# A Simplified Analysis Method for Seismic Response of
## 4 Example Verification  
### Model  
The superstructure of the calculation model used in this study is a system composed of beam elements and mass elements, and the superstructure and pile foundation are connected by rigid domains. Through calculation, the maximum base shear force generated by the superstructure is 870 MN. The horizontal dimension of the raft is 55 m \(\times\) 55 m, the thickness of the raft is 1.8 m, and 8 \(\times\) 8 foundation piles are evenly distributed under the raft, of which the pile diameter is 3 m, the pile length is 37 m, and the pile spacing is about 3D (D is the pile diameter). The parameters of piles and rafts are shown in Table 1. Solid elements are used for piles, rafts, and soil and the interface shares nodes. The pile adopts an elastic model, while the soil adopts the Duncan-Zhang constitutive model. The Duncan -Zhang constitutive model belongs to a nonlinear elastic model, and the elastic modulus and Poisson's ratio of the soil will change with the changes in the surrounding stress of the soil. An overview of the finite element model is shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Elastic Modulus/GPa** & **Poisson’s Ratio** & **Density/kg/m\({}^{3}\)** \\ \hline Pile & 30 & 0.17 & 2400 \\ Raft & 20 & 0.17 & 2400 \\ Bedrock & 15.49 & 0.28 & 2450 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical parameters of the pile, raft, and bedrock.  
The horizontal dimension of the calculation model is approximately four times the size of the raft [39] (460 m in the X direction and 260 m in the Y direction), and the vertical dimension is 53 m. In this study, both the soil and pile were simulated using solid elements with shared nodes at the pile-soil-raft interface. The nonlinear deformation characteristics of this part of the soil were simulated by refining the soil grid around the pile. The maximum horizontal and vertical grid sizes of the model are 8 m and 3 m, respectively. The minimum grid sizes are 0.9 m and 1 m, respectively. Due to the large size of the model and considering that comparative analysis is the main focus of this study, the selection of grid size is based on meeting the requirements of dynamic calculation [40], that is, the vertical grid size (1/5-1/8)/V\({}_{\mathrm{s}}\)/\(f_{\mathrm{max}}\), and the horizontal grid size is taken as 3-5 \(h_{\mathrm{max}}\), where \(f_{\mathrm{max}}\) is the highest frequency considered in the dynamic calculation, and \(h_{\mathrm{max}}\) is the maximum value among the vertical grid sizes obtained for different soil layers.  
The site class is III, the characteristic period is 0.65 s, and the rare earthquake action of 8 degrees seismic intensity is considered. According to the requirements of the specification, three sets of seismic waves, including the El Centro wave, Northridge wave, and artificial seismic wave (as shown in Figure 5), with the same size class and the corresponding peak value adjusted to 0.2 g, are used for the calculation to ensure that the actual ground motion is consistent with the one-dimensional wave theory solution. In the time history analysis, the model boundary adopts a viscoelastic artificial boundary [25].  
Figure 4: Overview of the model: (**a**) 3D model diagram of pile raft foundation; (**b**) cross section diagram of pile raft foundation.  
The basic parameters of the soil layer are shown in Table 2. The changes in the shear modulus and damping ratio of the soil with the shear strain are selected according to the sand, clay, and rock given by [PERSON] [41].