## 2 Finite Element Model of the Rock-Socketed Pile and Seismic Input  
### Finite Element Model  
The project is in the Plain of North China and passes through the high-water level seismic liquefaction area in a large range along the line. The karst develops in varied ways, the geological conditions are complex, and the construction is difficult. This paper mainly \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
\begin{tabular}{c} Propagation velocities of shear \\ wave \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of symbols in paper. studied the numerical simulation of a major bridge structure rock-socketed pile project in a karst area. According to the geological survey report of the actual working conditions, the material parameters of each soil layer are shown in Table 2.  
The finite element ABAQUS includes a rich library of linear and nonlinear material models, which can accurately simulate the nonlinear constitutive relationship between sand and bedrock under actual working conditions. Its powerful nonlinear analysis function can better simulate and solve the problems of material nonlinearity, geometric nonlinearity, and state nonlinearity in the pile-soil dynamic interaction [10]. Therefore, the numerical analysis modeling of the pile-soil system was established by using the large-scale general software of finite element ABAQUS in this paper [13]. The cross-section of the pile is circular with a radius of 1 m and the length of the pile is 100 m. Therefore, the soil foundation 20 m \(\times\) 20 m \(\times\) 120 m along three directions, which is great enough in the dynamic analysis. The finite element models of the soil and pile are shown in Figure 1. In this investigation, the karst cave was simulated by the equivalent materials, in which the unit weight and elastic modulus were approximately zero. The pile shaft damping ratio was 5%, which is widely adopted in the concrete material [14]. According to the actual engineering situation, the vertical load on the top of the pile foundation was calculated to be 18,000 kN. The nonlinear Interaction in ABAQUS was used to simulate complex nonlinear interaction features and displacement coordination between the pile and soil; the hard contact and thepenalty friction with a good value of 0.3 were employed in the normal direction and tangential direction, respectively [13]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & **Density** & **Elastic Modulus** & **Friction Angle (\({}^{\circ}\))** & **Cohesion** & **Thickness (m)** \\  & **(kg/m\({}^{3}\))** & **(MPa)** & **(MPa)** & **(kPa)** & **(kPa)** \\ \hline Layer-1 & 1950 & 7.39 & 10.4 & 30 & 6 \\ Layer-2 & 1850 & 5.60 & 8.4 & 29 & 14 \\ Layer-3 & 1920 & 5.68 & 9.6 & 36 & 6 \\ Layer-4 & 2000 & 4.64 & 12.3 & 15 & 2 \\ Layer-5 & 1870 & 4.61 & 14.0 & 43 & 6 \\ Layer-6 & 1920 & 7.27 & 29.5 & 43 & 28 \\ Layer-7 & 1930 & 7.79 & 28.8 & 32 & 18 \\ Bedrock & 2680 & 14,000 & 44.5 & 1 & 40 \\ Pile & 2400 & 30,000 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters of soil and pile. Figure 1: Finite element models of the soil and pile. (**a**) Soil, (**b**) pile. In this investigation, the Mohr-Columb elastoplastic constitutive model was adopted for the sandy soil foundation [13], the Drucker-Prager elastoplastic constitutive model was adopted for bedrock [13], and the linear elastic constitutive model was adopted for pile. The Mohr-Columb model is suitable for nonlinear, elastoplastic soil materials, and has been widely used in geotechnical engineering. In the stress space, the yield surface of the Mohr-Columb condition is an irregular hexagonal cone, and the corresponding yield curve is a closed irregular hexagon in the plane \(\pi\). The applicability of the Mohr-Columb yield criterion with constant cohesion c and internal friction angle \(\varphi\) becomes worse.