# Experimental and Numerical Studies of AP1000 Shield Building considering Fluid-Structure Interaction

Authors: Zhi Zhang; Chenning Song; Zhining Duan; Zhi Cheng
Journal: Science and Technology of Nuclear Installations
 (2022)

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## Abstract

The gravity cooling water tank is a remarkable structural feature of third-generation pressurized water reactor nuclear power plant. To investigate the influence of fluid-structure interaction (FSI) on the seismic response of the structure, this study designed two 1 : 50 simplified models of the AP1000 shield building. A series of shaking table tests were conducted to study the seismic responses with and without FSI effect. The natural frequency, acceleration, strain, and hydrodynamic pressure of the two models were analyzed, and the seismic reduction effect of the water tank was evaluated. Moreover, the test data were compared with the results of numerical analysis using the ABAQUS software. The results show that the presence of water and the sloshing of water reduce the natural frequency and seismic response of the model structure. Thus, the gravity cooling water tank has a certain seismic reduction effect. The simplified model of water sloshing can be used to analyze the seismic response of the shield building.


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## Body

## 1. Introduction

In recent years, the construction of nuclear power plants has accelerated, especially in China. Some of those currently under construction include third-generation nuclear power plants, represented by the AP1000, CAP1400, and HPR1000 [1–3]. As a representative plant, the AP1000 is developed by Westinghouse, and one of the obvious features is the passive containment cooling system (PCS). As the core component of nuclear power plant, the nuclear island building includes containment, shield building, and auxiliary building. As shown in Figure 1, the main function of the shield building is to protect the internal radioactive systems and equipment. A gravity cooling water tank at the top of the shield building is the salient feature of the AP1000 shield building. The fluid-structure interaction (FSI) induced by this water tank should be considered in the seismic design and safety analysis of nuclear power plants.Figure 1
Shield building and PCS of AP1000 nuclear island building [4].Despite considerable seismic research using AP1000 nuclear power plants, there have been few studies on the FSI effect. Liu [5] adopted the Coupled Eulerian Lagrange (CEL) method to analyze the FSI of the PCS water tank and studied the seismic performance of the shield building using the ABAQUS software. Zhao et al. [6–10] adopted the Arbitrary Lagrange Eulerian (ALE) method to simulate the FSI in terms of the fluid sloshing and oscillation of the water tank in the event of an earthquake. The influence of the water tank and the air intake were systematically studied, and some optimized design pieces of advice were proposed. Lu et al. [11] and Liu et al. [12] studied the dynamic response of the elevated water tank in the AP1000 PCS through experimental tests and numerical simulations and adopted an equivalent mechanical model to predict the seismic forces produced by the system. Xu et al. [13] adopted the smoothed particle hydrodynamics (SPH) and finite element method (FEM) coupling method to simulate the FSI between the cooling water and the shield building of AP1000. The research shows that the water tank can decrease the natural frequencies and seismic response of the shield building. Song et al. [14] and Li et al. [15] established finite element model with water, shield building, and auxiliary building and investigated the FSI effect of the nuclear island building using the ADINA software. A simplified model based on Housner’s model [16, 17] was proposed and proved to be reasonable to simulate FSI effect. Li et al. [18] adopted ALE method to study seismic fragile of shield building with different water levels; the seismic fragile curves were obtained for four kinds of baffle design schemes, and the effects on seismic fragile curves were analyzed. For the annular liquid containers used in third-generation nuclear power plants, Liu et al. [19] predicted the dynamic reactions of annular liquid containers and analyzed the characteristics of the hydrodynamic pressure response on the inner and outer walls. Wang et al. [20] conducted a series of numerical simulations on seven partially filled models in six natural and one artificial earthquake. The research shows that a reasonable design of the water level can reduce the structural responses and improve seismic safety. On this basis, Wang et al. [21] further studied the dynamic responses of the AP1000 shield building with base isolation using numerical simulations and derived the optimal water level ratio. Zhao et al. [22] considered eight water levels and three fragility analysis methods (linear regression, quadratic regression, and truncated maximum likelihood estimation) in assessing the seismic vulnerability of the shield building.Previous research on FSI in nuclear power plants has mainly used numerical analysis, with relatively few experimental studies being performed. Moreover, there have been fewer comparative experimental studies on the shield building with and without FSI effect. Compared with numerical analysis, shaking table tests provide a more intuitive and accurate reflection of the seismic response of a structure. Therefore, it is necessary to study the influence of FSI effect on the seismic response of the shield building through a series of comparative tests. This paper describes shaking table tests using two simplified models of the AP1000 shield building to study the influence of FSI. The seismic reduction effect of the water tank is evaluated through comparative analysis of the two models. In addition, numerical analysis of the shaking table tests is performed, and the rationality of the simplified model of water sloshing is verified.


## 2. Shaking Table Tests

### 2.1. Test Equipment

The shaking table tests were carried out at Shandong Jianzhu University using a shaking table measuring 3 m × 3 m with a maximum bearing capacity of 15 t. The specific parameters of the shaking table are listed in Table1.Table 1
Specific parameters of shaking table.
Size (m)Effective bearing capacity (t)Maximum bearing capacity (t)Vibration directionMaximum acceleration (g)Maximum velocity (mm/s)Maximum displacement (mm)3.00 × 3.001015X, Y, ZX: ±1.5X: ±1000X: ±1256 DOFY: ±1.5Y: ±1000Y: ±125Z: ±1.0Z: ±310Z: ±125Note:X and Y represent the horizontal directions, Z represents the vertical direction, and DOF represents degrees of freedom.


### 2.2. Similarity Relation

The shaking table test model is a scaled structure based on the proportions of a prototype structure. The test model needs to be subjected to similar working conditions as the prototype. Therefore, the model structure and the prototype structure should obey a certain similarity relation. This similarity relation is reflected by the similarity ratios, which are the ratios of the main physical parameters of the model structure to those of the prototype structure. Considering the size and bearing capacity of the shaking table, the geometric similarity ratio of the model structure to the prototype was 1 : 50. The shaking table tests were carried out under gravity of 1 g, so the acceleration similarity ratio was 1. As the geometric similarity ratio was very small, the model structure would have very thin walls if the substructure of the shield building was reduced by a ratio of 1 : 50. Such a thin-walled model was not easy to make and would be unsafe in the shaking table tests. Therefore, following previous research [11], the size of the water tank in the upper part of the shield building was reduced by a ratio of 1 : 50. The substructure was simulated by a simplified model, the size of which was determined by the stiffness equivalence principle. The similarity relation of the model structure is described in Table 2.Table 2
Similarity relation of model structure.
Physical typePhysical quantitySimilarity relationSimilarity ratioGeometryLength LSL1 : 50MaterialElastic modulus ESE—Equivalent densityρSρ = SE/(SLSa)—DynamicAcceleration aSa1Time tSt=SL/Sa1 : 7.07Frequency fSf = 1/St7.07Modal analysis of the prototype structure of the shield building was carried out using ABAQUS, and the fundamental frequency was found to be 2.90 Hz. As listed in Table2, the frequency similarity ratio (ratio of frequency of model structure to that of prototype structure) was 7.07, so the fundamental frequency of the model structure should be approximately 20.50 Hz. The test model was designed based on this frequency. To satisfy the similarity relation, the lower structure consisted of support shaft, top plate, bottom plate, and steel support. The support shaft was made of stainless-steel pipe, and the top plate, bottom plate, and steel support were made of steel plate. To ensure the integrity of the model structure, these components were welded together. This paper focuses on the influence law of FSI effect on the seismic response of the shield building. The influence of the cooling water on the acceleration response and strain response of the structure and the seismic reduction effect can be determined through the shaking table tests. The determination of the model material and section size based on the stiffness equivalence principle satisfies the test requirements. Therefore, the material similarity relation is not the major indicator and is not considered in the design of the similarity relation.


