## 3 Mechanical Analysis of Stress Optimization Induced by Roof Structure in GSER
### Analysis of Mechanical Model Results
There is a relatively clear negative correlation between the tensile stress in the uncut part of the main roof and the cutting angle within a certain range. As the cutting angle increases, the bending-section coefficient of the cantilever beam gradually increases, and the tensile stress gradually decreases to tend towards 0 MPa. The higher the cutting height, the more obvious the degree of reduction. When the cutting angle is 0\({}^{\circ}\), the tensile stress in the uncut part of the main roof reaches its peak. Therefore, during actual construction, increasing the cutting height or decreasing the cutting angle within a certain range can significantly increase the tensile stress in the uncut part of the main roof, which helps the cantilever roof on the goat side to break smoothly under the action of tensile stress. There is a negative correlation between the tensile stress in the uncut part of the main roof and the burial depth of the coal seam. From the analysis of the mechanical model, as the burial depth of the coal seam increases, the width of the limit equilibrium zone of the solid coal rib increases, leading to a decrease in the length of the cantilever roof and, subsequently, a decrease in tensile stress. Therefore, it is necessary to select suitable roof-cutting parameters in different coal seam burial depth areas to avoid problems such as difficulty in the collapse of the goat-side roof due to insufficient or excessive cutting or excessively long construction cycles for cutting-roof drilling. Analyzing Figure 5b, it can be seen that the tensile stress in the uncut part of the main roof is comprehensively influenced by the burial depth of the coal seam, the cutting height, and the cutting angle. Analyzing the state of sufficient roof cutting, where the tensile stress in the uncut part of the main roof is greater than the tensile strength, we find that when the cutting angle and cutting height are kept constant, for every 50 m decrease in the coal seam burial depth within the range from 350 m to 500 m, the tensile stress in the uncut part of the main roof will increase by approximately 16%-27%. The increase becomes more significant with the decrease in cutting angle and the increase in cutting height. When the coal seam burial depth and cutting angle are kept constant, for every 1 m increase in the cutting height within the range from 4 m to 8 m, the tensile stress in the uncut part of the main roof will increase by approximately 28%-54%. The increase becomes more significant with the decrease in coal seam burial depth and the increase in cutting angle. When the coal seam burial depth and cutting height are kept constant, for every 5\({}^{\circ}\) decrease in the cutting angle within the range from 0\({}^{\circ}\) to 50\({}^{\circ}\), the tensile stress in the uncut part of the main roof will increase by approximately 20%-35%. The increase becomes more significant with the decrease in coal seam burial depth and the increase in cutting height. Judging from the increase in tensile stress in the uncut part of the main roof, the overall impact of the cutting height is relatively significant, followed by the cutting angle. The burial depth of the coal seam also has an impact on the tensile stress of the main roof, especially in relatively shallow burial depth areas. Considering that actual cutting parameters are usually selected as integers, under different burial depth conditions, we can fix the cutting height and cutting angle as integers and further study the critical value and reasonable range of key cutting parameters by taking the tensile strength of the main roof as the threshold. By fixing the cutting angle and cutting height as integers and substituting them into Equation (14), with the tensile strength of the main roof as the threshold, the reasonable range of critical cutting parameters under different burial depths is obtained, as shown in Figure 6. By analyzing Figure 6, we can see that the reasonable range of cutting parameters exhibits a fan-shaped distribution characteristic.