## 3 Mechanical Analysis of Roadway Roof Shock Damage  
After the excavation of coal seam roadways, the overlying strata and supporting structure on the roadway's roof will form a structure similar to a rock beam. This structure undergoes deformation and damage under the influence of surrounding rock stress. Assuming the structure is a continuous and uniform medium, the mechanical model of roadway roof shock failure is established, as shown in Figure 4. In the figure, \(\sigma_{s}\) represents the static load of the surrounding rock; \(\sigma_{d}\) is the dynamic load generated by the seismic source; \(\sigma_{t}\) is the supporting force of the support structure; \(a\) is the width of the roadway; \(h\) is the thickness of the structure; and \(d\) is the distance between the seismic source and the structure. According to plate and shell theory [20; 21], the buckling equation of the roadway roof structure is as follows:  
\[K_{\rm c}\
abla^{4}\omega(x,y)=F_{x}\frac{\partial^{2}\omega(x,y)}{\partial x ^{2}}+2F_{xy}\frac{\partial^{2}\omega(x,y)}{\partial x\partial y}+F_{y}\frac{ \partial^{2}\omega(x,y)}{\partial y^{2}} \tag{1}\]  
where \(Kc\) is the flexural stiffness of the structure; \(\omega(x,y)\) is the deflection equation of the structure; and \(Fx\), \(Fxy\), and \(Fy\) are external forces acting on the structure in different directions, KN. For the calculation of the deflection of the structure, the action of only horizontal force Fx can be considered. Therefore, Equation (1) can be simplified to  
\[K_{\rm c}\
abla^{4}\omega(x)-F_{x}\frac{\partial^{2}\omega(x)}{\partial x^{2 }}=0 \tag{2}\]  
Figure 4: Mechanical model of shock failure of main roadway. Figure 3: Map of surface subsidence and microseismic distribution during stopping in 21205 and 21405 faces. (**a**) Distribution of microseismic activity on both sides of the main roadway after the mining of LW21405. (**b**) Microseismic distribution profile for working faces 21205 and 21405. (**c**) Surface settlement curves for working faces 21205 and 21405. The two ends of the structure on the roadway roof can be approximated as a simply supported beam structure. Substituting the boundary conditions at \(x=0\) and \(x=a\) into Equation (2), the buckling equation of the structure can be calculated as  
\[\sum_{\lambda=1}^{+\infty}A\sin(\frac{\lambda\pi x}{a})\left(\frac{K_{\rm c} \lambda^{4}\pi^{4}}{a^{4}}-\frac{F_{x}\lambda^{2}\pi^{2}}{a^{2}}\right)=0 \tag{3}\]  
where \(\lambda\) is a positive integer, \(\lambda\geq 1\). Simplifying Equation (3), \(Fx\) can be calculated as  
\[Fx=\frac{K_{\rm c}\lambda^{2}\pi^{2}}{a^{2}} \tag{4}\]  
For the structure, its flexural stiffness \(Kc\) is  
\[K_{\rm c}=\frac{E\hbar^{3}}{12(1-\mu^{2})} \tag{5}\]  
where \(E\) represents the elastic modulus of the structure in GPa, and \(\mu\) is Poisson's ratio of the structure. In conclusion, the failure load of the structure is  
\[\sigma_{b}=\frac{F_{x}}{h}=\frac{\lambda^{2}E\hbar^{2}\pi^{2}}{12a^{2}(1-\mu^ {2})} \tag{6}\]  
The stress on the roadway roof structure can be divided into static and dynamic loads. When the combined stress of both exceeds the failure load of the structure, the roadway roof will experience shock failure, and at this point:  
\[\sigma_{\rm s}+\sigma_{4}e^{-\eta d}-\frac{\lambda^{2}E\hbar^{2}\pi^{2}}{12a^ {2}(1-\mu^{2})}\geq 0 \tag{7}\]  
where \(\eta\) represents the attenuation indices for the dynamic load sources propagating in the coal-rock mass. Formula (7) shows that there are two main factors affecting the shock failure of the roadway roof. One is the strength of the structure, which includes the strength of the roof strata and the support strength. The other is the surrounding rock stress, mainly including the self-weight stress and the mining-induced stress. After the roadway is formed and supported, the critical factor for its failure will be the magnitude of the mining-induced stress.