## 3 Design for the Frozen Wall Thickness  
### Calculation Method and Formula  
For the frozen wall with its inner edge radially incompletely unloaded, the deformation is constrained and the displacement of the inner edge is limited to a part of that with the inner edge radially completely unloaded. To express the degree of being unloaded, parameter \(\alpha\) is introduced using the ratio of the displacement of the inner edge of the frozen wall with its inner edge radially incompletely unloaded to that of the inner edge of the same frozen wall with its inner edge radially completely unloaded; see Equation (38):  
\[\alpha=1-\frac{u^{\mathrm{I}}\big{|}_{r=r_{b}}}{U^{\mathrm{I}}\big{|}_{r=r_{b}}} \tag{38}\]  
where \(\alpha\) decreases with the degree of being unloaded, and \(\alpha=0\) represents that the inner edge of the frozen wall is radially completely unloaded. \(U^{\mathrm{I}}\big{|}_{r=r_{b}}\) is the displacement of the inner edge of the same frozen wall with its inner edge radially completely unloaded. Substituting \(r_{p}=R_{\mathrm{p}}\) and \(p=0\) into Equation (37), \(U^{\mathrm{I}}\big{|}_{r=r_{b}}\) is expressed as \[U^{I}\Big{|}_{r=r_{b}}=\frac{R_{p}}{E^{\Pi}}\Big{(}1+\mu^{\Pi}\Big{)}\left(\frac{R_ {p}}{r_{b}}\right)^{\beta}\Bigg{\{}\frac{-\sigma_{c}}{A-1}\Bigg{[}\bigg{(}\frac{ R_{p}}{r_{b}}\bigg{)}^{A-1}-1\Bigg{]}-p_{0}\Bigg{\}}\frac{\big{(}\frac{2}{M}-1 \big{)}\big{(}1-2\mu^{\Pi}\big{)}-\frac{r_{c}^{2}}{R_{p}^{2}}}{\frac{2}{M}-1+ \frac{r_{c}^{2}}{R_{p}^{2}}} \tag{39}\]  
where \(R_{p}\) is the outer radius of the plastic zone of the same frozen wall with its inner edge radially completely unloaded. Substituting \(r_{p}=R_{p}\) and \(p=0\) into Equation (30) \(R_{p}\) can be solved using  
\[\left(\frac{R_{p}}{r_{b}}\right)^{A-1}=\frac{-\frac{R_{0}}{\sigma_{c}}+\frac{ 1}{A-1}}{\Big{(}\frac{1}{M}-\frac{1}{2}\Big{)}\frac{R_{p}^{2}}{r_{c}^{2}}+ \frac{A+1}{2(A-1)}} \tag{40}\]  
Following [PERSON]'s limitation imposed on the outer radius of the frozen wall plastic zone [13], \(r_{p}=\sqrt{r_{b}r_{c}}\) is substituted into Equations (30) and (37). Combining Equation (38) with Equations (38)-(40), the formulas for the design thickness of the frozen wall with its inner edge radially incompletely unloaded are determined as follows:  
\[ \begin{array}{c}\Big{(}\frac{r_{c}}{r_{b}}\Big{)}^{\frac{A-1}{2}}=\frac{ \frac{p_{0}}{(A-1)-\sigma_{c}}-\frac{1}{A-1}\frac{\sigma_{c}}{p(A-1)-\sigma_{ c}}}{\big{(}\frac{M}{A-1}\big{)}\frac{r_{c}}{p}+\frac{A+1}{2(A-1)}}\\ \frac{\frac{1+\beta}{r_{c}^{2}}\frac{1-\beta}{r_{b}^{2}}\big{(}1+\mu^{\Pi} \big{)}}{E^{\Pi}}\bigg{[}\big{(}\frac{-\sigma_{c}}{A-1}+p\big{)}\Big{(}\frac{ r_{c}}{r_{b}}\Big{)}^{\frac{A-1}{2}}+\frac{\sigma_{c}}{A-1}-p_{0}\bigg{]}\frac{ \big{(}\frac{2}{M}-1\big{)}\big{(}1-2\mu^{\Pi}\big{)}-\frac{r_{c}}{p}}{\frac{2 }{M}-1+\frac{r_{c}}{p}}=\frac{p}{k}+r_{a}-r_{b}\\ =\frac{R_{p}(1-\alpha)\big{(}1+\mu^{\Pi}\big{)}}{E^{\Pi}}\Big{(}\frac{R_{p}}{ r_{b}}\Big{)}^{\beta}\Bigg{\{}\frac{-\sigma_{c}}{A-1}\bigg{[}\Big{(}\frac{R_{p}}{ r_{b}}\Big{)}^{A-1}-1\bigg{]}-p_{0}\Bigg{\}}\frac{\big{(}\frac{2}{M}-1\big{)} \big{(}1-2\mu^{\Pi}\big{)}-\frac{r_{c}^{2}}{R_{p}^{2}}}{\frac{2}{M}-1+\frac{r _{c}^{2}}{R_{p}^{2}}}\\ \Big{(}\frac{R_{p}}{r_{b}}\Big{)}^{A-1}=\frac{\frac{1}{A-1}-\frac{R_{0}}{\sigma _{c}}}{\big{(}\frac{M}{A-1}\big{)}\frac{R_{p}^{2}}{r_{c}^{2}}+\frac{A+1}{2(A-1 )}}\\ t=r_{c}-r_{b}\end{array}  \tag{41}\]  
\(r_{a}\) has a known value, and parameter \(\alpha\) in Equation (41) will be discussed later. The mechanical parameters of the shaft lining, the frozen wall, and the surrounding unfrozen alluvium are substituted into Equation (41) to obtain \(t\), \(p\), \(r_{b}\), \(r_{c}\), and \(R_{p}\).