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Seismic fragility analysis of AP1000 SB considering fluid-structure interaction effects
Structures 23 (2020) 103–110
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Seismic fragility analysis of AP1000 SB considering fluid-structure interaction effects ⁎
T
⁎
Chunfeng Zhaoa,c,d, , Na Yub, , Y.L. Mod a
Department of Civil Engineering, Hefei University of Technology, Hefei 230009, China College of Economics and Management, Hefei University, Hefei 230601, China c State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China d Department of Civil and Environmental Engineering, University of Houston, Houston 77021, USA b
A R T I C LE I N FO
A B S T R A C T
Keywords: Vulnerability assessment Fluid-structure interaction Quadratic regression Maximum likelihood estimation Shield building
AP1000 nuclear power plant has an advanced passive cooling system with a vast water tank located above the shield building, which may affect the seismic performance of the structure. This paper conducts a fragility assessment of shield building and considers the fluid-structure interaction (FSI) effect on the damage of the shield building. Arbitrary Lagrangian Eulerian algorithm is applied to calculate the FSI and eight cases of water levels are considered. Three different fragility analysis methods, namely linear regression, quadratic regression and truncated maximum likelihood estimation, are adopted for assessing the seismic vulnerability of shield building. It is indicated from the results that the regression and the truncated maximum likelihood estimation are close to each other. The maximum likelihood estimation not only can effectively save the computational cost, but also can assure the reasonable calculated results. It is observed that different water levels have the individual probabilities of failure, and case 6 has the smallest probability of failure with a more considerable seismic reduction, whereas the case 1 and case 2 have the higher failure probabilities. The fragility analysis framework can thus be applied for evaluating the vulnerability of shield building under earthquakes considering the FSI effect.
1. Introduction AP1000 nuclear power plant (NPP) has a series of advanced passive safety features that had been certified by the US. NRC, which relied on the nature circulation and spray watering system to prevent the steel containment vessel from overheating and ensure the safe shutdown of NPP [1,2]. The water tank of AP1000 is used to cool down the temperature of the containment by spraying water after the reactor shut down automatically. The water tank may affect the dynamic response and security of the shield building due to the heavy mass and huge volume when the structure encounters earthquakes. Simultaneously, the water slosh and water levels (WLs) may also influence the dynamic characteristic and the security of the NPP. Therefore, the study on the dynamic response and vulnerability assessment of AP1000 has attracted more and more attention [3]. The seismic probabilistic risk assessment (SPRA) is a powerful approach to assess the security and the regulatory requirement of NPPs in construction and operation. Seismic fragility analysis (SFA) is the most important and fundamental stage in the process of SPRA especially for
⁎
the seismic performance of structures [4]. The Nuclear Regulatory Commission (NRC) issued a guide that requested all the NPP must have a vulnerability safety review since 1992 [5,6].Up to now, the vulnerability assessment of NPPs consisting of components, systems or structures under earthquakes has been paid more attention and studied in some researches [7–10]. Firoozabad et al. carried out a seismic fragility assessment for a piping system and through the cyclic loading experiment to validate the numerical analysis [11]. The results provided the damage probability of piping system in the operation life of NPP. The seismic fragility analysis of a Korean NPP was conducted by an improved approach and compared with other methods by Cho and Joe [12]. The results indicated that the proposed method might overestimate the seismic resistance of Korean NPP facilities. Tran et al. applied the response surface method to analyze the vulnerability of the electric cabinet in NPP. The results showed that the boundary conditions played an essential role in the acceleration response of the cabinet in the NPP [13]. Han et al. conducted the fragility assessment of an old non-ductile RC building with or without retrofit using Lead Rubber Bearings (LRBs). The results
Corresponding authors. E-mail addresses: [email protected] (C. Zhao), [email protected] (N. Yu).
https://doi.org/10.1016/j.istruc.2019.11.003 Received 20 August 2019; Received in revised form 10 October 2019; Accepted 7 November 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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showed that the base isolation can effectively reduce the seismic risk of NPP under earthquakes [14]. Kim et al. used stress–strain criterion to assess the fragility of the base-isolated NPP piping system. The results showed that the failure criterion could be suitable for application in seismic fragility analysis of the pipes in NPP [4]. Saouma et al. conducted a fragility analysis to assess the seismic capacity of an ASR-affected nuclear containment vessel [15]. Pang et al. used the extended incremental dynamic analysis to assess the seismic behavior of the face rockfill dams. The results demonstrated that the method could evaluate the destruction and loss of the dams [16]. Zhao et al. performed a series of study on the seismic performance and seismic resistance of NPP under earthquakes. The influences of water level and ring baffles on the structural behavior and security of AP1000 shield building were also deeply investigated [17–21]. The result provided the pre-analysis and indicated that the AP1000 NPP should be carried out the fragility analysis. As mentioned before, most of the papers mainly focused on the seismic fragility of the generation II NPP or other structures. Few of researches paid attention to the seismic fragility of the advanced generation III+ AP1000 shield building with a large water tank. The water slosh may affect the seismic performance and vulnerability of the shield building, especially considering the FSI effects. The purpose of this study was to perform a fragility assessment of AP1000 shield building considering the FSI. Eight cases of water levels (WLs) were designed and simulated by the Arbitrary Lagrangian Eulerian algorithm. The incremental dynamic analysis of linear and quadratic regression, and truncated maximum likelihood estimation were also used to perform the fragility analysis.
Table 1 Geometric dimensions of SB and PCSWT. Components
Parameters
Symbol
Unit
Value
SB
Height Diameter
Hs Ds
m m
71 44.2
PCSWT
Height Outer diameter Inner diameter
Hw Dout Din
m m m
11.8 27.13 10.67
Containment vessel (CV)
Diameter Thickness
Dcv Wc
m m
39.6 0.045
Table 2 Material of shield building. Material
Parameters
Symbol
Unit
Value
Concrete
Density Elastic modulus Poisson ratio
ρ E γ
kg/m3 MPa –
2300 3.35 × 104 0.2
Rebar
Density Elastic modulus Poisson ratio
ρ E γ
kg/m3 MPa –
7800 2.06 × 105 0.3
Water
Density
ρ
kg/m3
1000
Air
Density
ρ
kg/m3
1
2.2. Finite element model In this study, eight cases of water levels (WLs) for PCSWT were proposed to investigate the FSI effects on the fragility of SB [2,20]. The heights of water levels were assumed as 2.8 m, 3.8 m, 4.8 m, 5.8 m, 6.8 m, 7.8 m, 8.8 m and 9.8 m, and represented by case 1, case 2, case 3, case 4, case 5, case 6, case 7, and case8, as shown in Fig. 2 and Table 3. It means that case 1 has water height of 2.8 m, case 2 has the water height of 3.8, and so on. A smear model of concrete and reinforcement bar was used to establish the three-dimensional finite element model (FEM) of the SB with PCSWT. A Solid element SOLID65 was applied to mesh all the components of SB, the water in PCSWT was modeled by the pressure-based fluid element FLUID30, which was suitable for modeling the FSI analysis. SOLID65 is used for the 3D modeling of solids with or without rebar and capable of cracking in tension and crushing in compression. The most important characteristic of the element is the treatment of
2. Numerical model 2.1. AP1000 PCSWT Shield building (SB) and air intake (AI) are the important parts of the passive cooling system (PCS) of AP1000 NPP. Passive cooling system water tank (PCSWT) located at the top of the SB is used to cool down the temperature of the steel containment through spray water by gravity in case of the reactor shut down [2,22]. The water is partially filled in PCSWT with air vacant height of about 1 m from the top roof of PCSWT to water surface in its intial design. The diameter, height and thickness of the SB are 44.2 m, 82.8 m and 0.92 m. The details of the geometry and material of SB and PCSWT are listed in Fig. 1, Table 1 and Table 2 [23].
Fig. 1. Geometric and numerical model of SB and PCSWT. 104
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Fig. 2. Various cases WLs for gravity water tank.
