## 3 Mechanism of Pressure Relief Caused by Roof Cutting  
Based on the key stratum principle, the excavation of adjacent long-wall panels induces characteristic \"O-X\" fracturing patterns, as depicted in Figure 2a. At the end of the long-wall face, an \"arc-shaped triangular block\" is generated referred to as the key block (Block B in Figure 2b). As the coal seam is mined, the immediate roof experiences irregular breakage and sagging, separating from the main roof above. Once the overburden strata stabilize, the main roof assumes a hinged structure composed of blocks A, B and C.  
The coal pillar is located beneath the cantilever beam. The long, thick cantilever typically results in deformation of the roadway. Therefore, block B is very important for the stability of the roadway structure, which can be defined as including three basic dimensions: the lateral rift size (\(D\)), the periodic weighting length (\(L\)) and thickness of the block B (\(h\)), with their interrelationship quantitatively described by the following empirical formulation [32]. \[D=\frac{2L}{17}\Bigg{[}\sqrt{(10\frac{L}{S})^{2}+102}-10\frac{L}{S}\Bigg{]} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Lithological Column** & **No.** & **Lithology** & **Thickness/m** & **Depth/m** & **Key Layer** \\ \hline \multirow{4}{*}{} & \multirow{2}{*}{7} & Medium & \multirow{2}{*}{26.12} & \multirow{2}{*}{579.34} & \multirow{2}{*}{No} \\  & & sandstone & & & \\ \cline{1-1} \cline{5-5}  & 6 & Silstone & & & \\ \cline{1-1} \cline{5-5}  & 5 & Sandy & & & \\ \cline{1-1} \cline{5-5}  & 4 & Fine sandstone & & & \\ \cline{1-1} \cline{5-5}  & 3 & 2-2 coal & & & \\ \cline{1-1} \cline{5-5}  & 2 & Sandy & & & \\ \cline{1-1} \cline{5-5}  & 1 & Silstone & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Lithology column of the strata. Figure 1: Configuration of the 21108 working face and deformation phenomenon. The force of block B is calculated and the structural diagram is depicted in Figure 3, presuming that the uniform roof load is \(q_{1}\), the distributed coal load is \(\sigma_{y}\) and the supporting resistance \(P_{2}\) acts at \(x_{0}\) (where \(s<x_{0}<s+a\)). The stress equilibrium zone for the coal rib (s) and \(\sigma_{y}\) can be defined as [33]  
\[s=\frac{mA_{1}}{2\tan\varphi_{0}}\ln\frac{k\gamma H+\frac{c_{0}}{\tan\varphi_{ 0}}}{\frac{c_{0}}{\tan\varphi_{0}}+\frac{P_{0}}{A_{1}}} \tag{2}\]  
\[\sigma_{y}=(\frac{c_{0}}{\tan\varphi_{0}}+\frac{P_{0}}{A_{1}})\exp\left[\frac{ 2\tan\varphi_{0}}{mA_{1}}(s-x_{0})\right]-\frac{c_{0}}{\tan\varphi_{0}} \tag{3}\]  
where \(P_{0}\) is the support strength [Pa] of the coal pillar; \(\varphi_{0}\) is the internal friction angle; \(A_{1}\) is lateral pressure coefficient; \(\gamma\) is the average unit weight of overlying layers [N/m\({}^{3}\)]; \(k\) is the stress concentration factor; \(H\) is the mining depth [m]; \(m\) is the mining height [m]; \(c_{0}\) is the cohesion [Pa]. The moment balance is applied at point A  
Figure 3: Simplified physical model of a lateral cantilever beam. Figure 2: Roof breaking law along the gob-side roadway. (**a**) “O-X” fracture; (**b**) Mechanical model. \[\sum M=0 \tag{4}\] \[\frac{1}{2}q_{1}(s+a+b+l)^{2}-P_{0}(s+a+\frac{1}{2}b)-P_{2}x_{0}-M_{1}=0 \tag{5}\]  
Thus, can be obtained  
\[M_{1}=\frac{1}{2}q_{1}(s+a+b+l)^{2}-P_{0}(s+a+\frac{1}{2}b)-P_{2}x_{0}-M^{\prime} \tag{6}\]  
among them  
\[M^{\prime}=\int_{0}^{s}\sigma_{y}xdx \tag{7}\]  
and  
\[ \begin{array}{c}M^{\prime}=\frac{m^{2}A_{1}^{2}c_{0}+m^{2}A_{1}P_{0}\tan \varphi_{0}}{4\tan^{3}\varphi_{0}}(e^{\frac{2\tan\varphi_{0}}{MA1}}-1)\\ -\frac{mA_{1}c_{0}+mP_{0}\tan\varphi_{0}}{2\tan^{2}\varphi_{0}}s-\frac{c_{0}} {2\tan\varphi_{0}}s^{2}\end{array}  \tag{8}\]  
For rectangular roof cross-section elements per unit width (\(b=1\) m), assuming cantilever tension there are:  
\[M=\frac{h^{2}}{6}R_{t} \tag{9}\]  
where \(R_{t}\) is the tensile strength (Pa).