## 3 Analysis of the Results
### Performance on the Support Schemes in the Tunnel Cross Section  
The support spacing should be determined first before analyzing the support effect, as the support spacing affects the support effect. As mentioned above, appropriate support spacing should be one that results in equal overall tunnel deformation, which means that the support effect of the support face is close to that of the middle face, i.e., the best support effect is close to the weakest one in the entire tunnel. We assumed this \"close\" criteria was whether the sum of the displacement reduction of the roof and the lateral wall was less than 2 cm. The plots in Figures 5 and 6 show the support effect of the support face and the middle face under different spacings of the radial support and non-radial support, \begin{table}
\begin{tabular}{c c c} \hline \hline
**System** & **Rock Mass Properties** & **Strain-Softening Parameter** \\ \hline stiff & 110\% of rock mass properties & \(\varnothing_{tr}=80\%\) of friction angle \\ medium & 100\% of rock mass properties & \(C_{r}=40\%\) of cohesion \\ soft & 90\% of rock mass properties & \(e^{\text{ps}}=0.004\), \(e^{\text{ps}}=0.006\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material property values used for rock bolt. Figure 4: Schematic diagram of the analysis unit of the tunnel: (**a**) supported tunnel and (**b**) analysis unit and two analysis face. respectively. Apparently, as one would intuitively expect, with the increase in support spacing, the difference between the displacement reduction of the two faces became bigger, which demonstrates that the support effect on the middle face weakened as the support spacing increased. The ranges of the support spacing that satisfied the \"close\" criteria (supposing the displacement reduction varies linearly) are presented in Figures 5 and 6. The range of the radial support was 3.24 m and of the non-radial support was 3.49 m. Therefore, 3 m was considered to be an appropriate support spacing to study the stability on the tunnel cross section. The distributions of stress, displacement, failure zone, and bolt force were investigated as follows. The underground excavation resulted in a redistribution of the surrounding rock. The analysis of the stress distribution was beneficial in order to study the behavior and stability of the rock mass and to check the authenticity of the numerical simulation results. Figures 7 and 8 show the stress distributions of the rock mass around the tunnel. The vertical and horizontal stresses approached zero at locations close to the roof and floor, and the lateral walls, respectively, and the far filed stresses agreed with the applied stresses  
Figure 5: Displacement reduction of the support face and the middle support under the radial support. Figure 6: Displacement reduction in the support face and the middle support under the non-radial support. on the model boundary, which indicate that inputs were applied correctly to simulate the tunnel excavation. The maximum vertical and horizontal stresses were 81.1 MPa and 62.1 MPa, respectively, under the unsupported condition. Under the radial and the non-radial support conditions, the vertical stresses increased to 92.8 MPa and 91 MPa, respectively, and the horizontal stresses increased to 67.5 MPa and 67.4 MPa, respectively. Note that the peak stress was observed a slight distance away from the boundary. The results describe that the change from one support scheme to the other had little influence on the stress distribution. The displacement distributions of the surrounding rock are presented in Figures 9 and 10. In general, the displacement distribution is a major concern because it shows the direct performance of the tunnel stability. Figure 9 shows the maximum vertical displacement at the boundary on the roof and the floor. Under an unsupported condition, the vertical displacement of the roof was 22 cm. Under the radial and the non-radial support conditions, the roof vertical displacements decreased to 13.99 cm and 15 cm, respectively.