DYNA, respectively An FSI algorithm was applied in the program of LS-DYNA to model the interaction among water, air and structure. The different physical domains of structure and fluid coupled interaction can be simulated by the Lagrangian and Eulerian methods, which are especially appropriate for the research about interaction problems including multiple materials of fluids and structures. The explicit nonlinear dynamic software LS-DYNA applied to Arbitrary Lagrange Eulerian (ALE) formulation has strong capacity to handle the FSI and water sloshing problem. ALE algorithm contains both Lagrangian and Eulerian formulations. In Lagrange description, the mesh moves with material and it can easily to capture material surface and boundary conditions. While for the Eulerian method, the mesh remains fixed and never changed when the material passes through mesh, the surface and boundary conditions of mesh are difficult to tract [24]. In the finite element model, the air was assumed as ideal gas that was modeled by linear internal energy and linear-polynomial equation of state (EOS). The material model of MAT_NULL and EOS were used to simulate the air and water, the density of water was 1000 kg/m3 [2,23].
Table 3 Cases of water levels (WLs). Height of WLs
h2 (m)
WLs Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
2.8
3.8
4.8
5.9
6.9
7.8
8.8
9.8
nonlinear material properties. The pressure-based fluid element FLUID30 is used for fluid-structure interaction analysis. The element also can be used with other 3D structural elements to perform unsymmetric or damped modal, full harmonic and full transient method analyses [24]. For the water in PCSWT, atmospheric pressure was implemented to the water surface and contacted to the air. Moreover, the FSI effects between the water and PCSWT were simulated by multiplematerial coupling algorithm. This method integrates the advantages of the Lagrangian and Eulerian elements but without the excessive mesh distortion problem, to the proposed method effectively, traces the movement of structural boundaries and observes the fluid’s pressure distribution in the medium [25]. This study mainly focused on the effects of FSI on the fragility, the reactor, internal equipment and pipeline were not established due to its complicated boundary or structural shape at a time. As an alternative, this internals were modeled by shell and mass elements. A firm foundation was used without considering the soil-structure interaction and reliable connection between the SB and the basement [1,2,22]. The detailed FSI model of AP1000 shield building is the same as our previous finite element model [2,20,26,27].
2.4. Mesh convergence analysis The mesh size of the finite element model has a significant influence on the accuracy of the numerical results. It is well known that the smaller size of the element is, the higher the accuracy. To get the more ideal results, the much smaller element size will be taken, which may require more computation performance and space. Notably, the Arbitrary Lagrangian Eulerian algorithm (ALE) approach will employ more massive CPU and memory space to solve the FSI coupling equations. Thus, mesh convergence analysis on different mesh size should be carried out to seek the suitable element size, which is acceptable for both the results accuracy and computation resources. In this study, three mesh sizes of 200, 500 and 1000 mm were used to calculate the acceleration responses of the SB, the errors of the results were 1.65% and 21.95% in the horizontal X direction. The results of convergence analysis were the same as the previous study [2], which had used the same finite element model. Lastly, the mesh size of 200 mm was applied to mesh the air intakes of the SB, and the PCSWT, steel containment vessel and cylinder wall of SB were meshed with 500 mm.
2.3. Material models To simply the analysis process and save the computation cost, the material model of MAT_PLASTIC_KINEMATIC from LS-DYNA was applied to simulate the reinforced concrete (RC). This model is capability of model isotropic and kinematic hardening plasticity with the option of including rate effects. It is a very cost-effective model and available for beam, shell, and solid elements [24]. Table 4 showed the parameters of the material model, the mass density, elastic modulus, Poisson ratio, yield stress, tangent modulus, hardening parameter, strain rate parameter C, strain rate parameter P, and failure strain were represented by the coefficients of RO, E, PR, SIGY, ETAN, BETA, SRC, SRP and FS in LSTable 4 Parameters of RC in LS-DYNA. Material RC
RO (Pa) 2.4 × 10
E(Pa) 3
3.35 × 10
PR 10
0.2
SIGY(Pa) 32.4 × 10
105
ETEN(Pa) 6
3.35 × 10
9
BETA
SRC
SRP
SC
0.0
0.4
0.5
0
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3. Preparation for fragility analysis 3.1. Ground motions selection The randomness and the uncertainty of the ground motions strongly influence the accuracy of the structural response. ATC-63 [28] guidelines are based on the fundamental principles that suggest selection of ground motion records considering the effects of the disperse characteristic of pan soil site. To achieve the more reliability of the calculated results, enough number of the ground motions should be chosen, which may increase the difficulty in the selection for the earthquakes and computation cost, especially enhance the saturation phenomenon of the computational accuracy. Thus, the peak ground acceleration (PGA) of ground motion should have preferably dispersion and satisfy the amount requirement, particularly the properties of the ground motions are consistent with the site characteristics. The NPP generally locates on the hard rock and the ground motion should obey the following principles: (1) the magnitudes of the ground motion records are greater than 6.0; (2) the site of the ground motion belongs to I0 classification, i.e. the velocity of shear wave is greater than 800 m/s; (3) ground motion pairs records at sites located farther than 10 km from fault rupture. Lastly, 20 groups of earthquakes were selected from the Peer Motion Database [29] and inputted in three directions to carry out the analysis, Table 5 showed the ground motions sequences and the elastic spectra with the damping of 5% for ground motion sequences were plotted in Fig. 3.
Fig. 3. Response spectrum of selected ground motions.
Lastly, we selected the PGA as the IM. 3.3. Definition of damage index The security performance of the NPP structures is much stricter than the safety request of the conventional reinforced concrete (CRC) structures as results of the unique requirement that the NPP structure should be in the linear status or slight nonlinearity under earthquakes. In view of the requirement of NPP, the selection of the damage index (DI) is significantly strict compared with the CRC structures. The junction between the cylinder wall of SB and the corner of base appeared stress concentration and was the weakest location, as shown in Fig. 4. Fig. 5 showed the relation between the maximum first principal and the elevation of SB, it was indicated the stress at the height of 14.82 had sharp change. Thus, the maximum first principal stress at the corner of SB was selected to be the DI to assess the damage of NPP subjected to the earthquake sequences. Lastly, the standard value of tensile strength of 2.51 MPa for concrete C45 was chosen to be the DI of the structure [33].
3.2. Intensity measure selection The intensity measure (IM) plays a significant role in the effectiveness of the vulnerability assessment and numerical simulation. Generally, the IM mainly consists of two types of earthquake records and the dynamic behavior of the structure. The parameters of earthquake records include peak ground velocity (PGV), peak ground displacement (PGD) and peak ground acceleration (PGA), while the parameters correlated to the structural response are peak velocity response spectrum (Sv), peak displacement response spectrum (Sd) and peak acceleration response spectrum (Sa). Elenas et al. [30] observed that the parameters of earthquake IM were mainly PGA, PGV and PGD, Shome et al. [31] conducted a correction study between the IM and the dynamic response of structures. The results showed that the IM of PGA would reflect the dynamic characteristic of structures better under the ground motion, which has been used in the previous studies [9,32].
4. Results of fragility analysis 4.1. Incremental Dynamic Analysis (IDA) based on regression curve Incremental Dynamic Analysis (IDA) is an approach to calculate the influences of increasing earthquake intensity on dynamic response up to collapse. It is assumed that there is a logarithmic correlation between the median value of engineering demand parameter (EDP) and IM. The mean and standard deviation of ln (EDP) can be calculated by
Table 5 List of earthquake ground motions used for the fragility analysis. No.
