## 3 Results of Numerical Analysis and Discussion
### Crown Deformation
Additionally, it increased immensely from 5.8 mm to 57.8 mm in the rectangular tunnel. Regarding the acceptable overcutting area and crown deformation in the practice of civil engineering, the case of E/10 cannot be implemented in actual scenarios. The E/10 case is considered the worst-case scenario in our further numerical analyses, as presented in the following figures. By grouting the surrounding rock of the tunnel at several diameters, the displacement is significantly decreased, as shown in Figure 12. The initial crown deformation of un-grouted rock is around 42.6 mm in the circular tunnel and 57.8 mm in the rectangular tunnel. Figure 10: General scheme of the grouting and jacking process: (a) scheme of grouted rock and concrete lining, and (b) grouting and tunneling steps. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Young’s Modulus** & **Poisson’s Ratio** & **Unit Weight** \\  & **E (MPa)** & **v** & **(MN/m\({}^{3}\))** \\ \hline Grouting material & 200 & 0.3 & 0.022 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Grouting material properties. After the grouting material was injected and the rock's strength improved, the amount of displacement was decreased to 24.7 mm, 17.05 mm, 13.7 mm, and 11.2 mm for each diameter of the circular tunnel and to 42.5 mm, 30.09 mm, 21.5 mm, and 15.8 mm for the diameters of the rectangular tunnel. It is seen that, as the displacement is reduced from 42 to 74% in the circular tunnels and from 26.5 to 73% in the rectangular tunnels, the grouting method is highly efficient, and the selected material significantly improves the surrounding rock masses' characteristics. Figures 13 and 14 clearly show the variance in the deformation contours and the reduction in the number of large deformations for both cases in which the grouting material was injected into a 5.6 m diameter tunnel and without grouting. The largest deformation occurred near the entrance of the main gate and in the surrounding rock from the roof at the end of the tunnel, with the tunnel inclination angle and highwall slope set as \(5^{\circ}\) and \(7^{\circ}\), respectively. According to the numerous numerical results obtained and based on the acceptable crown deformation and over-cutting area, the pipe-jacking method with a pre-cast concrete lining support can be applied at an actual site with weak geological structures, particularly in the weak/very weak rock strata. However, assuming the worst-case scenario, Young's modulus is reduced by ten times; the grouting  
Figure 11: Effect of decreasing Young’s modulus on crown deformation. Figure 12: The difference in crown deformation with grouted rock at various diameters (Dg—Total diameter of grouted rock and tunnel). method can be considered, and the surrounding rock mass can sufficiently be improved by alternative solutions. Beyond that, another achievement of this study is the low-cost, due to the optimization of the external diameter of the grouted rock. In Figure 12, the external diameters from 4.1 m to 4.6 m for the circular tunnel and from 5.1 m to 5.6 m for the rectangular tunnel are reasonable.