# Force Analysis of Circular Diaphragm Wall Based on Circular Cylindrical Shell Theory
## 4 Circular Cylindrical Shell Theory Calculation Method (CCSTCM)  
### Internal Force Expressions of Cylindrical Shell  
The cylindrical shell is a special case of the shell in elasticity, and the equilibrium differential equations of the cylindrical shell can be expressed by Equation (10) under arbitrary load [29]. In Equation (10), as shown in Figure 3, \(F_{\rm T1}\) represents the tension or pressure applied on the unit width of the middle face on face \(\alpha\); \(F_{\rm T12}\) represents the leveling force applied on the unit width of the middle surface on Surface \(\alpha\); \(F_{\rm T2}\) represents the tension or pressure applied on the unit width of the middle face on face \(\beta\); \(F_{\rm T21}\) represents the leveling force applied on the unit width of the middle face on face \(\beta\), and \(F_{\rm T21}\)=\(F_{\rm T12}\). These four internal forces are called membrane internal forces. \(M_{1}\) represents the bending moment applied on the unit width of the middle face on face \(\alpha\); \(M_{12}\) represents the torque applied on the unit width of the middle face on face \(\alpha\); \(F_{\rm S1}\) represents the transverse shear force applied on the unit width of the middle face on face \(\alpha\); \(M_{2}\) represents the bending moment applied on the unit width of the middle face on face \(\beta\); \(M_{21}\) represents the torque applied on the unit width of the middle face on face \(\beta\), and \(M_{21}\)=\(M_{12}\); and \(F_{\rm S2}\) represents the transverse shear force applied on the unit width of the middle face on face \(\beta\). These six internal forces are called bending internal forces. \(q_{1}\), \(q_{2}\) and \(q_{3}\) represent the distribution loads applied on the cylindrical shell in directions \(\alpha\), \(\beta\) and \(\gamma\), respectively.  
\[\left.\begin{array}{l}\frac{\partial F_{\rm T1}}{\partial\alpha}+\frac{ \partial F_{\rm T2}}{\partial\beta}+q_{1}=0\\ \frac{\partial F_{\rm T2}}{\partial\beta}+\frac{\partial F_{\rm T21}}{ \partial\alpha}+q_{2}=0\\ -\frac{F_{\rm T2}}{\partial\alpha}+\frac{\partial F_{\rm S1}}{\partial\alpha} +\frac{\partial F_{\rm S2}}{\partial\beta}+q_{3}=0\\ \frac{\partial M_{12}}{\partial\beta}+\frac{\partial M_{1}}{\partial\alpha} -F_{\rm S1}=0\\ \frac{\partial M_{21}}{\partial\alpha}+\frac{\partial M_{2}}{\partial\beta} -F_{\rm S2}=0\end{array}\right\} \tag{10}\]  
The geometric equations of the cylindrical shell are shown in Equation (11), as follows. In Equation (11), \(R\) represents the radius of the middle face of the cylindrical shell; \(\varepsilon_{1}\) and \(\varepsilon_{2}\) represent the normal strains along the directions \(\alpha\) and \(\beta\) on the middle face of the cylindrical shell, respectively; \(\varepsilon_{12}\) and \(\varepsilon_{21}\) represent the shear strains along the directions \(\alpha\) and \(\beta\) on the middle face of the cylindrical shell, respectively; \(\chi_{1}\) and \(\chi_{2}\) represent the change of the main curvature of each point along the directions \(\alpha\) and \(\beta\) on the middle face of the cylindrical shell, respectively; \(\chi_{12}\) and \(\chi_{21}\) represent the change of the twisting curvature of each point along the directions \(\alpha\) and \(\beta\) on the middle face of the cylindrical shell, respectively; and \(u\), \(v\) and \(w\) represent the displacement of the cylindrical shell in the directions \(\alpha\), \(\beta\) and \(\gamma\), respectively.