Ground motion
Time
Magnitude
Station
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
San Fernando Loma Prieta Loma Prieta Loma Prieta Northridge-01 Chi-Chi_Taiwan Chi-Chi_Taiwan Chi-Chi_Taiwan Chi-Chi_Taiwan Chi-Chi_Taiwan-04 Chi-Chi_Taiwan-04 Chi-Chi_Taiwan-05 Chi-Chi_Taiwan-06 Chi-Chi_Taiwan-06 Chi-Chi_Taiwan-06 Tottori_Japan Tottori_Japan Tottori_Japan Iwate_Japan San Simeon_CA
1971 1989 1989 1989 1994 1999 1999 1999 1999 1999 1999 1999 1999 1999 1999 2000 2000 2000 2008 2003
6.61 6.93 6.93 6.93 6.69 7.62 7.62 7.62 7.62 6.2 6.2 6.2 6.3 6.3 6.3 6.61 6.61 6.61 6.9 6.52
Cedar Springs_Allen Ranch Piedmont Jr High School Grounds SF – Rincon Hill So. San Francisco_Sierra Pt. Vasquez Rocks Park ILA063 CHY102 TCU085 TTN042 CHY102 TTN042 HWA003 CHY102 ILA063 TCU085 OKYH02 OKYH07 SMNH10 IWT010 Diablo Canyon Power Plant
Fig. 4. Example of maximum principal stress distribution in a two set of ground motions. 106
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Φ(x ) =
1 2π
2
∫−∞ exp ⎛− t2 ⎞ dx x
⎜
⎟
⎝
⎠
(6)
where Φ(·) represents the standard normal distribution function. 4.3. Fragility curves Fig. 7 and Table 6 showed the fragility curves and probabilities of failure for SB with various water levels. As shown in Fig. 7, it was observed that the results obtained using linear and quadratic fitting were close to each other with a difference ranging of 2%. The probabilities of failure for various water levels of SB were different, and the probability of failure increased with the increasing of PGA. The probability of failure expressed the variation trends of increase, decrease, increase and decrease with the increasing of water levels under the safe shutdown earthquake (SSE), the failure probability for the case 7 was larger with the value of 26.9%, while the case 5 and case 6 had the smaller probability of failure of 9.0% and 8.0%. Moreover, the case 6 had the higher seismic reduction with the smaller probability of failure under the earthquake ranging from 0.1 g to 0.6 g, the amplitudes of decrease for the failure probabilities were 0.1%, 6.4%, 18.9%, 18.8%, 9.55% and 0.8% respectively. On the contrary, the case 6 didn’t decrease the seismic response when the PGA was larger than 0.6 g, and its probability of failure was higher than that of the case 7. It was also indicated that the case 7 had the highest probability of failure when the PGA was less than 0.3 g, while the failure probability of case 2 was larger ranging from 0.3 g to 0.4 g, the case 1 had the highest value of the probability of failure when the PGA was greater than 0.4 g. The FSI effects of most of the water level cases ineffectively decreased the structural damage and the probabilities of failure of these cases were higher than the case of no water under the smaller PGA of 0.3 g. Simultaneously, the failure probability of the no water case was lower than the standard case 8 before the PGA of 0.45 g, but the variation trend turned reverse when the PGA was greater than 0.45 g.
Fig. 5. Relationship between maximum first principal stress and elevation.
regression analysis and the linear and quadratic curve are used to fit the data calculated by ln (EDP) and ln (IM).
ln(EDP ) = A + B ln(IM )
(1)
ln(EDP ) = C [ln(IM )]2 + D ln(IM ) + E
(2)
where A, B, C, D and E are constant coefficients. If ln (EDP) is assumed to have a constant variance for all IMs (a reasonable assumption over a small range of IM, but usually less appropriate for a wide range of IM), then its variance can be calculated as follows: n
ξIM1 =
∑i = 1 [ln(EDPi ) − (ln A + B ln IMi )]2 n−2
(3)
n
ξIM 2 =
∑i = 1 [ln(EDPi ) − (A (ln IMi )2 + B ln IMi + C )]2 n−2
4.4. IDA based on truncated maximum likelihood estimation (4) If the m ground motions cause collapse of the structure, the value of IM at the process of collapse has known. The likelihood function of collapse at IMi caused by the arbitrary ground motion can be expressed as the standard normal distribution probability density function (PDF).
where EDPi and IMi are engineering demand parameter EDP and IM values of record i, and n represents the number of records. 4.2. Fragility analysis of different water levels
Liklihood = ϕ ⎜⎛ ⎝
The dynamic performance of the SB with eight cases of water levels (WLs) and no water case were calculated by the IDA method using the 20 sets of ground motions. The regression analyses of linear and quadratic curves were performed on the condition of an independent variable of ln (PGA) and the dependent variable of ln (EDP), the fitting results were shown in Fig. 6. It could be observed from the Fig. 6 that the slope and intercept of the regression lines of different water levels were different, and the size and discretion of the data obtained from regression curves for different water levels were large. The results indicated that the water levels had a significant impact on vulnerability and seismic performance. The results of quadratic fitting and the linear fitting results coincided with a small difference. Generally, the probability density function of the EDP is expressed with the lognormal distribution, and the exceedance probability can be expressed by the logarithmic standard normal distribution in the fragility analysis. Thus, the probability of failure beyond the quantitative index of limit state (LS) for a given intensity measure (IM) can be defined as follows:
P [EDP ⩾ LS / IM ] = 1 − Φ ⎜⎛ ⎝
ln(LS) − ln(EDP ) ⎞ ⎟ ξIM ⎠
ln(IMi / θ) ⎞ ⎟ β ⎠
(7)
where ϕ () is the standard normal distribution probability density function. The likelihood functions for a known ground motion scaled to IMmax without causing collapse is the probability that IMi is higher than IMmax:
Liklihood = 1 − Φ ⎜⎛ ⎝
ln(IMi / θ) ⎞ ⎟ β ⎠
(8)
It is an assumption that IMi is independent for each ground motion, the likelihood function of the data is the product of the individual likelihoods. The likelihood is expressed as follows: m
Liklihood =
n−m
ln(IMi / θ) ⎞ ⎡ ln(IMi / θ) ⎞ ⎤ ⎟ 1 − Φ ⎜⎛ ⎟ ⎢ ⎥ β β ⎝ ⎠⎣ ⎝ ⎠⎦
∏ϕ ⎛
⎜
i=1
(9)
where Π is the product of i value from 1 to m. The parameters of fragility function are calculated through changing the parameters until the likelihood function obtained the maximum value using Eq. (9). The maximum value of the likelihood function is equivalent to the maximum value of the logarithm likelihood function. Then we can obtain the equation of maximum likelihood estimation of θ and β as follows:
(5) 107
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Fig. 6. IM- EDP data fitting of different WLs conditions.
Fig. 7. Fragility curves of the SB.
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Table 6 Damage probability of each PGA for various WLs. %
0.1 g
Linear fitting
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 No water Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 No water
Quadratic fitting
1.9 7.6 4.0 4.7 4.2 7.1 0.1 3.3 5.6 3.7 4.6 2.3 3.0 1.4 8.9 8.0 1.2 1.7
−5
× × × × × ×
10 10−3 10−2 10−2 10−7 10−5
× × × × × × × × × × ×
10−3 10−6 10−6 10−3 10−2 10−2 10−7 10−5 10−2 10−3 10−6
0.2 g
0.3 g
0.4 g
0.5 g
0.6 g
0.7 g
0.8 g
0.9 g
1.0 g
0.7 3.5 4.1 3.9 0.1 0.3 5.5 2.1 0.3 0.6 3.6 4.9 4.1 0.1 0.4 6.8 2.1 0.3
17.8 25.4 21.4 19.5 6.8 7.1 22.1 18.3 9.4 17.6 26.9 25.5 21.3 9.0 8.0 26.9 20.3 11.8
56.7 56.2 45.1 41.4 35.4 27.9 43.0 46.3 39.4 56.6 58.2 50.3 43.8 41.1 29.6 48.4 49.4 44.4
84.4 78.6 65.5 61.1 68.9 53.8 61.1 70.3 70.6 84.0 79.8 69.1 62.8 73.0 55.5 65.0 72.5 74.0
95.6 90.5 79.5 75.4 88.5 74.3 74.4 85.2 88.6 95.2 90.9 81.2 76.1 90.0 75.4 76.2 86.0 89.7
98.9 96.0 88.1 84.9 96.4 86.9 83.5 93.0 96.1 98.7 96.1 88.6 84.8 96.7 87.4 83.7 93.1 96.2
99.7 98.3 93.2 90.8 99.0 93.7 89.4 96.8 98.8 99.6 98.3 92.9 90.3 98.9 93.9 88.6 96.6 98.7
99.9 99.3 96.1 94.4 99.7 97.1 93.2 98.6 99.6 99.9 99.3 95.6 93.8 99.7 97.1 91.9 98.3 99.5
100 99.7 97.8 96.6 99.9 98.6 95.6 99.3 99.9 100 99.7 97.1 95.9 99.9 98.6 94.1 99.1 99.8
this section, the value of IMmax was chosen as 0.65 g, the probability of failure for each PGA were shown in Fig. 8 and Table 7. It was observed from the figure and table that the case 7 (origin design of AP1000 PCSWT) was not the optimal water levels for mitigating the seismic response and the failure probability was the highest when the value of PGA was less than 0.3 g. The probability of failure for case 6 decreased 3.6%, 14.4%, 17.6%, 11.9%, 5.5% and 1.5% with respect to the standard case 7 under PGA ranging from 0.2 g to 0.7 g, and the decease ratios were 92.3%, 65.5%, 37.7%, 17.5%, 6.7% and 1.7% respectively. In summary, case 6 had the smallest probability of failure compared with the original design case 7 and the no water case. 5. Conclusions This paper mainly carried out the vulnerability assessment by using IDA based on regression approach and the truncated maximum likelihood estimation considering the fluid-structure interaction effects. Eight types of water levels were considered, and Arbitrary Lagrange Eulerian coupling was applied to study the effects of FSI on the fragility of SB. The tensile strength of the concrete was selected as the damage index to assess the probability of failure for the SB. It was indicated that the results obtained from the regression curve and the truncated maximum likelihood estimation were consistent with each other, and the maximum likelihood estimation can effectively save the computational time on the condition of the reasonably calculated results. For the truncated maximum likelihood estimation, the probability of failure for case 6 decreased 3.6%, 14.4%, 17.6%, 11.9%, 5.5% and 1.5% with respect to the standard case 7 under the PGA ranging from 0.2 g to 0.7 g, and the decease ratios were 92.3%, 65.5%, 37.7%, 17.5%, 6.7% and 1.7% respectively. It was indicated that the failure probabilities of various water levels were different and case 6 had the smallest probability of failure with a larger seismic reduction, whereas the case 1 and
Fig. 8. Fragility curves of the SB with various WLs cases.
