# Study on Surface Damage Induced by High Heavy Layer Movement and Mining-Induced Earthquakes
## 2 Methodology
### Characteristics of Fracture and Movement in Heavy Layer
#### 2.2.2 Fracture Patterns of Heavy Layer Induced by Mining  
The Jurassic heavy conglomerate layer of the 7202 working face is simplified as a rectangular thick plate with orthotropic characteristics, the plate length of which is denoted by \(a\), the plate by \(b\), and the plate thickness by \(h\). A coordinate system is established based on the principal elastic directions, as shown in Figure 6. \(q\) represents the overburden load carried by the Jurassic heavy conglomerate layer. In this coordinate system, the open-off cut position of the 7202 working face is set at \(X=0\), and the advancing direction of the working face is aligned with the positive X-axis. The geometric similarity hypothesis based on cross-sectional deformation includes the following four assumptions [26]:  
Figure 5: Schematic of boundary condition changes in a heavy layer. Figure 6: Medium thick plate theory model for a heavy layer. 1. The heavy layer is continuous, with negligible influence from faults or joint surfaces, exhibiting characteristics of an integral rock mass. The layer can be treated as a uniformly distributed, isotropic, perfectly elastic body. 2. The neutral plane of the heavy layer experiences no extension or contraction during bending deformation. 3. Planes perpendicular to the neutral plane before bending remain perpendicular throughout the deformation process and maintain their original shape, meaning the strain on the neutral plane remains zero after bending. 4. The heavy layer plays a dominant role above the coal seam, controlling other weak discrete layers. Therefore, the load acting on these strata can be analyzed and calculated using the key stratum theory. The fundamental equation of the mechanical model for the Jurassic heavy layer is derived as shown in Equation (1). \[\left\{ \begin{array}{l}D\Delta w=q-\frac{2-\mu}{10(1-\mu)}h^{2}\Delta q\\ \frac{h^{2}}{10}\Delta Q_{x}-Q_{x}=D\frac{\partial}{\partial x}\Delta w+\frac{ h^{2}}{10(1-\mu)}-\frac{\partial q}{\partial x}\\ \frac{h^{2}}{10}\Delta Q_{y}-Q_{y}=D\frac{\partial}{\partial y}\Delta w+\frac{ h^{2}}{10(1-\mu)}-\frac{\partial q}{\partial y}\end{array} \right. \tag{1}\]  
Here, \(D\) represents the bending stiffness, calculated as \(D=\frac{Et^{3}}{12(1-\mu^{2})}\); \(q\) is the overburden load; \(w\) is the vertical deflection; \(\mu\) is poison's ratio; \(h\) is the thickness of the heavy layer; and \(Q_{x}\) and \(Q_{y}\) are the shear forces. By solving the above equation, the general solution for the bending moment of the thick plate is obtained as shown in Equation (2). \[ \begin{array}{l}M_{y}=\frac{qh^{4}G_{1}H_{1}}{\lambda_{c1}^{2}R_{1}S_{1}} \Big{\{}\mu\left(X_{1}{}^{\prime\prime}Y_{1}+\frac{h^{2}}{a^{2}}X_{1}Y_{1}{}^{ \prime\prime}\right)\\ +I_{2}h^{2}[X_{1}Y_{1}{}^{\prime\prime\prime}+(1+\mu)(X_{1}{}^{\prime\prime}Y ^{\prime\prime\prime}+\mu X_{1}{}^{\prime\prime\prime}Y_{1}{}^{\prime\prime}) ]\Big{\}}+qh^{2}I_{4}\end{array}  \tag{2}\]  
Here, \(\lambda_{c1}\) represents the parameter shown in Equation (3). \[\lambda_{c1}=\frac{\frac{M_{1}}{R_{1}}\left(\frac{b}{a}\right)^{4}+\frac{2\mu E _{1}F+2(1-\mu)X_{1}L_{1}}{R_{1}S}\Big{(}\frac{b}{a}\Big{)}^{4}+\frac{N_{1}}{ S_{1}}}{1+I_{2}\Big{(}\frac{h}{a}\Big{)}^{2}\left(\frac{K_{1}}{R_{1}} \right)+\frac{L_{1}}{S_{1}}} \tag{3}\]  
Here, \(a\) represents the working face advance, and \(b\) represents the working face length. The coefficients \(I_{2}=\frac{1}{S(1-\mu)}\) and \(I_{4}=\frac{\mu}{10(1-\mu)}\) are defined. The functions \(\lambda_{c1}\), \(L_{1}\), \(K_{1}\), \(E_{1}\), \(F_{1}\), \(G_{1}\), \(H_{1}\), \(R_{1}\), \(S_{1}\), \(M_{1}\), and \(N_{1}\) are determined by the deep beam function. Under consistent boundary support conditions for the thick plate in the XY plane, the following relationships hold: \(L_{1}=K_{1}\), \(E_{1}=F_{1}\), \(R_{1}=S_{1}\), \(M_{1}=N_{1}\)[27]. Based on the boundary conditions \(X(0)=X(a)=X^{\prime}(0)=X^{\prime}(a)=0\), the deep mode shape function is determined, as shown in Equation (4):  
\[\left\{