## 4 Mechanical Calculation and Analysis of Key Strata-Inclined-Bearing Structure
### Analysis of Surface Subsidence Based on Key Strata-Bearing Structure Control  
The settlement of the fractured bearing structure of the key strata can be obtained from Formulas (43) and (44). The subsidence curve of the key strata can be plotted based on each settlement point. It is known that the subsidence curve of the key strata and the overlying loose layer exhibit similarities; therefore, it is assumed that their subsidence curve is as depicted in Figure 10, as shown in the key strata and surface subsidence calculation model. In Figure 9, \(\delta_{0}\) is the boundary moving angle (\({}^{\circ}\)); \(\beta_{0}\) is the angle between the key stratum subsidence curve \(\text{D}_{1}^{\prime}\text{D}_{2}^{\prime}\text{D}_{5}^{\prime}\) and the surface subsidence curve \(\text{D}_{1}\text{D}_{2}\text{D}_{5}\) moving boundary in the vertical direction (\({}^{\circ}\)); \(H\) is the buried depth of the coal seam (m); \(H_{\text{Z}}\) is the key strata buried depth (m); and \(H_{\text{d}}\) is the depth of the connection between the key stratum subsidence curve and the surface subsidence curve and the maximum subsidence point of the subsidence curve (m). Based on the probability integration method, it can be determined that when a mining unit volume \(dv\) is extracted at a depth of \(H\), surface subsidence will inevitably occur, and the subsidence probability \(P(\text{d}v)\) is:  
\[P(\text{d}v)=\varsigma(x,y,z)\text{d}v \tag{47}\]  
where \(\varsigma(x,y,z)\) is the density function. At this point, the subsidence curves for the key strata and the surface are assumed to be \(\text{D}_{2}^{\prime}\text{O}_{1}\text{D}_{4}^{\prime}\) and \(\text{D}_{2}\text{O}_{2}\text{D}_{4}\), respectively. These subsidence curves are connected at the boundary of movement, with the intersection point denoted as O. For nearly horizontal coal seams, it can be assumed that the subsidence curve is symmetrical about its maximum subsidence point, so only half of it needs to be studied. Let us assume \(\text{D}_{1}\text{D}_{3}=l_{1}\), and \(\text{D}_{1}\text{O}=l_{2}\). Based on trigonometric geometry and functional relationships, we can derive:  
\[l_{1}=L/2+H\cot\delta_{0} \tag{48}\]  
Figure 10: Key strata and surface subsidence calculation model. \[l_{2}=l_{1}\sec\beta_{0} \tag{49}\]  
Through Formula (48), the calculation formula of \(H_{\text{d}}\) can be obtained as follows:  
\[H_{\text{d}}=(L/2+H\cot\delta_{0})\tan\beta_{0} \tag{50}\]  
When the mining depth is \(dv\) of the mining unit ore body at \(H\), the distance between the moving boundary of the key strata and the surface subsidence curve and the maximum subsidence point D\({}_{2}\)D\({}_{3}\) is set to \(x_{\text{d}}\), and D\({}_{2}^{\prime}\)D\({}_{3}^{\prime}\) is set to \(x_{\text{k}}\). At this time, the subsidence value D\({}_{2}\)D\({}_{28}=w_{\text{d}}(x_{\text{d}})\) of any point \(x_{\text{d}}\) in the surface subsidence curve D\({}_{1}\)D\({}_{28}\)D\({}_{5}\), and the subsidence value D\({}_{2}^{\prime}\)D\({}_{28}^{\prime}=w_{\text{k}}(x_{\text{k}})\) of any point \(x_{\text{k}}\) in the key strata subsidence curve D\({}_{1}^{\prime}\)D\({}_{28}^{\prime}\)D\({}_{5}^{\prime}\). The relationship between the surface abscissa D\({}_{2}\) and the key strata abscissa D\({}_{2}^{\prime}\) is:  
\[\frac{x_{\text{d}}}{x_{\text{k}}}=\frac{H_{\text{d}}}{H_{\text{d}}-H_{\text{Z }}} \tag{51}\]  
Substituting Formula (50) into Formula (51) results in the following:  
\[x_{\text{d}}=\frac{(L/2+H\cot\delta_{0})\tan\beta_{0}}{(L/2+H\cot\delta_{0}) \tan\beta_{0}-H_{\text{Z}}}x_{\text{k}} \tag{52}\]  
Due to the deformation and failure of the overlying rock and soil layers of the key strata, the change in the damaged volume within a certain range is referred to as volume deformation, denoted as \(\zeta\).