### 2.3. Experimental Model and Sensor Arrangement

The material parameters for each part of the model structure are listed in Table3, and the specific sizes are shown in Figure 2. To observe the sloshing of water, the water tank was made of plexiglass [23]. To analyze the influence law of the cooling water on the structural seismic response, two experimental models were constructed. The water tank of model I was filled with water, whereas the water tank of model II was left empty.Table 3
Material parameters of experimental model.
ComponentMaterialMaterial propertiesDensity (kg/m3)Elastic modulus (GPa)Poisson’s ratioWater tankPlexiglass12003.10.3Support shaftStainless steel7800201.00.3Top plateSteel7800201.00.3Bottom plateSteel7800201.00.3Steel supportSteel7800201.00.3Figure 2
Experimental model and specific sizes. (a) Dimensions of water tank. (b) Water tank model. (c) Dimensions of lower structure. (d) Lower model.
(a)(b)(c)(d)Three kinds of sensors were used to record data during the shaking table tests. The sensors and the data acquisition system are shown in Figure3. The sensors used in the tests included 13 acceleration sensors, 20 strain sensors, and 3 pore water pressure sensors. The DH5921 test and analysis system was used for data acquisition. The arrangement of the sensors is shown in Figure 4, where “A” denotes the acceleration sensor, “S” denotes the strain sensor, and “P” denotes the pore water pressure sensor. One acceleration sensor was placed on the shaking table to record the output of the table. Six acceleration sensors and ten strain sensors were arranged along the height of each model to study the variations in the seismic response of the model structure. Three pore water pressure sensors were arranged in the water tank of model I, with P1 and P2 placed on the outer wall and P3 placed on the inner wall. Based on the hydrodynamic pressure data, the sloshing frequency of the water could be measured and the differences in hydrodynamic pressure at different positions in the water tank would be studied. The test system is shown in Figure 5.Figure 3
Sensors and data acquisition system. (a) Acceleration sensor (model: CYY-JSD01). (b) Strain sensor. (c) Pore water pressure sensor (model: CYY2). (d) Data acquisition system (model: DH5921).
(a)(b)(c)(d)Figure 4
Sensor arrangement (units: mm).Figure 5
Test system. (a) Front view. (b) Side view.
(a)(b)


### 2.4. Natural Frequency of Model Structure

Before the shaking table tests, dynamic tests were conducted to verify whether the models satisfied the requirements of the dynamic similarity relation. The model structure is similar to a single degree of freedom system, and so the fundamental frequency has the greatest influence on the seismic response of the structure and the first-order mode plays a controlling role. The dynamic similarity relation was verified by comparing the fundamental frequencies of the prototype and model structure. Through the hammer striking method, the acceleration time histories were recorded, and Fourier spectra were calculated (Figures6 and 7). The frequencies of model I and model II were measured to be 16.51 Hz and 19.89 Hz, respectively. The fundamental frequency of the prototype structure without water is 2.90 Hz, and the designed similarity ratio of the frequency is 7.07. Therefore, the frequency of designed model II should be 20.50 Hz. The relative error between the frequency of the designed model and that of the actual model is about 3.0%, which is acceptable.Figure 6
Frequency identification of model I. (a) Acceleration time history. (b) Fourier spectrum.
(a)(b)Figure 7
Frequency identification of model II. (a) Acceleration time history. (b) Fourier spectrum.
(a)(b)


### 2.5. Input Ground Motion and Loading Conditions

For the tests, one natural ground motion (El Centro) and one artificial ground motion were chosen as inputs. The El Centro ground motion was recorded at El Centro Station in the Ms6.7 Imperial Valley earthquake on May 18, 1940, in California, USA. The artificial ground motion was fitted by the AP1000 design response spectrum in the seismic code [24, 25]. The normalized acceleration time histories and acceleration response spectra are shown in Figure 8. From the acceleration response spectra, it can be seen that the frequency contents of two ground motions are abundant in the main frequency band of structure. Based on the time similarity ratio, the two ground motions were compressed and used as the inputs for the shaking table tests. The peak ground accelerations (PGAs) were 0.10 g, 0.15 g, 0.20 g, and 0.30 g. White noise with a PGA of 0.05 g was used as an input to detect changes in the structural frequency. The loading conditions are listed in Table 4.Figure 8
Acceleration time histories and response spectra of input ground motion. (a) El Centro ground motion. (b) Response spectrum of El Centro ground motion. (c) Artificial ground motion. (d) Response spectrum of artificial ground motion.
(a)(b)(c)(d)Table 4
Loading conditions of shaking table tests.
Test numberGround motionPGA (g)Test duration (s)1White noise0.0510.002Scaled El Centro ground motion0.107.543Scaled artificial ground motion0.1010.004White noise0.0510.005Scaled El Centro ground motion0.157.546Scaled artificial ground motion0.1510.007White noise0.0510.008Scaled El Centro ground motion0.207.549Scaled artificial ground motion0.2010.0010White noise0.0510.0011Scaled El Centro ground motion0.307.5412Scaled artificial ground motion0.3010.0013White noise0.0510.00


## 2.1. Test Equipment

The shaking table tests were carried out at Shandong Jianzhu University using a shaking table measuring 3 m × 3 m with a maximum bearing capacity of 15 t. The specific parameters of the shaking table are listed in Table1.Table 1
Specific parameters of shaking table.
Size (m)Effective bearing capacity (t)Maximum bearing capacity (t)Vibration directionMaximum acceleration (g)Maximum velocity (mm/s)Maximum displacement (mm)3.00 × 3.001015X, Y, ZX: ±1.5X: ±1000X: ±1256 DOFY: ±1.5Y: ±1000Y: ±125Z: ±1.0Z: ±310Z: ±125Note:X and Y represent the horizontal directions, Z represents the vertical direction, and DOF represents degrees of freedom.


