Keywords

1 Introduction

The safe design of nuclear power plant structure and reasonable safety assessment provide important guarantee for the high efficient development of nuclear energy. In the seismic design of nuclear power structures, the nuclear building is usually buried below the ground surface in order to obtain higher seismic performance [1]. However, the seismic analysis of the building are usually performed without considering embedment effect for simplification. In this case, the support effect from the lateral backfill soil on the structure is ignored. Also, the stiffness of the whole SSI system is therefore reduced which further affect the dynamic characteristic of the system and its response. Additionally, the ground motion for seismic design is usually inputted at the bottom of the baseslab. The ground motion in a certain depth in the site is normally smaller than the motion on the ground surface. In other words, the assumption without considering embedment effect leads to a smaller seismic excitation. Hence, the results obtained from analysis without embedment tend to be conservative [2].

However, non-bedrock sites or soil sites are becoming the potential construction sites for nuclear power plant (NPP) in recent year due to the rapid development of nuclear power and scarce hard rock sites. Therefore, the soil-structure interaction (SSI) gradually draw the attention of nuclear structural engineers. For a soft site, non-uniform distribution of the foundation bearing capacity and the dynamic soil pressure caused by the embedment effect will affect the seismic response of the superstructure. Hence, the embedment effect shall be considered in the seismic design and seismic in-structure response spectra (ISRS) analysis for a nuclear building.

Many scholars have summarized the research on SSI effect [3]. Two widely-used methods are: direct method and sub-structuring method. The sub-structuring method is a frequency domain solution method. Its basic principle is to decompose the soil-structure interaction system into three parts: the super structure, the foundation and the excavated soil, and solve respectively. The load acting on the structure is obtained under the condition of displacement coordination at the interface between soil and structure. Compared with the direct method, the sub-structuring method has less degree of freedom due to that the soil layer is assumed to be an analytical model. In addition, the damping adopted by the sub-structuring method remains unchanged in the entire frequency range and is more stable. The sub-structuring method [4] is based on the superposition principle and is suitable only for linear analysis systems. Therefore, sub-structuring method is widely used in the field of nuclear engineering because the nuclear structure is required to maintain a linear state under the safe shutdown earthquake.

The ACS SASSI software used in this study is based on the sub-structuring method. The sensitivity analysis is conducted for two key parameters in the embedded effect analysis such as: the selection principle of the interaction node and the radius of the cylindrical central area. The conclusions provided technical reference for the seismic analysis and ISRS calculation of the deeply embedded nuclear buildings.

2 Analysis Model

2.1 Finite Element Model of the Superstructure

To avoid the influence from the adjacent buildings on the analysis results, a separate reactor building (including the internal structure and containment) in a Small Modular Reactor was selected as the analysis object in this analysis. The key information of the model of the reactor building is shown in Table 1, and the finite element model established in ANSYS is shown in Fig. 1.

In the model, the embedded part of the structure and the excavated volume are modeled separately but have common nodes at their interface. The characteristics of the excavated volume are consistent with the surrounding soil. And the vertical height of the element of the backfill soil is same to the thickness of the soil.

Table 1. Key information of the finite element model
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2.2 Parameters of the Site

According to the geological survey report, the target site in this study is a kind of soft soil site. The building is located on silty clay. The detailed parameters of silty clay are shown in Table 2.

Table 2. Key information of the finite element model
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2.3 Seismic Input

According to the safety assessment report of the site, the standard spectra RG1.60 is adopted in the sensitivity analysis. Although the target site of this paper is soft soil, the site amplification effect is not the focus of this study. Therefore, the input position of the RG1.60 spectra is assumed to be at the bottom of the building.

A single group of seismic acceleration time histories(shown in Fig. 2(a) –c)) compatible to RG1.60 spectra are used in ACS SASSI. The duration of the time history is 40 s and the time interval is 0.005 s. Both the peak ground accelerations (PGA) for horizontal direction and the vertical direction are 0.30 g.

Fig. 1.
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Finite element model for sensitivity analysis of deep embedded SSI parameters

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Fig. 2.
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The input acceleration time history curves

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2.4 Observation Nodes at Key Positions

In order to observe the influence of changing parameter on the ISRS, nodes at two key positions of the superstructure (elevations of 16 m and 6 m, respectively) were selected for output and comparison. The positions and numbers of the observation nodes are described in Table 3 and illustrated in Fig. 3.

Table 3. Observation nodes at key positions
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Fig. 3.
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The positions and numbers of the observation nodes

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3 Sensitivity Analysis Results

3.1 Selection of the Interaction Nodes

The solution of foundation impedance is the most time-consuming procedure in SSI analysis by using the sub-structuring method, and it is depended to the number of interaction nodes. According to the statistical data in Fig. 4, the time-consuming increases exponentially with the increase of the number of interaction nodes.

Fig. 4.
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Relationship between number of interaction nodes and calculation time

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For an SSI analysis with deeply embedment, if all nodes in the embedded part are specified as interaction nodes, it can be predicted that there will be a large amount of the interaction nodes when the embedded depth is large. Therefore, the calculation time cannot meet the requirement of the construction period. To provide an appropriate selection method for the interaction nodes, three interaction node selection methods (as shown in Fig. 5 (a) –c)) were tested: method A is to define all nodes in the embedded part as the interaction node; method B is to select the nodes around the embedded part; method C is based on the method B, but several layers of nodes in the embedded part are also defined as interaction nodes. According to these three interaction node selection methods, the SSI analyses for the deeply embedded reactor building using above-mentioned three methods are carried out respectively.

