## 3 Finite Element Models
### Results and Discussions
Based on these results, it can be seen that the tunnel-lining thickness affects the bending moment that occurs in the lining because the relative stiffness (i.e., the lining stiffness relative to the surrounding soil stiffness) varies with changes in the lining thickness. This means that the bending moment increases with the increasing lining thickness, indicating that the lining is able to withstand greater forces when it has a greater thickness. For instance, in the kinematic hardening model, with a tunnel depth of 12 m (\(D=12\) m) and an earthquake acceleration of 0.4 g, the maximum bending moment created on the lining thicknesses of 0.15 m, 0.30 m and 0.45 m are 22,990 N.m, 53,180 N.m, and 68,240 N.m, respectively. The earthquake forces of 0.4 g and 0.83 g are applied to the soil-tunnel system with a tunnel depth and thickness of 12 m and 0.3 m, respectively, to investigate the effect of the earthquake intensity on soil behavior. The distribution of equivalent plastic strain created near the tunnel for both the Mohr-Coulomb and kinematic hardening models is shown in Figure 15. Under the same conditions,  
Figure 12: Displacement of the top of the tunnel at three depths: \(\frac{C_{1}}{D}=1\), \(\frac{C_{2}}{D}=2\), and \(\frac{C_{3}}{D}=3\). Figure 13: Acceleration response at the soil surface for a tunnel at various embedment depths, according to the kinematic hardening model. the maximum equivalent plastic strain created in the soil elements in contact with the lining in the Mohr-Coulomb model is more than the kinematic hardening model, because the Mohr-Coulomb model cannot incorporate the damping caused by the soil plasticity. This means that the maximum value in the kinematic hardening model under 0.83 g (\(\varepsilon_{p}=0.0143\)) is approximately equal to the Mohr-Coulomb model under 0.4 g (\(\varepsilon_{p}=0.0126\)), but in the Mohr-Coulomb model, the equivalent plastic strain is zero in the soil elements away from the tunnel lining. Therefore, the Mohr-Coulomb model is not suitable for modeling soil under seismic loading, as it fails to generate loading and unloading loops, and it is unable to show the actual behavior of the soil. For instance, the equivalent plastic strain created during the earthquake loading (0.4 g and 0.83 g) for a node on the soil (above the tunnel crown) for both the Mohr-Coulomb and kinematic models is shown in Figure 16.  
Figure 14: Effect of tunnel lining thickness on the tunnel seismic response, where (\(\xi_{\rm D}=2\)), \(\frac{h}{D}=0.025\), \(\frac{h}{D}=0.05\) and \(\frac{h}{D}=0.075\). Figure 15: Distribution of equivalent plastic strain in: (**a**) the kinematic hardening model under 0.4 g peak ground acceleration (PGA), (**b**) the kinematic hardening model under 0.83 g PGA, (**c**) the Mohr–Coulomb model under 0.4 g PGA, and (**d**) the Mohr–Coulomb model under 0.83 g PGA. The choice of the shape and dimensions of a tunnel depends on its intended purpose. For example, circular, quadrilateral and horseshoe cross-section shapes are used for railways tunnels, and each of these tunnels exhibits different behavior under seismic loads. Here, we study two circular and square tunnels with the same embedment depth (D = 12 m) and lining thickness (t = 0.30 m), which are subject to the same earthquake intensity (0.4 g), to investigate the effect of tunnel shape on the seismic behavior. The difference between the results we obtain and those from the previous study [29] (Figure 17) are due to the different constitutive model used. Figure 9:0 Equivalent plastic strain for a soil node above the tunnel crown. Figure 9:1 Maximum bending moment on the tunnel lining, for: (**a**) a circular tunnel and (**b**) a square tunnel. The maximum bending moment generated at all nodes on the tunnel lining for both Mohr-Coulomb and kinematic hardening models are shown in Figure 17a,b, which are for the circular tunnel and square tunnel, respectively.