## 3 Results and Analysis
### Optimization Design of Stope Pillar Parameters
According to statistics, this formula yields more accurate results in calculating pillar strength compared to previous empirical formulas, and it has higher reliability. It has been applied to the analysis of pillar stability in some domestic hard rock mines. In this study, this formula was used to calculate the strength and safety factor of the pillar under investigation. Based on the calculation model shown in Figure 4, the relationship between the ratio of the pillar strength to the rock mass uniaxial compressive strength \(P_{s}/U\) and the width-to-height ratio, \(w/h\), of the pillar is shown in Figure 11a. It is observed that, overall, initially, with the increase in width-to-height ratio, the strength of the pillar shows a significant increasing trend then gradually stabilizes. This analysis helps to understand how the geometrical configuration of the pillars impacts their strength, providing a more comprehensive basis for optimizing the design of the pillar to ensure safety and stability during mining operations. In the stope analyzed in this study, the thickness of the gangue is about 8 m, and the stope height is 30 m. When only the gangue portion is retained as a pillar, the width-to-height ratio of the pillar is approximately 0.27, with a pillar strength of about 46.9 MPa. Based on the on-site experience in the mine, the thickness of the reserved pillar generally does not exceed 20 m. Therefore, the pillar design thickness in this study is between 8 m and 20 m, corresponding to a width-to-height ratio range of 0.27 to 0.67. The relationship between the \(P_{s}/U\) and \(w/h\) ratios within this range is shown in Figure 11b. The total length of the designed slope in this study is 116 m, with the center of the interlayer representing the center of the design pillar. When the thickness of the pillar changes, the span of the openings on both sides and the load borne by the pillar also change accordingly. In the pillar parameter optimization design scheme, 0.5 m is used as the increment of pillar thickness, and a total of 25 pillar parameter design schemes are included in this study. This research solved the mapping functions of the rectangular openings on both sides of the pillar in each scheme and used the improved Schwarz alternating method introduced earlier to analyze the stress state of the designed pillar, obtaining the safety factor of the designed pillar in each scheme. The mapping functions are determined using the polygonal method, with coefficients \(c_{2}\), \(c_{4}\), \(c_{6}\) and \(c_{8}\) being zero. The specific design schemes and solution results are shown in Table 1. The coefficients of the mapping functions for the openings of different design schemes are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & **Pillar** & **Left** & **Right Opening** & \multirow{2}{*}{_w/h_} & **Ps/U** & **Axial Stress in** & **Pillar Safety** \\  & **Thickness/m** & **Opening Span/m** & **Span/m** & & & **Pillar/MPa** & **Factor** \\ \hline
1 & 8.00 & 57.00 & 51.00 & 0.27 & 0.299 & 45.04 & 1.04 \\
2 & 8.50 & 56.75 & 50.75 & 0.28 & 0.299 &  
Figure 12b further illustrates the trend of the recovery rate of this stope with the increase in pillar design thickness. According to the relevant requirements of the mine, On the precondition of maintaining safety within the stope, the recovery rate of the stope should not be less than 80%, so the pillar design thickness should not exceed 15.5 m. Considering the actual conditions on site, a pillar thickness design of 15 m is relatively reasonable, at which point the safety factor of the pillar is 1.45.