Fig. 5.
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Schematic diagram of selection method for interaction nodes

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In method C, two different selection are made. The first conditions is to add one intermediate layer of nodes on the basis of method B, namely working condition 1–2. The second condition is to add two intermediate layers of nodes, namely working condition 1–3.

Table 4 showed the relationship between the number of interaction nodes and the calculation time.

Table 4. The interaction nodes and the calculation time of different methods
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Figure 6 and Fig. 7 showed the comparison of the ISRS from different working conditions. It can be found that the ISRS from four methods are nearly the same. The maximum relative error of the ISRS between methods is only about 3.75%. In other words, for the sensitivity analysis model, all the methods can provide stable and accurate analysis results. Hence, method B has the highest computational efficiency and the time consumption is only 15% of that of method A.

Fig. 6.
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Comparison of the ISRS of node A on the elevation of 16 m

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Fig. 7.
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Comparison of the ISRS of node F on the elevation of 6 m

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3.2 Radius of the Cylindrical Central Area

According to the basic principle of the sub-structuring method, the impedance matrix is obtained by inverting the flexibility matrix. For the three-dimensional SSI analysis of the nuclear island, the solution of the flexibility matrix [Ff] of the three-dimensional problem is the most time-consuming problem, and the solution of the flexibility matrix of the three-dimensional problem is to determine the displacement response of the horizontal layered system under the element simple harmonic point load, which can be solved by the axisymmetric model in Fig. 8.

Fig. 8.
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Axisymmetric model for impedance analysis [5]

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The model contains two areas, one is a cylindrical central area with a radius of R, which is composed of axisymmetric elements, and an axisymmetric transmission boundary outside the simulated central area. The bottom can be a fixed boundary, or a viscoelastic half space approximated by a variable thickness and a viscous boundary.

When solving the flexibility matrix, it is not necessary to apply the element harmonic load to all the interaction points to solve the dynamic displacement, but only to apply the load to a row of nodes on the soil interface covering the buried depth of the foundation in turn, and the element width is taken as the minimum transverse distance between the interaction nodes. The response of load on other nodes can be obtained simply by horizontal coordinate translation The radius R of the central area is taken as the smallest lateral distance between the interaction nodes. Changing R affects the flexibility matrix [Ff], and then affects the impedance matrix [S(ω)] of the SSI system, and finally changes the response of the whole system.

If the mesh of the embedded part or the base slab are relatively uniform, the value of R can be determined by the average value of the mesh size. However, in most practical projects, it is difficult to get an uniform mesh for the base slab. Figure 9 shows the mesh of the reactor building model.

To determine a relative reasonable value of R, five radius values of 1.80, 1.98, 1.56, 1.00, and 0.66 were selected in the analysis. Table 5 shows the selection principle for different radius R.

Fig. 9.
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Mesh generation of base plate for trial model

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Figures 10 and 11 show the comparison results of the ISRS in the horizontal X direction, the horizontal Y direction and the vertical Z direction, respectively.

Table 5. Selection principle for different radius R
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For both the horizontal X and Y direction, it can be seen that the change of R has slight effect on the shape of the ISRS. When R is taken as 1.80, 1.98 and 1.56 respectively, the ISRS are nearly the same. When R is taken as 1.00, the acceleration response of the containment and the internal structure increases. When R is taken as 0.66, the ISRS of the containment and the internal structure is slightly reduced. However, the response of the backfill soil in Y direction increases.

For Z direction, it can be seen that the ISRS from different radius are nearly the same. When R is taken as 0.66, the value of ISRS at the frequency higher than 20 Hz is amplified.

In general, the change of the radius R within a reasonable range has negligible effect on the shape and value of the ISRS. However, it can also be seen that radius of 1.80, 1.98 and 1.56 provided a relative stable results compared to radius of 1.00 and 0.66.

It is suggested that for non-uniform mesh, the value of the radius R can be estimated by weighted averaging, or take the grid sizes that accounts for a large proportion in the model. Deviation. It is not reasonable to estimate based using the grid sizes of a small percentage in the model or take the maximum or minimum grid size.

Fig. 10.
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Comparison of ISRS of node A for different radius R

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Fig. 11.
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Comparison of ISRS of node F for different radius R

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4 Conclusions

In this paper, sensitivity analysis for a deeply embedded reactor building is carried out. The sensitivity of changing selection method of the interaction nodes and the radius of the cylindrical central area are studied. The conclusion is as follows:

Three interaction node selection methods used in this study are all applicable. However, the calculation time of method B is much less than other two methods on the premise of enough accuracy. It is suggested to perform similar sensitivity analysis for selection interaction nodes before starting SSI analysis to determine an optimum interaction node number.

The central radius R has little effect on the shape and value of the ISRS. The value of the radius R obtained by weighted averaging, or taking the grid sizes that accounts for a large proportion in the model provides a relative stable response.

Apart from the two parameters studied in this paper, there are other ley influence parameters in deeply embedment analysis that might affect the result and calculation effectiveness of the SSI analysis, such as the thickness of the top layer and the element size of the superstructure. Further sensitivity analysis related to them shall be carried out to establish a suggestion principle guideline for each key parameters in deeply embedment analysis, ensuring the stability and rationality of the SSI analysis.