## 4 Numerical Simulation
### Numerical Model Establishment and Verification
In the same way, during the secondary lining construction, the secondary lining grid is activated by the activation plate to passivate the excavation module and the release of 70% of the ground stress. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Surrounding Rock** & \begin{tabular}{c} **Density** \\ \(\rho\)/(kg\(\cdot\)m\({}^{-3}\))** \\ \end{tabular} & \begin{tabular}{c} **Elastic Modulus** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio (\(\mu\))** \\ \end{tabular} & \
\begin{tabular}{c} **Internal Friction** \\ **Angle \(\varphi\)/(\({}^{\circ}\))** \\ \end{tabular} &
\begin{tabular}{c} **Cohesion** \\ \(c\)/kPa** \\ \end{tabular} \\ \hline Q\({}_{3}\) loess & 1420 & 12.6 & 0.38 & 15.1 & 20.1 \\ Q\({}_{2}\) loess & 1460 & 16.8 & 0.34 & 26.6 & 30.5 \\ Q\({}_{2}\) loess & 1630 & 18.9 & 0.41 & 30.1 & 41.3 \\ Reinforcement area & 2500 & 300 & 0.28 & 32.3 & 150.36 \\ Mudstone & 2010 & 48 & 0.32 & 33.1 & 66.2 \\ Primary support & 2500 & 50,500 & 0.28 & \\ Secondary lining & 2500 & 60,000 & 0.28 & \\ Pipe shed support & 4840 & 97,100 & 0.30 & \\ \hline \hline \end{tabular} Note: 1. The physical and mechanical parameters of Q\({}_{2}\) loess are divided into two layers, as shown in the table through field coring and laboratory tests. 2. The reinforcement area refers to the range of surrounding rock reinforced by advanced pre-grouting. This range is expressed by improving the elastic modulus, cohesion, and internal friction angle of the surrounding rock.  
\end{table} 
Table 1: Physical and mechanical parameters of surrounding rock and supporting structure. To simplify the calculation, an equivalent method was adopted to combine the influences of the section steel frame and shotcrete [39]. The formula for this method is as follows:  
\[E_{c}=E_{0}+\frac{A_{s}E_{s}}{A_{c}} \tag{5}\]  
where \(E_{c}\) is the converted concrete elastic modulus, \(E_{0}\) is the original concrete elastic modulus, \(A_{s}\) is the steel arch cross-sectional area, \(E_{s}\) is the steel elastic modulus and \(A_{c}\) is the concrete area. The surrounding rock and support parameters are shown in Table 1.  
In order to verify the correctness of the numerical model, the vault settlement and horizontal convergence of the field monitoring data (Figure 11) and the numerical simulation results after reinforcement (Figure 16) were compared and analyzed (Table 2). As shown in Table 2, the measured value of the surrounding rock vault settlement \(X_{\text{res}}\) was 65 mm and the simulated value \(X_{\text{res}}\) was 63.8 mm, corresponding to a relative error of 2.00%. The measured value of the horizontal convergence of the surrounding rock \(X_{\text{hc}}\) was 25 mm, the simulated value \(X_{\text{hc}}\) was 21.9 mm and the relative error was 12.4%, which were all within the allowable error range. This shows that the numerical model described in this article was reliable. Figure 19: Maps of shear stress of surrounding rock: (**a**) original design scheme; (**b**) scheme after collapse treatment. Figure 17: Maps of vertical stress of surrounding rock: (**a**) original design scheme; (**b**) scheme after collapse treatment. Figure 18: Maps of horizontal stress of surrounding rock: (**a**) original design scheme; (**b**) scheme after collapse treatment. Based on the results in Figures 17-19 and Table 3, tensile stress appeared at the top and bottom of the tunnel. After the collapse treatment, the vertical stress, horizontal stress, and shear stress decreased significantly. The stress of the surrounding rock above the arch waist decreased by more than 35% and that at the arch crown reached 50%. There was no stress concentration in various parts of the tunnel. Therefore, the treatment measures for the collapse section described in this paper were effective.