# Study on Fracture Behavior of Directly Covered Thick Hard
## 4 Analysis of Mining Instability of Roof Cut Falling Structure
### Analysis of the Bearing Capacity of the Roof Cut Falling Body--_Hydraulic Support_
Based on the principle of minimum potential energy and the Ritz method [29], the total potential energy of the CSCF plate is determined from Equation (5) as:  
\[E_{P}=\frac{D}{2}\iint\limits_{A}\left(\frac{\partial^{2}\omega}{\partial x^{2}}+ \frac{\partial^{2}\omega}{\partial y^{2}}\right)^{2}dxdy-(1-\mu)D\iint\limits_{ A}\frac{\partial^{2}\omega}{\partial x^{2}}\cdot\frac{\partial^{2}\omega}{ \partial y^{2}}-\left(\frac{\partial^{2}\omega}{\partial x\partial y}\right)^{2 }dxdy-\iint\limits_{A}q_{0}\omega dxdy \tag{6}\]  
where \(A\) is the integral boundary condition, \(x\in[-a,\,a]\), \(y\in[0,\,b]\). Equation (6) is used to obtain:  
\[E_{P}=\frac{DC^{2}\pi^{4}\big{(}3a^{4}+2a^{2}b^{2}+3b^{4}\big{)}}{16a^{3}b^{3} }-Cabq_{0} \tag{7}\]  
Deflection coefficient \(C\) is obtained thus:  
\[C=\frac{8a^{4}b^{4}q_{0}}{D\pi^{4}(3a^{4}+2a^{2}b^{2}+3b^{4})} \tag{8}\]  
where \(D\) is the bending stiffness of the plate, N/m. This is calculated from the following equation:  
\[D=\frac{EH^{3}}{12(1-\mu^{2})} \tag{9}\]  
where \(E\) is the modulus of rock elasticity, GPa; and \(H\) is roof thickness, m. The CSCF plate deflection surface equation is then:  
\[\omega=\frac{96\big{(}1-\mu^{2}\big{)}a^{4}b^{4}q_{0}}{EH^{3}\pi^{4}(3a^{4}+2a ^{2}b^{2}+3b^{4})}\cdot\Big{[}1+\cos(\frac{\pi x}{a})\Big{]}\cdot\sin^{2}( \frac{\pi y}{2b}) \tag{10}\]  
The hydraulic bracket acts as a form of working face support equipment, carrying the load of the overlying roof plate, and the overlying rock movement affects the working state and bearing characteristics of the bracket and roof after pre-cracking blasting; the roof plate bends and sinks under the action of multiple factors and acts directly on the hydraulic bracket. With the movement of the broken roof, combined with the deflection of the overlying roof and the change of the stiffness of the support, the support resistance can be obtained [30]. Considering the effective bracket support efficiency, the brace resistance can be expressed as:  
\[F=K\cdot\Delta\omega/\eta \tag{11}\]  
where \(F\) is hydraulic bracket support resistance, kN; \(K\) is bracket stiffness, N/m; \(\Delta\omega\) is the degree of overburden roof deflection change, m; and \(\eta\) is effective bracket support efficiency. According to Equation (10), the roof deflection surface can be derived during the excavation of the working face. The three-dimensional surface of roof deflection in Figure 6 shows that the deflection deformation of the clamped edge and simple support edge is small in the process of coal seam mining, and the maximum deflection area is concentrated in the middle of the free edge. According to the reflection of the top deflection surface, the maximum deflection position of the roof (\(x=0\)) is in the middle of the working face, as modelling and analysis of the thick hard top cut fall and hydraulic support shows that pressure sinks. In Figure 7, \(l\) is the hydraulic bracket control roof distance, m; \(b\) is the cut-off body length, m; and \(\omega_{m}\) is the degree of roof end movement, m. The relationship between hydraulic support resistance and roof cut falling body is then as follows:  
\[F=K\cdot\omega_{(0,l)}/\eta=\frac{192\big{(}1-\mu^{2}\big{)}a^{4}b^{4}Klq_{0} \sin^{2}\left(\frac{\pi l}{2b}\right)}{E\eta H^{3}\pi^{4}\big{[}3a^{4}+2a^{2}b^ {2}+3b^{4}\big{]}} \tag{12}\]  
In the above equation, hydraulic support resistance \(F\) is related to working face length \(2a\), cut falling body length \(b\), roof thickness \(h\), and the lithological parameter \(E\).