw pointing into the fluid and V is the vector gradient. Under earthquake, the dynamic response of the water tank is coupled through compatibility of velocity potential and prescribed normal velocity at the fluidstructure interface. The system dynamic equations of the tank filled with water can be obtained (Olson and Bathe, 1985):
€ 0 M ss C ss U € þ C ws 0 M ww F € g ðtÞ M ss 1u ¼ C ws 1u_ g ðtÞ
C sw 0
U_ þ K ss _ 0 F
0 K ww
U
F
(4) In addition
M ss ¼ rs
Z
N Ts N s dV; M ww ¼
Vs
K ss ¼ rs
Z
rw 2 Cw
Z Vw
N Ts DTs N s dV; K ww ¼ rw
Vs
C sw ¼ C Tws ¼ rw
N Tw N w dV
Z
BTw Bw dV
Vw
Z
N Ts nN w dS;
Vw
C ss ¼ aM ss þ bK ss ;
(5)
where N s and N w are standard isoparametric shape function matrices for shell and fluid elements respectively, rs and Vs are the density and volume of the tank, Ds is the elastic matrix of shell elements, M ss and K ss are the mass and stiffness matrices for the tank, M ww and K ww are the mass and stiffness matrices for water, 1 is the column vector which has the same dimension of nodal relative displacement U, C ws and C sw are the damping matrix accounting for FSI between water and tank, and C ss is the Rayleigh damping matrix, in which a and b are the Rayleigh damping coefficients. Using the first and second natural frequencies of nuclear island building, the damping coefficients a and b can be calculated by Eq. (6). The matrix Bw is given by Eq. (7)
C cr
2zu1 u2 2z ¼ 2M ss un ; a ¼ ; b¼ u1 þ u2 u1 þ u2
ð2Þ
vNw vx
,,,
ð2Þ
vNw vy
,,,
3 ðiÞ vNw 7 vx 7 7 ðiÞ 7 vNw 5
(7)
vy
where C cr is the critical damping matrix, un is the natural frequencies, u1 and u2 are the first and second natural frequencies. z is the damping ratio of structure, i is the number of nodes per fluid element.
2.2. Simplified model Considering the structure characteristics of water tank, the upper structure of shield building could be regarded as one rigid tankliquid system. So, Housner model might be adaptable. The basis assumptions are as follows: the structure and liquid motions remain linearly elastic, the tank’ sections are regular (rectangular or circular), the tank walls are rigid and the liquid is inviscid. In this model, the liquid mass is split accordingly into the impulsive mass and the convective mass. Impulsive component represents rigidbody motion of liquid. Under dynamic loading, a part of liquid moves synchronously with the tank as an added mass and is subjected to the same acceleration levels as the tank. Convective component represents sloshing of the liquid at the free surface. Under lateral excitation, oscillations of liquid occur and result in the generation of pressure on the wall, base and roof of the tank. As shown in Fig. 2, the impulsive mass m0 is assumed to be rigidly attached to the container walls while the convective mass is split into series of masses m1, m2, …,mn associated with the 1st, 2nd, …, nth sloshing masses respectively. These masses are attached to the container walls via springs of stiffness k1, k2, …, kn representing the 1st, 2nd, …, nth antisymmetric sloshing frequencies respectively. In this study, one simplified model considering the first sloshing mode is proposed to calculate the seismic response of the advanced nuclear island building. Based on Housner model, m1 can be considered to connect to the tank walls by some spring-dampers (sp1, …, spi, …spq) as shown in Fig. 3. m* ; k and c denote the discrete mass of m1 , the stiffness and damping coefficients of one spring-damper, dq is the angle between adjacent spring-dampers and qi is the angle between X-direction and spi. The following equations can be obtained:
dq ¼
2p m ; qi ¼ ði 1Þdq; m* ¼ 1 q q
(8)
where q is the number of spring-dampers. In Fig. 3 (b), d is the virtual displacement of m* assumed to occur along X-direction and some parameters can be calculated by the following equations:
di ¼ d cos qi fix ¼ 2kdi cos qi ¼ 2kdcos2 qi
(9)
where di is the deformation of spi along the axis direction, fix is the internal force of spi along X-direction. The bearing composite force of m1 is Fx which can be calculated below:
Fx ¼ (6)
ð1Þ
vN 6 w 6 vx Bw ¼ 6 6 ð1Þ 4 vNw vy
q X i¼1
fix ¼ 2kd
q X
cos2 qi
(10)
i¼1
The both sides of Eq. (10) are multiplied by dq and then Fx can be obtained:
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Fig. 2. Housner spring-mass model for rigid tanks.
Fig. 3. Lumped mass model of sloshing liquid. (a) The whole simplified model (b) One discrete spring-damper.
Fx dq ¼ 2kd
q X
Kx ¼ mn u2n Cx ¼ 2gmn un
cos2 qi dq
i¼1
Fx
2p ¼ 2kd q
Z2p cos2 qdq ¼ 2pkd
(11)
0
where g is sloshing damping ratio. The simplified method only considers the first sloshing mode in this study. So when n ¼ 1, k and c of one spring-damper can be calculated according to Eqs. (12)e(14) above:
Fx ¼ qkd So the total equivalent stiffness coefficient Kx can be obtained:
Kx ¼
Fx ¼ qk d
(12)
Simultaneously, the total equivalent damping coefficient Cx can be obtained:
Cx ¼ qc
(13)
According to the free vibration equation of structure with damping, Kx ; Cx can also be expressed by the following equation:
(14)
k¼
Kx m1 u21 ¼ q q
c¼
Cx 2 ¼ gm1 u1 q q
(15)
TID-7024 (1963) gives some equations of impulsive and convective mass for cylindrical tank based on Housner model. To get the key parameters of simplified model, the water tank is equivalent to regular cylindrical water tank (CWT) firstly. As shown in Fig. 4 (a) and (b), Ri ; Ro and h denote the inner radius, outer radius and the height of free fluid surface. The equivalent radius R of regular cylindrical tank can be calculated by keeping the volume of water constant. Then, the parameters can be obtained by Eqs. (16)
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Fig. 4. Models of water tank. (a) PCS water tank; (b) Equivalent CWT; (c) Equivalent CCWT.
and (17) (Housner, 1957).