(θ ,̂ β )̂ m
= arg max θ, β
∑ ⎡⎢ln ϕ ⎛ ln(IMi/θ) ⎞ ⎤⎥ + (n − m) ln ⎡⎢1 − Φ ⎛ ln(IMi/θ) ⎞ ⎤⎥ ⎜
j=1
⎝
⎣
β
⎟
⎜
⎠⎦
⎝
⎣
β
⎟
⎠⎦
(10) where θ ̂ and β ̂ are the θ and β of the maximum value of the logarithm likelihood function. 4.5. Fragility curves IDA based on truncated maximum likelihood estimation was used to calculate the fragility of the SB with different cases of water levels. In Table 7 Damage probability of each PGA under different WLs cases. %
0.1 g
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 No water
4.9 5.8 1.5 8.4 4.7 6.0 2.8 8.8 4.5
× × × × × × × × ×
10−10 10−8 10−2 10−3 10−9 10−5 10−2 10−3 10−5
0.2 g
0.3 g
0.4 g
0.5 g
0.6 g
0.7 g
0.8 g
0.9 g
1.0 g
4.2 × 10−2 0.1 3.7 2.2 5.5 × 10−2 0.3 3.9 3.3 0.7
9.6 11.1 23.3 16.4 8.3 7.6 22.0 23.8 14.9
55.4 52.1 50.9 40.2 47.8 29.5 47.1 53.3 48.9
89.5 85.1 72.8 62.5 83.6 56.2 68.1 75.8 77.6
98.5 96.8 86.1 78.2 96.6 76.5 82.0 88.6 91.9
99.8 99.4 93.2 87.9 99.4 88.6 90.1 94.8 97.4
100 99.9 96.7 93.4 99.9 94.7 94.7 97.7 99.2
100 100 98.4 96.5 100 97.7 97.1 99.0 99.8
100 100 99.2 98.1 100 99.0 98.5 99.6 99.9
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case 2 had the higher failure probabilities.
on the recorded earthquake data in Korea. Nucl Eng Des 2005;235:1867–74. [13] Tran T-T, Cao A-T, Nguyen T-H-X, Kim D. Fragility assessment for electric cabinet in nuclear power plant using response surface methodology. Nuclear. Eng Technol 2019;51:894–903. [14] Han R, Li Y, van de Lindt J. Seismic risk of base isolated non-ductile reinforced concrete buildings considering uncertainties and mainshock–aftershock sequences. Struct Saf 2014;50:39–56. [15] Saouma VE, Hariri-Ardebili MA. Seismic capacity and fragility analysis of an ASRaffected nuclear containment vessel structure. Nucl Eng Des 2019;346:140–56. [16] Pang R, Xu B, Kong X, Zou D, Zhou Y. Seismic reliability assessment of earth-rockfill dam slopes considering strain-softening of rockfill based on generalized probability density evolution method. Soil Dyn Earthquake Eng 2018;107:96–107. [17] Zhao C, Chen J, Wang J. Study on fluid structure interaction model and dynamical response of NPP shield building. China Civil Eng J 2016;49. [18] Zhao C, Chen J, Wang J. Dynamic performance of AP1000 nuclear power plant by parametrical analysis and optimization. Eng Mech 2017;34. [19] Zhao C, Chen J, Wang J, Yu N, Xu Q. Seismic mitigation performance and optimization design of NPP water tank with internal ring baffles under earthquake loads. Nucl Eng Des 2017;318:182–201. [20] Zhao C, Chen J, Xu Q, Wang J, Wang B. Investigation on sloshing and vibration mitigation of water storage tank of AP1000. Ann Nucl Energy 2016;90:331–42. [21] Zhao C, Chen J, Xu Q, Yang X. Sensitive analysis of water levels and air intakes on natural frequency of AP1000 nuclear island building considering FSI effects. Ann Nucl Energy 2015;78:1–9. [22] Wang D, Wu C, Huang W, Zhang Y. Vibration investigation on fluid-structure interaction of AP1000 shield building subjected to multi earthquake excitations. Ann Nucl Energy 2019;126:312–29. [23] Zhao C, Chen J, Xu Q. Dynamic analysis of AP1000 shield building for various elevations and shapes of air intakes considering FSI effects subjected to seismic loading. Prog Nucl Energy 2014;74:44–52. [24] Corporation LST. LS-DYNA Keyword User' s Manual 971. 2012. [25] Zhao CF, Chen JY, Wang Y, Lu SJ. Damage mechanism and response of reinforced concrete containment structure under internal blast loading. Theor Appl Fract Mech 2012;61:12–20. [26] Yu N, Zhao C, Peng T, Huang H, Roy SS, Mo YL. Numerical investigation of AP1000 NIB under mainshock-aftershock earthquakes. Prog Nucl Energy 2019;117:103096https://doi.org/10.1016/j.pnucene.2019.103096. [27] Zhao C, Yu N, Peng T, Lippolis V, Corona A, Mo YL. Study on the dynamic behavior of isolated AP1000 NIB under mainshock-aftershock sequences. Prog Nucl Energy 2019;103144. https://doi.org/10.1016/j.pnucene.2019.103144. [28] Deierlein GG, Liel AB, Haselton CB, Kircher C. ATC-63 methodology for evaluating seismic collapse safety of archetype buildings. ASCE-SEI Structures Congress, Vancouver, BC. 2008:24-6. [29] PEER Ground Motion Database. (http://ngawest2berkeleyedu/). [30] Elenas A, Meskouris K. Correlation study between seismic acceleration parameters and damage indices of structures. Eng Struct 2001;23:698–704. [31] Padgett JE, Nielson BG, DesRoches R. Selection of optimal intensity measures in probabilistic seismic demand models of highway bridge portfolios. Earthquake Eng Struct Dyn 2008;37:711–25. [32] Li C, Zhai C, Kunnath S, Ji D. Methodology for selection of the most damaging ground motions for nuclear power plant structures. Soil Dyn Earthquake Eng 2019;116:345–57. [33] Standard of China. Code for design of concrete structures GB50010. 2015.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is supported by National Natural Science of China (Grant No. 51508148), China Postdoctoral Science Foundation Funded Project (Grant Nos.:2015 M581980 and 2016 T90563), and State Key Laboratory of Structural Analysis for Industrial Equipment (Grant GZ19106). We also would like to thank the Chinese Scholarship Council (CSC) for his visiting scholarship stipend. References [1] Zhao C, Chen J. Dynamic characteristics of AP1000 shield building for various water levels and air intakes considering fluid-structure interaction. Prog Nucl Energy 2014;70:176–87. [2] Zhao C, Chen J, Xu Q. FSI effects and seismic performance evaluation of water storage tank of AP1000 subjected to earthquake loading. Nucl Eng Des 2014;280:372–88. [3] Zhao C, Yu N. Dynamic response of generation III+ integral nuclear island structure considering fluid structure interaction effects. Ann Nucl Energy 2018;112:189–207. [4] Kim S-W, Jeon B-G, Hahm D-G, Kim M-K. Seismic fragility evaluation of the baseisolated nuclear power plant piping system using the failure criterion based on stress-strain. Nucl Eng Technol 2019;51:561–72. [5] USNRC. Individual Plant Examination of External Events (IPEEE) for Severe Accident Vulnerabilities, Generic Letter No. 88-20. US Nuclear Regulatory Commission. 1991; Supplement 4. [6] USNRC. Procedural and Submittal Guidance for the Individual Plant Examination of External Events (IPEEE) for Severe Accident Vulnerabilities, NUREG-1407. US Nuclear Regulatory Commission, 1991. [7] Shin J, Kim J, Lee K. Seismic assessment of damaged piloti-type RC building subjected to successive earthquakes. Earthquake Eng Struct Dyn 2014;43:1603–19. [8] Ghosh J, Padgett JE, Sánchez-Silva M. Seismic damage accumulation in highway bridges in earthquake-prone regions. Earthquake Spectra. 2015;31:115–35. [9] Wen W, Zhai C, Ji D, Li S, Xie L. Framework for the vulnerability assessment of structure under mainshock-aftershock sequences. Soil Dyn Earthquake Eng 2017;101:41–52. [10] Zhao C, Chen J. Numerical simulation and investigation of the base isolated NPPC building under three-directional seismic loading. Nucl Eng Des 2013;265:484–96. [11] Salimi Firoozabad E, Jeon B-G, Choi H-S, Kim N-S. Seismic fragility analysis of seismically isolated nuclear power plants piping system. Nucl Eng Des 2015;284:264–79. [12] Cho SG, Joe YH. Seismic fragility analyses of nuclear power plant structures based
