## 4 Mechanical Calculation and Analysis of Key Strata-Inclined-Bearing Structure
### Mechanical Analysis of Load-Bearing Structure of Trapezoid-like Hinged Arch
\begin{array}{l}M_{\text{Y2-1}}=T_{\text{B2}}\big{(}h_{2}-L_{\text{ Y2}}\sin\theta_{\text{Y2}}-\frac{a}{2}\big{)}-T_{\text{B1}}\big{(}h_{2}-L_{ \text{Y1}}\sin\theta_{\text{Y2}}-\frac{a}{2}\big{)}\\ M_{\text{Y2-2}}=-Q_{\text{Y1}}(L_{\text{Y1}}\cos\theta_{\text{Y1}}+h_{2}\cot \beta_{\text{Y1}})\\ M_{\text{Y2-3}}=\frac{(W_{\text{C1}}+P_{\text{Y1}})(L_{\text{Y1}}\cos\theta_{ \text{Y1}}+h_{2}\cot\beta_{\text{Y1}})+(P_{\text{R}}-W_{\text{C4}}-P_{\text{ Y4}})L_{\text{Y4}}}{2}\\ M_{\text{Y2-4}}=-(W_{\text{C3}}+P_{\text{Y3}})\Big{(}L_{\text{Y4}}+\frac{L_{ \text{Y3}}\cos\theta_{\text{Y3}}+h_{2}\cot\beta_{\text{Y2}}}{2}\Big{)}\\ M_{\text{Y2-5}}=-(W_{\text{C2}}+P_{\text{Y2}})\Big{(}L_{\text{Y4}}+L_{\text{Y3}} \cos\theta_{\text{Y3}}+h_{2}\cot\beta_{\text{Y2}}+\frac{L_{\text{Y2}}\cos \theta_{\text{Y2}}-d\cot\beta_{\text{Y2}}}{2}\Big{)}\\ M_{\text{Z2-6}}=Q_{\text{Y2}}(L_{\text{Y4}}+L_{\text{Y3}}\cos\theta_{\text{Y3}} +h_{2}\cot\beta_{\text{Y2}}+L_{\text{Y2}}\cos\theta_{\text{Y2}}-d\cot\beta_{ \text{Y2}})\end{array} \right. \tag{26}\]  
According to Formula (20), \(\sum F_{\text{YX}}=0\) and \(\sum F_{\text{YY}}=0\) can be derived as follows:  
\[T_{\text{B1}}+T_{\text{B2}}=0 \tag{27}\]  
\[Q_{\text{Y1}}+Q_{\text{Y2}}+P_{\text{R1}}=W_{\text{C1}}+W_{\text{C2}}+W_{\text{ C3}}+W_{\text{C4}}+P_{\text{Y1}}+P_{\text{Y2}}+P_{\text{Y3}}+P_{\text{Y4}} \tag{28}\]  
Based on the characteristics of rock fracture, it can be inferred that key block C\({}_{2}\) has completely fractured and collapsed. \[P_{\text{R}}=W_{\text{C4}}+P_{\text{Y4}} \tag{29}\]  
Substituting Formula (29) into Formulas (24) and (26) yields:  
\[M_{\text{Y1-2}}=0 \tag{30}\]  
\[M_{\text{Y2-3}}=\frac{(W_{\text{C1}}+P_{\text{Y1}})(L_{\text{Y1}}\cos\theta_{ \text{Y1}}+h_{2}\cot\beta_{\text{Y1}})}{2} \tag{31}\]  
By substituting Formulas (23) and (29) into Formula (28), \(Q_{\text{Y1}}\) can be determined as follows:  
\[Q_{\text{Y1}}=W_{\text{C1}}+W_{\text{C2}}+W_{\text{C3}}+P_{\text{Y1}}+P_{\text{ Y2}}+P_{\text{Y3}}-\frac{M_{\text{Y1-1}}+M_{\text{Y1-3}}+M_{\text{Y1-4}}}{L_{\text{Y1}} \cos\theta_{\text{Y1}}+h_{2}\cot\beta_{\text{Y1}}+L_{\text{Y4}}+L_{\text{Y3}} \cos\theta_{\text{Y3}}+L_{\text{Y2}}\cos\theta_{\text{Y2}}-d\cot\beta_{\text{Y2 }}+L_{\text{Y2}}\cot\theta_{\text{Y2}}} \tag{32}\]  
By substituting Formulas (26) and (27) into Formula (25), \(T_{\text{B1}}\) and \(T_{\text{B2}}\) are obtained as:  
\[\left\{ \begin{array}{l}T_{\text{B1}}=\frac{M_{\text{Y2-2}}+M_{\text{Y2-3}}+M_{ \text{Y2-4}}+M_{\text{Y2-5}}}{L_{\text{Y1}}\sin\theta_{\text{Y1}}-L_{\text{Y2}} \sin\theta_{\text{Y2}}}\\ T_{\text{B2}}=-\frac{M_{\text{Y2-3}}+M_{\text{Y2-3}}+M_{\text{Y2-4}}+M_{\text{Y2-5 }}}{L_{\text{Y1}}\sin\theta_{\text{Y1-1}}-L_{\text{Y2}}\sin\theta_{\text{Y2}}} \end{array} \right. \tag{33}\]According to the masonry beam theory, step rock beam structure, and S-R criteria, it can be inferred that to ensure the stability of the trapezoidal-like hinged arch structure in advance, the arch base must meet the following requirements:  
\[T_{\text{B1}}\tan\varphi\geq Q_{\text{Y1}} \tag{34}\]  
\[T_{\text{B2}}\tan\varphi\geq Q_{\text{Y2}} \tag{35}\]  
Based on Formulas (16) and (33), it is evident that when the denominator of the equation approaches zero, the horizontal stress \(T\) tends toward infinity. Clearly, achieving this condition under practical circumstances is challenging. Therefore, the structure must still satisfy the criterion for lateral-torsional buckling:  
\[T\leq a\eta\sigma_{\text{c}} \tag{36}\]  
where \(\eta\sigma_{\text{c}}\) is the extrusion strength of the key block at the corner end of the key strata (MPa). At this time, the arch height \(h_{l}\) of the trapezoidal-like hinged arch structure is determined as:  
\[h_{l}=\max\{L_{\text{Y1}}\sin\theta_{\text{Y1}},L_{\text{Y2}}\sin\theta_{\text {Y2}}+L_{\text{Y3}}\sin\theta_{\text{Y3}}+d\} \tag{37}\]  
A semi-trapezoidal hinged arch structure can be deduced using the trapezoid-like hinged arch structure, but it is not deduced in this paper.