m1 R h ¼ 0:318 tanh 1:84 h R m cosh 1:84 Rh 1 h1 ¼1 h 1:84 Rh sinh 1:84 Rh
(16)
(17)
where m is the total mass, m1 and h1 are the 1st sloshing mass and height of 1st sloshing mass. To get the sloshing frequency more accurately, one method is proposed for circular cylindrical water tank (CCWT) (Meserole and Fortini, 1987). As shown in Fig. 4 (c), the water tank is equivalent to regular CCWT and equivalent height of free surface ha can be calculated by keeping Ro ; Ri and V constant. The 1st sloshing frequency u1 can be got by the following equation:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 ¼ ðg=Ro Þx1 tanh k1 k1 ¼ ðha =Ro Þx1
(18)
where g is the acceleration due to gravity. And x1 is the root of Eq. (19) 0
0
0
0
J1 ðx1 ÞY1 ðkx1 Þ J1 ðkx1 ÞY1 ðkx1 Þ ¼ 0 k ¼ Ri =Ro
(19)
where J1 and Y1 denote the Bessel function of the first and second 0 0 kinds and first order. J1 and Y1 denote the first derivative of J1 and Y1 . 3. Numerical analysis and discussion 3.1. Description of numerical models In the previous study, Zhao et al. (2014a,b) proposed that the variation of stiffness as location of air intake was very small compared to the integral rigidity of overall nuclear island building. So the air intake and the steel containment vessel are not considered in this study. To analyze the FSI effect for nuclear island building and test the reasonableness of simplified model, three models are established: (a) Model one: Model considering FSI effect; (b) Model two: Simplified model; (c) Model three: Model with empty water tank. The software ADINA is used in the numerical analysis. The baseboard of nuclear island building is modeled using 3D-solid
finite elements while the main body of shield building and auxiliary building is modeled using shell finite element. The steel frames are simulated using beam elements and the water in the tank is simulated using 3D PBFE. The spring-dampers are simulated using spring elements and concentrate masses in model two. The main structure is shown in Fig. 5 and the differences between three models are the water tanks. As shown in Fig. 6, the water tank in model one is filled with the designed cooling water. In model two, the water is replaced by spring-dampers according to the theoretical derivation in the second section. The water tank is empty in model three. The numerical models are shown in Fig. 5 and the material parameters are listed in Table 1. The geometry sizes of previous papers (Zhao et al., 2014a,b, 2015) are referenced in this study. 3.2. Dynamic characteristics of numerical models 3.2.1. Parameters of simplified model To avoid the local stress of tank walls, the discrete models are used in the numerical models. As shown in Fig. 6 (b), the first sloshing mass is divided to one hundred and twenty-eight lumped masses at two layers. The lumped masses in each layer are coupled at one rigid point respectively. Based on ASCE 4-98, the convective mass m1 is divided into two parts: horizontal components and vertical components. The horizontal components are connected to tank walls by spring-dampers in which the stiffness and damping coefficient are calculated by Eqs. (15)e(18) and the vertical components are added to the bottom of water tank discretely. The impulsive mass m0 is added to the bottom and walls of the tank discretely. According to the geometry sizes of nuclear island building, the key parameters of one spring-damper can be calculated. 3.2.2. Natural frequencies and 1st sloshing frequency Natural frequencies are important dynamic characteristics for structures. In this part, the natural frequencies of nuclear island building with different water tanks are studied. On one hand, the dynamic characteristics of models can be obtained, on the other hand, the reasonableness of simplified model can be verified preliminary. The 1st natural frequency and 1st sloshing frequency of models are list in Table 2. The errors of 1st natural frequencies and 1st sloshing frequencies between model one and model two are 1.402% and 0.794% separately. Moreover, the first ten order natural frequencies are compared in Fig. 7. The first two modes of structures are integral vibrations in the two horizontal directions and the 3rd to 10th modes are local vibrations of floors and walls in auxiliary building. Therefore, the FSI only has obvious influence on the first two modes and it is similar to results from the previous paper (Zhao et al., 2015). The “mode-frequencies” curve of model two is in good agreement with model one. It indicates that the proposed simplified model considering 1st sloshing mass is
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
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Fig. 5. Elevation drawing and marking of numerical model. (a) S-N section (b) E-W section. Ls-n: Width of south-north. Le-w: Width of east-west. Ds: Diameter of shield building. Di: Inner diameter of water tank. Do: Outer diameter of water tank. Ha1: Southern height of auxiliary building. Ha2: Northern height of auxiliary building. Hs: Height of shield building. Ht: Height of water tank. Hw: Height of water surface.
Fig. 6. Three different water tanks in numerical models. (a) Water tank in model one; (b) Water tank in model two; (c) Water tank in model three.
3.3. Seismic response of numerical models
Table 1 Material parameters of nuclear island building. Material parameters 3
Density (kg/m ) Young's modulus (Pa) Bulk modulus (Pa) Poisson’ ratio
Concrete
Steel
Water
2400 3.00 1010 e 0.17
7800 2.00 1011 e 0.30
1000 e 2.30 109 e
Table 2 1st natural frequency and 1st sloshing frequency. Parameters
1st natural frequency (Hz) (error)
1st sloshing frequency (Hz) (error)
Model one Model two Model three
3.183 3.139(-1.382%) 3.402
0.126 0.127(0.794%) e
suitable to simulate the seismic response of structure approximately in this study.
This study analyzes the FSI effect of nuclear island building subjected to seismic loads. The special acceleration time histories (ATH) used as inputs at the foundation is fitted by code response spectrum. The peak ground accelerations (PGA) of safe shutdown earthquake (SSE) in three directions are 0.3 g and the inputs ratio of three-direction is 1: 1: 1 (Lin, 2012; Xia, 2013; Westinghouse Electric Co. LLC, 2011). Fig. 8 describes the ATHs and the corresponding spectral acceleration with the damping ratio of 5%. X, Y and Z represent the S-N, E-W and vertical direction separately. The dynamic responses of three numerical models are calculated in this part. Three groups of reference points are selected to investigate the FSI effect and the reasonableness of simplified model. The first two groups are located at shield building, while the third group is located at auxiliary building.
3.3.1. Response of shield building Four reference points in the first group are chosen at the north
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C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
Fig. 7. First ten order natural frequencies of three models.
(model one) is less than that of the model with empty water tank (model three) in period range from 0.1s to 0.3s basically. This period range corresponds to the main frequency band of structure from 3.0 Hz to 10 Hz. The amplitudes of spectra do not have much difference and the corresponding periods of amplitudes do not change almost between the two models. In the period range from 0.3s to 0.6s, the seismic response of model one is slightly larger than that of model three. The FSI effect may reduce the horizontal seismic response in some degree. In the vertical direction, the seismic response of model one at the water tank (P1 and P2) is less than that of model three when the period is smaller than 0.1s. When the period exceeds 0.1s, the seismic response of model one is more than that of model three. Moreover, the amplitudes of spectra move to long period band when considering FSI effect. The influence of FSI effect at shield building (P3 and P4) is complex. The amplitudes of spectra also do not have much difference but the corresponding periods of amplitudes move to long period band in model one. The FSI effect may change the dynamic characteristic and amplify the seismic response in the vertical direction.
