## 5 Parametric Study  
To validate the proposed methodology, the cases with different lithological characters and lateral pressure coefficients in Table 1 are implemented to validate the regularity of the proposed method. The case 0 has been analyzed in Section 4 Numerical Analysis of Case Study. The material parameters in case 1 and case 2 remain unchanged with case 0, while the lateral pressure coefficients in case 3 and case 4 are same as that of case 0. Differing from case 0, the lateral pressure coefficient is 1.0 in case 1 and 1.5 in case 2; the rock mass in case 3 is softer than that of case 0 under the geological condition of cretaceous clay rock, while the rock mass in case 4 is harder quartz diorite than that of case 0. According to the above procedures, the monitoring data in four separate cases from the numerical simulations are recorded in Figures 12-14, including the evolution of \(\mathbf{F}\) with \(\mathbf{r}\), the evolution of \(\mathbf{\Lambda}\) with \(\mathbf{r}\), and the evolution of \(\mathbf{\Lambda}\) with \(\mathbf{x}\). Then, the geometric mean at each calculation step is calculated to obtain the curve of \(\mathbf{r}-\overline{\mathbf{F}}\), \(\mathbf{r}-\overline{\mathbf{\lambda}}\) and \(\mathbf{x}-\overline{\mathbf{\lambda}}\) and establish the graphic correspondence between \(\overline{\mathbf{F}}\) and \(\mathbf{x}\) in Figure 15. The curve of \(\mathbf{x}-\overline{\mathbf{\lambda}}\) can be fitted by Equation (8) for the explicit mathematical expression and the fitting results are listed in Table 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline Case & \begin{tabular}{c} **Density** \\ \(\rho\) (g/cm\({}^{3}\)) \\ \end{tabular} & \begin{tabular}{c} **Deformation** \\ **Modulus** \\ \(\mathbf{E_{0}}\) **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} & \begin{tabular}{c} **Friction** \\ **Angle** \\ \(\mathbf{\psi}\) **(\({}^{\circ}\))** \\ \end{tabular} & \begin{tabular}{c} **Cohesion** \\ **Angle** \\ \(\mathbf{\phi}\) **(\({}^{\circ}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Burial** \\ **Depth** \\ **\(\mathbf{\phi}\) **(\({}^{\circ}\))** \\ \end{tabular} &
\begin{tabular}{c} **Lateral** \\ **Pressure** \\ **\(\mathbf{\Lambda}\) **(MPa)** \\ \end{tabular} \\ \hline
0 & 2.70 & 8.5 & 0.27 & 40 & 0.8 & 300 & 0.4 \\
1 & 2.70 & 8.5 & 0.27 & 40 & 0.8 & 300 & 1.0 \\
2 & 2.70 & 8.5 & 0.27 & 40 & 0.8 & 300 & 1.5 \\
3 & 2.43 & 3.0 & 0.31 & 30 & 0.35 & 300 & 0.4 \\
4 & 2.72 & 14.0 & 0.20 & 60 & 1.6 & 300 & 0.4 \\ \hline \end{tabular}
\end{table} 
Table 1: Computational parameters of discussions. Figure 11: Installation location of initial ground support. Figure 12: Evolution of safety factor: (**a**) case 1, (**b**) case 2, (**c**) case 3, (**d**) case 4. Figure 13: Evolution of dimensionless displacement: (**a**) case 1, (**b**) case 2, (**c**) case 3, (**d**) case 4. Figure 14: Longitudinal displacement profile: **(a)** case 1, **(b)** case 2, **(c)** case 3, **(d)** case 4. Figure 15: Installation location for the initial support: **(a)** case 1, **(b)** case 2, **(c)** case 3. As the procedures mentioned above, the admissible value of global safety factor, \([\overline{\mathbf{F}}]\), is defined to indicate the stability of the surrounding rock and determine the corresponding appropriate installation location of the initial support. The value of \([\overline{\mathbf{F}}]\) will be set as 1.0 and 1.2, respectively. The results under the condition that \([\overline{\mathbf{F}}]\) is equal to 1.0 are shown in Figure 15. In case 1, with the lateral pressure coefficient of 1.0, when \(\overline{\mathbf{F}}\) decreases to \([\overline{\mathbf{F}}]=1.0\), the corresponding excavation load releasing ratio \(\mathbf{r}=\mathbf{r_{s}}\) is 90.9% and the global dimensionless displacement \(\overline{\lambda}=\overline{\lambda_{s}}\) is 0.698.