## 3 Finite Element Models  
### Model Descriptions  
As previously mentioned, many factors--such as the type and thickness of soil, shape, dimensions, and depth of a tunnel, as well as earthquake intensity--affect the seismic response of a tunnel to an earthquake. In this study, we aim to examine the effect of these factors on the seismic behavior of a tunnel and compare the results with those from previous studies performed using a linear elastic-perfectly plastic Mohr-Coulomb model [29]. In all of the studied models, five layers of soft soil are used with different characteristics and various thicknesses of 29.2 m, 19.6 m, 16 m, 7.2 m, and 3 m, corresponding to structural layers from bedrock to the soil surface, respectively. The schematic configuration of the models, including the soil layers and tunnel used in all of the analyses, is given in Figure 8, and the properties of the clay soils used in this study are summarized in Table 2.  
A circular tunnel with a diameter of 6 m and a square tunnel with a length of 5.3 m are used to investigate the effect of the shape of the tunnel on the dynamic response. Different tunnel-lining thicknesses of the circular tunnel are also considered to study the effect of lining thickness on the seismic response. The properties of the concrete used in the tunnel lining are given in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Layer No.**} & \multirow{2}{*}{\begin{tabular}{c} **Layer** \\ **Thickness (m)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Unit Weight** \\ **\(\gamma\) (kN/m\({}^{3}\))** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Undrained Shear** \\ **Strength Su (kPa)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Poisson’s** \\ **Ratio \(\
u\)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Damping** \\ **Ratio \(\alpha\)** \\ \end{tabular} } \\ \hline
1 (top) & 3.00 & 18.40 & 29.90 & 0.49 & 9.6600 & 0.7764 \(\times\) 10\({}^{-3}\) \\ \hline
2 & 7.20 & 17.50 & 27.40 & 0.49 & 3.8930 & 1.9260 \(\times\) 10\({}^{-3}\) \\ \hline
3 & 16.00 & 16.90 & 19.80 & 0.49 & 1.7710 & 4.2380 \(\times\) 10\({}^{-3}\) \\ \hline
4 & 19.60 & 18.00 & 26.30 & 0.49 & 1.7440 & 4.3010 \(\times\) 10\({}^{-3}\) \\ \hline
5 & 29.20 & 18.10 & 30.00 & 0.48 & 1.7060 & 4.3970 \(\times\) 10\({}^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Properties of the clay soils.  
Figure 8: The soil–tunnel system.  
One of the factors affecting the seismic behavior of a tunnel is the intensity of the earthquake force imposed on the soil-tunnel model. The earthquake load applied to the base of models is that of the 1995 Kobe earthquake, i.e., 0.83 g PGA (peak ground acceleration) (as recorded at KJMA station); then, this is scaled down to 0.4 g to investigate the effect of this parameter on the dynamic soil-tunnel behavior. There are various methods [35; 36; 37; 38; 39] used for the boundary conditions of models to ensure that these conditions prevent the return of earthquake waves, and we use the viscous spring boundary method to surround our soil models. The acceleration time history of the Kobe earthquake, and the power amplitude and Fourier amplitude spectra used in this study, are presented in Figure 9.