# Parameter Analysis on Seismic Response of Long Lined Tunnel by 2.5D Substructure Method
## 4 Method Verification
### Response of Lining Structure in Elastic Half-Space  
For the soil-tunnel interaction problem, firstly, it is compared with the vertical incident seismic wave in the plane. As shown in Figure 6, the radius of concrete circular lining  
Figure 4: Displacement\(-\)time curve of point A when P wave is incident at an angle of 60\({}^{\circ}\). (**a**) Horizontal displacement and (**b**) vertical displacement. Figure 5: Displacement\(-\)time curve of point A when SV wave is incident at an angle of 30\({}^{\circ}\). (**a**) Horizontal displacement and (**b**) vertical displacement. Figure 3: The velocity\(-\)time history. \(r=5\) m. The dimensionless parameter is \(\eta=\omega\tau_{1}/\pi c_{s}=0.132\). The buried depth of lining \(H=8.33\)\(r_{1}\). Lining thickness \(d=0.1\)\(r_{1}\). The material parameters of concrete lining and uniform half-space site are listed in Table 1. When the seismic wave is obliquely incident as a P-wave, the incident angle \(\theta=60^{\circ}\). When the seismic wave is an SV wave with oblique incidence, the incidence angle \(\theta=15^{\circ}\). For the problem of tunnel wave scattering in a 2D homogeneous bedrock half-space, we use the ABAQUS large model as a reference solution in the study, with a finite element model size of 300 m \(\times\) 140 m. The dotted line in Figure 6 represents a 2D finite element viscous spring artificial boundary. As shown in Figure 7, the radial displacement of the substructure model is compared with the normalized radial displacement of the structure under the soil-structure interaction of the 2D finite element model. From the results, it can be seen that the 2.5D finite element substructure method is highly consistent with the reference scheme under 2D oblique incidence, regardless of whether the P wave or SV wave are obliquely incident. The effectiveness of the proposed finite element substructure method has been verified under the condition of oblique incidence of seismic waves. \begin{table}
\begin{tabular}{c c c c} \hline  & **Density** & **The First Lamé Constant** & **The Second Lamé** \\  & \(\rho\) (kg/m\({}^{3}\)) & \(\lambda_{1}\) (GPa)** & **Constant G\({}_{1}\) (GPa)** \\ \hline Soil & 2665 & 18 & 0.25 \\ Lining & 2240 & 22 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Site conditions. Figure 6: Amplification of radial displacement along the external tunnel surface: (**a**) P wave vertically incident and (**b**) SV wave vertically incident. Figure 7: Amplification of radial displacement along the external tunnel surface: (**a**) P wave incident and (**b**) SV wave incident. So far, the accuracy of the in-plane substructure method has been verified. In order to further verify the accuracy of the tunnel along the axis direction, the oblique incidence angle of seismic wave is taken as \(\theta_{v}=30^{\circ}\), as shown in Figure 8a. As in the plane, the radius of concrete circular lining \(r=5\) m, lining thickness \(d=0.1\)\(r_{1}\), and lining buried depth \(H=5\)\(r_{1}\). The material parameters of circular lining and single half-space site are listed in Table 2. The dimensionless parameter is \(\eta=\omega r_{1}/\pi c_{5}=0.105\). The results are compared with the analytical solutions of [PERSON] and [PERSON] [35]. By comparing the results, it can be found that the 2.5D substructure calculation method used in this paper is in good agreement with the normalized radial displacement amplitude of the analytical solution under the vertical incidence of seismic wave, whether P waves or SV waves are used, as shown in Figure 9. In the direction of the tunnel axis, Figure 10 shows that the normalized axial displacement amplitude of the substructure model is consistent with that of the analytical solution. The accuracy of the proposed substructure method in the longitudinal direction of the tunnel is verified.