## 3 Determination of Key Parameters
### Control Mechanism of Aperture on Roof Breaking  
Increasing the aperture can increase the radius of the pressure relief zone, which is conducive to the formation of a weakening zone in the coal body. Reducing the hole spacing results in fuller crack propagation around the hole [28; 29]. Based on the theory of material mechanics, the mechanical model of the roof cantilever beam under the influence of dense drilling is constructed for stress analysis. According to the mechanical model shown in Figure 6, the bending moment equation \(M_{2}(x)\) and the shear force equation \(F_{2}S(x)\) of the cantilever beam are:  
\[M_{2}(x)=\left\{ \begin{array}{ll}qb\frac{(l_{1}+l_{2}-x)^{2}}{2}&l_{0}\leq x \leq l_{1}+l_{2}\\ qb\frac{(l_{1}+l_{2}-x)^{2}}{2}-F_{A}(l_{0}-x)&0\leq x\leq l_{0}\end{array} \right. \tag{11}\]  
\[F_{25}(x)=\left\{ \begin{array}{ll}qb(l_{1}+l_{2}-x)&l_{0}\leq x\leq l_{1}+l_ {2}\\ qb(l_{1}+l_{2}-x)-F_{A}&0\leq x\leq l_{0}\end{array} \right. \tag{12}\]  
In the formula: \(b\) is the thickness of the beam (m); \(l_{1}\) is the roadway width (m); \(l_{0}\) is the width of the gob-side roadway (m); \(l_{2}\) is the lateral cantilever length of the roof (m). According to the theory of material mechanics, the corresponding normal stress and shear stress can be expressed as:  
\[\sigma(x,z)=\frac{M(x)}{I_{\rm y}}z \tag{13}\]  
\[\tau(x,z)=\frac{F_{\rm S}(x)S_{\rm y}^{*}}{I_{\rm y}b^{*}} \tag{14}\]  
In the formula, \(I_{\rm y}\) is the moment of inertia of the beam (m4); \(S^{*}_{\rm y}\) is the static moment of the beam (m3); \(b^{*}\) is the width (m) at the neutral axis of the beam. From the bending moment equation and shear force equation of the cantilever beam shown in Equations (11) and (12), it can be seen that the bending moment and shear force at the O section of the fixed end are the largest. According to the normal stress and shear stress equations shown in Equations (15) and (16), when the roof is not cut by dense drilling,  
Figure 6: Mechanical model of roof of dense-drilling roof cutting roadway. the normal stress and shear stress at the O section of the fixed end are also the largest, and the roof is destroyed first. Therefore, in order to ensure the success of roof cutting, the maximum normal stress and maximum shear stress on the B section should be greater than the maximum normal stress and maximum shear stress on the O section:  
\[\frac{\sigma_{B\max}}{\sigma_{O\max}}>1,\ \frac{\tau_{B\max}}{\tau_{O\max}}>1 \tag{15}\]  
At the same time, it is ensured that the maximum normal stress and maximum shear stress are greater than the strength limit of the rock beam, that is:  
\[\sigma_{B\max}>\sigma_{c},\ \sigma_{B\max}>\sigma_{c} \tag{16}\]  
After roof cutting, the moment of inertia \(I_{By}\) and static moment \(S_{By}^{*}\) of the roof borehole can be expressed as  
\[I_{By}=\int_{A}z^{2}dA=\frac{bh^{3}}{12}-\left[\frac{Rh_{1}^{3}}{12}+2 Rh_{1} \left(\frac{h-h_{1}}{2}\right)^{2}\right] \tag{17}\]  
\[S_{By}^{*}=\left\{ \begin{array}{ll}b\left(\frac{h}{2}-z\right)\left(\frac{h }{4}+\frac{z}{2}\right)&h_{1}-\frac{h}{2}\leq z\leq\frac{h}{2}\\ \frac{1}{8}(b-2R)h^{2}-2R\left(h_{1}^{2}-hh_{1}\right)-\frac{1}{2}bz^{2}&0 \leq z\leq h_{1}-\frac{h}{2}\\ (b-2R)\left(\frac{h^{2}}{4}-z^{2}\right)&-\frac{h}{2}\leq z\leq 0\end{array} \right. \tag{18}\]  
In the formula: \(h\) is the thickness of the roof (m); \(h_{1}\) is the drilling depth (m); \(R\) is the diameter of the drilling (m).