## 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
The maximum bending moment decreases linearly with increasing immediate roof thickness: when the immediate roof thickness increases from 4 m to 12 m, the maximum bending moment decreases by 46 MN \(\cdot\) m.  
Figure 6: Compressive Stress Distribution of the Main Roof on Underlying Coal and Rock Layers under Different Coal Seam Thickness Conditions: (**a**) Variation in Compressive Stress with Distance from Goal; (**b**) Variation in Maximum Compressive Stress with Coal Seam Thickness. Figure 7: Bending Moment Distribution of the Main Roof under Different Coal Seam Thickness Conditions: (**a**) Variation in Bending Moment with Distance from Goal; (**b**) Variation in Maximum Bending Moment with Coal Seam Thickness. (3) Coal seam elastic moduli were set to 0.5 \(\times\) 10\({}^{3}\) MPa, 1 \(\times\) 10\({}^{3}\) MPa, 1.5 \(\times\) 10\({}^{3}\) MPa, 2 \(\times\) 10\({}^{3}\) MPa, and 2.5 \(\times\) 10\({}^{3}\) MPa, with other conditions held constant. The compressive stress distribution curves of the main roof on underlying coal and rock strata under different coal seam elastic moduli and the variation curves of maximum compressive stress with elastic modulus are shown in \[M=\frac{E_{j}{h_{j}}^{3}}{12}{\omega^{\omega}}_{2}(x) \tag{15}\]0 Additionally, the bending moment curves of the main roof and the variation curves of maximum bending moment with elastic modulus are presented in \[M=\frac{E_{j}{h_{j}}^{3}}{12}{\omega^{\omega}}_{2}(x) \tag{15}\]1 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 increases linearly with increasing coal seam elastic modulus: when the elastic modulus increases from 0.5 \(\times\) 10\({}^{3}\) MPa to 2.5 \(\times\) 10\({}^{3}\) MPa, the maximum compressive stress increases by 0.62 MPa. The position of maximum bending moment in the main roof is close to the goat edge. The maximum bending moment increases linearly with increasing coal seam elastic modulus: when the elastic modulus increases from 0.5 \(\times\) 10\({}^{3}\) MPa to \(\times\)10\({}^{3}\) MPa, the maximum bending moment increases by 28.6 MN \(\cdot\) m.  
\[M=\frac{E_{j}{h_{j}}^{3}}{12}{\omega^{\omega}}_{2}(x) \tag{15}\]2 Compressive Stress Distribution of the Main Roof on Underlying Coal and Rock Layers under Different Immediate Roof Thickness Conditions: (**a**) Variation in Compressive Stress with Distance from Goaf; (**b**) Variation in Maximum Compressive Stress with Immediate Roof Thickness. \[M=\frac{E_{j}{h_{j}}^{3}}{12}{\omega^{\omega}}_{2}(x) \tag{15}\]3 Bending Moment Distribution of the Main Roof under Different Immediate Roof Thickness Conditions: (**a**) Variation in Bending Moment with Distance from Goaf; (**b**) Variation in Maximum Bending Moment with Immediate Roof Thickness. \(5\times 10^{3}\) MPa, the maximum bending moment increases by 57.7 MN\(\cdot\) m; when increasing from \(5\times 10^{3}\) MPa to \(7\times 10^{3}\) MPa, the increase is 34.1 MN\(\cdot\) m.  
In summary, the quantitative study on factors influencing roof stability during coal seam mining using the range analysis method shows that the immediate roof elastic modulus (range 2.38/91.8) is the most critical parameter affecting maximum compressive stress and maximum bending moment. As the immediate roof elastic modulus varies from \(3\times 10^{3}\) MPa to \(7\times 10^{3}\) MPa, the corresponding maximum compressive stress ranges from 11.46 MPa to 13.84 MPa, and the maximum bending moment ranges from 5491.1 MN\(\cdot\) m to 5582.9 MN\(\cdot\) m, with ranges reaching 2.38 and 91.8, respectively.