## 2 Numerical Approach
### The Numerical Model
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **Volumetric** \\ **Weight (kN/m3)** \\ \end{tabular} & \begin{tabular}{c} **Internal Friction** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} & \begin{tabular}{c} **Cohesive** \\ **Force (kPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Elastic** \\ **Modulus (MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Poisson Ratio** \\ \end{tabular} \\ \hline
18.1 & 30 & 10 & 26.18 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical and mechanical parameters of stratum. Figure 4: (a–d) Dimensions and boundary conditions of the tunnel model. Figure 5: The assembly method and mesh division of the double-layered shield tunnel model. The refined numerical calculation model was calculated based on the finite difference method, and the segment lining was simulated using the Liner element (a three-node flat finite element that can resist the shear force, axial force, and bending moment, as well as simulate the effects of tangential friction and the tension and compression in the normal direction on the contact surface between the segment lining and the secondary lining). The thickness, stiffness, and strength of the secondary lining are relatively small, and the mechanical mechanism is in accordance with the characteristics of thin shell structure; therefore, the shell element was utilized to simulate the secondary lining. The parameters of segment lining and secondary lining were determined based on the engineering data shown in Figure 2, as listed in Table 2.  
For the liner element used to simulate the segment, a reasonable method was used to divide the mesh so that the structural nodes were located exactly where the longitudinal and circumferential bolts are. This treatment allowed for the better implementation of the refined simulation of the bolts later. In addition, for shell elements used to simulate the secondary lining, the nodes of the mesh needed to coincide with the liner element. In this way, a reasonable interface element could be established between the liner element and shell element, so as to realize a reasonable simulation of the contact surface between the segment lining and the secondary lining. For convenience, the same meshing as used for the liner element was used for the shell element. The specific meshing of the model is shown in Figures 4 and 5. The entire model comprised a total of 536,400 solid elements, 556,187 grid points, 70,400 structural elements, and 43,644 structural nodes. The segments were assembled in a staggered manner in the refined numerical calculation model. They were joined into a segment ring using circumferential bolts, then connected into a longitudinal 3D tunnel structure using longitudinal bolts. Link elements were employed to simulate the bolts. Making the connection position and connection characteristics of the link element align with bolts used in practice helped to reflect the deformation mechanisms of the shield tunnel realistically. Each segment was uniformly divided into four pieces along the width direction, and two adjacent segments in same segment ring had five pairs of nodes with overlapping positions at the connection location of segments, numbered 1 to 5. The link element was established at nodes 2 and 4 to simulate the circumferential bolts (Figure 6a). Then, each segment ring was uniformly divided into 50 pieces along the circumferential direction and there were 50 nodes with an overlapping position between two adjacent segment rings. The link element was established at every other node to simulate the longitudinal bolts, with a total of 25 link elements, as shown by the red dots in Figure 6b. There were three link elements on each standard and adjacent block segment and only one link element on each capping block segment. Each link element has six degrees of freedom in six directions, including three translational degrees of freedom (one in each axial direction) and three rotational degrees of freedom (one around each axis).