# Mechanical Behavior of Hollow Corrugated Sandwich Cylinders Under Inner Pressure Loading
## 3 Structural Analysis
### Closed-Form Solution for Corrugated Core Panel  
For a long, hollow sandwich cylinder with no column caps, the problem can be treated as plane axisymmetric. Based on the elastic solution of the anisotropic hollow cylinder under inner and outer pressure, the closed-form stress solution of the sandwich can be obtained [15].  
Figure 3: Forces on panels of the corrugated sandwich cylinder shell.  
\[\left\{\begin{array}{rl}\sigma_{r}^{c}&=\frac{P_{a}c^{k+1}-P_{b}}{1-c^{2k}}\rho ^{k-1}+\frac{P_{b}c^{k-1}-P_{a}}{1-c^{2k}}ck^{k+1}\rho^{-k-1}\\ &+B_{3}\eta_{1}\Big{(}1+\frac{1-c^{k+1}}{c^{2k}}\rho^{k-1}+\frac{1-c^{k+1}}{1- c^{2k}}ck^{k+1}\rho^{-k-1}\Big{)}\\ &+\theta\mu_{1}R_{0}\Big{(}2\rho-\frac{1-c^{k+1}}{1-c^{2k}}\rho^{k-1}+\frac{1- c^{k-2}}{1-c^{2k}}ck^{k+1}\rho^{-k-1}\Big{)}\\ \sigma_{\theta}^{c}&=\frac{P_{c}ck^{k+1}-P_{b}}{1-c^{2k}}k\rho^{k-1}-\frac{P_{ b}c^{k-1}-P_{a}}{1-c^{2k}}ck^{k+1}\rho^{-k-1}\\ &+B_{3}\eta_{1}\Big{(}1+\frac{1-c^{k+1}}{c^{2k}}k\rho^{k-1}+\frac{1-c^{k+1}}{1 -c^{2k}}ck^{k+1}\rho^{-k-1}\Big{)}\\ &+\theta\mu_{1}R_{0}\Big{(}2\rho-\frac{1-c^{k+2}}{1-c^{2k}}k\rho^{k-1}+\frac{1 -c^{k-2}}{1-c^{2k}}ck^{k+2}\rho^{-k-1}\Big{)}\\ \sigma_{z}^{c}&=&-\frac{1}{s_{33}}\big{(}s_{13}\sigma_{RR}^{c}+s_{23}\sigma_{ \theta\theta}^{c}+s_{34}\sigma_{ZZ}^{c}\big{)}\\ &=&-\frac{1}{s_{33}}\big{[}\frac{P_{a}c^{k+1}-P_{b}}{1-c^{2k}}(s_{13}+ks_{23}) \rho^{k-1}+\frac{P_{b}c^{k-1}-P_{a}}{1-c^{2k}}(s_{13}-ks_{23})c^{k+1}\rho^{-k-1 }\big{]}\\ &+B_{3}\left[-\eta_{2}+\eta_{1}\Big{(}1+\frac{1-c^{k+1}}{c^{2k}}sk_{1}\rho^{k -1}+\frac{1-c^{k-1}}{1-c^{2k}}sk_{2}ck^{k+1}\rho^{-k-1}\Big{)}\right]\\ &+\theta R_{0}\Big{(}\mu_{2}\rho+\mu_{1}\frac{1-c^{k+2}}{1-c^{2k}}g_{k1}\rho^{ k-1}+\frac{1-c^{k-2}}{1-c^{2k}}sk_{2}ck^{k+2}\rho^{-k-1}\Big{)}\end{array},\right. \tag{8}\]  
where  
\[\mu_{1}=\frac{\beta_{14}-2\beta_{24}}{4(\beta_{22}\beta_{44}-\beta_{24}^{2})- \big{(}\beta_{11}\beta_{44}-\beta_{14}^{2}\big{)}}, \tag{9}\]  
\[\mu_{2}=\frac{\beta_{11}-4\beta_{22}}{4(\beta_{22}\beta_{44}-\beta_{24}^{2})- \big{(}\beta_{11}\beta_{44}-\beta_{14}^{2}\big{)}}, \tag{10}\]  
\[\eta_{1}=\frac{(s_{13}-s_{23})\beta_{44}-s_{34}(\beta_{14}-\beta_{24})}{\beta _{22}\beta_{44}-\beta_{24}^{2}-(\beta_{11}\beta_{44}-\beta_{14}^{2})}, \tag{11}\]  
\[\eta_{2}=\frac{(s_{13}-s_{23})(\beta_{14}+\beta_{24})-s_{34}(\beta_{11}-\beta _{22})}{\beta_{22}\beta_{44}-\beta_{24}^{2}-(\beta_{11}\beta_{44}-\beta_{14}^{ 2})}, \tag{12}\]  
\[k=\sqrt{\frac{\beta_{44}\beta_{11}-\beta_{14}^{2}}{\beta_{22}\beta_{44}-\beta_ {24}^{2}}}, \tag{13}\]  
\[g_{k1}=\frac{\beta_{14}+k\beta_{24}}{\beta_{44}}, \tag{14}\]  
\[g_{k2}=\frac{\beta_{14}-k\beta_{24}}{\beta_{44}}, \tag{15}\]  
\[c=\frac{R_{i}+T_{i}}{R_{o}}, \tag{16}\]  
\[\beta_{ij}=s_{ij}-\frac{s_{i3}s_{j3}}{s_{33}},(i,j=1,\ldots,6), \tag{17}\]  
\[\rho=\frac{R}{R_{o}-T_{o}/2},R\in[R_{i}+T_{i},R_{o}], \tag{18}\]  
where \(s_{11},s_{12},s_{22},s_{13}\), and \(s_{23}\) are the homogenized effective compliance tensors of the core. \(\sigma_{RR}^{c},\sigma_{\theta\theta}^{c},\) and \(\tau_{ZZ}^{c}\) are stresses in an isotropic core. \(c,k\), and \(\rho\) are constants.  
The stress and displacement are independent of the coordinate \(Z\), thus the boundary condition of both ends can be treated as fixed [20], which is as follows:  
\[B_{3}=\vartheta=0. \tag{19}\]  
Thus, the closed-form stress solution of the sandwich can be written as \[\left\{\begin{array}{l}\sigma_{r}^{c}=\frac{P_{a}c^{k+1}-P_{b}}{1-c^{2k}}\rho^{k -1}+\frac{P_{b}c^{k-1}-P_{a}}{1-c^{2k}}c^{k+1}\rho^{-k-1}\\ \sigma_{\theta}^{c}=\frac{P_{a}c^{k+1}-P_{b}}{1-c^{2k}}kp^{k-1}-\frac{P_{b}c^{k -1}-P_{a}}{1-c^{2k}}kc^{k+1}\rho^{-k-1}\\ \sigma_{z}^{c}=-\frac{1}{s_{33}}\big{(}s_{13}\sigma_{RR}^{c}+s_{23}\sigma_{ \theta\theta}^{c}+s_{34}\sigma_{6Z}^{c}\big{)}=-\frac{1}{s_{33}}\big{[}\frac{P_ {a}c^{k+1}-P_{b}}{1-c^{2k}}(s_{13}+ks_{23})\rho^{k-1}+\frac{P_{b}c^{k-1}-P_{a}} {1-c^{2k}}(s_{13}-ks_{23})c^{k+1}\rho^{-k-1}\big{]}\end{array}\right. \tag{20}\]