Fig. 8. Input ground motions in this study. (a) ATH in X-direction (b) ATH in Y-direction (c) ATH in Z-direction (d) Response spectra with the damping ratio of 5%.
of shield building. The heights of P1, P2, P3 and P4 are 91.25 m, 79.27 m, 72.54 m and 63.54 m separately. P1 and P2 are at the water tank, while P3 and P4 are at shield building. Fig. 9 shows the floor response spectra at the chosen points in the three directions. In the horizontal directions, the seismic response of FSI model
Another four points in the second group are chosen at the east of shield building as supplementary. The heights of P5, P6, P7 and P8 are 91.25 m, 79.27 m, 72.54 m and 63.54 m which are the same as the first four points. The floor response spectra at the chosen points in the three directions are shown in Fig. 10. It can be noted that the
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
Fig. 9. Floor response spectra at reference points in group one with damping ratio of 5%.
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Fig. 10. Floor response spectra at reference points in group two with damping ratio of 5%.
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Fig. 11. Floor response spectra at reference points in group three with damping ratio of 5%.
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results are similar to group one. While the amplitudes of spectra are larger than those in group one because of the different characteristics in the two horizontal directions.
smaller when FSI effect is considered, however the corresponding periods of amplitude have no change. So the water tank may reduce the seismic response of auxiliary building in the vertical direction.
3.3.2. Response of auxiliary building Five reference points are chosen for analysis at the auxiliary building. P9 is the middle point at the top of auxiliary building and the height is 51.20 m. P10, P11, P12 and P13 are the middle points on the floors with heights of 31.65 m, 28.55 m, 25.23 m and 21.90 m separately. Fig. 11 shows the floor response spectra at the chosen points in the three directions. In the horizontal directions, the X-direction is the weak-axis and the Y-direction is the strong-axis of P9, while it is opposite to P10, P11, P12 and P13. Fig. 11(a), (e), (h), (k) and (n) show that FSI effect may reduce the seismic response at the weak-axis of floor when the period is close to the 1st natural period but it may amplify the seismic response when the period is close to 0.15s. At the strongaxis of floor, the influence is not obvious as shown in Fig. 11(b), (d), (g), (j) and (m). In the vertical direction, the amplitudes of spectra become
3.3.3. Reasonableness of simplified model The floor response spectra of three models are compared in Fig 9e11. It can be seen that although there are some difference of amplitudes between the FSI model (model one) and the simplified model (model two) in some periods, the response spectral curves of two models are in good agreement. To analyze the error between two models and identify the reasonableness of simplified model more accurately, the amplitudes of spectra and corresponding periods are listed in Tables 3e5. The corresponding periods of amplitudes are all the same in the three directions. The maximum absolute errors of amplitude are 6.32%, 6.86% and 4.94% in three directions. In Table 3 and Table 4, the maximal mean absolute error of amplitude occurs in X-direction which is the strong-axis of nuclear island building. In Table 5, the maximal mean absolute error of amplitude occurs in Y-direction which is the weak-axis. In Z-
Table 3 Comparison of amplitude and corresponding period between model one and model two in group one. Points
Models
P1
Model one Model two P2 Model one Model two P3 Model one Model two P4 Model one Model two Mean absolute error
X-direction
Y-direction
Z-direction
Amplitude (m/s2) (error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
86.31 81.16 (5.97%) 66.68 64.68 (3.00%) 64.53 61.56 (4.60%) 45.01 43.43 (3.51%) 4.27%
0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278
73.46 70.58 (3.92%) 58.12 57.26 (1.48%) 50.42 49.70 (1.43%) 41.93 41.43 (1.19%) 2.01%
0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290
50.76 50.74 (0.04%) 47.08 48.85 (3.76%) 22.92 22.21 (3.10%) 21.73 21.14 (2.72%) 2.41%
0.111 0.111 0.111 0.111 0.278 0.278 0.278 0.278
(0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%)
Table 4 Comparison of amplitude and corresponding period between model one and model two in group two. Points
Models
P5
Model one Model two P6 Model one Model two P7 Model one Model two P8 Model one Model two Mean absolute error
X-direction
Y-direction
Z-direction
Amplitude (m/s2) (error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
85.43 80.03 (6.32%) 65.26 62.74 (3.86%) 54.44 52.38 (3.78%) 44.63 43.27 (3.05%) 4.25%
0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278
72.53 69.82 (3.74%) 55.86 55.43 (0.77%) 55.41 53.99 (2.56%) 42.46 41.91 (1.30%) 2.09%
0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290
59.80 58.70 (1.84%) 57.12 56.75 (0.65%) 23.93 25.06 (4.72%) 21.24 22.28 (4.90%) 3.03%
0.138 0.138 0.138 0.138 0.138 0.138 0.138 0.138
(0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%)
Table 5 Comparison of amplitude and corresponding period between model one and model two in group three. Points
P9
Models
Model one Model two P10 Model one Model two P11 Model one Model two P12 Model one Model two P13 Model one Model two Mean absolute error
X-direction
Y-direction
Z-direction
Amplitude (m/s2) (error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
44.49 42.85 (3.69%) 19.01 18.99 (0.11%) 17.96 17.95 (0.06%) 15.28 15.29 (0.07%) 13.91 13.92 (0.07%) 0.80%
0.154 0.154 0.074 0.074 0.074 0.074 0.074 0.074 0.074 0.074
37.43 36.18 (3.34%) 20.68 21.96 (6.19%) 26.23 28.03 (6.86%) 16.07 17.06 (6.16%) 17.10 18.21 (6.49%) 5.81%
0.133 0.133 0.118 0.118 0.118 0.118 0.118 0.118 0.118 0.118
69.56 69.90 (0.49%) 55.89 56.91 (1.83%) 64.78 65.41 (0.97%) 52.67 53.86 (2.26%) 59.11 62.03 (4.94%) 2.10%
0.174 0.174 0.138 0.138 0.167 0.167 0.133 0.133 0.148 0.148
(0%) (0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%) (0%)
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
direction, the mean absolute error is small between the two models. Reference points in group one and group two are at the shield building and reference points in group three are at the auxiliary building. To the shield building, the simplified method might be not accurate enough in the strong-axis. On the other hand, to the auxiliary building, this method might be not accurate enough in the weak-axis. 4. Conclusion The FSI effect of nuclear island building is investigated. Three 3D finite element models with water tank, shield building and auxiliary building are established based on ADINA. These calculation results can provide reference for structure design and simplified calculation. One simplified method is proposed in this study. The liquid mass is split into the impulsive mass and convective mass based on Housner model. To simplify analysis, the 1st sloshing mass is considered only. The discrete convective masses are connected to tank walls by spring-dampers in the horizontal directions, while the discrete vertical components and the impulsive masses are added to the bottom and walls. The structural