## 3 Finite Element Models
### Results and Discussions  
In this study, the linearly elastic-perfectly plastic model and the kinematic hardening constitutive model are used to model soil behavior under earthquake load and some of the parameters that affect the seismic behavior of tunnels are investigated. The results that we obtain from a nonlinear kinematic hardening model are compared with those of a previous study (performed using a Mohr-Coulomb model) [29]. It is noteworthy that in this research, all the soil-tunnel models were simulated in addition to the kinematic hardening constitutive model using the Mohr-Coulomb model, and the results were in good agreement with the results of a previous study that considered the Mohr-Coulomb model using PLAXIS software. The results we obtain are discussed below. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Density** & **Elastic Modulus** & **Poisson’s** & **Damping** & **Damping** \\  & **(kN/m3)** & **(GPa)** & **Ratio v** & **Ratio \(\alpha\)** & **Ratio \(\beta\)** \\ \hline Concrete & 25 & 30 & 0.20 & 2.2544 & 0.000908 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Properties of the concrete used in the tunnel lining. Figure 9: Characteristics of the 1995 Kobe earthquake: (**a**) the acceleration time history; (**b**) the power amplitude; (**c**) the Fourier amplitude spectrum. To investigate the effect of tunnel embedment depth on the seismic response of a soil-tunnel system, a circular tunnel with a diameter of 6 m (D = 6 m) at depths of 6, 12, and 18 m (C1 = 6 m, C2 = 12 m, and C3 = 18 m) is subject to Kobe earthquake force with a maximum acceleration of 0.4 g to examine the effect of the interaction between soil and tunnel. Figure 10 shows the maximum acceleration response of the earthquake to the soil depth; as can be seen, the earthquake acceleration response decreases from the bedrock to the soil surface. Therefore, the forces that are exerted on deeper tunnels are larger. The bending moment created in the tunnel lining is also investigated to determine the effect of tunnel depth on the seismic response. Figure 11 shows the bending moment created in a node on the lining at three different depths (6 m, 12 m, and 18 m) by considering the kinematic hardening model. The bending moment increases with increasing tunnel depth, which indicates an increase in the forces applied to the tunnel lining, i.e., the earthquake force increases with depth (see Figure 10). Figure 12 shows the comparison of tunnel displacement (considering the top node of the tunnel lining) at three different depths during the earthquake loading. As the tunnel depth increases, its displacement decreases, which means that the shallow tunnel experiences the largest displacement. The acceleration response curve on the soil surface (at the center line of the model) for a tunnel at different depths (C1 = 6 m, C2 = 12 m, and C3 = 18 m) is shown in Figure 13, illustrating the effect of  
Figure 11: Bending moment created at a node on the tunnel lining. Figure 10: Maximum acceleration response with depth. tunnel depth on the acceleration response of the soil surface. This graph results from a model in which the thickness of the tunnel lining is 0.3 m and the input motion of earthquake is 0.4 g. According to Figure 13, the maximum acceleration response on the soil surface does not change with increasing depth, and the value of maximum acceleration for all depths is approximately 0.06 g.  
Three lining thicknesses (t1 = 0.15 m, t2 = 0.30 m, and t3 = 0.45 m) are considered for a circular tunnel of diameter 6 m to study the effect of the tunnel-lining thickness on its seismic behavior under the Kobe earthquake load. Figure 14 is based on the bending moment (normalized to the radius of the tunnel and the free-field shear stress (\(\frac{M}{\tau R^{2}}\)), (\(\tau=G\gamma_{max}\))) in terms of the lining thickness ratio (\(\frac{t}{D}\)), and it shows a comparison of the results of this study using the nonlinear kinematic hardening and Mohr-Coulomb models with those of a previous study [29].