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Structures journal homepage: www.elsevier.com/locate/structures
Seismic fragility analysis of AP1000 SB considering fluid-structure interaction effects ⁎
T
⁎
Chunfeng Zhaoa,c,d, , Na Yub, , Y.L. Mod a
Department of Civil Engineering, Hefei University of Technology, Hefei 230009, China College of Economics and Management, Hefei University, Hefei 230601, China c State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China d Department of Civil and Environmental Engineering, University of Houston, Houston 77021, USA b
A R T I C LE I N FO
A B S T R A C T
Keywords: Vulnerability assessment Fluid-structure interaction Quadratic regression Maximum likelihood estimation Shield building
AP1000 nuclear power plant has an advanced passive cooling system with a vast water tank located above the shield building, which may affect the seismic performance of the structure. This paper conducts a fragility assessment of shield building and considers the fluid-structure interaction (FSI) effect on the damage of the shield building. Arbitrary Lagrangian Eulerian algorithm is applied to calculate the FSI and eight cases of water levels are considered. Three different fragility analysis methods, namely linear regression, quadratic regression and truncated maximum likelihood estimation, are adopted for assessing the seismic vulnerability of shield building. It is indicated from the results that the regression and the truncated maximum likelihood estimation are close to each other. The maximum likelihood estimation not only can effectively save the computational cost, but also can assure the reasonable calculated results. It is observed that different water levels have the individual probabilities of failure, and case 6 has the smallest probability of failure with a more considerable seismic reduction, whereas the case 1 and case 2 have the higher failure probabilities. The fragility analysis framework can thus be applied for evaluating the vulnerability of shield building under earthquakes considering the FSI effect.
1. Introduction AP1000 nuclear power plant (NPP) has a series of advanced passive safety features that had been certified by the US. NRC, which relied on the nature circulation and spray watering system to prevent the steel containment vessel from overheating and ensure the safe shutdown of NPP [1,2]. The water tank of AP1000 is used to cool down the temperature of the containment by spraying water after the reactor shut down automatically. The water tank may affect the dynamic response and security of the shield building due to the heavy mass and huge volume when the structure encounters earthquakes. Simultaneously, the water slosh and water levels (WLs) may also influence the dynamic characteristic and the security of the NPP. Therefore, the study on the dynamic response and vulnerability assessment of AP1000 has attracted more and more attention [3]. The seismic probabilistic risk assessment (SPRA) is a powerful approach to assess the security and the regulatory requirement of NPPs in construction and operation. Seismic fragility analysis (SFA) is the most important and fundamental stage in the process of SPRA especially for
⁎
the seismic performance of structures [4]. The Nuclear Regulatory Commission (NRC) issued a guide that requested all the NPP must have a vulnerability safety review since 1992 [5,6].Up to now, the vulnerability assessment of NPPs consisting of components, systems or structures under earthquakes has been paid more attention and studied in some researches [7–10]. Firoozabad et al. carried out a seismic fragility assessment for a piping system and through the cyclic loading experiment to validate the numerical analysis [11]. The results provided the damage probability of piping system in the operation life of NPP. The seismic fragility analysis of a Korean NPP was conducted by an improved approach and compared with other methods by Cho and Joe [12]. The results indicated that the proposed method might overestimate the seismic resistance of Korean NPP facilities. Tran et al. applied the response surface method to analyze the vulnerability of the electric cabinet in NPP. The results showed that the boundary conditions played an essential role in the acceleration response of the cabinet in the NPP [13]. Han et al. conducted the fragility assessment of an old non-ductile RC building with or without retrofit using Lead Rubber Bearings (LRBs). The results
Corresponding authors. E-mail addresses: [email protected] (C. Zhao), [email protected] (N. Yu).
https://doi.org/10.1016/j.istruc.2019.11.003 Received 20 August 2019; Received in revised form 10 October 2019; Accepted 7 November 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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showed that the base isolation can effectively reduce the seismic risk of NPP under earthquakes [14]. Kim et al. used stress–strain criterion to assess the fragility of the base-isolated NPP piping system. The results showed that the failure criterion could be suitable for application in seismic fragility analysis of the pipes in NPP [4]. Saouma et al. conducted a fragility analysis to assess the seismic capacity of an ASR-affected nuclear containment vessel [15]. Pang et al. used the extended incremental dynamic analysis to assess the seismic behavior of the face rockfill dams. The results demonstrated that the method could evaluate the destruction and loss of the dams [16]. Zhao et al. performed a series of study on the seismic performance and seismic resistance of NPP under earthquakes. The influences of water level and ring baffles on the structural behavior and security of AP1000 shield building were also deeply investigated [17–21]. The result provided the pre-analysis and indicated that the AP1000 NPP should be carried out the fragility analysis. As mentioned before, most of the papers mainly focused on the seismic fragility of the generation II NPP or other structures. Few of researches paid attention to the seismic fragility of the advanced generation III+ AP1000 shield building with a large water tank. The water slosh may affect the seismic performance and vulnerability of the shield building, especially considering the FSI effects. The purpose of this study was to perform a fragility assessment of AP1000 shield building considering the FSI. Eight cases of water levels (WLs) were designed and simulated by the Arbitrary Lagrangian Eulerian algorithm. The incremental dynamic analysis of linear and quadratic regression, and truncated maximum likelihood estimation were also used to perform the fragility analysis.
