# Force Analysis of Circular Diaphragm Wall Based on Circular Cylindrical Shell Theory
## 4 Circular Cylindrical Shell Theory Calculation Method (CCSTCM)
### Internal Force Expressions of Circular Cylindrical Shell  
Equation (15) is very difficult to solve because the coefficients include \(R\) and \(\frac{\mathrm{d}R}{\mathrm{d}\beta}\), which are usually functions of \(\beta\), so the equilibrium differential equations are not constant coefficients but variable coefficients. However, if the middle face is set to a circular cylindrical surface, the cylindrical shell will be a circular cylindrical shell, and as a result \(R\) will become a constant and \(\frac{\mathrm{d}R}{\mathrm{d}\beta}\) will become 0. Therefore, the equilibrium differential Equation (15) variable coefficients become differential equations with constant coefficients, which are relatively easy to solve. The circular diaphragm wall of the north anchorage foundation pit in the engineering case can be approximately simplified into a thin circular cylindrical shell model. Moreover, it has been noted that the loads applied on the circular diaphragm wall are mainly in the horizontal direction (the normal direction) in the excavation process of the foundation pit, and the impact of the loads in other directions is slight; as a consequence, the loads in other directions can be ignored. Thus, Equation (15) can be further simplified to derive the basic equilibrium differential equations of the circular cylindrical shell under only a normal load, as shown in Equation (16).  
\[\begin{array}{c}\left(\frac{\partial^{2}}{\partial\alpha^{2}}+\frac{1-\mu} {2}\frac{\partial^{2}}{\partial\beta^{2}}\right)u+\frac{1+\mu}{2}\frac{ \partial^{2}v}{\partial\alpha\beta}+\frac{\mu}{R}\frac{\partial v}{\partial \alpha}=0\\ \frac{1+\mu}{2}\frac{\partial^{2}u}{\partial\alpha\beta}+\left(\frac{\partial ^{2}}{\partial\beta^{2}}+\frac{1-\mu}{2}\frac{\partial^{2}}{\partial\alpha^{2 }}\right)v+\frac{1}{R}\frac{\partial v}{\partial\beta}=0\\ \frac{u}{R}\frac{\partial u}{\partial\alpha}+\frac{1}{R}\frac{\partial v}{ \partial\beta}+\frac{w}{R^{2}}+\frac{\partial^{2}}{12}\
abla^{4}w=\frac{1-\mu ^{2}}{E\delta}q3\end{array} \tag{16}\]  
By introducing the displacement function \(H=H(\alpha,\beta)\), the displacements of the middle face can be expressed as Equation (17).  
\[\begin{array}{c}u=\frac{\partial}{\partial\alpha}\left(\frac{\partial^{2}}{ \partial\beta^{2}}-\mu\frac{\partial^{2}}{\partial\alpha^{2}}\right)H\\ v=-\frac{\partial}{\partial\beta}\left[\frac{\partial^{2}}{\partial\beta^{2 }}+(2+\mu)\frac{\partial^{2}}{\partial\alpha^{2}}\right]H\\ w=R\
abla^{4}H\end{array} \tag{17}\]  
Then, the first two equations of Equation (11) can be always satisfied, while the third equation must be satisfied with Equation (18).  
\[\
abla^{8}H+\frac{E\delta}{R^{2}D}\frac{\partial^{4}H}{\partial\alpha^{4}}= \frac{q_{3}}{RD} \tag{18}\]  
Then, the internal forces expressed by the displacement function \(H\) can be obtained by substituting Equation (17) into Equations (13) and (14), producing Equation (19), as follows. Thus, the boundary conditions can always be expressed by \(H\), and \(H\) can be solved from Equation (18) in the boundary conditions. Then, the displacements of the middle face can be solved from Equation (17), and the internal forces can be obtained from Equation (19).  
\[\begin{array}{c}F_{\mathrm{T1}}=E\frac{\delta^{4}H}{\partial\alpha^{2} \delta\beta^{2}}\\ F_{\mathrm{T2}}=E\frac{\delta^{4}H}{\partial\alpha^{4}}\\ F_{\mathrm{T12}}=F_{\mathrm{T21}}=-E\delta\frac{\partial^{4}H}{\partial a^{3} \partial\beta}\\ M_{1}=-RD\left(\frac{\partial^{2}}{\partial\alpha^{2}}+\mu\frac{\partial^{2}} {\partial\beta^{2}}\right)\
abla^{4}H\\ M_{2}=-RD\left(\frac{\partial^{2}}{\partial\beta^{2}}+\mu\frac{\partial^{2}} {\partial\alpha^{2}}\right)\
abla^{4}H\\ M_{12}=M_{21}=-(1-\mu)RD\frac{\partial^{2}}{\partial a\partial\beta}\
abla^{4 }H\\ F_{\mathrm{S1}}=-RD\frac{\partial}{\partial a\
abla}\
abla^{6}H\\ F_{\mathrm{S2}}=-RD\frac{\partial}{\partial\beta}\
abla^{6}H\end{array} \tag{19}\]