## 3 Numerical Modeling Calculation
### Numerical Simulation-Based Analysis of the Arch Settlement of a V-Class Surrounding Rock Tunnel  
Numerical simulations were performed on a V-shaped three-step tunnel excavation model for surrounding rock using the ideal elastic-plastic model based on the Mohr-Coulomb criterion. The inverted arch had a closed distance of 30 m, and a one-time excavation of 6 m was conducted for the inverted arch. The three-dimensional simulation results can be observed in Figure 5.  
The simulation was conducted for tunnel excavations with closed distances of the inverted arch at 30 m, 40 m, 45 m, 50 m, 55 m, and 60 m. The maximum settlement curve of the arch crown is presented in Figure 6 with a maximum settlement deformation of only 2.5 mm. However, there is no correlation between settlement and time and space, which differs somewhat from the actual situation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material Science** & **Shear-Maxwell** & **Viscosity-Maxwell** & **Shear-Kelvin** & **Viscosity-Kelvin** \\ \hline V-grade surrounding rock & \(3.333\times 10^{9}\) & \(7.2\times 10^{12}\) & \(3.8\times 10^{9}\) & \(1.8\times 10^{11}\) \\ IV-grade surrounding rock & \(3.67\times 10^{9}\) & \(7.2\times 10^{12}\) & \(4.5\times 10^{9}\) & \(1.8\times 10^{11}\) \\ III-grade surrounding rock & \(4.2\times 10^{9}\) & \(7.2\times 10^{12}\) & \(5.0\times 10^{9}\) & \(1.8\times 10^{11}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stratum material parameters. Figure 4: Modified Burgers creep model. Figure 5: Deformation nephogram of three-step excavation tunnel with Mohr–Coulomb criterion. (**a**) Excavation and lining of upper steps. (**b**) Excavation of upper bench for 24 m. (**c**) Excavation of middle bench 7.2 m. (**d**) Excavation of lower bench 10.8 m. (**e**) Invert excavation. (**f**) Invert closure distance of 30 m excavation completed. Figure 6: Settlement curve with Mohr–Coulomb model. The modified Burgers creep model was employed to numerically calculate the excavation model of a V-level surrounding rock three-step tunnel, with a 30 m closed distance for the inverted arch and a one-time excavation of 6 m for the inverted arch. The simulation results in three dimensions are presented in Figure 7.  
Figure 8 presents the maximum settlement curve of the arch crown for various inverted arch closure distances. It shows that the maximum settlement of the arch crown reaches 24 mm when the inverted arch closure distance is 60 m. The settlement distribution in this case is more realistic and better reflects the actual deformation of the surrounding rock during on-site construction compared to the Mohr-Coulomb criterion-based three-step excavation model. The numerical analysis results emphasize the importance of considering the effects of stress release and timeliness on the mechanical and deformation properties of surrounding rocks. Failure to account for these factors in the Mohr-Coulomb constitutive model leads to underestimation of settlement and deformation of the arch crown during excavation. In contrast, the modified Burgers constitutive model addresses the strength reduction resulting from stress release and the time-dependent nature of construction steps. As a result, it provides a more accurate representation of the excavation process. Figure 7: Deformation nephogram of three-step excavation tunnel with modified Burgers creep model. (**a**) Excavation and lining of upper steps. (**b**) Excavation of upper bench for 24 m. (**c**) Excavation of middle bench 7.2 m. (**d**) Excavation of lower bench 10.8 m. (**e**) Invert excavation. (**f**) Invert closure distance of 30 m excavation completed. Simulation results using the modified Burgers model show that the deformation at the arch waist of the unclosed section progressively increases with each excavation step. This behavior aligns with the actual monitoring data. However, when using the Mohr-Coulomb model, the deformation at the arch waist of the unclosed section remains stable, which significantly differs from the observed monitoring results.