# Seismic Response of Shield Tunnel with Double-Layer Lining
## 3 Finite Element Model of Shield Tunnel
### Input Seismic Motion and Dynamic Boundary  
The Shiziyang Tunnel is designed to withstand seismic forces calculated based on earthquake ground motion parameters at a 2% exceedance probability level over a 100-year return period. Scholars investigated the seismic response of a subway metro station and tunnel through a large set of numerical simulations using different real and artificial accelerometers and found that the structural seismic response may be significantly influenced by the main characteristics of the accelerogram considered [29]. Due to the lack of locally measured earthquake motion, according to the \"Design Specification for the Shiziyang Tunnel of the Guangzhou-Shenzhen-Hong Kong Passenger Dedicated Line\" [27], an artificially synthesized motion with a peak acceleration of 0.15 g was chosen as the seismic load, applied laterally along the tunnel, with a duration of 30 s (Figure 3). \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Material**} & **Density** & **Young’s Modulus** & **Poisson’s** & **Friction** & **Cohesion** \\  & **(kg/m\({}^{3}\))** & **(Mpa)** & **Ratio** & **Angle (\({}^{\circ}\))** & **(Mpa)** \\ \hline Stratum & 2000 & 450 & 0.3 & 30 & 0.3 \\ Segmental lining & 2500 & 34,500 & 0.2 & \(\backslash\) & \(\backslash\) \\ Secondary lining & 2500 & 28,000 & 0.2 & \(\backslash\) & \(\backslash\) \\ Bolt & 7850 & 210,000 & 0.2 & \(\backslash\) & \(\backslash\) \\ \hline \end{tabular}
\end{table} 
Table 1: Material parameters. Figure 2: Double-layer lining shield tunnel numerical model (Unit: m). To reduce the reflection of seismic waves at the truncated boundaries of the model, equivalent viscoelastic artificial boundary elements are applied to the side and bottom boundaries. The seismic loads are converted into forces applied to the boundary nodes, enabling the study of the seismic response characteristics of the shield tunnel (Figure 4). Except for the surface being free, equivalent viscoelastic boundary elements are set up in both the normal and bottom directions of the model. The application method of equivalent viscoelastic boundary elements is to extend a layer element outward from the boundary elements of the finite element model. The parameter calculation formulas for the equivalent viscoelastic boundary element are  
Figure 4: Equivalent viscoelastic boundary element artificial dynamic boundary. Figure 3: Input seismic motion: (**a**) Acceleration time–history; (**b**) Response spectrum. \[\widetilde{G}=hK_{BT}=\alpha_{T}h\frac{G}{R} \tag{1}\]  
\[\widetilde{E}=\frac{(1+\widetilde{\
u})(1-2\widetilde{\
u})}{(1-\widetilde{\
u })}hK_{BN}=\alpha_{N}h\frac{G}{R}\times\frac{(1+\widetilde{\
u})(1-2\widetilde {\
u})}{(1-\widetilde{\
u})} \tag{2}\]  
\[\widetilde{\
u}=\left\{ \begin{array}{l}\frac{\alpha-2}{2(\alpha-1)},\alpha= \frac{\alpha_{N}}{\alpha_{T}},\alpha\geq 2\\ 0,\alpha<2\end{array} \right. \tag{3}\]  
\[\widetilde{\eta}=\frac{\rho R}{3G}\bigg{(}2\frac{c_{s}}{\alpha_{T}}+\frac{c_{ p}}{\alpha_{N}}\bigg{)} \tag{4}\]  
Here,  
\(\widetilde{G}\)--Equivalent shear modulus of artificial boundary elements;  
\(G\)--Medium shear modulus;  
\(R\)--Distance from wave source to artificial boundary point;  
\(K_{BT}\)--Tangential spring stiffness;  
\(K_{BN}\)--Normal spring stiffness;  
\(\widetilde{\
u}\)--Equivalent elastic modulus of artificial boundary elements;  
\(h\)--Artificial boundary element thickness;  
\(\widetilde{\
u}\)--Equivalent Poisson's ratio of strata;  
\(\widetilde{\eta}\)--Equivalent damping coefficient of artificial boundary element material;  
\(\rho\)--Medium mass density;  
\(c_{s}\)--S-wave velocity;  
\(c_{p}\)--P-wave velocity. A reasonable seismic load application method needs to be matched with a dynamic boundary. Here, the seismic load is transformed into equivalent nodal forces and applied to the boundary points of the model.