## 2 Materials and Methods
### Response Surface Methodology
However, when the roof thickness is fixed at 14 m, the compressive stress decreases with an increase in pillar span but increases with an increase in stope span. Figure 3e,f illustrate that the maximum displacement of the stope shows a positive correlation with both the stope span and pillar span, meaning that the displacement increases with the stope span and mild span and also increases with the roof thickness. The relationships between different variables are explained in points 1, 2, 3, and 4 below. 1. Effect of Stope Span on Compressive and Tensile Stresses in the Stope: As the stope span increases, the stability of the surrounding rock gradually decreases, weakening the load-bearing capacity of the roof and causing a redistribution of stresses above the stope. A larger span increases tensile stress in the middle of the roof, potentially triggering bending and tensile effects, similar to the \"arching effect\" in structural mechanics. With a larger span, stress concentrates in the central region, increasing tensile stress, especially in rock structures where cracks and fractures are more likely to form, leading to stress concentration. As the span grows, the roof's load-bearing capacity further diminishes, reducing overall stiffness and increasing compressive stress in the support zones. Larger spans also lead to compressive stress concentration near the center and support points, potentially causing local stress instability and rock failure. In general, the greater the span, the more likely bending and collapse occur in the roof, increasing maximum compressive stress and reducing structural stability overall. 2. Effects of Pillar Width on Compressive and Tensile Stress in the Stope: When the pillar width is small, the support strength and stiffness of the pillar are insufficient to effectively resist the self-weight of the roof and external loads, leading to significant deformation of the roof and concentration of tensile stress in certain areas. As the size of the pillar increases, their load-bearing capacity and stiffness improve, allowing for better resistance against roof subsidence and deformation, which in turn reduces the deflection of the roof and the tensile stress resulting from that deflection. However, when the pillar reaches a critical size, the support strength exhibits a saturation effect, and further increases in pillar size yield minimal improvement in load-bearing capacity. At the same time, the increase in the weight of the pillar causes the roof to bear greater downward forces, thereby increasing tensile stress in the roof. When the pillar width is small, the support effect is relatively good, resulting in a more uniform distribution of compressive stress in the roof. However, as the pillar width increases, the support capacity gradually weakens, leading to the concentration of compressive stress near the support points and increasing the risk of localized compressive stress. When the pillar width reaches its critical size, the support stiffness is significantly insufficient, causing further increases in compressive stress in the central area of the roof, which may trigger local instability and rock failure. 3. Effects of Roof Thickness on Compressive and Tensile Stress: Thicker roofs possess greater stiffness, enabling them to better resist bending deformation and reduce the concentration of tensile stress, particularly diminishing the tensile effects in the central region of the roof. Consequently, thick roofs can effectively distribute tensile stress under support conditions, lowering the risk of structural instability. In contrast, thinner roofs have lower stiffness, making them more susceptible to larger bending and tensile stress concentrations, which increases the risk of fissures and cracking in the central area. Additionally, the increased stiffness of thicker roofs helps to minimize stope deformation and decrease compressive stress concentrations, resulting in a more uniform distribution of maximum compressive stress and an overall reduction in compressive stress. 4.