## 6 Discussion  
### Surface Subsidence Analysis of the Mining Area  
The probability integral method (PIM) is one of the most representative and mature methods for calculating the influence function in mining subsidence analysis [33]. This method decomposes the mining area into an infinite number of infinitesimal mining units from a statistical perspective, wherein the overall influence of mining on the overlying strata and the ground surface is considered to be equal to the cumulative influence of each mining unit. Based on stochastic medium theory, the ground surface subsidence basin  
Figure 13: Soil pressure concentration. induced by mining follows a normal distribution curve. Therefore, the entire subsidence profile caused by mining can be represented by the integral of a probability density function, as follows:  
\[r=H/\tan\beta \tag{6}\]  
\[W_{e}(x)=\frac{1}{r}e^{-\pi\frac{x^{2}}{r^{2}}} \tag{7}\]  
where \(r\) is the radius of influence, \(H\) is the mining depth, and \(\beta\) is the principal angle of draw. Assuming the goaf extends from \(s=0\) to \(s=\infty\), the subsidence can be obtained by integrating the influence of each infinitesimal unit:  
\[W(x)=\int_{0}^{\infty}W_{e}(x-s)ds=W_{0}\int_{0}^{\infty}\frac{1}{r}\exp\left[ -\pi\frac{(x-s)^{2}}{r^{2}}\right]ds \tag{8}\]  
For a finite goaf, the surface subsidence at any given point \(x\) can be represented as the subsidence influence from the left-side semi-infinite mining subtracting the subsidence influence from the right-side semi-infinite mining. The equation for surface subsidence induced by finite excavation can be expressed as follows:  
\[W(x)=W_{0}\int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{1}{r}\exp\left[-\pi\frac{(x- z)^{2}}{r^{2}}\right]ds \tag{9}\]  
In the initial stages of shallow room-and-pillar mining, the overlying rock structure is stable and the development of surface subsidence is relatively minor. The traditional probability integral method (PIM) model demonstrates a high degree of accuracy. As the mining ratio gradually increases, the coal pillars are subjected to accumulating stress from the goaf. Once the pillars lose stability, surface subsidence within the fracture line region accelerates, while surface displacement outside the fracture line remains minimal. The traditional probability integral method, which calculates the subsidence range based on the angle of draw, cannot accurately reflect the actual subsidence distribution on both sides of the fracture line. To further refine the depiction of the subsidence distribution profile, a correction factor \(\sigma\)(s) is introduced [34]:  
\[\sigma(s)=1-\frac{1}{1+\mathrm{e}^{\frac{s-0.5I}{d}}} \tag{10}\]  
\[W(s)=W_{0}\int_{-\frac{1}{2}}^{\frac{L}{2}}\frac{\sigma(s)}{r}\exp\left[-\pi \frac{(x-s)^{2}}{r}\right]ds \tag{11}\]  
where \(I\) is the length of the masonry beam block; \(a\) is a coefficient related to the block size of the masonry beam and the stiffness of the coal body, typically taken as \(0.25\;I\). Based on the solutions derived from elasticity theory and rock mechanics, the stress state of any arbitrary element within the key stratum rock beam is illustrated in Figure 14. The stress acting on a plane with an angle of \(\alpha\) to the \(\sigma_{x}\) plane can be expressed by Equations (12) and (13). \[\sigma_{n}=\frac{\sigma_{x}+\sigma_{y}}{2}+\frac{\sigma_{x}-\sigma_{y}}{2} \cos 2\alpha-\tau_{xy}\sin 2\alpha \tag{12}\]  
\[\tau_{n}=\frac{\sigma_{x}-\sigma_{y}}{2}\sin 2\alpha+\tau_{xy}\cos 2\alpha \tag{13}\]The stress components in the rock layer can be expressed by the following Equations (14)-(16), based on [PERSON]'s [35] solution for the stress components in the rock beam under the assumption of homogeneity. \[r=H/\tan\beta \tag{6}\]0  
\[r=H/\tan\beta \tag{6}\]1  
\[r=H/\tan\beta \tag{6}\]2  
where \(q_{c}\) represents the uniformly distributed load acting on the rock beam, \(Q^{\prime}\) and \(N^{\prime}\) represent the forces acting on the free ends of the rock beam, and \(\Delta\) refers to the bending deflection of the rock beam.