# Mechanical Behavior of Hollow Corrugated Sandwich Cylinders Under Inner Pressure Loading
## 4 The Minimum Weight Design Model
### Failure Mode  
The deformation of the incline core panel of the hollow corrugated sandwich cylinder is along the axial direction; thus, the maximum membrane stress can be described as follows:  
\[\left\{\begin{array}{ll}\sigma_{y}^{c}=\sigma_{r}^{c}(s_{11}\sin^{2}\alpha+ s_{12}\cos^{2}\alpha)+\sigma_{\theta}^{c}(s_{12}\sin^{2}\alpha+s_{22}\cos^{2} \alpha)+\sigma_{z}^{c}(s_{13}\sin^{2}\alpha+s_{23}\cos^{2}\alpha)\\ \sigma_{x}^{c}=\frac{\sigma_{z}^{c}p(2R_{o}-3t_{c})\sin\alpha}{2(R_{o}-2t_{c}) T_{c}}\end{array}\right. \tag{29}\]  
Considering three failure modes, i.e., facesheet yielding, core yielding, and core buckling,  
\[\left\{\begin{array}{ll}\frac{\sqrt{2}}{2}\left\{\left(\frac{\sigma_{x}^{c}} {E}-\frac{\sigma_{x}^{c}}{E}\right)^{2}+\left(\frac{\sigma_{x}^{c}}{E}-\frac {\sigma_{x}^{c}}{E}\right)^{2}+\left(\frac{\sigma_{x}^{c}}{E}-\frac{\sigma_{x }^{c}}{E}\right)^{2}\right\}^{1/2}\leq\frac{\sigma_{y}}{E}&\text{facesheet yielding}\\ \frac{\sqrt{2}}{2}\left\{\left(\frac{\sigma_{x}^{c}}{E}-\frac{\sigma_{y}^{c}} {E}\right)^{2}+\left(\frac{\sigma_{x}^{c}}{E}\right)^{2}+\left(\frac{\sigma_{ y}^{c}}{E}\right)^{2}\right\}^{1/2}\leq\frac{\sigma_{y}}{E}&\text{core yielding}\\ \frac{\sigma_{x}^{c}}{E}\leq\frac{\pi^{2}T_{c}^{2}\sin^{2}\alpha}{12(1-\mu^{2 })h_{c}^{2}}&\text{core buckling}\end{array}\right. \tag{30}\]  
where  
\[\lambda=i,o. \tag{31}\]  
The length of the incline cell is far greater than its width; thus, the effect of the axial normal stress \(\sigma_{x^{*}}\) on the core member buckling can be neglected. Because the horizontal  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Case** & \(R_{o}\) **(m)** & \(p/R_{o}\) & \(h_{c}/R_{o}\) & \(f/R_{o}\) & \(T_{o}/R_{o}\) & \(T_{i}/R_{o}\) & \(T_{c}/R_{o}\) & \(P_{i}/E\) \\ \hline \(1\) & \(0.15\) & \(0.0837\) & \(0.333\) & \(0.0533\) & \(0.00367\) & \(0.00867\) & \(0.0026\) & \(15\times 10^{-5}\) \\ \hline \end{tabular}
\end{table}
Table 1: Structural parameters of hollow corrugated sandwich cylinder shell.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Case** & & **Results in [(23)]** & \begin{tabular}{c} **Closed-Form Solution** \\ **in This Paper** \\ \end{tabular} & **Error (\%)** \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{\begin{tabular}{c} Inner \\ Facesheet \\ \end{tabular} } & \(\sigma_{r}^{i}\) (Pa) & \(-3.01\times 107\) & \(-3.09\times 107\) & \(2.66\) \\  & & \(\sigma_{\theta}^{c}\) (Pa) & \(1.89\times 109\) & \(1.81\times 109\) & \(4.23\) \\  & & \(\sigma_{z}^{c}\) (Pa) & \(5.54\times 108\) & \(5.32\times 108\) & \(3.97\) \\  & & \(\sigma_{z}^{c}\) (Pa) & \(1.69\times 109\) & \(1.63\times 109\) & \(3.55\) \\ \hline \multirow{3}{*}{2} & \multirow{3}{*}{\
\begin{tabular}{c} Outer \\ Facesheet \\ \end{tabular} } & \(\sigma_{r}^{o}\) (Pa) & \(-4.3\times 106\) & \(-4.22\times 106\) & \(1.86\) \\  & & \(\sigma_{\theta}^{c}\) (Pa) & \(1.17\times 109\) & \(1.15\times 109\) & \(1.71\) \\  & & \(\sigma_{z}^{c}\) (Pa) & \(3.5\times 108\) & \(3.45\times 108\) & \(1.43\) \\  & & \(\sigma_{z}^{c}\) (Pa) & \(1.05\times 109\) & \(1.03\times 109\) & \(1.90\) \\ \hline \multirow{3}{*}{3} & \multirow{3}{*}{
\begin{tabular}{c} Core \\ \end{tabular} } & \(\sigma_{\theta}^{c}\) (Pa) & \(-6.23\times 107\) & \(-6.58\times 107\) & \(5.62\) \\  & & \(\sigma_{z}^{c}\) (Pa) & \(-2.08\times 108\) & \(-1.82\times 108\) & \(12.50\) \\ \cline{1-1}  & & \(\sigma_{z}^{c}\) (Pa) & \(1.84\times 108\) & \(1.60\times 108\) & \(13.04\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Stress results of hollow corrugated sandwich cylinder shell.  
component of the corrugate core cell has been merged with the facesheets, the failure mode of the cell is considered to be the same as that of the facesheets.  
When failure occurs in the facesheet or the corrugate core, the relevant limit inner pressure load can be obtained as follows:  
\[P_{\text{lim}}=P_{ik},\;if\;\sigma_{k}=[\sigma],\;k\in(0,\infty). \tag{32}\]