# Design Method Using Response Surface Model for CFRP Corrugated Structure under Quasistatic Crushing
## 4 Analysis and Discussion
### Mathematical Model of Crushing Force per Unit Length  
In developing the part that is this study's subject, the targeted energy absorption and displacement were determined, and these define the generated load and decide the specifications of the parts. It is, therefore, necessary to derive a response surface to be used to obtain the generated load. In obtaining the crushing force per unit length (CFL), the mean crushing force (MCF) of a specimen is used. We used the average crushing force in the displacement range between 10 and 60 mm as the MCF. CFL is obtained by dividing MCF by the cross-sectional circumferential length of the corrugated shape. CFL is represented by Expression (4), where L is the cross-sectional circumferential length. \[\text{CFL}\ =\frac{MCF}{L} \tag{4}\]  
Multiple regression analysis was conducted with the number of stacks and pitch as explanatory variables, and CFL as a response variable. Table 6 shows the explanatory variables and response variable. Figure 17: Response surface and test results (EAE). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multicolumn{2}{c}{**Partial Regression**} & \multirow{2}{*}{**Std. Error**} & \multirow{2}{*}{**
\begin{tabular}{c} **95\% Confidence Interval** \\ **Lower** \\ \end{tabular} } \\ \cline{1-1}  & **Coefficient** & & & \\  & **(\(\beta_{\text{o}}\), \(\beta_{\text{L}}\), \(\beta_{\text{2}}\))** & & \\ \hline \hline \multicolumn{4}{c}{Multiple} & \multirow{2}{*}{0.925} \\ R-squared (R\({}^{2}\)) & & & \\ Adjusted & & & \\ R-squared (R\({}^{2}\)) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._Multiple regression analysis was conducted on both of the following patterns: all experimental results, and only the results in progressive crushing mode. On the basis of Expression (1), CFL, 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 CFL. Expression (5) shows the model expression for obtaining CFL. \[\text{CFL}=\beta_{0}+\beta_{1}\text{S}+\beta_{2}\text{P} \tag{5}\]  
where S is the number of stacks, P is pitch, \(\beta_{0}\) is intercept, \(\beta_{1}\) is partial regression coefficient of the number of stacks, and \(\beta_{2}\) is partial regression coefficient of the pitch. Table 7 shows the multiple regression analysis results conducted by using all experiments; results. In this model, since the \(p\) values of the pitch and intercept were larger than 0.05, and 0 was included in the 95% confidence interval, the response variable was not explained by the explanatory variables. Subsequently, multiple regression analysis was conducted by excluding the experimental results of 16 ply-P5 that had not resulted in progressive crushing mode. Table 8 shows the multiple regression analysis results conducted by excluding the experimental results of 16 ply-P5. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Explanatory Variable**} & \multicolumn{2}{c}{**Response Variable**} \\ \hline
**Number of Stacks (ply)** & **Pitch (mm)** & **CFL (kN/mm)** \\ \hline \multirow{4}{*}{8} & 5 & 0.275 \\ \cline{2-3}  & 10 & 0.255 \\ \cline{2-3}  & 15 & 0.249 \\ \cline{2-3}  & 20 & 0.234 \\ \hline \multirow{4}{*}{12} & 5 & 0.506 \\ \cline{2-3}  & 10 & 0.456 \\ \cline{2-3}  & 15 & 0.437 \\ \cline{2-3}  & 20 & 0.412 \\ \hline \multirow{4}{*}{16} & 5 & 0.527 \\ \cline{2-3}  & 10 & 0.710 \\ \cline{2-3}  & 15 & 0.635 \\ \cline{2-3}  & 20 & 0.605 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Explanatory variable and response variable (CFL).