# Dynamic Response of Rectangular Tunnels Embedded at Various Depths in Spatially Variable Soils
## 3 Finite Difference Model  
In this paper, the finite difference program FLAC3D version 5.0 is used to establish the numerical model of rectangular tunnels in the liquefied soil layer [43], and the pore water pressure variation and deformation of soil under seismic loading, as well as the dynamic response of the tunnel, such as uplift displacement, are studied. The soil is a rectangular domain of 50 m in both horizontal and vertical directions. Since the plane strain condition is simulated in this paper, the length in the depth direction is 1 m; the unit length. The tunnel size in this study is 6 m \(\times\) 6 m, and the distance between the center point of the tunnel and the ground is \(H\) = 6, 8, 10 and 12 m, respectively. The central axis of the tunnel coincides with the central axis of the soil, and the groundwater level is located at the surface. In order to consider the calculation accuracy requirement and the cost of calculation efficiency, according to the selected seismic load characteristics and soil parameters, through a series of trial calculations, this paper finally chooses to divide the model into 1 m\(\times\) 1 m \(\times\) 1 m cubic grid. As shown in Figure 1, the model is divided into 2500 elements when the tunnel is embedment at 6 m depth. Shell element is used to simulate tunnel lining. It can simulate the interaction between the soil and the tunnel by directly adhering to the stratum element. Referring to the research of [PERSON] et al. [16], the physical and mechanical parameters of the soil and tunnel structure for deterministic analysis in this paper are selected as shown in Table 1. Since the spatial variability of soil shear modulus is considered in this paper, random field data are used for shear modulus in stochastic analysis, and other parameters are the same as those in deterministic analysis. The fluid calculation parameters adopted in this paper are shown in Table 2. In the dynamic analysis, the Finn model is adopted to describe the effect of pore pressure accumulation and simulate the process of pore water pressure rising until liquefaction of soil under seismic loading. The parameters used in the Finn model are as follows: \(C_{1}=0.79\), \(C_{2}=0.52\), \(C_{3}=0.2\), \(C_{4}=0.5\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Soil Parameters** & **Value** & **Lining Parameters** & **Value** \\ \hline Thickness (m) & 30 & Thickness (m) & 0.3 \\ Unit weight (kN/m\({}^{2}\)) & 15 & Unit weight (kN/m\({}^{2}\)) & 24 \\ Shear modulus (MPa) & 20 & Young’s modulus (GPa) & 30 \\ Bulk modulus (MPa) & 30 & Poisson ratio & 0.25 \\ Friction angle (\({}^{\circ}\)) & 25 & \\ Cohesion (kPa) & 0 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the soil and tunnel parameters. Figure 1: The tunnel with an embedment depth of 6 m.  
In the process of static analysis, the \(Y\) coordinate plane of the model is set to constrain the horizontal (\(x\) direction) and vertical (\(z\) direction) displacements due to the simulated plane strain situation. The \(X\) coordinate planes of the model (i.e., left and right) are set to constrain the horizontal displacement; the bottom of the model is set to constrain displacements in all directions; and the top of the model is set as the free surface. In the dynamic analysis, the free field boundary is set to reduce the influence of reflected waves on the calculation results. Since this paper is biased toward regularity research, the seismic excitation adopted is 8 degrees of seismic intensity, which corresponds to the east-west component of the Kobe seismic wave with a peak acceleration of 0.3 g; the duration of the seismic wave is 30 s. Seismic waves are input from the bottom of the model as shear waves, and the time history of seismic waves is shown in Figure 2.  
The maximum uplift displacement of the tunnel with embedment depth of 10 m obtained by the deterministic calculation is 23.9 cm, which is similar to [PERSON] et al. (2018) (about 25.8 cm), and the error is 7.36%.