## 2.2. Similarity Relation

The shaking table test model is a scaled structure based on the proportions of a prototype structure. The test model needs to be subjected to similar working conditions as the prototype. Therefore, the model structure and the prototype structure should obey a certain similarity relation. This similarity relation is reflected by the similarity ratios, which are the ratios of the main physical parameters of the model structure to those of the prototype structure. Considering the size and bearing capacity of the shaking table, the geometric similarity ratio of the model structure to the prototype was 1 : 50. The shaking table tests were carried out under gravity of 1 g, so the acceleration similarity ratio was 1. As the geometric similarity ratio was very small, the model structure would have very thin walls if the substructure of the shield building was reduced by a ratio of 1 : 50. Such a thin-walled model was not easy to make and would be unsafe in the shaking table tests. Therefore, following previous research [11], the size of the water tank in the upper part of the shield building was reduced by a ratio of 1 : 50. The substructure was simulated by a simplified model, the size of which was determined by the stiffness equivalence principle. The similarity relation of the model structure is described in Table 2.Table 2
Similarity relation of model structure.
Physical typePhysical quantitySimilarity relationSimilarity ratioGeometryLength LSL1 : 50MaterialElastic modulus ESE—Equivalent densityρSρ = SE/(SLSa)—DynamicAcceleration aSa1Time tSt=SL/Sa1 : 7.07Frequency fSf = 1/St7.07Modal analysis of the prototype structure of the shield building was carried out using ABAQUS, and the fundamental frequency was found to be 2.90 Hz. As listed in Table2, the frequency similarity ratio (ratio of frequency of model structure to that of prototype structure) was 7.07, so the fundamental frequency of the model structure should be approximately 20.50 Hz. The test model was designed based on this frequency. To satisfy the similarity relation, the lower structure consisted of support shaft, top plate, bottom plate, and steel support. The support shaft was made of stainless-steel pipe, and the top plate, bottom plate, and steel support were made of steel plate. To ensure the integrity of the model structure, these components were welded together. This paper focuses on the influence law of FSI effect on the seismic response of the shield building. The influence of the cooling water on the acceleration response and strain response of the structure and the seismic reduction effect can be determined through the shaking table tests. The determination of the model material and section size based on the stiffness equivalence principle satisfies the test requirements. Therefore, the material similarity relation is not the major indicator and is not considered in the design of the similarity relation.


## 2.3. Experimental Model and Sensor Arrangement

The material parameters for each part of the model structure are listed in Table3, and the specific sizes are shown in Figure 2. To observe the sloshing of water, the water tank was made of plexiglass [23]. To analyze the influence law of the cooling water on the structural seismic response, two experimental models were constructed. The water tank of model I was filled with water, whereas the water tank of model II was left empty.Table 3
Material parameters of experimental model.
ComponentMaterialMaterial propertiesDensity (kg/m3)Elastic modulus (GPa)Poisson’s ratioWater tankPlexiglass12003.10.3Support shaftStainless steel7800201.00.3Top plateSteel7800201.00.3Bottom plateSteel7800201.00.3Steel supportSteel7800201.00.3Figure 2
Experimental model and specific sizes. (a) Dimensions of water tank. (b) Water tank model. (c) Dimensions of lower structure. (d) Lower model.
(a)(b)(c)(d)Three kinds of sensors were used to record data during the shaking table tests. The sensors and the data acquisition system are shown in Figure3. The sensors used in the tests included 13 acceleration sensors, 20 strain sensors, and 3 pore water pressure sensors. The DH5921 test and analysis system was used for data acquisition. The arrangement of the sensors is shown in Figure 4, where “A” denotes the acceleration sensor, “S” denotes the strain sensor, and “P” denotes the pore water pressure sensor. One acceleration sensor was placed on the shaking table to record the output of the table. Six acceleration sensors and ten strain sensors were arranged along the height of each model to study the variations in the seismic response of the model structure. Three pore water pressure sensors were arranged in the water tank of model I, with P1 and P2 placed on the outer wall and P3 placed on the inner wall. Based on the hydrodynamic pressure data, the sloshing frequency of the water could be measured and the differences in hydrodynamic pressure at different positions in the water tank would be studied. The test system is shown in Figure 5.Figure 3
Sensors and data acquisition system. (a) Acceleration sensor (model: CYY-JSD01). (b) Strain sensor. (c) Pore water pressure sensor (model: CYY2). (d) Data acquisition system (model: DH5921).
(a)(b)(c)(d)Figure 4
Sensor arrangement (units: mm).Figure 5
Test system. (a) Front view. (b) Side view.
(a)(b)


## 2.4. Natural Frequency of Model Structure

Before the shaking table tests, dynamic tests were conducted to verify whether the models satisfied the requirements of the dynamic similarity relation. The model structure is similar to a single degree of freedom system, and so the fundamental frequency has the greatest influence on the seismic response of the structure and the first-order mode plays a controlling role. The dynamic similarity relation was verified by comparing the fundamental frequencies of the prototype and model structure. Through the hammer striking method, the acceleration time histories were recorded, and Fourier spectra were calculated (Figures6 and 7). The frequencies of model I and model II were measured to be 16.51 Hz and 19.89 Hz, respectively. The fundamental frequency of the prototype structure without water is 2.90 Hz, and the designed similarity ratio of the frequency is 7.07. Therefore, the frequency of designed model II should be 20.50 Hz. The relative error between the frequency of the designed model and that of the actual model is about 3.0%, which is acceptable.Figure 6
Frequency identification of model I. (a) Acceleration time history. (b) Fourier spectrum.
(a)(b)Figure 7
Frequency identification of model II. (a) Acceleration time history. (b) Fourier spectrum.
(a)(b)


## 2.5. Input Ground Motion and Loading Conditions

For the tests, one natural ground motion (El Centro) and one artificial ground motion were chosen as inputs. The El Centro ground motion was recorded at El Centro Station in the Ms6.7 Imperial Valley earthquake on May 18, 1940, in California, USA. The artificial ground motion was fitted by the AP1000 design response spectrum in the seismic code [24, 25]. The normalized acceleration time histories and acceleration response spectra are shown in Figure 8. From the acceleration response spectra, it can be seen that the frequency contents of two ground motions are abundant in the main frequency band of structure. Based on the time similarity ratio, the two ground motions were compressed and used as the inputs for the shaking table tests. The peak ground accelerations (PGAs) were 0.10 g, 0.15 g, 0.20 g, and 0.30 g. White noise with a PGA of 0.05 g was used as an input to detect changes in the structural frequency. The loading conditions are listed in Table 4.Figure 8
Acceleration time histories and response spectra of input ground motion. (a) El Centro ground motion. (b) Response spectrum of El Centro ground motion. (c) Artificial ground motion. (d) Response spectrum of artificial ground motion.
(a)(b)(c)(d)Table 4
Loading conditions of shaking table tests.
Test numberGround motionPGA (g)Test duration (s)1White noise0.0510.002Scaled El Centro ground motion0.107.543Scaled artificial ground motion0.1010.004White noise0.0510.005Scaled El Centro ground motion0.157.546Scaled artificial ground motion0.1510.007White noise0.0510.008Scaled El Centro ground motion0.207.549Scaled artificial ground motion0.2010.0010White noise0.0510.0011Scaled El Centro ground motion0.307.5412Scaled artificial ground motion0.3010.0013White noise0.0510.00


