## 3 Numerical Simulation
### Tunnel Excavation Simulation  
The shield tunneling process induces disturbances in the soil at the excavation face, leading to a reduction in its strength. Hence, a modulus reduction method was employed for simulation, where the elastic modulus of the soil to be excavated in the next ring is reduced to 50% of its original value in the simulation. A simplified approach was adopted, assuming that the support pressure is equal to the initial horizontal stress at the tunnel axis of the excavation face. A linear shrinkage of the shield diameter with a contraction rate of 0.4% is considered in this study. When the excavation diameter is 6.7 m, the over-excavation gap is approximately 13 mm. The state where the shield machine has fully entered the strata is taken as the initial condition, and the simulation of shield excavation and shield support is performed following the steps below: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Steel** & **Material** & **Equivalent** & **Equivalent** & **Elastic** & **Poisson** \\
**Corrugated Pipe** & **Density** & **Density** & **Thickness** & **Modulus** & **Ratio** \\  & **\(\rho\) (g/cm\({}^{3}\))** & **\(\rho_{1}\) (g/cm\({}^{3}\))** & **\(t\) (mm)** & **\(E\) (GP)** & **Ratio** \(v\) \\ \hline
4–9 holes & 7.85 & 1.411 & 70 & 206 & 0.3 \\ Other holes & 7.85 & 1.171 & 65 & 206 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation parameters of steel corrugated pipes. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Stata** & **Dry Density** & **Cohesion** & **Internal Friction** & **Compression** & **Deformation** & **Poisson** & **Permeability** & **Porosity** \\  & **\(\rho_{d}\) (g/cm\({}^{3}\))** & **C (kPa)** & **Argle \(\rho\) (\({}^{\circ}\))** & **\(E_{s}\) (MPa)** & **\(E_{0}\) (MPa)** & **Ratio** & **Coefficient** & **\(n\)** \\ \hline \multicolumn{8}{c}{Natural strata} \\ \(\backslash\)1 & 1.60 & 20.0 & 10.0 & 3.5 & / & 0.40 & \(2.0\times 10^{-3}\) & 0.33 \\ \(\backslash\)1 & 1.10 & 6.8 & 3.40 & 2.4 & / & 0.35 & \(1.0\times 10^{-7}\) & 0.50 \\ \(\backslash\)2 & 1.50 & 24.7 & 11.2 & 5.7 & / & 0.32 & \(6.0\times 10^{-1. Excavate the soil and create liner structural elements attached to the tunnel linings to simulate the shield shells. 2. Apart from the liner structural elements, fix the displacements of all nodes of the solid and structural elements, and only assign a radial inward velocity of \(1\times 10^{-4}\) (m/step) to the liner structural element nodes. Iterate for 130 steps to achieve a uniform radial contraction of 13 mm, simulating the gap between of shield tunnel linings and shield shells. 3. Fix the displacements of liner structural element nodes and release the node constraints of other solid and structural elements (except for boundaries), and then solve for equilibrium. 4. After each shield excavation of one ring by 2 m, repeat steps (1) to (3) for a new liner structural element. The implementation of the above method is based on the contact interface characteristics between liner structural elements and zone solid elements. As shown in Figure 5, when there is a gap between the shield shell and the tunnel lining, it is equivalent to the contact surface being subjected to tension and separation, and the interface tensile strength becomes 0. When the soil undergoes radial contraction and recontacts the shield shell, the contact surface is closed again, and the normal stress is generated. A simplified measure was performed for simulating simultaneous grouting at the shield tail. The density of the grout obtained from tests was 20 kN/m\({}^{3}\). When the grout is in a liquid state, it exerts radial outward grouting pressure on the tunnel linings. The grouting pressure was applied as a uniformly distributed surface load. After the grout has hardened, the simulation of the grouting pressure surface load was removed, and liner structural elements representing the grout layer were then established on the tunnel linings for support.