## 3 Mechanical Analysis of Initial Fracture  
The 2102 working face of the Balasu coal mine is the first mining face. Before the initial collapse, it can be regarded as a beam structure with fixed supports at both ends, and there are multi-layer hard and stable sandstones in the upper part of the coal seam. During the mining process, it is not easy to collapse the roof, and there is a risk for a large, suspended roof area, which may lead to strong-mine-pressure dynamic disasters. According to the method of determining the load of the key layer and its upper part [33], the ultimate caving step of the basic roof can be further calculated. The determination of the key layer and the determination of the upper load layer can be based on the following formula:  
\[(\mathrm{q_{n}})_{\mathrm{i}}=\frac{E_{\mathrm{1}}\mathrm{h_{1}}^{3}(\gamma_{ \mathrm{1}}\mathrm{h_{1}}+\gamma_{\mathrm{2}}\mathrm{h_{2}}+\cdots+\gamma_{ \mathrm{n}}\mathrm{h_{n}})}{E_{\mathrm{1}}\mathrm{h_{1}}^{3}+E_{\mathrm{2}} \mathrm{h_{2}}^{3}+\cdots+E_{\mathrm{n}}\mathrm{h_{n}}^{3}} \tag{1}\]  
where \(\mathrm{q_{n}}\) is the load of each layer, kN; \((\mathrm{q_{n}})_{\mathrm{i}}\) is the action of the nth layer of rock on the layer of rock, kN; \(E\) is the elastic modulus of each rock layer, MPa; h is the thickness of each rock layer, m; and \(\gamma\) is the unit weight of each rock layer, kN/m\({}^{3}\). By measuring the physical and mechanical parameters and inputting them into Formula (1), the medium-grained sandstone in the upper 4.8-22.43 m layer of the roof of the 2102 working face can be found to be the basic roof, which determines the first weighting interval of the working face. Figure 1: Layout of longwall panel 2102. (**a**) Panel layout. (**b**) Generalized stratigraphy column. Before the initial collapse of the 2102 working face, the upper roof should be a fixed-beam mechanical structure. According to the limit collapse step of the fixed beam [34], it can be seen that  
\[L=\text{h}\sqrt{\frac{2R_{T}}{\text{q}}} \tag{2}\]  
where \(L\) is the limit collapse step, m; \(R_{T}\) is the tensile strength of the key layer, MPa; q is the upper load of the key layer, kN; and h is the thickness of the key stratum. It can be seen from Equation (2) that under the condition of no fracturing, the first weighting interval of the 2102 first mining face is 77 m.  
According to the asymmetric three-hinged arch structure [35] of the initial collapse of the roof of the working face, during the advancing process of this face, the damage degree of the roof is different due to the time effect. According to damage mechanics analysis, the initial collapse of the roof of the working face is asymmetric. This collapse is easier to accomplish due to the high degree of damage accumulation near the cut side. The asymmetric three-hinged arch structure of this collapse is shown in Figure 2.  
Considering the accumulation of rock mass excavation damage, the ratio of the length of the rock mass near the working face l\({}_{1}\) to the length of the rock mass on the side of the open-off cut l\({}_{2}\) is \(K=1.5\), so the easy fracture distance of the roof under unfractured conditions is  
\[\text{l}_{0}=\text{l}_{2}=L\frac{1}{K+1} \tag{3}\]  
where \(K\) is the ratio of l\({}_{1}\) to l\({}_{2}\), and \(K\) is found to be 1.5 via analysis. Equation (3) shows that the distance of l\({}_{0}\) is 30.8 m, so the horizontal projection distance of the effective fracturing section of the hydraulic fracturing borehole should be within the range of 30.8 m in combination with the geological conditions. Based on the above analysis, in the case of the 2102 working face without hydraulic fracturing roof weakening measures, the mechanical model calculates the first weighting interval to be 77 m, which makes it very easy to cause strong pressure and even dynamic disasters of the working face.