## 4 Mechanical Calculation and Analysis of Key Strata-Inclined-Bearing Structure
### Mechanical Analysis of Full-Trapezoidal Hinged Arch Load-Bearing Structure  
The stability and mechanical characteristics of the key strata, which serve as critical load-bearing rock layers controlling surface subsidence, have a significant impact on mining operations. According to the results of physical similarity simulation experiments, the fracture-bearing structure of the key strata has a favorable control effect on rock layer subsidence. Therefore, the stability of the load-bearing structure determines the subsidence of the overlying rock and soil layers. The mechanical characteristics of the key strata fracture-bearing structure were simplified into a mechanical calculation model of a full-trapezoidal hinged arch structure, as shown in Figure 7.  
According to the action relationship of force, the key blocks B, C, and B\({}_{1}\) form a unified force system with hinge points F\({}_{1}\), F\({}_{2}\), F\({}_{3}\), and F\({}_{4}\). Based on the stress balance condition, the following can be obtained:  
\[\left\{ \begin{array}{l}\sum M_{\text{F1}}=0\\ \sum M_{\text{F2}}=0\\ \sum F_{\text{Zy}}=0\\ \sum F_{\text{Zx}}=0\end{array} \right. \tag{1}\]  
Considering the difficulty of calculation, the uniform load of the mechanical structure is simplified as the point load acting on the middle of the key block, and then, \(P_{\text{Z1}}\) and \(P_{\text{Z2}}\) are:  
\[\left\{ \begin{array}{l}P_{\text{Z1}}=\gamma_{5}h_{5}L_{\text{Z1}}\cos\theta _{\text{Z1}}\\ P_{\text{Z2}}=\gamma_{5}h_{5}L_{\text{Z2}}\cos\theta_{\text{Z2}}\end{array} \right. \tag{2}\]  
where \(\gamma_{5}\) is the bulk density of the loess layer (kN/m\({}^{3}\)). In the full-trapezoidal hinged arch structure, all the rock masses compacted and bearing the fractured strata are considered key block C; thus the point load \(P_{\text{Z3}}\) is given by:  
\[P_{\text{Z3}}=\gamma_{5}h_{5}L_{\text{Z3}} \tag{3}\]  
From the fracture angle of the main key stratum rock layers and the fracture length of the key block, \(L_{\text{Z3}}\) can be determined as follows:  
\[L_{\text{Z3}}=L-\sum_{n=t}^{4}h_{\text{t}}\bigg{(}\cot\frac{\beta_{\text{Y1}} +\beta_{\text{Z1}}}{2}+\cot\frac{\beta_{\text{Y2}}+\beta_{\text{Z2}}}{2} \bigg{)}-(L_{\text{Z1}}\cos\theta_{\text{Z1}}+L_{\text{Z2}}\cos\theta_{\text{ Z2}}) \tag{4}\]  
where \(L\) is the inclined length of the working face (m). Based on the theory of masonry beams, it can be determined that the contact area between key blocks is 0.5a. From the key block rotation and thickness, the value of \"a\" can be obtained as follows:  
\[a=0.5(h_{4}-L_{\text{Z1}}\sin\theta_{\text{Z1}}) \tag{5}\]  
Figure 7: Mechanical calculation model of full-trapezoidal hinged arch structure. According to Formula (1), \(\sum M_{\rm FI}=0\) can be derived as follows:  
\[Q_{\rm Z2}=\frac{M_{\rm Z1\text{-}1}+M_{\rm Z1\text{-}2}+M_{\rm Z1\text{-}3}}{L_{ \rm Z1}\cos\theta_{\rm Z1}+h_{4}\cot\beta_{\rm Z1}+L_{\rm Z3}+L_{\rm Z2}\cos \theta_{\rm Z2}+h_{4}\cot\beta_{\rm Z2}} \tag{6}\]  
wherein:  
\[\left\{ \begin{array}{l}M_{\rm Z1\text{-}1}=(P_{\rm Z1}+P_{\rm Z1})\Big{(} \frac{L_{\rm Z1}\cos\theta_{\rm Z1}}{2}+h_{4}\cot\beta_{\rm Z1}\Big{)}\\ M_{\rm Z1\text{-}2}=(P_{\rm Z3}+P_{\rm Z3}+P_{\rm Z2}+P_{\rm Z2}-P_{\rm Z4}) \Big{(}L_{\rm Z1}\cos\theta_{\rm Z1}+h_{4}\cot\beta_{\rm Z1}+\frac{L_{\rm Z3} }{2}\Big{)}\\ M_{\rm Z1\text{-}3}=-\frac{(P_{\rm Z2}+P_{\rm Z2})L_{\rm Z2}\cos\theta_{\rm Z2 }}{2}\end{array} \right. \tag{7}\]  
where \(P_{\rm Z3}\) is the load transferred from the loess layer to the key block C (kN); and \(P_{\rm Z4}\) is the supporting load of the interlayer rock stratum after key block C is broken (kN). According to Formula (1), \(\sum M_{\rm F2}=0\) can be derived as follows:  
\[M_{\rm Z2\text{-}1}=M_{\rm Z2\text{-}2}+M_{\rm Z2\text{-}3}+M_{\rm Z2\text{-}4 }+M_{\rm Z2\text{-}5} \tag{8}\]  
wherein:  
\[\left\{