stiffness is less in the horizontal directions than in the vertical direction, and 1st sloshing frequency of water is far less than structural frequencies. By comparing the responses of model with designed water and the model with empty tank, water tank do not have obvious seismic reduction effect in the horizontal directions. In the vertical direction, FSI effect may change corresponding period of spectral amplitude in shield building and reduce the spectral amplitude in auxiliary building. In some period bands, FSI effect may increase the seismic response of structure. So FSI effect could not be regarded to decrease the seismic response simply but it should be considered in structure design and analysis. The simplified model has a good agreement with the FSI model in the corresponding period of spectrum and the error of spectral amplitude between two models is acceptable. In further research, multiple sloshing modes could be considered and the soil-structure interaction (SSI) effect for nuclear island building could be calculated with the simplified method. Acknowledgements This research is supported by National Science and Technology Major Project (2013ZX06002001-9) and Natural Science Foundation of China (51408255, 51421005). References ADINA R&D, 2010. Theory and Modeling Guide. Rep. ARD 10-7. ADINA R&D, Watertown, MA. American society of civil engineers. Seismic analysis of safety-related nuclear
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structures and commentary. ASCE 4e98. Bouaanani, N., Lu, F.Y., 2009. Assessment of potential-based fluid finite element for seismic analysis of dam-reservoir systems. Comput. Struct. 87 (3), 206e224. Choi, Y., Lim, S., Ko, B.H., Park, K.S., et al., 2013. Dynamic characteristics identification of reactor internals in SMART considering fluid-structure interaction. Nucl. Eng. Des. 255, 202e211. Haroun, M.A., 1980. Dynamic Analyses of Liquid Storage Tanks. California Institute of Technology. Housner, G.W., 1957. Dynamic pressures on accelerated fluid containers. Bull. Seismol. Soc. Am. 47 (1). Housner, G.W., 1963. The dynamic behavior of water tanks. Bull. Seismol. Soc. Am. 53 (2), 381e387. Lee, D.S., Liu, M.L., Hung, T.C., et al., 2013. Optimal structural analysis with associated passive heat removal for AP1000 shield building. Appl. Therm. Eng. 50 (1), 207e216. Leonardo, T.S., Richard, S.O., et al., 2007. Finite element modeling of AP1000 nuclear island for seismic analyses at genetic soil and rock sites. Nucl. Eng. Des. 237, 1474e1485. Lin, C.G., 2012. Advanced Passive Safety PWR Nuclear Power Technology. Atom Energy Press, Beijing, pp. 904e905. Liu, G.Q., 2014. Comparison of Nuclear Power Plant Seismic Design in Chinese and Foreign Code. Institute of Engineering Mechanics, China Earthquake Administration. Liu, J., Kong, J.W., Kong, X.J., 2016. Shaking table model tests of concrete containment vessel (CCV) for CPR1000 nuclear power plant. Prog. Nucl. Energy 93, 186e204. Lo Frano, R., 2015. Evaluation of the fluid-structure interaction effects in a leadcooled fast reactor. Nucl. Technol. 33, 491e499. Lo Frano, R., Forasassi, G., 2009. Conceptual evaluation of fluid-structure interaction effects coupled to a seismic event in an innovative liquid metal nuclear reactor. Nucl. Eng. Des. 239 (11), 2333e2342. Lo Frano, R., Forasassi, G., 2011. Preliminary evaluation of seismic isolation effects in a generation IV reactor. Energy 36 (4), 2278e2284. Lo Frano, R., Forasassi, G., 2012. Preliminary evaluation of structural response of ELSY reactor in the after shutdown condition. Nucl. Eng. Des. 246 (4), 298e305. Manos, G.C., Clough, R.W., 1983. The Measured and Predicted Shaking Table Response of a Broad Tank Model. Meserole, J.S., Fortini, A., 1987. Sloshing dynamics in a toroidal tank. J. Spacecr. Rock. 24 (6), 523e531. Olson, L.G., Bathe, K.J., 1985. Analysis of fluidestructure interactions: a direct symmetric coupled formulation based on the fluid velocity potential. Comput. Struct. 21 (1e2), 21e32. Schulz, T.L., 2006. Westinghouse AP1000 advanced passive plant. Nucl. Eng. Des. 236 (14e16), 1547e1557. US Atomic Energy Commission, 1963. Nuclear Reactors and Earthquakes. TID-7024. Office of Technical Service, Washington DC. Wei, K., Yuan, W.C., Bouaanani, N., 2013. Experimental and numerical assessment of the three-dimensional modal dynamic response of bridge pile foundations submerged in water. J. Bridge Eng. 18 (10), 1032e1041. Westinghouse Electric Co. LLC, 2011. AP1000 Design Control Document, Revision 19. Westinghouse Electric Co. LLC, Butler County, Pennsylvania. Xia, Z.F., 2013. Seismic input of NPP & topic of seismic-isolated research for AP1000 nuclear island. Eng. Sci. 15 (4), 52e56. Xu, Q., Chen, J.Y., Zhang, C.B., Li, J., Zhao, C.F., 2016. Dynamic analysis of AP1000 shield building considering fluid and structure interaction effects. Nucl. Eng. Technol. 48 (1), 246e258. Zhao, C.F., Chen, J.Y., Xu, Q., 2014a. Dynamic analysis of AP1000 shield building for various elevations and shapes of air intakes considering FSI effect subjected to seismic loading. Prog. Nucl. Energy 74 (3), 44e52. Zhao, C.F., Chen, J.Y., Xu, Q., 2014b. FSI effects and seismic performance evaluation of water storage tank of AP1000 subjected to earthquake loading. Nucl. Eng. Des. 280, 372e388. Zhao, C.F., Chen, J.Y., Xu, Q., Yang, X.Y., 2015. Sensitive analysis of water levels and air intakes on natural frequency of AP1000 nuclear island building considering FSI effects. Ann. Nucl. Energy 78, 1e9.
Contents lists available at ScienceDirect
Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene
Research on FSI effect and simplified method of PCS water tank of nuclear island building under earthquake Chenning Song a, Xiaojun Li a, b, *, Guoliang Zhou c, Chao Wei c a
The College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China Institute of Geophysics, China Earthquake Administration, Beijing 100081, China c Nuclear and Radiation Safety Center, Ministry of Environmental Protection of the People's Republic of China, Beijing, 100082, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 November 2016 Received in revised form 17 April 2017 Accepted 25 May 2017
Water tank is one of the most important components of passive containment cooling system (PCS) in advanced nuclear power plant. Moreover, the dynamic characteristics of shield building and auxiliary building might be changed because of fluid-structure interaction (FSI) of the PCS. Therefore, FSI between cooling water and the shield building should be considered in the dynamic analysis of the nuclear island building. To study the FSI effect and propose an effective and simplified method, three numerical models were established and analyzed under three-dimensional seismic loadings. Acceleration time histories of reference points in the shield building and auxiliary building were got and floor response spectra were calculated. The results show that FSI effect could not be regarded to decrease the seismic response simply but it should be considered in structure design and analysis. A simplified model is proposed in this study and proved to have a good agreement with the FSI model in the curves of floor response spectra. The conclusions could provide reference for further study in the field of nuclear island building. © 2017 Published by Elsevier Ltd.
