## 4 Frozen Wall Elastic-Plastic Problem
### Modified Solution for the Plastic Zone Radius  
The plastic zone contour equation given by Equation (22) does not consider the redistribution of stress in the plastic zone. In fact, when the frozen wall enters the plastic zone, its stress will be constantly adjusted and redistributed, resulting in continuous expansion of the plastic zone. In order to obtain a more accurate contour equation for the plastic zone than the previous one, a second approximation correction of the contour line of the plastic zone is performed considering stress redistribution of the plastic zone. In particular, for this second approximation, we use the Castner calculation method twice, wherein the initial plastic zone radius is calculated using the Castner method first, and then the stress values \(\sigma_{r}^{p}\) and \(\tau_{r\theta}^{p}\) of the calculated points are obtained using the stress equation of the plastic zone; in this case, the contour line of the plastic zone is assumed to be a circle (the circle is made per the radius of the plastic zone for the calculated point). Finally, a new contour line equation for the plastic zone considering the action of \(\sigma_{r}^{p}\) and \(\tau_{r\theta}^{p}\) is derived in a manner similar to [PERSON]'s method to obtain a more accurate plastic zone radius compared with that of the previous case. After the approximation correction similar to [PERSON]'s method, the elastic stress field of the frozen wall with \(\sigma_{r}^{p}\) and \(\tau_{r\theta}^{p}\) acting on the inner edge of the frozen wall is obtained as follows:  
\[\left\{ \begin{array}{l}\sigma_{r1}^{e}=-(2d_{1}+\frac{4e_{1}}{r^{2}}+\frac{6g _{1}}{r^{4}})\cos 2\theta+2a_{1}+2b_{1}\ln r+b_{1}+\frac{c_{1}}{r^{2}}+ \sigma_{r}^{p}\frac{r_{r}^{2}}{r_{r}^{2}}\\ \sigma_{\theta 1}^{e}=(2d_{1}+12f_{1}r^{2}+\frac{6g_{1}}{r^{4}})\cos 2\theta+2a_{1 }+2b_{1}\ln r+3b_{1}-\frac{c_{1}}{r^{2}}-\sigma_{r}^{p}\frac{r_{r}^{2}}{r^{2}} \\ \tau_{r\theta 1}^{e}=(6f_{1}r^{2}+2d_{1}-\frac{2e_{1}}{r^{2}}-\frac{6g_{1}}{r^{4 }})\sin 2\theta\quad+\tau_{r\theta}^{p}\frac{r_{r}^{2}}{r^{2}}\end{array} \right. \tag{26}\]  
By substituting Equation (26) into Equation (21), the modified implicit equation for the radius of the plastic zone can be obtained as follows:  
\[ \begin{array}{l}12f_{1}\cos 2\theta\Big{(}2a_{1}-\frac{B}{1-A}\Big{)}r^{2}-4e _{1}\cos 2\theta\Big{(}2a_{1}-\frac{B}{1-A}\Big{)}\frac{1}{r^{2}}+\left(36f_{1}^ {2}r^{4}+\frac{4e_{1}^{2}}{r^{4}}-24f_{1}e_{1}\right)\cos^{2}2\theta\\ =\Big{(}\frac{11+A}{1-A}\Big{)}^{2}\left[4d_{1}^{2}+36f_{1}^{2}r^{4}+24d_{1}f _{1}r^{2}+\frac{4e_{1}^{2}}{r^{4}}+\frac{36g_{1}^{2}}{r^{8}}+\frac{24e_{1}g_{1 }}{r^{6}}+\left(\frac{c_{1}}{r^{2}}+\sigma_{r}^{p}\frac{r_{r}^{2}}{r^{2}} \right)^{2}+\left(\tau_{r\theta}^{p}\right)^{2}\frac{r_{\theta}^{4}}{r^{2}} \right]\\ \qquad\qquad\qquad\qquad+\left(\frac{8d_{1}e_{1}}{r^{2}}+\frac{24d_{1}g_{1}} {r^{4}}+24f_{1}e_{1}+\frac{72f_{1}g_{1}}{r^{2}}\right)\cos 4\theta\\ \qquad\qquad\qquad\qquad-4\Big{(}d_{1}+3f_{1}r^{2}+\frac{c_{1}}{r^{4}}\Big{)} \bigg{(}\frac{c_{1}}{r^{2}}+\sigma_{r}^{p}\frac{r_{r}^{2}}{r^{2}}\bigg{)}\cos 2 \theta\\ \qquad\qquad\qquad+\left(12f_{1}r_{r\theta}^{p}r_{0}^{2}+\frac{4d_{1}r_{r \theta}^{p}r_{0}^{2}}{r^{2}}-\frac{4e_{1}r_{r\theta}^{p}r_{0}^{2}}{r^{4}}- \frac{12g_{1}r_{r\theta}^{p}r_{0}^{2}}{r^{6}}\right)\sin 2\theta\end{array}  \right]-\Big{(}2a_{1}-\frac{B}{1-A}\Big{)}^{2}\end{array} \tag{27}\]  
Engineering Examples and Analysis of the Evolution Law of the Elastoplastic Zone of the Frozen Wall  
In the freezing design of the horizontal freezing method, the size of the plastic zone is an important basis for evaluating the stability of the frozen wall. In order to study the evolution law of the elastoplastic zone of the frozen wall under nonuniform load, we taking a horizontal frozen tunnel project as the background; in particular, the effects of the vertical external load \(p_{0}\) and lateral pressure coefficient \(\lambda\) on the plastic zone contour and range of the tensile stress zone of the horizontal frozen wall are analyzed.