## 3 Analysis
### Free Ring and Thin-Shell Theory
The rotation of side AB about the \(z\)-axis of side OC is caused by the bending of the generatrix in the tangent plane, and it is equal to the following: \[\bigg{(}\frac{\partial^{2}v}{\partial\theta}\frac{-\partial w}{\partial x}\bigg{)}d\theta\] (7) By projecting all forces depicted in Figure 14 on the \(x\)-, \(y\)- and \(z\)-axis, three equilibrium equations of volume element OABC can be obtained as follows: \[R\frac{\partial N_{x}}{\partial x}+\frac{\partial N_{yx}}{\partial\theta}-N_{y }\bigg{(}\frac{\partial^{2}v}{\partial\theta\cdot\partial x}-\frac{\partial w }{\partial x}\bigg{)}-RN_{xy}\frac{\partial^{2}v}{\partial x^{2}}-RQ_{x}\frac {\partial^{2}w}{\partial x^{2}}-Q_{y}\bigg{(}\frac{\partial^{2}w}{\partial \theta\cdot\partial x}-\frac{\partial v}{\partial x}\bigg{)}=0\] (8a) \[\frac{\partial N_{y}}{\partial\theta}+R\frac{\partial N_{xy}}{\partial x}+RN _{z}\frac{\partial^{2}v}{\partial x^{2}}-Q_{x}\bigg{(}\frac{\partial v}{ \partial x}+\frac{\partial^{2}w}{\partial x\cdot\partial\theta}\bigg{)}+N_{ yx}\bigg{(}\frac{\partial^{2}v}{\partial x\cdot\partial\theta}-\frac{\partial^{2}w}{ \partial x\cdot\partial\theta}\bigg{)}-Q_{y}\bigg{(}1+\frac{\partial v}{R \cdot\partial\theta}+\frac{\partial^{2}w}{R\cdot\partial\theta}\bigg{)}=0\] (8b) \[R\frac{\partial Q_{x}}{\partial x}+\frac{\partial Qy}{\partial\theta}+N_{xy} \bigg{(}\frac{\partial v}{\partial x}+\frac{\partial^{2}w}{\partial x\cdot \partial\theta}\bigg{)}+RN_{x}\frac{\partial^{2}w}{\partial x^{2}}+N_{y} \bigg{(}1+\frac{\partial v}{R\cdot\partial\theta}+\frac{\partial^{2}w}{R\cdot \partial\theta^{2}}\bigg{)}+N_{yx}\bigg{(}\frac{\partial v}{\partial x}+\frac {\partial^{2}w}{\partial x\cdot\partial\theta}\bigg{)}+pR=0\] (8c)  
For the three moment equations of \(x\)-, \(y\)- and \(z\)-axis, considering the slight displacement of side BC and side AB for side OA and side OC, respectively, it can be obtained as follows: \[-\frac{\partial^{2}w}{\partial x^{2}}dx\]0 (9a) \[-\frac{\partial^{2}w}{\partial x^{2}}dx\]1 (9b) \[-\frac{\partial^{2}w}{\partial x^{2}}dx\]2 (9c)  
The resultant force \(Nx\), \(Ny\) and \(Nxy\) and torque \(Mx\), \(My\) and \(Mxy\) in the above two equations can be represented by the three strain components \(\varepsilon_{1}\), \(\varepsilon_{2}\), \(\gamma\), and the changes in curvature \(\chi_{x}\), \(\chi_{y}\), and \(\chi_{xy}\). Moreover, the variation of strain components and curvature can be represented by displacements \(u\), \(v\) and \(w\):  
\[-\frac{\partial^{2}w}{\partial x^{2}}dx\]3 \[-\frac{\partial^{2}w}{\partial x^{2}}dx\]4 (10b) \[-\frac{\partial^{2}w}{\partial x^{2}}dx\]5 (10c) \[-\frac{\partial^{2}w}{\partial x^{2}}dx\]6 (11a) \[-\frac{\partial^{2}w}{\partial x^{2}}dx\]7  
\[-\frac{\partial^{2}w}{\partial x^{2}}dx\]8 An integral element OABC from the thin shell. \[-\frac{\partial^{2}w}{\partial x^{2}}dx\]9  
When the ratio of length to the diameter of a cylinder is not large enough, the boundary conditions cannot be ignored. Assuming that all the resultant forces except \(N_{y}\) in Equation (8a-c) are small for the uniform transverse pressure, and the product of the derivatives of the displacements of \(u\), \(v\) and \(w\) of these resultant forces can be omitted, the following equation can be obtained:  
\[\frac{\partial^{2}v}{\partial x^{2}}dx\]0  
\[\frac{\partial^{2}v}{\partial x^{2}}dx\]1  
\[\frac{\partial^{2}v}{\partial x^{2}}dx\]2  
Assuming that the bending moment and the torque are very small, and omitting the product term of the derivatives of these torques and the derivatives of the displacements \(u\), \(v\) and \(w\), we get the following:  
\[\frac{\partial^{2}v}{\partial x^{2}}dx\]3  
\[\frac{\partial^{2}v}{\partial x^{2}}dx\]4  
By substituting Equation (13a,b) into Equation (12a-c), respectively, then we get the following:  
\[\frac{\partial^{2}v}{\partial x^{2}}dx\]5  
\[\frac{\partial^{2}v}{\partial x^{2}}dx\]6  
\[\frac{\partial^{2}v}{\partial x^{2}}dx\]7  
The cylindrical shell remains circular under uniform distribution pressure and suffers uniform circumferential compression.