## 3. Test Results

### 3.1. Natural Frequency of Model Structure

Before starting the tests, the fundamental frequencies of model I and model II were found to be 16.51 Hz and 19.89 Hz by the hammer striking method. Through the white noise tests, the changes of fundamental frequencies are presented in Table5. There is no obvious change in the fundamental frequency of either model. After the input with the PGA of 0.30 g, the frequencies of two models decrease by 0.36% and 0.55%, respectively, which is negligible. The two models remain intact and there is no degradation in their structural stiffness.Table 5
Loading conditions of shaking table test.
Test numberModel I (with water)Model II (without water)Frequency (Hz)Frequency variation (%)Frequency (Hz)Frequency variation (%)116.51—19.89—416.49−0.1219.88−0.05716.49−0.1219.85−0.201016.47−0.2419.81−0.401316.45−0.3619.78−0.55


### 3.2. Acceleration Response of Model Structure

Based on the acceleration time history records measured in the shaking table tests, the acceleration amplification factor (defined as the ratio of the peak acceleration response of the structure to the PGA of the input) is used to study the acceleration response law. The acceleration amplification factor curves of the two models with height are shown in Figure9.Figure 9
Acceleration amplification factor curves under different inputs. (a) El-M1. (b) El-M2. (c) AG-M1. (d) AG-M2 (El: scaled El Centro ground motion; AG: scaled artificial ground motion; M1: model I; and M2: model II).
(a)(b)(c)(d)It can be seen that the acceleration amplification factors gradually increase from the bottom of the support shaft to the top of the water tank. As the input ground motion PGA increases, the acceleration amplification factors have no obvious changes. It is similar to the change rule of model structure frequency, and the reason is still that the stiffness of the structure does not decrease significantly.To study the influence of water on the seismic response of the model structure, the acceleration amplification factors of the two models are compared in Figure10. One phenomenon is that the acceleration responses of the two models are greater under the scaled artificial ground motion than those under the scaled El Centro ground motion. The reason is that the model structure is similar to a single degree of freedom system, and the seismic response is affected by the acceleration response spectrum of the input. From Figure 8, it can be seen that the response spectrum value of the artificial ground motion is greater than that of the El Centro ground motion in the main frequency band of the structure. Therefore, the acceleration responses of the model structure under scaled artificial ground motion are larger. Another phenomenon is that the acceleration responses of model I are smaller than those of model II, and the reduction ratio increases with model height. The reason is that the presence of water reduces the acceleration of the model structure.Figure 10
Comparison of acceleration amplification factors. (a) PGA of 0.10 g. (b) PGA of 0.15 g. (c) PGA of 0.20 g. (d) PGA of 0.30 g.
(a)(b)(c)(d)To study the acceleration reduction ratio quantitatively, the acceleration amplification factors of the measuring point at the top of the water tank are listed in Table6. The acceleration reduction ratio is about 5–20%, and the spectral characteristics of the input ground motions have no decisive effect on the value. As the PGA of the input ground motion increases, the acceleration reduction ratio tends to decrease overall.Table 6
Comparison of acceleration amplification factors.
PGA (g)Ground motionAcceleration amplification factorAcceleration reduction ratio (%)Model-IModel-II0.10Scaled El ground motion2.5963.204−18.98Scaled artificial ground motion3.0883.741−17.460.15Scaled El ground motion2.8083.320−15.42Scaled artificial ground motion3.2503.638−10.670.20Scaled El ground motion2.8783.099−7.13Scaled artificial ground motion3.3823.636−6.990.30Scaled El ground motion2.6593.034−12.36Scaled artificial ground motion3.1283.512−10.93


### 3.3. Strain Response of Model Structure

The peak strain curves of the two models with height are shown in Figure11. The peak strains increase with the increasing PGA of input ground motion. The strain is largest at the bottom of the model structure and smallest at the top. The reason is that the shear force and bending moment are highest at the bottom of the structure and gradually decrease with the increase of height. Under the scaled artificial ground motions, the peak strains of the model structure are larger than those under the El Centro ground motion. This law is similar to that for the acceleration, and the reason is also the same. Overall, the curves of peak strain are approximately linear. Even under the 0.30 g input, the strains of model structure are still small.Figure 11
Peak strain curves under different inputs. (a) El-M1. (b) El-M2. (c) AG-M1. (d) AG-M2.
(a)(b)(c)(d)Based on the structural strains under the 0.10 g input, the influence law of increasing PGA on the structural strain is now quantitatively studied. The strain variation factor (ratio of strain under 0.15 g, 0.20 g, or 0.30 g input to that under 0.10 g input) is introduced to study the variation law. Taking the scaled El Centro ground motion as an example, the peak strains and strain variation factors (values in brackets) are listed in Table7. Compared with the strains under the 0.10 g input, the peak strains increase by about 1.5, 2.0, and 3.0 times, respectively, under the 0.15 g, 0.20 g, and 0.30 g inputs. The strains show linear growth. It can be seen that the support shaft is always in the elastic state during the shaking table tests. The reason is that the stiffness of model structure is very large, and the strain is relatively small.Table 7
Peak strain and strain variation factor under scaled El Centro ground motion.
ModelHeight of measuring point (m)Peak strain (με) (strain variation factor)0.10 g0.15 g0.20 g0.30 gModel I0.1820.7131.61 (1.53)41.33 (2.00)61.28 (2.96)0.3915.6123.27 (1.49)31.49 (2.02)48.10 (3.08)0.6012.6318.84 (1.49)26.61 (2.11)37.14 (2.94)0.818.9213.35 (1.50)18.56 (2.08)26.87 (3.01)1.024.757.21 (1.52)9.37 (1.97)14.59 (3.07)Model II0.1821.6733.31 (1.54)44.86 (2.07)68.23 (3.15)0.3916.0524.24 (1.51)34.94 (2.18)52.82 (3.29)0.6012.8119.72 (1.54)28.77 (2.25)39.46 (3.08)0.818.7113.26 (1.52)20.48 (2.35)27.34 (3.14)1.024.737.36 (1.56)9.50 (2.01)15.27 (3.23)The peak strains in model I and model II are compared in Figure12 and Table 8. The strains of model I are slightly smaller than those of model II. The strain differences are obvious at the middle and lower parts of the structures, but those at the top are small. The presence of water in the tank reduces the strain responses of the model structure, and the strain reduction ratio is about 5–10% at the bottom of the structure.Figure 12
Peak strain curves under different inputs. (a) PGA of 0.10 g. (b) PGA of 0.15 g. (c) PGA of 0.20 g. (d) PGA of 0.30 g.
(a)(b)(c)(d)Table 8
Peak strain at the bottom and strain reduction ratio.
PGA (g)Ground motionPeak strain (με)Strain reduction ratio (%)Model IModel II0.10Scaled El ground motion20.7121.67−4.43Scaled artificial ground motion39.1941.26−5.020.15Scaled El ground motion31.6133.31−5.10Scaled artificial ground motion60.3567.64−10.780.20Scaled El ground motion41.3344.86−7.87Scaled artificial ground motion79.2389.44−11.420.30Scaled El ground motion61.2868.23−10.19Scaled artificial ground motion129.94148.09−12.26


