# Design Method Using Response Surface Model for CFRP Corrugated Structure under Quasistatic Crushing
## 4 Analysis and Discussion
### Understanding the Phenomena Using the Stress Model of Corner Portions and Linear Portions  
To understand the phenomena, multiple regression analysis was conducted by changing the explanatory variables. The cross-sectional areas of the corner and linear portions could be obtained by replacing the explanatory variables of pitch and of the number of stacks with the geometric elements of corrugated cross-section. As shown in Figure 20, the cross-sectional area of a corner portion is defined as the area between the lines drawn by connecting the inner and outer side ends of the radius. A partial regression coefficient, obtained by conducting multiple regression analysis without intercept and with the use of cross-sectional area of the corner portion and cross-sectional area of the linear portion as explanatory variables and MCF as the response variable, represents the stress. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multicolumn{2}{c}{**Partial Regression**} & \multirow{2}{*}{**Std.**} & \multirow{2}{*}{**95\% Confidence Interval**} \\  & **Coefficient** & & & & **Error** \\  & **(\(\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text{\text{\text{\text{\text{\text{ \text{     }}}}}}}}}}}}}}}\)) & & & **Lower** & **Upper** \\ \hline Intercept & \(-0.0919\)** & \(0.00594\) & \(0.0248\) & \(-0.149\) & \(-0.0348\) \\ Number of stacks & \(0.0512\)** & \(p<0.001\) & \(0.00188\) & \(0.0468\) & \(0.0555\) \\ Pitch & \(-0.00528\)** & \(0.00146\) & \(0.00111\) & \(-0.00784\) & \(-0.00271\) \\ \hline Multiple & & & & & \\ R-squared (R2) & & & & & \\ Adjusted & & & & & \\ R-squared (R2) & & & & & \\ \hline \hline \end{tabular}  
* Coefficient estimated with statistical significance of 0.001; ** coefficient estimated with statistical significance of 0.01.  
\end{table} 
Table 8: Multiple regression analysis results using progressive crushing mode results (CFL). Figure 18: Response surface and test results (CFL). On the basis of Expression (1), MCF, which was proportionate to F, was proportionate to EAE, which was proportionate to EA. Since a linear model was used for EAE because the relation between the number of stacks and the pitch mostly linearly varied, a linear model was also used for MCF. Expression (6) shows the model expression for obtaining MCF. \[\text{MCF}=\beta_{1}\text{CA}+\beta_{2}\text{LA} \tag{6}\]  
Figure 19: Designable area of corrugate structure. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Case**} & \multicolumn{2}{c}{**Explanatory Variable**} & \multicolumn{2}{c}{**Response Variable**} \\ \hline \multirow{3}{*}{\begin{tabular}{c} **Number of** \\ **Stacks (ply)** \\ \end{tabular} } & **Pitch** & **Corner Area (CA)** & **Straight-Line Area (LA)** & **MCF** \\  & **(mm)** & **(mm\({}^{2}\))** & **(mm\({}^{2}\))** & **(km\({}^{2}\))** & **(km)** \\ \hline \multirow{3}{*}{\begin{tabular}{c} 8 \\ \end{tabular} } & 5 & 72.4 & 242 & 48.8 \\ \cline{2-5}  & 10 & 45.2 & 204 & 35.7 \\ \cline{2-5}  & 15 & 36.2 & 209 & 34.4 \\ \cline{2-5}  & 20 & 27.1 & 196 & 29.3 \\ \hline \multirow{3}{*}{\
\begin{tabular}{c} 12 \\ \end{tabular} } & 5 & 108.0 & 363 & 89.6 \\ \cline{2-5}  & 10 & 67.8 & 306 & 63.8 \\ \cline{2-5}  & 15 & 54.3 & 313 & 60.2 \\ \cline{2-5}  & 20 & 40.7 & 294 & 51.8 \\ \hline \multirow{3}{*}{
\begin{tabular}{c} 16 \\ \end{tabular} } & 10 & 90.5 & 408 & 99.4 \\ \cline{2-5}  & 15 & 72.4 & 418 & 87.5 \\ \cline{2-5}  & 20 & 54.3 & 393 & 75.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Explanatory variable and response variable (MCF). Figure 20: Straight-line and corner areas of corrugate structure. where CA is the corner portion area, LA is the linear portion area, \(\beta_{1}\) is the partial regression coefficient of the corner portion area, and \(\beta_{2}\) is the partial regression coefficient of the linear portion area. Table 10 shows the multiple regression analysis results conducted by using all experiment results.