Keywords: Water tank Nuclear island building Fluid-structure interaction (FSI) Floor response spectrum
1. Introduction The nuclear island building studied in this paper is one of the third-generation nuclear power plants (Schulz, 2006). One of the most obvious characteristics is the passive containment cooling system (PCS). Shield building, steel containment vessel and auxiliary building are the three important components of nuclear island building. Shield building is used to protect the steel containment vessel and internal structure. Auxiliary building is used to protect and shield the mechanical equipment and electrical equipment outside of the safety containment. The PCS and the numerical model of nuclear island building are shown in Fig. 1. Because of FSI between cooling water and shield building, the dynamic characteristics of nuclear island building would be changed. In recent years, the research around the FSI of nuclear power plant has been done to evaluate the security of structure. Lo Frano and Forasassi (2009, 2011, 2012, 2015) did some research on liquid metal nuclear reactor considering the FSI effect subjected to seismic loads. Choi et al. (2013) designed a 1/12 scale model of the
* Corresponding author. The College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.pnucene.2017.05.025 0149-1970/© 2017 Published by Elsevier Ltd.
System-Integrated Modular Advanced Reactor (SMART) and established the three-dimensional finite element model based on ANSYS. The numerical results considering FSI effect are well matched with the test. Leonardo et al. (2007) established finite element model of AP1000 nuclear island and analyzed the seismic response of the structure at generic and rock sites. Liu et al. (2016) designed a 1/20 scale concrete containment vessel (CCV) and did the shaking table tests of CCV for CPR1000 nuclear power plant. It resulted that the prototype PCCV for CPR nuclear power plant has sufficient seismic safety margin. Lee et al. (2013) did some thermodynamic and seismic analysis for AP1000 structure considering water levels and the location of air intake. Moreover, some optimizing opinions were proposed. In the field of FSI effect of AP1000 nuclear power plant, some advanced numerical methods were used. Liu (2014) used the Coupled Eulerian Lagrange (CEL) technique to analyze the FSI of the PCS water tank and the seismic response of the shield building based on ABAQUS. Zhao et al. (2014a,b) used the Arbitrary Lagrange Eulerian (ALE) algorithm to simulate the FSI, fluid sloshing and oscillation of water tank under earthquake. The influence of water tank and the air intake were systematically studied. The research shows that water levels and the location of air intake could change the dynamic characteristics of shield building. In addition, some
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
Acronyms PCS FSI SSI SMART CCV PCCV CEL ALE SPH PBFE CWT CCWT ATH PGA
Passive Containment Cooling System Fluid-Structure Interaction Soil-Structure Interaction System-Integrated Modular Advanced Reactor Concrete Containment Vessel Prestressed Concrete Containment Vessel Coupled Eulerian Lagrange Arbitrary Lagrange Eulerian Smoothed Particle Hydrodynamics Potential-Based Fluid Elements Cylindrical Water Tank Circular Cylindrical Water Tank Acceleration Time History Peak Ground Acceleration
optimized design advices were proposed. On the basis, Xu et al. (2016) used the smoothed particle hydrodynamics (SPH) and finite element method (FEM) coupling method to simulate the FSI between cooling water and shield building of AP1000. The research shows that the water tank can decrease the natural frequency and response of the shield building. But the auxiliary building was not considered in these studies. The auxiliary building is another important component of nuclear island building and there is also some important equipment in it, so it is necessary to do some study on auxiliary building. The existing research mainly paid attention to the shield building, but there was little study on auxiliary building and the FSI effect of the whole nuclear island building has not been studied systematically. Zhao et al. (2015) established the numerical model and did some research on the natural frequency of nuclear island building considering FSI effect. The frequency characteristics of structure considering various water levels and the elevation of air intake were got from analysis. However, this study was not involving the seismic response. In the field of simplified method, researchers have proposed
49
some models to analyze the dynamic response of liquid containers in the past decades and these models are almost based on Housner model. Housner (1957, 1963) developed an approximate method for regular containers on the basis of some assumption and the core idea was that the water could be split into impulsive mass and convective mass. This method was based on results from experimental as well as analytical method. Haroun (1980) developed the model by considering the flexibility of the storage. Aimed at the influence of high modes of sloshing, Manos and Clough (1983) did shaking table tests to study seismic response of the storage model. The previous researches show that the water tank has an influence on the seismic response of shield building. But the auxiliary building was not considered in these studies. In addition, the simplified model for sloshing liquid has not been used to analyze the seismic response of nuclear island building. In this study, one simplified method is proposed based on Housner model. Then, numerical models of shield building and auxiliary building are established to analyze the seismic response of structure. The FSI effect of nuclear island building is investigated. By comparing the FSI model and simplified model, the simplified method is proved reasonably and could be used for further study of structure design and analysis of nuclear power plant.
2. Analysis methods The formulation of potential-based fluid elements (PBFE) and simplified model of water are reviewed in this section.
2.1. Potential e based fluid elements The equations of the f u formulation in this study are briefly introduced in this part. As the special fluid element in ADINA (ADINA R&D 2010), the PBFE can be used for frequency and timedomain analyses of structure (Wei et al., 2013; Bouaanani and Lu, 2009). There are two basic parameters in the formulation: velocity potential f and displacement u. The velocity potential f satisfies the wave equation:
Fig. 1. The passive containment cooling system and profile of numerical model.
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C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
2
1 v2 f V f¼ 2 Cw vt 2 2
(1)
2
where V is the Laplace differential operator, t is the time variable, Cw is the wave velocity in fluid. And Cw is given by
Cw ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi kw =rw
(2)
where rw is water density, kw is the bulk modulus. Based on the standard theories, the variational form of Eq. (1) can be got as
rw 2 Cw
Z Vw
v2 f dfdV þ rw vt 2
Z Sw
vu ndfdS þ rw vt
Z Vf$VdfdV ¼ 0
(3)
Vw
where Vw is the volume of water, Sw is water boundary where normal velocity is prescribed, n is unit normal on Sw pointing into the fluid and V is the vector gradient. Under earthquake, the dynamic response of the water tank is coupled through compatibility of velocity potential and prescribed normal velocity at the fluidstructure interface. The system dynamic equations of the tank filled with water can be obtained (Olson and Bathe, 1985):
€ 0 M ss C ss U € þ C ws 0 M ww F € g ðtÞ M ss 1u ¼ C ws 1u_ g ðtÞ
C sw 0
U_ þ K ss _ 0 F
0 K ww
U
F
(4) In addition
M ss ¼ rs
Z
N Ts N s dV; M ww ¼
Vs
K ss ¼ rs
Z
rw 2 Cw
Z Vw
N Ts DTs N s dV; K ww ¼ rw
Vs
C sw ¼ C Tws ¼ rw
N Tw N w dV
Z
BTw Bw dV
Vw
Z
N Ts nN w dS;
Vw
C ss ¼ aM ss þ bK ss ;
(5)
where N s and N w are standard isoparametric shape function matrices for shell and fluid elements respectively, rs and Vs are the density and volume of the tank, Ds is the elastic matrix of shell elements, M ss and K ss are the mass and stiffness matrices for the tank, M ww and K ww are the mass and stiffness matrices for water, 1 is the column vector which has the same dimension of nodal relative displacement U, C ws and C sw are the damping matrix accounting for FSI between water and tank, and C ss is the Rayleigh damping matrix, in which a and b are the Rayleigh damping coefficients. Using the first and second natural frequencies of nuclear island building, the damping coefficients a and b can be calculated by Eq. (6). The matrix Bw is given by Eq. (7)
C cr
2zu1 u2 2z ¼ 2M ss un ; a ¼ ; b¼ u1 þ u2 u1 þ u2
ð2Þ
vNw vx
,,,
ð2Þ
vNw vy
,,,
3 ðiÞ vNw 7 vx 7 7 ðiÞ 7 vNw 5
(7)
vy
where C cr is the critical damping matrix, un is the natural frequencies, u1 and u2 are the first and second natural frequencies. z is the damping ratio of structure, i is the number of nodes per fluid element.
