## 7 Conclusions  
Based on unified strength theory, the elastic-plastic analysis of the double-row-pipe frozen wall is carried out, and the analytical solutions of the elastic-plastic stress field distribution, the elastic ultimate bearing capacity, the plastic ultimate bearing capacity, and the relative radius of the plastic zone of the frozen wall are derived. The stress field distribution of the frozen wall in the control layer of the surface soil layer in Qingdong Mine is analyzed using the analytical solution, and the numerical simulation is carried out based on COMSOL. This study determined whether the frozen wall of this layer meets the strength requirements and analyzed the influence of strength theoretical parameters on the stress characteristics of heterogeneous and homogeneous frozen walls. The following conclusions were obtained:  
1. Considering the FGM characteristics of the frozen wall, the temperature field of the double-row-pipe frozen wall is equivalent to a trapezoidal distribution along the radial direction, and the elastic modulus \(E\) and cohesion \(c\) of the frozen wall are considered to be linearly distributed with the radius. Based on unified strength theory, the analytical solutions of the elastic-plastic stress field, the elastic ultimate bearing capacity, the plastic ultimate bearing capacity, and the relative radius of the plastic zone of the double-row-pipe frozen wall are derived; the results can provide a theoretical reference for the design of frozen wall. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(b\)} & \multirow{2}{*}{\(\mathbf{P}\) (MPa)} & \multicolumn{4}{c}{**Heterogeneous Frozen Wall**} & \multicolumn{4}{c}{**Homogeneous Frozen Wall**} \\ \cline{2-9}  & & \(r_{p}\) & \(P_{e}\) (MPa) & \(P_{p}\) (MPa) & \(K\) & \(r_{p}\) & \(P_{e}\) (MPa) & \(P_{p}\) (MPa) & \(K\) \\ \hline
0 & & 1.63 & 1.20 & 3.11 & 1.06 & 1.94 & 1.35 & 2.93 & 2.17 \\
0.2 & & 1.44 & 1.31 & 3.44 & 1.18 & 1.57 & 1.47 & 3.28 & 2.23 \\
0.4 & & 1.35 & 1.40 & 3.73 & 1.28 & 1.45 & 1.57 & 3.52 & 2.24 \\
0.6 & & 1.30 & 1.47 & 3.97 & 1.36 & 1.37 & 1.66 & 3.75 & 2.26 \\
0.8 & & 1.26 & 1.54 & 4.19 & 1.43 & 1.32 & 1.73 & 3.95 & 2.28 \\
1 & & 1.23 & 1.60 & 4.38 & 1.50 & 1.28 & 1.80 & 4.13 & 2.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical characteristics of heterogeneous and homogeneous frozen walls with different values of strength theory parameter \(b\). Figure 9: The influence of different strength theory parameter \(b\) values on the stress fields of heterogeneous and homogeneous frozen walls. 2. We analyzed a frozen wall with a vertical depth of 225 m in Qingdong Mine. When \(b\) is 0.5, the elastic ultimate bearing capacity of the frozen wall is 1.44 MPa, and the plastic ultimate bearing capacity of the frozen wall is 3.55 MPa. The external load \(p=2.925\) MPa of the frozen wall is between the elastic ultimate bearing capacity and the plastic ultimate bearing capacity, indicating that the stress state of the frozen wall is safe. Under the action of external load, the relative radius \(r_{\rho}\) of the plastic zone of the frozen wall is 1.32, which is located between the inner and outer frozen pipes. 3. Based on the Qingdong Mine, the analytical solution was calculated and compared with the numerical solution, and it is concluded that the stress distribution of the two solutions is consistent, which mutually verifies the accuracy of the two solutions. 4. The strength theory parameter \(b\) has an important influence on the stress state of heterogeneous and homogeneous frozen walls. The radial stress increases with the increase in \(b\) and the relative radius \(r\), the tangential stress increases with the increase in \(b\), and the tangential stress first increases and then decreases with the increase in \(r\). The elastic ultimate bearing capacity \(P_{e}\) and \(P_{p}\) of the frozen wall increase with the increase in \(b\).