## 4 Frozen Wall Elastic-Plastic Problem
### Safety Analysis of Freezing Engineering  
By substituting the relevant engineering and geological parameters into Equations (15)-(18) and solving the simultaneous equations for values of \(a_{i}\), \(b_{i}\), \(c_{i}\), \(d_{i}\), \(e_{i}\), \(f_{i}\), and \(g_{i}\) (\(i=1,2\)) using MAPLE, we obtain the values listed in the Table 2.  
The stress values of the frozen wall under the elastic state can be obtained by substituting the undetermined parameters into Equation (12); the corresponding values for principal stresses, principal stress direction, and yield state for \(k\pi/12\) (\(k=0\), \(1\), \(2\), \(3\), \(4\), \(5\), \(6\)) directions are listed in Table 3. Verification using Equation (21) indicates that plastic yield does not occur at the inner edge of the frozen wall of the tunnel; furthermore, the maximum circumferential stress at the inner edge of the frozen wall is \(2.76\) MPa, which meets the safety factor requirement of the compressive strength being twice the maximum circumferential stress. Under the actual working conditions for the tunnel considered in this study, for a frozen wall thickness of \(3\) m, it was calculated that, when the vertical external load increases to \(1.83\) MPa, the frozen wall reaches the elastic limit at the horizontal inner edge, and the vertical external load of the elastic limit is \(2.03\) times the design load, indicating that the tunnel freezing methodology leads to safe and reliable results. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline \(a_{1}\) & \(0.52\) & \(a_{2}\) & \(0.41\) \\ \(b_{1}\) & \(0.00\) & \(b_{2}\) & \(0.00\) \\ \(c_{1}\) & \(-11.36\) & \(c_{2}\) & \(-2.11\) \\ \(d_{1}\) & \(0.106\) & \(d_{2}\) & \(0.045\) \\ \(e_{1}\) & \(-2.06\) & \(e_{2}\) & \(-0.39\) \\ \(f_{1}\) & \(-6.60\times 10^{-4}\) & \(f_{2}\) & \(0.00\) \\ \(g_{1}\) & \(10.80\) & \(g_{2}\) & \(-1.72\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stress function parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Angle** & **First Principal** & **Second Principal** & **Principal Stress** \\
**0**\({}^{\circ}\) & **Stress** & **Stress** & **Direction** \\  & **\(\sigma_{1}\)/MPa** & **\(\sigma_{2}\)/MPa** & **A+0**\({}^{\circ}\) \\ \hline \(0^{\circ}\) & \(2.76\) & \(0\) & \(0^{\circ}\) & Not yielding \\ \(15^{\circ}\) & \(2.67\) & \(0\) & \(15^{\circ}\) & Not yielding \\ \(30^{\circ}\) & \(2.42\) & \(0\) & \(30^{\circ}\) & Not yielding \\ \(45^{\circ}\) & \(2.09\) & \(0\) & \(45^{\circ}\) & Not yielding \\ \(60^{\circ}\) & \(1.75\) & \(0\) & \(60^{\circ}\) & Not yielding \\ \(75^{\circ}\) & \(1.51\) & \(0\) & \(75^{\circ}\) & Not yielding \\ \(90^{\circ}\) & \(1.42\) & \(0\) & \(90^{\circ}\) & Not yielding \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Stress and elastic-plastic state of the inner edge of the frozen wall. Figure 9: Grid division and frozen temperature field distribution cloud map. By substituting the parameters listed in Tables 1 and 2 in Equations (12) and (14), the stress and displacement distribution of the frozen wall in the No. 7 tunnel using the freezing method construction can be obtained as shown in Figures 10 and 11. As shown in Figure 10, because the frozen wall is subject to a nonuniform stress field, its radial stress increases at different gradients with the radius in each direction. In particular, when \(\lambda<1\), the radial stress in the horizontal direction increases faster than that in the vertical direction near the inner edge of the frozen wall; in addition, the magnitude of horizontal radial stress is larger than vertical radial stress. In contrast, near the outer edge, the radial stress in the vertical direction increases faster than that in the horizontal direction, and its value is also greater than that in the horizontal direction. However, for \(r=5.31m\), the radial stresses in all directions are equal to 0.64 MPa. Furthermore, the circumferential stress in the horizontal direction is significantly greater than that in the vertical direction.