2.2. Simplified model Considering the structure characteristics of water tank, the upper structure of shield building could be regarded as one rigid tankliquid system. So, Housner model might be adaptable. The basis assumptions are as follows: the structure and liquid motions remain linearly elastic, the tank’ sections are regular (rectangular or circular), the tank walls are rigid and the liquid is inviscid. In this model, the liquid mass is split accordingly into the impulsive mass and the convective mass. Impulsive component represents rigidbody motion of liquid. Under dynamic loading, a part of liquid moves synchronously with the tank as an added mass and is subjected to the same acceleration levels as the tank. Convective component represents sloshing of the liquid at the free surface. Under lateral excitation, oscillations of liquid occur and result in the generation of pressure on the wall, base and roof of the tank. As shown in Fig. 2, the impulsive mass m0 is assumed to be rigidly attached to the container walls while the convective mass is split into series of masses m1, m2, …,mn associated with the 1st, 2nd, …, nth sloshing masses respectively. These masses are attached to the container walls via springs of stiffness k1, k2, …, kn representing the 1st, 2nd, …, nth antisymmetric sloshing frequencies respectively. In this study, one simplified model considering the first sloshing mode is proposed to calculate the seismic response of the advanced nuclear island building. Based on Housner model, m1 can be considered to connect to the tank walls by some spring-dampers (sp1, …, spi, …spq) as shown in Fig. 3. m* ; k and c denote the discrete mass of m1 , the stiffness and damping coefficients of one spring-damper, dq is the angle between adjacent spring-dampers and qi is the angle between X-direction and spi. The following equations can be obtained:
dq ¼
2p m ; qi ¼ ði 1Þdq; m* ¼ 1 q q
(8)
where q is the number of spring-dampers. In Fig. 3 (b), d is the virtual displacement of m* assumed to occur along X-direction and some parameters can be calculated by the following equations:
di ¼ d cos qi fix ¼ 2kdi cos qi ¼ 2kdcos2 qi
(9)
where di is the deformation of spi along the axis direction, fix is the internal force of spi along X-direction. The bearing composite force of m1 is Fx which can be calculated below:
Fx ¼ (6)
ð1Þ
vN 6 w 6 vx Bw ¼ 6 6 ð1Þ 4 vNw vy
q X i¼1
fix ¼ 2kd
q X
cos2 qi
(10)
i¼1
The both sides of Eq. (10) are multiplied by dq and then Fx can be obtained:
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
51
Fig. 2. Housner spring-mass model for rigid tanks.
Fig. 3. Lumped mass model of sloshing liquid. (a) The whole simplified model (b) One discrete spring-damper.
Fx dq ¼ 2kd
q X
Kx ¼ mn u2n Cx ¼ 2gmn un
cos2 qi dq
i¼1
Fx
2p ¼ 2kd q
Z2p cos2 qdq ¼ 2pkd
(11)
0
where g is sloshing damping ratio. The simplified method only considers the first sloshing mode in this study. So when n ¼ 1, k and c of one spring-damper can be calculated according to Eqs. (12)e(14) above:
Fx ¼ qkd So the total equivalent stiffness coefficient Kx can be obtained:
Kx ¼
Fx ¼ qk d
(12)
Simultaneously, the total equivalent damping coefficient Cx can be obtained:
Cx ¼ qc
(13)
According to the free vibration equation of structure with damping, Kx ; Cx can also be expressed by the following equation:
(14)
k¼
Kx m1 u21 ¼ q q
c¼
Cx 2 ¼ gm1 u1 q q
(15)
TID-7024 (1963) gives some equations of impulsive and convective mass for cylindrical tank based on Housner model. To get the key parameters of simplified model, the water tank is equivalent to regular cylindrical water tank (CWT) firstly. As shown in Fig. 4 (a) and (b), Ri ; Ro and h denote the inner radius, outer radius and the height of free fluid surface. The equivalent radius R of regular cylindrical tank can be calculated by keeping the volume of water constant. Then, the parameters can be obtained by Eqs. (16)
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C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
Fig. 4. Models of water tank. (a) PCS water tank; (b) Equivalent CWT; (c) Equivalent CCWT.
and (17) (Housner, 1957).
