## 2 Materials and Methods
### Response Surface Methodology  
Based on the calculation results of the stope parameters from Table 6, a response surface model is established for the relationships between the stope span X\({}_{1}\), pillar span X\({}_{2}\), roof thickness X\({}_{3}\), maximum tensile stress Y\({}_{1}\), maximum compressive stress Y\({}_{2}\), and maximum vertical displacement Y\({}_{3}\). To improve the accuracy of the response surface model, a second-order nonlinear response surface model is used to map the relationships between the stope parameters and each mechanical response. The basic expression of the model is as follows [36]:  
\[\hat{y}=f(x_{1},x_{2},\ldots,x_{n})=a_{0}+\sum_{i=1}^{n}a_{i}x_{i}+\sum_{i=1}^ {n}a_{ii}x_{i}^{2}+\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{ij}x_{i}x_{j} \tag{5}\]  
Here, \(\hat{y}\) is the approximated response coefficient; a\({}_{0}\), a\({}_{\text{i}}\), a\({}_{\text{ii}}\), and a\({}_{\text{ij}}\) (i = 1, 2,\(\ldots\), \(n\); j = 1, 2,\(\ldots\), \(n\)) are the coefficients to be determined; and x\({}_{\text{i}}\) represents the random variables (e.g., stope span, pillar span, and roof thickness). Using the least squares method and MATLAB R2023a software, the unbiased estimates of the coefficients a\({}_{0}\), a\({}_{\text{i}}\), a\({}_{\text{ii}}\), and a\({}_{\text{ij}}\) are determined. For the fitting of the three response surface models, the resulting models are as follows:  
\[ \begin{split} Y_{1}=-1.5304+0.14849x_{1}-0.090238x_{2}+0.13829x_ {3}-0.0037067x_{1}^{2}+0.0043333x_{2}^{2}-\\ 0.0024167x_{3}^{2}+0.002475x_{1}x_{2}-0.002075x_{1}x_{3}-0.0040312 x_{2}x_{3}\end{split}  \tag{6}\]  
\[ \begin{split} Y_{2}=6.7801+0.16224x_{1}-1.6968x_{2}+1.0459x_{3}-0. 0044x_{1}^{2}+0.04225x_{2}^{2}-\\ 0.056x_{3}^{2}+0.0151x_{1}x_{2}-0.00605x_{1}x_{3}+0.032313x_{2}x_ {3}\end{split}  \tag{7}\]  
\[ \begin{split} Y_{3}=-11.5338+1.0938x_{1}+0.5934x_{2}+0.46357x_{3}- 0.019636x_{1}^{2}-0.014681x_{2}^{2}+\\ 0.002444x_{3}^{2}-0.010425x_{1}x_{2}-0.012975x_{1}x_{3}-0.021031 x_{2}x_{3}\end{split}  \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Experimental** & **Stope Span** & **Pillar Span** & **Roof** & **Maximum Tensile** & **Maximum Tensile** & **Maximum Vertical** \\
**Schemes** & **/m** & **/m** & **Thickness** & **Stress/Mpa** & **Stress/Mpa** & **Displacement/mm** \\ \hline
1 & 15 & 10 & 10 & 0.183 & 4.873 & 4.154 \\
2 & 15 & 10 & 14 & 0.238 & 4.662 & 4.665 \\
3 & 15 & 12 & 12 & 0.198 & 4.886 & 4.126 \\
4 & 15 & 14 & 10 & 0.243 & 4.299 & 3.728 \\
5 & 15 & 14 & 14 & 0.204 & 4.636 & 3. The coefficient of determination (R\({}^{2}\)) reflects the goodness of fit of the model. For the quadratic response surface models described in Equations (10) to (12), which include interaction terms, the R\({}^{2}\) values are as follows:  
Maximum Tensile Stress (Y\({}_{1}\)): R\({}^{2}\) = 0.98. Maximum Compressive Stress (Y\({}_{2}\)): R\({}^{2}\) = 0.91. Maximum Vertical Displacement (Y\({}_{3}\)): R\({}^{2}\) = 0.96. These R\({}^{2}\) values indicate a good fit of the response surface models to the data [37; 38], suggesting that the models are well-suited to meet the needs of practical engineering applications. The regression models Y\({}_{1}\), Y\({}_{2}\), and Y\({}_{3}\) indicate that the stope span, pillar span, and roof thickness have significant effects on the compressive stress, tensile stress, and displacement in the stope. In Figure 3a,b, it can be observed that when the roof thickness is 10 m and 14 m, there is a positive correlation between the maximum tensile stress and the stope span, meaning that as the stope span increases, the maximum tensile stress also increases. Additionally, with the increase in pillar span, the maximum tensile stress first decreases and then increases. Figure 3c,d show that when the roof thickness is 10 m, the maximum compressive stress exhibits a negative correlation with both the stope span and pillar span, meaning that the maximum compressive stress decreases as both the pillar span and stope span increase.