## 5 The Application of Trapezoidal Freezing Soil Mass in the Wuhu River-Crossing Tunnel  
In order to verify the reinforcement effects of the trapezoidal freezing soil mass, it was applied in the Wuhu River-Crossing Tunnel. As mentioned in Section 3, the optimized freezing soil mass dimensions are as follows. The upper and lower side lengths of the trapezoidal freezing soil mass were 2 m and 4 m, respectively. The height and width of the freezing soil mass were unchanged. The mixing pile dimensions were also the same as those in real engineering. The portal displacement was analyzed by the numerical simulation method.  
Figure 16 shows the displacement contour of the tunnel portal with the optimization scheme. Figure 16a shows that x-direction displacement is relatively larger on the left and right sides of the portal. The maximum x-direction displacements are \(-\)1.09 mm and 1.14 mm, respectively, on the left and right sides of the tunnel portal. Figure 16b shows that the z-direction displacement is relatively larger on the top and bottom of the tunnel portal. The maximum z-direction displacements of the portal are \(-\)1.12 mm and 1.14 mm, respectively, on the top and bottom of the tunnel portal.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Maximum y-Direction** & **Maximum x-Direction** & **Maximum z-Direction** \\  & **Displacement** & **Displacement** & **Displacement** \\ \hline trapezoidal freezing soil mass & 6.64 mm & 1.27 mm & 1.30 mm \\ rectangular freezing soil mass & 9.24 mm & 1.86 mm & 1.67 mm \\ \hline \hline \end{tabular} Note: Longitudinal length of the rectangular freezing soil mass is 3 m; the upper and lower side lengths of the trapezoidal freezing soil mass are 2 m and 4 m, respectively.  
\end{table}
Table 4: Displacement of the tunnel portal with trapezoidal freezing soil mass or rectangular freezing soil mass.  
Figure 15: Maximum displacement of tunnel portal corresponding to different upper side lengths of the trapezoidal freezing soil mass.  
Figure 17 is the y-direction displacement contour on the longitudinal section of the tunnel with the optimization scheme. From the figure, it can be seen that the portal's central displacement is the highest. The maximum y-direction displacement is \(-5.66\) mm. The maximum displacement value is much less than the allowable value of \(15\) mm [24; 25]. This means the optimized scheme can also meet the safety requirements of real engineering.  
Figure 18 shows the monitoring point displacement of the tunnel portal with the optimization scheme or the original scheme. From the figure, it can be seen that the displacement of the portal with the optimization scheme is always lower than that of the portal with the original scheme. The maximum reduction ratio is \(37.4\%\) and the minimum reduction ratio is \(14.2\%\).  
Figure 16: The displacement contour of the portal with the optimization scheme (unit: m).  
Figure 17: The y-direction displacement contour on the longitudinal section of the tunnel with the optimization scheme (unit: m).  
The portal's central displacement was 9.91 mm for the original scheme, while it was 5.56 mm for the optimization scheme. The reduction rate was 42.9%. This means that the optimization scheme is more effective than the original scheme in controlling the portal displacement. It can be concluded that the trapezoidal freezing soil mass is more effective for the tunnel end soil reinforcement compared with the rectangular freezing soil mass.