m1 R h ¼ 0:318 tanh 1:84 h R m cosh 1:84 Rh 1 h1 ¼1 h 1:84 Rh sinh 1:84 Rh
(16)
(17)
where m is the total mass, m1 and h1 are the 1st sloshing mass and height of 1st sloshing mass. To get the sloshing frequency more accurately, one method is proposed for circular cylindrical water tank (CCWT) (Meserole and Fortini, 1987). As shown in Fig. 4 (c), the water tank is equivalent to regular CCWT and equivalent height of free surface ha can be calculated by keeping Ro ; Ri and V constant. The 1st sloshing frequency u1 can be got by the following equation:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 ¼ ðg=Ro Þx1 tanh k1 k1 ¼ ðha =Ro Þx1
(18)
where g is the acceleration due to gravity. And x1 is the root of Eq. (19) 0
0
0
0
J1 ðx1 ÞY1 ðkx1 Þ J1 ðkx1 ÞY1 ðkx1 Þ ¼ 0 k ¼ Ri =Ro
(19)
where J1 and Y1 denote the Bessel function of the first and second 0 0 kinds and first order. J1 and Y1 denote the first derivative of J1 and Y1 . 3. Numerical analysis and discussion 3.1. Description of numerical models In the previous study, Zhao et al. (2014a,b) proposed that the variation of stiffness as location of air intake was very small compared to the integral rigidity of overall nuclear island building. So the air intake and the steel containment vessel are not considered in this study. To analyze the FSI effect for nuclear island building and test the reasonableness of simplified model, three models are established: (a) Model one: Model considering FSI effect; (b) Model two: Simplified model; (c) Model three: Model with empty water tank. The software ADINA is used in the numerical analysis. The baseboard of nuclear island building is modeled using 3D-solid
finite elements while the main body of shield building and auxiliary building is modeled using shell finite element. The steel frames are simulated using beam elements and the water in the tank is simulated using 3D PBFE. The spring-dampers are simulated using spring elements and concentrate masses in model two. The main structure is shown in Fig. 5 and the differences between three models are the water tanks. As shown in Fig. 6, the water tank in model one is filled with the designed cooling water. In model two, the water is replaced by spring-dampers according to the theoretical derivation in the second section. The water tank is empty in model three. The numerical models are shown in Fig. 5 and the material parameters are listed in Table 1. The geometry sizes of previous papers (Zhao et al., 2014a,b, 2015) are referenced in this study. 3.2. Dynamic characteristics of numerical models 3.2.1. Parameters of simplified model To avoid the local stress of tank walls, the discrete models are used in the numerical models. As shown in Fig. 6 (b), the first sloshing mass is divided to one hundred and twenty-eight lumped masses at two layers. The lumped masses in each layer are coupled at one rigid point respectively. Based on ASCE 4-98, the convective mass m1 is divided into two parts: horizontal components and vertical components. The horizontal components are connected to tank walls by spring-dampers in which the stiffness and damping coefficient are calculated by Eqs. (15)e(18) and the vertical components are added to the bottom of water tank discretely. The impulsive mass m0 is added to the bottom and walls of the tank discretely. According to the geometry sizes of nuclear island building, the key parameters of one spring-damper can be calculated. 3.2.2. Natural frequencies and 1st sloshing frequency Natural frequencies are important dynamic characteristics for structures. In this part, the natural frequencies of nuclear island building with different water tanks are studied. On one hand, the dynamic characteristics of models can be obtained, on the other hand, the reasonableness of simplified model can be verified preliminary. The 1st natural frequency and 1st sloshing frequency of models are list in Table 2. The errors of 1st natural frequencies and 1st sloshing frequencies between model one and model two are 1.402% and 0.794% separately. Moreover, the first ten order natural frequencies are compared in Fig. 7. The first two modes of structures are integral vibrations in the two horizontal directions and the 3rd to 10th modes are local vibrations of floors and walls in auxiliary building. Therefore, the FSI only has obvious influence on the first two modes and it is similar to results from the previous paper (Zhao et al., 2015). The “mode-frequencies” curve of model two is in good agreement with model one. It indicates that the proposed simplified model considering 1st sloshing mass is
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
53
Fig. 5. Elevation drawing and marking of numerical model. (a) S-N section (b) E-W section. Ls-n: Width of south-north. Le-w: Width of east-west. Ds: Diameter of shield building. Di: Inner diameter of water tank. Do: Outer diameter of water tank. Ha1: Southern height of auxiliary building. Ha2: Northern height of auxiliary building. Hs: Height of shield building. Ht: Height of water tank. Hw: Height of water surface.
Fig. 6. Three different water tanks in numerical models. (a) Water tank in model one; (b) Water tank in model two; (c) Water tank in model three.
3.3. Seismic response of numerical models
Table 1 Material parameters of nuclear island building. Material parameters 3
Density (kg/m ) Young's modulus (Pa) Bulk modulus (Pa) Poisson’ ratio
Concrete
Steel
Water
2400 3.00 1010 e 0.17
7800 2.00 1011 e 0.30
1000 e 2.30 109 e
Table 2 1st natural frequency and 1st sloshing frequency. Parameters
1st natural frequency (Hz) (error)
1st sloshing frequency (Hz) (error)
Model one Model two Model three
3.183 3.139(-1.382%) 3.402
0.126 0.127(0.794%) e
suitable to simulate the seismic response of structure approximately in this study.
This study analyzes the FSI effect of nuclear island building subjected to seismic loads. The special acceleration time histories (ATH) used as inputs at the foundation is fitted by code response spectrum. The peak ground accelerations (PGA) of safe shutdown earthquake (SSE) in three directions are 0.3 g and the inputs ratio of three-direction is 1: 1: 1 (Lin, 2012; Xia, 2013; Westinghouse Electric Co. LLC, 2011). Fig. 8 describes the ATHs and the corresponding spectral acceleration with the damping ratio of 5%. X, Y and Z represent the S-N, E-W and vertical direction separately. The dynamic responses of three numerical models are calculated in this part. Three groups of reference points are selected to investigate the FSI effect and the reasonableness of simplified model. The first two groups are located at shield building, while the third group is located at auxiliary building.
3.3.1. Response of shield building Four reference points in the first group are chosen at the north
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C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
Fig. 7. First ten order natural frequencies of three models.
(model one) is less than that of the model with empty water tank (model three) in period range from 0.1s to 0.3s basically. This period range corresponds to the main frequency band of structure from 3.0 Hz to 10 Hz. The amplitudes of spectra do not have much difference and the corresponding periods of amplitudes do not change almost between the two models. In the period range from 0.3s to 0.6s, the seismic response of model one is slightly larger than that of model three. The FSI effect may reduce the horizontal seismic response in some degree. In the vertical direction, the seismic response of model one at the water tank (P1 and P2) is less than that of model three when the period is smaller than 0.1s. When the period exceeds 0.1s, the seismic response of model one is more than that of model three. Moreover, the amplitudes of spectra move to long period band when considering FSI effect. The influence of FSI effect at shield building (P3 and P4) is complex. The amplitudes of spectra also do not have much difference but the corresponding periods of amplitudes move to long period band in model one. The FSI effect may change the dynamic characteristic and amplify the seismic response in the vertical direction.
Fig. 8. Input ground motions in this study. (a) ATH in X-direction (b) ATH in Y-direction (c) ATH in Z-direction (d) Response spectra with the damping ratio of 5%.
of shield building. The heights of P1, P2, P3 and P4 are 91.25 m, 79.27 m, 72.54 m and 63.54 m separately. P1 and P2 are at the water tank, while P3 and P4 are at shield building. Fig. 9 shows the floor response spectra at the chosen points in the three directions. In the horizontal directions, the seismic response of FSI model
Another four points in the second group are chosen at the east of shield building as supplementary. The heights of P5, P6, P7 and P8 are 91.25 m, 79.27 m, 72.54 m and 63.54 m which are the same as the first four points. The floor response spectra at the chosen points in the three directions are shown in Fig. 10. It can be noted that the
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
Fig. 9. Floor response spectra at reference points in group one with damping ratio of 5%.
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C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
Fig. 10. Floor response spectra at reference points in group two with damping ratio of 5%.
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
Fig. 11. Floor response spectra at reference points in group three with damping ratio of 5%.
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C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
results are similar to group one. While the amplitudes of spectra are larger than those in group one because of the different characteristics in the two horizontal directions.
smaller when FSI effect is considered, however the corresponding periods of amplitude have no change. So the water tank may reduce the seismic response of auxiliary building in the vertical direction.