### 3.4. Hydrodynamic Pressure

#### 3.4.1. Sloshing Frequency of Water

A time history of hydrodynamic pressure under the scaled El Centro ground motion is shown in Figure13(a). During the period of external excitation, the hydrodynamic pressure is greatly affected by the input ground motion, and this part of hydrodynamic pressure data cannot reflect the sloshing characteristics of the water. After the external excitation has ended, the water in the tank continues sloshing. At this time, the recorded hydrodynamic pressure only reflects the sloshing characteristics of the water in the tank, with no additional external excitation. As shown in Figure 13(b), the sloshing frequency of the water can be identified by taking a fast Fourier transform (FFT) of the hydrodynamic pressure data in the free sloshing stage.Figure 13
Data processing of hydrodynamic pressure record. (a) Time history of hydrodynamic pressure. (b) Fourier spectrum.
(a)(b)Twenty time histories of hydrodynamic pressure were selected for sloshing frequency identification, and the average value of the first-order sloshing frequency was found to be 0.879 Hz. In the previous study [14], the first-order sloshing frequency of a prototype shield building was found to be 0.126 Hz. The designed similarity ratio of frequency is 7.07, which suggests that the first-order sloshing frequency of the designed model should be 0.891 Hz. The relative error between the sloshing frequency of the designed model and that of the actual model is about −1.3%, which is acceptable. Considering the dynamic characteristics of the model structure and the water sloshing comprehensively, the test model has a reasonable design.


#### 3.4.2. Peak Hydrodynamic Pressure

The peak hydrodynamic pressures at the three measuring points (shown in Figure4) are listed in Table 9. The peak hydrodynamic pressure gradually increases with increasing PGA. The water depth at the outer wall and the violent sloshing of water near the free surface mean that P1 encounters the largest hydrodynamic pressure. As the water depth increases, the hydrodynamic pressure gradually decreases.Table 9
Statistics of peak hydrodynamic pressure.
PGA (g)Ground motionPeak hydrodynamic pressure (kPa)P1P2P30.10Scaled El ground motion0.270.080.07Scaled artificial ground motion0.340.140.120.15Scaled El ground motion0.430.170.09Scaled artificial ground motion0.560.220.150.20Scaled El ground motion0.610.310.13Scaled artificial ground motion0.720.370.210.30Scaled El ground motion0.840.390.19Scaled artificial ground motion0.980.430.28Because the input ground motions are compressed by a ratio of 1 : 7.07 and the stiffness of model structure is large, the low-frequency sloshing of water is not especially violent and the hydrodynamic pressures are not particularly high. Different from the approximate linear growth of the strain response, the peak hydrodynamic pressure does not increase in proportion to the PGA. The reason is that the model structure is in the elastic state during the tests, but the sloshing of water is highly nonlinear [26, 27].


## 3.1. Natural Frequency of Model Structure

Before starting the tests, the fundamental frequencies of model I and model II were found to be 16.51 Hz and 19.89 Hz by the hammer striking method. Through the white noise tests, the changes of fundamental frequencies are presented in Table5. There is no obvious change in the fundamental frequency of either model. After the input with the PGA of 0.30 g, the frequencies of two models decrease by 0.36% and 0.55%, respectively, which is negligible. The two models remain intact and there is no degradation in their structural stiffness.Table 5
Loading conditions of shaking table test.
Test numberModel I (with water)Model II (without water)Frequency (Hz)Frequency variation (%)Frequency (Hz)Frequency variation (%)116.51—19.89—416.49−0.1219.88−0.05716.49−0.1219.85−0.201016.47−0.2419.81−0.401316.45−0.3619.78−0.55


## 3.2. Acceleration Response of Model Structure

Based on the acceleration time history records measured in the shaking table tests, the acceleration amplification factor (defined as the ratio of the peak acceleration response of the structure to the PGA of the input) is used to study the acceleration response law. The acceleration amplification factor curves of the two models with height are shown in Figure9.Figure 9
Acceleration amplification factor curves under different inputs. (a) El-M1. (b) El-M2. (c) AG-M1. (d) AG-M2 (El: scaled El Centro ground motion; AG: scaled artificial ground motion; M1: model I; and M2: model II).
(a)(b)(c)(d)It can be seen that the acceleration amplification factors gradually increase from the bottom of the support shaft to the top of the water tank. As the input ground motion PGA increases, the acceleration amplification factors have no obvious changes. It is similar to the change rule of model structure frequency, and the reason is still that the stiffness of the structure does not decrease significantly.To study the influence of water on the seismic response of the model structure, the acceleration amplification factors of the two models are compared in Figure10. One phenomenon is that the acceleration responses of the two models are greater under the scaled artificial ground motion than those under the scaled El Centro ground motion. The reason is that the model structure is similar to a single degree of freedom system, and the seismic response is affected by the acceleration response spectrum of the input. From Figure 8, it can be seen that the response spectrum value of the artificial ground motion is greater than that of the El Centro ground motion in the main frequency band of the structure. Therefore, the acceleration responses of the model structure under scaled artificial ground motion are larger. Another phenomenon is that the acceleration responses of model I are smaller than those of model II, and the reduction ratio increases with model height. The reason is that the presence of water reduces the acceleration of the model structure.Figure 10
Comparison of acceleration amplification factors. (a) PGA of 0.10 g. (b) PGA of 0.15 g. (c) PGA of 0.20 g. (d) PGA of 0.30 g.
(a)(b)(c)(d)To study the acceleration reduction ratio quantitatively, the acceleration amplification factors of the measuring point at the top of the water tank are listed in Table6. The acceleration reduction ratio is about 5–20%, and the spectral characteristics of the input ground motions have no decisive effect on the value. As the PGA of the input ground motion increases, the acceleration reduction ratio tends to decrease overall.Table 6
Comparison of acceleration amplification factors.
PGA (g)Ground motionAcceleration amplification factorAcceleration reduction ratio (%)Model-IModel-II0.10Scaled El ground motion2.5963.204−18.98Scaled artificial ground motion3.0883.741−17.460.15Scaled El ground motion2.8083.320−15.42Scaled artificial ground motion3.2503.638−10.670.20Scaled El ground motion2.8783.099−7.13Scaled artificial ground motion3.3823.636−6.990.30Scaled El ground motion2.6593.034−12.36Scaled artificial ground motion3.1283.512−10.93


