## 2 Quantitative Relationship of Roof-Cutting Parameters for the Retaining Roadway Roof
### Quantitative Relationship Analysis of Roof Cutting Parameters for Retaining Roadway  
According to [23], it can be concluded that the pre-splitting blasting of the main roof-cutting seam section of the top-cutting and retaining roadway not only ensures the formation of seams along the direction of the blast hole connection but also ensures the stability of the roadway roof. A main dynamic static coupling mechanical model was established, and with the 7135 working face of Qidong Coal Mine as the engineering background, the minimum quantitative relationship between the length of the charge and the spacing between the blast holes during the basic roof formation and the maximum quantitative relationship between the length of the charge and the spacing between the blast holes during the basic roof stability of the roadway were obtained. Based on this, the parameters for advanced pre-splitting blasting in the return airway of the 7135 working face were designed, including the length of the charge and the spacing between the blast holes. The parameters of roof-cutting angle and height directly affect the distribution of mining-induced stress on the roof during the retention period and the collapse of the goal roof. Due to the variability in the thickness of the roadway roof, the design of roof-cutting parameters depends not only on the rock type of the roof but also on its thickness. In areas where the immediate roof is thicker and the main roof is thinner, based on the crushing and swelling characteristics of the immediate roof, the roadway roof should be supported before the main roof reaches the maximum span, and the cutting height should equal the thickness of the immediate roof. In areas where the immediate roof is thinner and the main roof is thicker, it is necessary to consider the cutting height and angle to ensure that the main roof completely collapses along the cutting seam surface. In this case, the activity of the rock strata in the goat includes the fragmentation and subsidence of the direct roof and the breaking and sinking of the basic roof. The main role in stabilizing the roof of the retained roadway is whether the main roof is completely fractured along the cutting line direction. Consider the roof plate as a \"cantilever beam\" structure [24] with an angle of \(\theta\) between the cutting seam and the vertical direction, and a height of \(h\) to cut off the roadway roof. The load \(q_{1}\) of the overlying rock layer borne by the main suspended roof acts as a uniformly distributed force on the roof \(q_{1}\) of the cantilever beam, and a mechanical model is established as shown in Figure 2.  
Figure 2: Roof mechanics model of gob-side entry retaining by roof cutting. After mining the working face, the advanced pre-cracked roof begins to collapse under the action of mining stress and self-weight. In order to ensure that the main roof of the uncut joint collapses along the direction of the cutting roof line, the tensile strength of the rock mass is used as the fracture criterion for the main roof of the uncut joint along the cutting roof line. This criterion is:  
\[\sigma_{\text{nt}}>\sigma_{\text{t}} \tag{1}\]  
\[\sigma_{\text{nt}}=\frac{M_{\text{nt}}}{W_{\text{nt}}} \tag{2}\]  
where \(M_{nt}\) is the bending moment at the main roof of the uncut joint, N\(\cdot\)m; and \(W_{nt}\) is the bending section coefficient at the main roof of the uncut joint. This is expressed as:  
\[\left\{ \begin{array}{l}M_{\text{nt}}=\frac{q_{1}l_{4}^{2}+\rho_{2}gm_{2}l_{4 }^{2}}{2}\\ W_{\text{nt}}=\frac{{h_{1}}^{2}}{6}\end{array} \right. \tag{3}\]  
where \(q_{1}\) is the load on the overlying rock layer borne by the basic roof, N\(/\)m; \(l_{4}\) is the distance from the uncut part to the end of the cantilever beam, m; \(\rho_{2}\) is the density of the basic roof rock layer, kg\(/\)m\({}^{3}\); \(m_{2}\) is the thickness of the basic roof, m; and \(h_{1}\) is the height of the basic roof of the uncut part, m.  
According to Equation (4), \(l_{4}\) and \(h_{1}\) can be calculated as:  
\[\left\{