# Behaviour of Horseshoe-Shaped Tunnel Subjected to Different In Situ Stress Fields
## 3 Results
### Case I-Impact of In Situ Stress Ratio
#### 3.1.1 Wall Convergence Ratio  
The ratios of walls (e.g., right and left) convergence are calculated as given in Tables 3 and 4 and are plotted as shown in Figures 3 and 4, respectively. It can be shown that the ratio of wall closure/convergence increases as the in situ stress ratio, K, increases. After excavating a tunnel in the rock, the tunnel walls tend to move towards the entire tunnel opening, see Figure 5 (e.g., vectors of horizontal displacements contours around tunnel opening at in situ stress ratio, K, of 2.5). Additionally, the maximum left (e.g., point #2) and right (e.g., point #3) walls convergence occurs at the tunnel perimeter (e.g., at zero m lateral distances from the tunnel boundary) and in situ stress ratio, K, of 2.5 are 0.0046% and \(-0.0148\%\), respectively.  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter/Analysis** & **Case I** & **Case II** \\ \hline In situ stress ratio, K & K = 0.5, 1, 1.5, 1.65, 1.80, 2, 2.13 \& 2.5 & 2.13 \\ \hline Tunnel depth, m & 50 & Z = 50, 100, 150, 200 \& 250 \\ \hline Joint dip angle, \(\alpha\) (deg.) & 30 & 30 \\ \hline Joint spacing, m & 5 & 5 \\ \hline \end{tabular}
\end{table}
Table 1: Different factors considered during the sensitivity analysis.  
\begin{table}
\begin{tabular}{l c c} \hline
**Rock Mass Property** & **Limestone** & **Slate** \\ \hline Density (kg/m\({}^{3}\)) & 2600 & 2500 \\ \hline UCS (MPa) & 30–250 & 100–200 \\ \hline Poisson’s ratio, \(\upsilon\) & 0.30 & 0.25 \\ \hline Tensile strength, \(\sigma\)t (Mpa) & 0.60 & 0.10 \\ \hline Friction angle, \(\phi\) (deg.) & 40 & 35 \\ \hline Young’s Modulus, E (Gpa) & 35 & 2.60 \\ \hline Cohesion, C (Mpa) & 1 & 0.25 \\ \hline Dilation angle, \(\Psi\) (deg.) & 20 & 15 \\ \hline
**Rock joints properties**[112] & & \\ \hline Normal stiffness, Kn (Gpa) & 21 & \\ \hline Shear stiffness, Ks (Gpa) & 15 & \\ \hline Friction angle (deg.) & 39 [113] & \\ \hline \end{tabular}
\end{table}
Table 2: Geomechanical properties of rock mass and joints [111,112].  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Left Wall Convergence Ratio (LWCR), \%**} \\ \cline{2-7}
**K, Value** & \multicolumn{6}{c}{**Distance from the Left Wall of the Tunnel (e.g., from Reference Point \#2)**} \\ \cline{2-7}  & **0.0 m** & **0.5 m** & **1.0 m** & **1.5 m** & **2.0 m** & **2.5 m** \\ \hline
0.5 & \(-\)0.00096 & \(-\)0.00124 & \(-\)0.00152 & \(-\)0.00152 & \(-\)0.0018 & \(-\)0.0018 \\ \hline
1.0 & 0.00024 & \(-\)0.00014 & \(-\)0.00052 & \(-\)0.0009 & \(-\)0.0009 & \(-\)0.00128 \\ \hline
1.5 & 0.0016 & 0.00107 & 0.00054 & 0.00001 & \(-\)0.00052 & \(-\)0.00052 \\ \hline
1.65 & 0.002 & 0.00141 & 0.00082 & 0.00023 & \(-\)0.00036 & \(-\)0.00036 \\ \hline
1.80 & 0.00175 & 0.00175 & 0.0011 & 0.00045 & \(-\)0.0002 & \(-\)0.0002 \\ \hline
2.0 & 0.0028 & 0.00208 & 0.00134 & 0.00061 & 0.00061 & \(-\)0.00012 \\ \hline
2.13 & 0.0032 & 0.0024 & 0.00162 & 0.00083 & 0.00083 & 0.00004 \\ \hline
2.50 & 0.0046 & 0.00364 & 0.00266 & 0.00169 & 0.00169 & 0.00072 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Ratio of left wall convergence (LWCR) at various in situ stresses and different distances from the left wall of the tunnel (e.g., from reference point #2).  
Figure 3: Ratio of convergence occurs in the tunnel right wall (RWCR) at various in situ stress ratios monitored at different lateral distances from tunnel boundary (e.g., point #3).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Right Wall Convergence Ratio (RWCR), \%**} \\ \cline{2-7}
**K, Value** & \multicolumn{6}{c}{**Distance from the Right Wall of the Tunnel (e.g., from Reference Point \#3)**} \\ \cline{2-7}  & **0.0 m** & **0.5 m** & **1.0 m** & **1.5 m** & **2.0 m** & **2.5 m** \\ \hline
0.5 & \(-\)0.00376 & \(-\)0.00292 & \(-\)0.00264 & \(-\)0.00264 & \(-\)0.00236 & \(-\)0.00236 \\ \hline
1.0 & \(-\)0.00622 & \(-\)0.00546 & \(-\)0.00508 & \(-\)0.0047 & \(-\)0.0047 & \(-\)0.00432