## 3.3. Strain Response of Model Structure

The peak strain curves of the two models with height are shown in Figure11. The peak strains increase with the increasing PGA of input ground motion. The strain is largest at the bottom of the model structure and smallest at the top. The reason is that the shear force and bending moment are highest at the bottom of the structure and gradually decrease with the increase of height. Under the scaled artificial ground motions, the peak strains of the model structure are larger than those under the El Centro ground motion. This law is similar to that for the acceleration, and the reason is also the same. Overall, the curves of peak strain are approximately linear. Even under the 0.30 g input, the strains of model structure are still small.Figure 11
Peak strain curves under different inputs. (a) El-M1. (b) El-M2. (c) AG-M1. (d) AG-M2.
(a)(b)(c)(d)Based on the structural strains under the 0.10 g input, the influence law of increasing PGA on the structural strain is now quantitatively studied. The strain variation factor (ratio of strain under 0.15 g, 0.20 g, or 0.30 g input to that under 0.10 g input) is introduced to study the variation law. Taking the scaled El Centro ground motion as an example, the peak strains and strain variation factors (values in brackets) are listed in Table7. Compared with the strains under the 0.10 g input, the peak strains increase by about 1.5, 2.0, and 3.0 times, respectively, under the 0.15 g, 0.20 g, and 0.30 g inputs. The strains show linear growth. It can be seen that the support shaft is always in the elastic state during the shaking table tests. The reason is that the stiffness of model structure is very large, and the strain is relatively small.Table 7
Peak strain and strain variation factor under scaled El Centro ground motion.
ModelHeight of measuring point (m)Peak strain (με) (strain variation factor)0.10 g0.15 g0.20 g0.30 gModel I0.1820.7131.61 (1.53)41.33 (2.00)61.28 (2.96)0.3915.6123.27 (1.49)31.49 (2.02)48.10 (3.08)0.6012.6318.84 (1.49)26.61 (2.11)37.14 (2.94)0.818.9213.35 (1.50)18.56 (2.08)26.87 (3.01)1.024.757.21 (1.52)9.37 (1.97)14.59 (3.07)Model II0.1821.6733.31 (1.54)44.86 (2.07)68.23 (3.15)0.3916.0524.24 (1.51)34.94 (2.18)52.82 (3.29)0.6012.8119.72 (1.54)28.77 (2.25)39.46 (3.08)0.818.7113.26 (1.52)20.48 (2.35)27.34 (3.14)1.024.737.36 (1.56)9.50 (2.01)15.27 (3.23)The peak strains in model I and model II are compared in Figure12 and Table 8. The strains of model I are slightly smaller than those of model II. The strain differences are obvious at the middle and lower parts of the structures, but those at the top are small. The presence of water in the tank reduces the strain responses of the model structure, and the strain reduction ratio is about 5–10% at the bottom of the structure.Figure 12
Peak strain curves under different inputs. (a) PGA of 0.10 g. (b) PGA of 0.15 g. (c) PGA of 0.20 g. (d) PGA of 0.30 g.
(a)(b)(c)(d)Table 8
Peak strain at the bottom and strain reduction ratio.
PGA (g)Ground motionPeak strain (με)Strain reduction ratio (%)Model IModel II0.10Scaled El ground motion20.7121.67−4.43Scaled artificial ground motion39.1941.26−5.020.15Scaled El ground motion31.6133.31−5.10Scaled artificial ground motion60.3567.64−10.780.20Scaled El ground motion41.3344.86−7.87Scaled artificial ground motion79.2389.44−11.420.30Scaled El ground motion61.2868.23−10.19Scaled artificial ground motion129.94148.09−12.26


## 3.4. Hydrodynamic Pressure

### 3.4.1. Sloshing Frequency of Water

A time history of hydrodynamic pressure under the scaled El Centro ground motion is shown in Figure13(a). During the period of external excitation, the hydrodynamic pressure is greatly affected by the input ground motion, and this part of hydrodynamic pressure data cannot reflect the sloshing characteristics of the water. After the external excitation has ended, the water in the tank continues sloshing. At this time, the recorded hydrodynamic pressure only reflects the sloshing characteristics of the water in the tank, with no additional external excitation. As shown in Figure 13(b), the sloshing frequency of the water can be identified by taking a fast Fourier transform (FFT) of the hydrodynamic pressure data in the free sloshing stage.Figure 13
Data processing of hydrodynamic pressure record. (a) Time history of hydrodynamic pressure. (b) Fourier spectrum.
(a)(b)Twenty time histories of hydrodynamic pressure were selected for sloshing frequency identification, and the average value of the first-order sloshing frequency was found to be 0.879 Hz. In the previous study [14], the first-order sloshing frequency of a prototype shield building was found to be 0.126 Hz. The designed similarity ratio of frequency is 7.07, which suggests that the first-order sloshing frequency of the designed model should be 0.891 Hz. The relative error between the sloshing frequency of the designed model and that of the actual model is about −1.3%, which is acceptable. Considering the dynamic characteristics of the model structure and the water sloshing comprehensively, the test model has a reasonable design.


### 3.4.2. Peak Hydrodynamic Pressure

The peak hydrodynamic pressures at the three measuring points (shown in Figure4) are listed in Table 9. The peak hydrodynamic pressure gradually increases with increasing PGA. The water depth at the outer wall and the violent sloshing of water near the free surface mean that P1 encounters the largest hydrodynamic pressure. As the water depth increases, the hydrodynamic pressure gradually decreases.Table 9
Statistics of peak hydrodynamic pressure.
PGA (g)Ground motionPeak hydrodynamic pressure (kPa)P1P2P30.10Scaled El ground motion0.270.080.07Scaled artificial ground motion0.340.140.120.15Scaled El ground motion0.430.170.09Scaled artificial ground motion0.560.220.150.20Scaled El ground motion0.610.310.13Scaled artificial ground motion0.720.370.210.30Scaled El ground motion0.840.390.19Scaled artificial ground motion0.980.430.28Because the input ground motions are compressed by a ratio of 1 : 7.07 and the stiffness of model structure is large, the low-frequency sloshing of water is not especially violent and the hydrodynamic pressures are not particularly high. Different from the approximate linear growth of the strain response, the peak hydrodynamic pressure does not increase in proportion to the PGA. The reason is that the model structure is in the elastic state during the tests, but the sloshing of water is highly nonlinear [26, 27].


