## 4 Mechanical Calculation and Analysis of Key Strata-Inclined-Bearing Structure
### Analysis of Surface Subsidence Based on Key Strata-Bearing Structure Control
Therefore, the relationship between the area under the subsidence curve of the key strata and the undisturbed subsidence curve of the surface can be expressed as:  
\[\int\limits_{0}^{|D_{2}D_{3}|}w_{\text{d}}(x_{\text{d}})dx_{\text{d}}=\int \limits_{0}^{|D_{2}^{\prime}D_{3}^{\prime}|}\zeta(x_{\text{k}})w_{\text{k}}( x_{\text{k}})dx_{\text{k}} \tag{53}\]  
Substituting Formula (52) into Formula (53), we obtain:  
\[\int\limits_{0}^{|D_{2}D_{3}|}w_{\text{d}}(x_{\text{d}})dx_{\text{d}}=\int \limits_{0}^{|D_{2}^{\prime}D_{3}^{\prime}|}\zeta(x_{\text{k}})\frac{(L/2+H \cot\delta_{0})\tan\beta_{0}}{(L/2+H\cot\delta_{0})\tan\beta_{0}-H_{\text{Z}}} x_{\text{k}}dx_{\text{k}} \tag{54}\]  
In order to simplify the complexity of calculations, the overlying rock strata above the key strata are considered homogeneous, isotropic, and volume-conserving ideal bodies. Using the integral calculation method, the subsidence model of the key strata and the surface can be expressed as follows:  
\[w_{\text{d}}(x_{\text{d}})=\frac{(L/2+H\cot\delta_{0})\tan\beta_{0}}{(L/2+H \cot\delta_{0})\tan\beta_{0}-H_{\text{Z}}}w_{\text{k}}(x_{\text{k}}) \tag{55}\]  
The connection between the key stratum and the surface subsidence points is the subsidence curve; that is, the subsidence curve of the key stratum can be expressed as \(w_{\text{k}}(x_{\text{k}})\), and the surface subsidence curve is \(w_{\text{drock strata have an average fragmentation coefficient of 1.05. The average bulk density of the bedrock layer is 26 kN/m\({}^{3}\), and for the loess layer, it is 23 kN/m\({}^{3}\). The boundary movement angle is set at 63\({}^{\circ}\), and the angle between the sinking curve \(\rm D_{1}^{\prime}D_{2k}^{\prime}D_{5}^{\prime}\) of the key stratum and the subsurface sinking curves \(\rm D_{1}D_{2s}D_{5}\) connecting the boundary movement is 52\({}^{\circ}\) in the vertical direction. The horizontal thrust \(T\) and the overlying load \(Q\) can be obtained by substituting the above geological conditions and related parameters into calculation Formulas (6), (15), (16), (23), (32), and (33) of the trapezoidal hinged arch structure. The subsidence of each key block can be obtained from calculation Formulas (38)-(42) of the subsidence of the key strata, and the subsidence curve can be drawn. The surface subsidence curve can be obtained by calculation Formula (55) of the key strata subsidence curve and the surface subsidence curve. Due to the complexity and difficulty of the aforementioned calculations, this study employs Matlab and Origin 2022 software to perform the calculations and create the subsidence curve, as illustrated in \[l_{1}=L/2+H\cot\delta_{0} \tag{48}\]0  
The theoretically predicted key strata and surface subsidence curve indicate that the working face between 97.5 and 196.5 m falls within the zone of excessive mining stability, where the maximum subsidence value has been reached. Due to the influence of fractured key blocks in the key strata, the primary subsidence zones are from 0 to 97.5 m and from 196.5 to 294 m. The subsidence curves of the main key strata and the sub-key strata exhibit strong similarities; however, the supporting effect of the sub-key strata significantly reduces the subsidence value of the main key strata, with the subsidence points of the curves aligning. The surface subsidence curve is flatter compared to that of the main key strata, indicating a reduction in subsidence and suggesting that the surface plays a role in mitigating and weakening subsidence. Field measurements show a maximum surface subsidence of 1.93 m, with a subsidence coefficient of 0.65. Since this study did not consider the volumetric deformation rate \(\zeta\) of the loess layer, the measured maximum subsidence is greater than the theoretical predicted value. However, the overall predicted results are similar, validating the scientific and rational basis of this study on the interaction between the key strata's bearing structure and surface subsidence.