## 4 Mechanical Calculation and Analysis of Key Strata-Inclined-Bearing Structure
### Mechanical Analysis of Load-Bearing Structure of Trapezoid-like Hinged Arch  
For the dual key strata load-bearing structure, they jointly bear the load of overlying rock and soil layers, controlling surface subsidence. Based on the results of physical similarity simulations and the mechanical characteristics of key strata fracture-bearing structures, the load-bearing structure of the sub-key strata is simplified into a trapezoidal-like hinged arch mechanical calculation model, as shown in Figure 8.  
With the same applied force as the main key strata, the sub-key strata-fractured key blocks B\({}_{\text{2}}\), C\({}_{\text{2}}\), C\({}_{\text{3}}\), and B\({}_{\text{3}}\) form a unified force system at hinge points F\({}_{\text{5}}\), F\({}_{\text{6}}\), F\({}_{\text{7}}\), and F\({}_{\text{8}}\). Based on the stress equilibrium condition, it can be derived that:  
\[\left\{ \begin{array}{l}\sum M_{\text{F5}}=0\\ \sum M_{\text{F6}}=0\\ \sum F_{\text{Yy}}=0\\ \sum F_{\text{Yx}}=0\end{array} \right. \tag{20}\]  
Based on the relationship between the uniform load and point load on the key blocks, the structural loads are simplified as point loads acting at the center of the key blocks, resulting in \(W_{\text{C1}}\), \(W_{\text{C2}}\), \(W_{\text{C3}}\), and \(W_{\text{C4}}\):  
\[\left\{ \begin{array}{l}W_{\text{C1}}=(Q_{\text{Z1}}+P_{\text{C1}})\cos \theta_{\text{Y1}}\\ W_{\text{C2}}=(Q_{\text{Z2}}+P_{\text{C2}})\cos\theta_{\text{Y2}}\\ W_{\text{C3}}=\gamma_{3}h_{\text{JX3}}\cos\theta_{\text{Y3}}\\ W_{\text{C4}}=\gamma_{3}h_{\text{JX4}}+P_{\text{Z3}}+P_{\text{ZJ3}}\end{array} \right. \tag{21}\]  
where \(\gamma_{3}\) is the interlayer rock bulk density (kN/m\({}^{3}\)); \(P_{\text{C1}}\) is the load of interlayer rock to the key stratum fracture block B\({}_{\text{2}}\) (kN); and \(P_{\text{C2}}\) is the load of the interlayer rock layer on the key layer fracture block B\({}_{\text{3}}\) (kN). Based on the fracture angle of the sub-key strata rock layers and the fracture length of the key block, \(L_{\text{Y4}}\) can be determined as follows. \[L_{\text{Y4}}=L-\sum_{n=t}^{3}h_{\text{t}}(\cot\beta_{\text{Y1}}+\cot\beta_{ \text{Y2}})-(L_{\text{Y1}}\cos\theta_{\text{Y1}}+(L_{\text{Y2}}-d)\cos\theta_ {\text{Y2}}+L_{\text{Y3}}\cos\theta_{\text{Y3}}) \tag{22}\]  
According to Formula (20), \(\sum M_{\text{F5}}=0\) can be derived as follows:  
Figure 8: Mechanical calculation model of trapezoidal-like hinged arch structure. \[Q_{\text{Y2}}=\frac{M_{\text{Y1-1}}+M_{\text{Y1-2}}+M_{\text{Y1-3}}+M_{\text{Y1-4}} }{L_{\text{Y1}}\cos\theta_{\text{Y1}}+h_{2}\cot\beta_{\text{Y1}}+L_{\text{Y4}}+ L_{\text{Y3}}\cos\theta_{\text{Y3}}+L_{\text{Y2}}\cos\theta_{\text{Y2}}-d\cot \beta_{\text{Y2}}+L_{\text{Y2}}\cot\theta_{\text{Y2}}} \tag{23}\]  
wherein:  
\[\left\{ \begin{array}{l}M_{\text{Y1-1}}=(P_{\text{Y1}}+W_{\text{C1}})\Big{(} \frac{L_{\text{Y1}}\cos\theta_{\text{Y1}}}{2}+h_{2}\cot\beta_{\text{Y1}}\Big{)} \\ M_{\text{Y1-2}}=(W_{\text{C3}}+P_{\text{Y4}}-P_{\text{R}})\Big{(}L_{\text{Y1}} \cos\theta_{\text{Y1}}+h_{2}\cot\beta_{\text{Y1}}+\frac{L_{\text{Y4}}}{2} \Big{)}\\ M_{\text{Y1-3}}=-(W_{\text{C3}}+P_{\text{Y3}}+W_{\text{C2}}+P_{\text{Y2}}) \Big{(}L_{\text{Y1}}\cos\theta_{\text{Y1}}+h_{2}\cot\beta_{\text{Y1}}+L_{ \text{Y4}}+\frac{L_{\text{Y3}}\cos\theta_{\text{Y3}}}{2}\Big{)}\\ M_{\text{Y1-4}}=(W_{\text{C2}}+P_{\text{Y2}})\Big{(}\frac{d\cot\beta_{\text{Y2 }}-L_{\text{Y3}}\cos\theta_{\text{Y3}}-L_{\text{Y2}}\cos\theta_{\text{Y2}}}{2 }\Big{)}\end{array} \right. \tag{24}\]  
where \(P_{\text{R}}\) is the supporting load of the interlayer rock stratum after the key block C\({}_{2}\) is broken (kN); and \(d\) is the vertical height of key blocks B\({}_{3}\) and C\({}_{3}\) (m). According to Formula (20), \(\sum M_{\text{F6}}=0\) can be derived as follows:  
\[M_{\text{Y2-1}}=M_{\text{Y2-2}}+M_{\text{Y2-3}}+M_{\text{Y2-4}}+M_{\text{Y2-5}} \tag{25}\]  
wherein:  
\[\left\{