3.3.2. Response of auxiliary building Five reference points are chosen for analysis at the auxiliary building. P9 is the middle point at the top of auxiliary building and the height is 51.20 m. P10, P11, P12 and P13 are the middle points on the floors with heights of 31.65 m, 28.55 m, 25.23 m and 21.90 m separately. Fig. 11 shows the floor response spectra at the chosen points in the three directions. In the horizontal directions, the X-direction is the weak-axis and the Y-direction is the strong-axis of P9, while it is opposite to P10, P11, P12 and P13. Fig. 11(a), (e), (h), (k) and (n) show that FSI effect may reduce the seismic response at the weak-axis of floor when the period is close to the 1st natural period but it may amplify the seismic response when the period is close to 0.15s. At the strongaxis of floor, the influence is not obvious as shown in Fig. 11(b), (d), (g), (j) and (m). In the vertical direction, the amplitudes of spectra become
3.3.3. Reasonableness of simplified model The floor response spectra of three models are compared in Fig 9e11. It can be seen that although there are some difference of amplitudes between the FSI model (model one) and the simplified model (model two) in some periods, the response spectral curves of two models are in good agreement. To analyze the error between two models and identify the reasonableness of simplified model more accurately, the amplitudes of spectra and corresponding periods are listed in Tables 3e5. The corresponding periods of amplitudes are all the same in the three directions. The maximum absolute errors of amplitude are 6.32%, 6.86% and 4.94% in three directions. In Table 3 and Table 4, the maximal mean absolute error of amplitude occurs in X-direction which is the strong-axis of nuclear island building. In Table 5, the maximal mean absolute error of amplitude occurs in Y-direction which is the weak-axis. In Z-
Table 3 Comparison of amplitude and corresponding period between model one and model two in group one. Points
Models
P1
Model one Model two P2 Model one Model two P3 Model one Model two P4 Model one Model two Mean absolute error
X-direction
Y-direction
Z-direction
Amplitude (m/s2) (error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
86.31 81.16 (5.97%) 66.68 64.68 (3.00%) 64.53 61.56 (4.60%) 45.01 43.43 (3.51%) 4.27%
0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278
73.46 70.58 (3.92%) 58.12 57.26 (1.48%) 50.42 49.70 (1.43%) 41.93 41.43 (1.19%) 2.01%
0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290
50.76 50.74 (0.04%) 47.08 48.85 (3.76%) 22.92 22.21 (3.10%) 21.73 21.14 (2.72%) 2.41%
0.111 0.111 0.111 0.111 0.278 0.278 0.278 0.278
(0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%)
Table 4 Comparison of amplitude and corresponding period between model one and model two in group two. Points
Models
P5
Model one Model two P6 Model one Model two P7 Model one Model two P8 Model one Model two Mean absolute error
X-direction
Y-direction
Z-direction
Amplitude (m/s2) (error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
85.43 80.03 (6.32%) 65.26 62.74 (3.86%) 54.44 52.38 (3.78%) 44.63 43.27 (3.05%) 4.25%
0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278
72.53 69.82 (3.74%) 55.86 55.43 (0.77%) 55.41 53.99 (2.56%) 42.46 41.91 (1.30%) 2.09%
0.290 0.290 0.290 0.290 0.290 0.290 0.290 0.290
59.80 58.70 (1.84%) 57.12 56.75 (0.65%) 23.93 25.06 (4.72%) 21.24 22.28 (4.90%) 3.03%
0.138 0.138 0.138 0.138 0.138 0.138 0.138 0.138
(0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%)
Table 5 Comparison of amplitude and corresponding period between model one and model two in group three. Points
P9
Models
Model one Model two P10 Model one Model two P11 Model one Model two P12 Model one Model two P13 Model one Model two Mean absolute error
X-direction
Y-direction
Z-direction
Amplitude (m/s2) (error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
Amplitude (m/s2)(error)
Period(s) (error)
44.49 42.85 (3.69%) 19.01 18.99 (0.11%) 17.96 17.95 (0.06%) 15.28 15.29 (0.07%) 13.91 13.92 (0.07%) 0.80%
0.154 0.154 0.074 0.074 0.074 0.074 0.074 0.074 0.074 0.074
37.43 36.18 (3.34%) 20.68 21.96 (6.19%) 26.23 28.03 (6.86%) 16.07 17.06 (6.16%) 17.10 18.21 (6.49%) 5.81%
0.133 0.133 0.118 0.118 0.118 0.118 0.118 0.118 0.118 0.118
69.56 69.90 (0.49%) 55.89 56.91 (1.83%) 64.78 65.41 (0.97%) 52.67 53.86 (2.26%) 59.11 62.03 (4.94%) 2.10%
0.174 0.174 0.138 0.138 0.167 0.167 0.133 0.133 0.148 0.148
(0%) (0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%) (0%)
(0%) (0%) (0%) (0%) (0%)
C. Song et al. / Progress in Nuclear Energy 100 (2017) 48e59
direction, the mean absolute error is small between the two models. Reference points in group one and group two are at the shield building and reference points in group three are at the auxiliary building. To the shield building, the simplified method might be not accurate enough in the strong-axis. On the other hand, to the auxiliary building, this method might be not accurate enough in the weak-axis. 4. Conclusion The FSI effect of nuclear island building is investigated. Three 3D finite element models with water tank, shield building and auxiliary building are established based on ADINA. These calculation results can provide reference for structure design and simplified calculation. One simplified method is proposed in this study. The liquid mass is split into the impulsive mass and convective mass based on Housner model. To simplify analysis, the 1st sloshing mass is considered only. The discrete convective masses are connected to tank walls by spring-dampers in the horizontal directions, while the discrete vertical components and the impulsive masses are added to the bottom and walls. The structural stiffness is less in the horizontal directions than in the vertical direction, and 1st sloshing frequency of water is far less than structural frequencies. By comparing the responses of model with designed water and the model with empty tank, water tank do not have obvious seismic reduction effect in the horizontal directions. In the vertical direction, FSI effect may change corresponding period of spectral amplitude in shield building and reduce the spectral amplitude in auxiliary building. In some period bands, FSI effect may increase the seismic response of structure. So FSI effect could not be regarded to decrease the seismic response simply but it should be considered in structure design and analysis. The simplified model has a good agreement with the FSI model in the corresponding period of spectrum and the error of spectral amplitude between two models is acceptable. In further research, multiple sloshing modes could be considered and the soil-structure interaction (SSI) effect for nuclear island building could be calculated with the simplified method. Acknowledgements This research is supported by National Science and Technology Major Project (2013ZX06002001-9) and Natural Science Foundation of China (51408255, 51421005). References ADINA R&D, 2010. Theory and Modeling Guide. Rep. ARD 10-7. ADINA R&D, Watertown, MA. American society of civil engineers. Seismic analysis of safety-related nuclear
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