Table 1 Geometric dimensions of SB and PCSWT. Components
Parameters
Symbol
Unit
Value
SB
Height Diameter
Hs Ds
m m
71 44.2
PCSWT
Height Outer diameter Inner diameter
Hw Dout Din
m m m
11.8 27.13 10.67
Containment vessel (CV)
Diameter Thickness
Dcv Wc
m m
39.6 0.045
Table 2 Material of shield building. Material
Parameters
Symbol
Unit
Value
Concrete
Density Elastic modulus Poisson ratio
ρ E γ
kg/m3 MPa –
2300 3.35 × 104 0.2
Rebar
Density Elastic modulus Poisson ratio
ρ E γ
kg/m3 MPa –
7800 2.06 × 105 0.3
Water
Density
ρ
kg/m3
1000
Air
Density
ρ
kg/m3
1
2.2. Finite element model In this study, eight cases of water levels (WLs) for PCSWT were proposed to investigate the FSI effects on the fragility of SB [2,20]. The heights of water levels were assumed as 2.8 m, 3.8 m, 4.8 m, 5.8 m, 6.8 m, 7.8 m, 8.8 m and 9.8 m, and represented by case 1, case 2, case 3, case 4, case 5, case 6, case 7, and case8, as shown in Fig. 2 and Table 3. It means that case 1 has water height of 2.8 m, case 2 has the water height of 3.8, and so on. A smear model of concrete and reinforcement bar was used to establish the three-dimensional finite element model (FEM) of the SB with PCSWT. A Solid element SOLID65 was applied to mesh all the components of SB, the water in PCSWT was modeled by the pressure-based fluid element FLUID30, which was suitable for modeling the FSI analysis. SOLID65 is used for the 3D modeling of solids with or without rebar and capable of cracking in tension and crushing in compression. The most important characteristic of the element is the treatment of
2. Numerical model 2.1. AP1000 PCSWT Shield building (SB) and air intake (AI) are the important parts of the passive cooling system (PCS) of AP1000 NPP. Passive cooling system water tank (PCSWT) located at the top of the SB is used to cool down the temperature of the steel containment through spray water by gravity in case of the reactor shut down [2,22]. The water is partially filled in PCSWT with air vacant height of about 1 m from the top roof of PCSWT to water surface in its intial design. The diameter, height and thickness of the SB are 44.2 m, 82.8 m and 0.92 m. The details of the geometry and material of SB and PCSWT are listed in Fig. 1, Table 1 and Table 2 [23].
Fig. 1. Geometric and numerical model of SB and PCSWT. 104
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Fig. 2. Various cases WLs for gravity water tank.
DYNA, respectively An FSI algorithm was applied in the program of LS-DYNA to model the interaction among water, air and structure. The different physical domains of structure and fluid coupled interaction can be simulated by the Lagrangian and Eulerian methods, which are especially appropriate for the research about interaction problems including multiple materials of fluids and structures. The explicit nonlinear dynamic software LS-DYNA applied to Arbitrary Lagrange Eulerian (ALE) formulation has strong capacity to handle the FSI and water sloshing problem. ALE algorithm contains both Lagrangian and Eulerian formulations. In Lagrange description, the mesh moves with material and it can easily to capture material surface and boundary conditions. While for the Eulerian method, the mesh remains fixed and never changed when the material passes through mesh, the surface and boundary conditions of mesh are difficult to tract [24]. In the finite element model, the air was assumed as ideal gas that was modeled by linear internal energy and linear-polynomial equation of state (EOS). The material model of MAT_NULL and EOS were used to simulate the air and water, the density of water was 1000 kg/m3 [2,23].
Table 3 Cases of water levels (WLs). Height of WLs
h2 (m)
WLs Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
2.8
3.8
4.8
5.9
6.9
7.8
8.8
9.8
nonlinear material properties. The pressure-based fluid element FLUID30 is used for fluid-structure interaction analysis. The element also can be used with other 3D structural elements to perform unsymmetric or damped modal, full harmonic and full transient method analyses [24]. For the water in PCSWT, atmospheric pressure was implemented to the water surface and contacted to the air. Moreover, the FSI effects between the water and PCSWT were simulated by multiplematerial coupling algorithm. This method integrates the advantages of the Lagrangian and Eulerian elements but without the excessive mesh distortion problem, to the proposed method effectively, traces the movement of structural boundaries and observes the fluid’s pressure distribution in the medium [25]. This study mainly focused on the effects of FSI on the fragility, the reactor, internal equipment and pipeline were not established due to its complicated boundary or structural shape at a time. As an alternative, this internals were modeled by shell and mass elements. A firm foundation was used without considering the soil-structure interaction and reliable connection between the SB and the basement [1,2,22]. The detailed FSI model of AP1000 shield building is the same as our previous finite element model [2,20,26,27].
2.4. Mesh convergence analysis The mesh size of the finite element model has a significant influence on the accuracy of the numerical results. It is well known that the smaller size of the element is, the higher the accuracy. To get the more ideal results, the much smaller element size will be taken, which may require more computation performance and space. Notably, the Arbitrary Lagrangian Eulerian algorithm (ALE) approach will employ more massive CPU and memory space to solve the FSI coupling equations. Thus, mesh convergence analysis on different mesh size should be carried out to seek the suitable element size, which is acceptable for both the results accuracy and computation resources. In this study, three mesh sizes of 200, 500 and 1000 mm were used to calculate the acceleration responses of the SB, the errors of the results were 1.65% and 21.95% in the horizontal X direction. The results of convergence analysis were the same as the previous study [2], which had used the same finite element model. Lastly, the mesh size of 200 mm was applied to mesh the air intakes of the SB, and the PCSWT, steel containment vessel and cylinder wall of SB were meshed with 500 mm.
2.3. Material models To simply the analysis process and save the computation cost, the material model of MAT_PLASTIC_KINEMATIC from LS-DYNA was applied to simulate the reinforced concrete (RC). This model is capability of model isotropic and kinematic hardening plasticity with the option of including rate effects. It is a very cost-effective model and available for beam, shell, and solid elements [24]. Table 4 showed the parameters of the material model, the mass density, elastic modulus, Poisson ratio, yield stress, tangent modulus, hardening parameter, strain rate parameter C, strain rate parameter P, and failure strain were represented by the coefficients of RO, E, PR, SIGY, ETAN, BETA, SRC, SRP and FS in LSTable 4 Parameters of RC in LS-DYNA. Material RC
RO (Pa) 2.4 × 10
E(Pa) 3
3.35 × 10
PR 10
0.2
SIGY(Pa) 32.4 × 10
105
ETEN(Pa) 6
3.35 × 10
9
BETA
SRC
SRP
SC
0.0
0.4
0.5
0
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3. Preparation for fragility analysis 3.1. Ground motions selection The randomness and the uncertainty of the ground motions strongly influence the accuracy of the structural response. ATC-63 [28] guidelines are based on the fundamental principles that suggest selection of ground motion records considering the effects of the disperse characteristic of pan soil site. To achieve the more reliability of the calculated results, enough number of the ground motions should be chosen, which may increase the difficulty in the selection for the earthquakes and computation cost, especially enhance the saturation phenomenon of the computational accuracy. Thus, the peak ground acceleration (PGA) of ground motion should have preferably dispersion and satisfy the amount requirement, particularly the properties of the ground motions are consistent with the site characteristics. The NPP generally locates on the hard rock and the ground motion should obey the following principles: (1) the magnitudes of the ground motion records are greater than 6.0; (2) the site of the ground motion belongs to I0 classification, i.e. the velocity of shear wave is greater than 800 m/s; (3) ground motion pairs records at sites located farther than 10 km from fault rupture. Lastly, 20 groups of earthquakes were selected from the Peer Motion Database [29] and inputted in three directions to carry out the analysis, Table 5 showed the ground motions sequences and the elastic spectra with the damping of 5% for ground motion sequences were plotted in Fig. 3.
Fig. 3. Response spectrum of selected ground motions.
Lastly, we selected the PGA as the IM. 3.3. Definition of damage index The security performance of the NPP structures is much stricter than the safety request of the conventional reinforced concrete (CRC) structures as results of the unique requirement that the NPP structure should be in the linear status or slight nonlinearity under earthquakes. In view of the requirement of NPP, the selection of the damage index (DI) is significantly strict compared with the CRC structures. The junction between the cylinder wall of SB and the corner of base appeared stress concentration and was the weakest location, as shown in Fig. 4. Fig. 5 showed the relation between the maximum first principal and the elevation of SB, it was indicated the stress at the height of 14.82 had sharp change. Thus, the maximum first principal stress at the corner of SB was selected to be the DI to assess the damage of NPP subjected to the earthquake sequences. Lastly, the standard value of tensile strength of 2.51 MPa for concrete C45 was chosen to be the DI of the structure [33].
