## 3 Stability Analysis of an Integral Load-Bearing Structure of GER
### Roof Structure of GER in a Fully Mechanized Top Coal Caving Longwall
The displacement of the main roof at the point above the roadside support under the combined action of two external forces is \(\Delta_{c}\), the deflection w curve equation of the cantilever beam is as follows:  
\[w(x)=-\frac{q_{j}x^{2}}{24 EI}\Big{(}x^{2}-4 lx+6l^{2}\Big{)} \tag{1}\]  
Substituting the main roof weight \(q_{z}\), overburden pressure \(q_{j}\), and cantilever distance \(l_{j}=x_{1}+x_{2}+x_{3}\) into Formula (1), the displacement \(w_{j}\) of rock mass above the roadside support when \(q_{j}\) acts alone is given by Formula (2) [27, 28]:  
\[w_{j}=-\frac{q_{j}(x_{1}+x_{2})^{2}}{24 EI}\Big{[}(x_{1}+x_{2})^{2}-4(x_{1}+x_{ 2}+x_{3})x_{1}+6(x_{1}+x_{2}+x_{3})^{2}\Big{]} \tag{2}\]  
Substituting the supporting force of the immediate roof \(q_{m}\) and cantilever distance \(l_{z}=x_{1}+x_{2}\) into Formula (1), the displacement \(w_{m}\) of rock mass above the roadside support when \(q_{m}\) acts alone can be calculated by Formula (3):  
\[w_{m}=\frac{q_{m}(x_{1}+x_{2})^{4}}{8 EI} \tag{3}\]  
According to the deformation compatibility relation of the main roof, the displacement \(\Delta_{c}\) equal to the algebraic sum of the displacement caused by \(q_{j}\) and \(q_{m}\) acting alone on the rock mass above the roadside support is calculated as follows:  
\[\Delta_{c}=w_{j}+w_{m} \tag{4}\]  
There are unknown parameters in Formula (4); therefore, it is necessary to further analyze the force of self-weight \(q_{z}\), the support resistance in the roadway \(q_{s}\), the roadside support resistance \(q_{b}\), and the reaction force interacting with the main roof \(q_{m}\). The moment of each force to the coordinate origin O is calculated using Formula (5):  
\[\left\{ \begin{array}{l}R_{P}=\int_{x_{1}}^{x_{1}+x_{2}}P_{n}xdx\\ R_{s}=\int_{0}^{x_{1}}q_{s}xdx\\ R_{z}=\int_{0}^{x_{1}+x_{2}}q_{z}xdx\\ R_{m}=\int_{0}^{x_{1}+x_{2}}q_{m}xdx\end{array} \right. \tag{5}\]  
The distribution intensity of each external force acting on the immediate roof is simplified as uniform load, and by shifting the term, the following can be obtained:  
\[q_{m}=\frac{q_{s}x_{1}^{2}+P_{n}x_{2}(2x_{1}+x_{2})-q_{z}(x_{1}+x_{2})^{2}}{ \left(x_{1}+x_{2}\right)^{2}} \tag{6}\]  
By substituting Formula (6) into Formula (4), the displacement of the main roof above the roadside support under the action of various external forces can be obtained using Formula (7):  
\[\Delta_{C}=\frac{\Big{[}q_{s}x_{1}^{2}+P_{n}x_{2}(2x_{1}+x_{2})-q_{z}(x_{1}+x _{2})^{2}\Big{]}}{8 EI}-\frac{q_{j}\Big{[}\left(x_{1}+x_{2}\right)^{2}-4x_{1}( x_{1}+x_{2}+x_{3})+6(x_{1}+x_{2}+x_{3})^{2}\Big{]}}{24 EI} \tag{7}\]  
According to Formula (7), the rotary sinking of the main roof is related to the resistance of the roadside support, the support strength of the anchor bolt and anchor cable in the roadway, the overburden pressure, and the material and geometric properties of the main roof. Among these factors, except for the resistance of the roadside support and the support strength in the roadway, the other parameters cannot be controlled artificially. When the overlying rock pressure and the material properties of the main roof are kept constant, the deformation of the main roof is negatively correlated with the roadside support resistance. The reduction in roadside support resistance will cause the internal bending force in the main roof above the solid coal wall to increase sharply. In the whole system of the rock beam, the dangerous section of internal bending force is located above the solid coal wall and roadside support body, and the passive roof cutting of the main roof outside the roadway requires the roadside support body to have a high support resistance.