## 4 Numerical Modeling
A horseshoe-shaped tunnel was selected because it is a better shape for the drilling and blasting methods that are used in such circumstances, and is the most commonly used tunnel shape in Pakistan. The two excavation stages (top heading and bench) and three construction steps (excavation, applying soft shotcrete and rock bolt, and hard shotcrete) in each stage are simulated for analysis. The model dimensions are 90 m \(\times\) 80 m for both projects 5 and 6. The modified Hoek-Brown model was used for the analysis. This model is based on the nonlinear relation between major and minor principal stresses, \(\sigma_{1}\) and \(\sigma_{3}\), and the criterion is used for plastic yielding when \(\sigma_{3}\) is compressive. The fine mesh was simulated around the tunnel boundary for better results. The model was fixed at the sides and bottom, and vertical stresses (\(\sigma_{\text{yy}}\)) were applied at the top of the model. The in situ stress environment was created using gravity, \(\sigma_{\text{yy}}\), and the FISH (a programming language embedded within FLAC that enables the user to define new variables and functions) function. During the three construction steps for each excavation stage, 40% 30%, and 30% relaxation were used. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Case No.** & **Tunnel Span (m)** & \(RQD/J_{n}\) & \(J/J_{a}\) & \(\sigma_{c}/\sigma_{1}\) & \(Q\) **or \(Q_{c}\)** & \begin{tabular}{c} **Rock Bolt** \\ **Length (m)** \\ \end{tabular} & \
\begin{tabular}{c} **Rock Bolt** \\ **Spacing (m)** \\ \end{tabular} &
\begin{tabular}{c} **Rock Bolt** \\ **Thickness (cm)** \\ \end{tabular} \\ \hline
1 & 15.4 & 11 & 3 & 5 & 1.5 & 4 & 1.7–1.8 & 9–12 \\
2 & 15.4 & 8.5 & 3 & 4 & 1.7 & 4 & 1.7–1.8 & 9–12 \\
3 & 8.5 & 6 & 3 & 3 & 1.6 & 2.9 & 1.7–1.8 & 6–9 \\
4 & 8.5 & 3 & 3 & 2.2 & 0.9 & 2.9 & 1.7 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Recommended empirical supports for two tunnel projects using the \(Q\) system. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Case No.** & **GSI** & \(\mathbf{m_{b}}\) & **s** & **a** & **c (MPa)** & \(\Phi^{\circ}\) & **E (GPa)** & \(\sigma_{1}\) **(MPa)** \\ \hline
1 & 75 & 9.418 & 0.0622 & 0.501 & 5.374 & 52.29 & 34.29 & 0.66 \\
2 & 72 & 9.197 & 0.0446 & 0.501 & 4.565 & 50.69 & 18.445 & 0.388 \\
3 & 68 & 5.421 & 0.0286 & 0.502 & 4.6 & 44.31 & 26.545 & 0.422 \\
4 & 58 & 3.793 & 0.0094 & 0.503 & 2.899 & 39.09 & 14.24 & 0.124 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Rock mass properties for different rocks units of the four projects. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Case No. ** & **a** & \(\mathbf{a^{\prime}}\) & **b** & \(\mathbf{b^{\prime}}\) & **c** & \(\mathbf{c^{\prime}}\) & **d** & \(\mathbf{d^{\prime}}\) & **e** & \(\mathbf{e^{\prime}}\) \\ \hline
1 & 9.9 & 9.6 & 6.8 & 6.4 & 2.8 & 2.6 & 7.5 & 7.5  
To calculate the total displacement at points along the tunnel perimeter, the x and y displacement components were obtained from the grid points data. The plastic zones before and after support were analyzed for all four cases of the two projects. The plastic state zone was retrieved from the plot command using the zone condition in the model. For the SS-1 rock case of Project 6, the yield in shear was still the comparative maximum of all the cases, as shown in Figure 10; however, with the application of support, this decreased enormously. Capacity diagrams, which graphically represent the relationships between thrust-bending moments and thrust-shear forces, are used to evaluate the stability of liners [55]. For this, the structure's liner information was obtained from the utility function command. Based on the proposed shotcrete thickness from Table 10, thrust-bending diagrams were plotted (Figure 11) for shotcrete thicknesses of 12 cm and 9 cm for factor of safety (FOS) values equal to 1.5, 2, and 2.5.  
developed by [PERSON] in 1946, which was known as rock load classification.