## 2 Methodology
### Numerical Simulation  
#### 2.1.1 Numerical Configuration  
The analyses of static stress distribution and the dynamic stress increment were carried out using the finite MIDAS/GTS 12 in this paper. A typical tunnel prototype was deduced from the standard tunnel for an HSR double-line in accordance with the code TB10621-2014 (Figure 1). The length of the soil grid was 50 m from each left, right, front, and back portion along the upper tunnel up to the surface for all investigated cases. These cases were based on the results of a sensitivity analysis that was conducted to investigate potential boundary effects on the computed response at the central area of the numerical models. The surface was buried deep, according to the  
To determine the static stress and dynamic responses, two phases of the numerical simulation must be undergone. The first phase refers to the simulation of construction excavation under the stratum load, wherein the initial static stresses, which involve the surrounding rock, primary lining, and secondary lining, are determined. A train load is implemented to simulate the tunnel lining response, wherein the dynamic responses, which include the tensile and compressive stresses of the secondary lining, are obtained. The stress distribution of the tunnel lining was the focus of this paper. Therefore, in the first stage, an equivalent simulation method was adopted to simulate construction excavation, which would simplify the difficulty and computational complexity of the numerical simulation. The detail components of a steel arch model, including the primary lining [24] and rock bolt in surrounding mass [25; 26; 27], mainly consist of surrounding rock, primary lining, and secondary lining. The surrounding rock was divided into eight-node solid units, and the material properties were described by the elastoplastic model and the Mohr-Coulomb yield criterion. The reinforcement effect of an anchor rod on surrounding rock was simulated by improving the mechanical parameters of the surrounding rock in the anchor zone (Table 1). The initial support, secondary lining, track plate, concrete support layer, and inverted arch filling layer were divided into eight-node solid units and simulated by linear elastic materials (Table 2). The surrounding rock was simulated by a spring, and the boundary conditions were set (Figure 4). The outer nodes of the spring set the degrees of freedom (DOF) constraints in each direction. The nodes and lining in the spring were set as binding constraints. The DOF constraints were set at both ends \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rock Level** & \(\gamma\)**/**kN**-**m\({}^{-3}\)** & **E/GPa** & \(\mu\) & **C/kPa** & \(\Phi\)/\({}^{\circ}\) \\ \hline III & 25 & 13.5 & 0.30 & 763.4 & 39 \\ IV & 22 & 3.1 & 0.35 & 153.4 & 30 \\ V & 19.5 & 0.9 & 0.35 & 93.8 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Stratum mechanical parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Structural Component** & \(\gamma\)**/**kN**-**m\({}^{-3}\)** & **E/GPa** & \(\mu\) & \(f_{t}\)**/MPa** & \(f_{c}\)**/MPa** \\ \hline Secondary lining (C35) & 26.3 & 31.5 & 0.2 & 1.65 & 17.5 \\ Concrete foundation (C30) & 25 & 30 & 0.2 & 1.5 & 11.0 \\ Filling layer (C25) & 23 & 28 & 0.2 & 1.3 & 12.5 \\ Primary lining (C25) & 25 & 33.4 & 0.2 & 1.3 & 12.5 \\ Track plate (C45) & 27 & 33.5 & 0.2 & 1.9 & 21.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Structural mechanical parameters (e.g., lining and track plates). Figure 3: Mesh grid. of the model. The viscous damper was arranged in the normal and tangential boundaries, and the normal damping ratio was 1.0. The tangential damping ratio for absorbing the dynamic wave was 0.5. Before the dynamic calculation, the initial stress was already available in the tunnel lining and the surrounding rock. Thus, the initial stress field and the construction excavation process were simulated first to obtain the initial stress field before the dynamic analysis.