## 3.4.1. Sloshing Frequency of Water

A time history of hydrodynamic pressure under the scaled El Centro ground motion is shown in Figure13(a). During the period of external excitation, the hydrodynamic pressure is greatly affected by the input ground motion, and this part of hydrodynamic pressure data cannot reflect the sloshing characteristics of the water. After the external excitation has ended, the water in the tank continues sloshing. At this time, the recorded hydrodynamic pressure only reflects the sloshing characteristics of the water in the tank, with no additional external excitation. As shown in Figure 13(b), the sloshing frequency of the water can be identified by taking a fast Fourier transform (FFT) of the hydrodynamic pressure data in the free sloshing stage.Figure 13
Data processing of hydrodynamic pressure record. (a) Time history of hydrodynamic pressure. (b) Fourier spectrum.
(a)(b)Twenty time histories of hydrodynamic pressure were selected for sloshing frequency identification, and the average value of the first-order sloshing frequency was found to be 0.879 Hz. In the previous study [14], the first-order sloshing frequency of a prototype shield building was found to be 0.126 Hz. The designed similarity ratio of frequency is 7.07, which suggests that the first-order sloshing frequency of the designed model should be 0.891 Hz. The relative error between the sloshing frequency of the designed model and that of the actual model is about −1.3%, which is acceptable. Considering the dynamic characteristics of the model structure and the water sloshing comprehensively, the test model has a reasonable design.


## 3.4.2. Peak Hydrodynamic Pressure

The peak hydrodynamic pressures at the three measuring points (shown in Figure4) are listed in Table 9. The peak hydrodynamic pressure gradually increases with increasing PGA. The water depth at the outer wall and the violent sloshing of water near the free surface mean that P1 encounters the largest hydrodynamic pressure. As the water depth increases, the hydrodynamic pressure gradually decreases.Table 9
Statistics of peak hydrodynamic pressure.
PGA (g)Ground motionPeak hydrodynamic pressure (kPa)P1P2P30.10Scaled El ground motion0.270.080.07Scaled artificial ground motion0.340.140.120.15Scaled El ground motion0.430.170.09Scaled artificial ground motion0.560.220.150.20Scaled El ground motion0.610.310.13Scaled artificial ground motion0.720.370.210.30Scaled El ground motion0.840.390.19Scaled artificial ground motion0.980.430.28Because the input ground motions are compressed by a ratio of 1 : 7.07 and the stiffness of model structure is large, the low-frequency sloshing of water is not especially violent and the hydrodynamic pressures are not particularly high. Different from the approximate linear growth of the strain response, the peak hydrodynamic pressure does not increase in proportion to the PGA. The reason is that the model structure is in the elastic state during the tests, but the sloshing of water is highly nonlinear [26, 27].


## 4. Numerical Analysis

### 4.1. Numerical Model

The numerical model for analysis was established using the ABAQUS finite element software and shown in Figure14(a). The ideal elastoplastic model was used to simulate the steel and stainless steel, and the damage model was used to simulate the plexiglass. For the simulation of sloshing water, one simplified model was proposed in the previous study [14]. This simplified model is based on the Housner model and allows the water mass to be split into impulsive mass and convective mass components [16,17]. As shown in Figure 14(b), the impulsive mass m0 is assumed to be rigidly attached to the container walls. The convective mass is split into a series of masses m1, m2, …, mn associated with the 1st, 2nd, …, nth sloshing masses, respectively. The sloshing masses other than m1 constitute a very small proportion. Therefore, only the 1st sloshing mass is considered in the simplified model. The horizontal components of m1 are connected to the tank walls by spring-dampers, and the vertical components of m1 are added to the bottom of the water tank discretely. This simplified method was used to simulate the sloshing of water in the tank.Figure 14
Numerical model of the test model. (a) Numerical model. (b) FSI model. (c) Simplified model.
(a)(b)(c)


### 4.2. Modal Analysis

Modal analysis of test model I (the model with water) was performed, and the first-order mode is shown in Figure15. The fundamental frequencies of the test model and the numerical model are listed in Table 10. The fundamental frequency of the numerical model is 16.37 Hz, and the relative error is −0.85% compared with the test model. This demonstrates that the numerical model is reasonable.Figure 15
First-order mode of model I.Table 10
Comparison of fundamental frequency between test and simulation.
Model numberFundamental frequency of test (Hz)Fundamental frequency of simulation (Hz)Relative error (%)Model I16.5116.37−0.85%


### 4.3. Acceleration Response and Comparison

According to the test data, the test model remained in the elastic state during the tests. As an example, the scaled El Centro ground motion with a PGA of 0.30 g was used as the input for time history analysis. Two measuring points (A8 at the top of the model and A5 at the middle of the model) were chosen to study the seismic response of the test model. The acceleration time histories and Fourier spectra are compared in Figures16 and 17. It can be seen that the numerical simulation results are in good agreement with the test results.Figure 16
Comparison of acceleration of A8 under 0.30 g scaled El Centro ground motion. (a) Acceleration time history. (b) Fourier spectrum.
(a)(b)Figure 17
Comparison of acceleration of A5 under 0.30 g scaled El Centro ground motion. (a) Acceleration time history. (b) Fourier spectrum.
(a)(b)To study the relative error between the test results and the simulation results quantitatively, the peak accelerations at points A4, A5, A6, A7, and A8 on the model structure under the scaled El Centro ground motion with the PGA of 0.30 g are listed in Table11. The relative errors in peak acceleration are also calculated. It can be seen that the peak accelerations of the simulations are slightly smaller than those of the tests. The relative errors in peak accelerations are about 5%, which are acceptable.Table 11
Comparison of peak acceleration between test and simulation.
PGA (g)Ground motionAcceleration numberPeak acceleration of test (m/s2)Peak acceleration of simulation (m/s2)Relative error (%)0.30Scaled El Centro ground motionA43.012.91−3.07A53.733.54−5.03A65.224.98−4.52A76.366.07−4.51A87.827.35−5.98


### 4.4. Strain Response and Comparison

The strain time histories at points S6, S7, S8, and S9 on the support shaft are compared in Figure18. The shape and peak strain of time history curve given by the simulation are similar to those in the test. Similar to the comparison of acceleration, the relative errors between the tests and the simulations are calculated, and the results are listed in Table 12.Figure 18
Comparison of strain under 0.30 g scaled El Centro ground motion. (a) Strain time history of S6. (b) Strain time history of S7. (c) Strain time history of S8. (d) Strain time history of S9.
(a)(b)(c)(d)Table 12
Comparison of peak strain between test and simulation.
PGA (g)Ground motionAcceleration numberPeak strain of test (με)Peak strain of simulation (με)Relative error (%)0.30Scaled El Centro ground motionS661.2857.64−5.94S748.1045.17−6.09S837.1435.02−5.71S926.8724.94−7.18It can be seen that the peak strains of the simulations are smaller than those of the tests, and the relative errors are about 5–10%. This is similar to the law in the acceleration response, where the numerical simulation results are still smaller. One reason for this is that the convective mass of water is concentrated in the upper part of the water body in the simplified model, and this assumption underestimates the influence of water sloshing on the seismic response of the structure. Another reason is that the fundamental frequency of the numerical model is slightly different from that of the test model. The frequency of the model is not completely consistent, which will also lead to differences in the seismic response.Considering the comparisons of acceleration and strain comprehensively, the maximum error is less than 10% and most errors are concentrated near 5%. Thus, the simplified model proposed in the previous study can be used to simulate the FSI effect of the water tank, as the simulation results are in good agreement with the test results. The numerical simulation results are reasonable and feasible.


