## 5 Analysis of Influencing Factors and Engineering Example
### Engineering Example  
On the basis of the above research, we analyzed the catastrophe instability condition of no. 12401 working face in Shendong mining area, calculating its reasonable support resistance to verify the theoretical analysis results. According to Equation (19), the necessary and sufficient conditions for system instability can be determined when \(h_{\text{j}}=17.6\) m, \(H=31.3\) m, \(\varphi^{\prime}=31^{\circ}\), \(E=3.8\) GPa, \(L_{\text{Z}}=14\) m, \(h_{1}=50\) m, \(l_{\text{A}}=30.4\) m, \(k_{1}=2.3\times 10^{5}\) kN/m, \(q=1.33\) MPa, \(u_{0}=35\) mm, \(\rho g=22.3\) kN/m\({}^{3}\), \(\rho_{1}\)g = 16.3 kN/m\({}^{3}\), \(\varphi=27^{\circ}\), \(i=0.67\), tan\(\varphi^{\prime}=0.6\), tan\(\varphi=0.5\), \(\lambda_{z}=0.54\). Combined with the calculation equation of the thickness of an immediate roof rock stratum [3]  
\[\sum h=(1-P)M/(1-k_{p}) \tag{29}\]  
where \(\sum h\) is the thickness of an immediate roof rock stratum, m; \(M\) is the thickness of coal seam, 6.2 m; \(P\) is the recovery rate of the working face, 80%; and \(k_{\text{p}}\) is the bulking coefficient of the immediate roof rock stratum, 1.15-1.30, 1.25 in this study. According to Equation (29), the thickness of the immediate roof was 31.3 m. Combined with the mining conditions of no. 12401 working face, the thickness of the immediate roof was 31.3 m when the collapsed gangue filled the goat. Therefore, in the process of collapse, the mudstone with a thickness of 5.86 m and the fine sandstone with a thickness of 4.72 m above the coal seam will be transformed into an immediate roof and enter the goat, forming a combined cantilever beam structure. Calculation of main roof step sinking  
\[ \begin{array}{l}\Delta h=2u_{0}+\frac{2u_{0}}{K}[1+\frac{hj\rho g+\frac{P_{ 1}g^{2}}{2\lambda_{z}\tan\varphi}-\frac{\text{tan}\varphi^{\prime}}{2}hj\rho g }{4\lambda_{z}\alpha_{0}e^{-2}}+\frac{\sqrt{2}}{3}([1-K])^{3/2}]\\ =0.07+\frac{0.07}{1.61}[1+1.87+0.224]\\ =0.205m\end{array} \]  
At the time of the working face first weighting, the step sinking of the main roof was between 210 mm and 235 mm, with an average of 224 mm. The theoretical calculation results were close to the monitoring data of the step sinking, which proved the rationality of the theoretical analysis. However, there was also a difference of 0.021 m, indicating that the immediate roof had undergone a catastrophic instability, resulting in the step sinking of the main roof, which was consistent with the measured results. Judgment of system stability  
Necessary conditions for catastrophic instability:From \(\lambda\)= \(EA/L\), \(\lambda_{1}=\lambda e^{-2}=\frac{E_{Z}A}{H_{Z}}e^{-2}\), and substituted into Equation (22)  
\[ \begin{array}{l}\frac{k_{1}}{\lambda_{1}}=\frac{k_{1}H_{Z}e^{2}}{E_{Z}A}\\ =\frac{2.3\times 10^{6}\times 31.3\times 2.72^{2}}{3.8\times 10^{6}\times 14 \times 1}\\ =0.986<1\end{array} \]  
Analysis of sufficient conditions for catastrophic instability  
\[ \begin{array}{l}-\frac{h_{i}l_{A}\rho g+\frac{p_{1}g_{A}^{2}}{2\lambda_{1} \tan\varphi}-\frac{\tan\varphi^{\prime}}{2}h_{j}\rho g}{4\lambda\mu e^{-2}}+K \xi-1\\ =-1.866+2.78-1\\ =-0.086<0\end{array} \]  
It can be found from the calculation results of the catastrophe instability criterion that the mechanical system composed of the no. 12401 working face can meet the necessary and sufficient conditions of instability. Under the current support conditions, the catastrophic instability of the system is prone to occur and induce roof accidents. It can be found from the in-site monitoring that the roof had different degrees of step sinking with the average subsidence of 224 mm during the first weighting. The rib fall of coal wall is serious, which proves that the theoretical calculation is consistent with the actual situation. Therefore, it is necessary to adjust the external conditions, such as increasing the support resistance or improving the advance speed, in order to prevent the roof cutting along the coal wall.