## 5 Stress Field Distribution of Frozen Wall and Numerical Verification: A Case Study  
The Dongfeng shaft of Qingdong Mine in the Huaibei mining area is constructed by the artificial ground freezing method. The clay layer with a vertical depth of 225 m is frozen by double-row-pipe. The inner radius of the frozen wall is \(R_{B}=4.053\) m, the wall temperature is \(-9\)\({}^{\circ}\)C, the outer radius is \(R_{e}=8.147\) m, and the corresponding temperature is \(-2.11\)\({}^{\circ}\)C; \(r_{e}=R_{e}/R_{B}=2.01\). The minimum temperature of the trapezoidal equivalent temperature field is \(-22.32\)\({}^{\circ}\)C. The material is soil taken from the frozen wall. The internal friction angle of soil is measured by a triaxial shear test, and the Poisson's ratio of the soil is measured by a triaxial compression test. The internal friction angle of the frozen soil is \(10.5^{\circ}\), and the Poisson's ratio is \(0.29\). The temperature field distribution of the frozen wall and the material parameters obtained from the thermophysical and mechanical test report of the frozen soil are shown in Equations (25)-(27). \[T(r)= \begin{cases}-53.27r+44.27&r\in[1,1.25]\\ -22.32&r\in[1.25,1.63]\\ 53.27r-109.15&r\in[1.63,2.01]\end{cases}  \tag{25}\]  
\[E(r)=\begin{cases} \begin{array}{ll}359.04r-284.26&r\in[1,1.25]\\ 164.54&r\in[1.25,1.63]\\ -359.04r+749.78&r\in[1.63,2.01]\end{array} \\ C(r)= \begin{cases}\begin{array}{ll}5.27r-4.51&r\in[1,1.25]\\ 2.08&r\in[1.25,1.63]\\ -5.27r+10.67&r\in[1.63,2.01]\end{array}\end{cases}  \tag{27}\]  
The external load of the frozen wall is generally calculated by the heavy liquid equation [36]:  
\[P=0.013h \tag{28}\]  
where \(h\) is the depth, m.  
The calculation shows that the external load borne by the frozen wall at the vertical depth of 225 m is 2.925 MPa. On the premise that \(b=0.5\), according to the equations derived above, it can be obtained that the relative plastic radius of the frozen wall under external load is \(r_{\rho}=1.32\). The radial stress \(\sigma_{r}\) and tangential stress \(\sigma_{\theta}\) distribution of the frozen wall are shown in Figure 3. Figure 4 is drawn by Origin software. The radial stress \(\sigma_{r}\) of the frozen wall increases with the increase in relative radius \(r\) and reaches the maximum at the outer frozen wall. The tangential stress \(\sigma_{\theta}\) first increases and then decreases with the increase in the relative radius \(r\). When \(r\in[1,r_{\rho}]\), in the plastic zone, the tangential stress \(\sigma_{\theta}\) of the frozen wall increases with the increase in the relative radius \(r\), and the rate of increase in the tangential stress between the inner wall of the frozen wall and the inner frozen pipe is greater than the rate of increase between the inner frozen pipe and the radius of the plastic zone; the tangential stress reaches the maximum at the plastic radius. When \(r\in[r_{\rho},r_{e}]\), in the elastic zone, the tangential stress of the frozen wall decreases with the increase in the relative radius, and the reduction rate of the tangential stress between the plastic zone radius and the outer frozen pipe is less than the reduction rate between the outer frozen pipe and the outer wall of the frozen wall. According to the equation derived above, when the strength theoretical parameter \(b=0.5\), the elastic limit load and plastic limit load of the frozen wall are 1.44 MPa and 3.55 MPa, respectively. The external load of the frozen wall is greater than its elastic ultimate bearing capacity, and the external load of the frozen wall is less than its plastic ultimate bearing capacity, indicating that the stress state of the frozen wall is safe. In order to verify the rationality of the analytical solution, COMSOL software is used for numerical simulation. A numerical calculation model is established based on the Drucker-Prager strength criterion.