## 3 Regression Analysis of Neutral Layer Shifting
### Regression Model and Solution  
In a certain range, there is an unknown law of the neutral layer shifting with the factors. The purpose of regression analysis is to get the approximation function describing the complex unknown law. Besides the sufficient fitting accuracy and wide application, the response surface method (RSM) also has high-efficiency and its calculation is fast. In the further actual production and simulation, new data can be added into the model and the modified results can be induced instantly. Therefore, RSM is selected as the function construction tool.  
Taking each experimental condition as a sample point \(x^{(k)}\), the corresponding result is the response value \(y^{(k)}=y(x^{(k)})\). There is an approximate function for the actual performance function \(y=y(x)\):  
\[\overline{y}=f(x) \tag{18}\]  
whose specific form can be obtained by the undetermined coefficient method. This approximate function \(\overline{y}\) is the response surface function to be constructed. According to the engineering experience, the quadratic form with the cross term is selected as the undetermined form of \(\overline{y}\):  
\[\overline{y}=a+\sum_{i=1}^{n}b_{i}x_{i}+\sum_{i=1}^{n}\sum_{j=i}^{n}c_{ij}x_{i} y_{j},n=6 \tag{19}\]  
To solve the undetermined coefficients, the least square method is used to minimize the error \(\varepsilon^{(k)}\) between the actual performance function and the response surface function. With the number of experiments \(m=47\), the process and conditions of minimization can be expressed as follows:  
\[\varepsilon^{(k)}=\overline{y}^{(k)}-y^{(k)} \tag{20}\]  
\[S=\sum_{k=1}^{m}\left(\varepsilon^{(k)}\right)^{2}\xrightarrow{}min \tag{21}\]  
\[\left\{\begin{array}{c}\frac{\partial\mathcal{S}}{\partial a}=0\\ \frac{\partial\mathcal{S}}{\partial b_{i}}=0\\ \frac{\partial\mathcal{S}}{\partial c_{ij}}=0\end{array}\right. \tag{22}\]  
By substituting the data of all sample points into the conditions above, the linear equations about undetermined coefficients can be sorted out. Solving the equations, those coefficients can be obtained.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{**No.**} & \multirow{3}{*}{**Level**} & \multirow{3}{*}{**Combinations**} & \multirow{3}{*}{**Bending**} & \multirow{3}{*}{**Outer**} & \multicolumn{5}{c}{**Experimental Conditions**} & \\ \cline{3-3} \cline{5-10}  & & & & **Relative** & **Diameter-** & **Bending** & **Diameters-** & **Layered** & **Interlaminar** & **Results/mm** \\  & & & **Angle/\({}^{c}\)** & **Diameter/mm** & **Radius** & **Ratio** & **Ratio** & **Coefficient** & \\ \hline
36 & A\({}_{\text{B}}\)B\({}_{\text{C}}\)C\({}_{\text{D}}\)B\({}_{\text{E}}\)F\({}_{2}\) & 75 & 14 & 1.5 & 8 & 4 & 0.4 & 0.6551 \\
37 & A\({}_{\text{B}}\)C\({}_{\text{C}}\)D\({}_{\text{2}  
The matrices \(B_{i}\) and \(C_{ij}\) are used to express the coefficients. The values of their elements are listed as follow:  
\[B_{i}=\left[\begin{array}{ccccc}-0.0677&0.1583&0.8710&0.5016&0.1617&-5.8006\\ \end{array}\right]\] \[C_{ij}=\left[\begin{array}{ccccc}0.0004&0.0006&-0.0023&0.0023&0.0032&0. 0045\\ &-0.0019&-0.0267&0.0061&-0.0089&-0.1106\\ &&-0.0870&-0.0572&0.3181&2.0120\\ &&&-0.0368&-0.0262&-0.0439\\ &&&&&-0.0422&-1.0555\\ &&&&&&&5.6157\\ \end{array}\right]\]