## 4 Theoretical Analysis of Roof Influence on Underlying Coal and Rock Strata After Coal Seam Mining
### Fracture Laws of Overburden Roof
#### 4.2.2 Analysis of Roof Fracture Characteristics  
Typically, the failure position of the main roof occurs at the locations with maximum bending moments, namely, the upper and lower surfaces of the main roof. This is because  
Figure 4: Compressive Stress Curve of the Main Roof on Underlying Coal and Rock Layers. transverse stress reaches its peak at these positions, exceeding the tensile strength of rock-like materials and leading to failure. Thus, the relationship between the bending moment and fracture position of the main roof can be summarized as follows: the position with the maximum bending moment generally corresponds to the fracture position of the main roof. Analyzing the bending moment distribution of the main roof can help predict its failure location, thereby enabling rational support design [23; 24]. The calculation formula of the bending moment of the main roof is as follows:  
\[M=\frac{E_{j}{h_{j}}^{3}}{12}{\omega^{\omega}}_{2}(x) \tag{15}\]  
Substituting the geological and mining conditions of the 22,031 working face of Xin-Deng (Zhengzhou, China) Coal Mine into Equation (15), the bending moment curve of the main roof is obtained as shown in Figure 5. It can be seen from the figure that after the No. 2-1 coal seam is mined, under the action of the overlying strata load and the support of the underlying coal and rock strata, bending tensile stress appears in the main roof. The position with the largest bending moment is near the goat, about 0.5 m away from the edge of the goat, indicating that if the main roof fractures, the fracture position is about 0.5 m away from the edge of the goat. (1) Coal seam thicknesses were set to 3 m, 4 m, 5 m, 6 m, and 7 m, with other conditions held constant. The compressive stress distribution curves of the main roof on underlying coal and rock strata under different seam thicknesses and the variation curves of maximum compressive stress with seam thickness are shown in Figure 6. Additionally, the bending moment curves of the main roof and the variation curves of maximum bending moment with seam thickness are presented in Figure 7. The figures indicate that the maximum compressive stress of the main roof on the underlying strata occurs at the goat edge. The maximum compressive stress decreases exponentially with increasing seam thickness: when the seam thickness increases from 3 m to 5 m, the maximum compressive stress decreases by 1.2 MPa; when increasing from 5 m to 7 m, the reduction is 0.3 MPa. The position of maximum bending moment in the main roof is close to the goat edge. The maximum bending moment decreases linearly with increasing seam thickness: when the seam thickness increases from 3 m to 7 m, the maximum bending moment decreases by 66.6 MN \(\cdot\) m.  
Figure 5: The basic top bending moment curve. (2) Immediate roof thicknesses were set to 4 m, 6 m, 8 m, 10 m, and 12 m, with other conditions held constant. The compressive stress distribution curves of the main roof on underlying coal and rock strata under different immediate roof thicknesses and the variation curves of maximum compressive stress with immediate roof thickness are shown in Figure 8. Additionally, the bending moment curves of the main roof and the variation curves of maximum bending moment with immediate roof thickness are presented in Figure 9. The figures indicate that the maximum compressive stress of the main roof on the underlying strata occurs at the goaf edge. The maximum compressive stress decreases linearly with increasing immediate roof thickness: when the immediate roof thickness increases from 4 m to 12 m, the maximum compressive stress decreases by 1 MPa. The position of maximum bending moment in the main roof is close to the goaf edge.