3.2. Intensity measure selection The intensity measure (IM) plays a significant role in the effectiveness of the vulnerability assessment and numerical simulation. Generally, the IM mainly consists of two types of earthquake records and the dynamic behavior of the structure. The parameters of earthquake records include peak ground velocity (PGV), peak ground displacement (PGD) and peak ground acceleration (PGA), while the parameters correlated to the structural response are peak velocity response spectrum (Sv), peak displacement response spectrum (Sd) and peak acceleration response spectrum (Sa). Elenas et al. [30] observed that the parameters of earthquake IM were mainly PGA, PGV and PGD, Shome et al. [31] conducted a correction study between the IM and the dynamic response of structures. The results showed that the IM of PGA would reflect the dynamic characteristic of structures better under the ground motion, which has been used in the previous studies [9,32].
4. Results of fragility analysis 4.1. Incremental Dynamic Analysis (IDA) based on regression curve Incremental Dynamic Analysis (IDA) is an approach to calculate the influences of increasing earthquake intensity on dynamic response up to collapse. It is assumed that there is a logarithmic correlation between the median value of engineering demand parameter (EDP) and IM. The mean and standard deviation of ln (EDP) can be calculated by
Table 5 List of earthquake ground motions used for the fragility analysis. No.
Ground motion
Time
Magnitude
Station
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
San Fernando Loma Prieta Loma Prieta Loma Prieta Northridge-01 Chi-Chi_Taiwan Chi-Chi_Taiwan Chi-Chi_Taiwan Chi-Chi_Taiwan Chi-Chi_Taiwan-04 Chi-Chi_Taiwan-04 Chi-Chi_Taiwan-05 Chi-Chi_Taiwan-06 Chi-Chi_Taiwan-06 Chi-Chi_Taiwan-06 Tottori_Japan Tottori_Japan Tottori_Japan Iwate_Japan San Simeon_CA
1971 1989 1989 1989 1994 1999 1999 1999 1999 1999 1999 1999 1999 1999 1999 2000 2000 2000 2008 2003
6.61 6.93 6.93 6.93 6.69 7.62 7.62 7.62 7.62 6.2 6.2 6.2 6.3 6.3 6.3 6.61 6.61 6.61 6.9 6.52
Cedar Springs_Allen Ranch Piedmont Jr High School Grounds SF – Rincon Hill So. San Francisco_Sierra Pt. Vasquez Rocks Park ILA063 CHY102 TCU085 TTN042 CHY102 TTN042 HWA003 CHY102 ILA063 TCU085 OKYH02 OKYH07 SMNH10 IWT010 Diablo Canyon Power Plant
Fig. 4. Example of maximum principal stress distribution in a two set of ground motions. 106
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Φ(x ) =
1 2π
2
∫−∞ exp ⎛− t2 ⎞ dx x
⎜
⎟
⎝
⎠
(6)
where Φ(·) represents the standard normal distribution function. 4.3. Fragility curves Fig. 7 and Table 6 showed the fragility curves and probabilities of failure for SB with various water levels. As shown in Fig. 7, it was observed that the results obtained using linear and quadratic fitting were close to each other with a difference ranging of 2%. The probabilities of failure for various water levels of SB were different, and the probability of failure increased with the increasing of PGA. The probability of failure expressed the variation trends of increase, decrease, increase and decrease with the increasing of water levels under the safe shutdown earthquake (SSE), the failure probability for the case 7 was larger with the value of 26.9%, while the case 5 and case 6 had the smaller probability of failure of 9.0% and 8.0%. Moreover, the case 6 had the higher seismic reduction with the smaller probability of failure under the earthquake ranging from 0.1 g to 0.6 g, the amplitudes of decrease for the failure probabilities were 0.1%, 6.4%, 18.9%, 18.8%, 9.55% and 0.8% respectively. On the contrary, the case 6 didn’t decrease the seismic response when the PGA was larger than 0.6 g, and its probability of failure was higher than that of the case 7. It was also indicated that the case 7 had the highest probability of failure when the PGA was less than 0.3 g, while the failure probability of case 2 was larger ranging from 0.3 g to 0.4 g, the case 1 had the highest value of the probability of failure when the PGA was greater than 0.4 g. The FSI effects of most of the water level cases ineffectively decreased the structural damage and the probabilities of failure of these cases were higher than the case of no water under the smaller PGA of 0.3 g. Simultaneously, the failure probability of the no water case was lower than the standard case 8 before the PGA of 0.45 g, but the variation trend turned reverse when the PGA was greater than 0.45 g.
Fig. 5. Relationship between maximum first principal stress and elevation.
regression analysis and the linear and quadratic curve are used to fit the data calculated by ln (EDP) and ln (IM).
ln(EDP ) = A + B ln(IM )
(1)
ln(EDP ) = C [ln(IM )]2 + D ln(IM ) + E
(2)
where A, B, C, D and E are constant coefficients. If ln (EDP) is assumed to have a constant variance for all IMs (a reasonable assumption over a small range of IM, but usually less appropriate for a wide range of IM), then its variance can be calculated as follows: n
ξIM1 =
∑i = 1 [ln(EDPi ) − (ln A + B ln IMi )]2 n−2
(3)
n
ξIM 2 =
∑i = 1 [ln(EDPi ) − (A (ln IMi )2 + B ln IMi + C )]2 n−2
4.4. IDA based on truncated maximum likelihood estimation (4) If the m ground motions cause collapse of the structure, the value of IM at the process of collapse has known. The likelihood function of collapse at IMi caused by the arbitrary ground motion can be expressed as the standard normal distribution probability density function (PDF).
where EDPi and IMi are engineering demand parameter EDP and IM values of record i, and n represents the number of records. 4.2. Fragility analysis of different water levels
Liklihood = ϕ ⎜⎛ ⎝
The dynamic performance of the SB with eight cases of water levels (WLs) and no water case were calculated by the IDA method using the 20 sets of ground motions. The regression analyses of linear and quadratic curves were performed on the condition of an independent variable of ln (PGA) and the dependent variable of ln (EDP), the fitting results were shown in Fig. 6. It could be observed from the Fig. 6 that the slope and intercept of the regression lines of different water levels were different, and the size and discretion of the data obtained from regression curves for different water levels were large. The results indicated that the water levels had a significant impact on vulnerability and seismic performance. The results of quadratic fitting and the linear fitting results coincided with a small difference. Generally, the probability density function of the EDP is expressed with the lognormal distribution, and the exceedance probability can be expressed by the logarithmic standard normal distribution in the fragility analysis. Thus, the probability of failure beyond the quantitative index of limit state (LS) for a given intensity measure (IM) can be defined as follows:
P [EDP ⩾ LS / IM ] = 1 − Φ ⎜⎛ ⎝
ln(LS) − ln(EDP ) ⎞ ⎟ ξIM ⎠
ln(IMi / θ) ⎞ ⎟ β ⎠
(7)
where ϕ () is the standard normal distribution probability density function. The likelihood functions for a known ground motion scaled to IMmax without causing collapse is the probability that IMi is higher than IMmax:
Liklihood = 1 − Φ ⎜⎛ ⎝
ln(IMi / θ) ⎞ ⎟ β ⎠
(8)
It is an assumption that IMi is independent for each ground motion, the likelihood function of the data is the product of the individual likelihoods. The likelihood is expressed as follows: m
Liklihood =
n−m
ln(IMi / θ) ⎞ ⎡ ln(IMi / θ) ⎞ ⎤ ⎟ 1 − Φ ⎜⎛ ⎟ ⎢ ⎥ β β ⎝ ⎠⎣ ⎝ ⎠⎦
∏ϕ ⎛
⎜
i=1
(9)
where Π is the product of i value from 1 to m. The parameters of fragility function are calculated through changing the parameters until the likelihood function obtained the maximum value using Eq. (9). The maximum value of the likelihood function is equivalent to the maximum value of the logarithm likelihood function. Then we can obtain the equation of maximum likelihood estimation of θ and β as follows:
(5) 107
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Fig. 6. IM- EDP data fitting of different WLs conditions.
Fig. 7. Fragility curves of the SB.
