# Study on Load Transfer Mechanism of Pile-Supported Embankment Based on Response Surface Method
## 3 Analysis of the Numerical Simulation Results
### Response Surface Optimization Test and Results Analysis  
#### 3.2.1 Response Surface Test Design  
The main variation range of each influencing factor was determined by the single-factor test. Based on the Box-Behnken design method of Design-Expert software, the response surface experimental design was carried out, as shown in Table 4. According to the test design, the total number of tests was twenty-nine, of which twenty-four test points were external analysis and five test points were the center of the region. Center point repeat calculations were used to estimate the experimental error. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Influencing Factor** & **Evaluation Scope** & **Sensitivity Coefficient 15\({}_{AF}\)** \\ \hline Cohesion [kPa] & 6–25 & 0.175 \\ Internal Friction Angle [\({}^{\circ}\)] & 25–45 & 0.253 \\ Elastic Modulus of Embarknment [MPa] & 10–45 & 0.24 \\ Geogrid Strength [kN/m] & 30–120 & 0.048 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Sensitivity coefficient \(\mid S_{AF}\mid\) of influencing factors. The regression equation of the pile-soil stress ratio can be obtained by fitting the test results with a second-order polynomial. In order to further optimize the regression equation, the insignificant items in the variance analysis can be removed and refitted. The optimized pile-soil stress ratio regression equation is as follows:  
\[n=9.30\,-0.24A\,-0.033B+0.056C+0.089D+0.078 AB+0.14 AC+0.26A^{2}+0.14C^{2} \tag{3}\]  
To ensure the accuracy and adaptability of the model, its predictive ability needed to be evaluated. For the significance test of the response surface equation, the correlation \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Influencing Factors** & \multicolumn{2}{c}{**Coding Level**} \\  & \(-\mathbf{1}\) & \(\mathbf{0}\) & \(\mathbf{1}\) \\ \hline Cohesion \(A\) [kPa] & 2 & 15 & 27 \\ Internal Friction Angle \(B\) [\({}^{\circ}\)] & 25 & 35 & 45 \\ Elastic Modulus of Embankment C [MPa] & 10 & 27.5 & 45 \\ Geogrid Strength \(D\) [kN/m] & 30 & 75 & 120 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Test design factors and levels. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**NO.** & **Cohesion \(A\) [kPa]** & \begin{tabular}{c} **Internal Friction** \\ **Angle \(B\) [\({}^{\circ}\)]** \\ \end{tabular} & \begin{tabular}{c} **Elastic Modulus of** \\ **Embankment C [MPa]** \\ \end{tabular} & \
\begin{tabular}{c} **Geogrid Strength** \\ **D [kN/m]** \\ \end{tabular} &
\begin{tabular}{c} **Pile–Soil Stress** \\ **Ratio \(n\)** \\ \end{tabular} \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 5: Response surface design test results. coefficient, \(R^{2}\), and the determination coefficient, \(R^{2}\), were generally used to express its approximation degree. \[ \begin{array}{l}R^{2}=\frac{S_{R}}{S_{T}}=\frac{S_{T}-S_{E}}{S_{T}}=1-\frac{S_{ E}}{S_{T}}\\ S_{T}=\sum\limits_{i=1}^{n}{{{\left({y_{i}-\overline{y}}\right)}^{2}}}\\ S_{E}=\sum\limits_{i=1}^{n}{{{\left({\widehat{y}-y_{i}}\right)}^{2}}}\end{array}  \tag{4}\]  
where \(S_{T}\) is the total sum of the squares; \(S_{R}\) is the regression sum of the squares; \(S_{E}\) is the residual sum of the squares; \(\overline{y}\) is the mean of the response values; and \(\widehat{\mathrm{y}}\) is the predicted value of the response value. The correlation coefficient, \(R^{2}\), is a measure of complete fit, reflecting the degree of conformity between the response surface and the experimental data; \(R^{2}\) is greater than 0.9 to ensure the fitting effect:  
\[R^{2}_{adj}=1-\frac{S_{E}/\left(n-n_{p}-1\right)}{S_{T}/\left(n-1\right)}=1- \frac{n-1}{n_{p}-1}(1-R^{2}) \tag{5}\]  
where \(n\) is the sample size, and \(n_{p}\) is the number of optimization variables. The determination coefficient, \(R^{2}_{adj}\), represents the correlation between the independent variables and dependent variables, and the closer the value is to 1, the better the fitting effect is.