## 4.1. Numerical Model

The numerical model for analysis was established using the ABAQUS finite element software and shown in Figure14(a). The ideal elastoplastic model was used to simulate the steel and stainless steel, and the damage model was used to simulate the plexiglass. For the simulation of sloshing water, one simplified model was proposed in the previous study [14]. This simplified model is based on the Housner model and allows the water mass to be split into impulsive mass and convective mass components [16,17]. As shown in Figure 14(b), the impulsive mass m0 is assumed to be rigidly attached to the container walls. The convective mass is split into a series of masses m1, m2, …, mn associated with the 1st, 2nd, …, nth sloshing masses, respectively. The sloshing masses other than m1 constitute a very small proportion. Therefore, only the 1st sloshing mass is considered in the simplified model. The horizontal components of m1 are connected to the tank walls by spring-dampers, and the vertical components of m1 are added to the bottom of the water tank discretely. This simplified method was used to simulate the sloshing of water in the tank.Figure 14
Numerical model of the test model. (a) Numerical model. (b) FSI model. (c) Simplified model.
(a)(b)(c)


## 4.2. Modal Analysis

Modal analysis of test model I (the model with water) was performed, and the first-order mode is shown in Figure15. The fundamental frequencies of the test model and the numerical model are listed in Table 10. The fundamental frequency of the numerical model is 16.37 Hz, and the relative error is −0.85% compared with the test model. This demonstrates that the numerical model is reasonable.Figure 15
First-order mode of model I.Table 10
Comparison of fundamental frequency between test and simulation.
Model numberFundamental frequency of test (Hz)Fundamental frequency of simulation (Hz)Relative error (%)Model I16.5116.37−0.85%


## 4.3. Acceleration Response and Comparison

According to the test data, the test model remained in the elastic state during the tests. As an example, the scaled El Centro ground motion with a PGA of 0.30 g was used as the input for time history analysis. Two measuring points (A8 at the top of the model and A5 at the middle of the model) were chosen to study the seismic response of the test model. The acceleration time histories and Fourier spectra are compared in Figures16 and 17. It can be seen that the numerical simulation results are in good agreement with the test results.Figure 16
Comparison of acceleration of A8 under 0.30 g scaled El Centro ground motion. (a) Acceleration time history. (b) Fourier spectrum.
(a)(b)Figure 17
Comparison of acceleration of A5 under 0.30 g scaled El Centro ground motion. (a) Acceleration time history. (b) Fourier spectrum.
(a)(b)To study the relative error between the test results and the simulation results quantitatively, the peak accelerations at points A4, A5, A6, A7, and A8 on the model structure under the scaled El Centro ground motion with the PGA of 0.30 g are listed in Table11. The relative errors in peak acceleration are also calculated. It can be seen that the peak accelerations of the simulations are slightly smaller than those of the tests. The relative errors in peak accelerations are about 5%, which are acceptable.Table 11
Comparison of peak acceleration between test and simulation.
PGA (g)Ground motionAcceleration numberPeak acceleration of test (m/s2)Peak acceleration of simulation (m/s2)Relative error (%)0.30Scaled El Centro ground motionA43.012.91−3.07A53.733.54−5.03A65.224.98−4.52A76.366.07−4.51A87.827.35−5.98


## 4.4. Strain Response and Comparison

The strain time histories at points S6, S7, S8, and S9 on the support shaft are compared in Figure18. The shape and peak strain of time history curve given by the simulation are similar to those in the test. Similar to the comparison of acceleration, the relative errors between the tests and the simulations are calculated, and the results are listed in Table 12.Figure 18
Comparison of strain under 0.30 g scaled El Centro ground motion. (a) Strain time history of S6. (b) Strain time history of S7. (c) Strain time history of S8. (d) Strain time history of S9.
(a)(b)(c)(d)Table 12
Comparison of peak strain between test and simulation.
PGA (g)Ground motionAcceleration numberPeak strain of test (με)Peak strain of simulation (με)Relative error (%)0.30Scaled El Centro ground motionS661.2857.64−5.94S748.1045.17−6.09S837.1435.02−5.71S926.8724.94−7.18It can be seen that the peak strains of the simulations are smaller than those of the tests, and the relative errors are about 5–10%. This is similar to the law in the acceleration response, where the numerical simulation results are still smaller. One reason for this is that the convective mass of water is concentrated in the upper part of the water body in the simplified model, and this assumption underestimates the influence of water sloshing on the seismic response of the structure. Another reason is that the fundamental frequency of the numerical model is slightly different from that of the test model. The frequency of the model is not completely consistent, which will also lead to differences in the seismic response.Considering the comparisons of acceleration and strain comprehensively, the maximum error is less than 10% and most errors are concentrated near 5%. Thus, the simplified model proposed in the previous study can be used to simulate the FSI effect of the water tank, as the simulation results are in good agreement with the test results. The numerical simulation results are reasonable and feasible.


## 5. Conclusions

The AP1000 shield building was taken as the research object in this study. Two simplified models of the shield building (one with a tank containing water and one with an empty tank) were designed based on similarity theory. A series of shaking table tests were conducted to study the seismic responses of the model structures and evaluate the influence of water in the tank. Modal analysis and time history analysis of the test model were then performed. The following conclusions have been obtained:(1)
The model structure is similar to a single degree of freedom system, and the seismic response is affected by the acceleration response spectrum of the input ground motion. The seismic responses of the model structure under scaled artificial ground motion are larger than those under scaled El Centro ground motion.(2)
The presence of water in the tank and the sloshing of the water can reduce the natural frequency and seismic response of the model structure. The gravity cooling water tank has a certain seismic reduction effect. The acceleration reduction ratio is about 5–20%, and the strain reduction ratio is about 5–10%.(3)
The simplified model of water sloshing proposed in the previous study can be used for seismic response analysis of the shield building, and the numerical results are reasonable and feasible.(4)
The limitation of the simplified model means that the simulation results are slightly smaller than the test results. Using an FSI model would improve the calculation accuracy but may also lead to lower computational efficiency. The calculation accuracy of the simplified model satisfies the requirements of general engineering projects.



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*Source: 6458549-2022-09-02.xml*