## 2 Analysis of Roof Instability Effect
### Instability Analysis of the Upper Immediate Roof  
According to the location and lithology of the roof, the instability of the upper immediate roof is the main factor causing the pressure on the working face. The different crushing conditions of the lower strata directly affect the stability and rotation angle of the upper immediate roof [42]. In addition, the instability of the roof has a certain lag. Therefore, the fracture line of the roof is located on the extension line of the coal wall and roof cutting, respectively. There are three forms of the upper immediate roof instability: sliding instability at the coal wall, rotary instability at the coal wall, and rotary instability at the roof cutting. The instability model of the upper immediate roof is established, as shown in Figure 4.  
The horizontal force cannot be transmitted after the rock strata breaks, which results in the upper immediate roof presenting a cantilever beam state. Different crushing conditions directly affect the cantilever beam's instability forms [43; 44; 45]. Figure 4a shows that the cantilever beam showed sliding instability at the coal wall. The working resistance of the hydraulic support can be defined as Equation (4):  
\[ \begin{array}{c}F_{ZA}=KB[(L_{1}+\frac{1}{2}T\cot\beta)T\gamma_{C}+(L_{1}+T \cot\beta+\frac{1}{2}H_{L}\cot\beta)H_{L}\gamma_{S}\\ +(L_{1}+T\cot\beta+H_{L}\cot\beta+L_{X}+\frac{1}{2}H_{U}\cot\alpha)H_{U}\gamma _{F}]+Q_{m}\end{array}  \tag{4}\]  
where \(F_{ZA}\) is the working resistance of the hydraulic support under the sliding instability at the coal wall, kN; \(K\) is the safety factor, which is set to 1.4; \(B\) is the width of the hydraulic support; \(L_{1}\) is the roof control distance, and length is equal to the sum of end face distance and canopy length, m; \(\beta\) is the caving angle of the top coal, which is set to 75\({}^{\circ}\); \(L_{X}\) is the  
Figure 3: Coupling mechanical model of roof-top coal-hydraulic support. length of the hanging roof, which is set to 2 m; \(\gamma_{C}\) is the bulk density of the top coal, which is set to 14 kN/m\({}^{3}\); \(\gamma_{S}\) is the bulk density of the lower immediate roof, which is set to 24.1 kN/m\({}^{3}\); \(\gamma_{F}\) is the bulk density of the upper immediate roof, which is set to 24.8 kN/m\({}^{3}\); \(Q_{m}\) is the load of the overlying strata, kN. According to calculations, the hydraulic support has a working resistance of 18,993.5 kN. In the 08 mining area of Shilawusu, the hydraulic support type is ZF22000/29/55. Therefore, the working resistance reaches 86% of the rated resistance, which meets the working face's support requirements. Because of rotary instability, the impact load will be transferred to the lower immediate roof and top coal. This will make the support resistance of the hydraulic support go up. Usually, the fracture length of the upper immediate roof is the same, but fracture positions differ [6]. Figure 4b shows that the cantilever beam rotated with instability at the coal wall. In the initial state of the rotary, the working resistance of the hydraulic support can be defined as Equation (5):  
\[F_{Zb}{}^{1}=K(F+G_{C}+G_{L}+G_{U1}+G_{U2})+Q_{m} \tag{5}\]  
where \(F_{Zb}{}^{1}\) is the working resistance of the hydraulic support under the initial state of the rotary instability at the coal wall, kN; \(F\) is the additional force of the overlying strata, kN; \(G_{C}\) is the gravity of the top coal, kN; \(G_{L}\) is the gravity of the lower immediate roof, kN; \(G_{U1}\) is the gravity of unbroken upper immediate roof, kN; \(G_{U2}\) is the gravity of the cantilever beam, kN. According to Equation (5), the working resistance is impacted by the additional force of overlying strata. Figure 4b demonstrates that the support force of the lower immediate roof acts at the \(A\).