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Table 6 Damage probability of each PGA for various WLs. %
0.1 g
Linear fitting
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 No water Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 No water
Quadratic fitting
1.9 7.6 4.0 4.7 4.2 7.1 0.1 3.3 5.6 3.7 4.6 2.3 3.0 1.4 8.9 8.0 1.2 1.7
−5
× × × × × ×
10 10−3 10−2 10−2 10−7 10−5
× × × × × × × × × × ×
10−3 10−6 10−6 10−3 10−2 10−2 10−7 10−5 10−2 10−3 10−6
0.2 g
0.3 g
0.4 g
0.5 g
0.6 g
0.7 g
0.8 g
0.9 g
1.0 g
0.7 3.5 4.1 3.9 0.1 0.3 5.5 2.1 0.3 0.6 3.6 4.9 4.1 0.1 0.4 6.8 2.1 0.3
17.8 25.4 21.4 19.5 6.8 7.1 22.1 18.3 9.4 17.6 26.9 25.5 21.3 9.0 8.0 26.9 20.3 11.8
56.7 56.2 45.1 41.4 35.4 27.9 43.0 46.3 39.4 56.6 58.2 50.3 43.8 41.1 29.6 48.4 49.4 44.4
84.4 78.6 65.5 61.1 68.9 53.8 61.1 70.3 70.6 84.0 79.8 69.1 62.8 73.0 55.5 65.0 72.5 74.0
95.6 90.5 79.5 75.4 88.5 74.3 74.4 85.2 88.6 95.2 90.9 81.2 76.1 90.0 75.4 76.2 86.0 89.7
98.9 96.0 88.1 84.9 96.4 86.9 83.5 93.0 96.1 98.7 96.1 88.6 84.8 96.7 87.4 83.7 93.1 96.2
99.7 98.3 93.2 90.8 99.0 93.7 89.4 96.8 98.8 99.6 98.3 92.9 90.3 98.9 93.9 88.6 96.6 98.7
99.9 99.3 96.1 94.4 99.7 97.1 93.2 98.6 99.6 99.9 99.3 95.6 93.8 99.7 97.1 91.9 98.3 99.5
100 99.7 97.8 96.6 99.9 98.6 95.6 99.3 99.9 100 99.7 97.1 95.9 99.9 98.6 94.1 99.1 99.8
this section, the value of IMmax was chosen as 0.65 g, the probability of failure for each PGA were shown in Fig. 8 and Table 7. It was observed from the figure and table that the case 7 (origin design of AP1000 PCSWT) was not the optimal water levels for mitigating the seismic response and the failure probability was the highest when the value of PGA was less than 0.3 g. The probability of failure for case 6 decreased 3.6%, 14.4%, 17.6%, 11.9%, 5.5% and 1.5% with respect to the standard case 7 under PGA ranging from 0.2 g to 0.7 g, and the decease ratios were 92.3%, 65.5%, 37.7%, 17.5%, 6.7% and 1.7% respectively. In summary, case 6 had the smallest probability of failure compared with the original design case 7 and the no water case. 5. Conclusions This paper mainly carried out the vulnerability assessment by using IDA based on regression approach and the truncated maximum likelihood estimation considering the fluid-structure interaction effects. Eight types of water levels were considered, and Arbitrary Lagrange Eulerian coupling was applied to study the effects of FSI on the fragility of SB. The tensile strength of the concrete was selected as the damage index to assess the probability of failure for the SB. It was indicated that the results obtained from the regression curve and the truncated maximum likelihood estimation were consistent with each other, and the maximum likelihood estimation can effectively save the computational time on the condition of the reasonably calculated results. For the truncated maximum likelihood estimation, the probability of failure for case 6 decreased 3.6%, 14.4%, 17.6%, 11.9%, 5.5% and 1.5% with respect to the standard case 7 under the PGA ranging from 0.2 g to 0.7 g, and the decease ratios were 92.3%, 65.5%, 37.7%, 17.5%, 6.7% and 1.7% respectively. It was indicated that the failure probabilities of various water levels were different and case 6 had the smallest probability of failure with a larger seismic reduction, whereas the case 1 and
Fig. 8. Fragility curves of the SB with various WLs cases.
(θ ,̂ β )̂ m
= arg max θ, β
∑ ⎡⎢ln ϕ ⎛ ln(IMi/θ) ⎞ ⎤⎥ + (n − m) ln ⎡⎢1 − Φ ⎛ ln(IMi/θ) ⎞ ⎤⎥ ⎜
j=1
⎝
⎣
β
⎟
⎜
⎠⎦
⎝
⎣
β
⎟
⎠⎦
(10) where θ ̂ and β ̂ are the θ and β of the maximum value of the logarithm likelihood function. 4.5. Fragility curves IDA based on truncated maximum likelihood estimation was used to calculate the fragility of the SB with different cases of water levels. In Table 7 Damage probability of each PGA under different WLs cases. %
0.1 g
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 No water
4.9 5.8 1.5 8.4 4.7 6.0 2.8 8.8 4.5
× × × × × × × × ×
10−10 10−8 10−2 10−3 10−9 10−5 10−2 10−3 10−5
0.2 g
0.3 g
0.4 g
0.5 g
0.6 g
0.7 g
0.8 g
0.9 g
1.0 g
4.2 × 10−2 0.1 3.7 2.2 5.5 × 10−2 0.3 3.9 3.3 0.7
9.6 11.1 23.3 16.4 8.3 7.6 22.0 23.8 14.9
55.4 52.1 50.9 40.2 47.8 29.5 47.1 53.3 48.9
89.5 85.1 72.8 62.5 83.6 56.2 68.1 75.8 77.6
98.5 96.8 86.1 78.2 96.6 76.5 82.0 88.6 91.9
99.8 99.4 93.2 87.9 99.4 88.6 90.1 94.8 97.4
100 99.9 96.7 93.4 99.9 94.7 94.7 97.7 99.2
100 100 98.4 96.5 100 97.7 97.1 99.0 99.8
100 100 99.2 98.1 100 99.0 98.5 99.6 99.9
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case 2 had the higher failure probabilities.
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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is supported by National Natural Science of China (Grant No. 51508148), China Postdoctoral Science Foundation Funded Project (Grant Nos.:2015 M581980 and 2016 T90563), and State Key Laboratory of Structural Analysis for Industrial Equipment (Grant GZ19106). We also would like to thank the Chinese Scholarship Council (CSC) for his visiting scholarship stipend. References [1] Zhao C, Chen J. Dynamic characteristics of AP1000 shield building for various water levels and air intakes considering fluid-structure interaction. Prog Nucl Energy 2014;70:176–87. [2] Zhao C, Chen J, Xu Q. FSI effects and seismic performance evaluation of water storage tank of AP1000 subjected to earthquake loading. Nucl Eng Des 2014;280:372–88. [3] Zhao C, Yu N. Dynamic response of generation III+ integral nuclear island structure considering fluid structure interaction effects. Ann Nucl Energy 2018;112:189–207. [4] Kim S-W, Jeon B-G, Hahm D-G, Kim M-K. Seismic fragility evaluation of the baseisolated nuclear power plant piping system using the failure criterion based on stress-strain. Nucl Eng Technol 2019;51:561–72. [5] USNRC. Individual Plant Examination of External Events (IPEEE) for Severe Accident Vulnerabilities, Generic Letter No. 88-20. US Nuclear Regulatory Commission. 1991; Supplement 4. [6] USNRC. Procedural and Submittal Guidance for the Individual Plant Examination of External Events (IPEEE) for Severe Accident Vulnerabilities, NUREG-1407. US Nuclear Regulatory Commission, 1991. [7] Shin J, Kim J, Lee K. Seismic assessment of damaged piloti-type RC building subjected to successive earthquakes. Earthquake Eng Struct Dyn 2014;43:1603–19. [8] Ghosh J, Padgett JE, Sánchez-Silva M. Seismic damage accumulation in highway bridges in earthquake-prone regions. Earthquake Spectra. 2015;31:115–35. [9] Wen W, Zhai C, Ji D, Li S, Xie L. Framework for the vulnerability assessment of structure under mainshock-aftershock sequences. Soil Dyn Earthquake Eng 2017;101:41–52. [10] Zhao C, Chen J. Numerical simulation and investigation of the base isolated NPPC building under three-directional seismic loading. Nucl Eng Des 2013;265:484–96. [11] Salimi Firoozabad E, Jeon B-G, Choi H-S, Kim N-S. Seismic fragility analysis of seismically isolated nuclear power plants piping system. Nucl Eng Des 2015;284:264–79. [12] Cho SG, Joe YH. Seismic fragility analyses of nuclear power plant structures based
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