# 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
\begin{array}{l}A+C=0\\ \beta B+\beta D=0\\ A\cosh\beta a+B\sinh\beta a+C\cos\beta a+D\sin\beta a=0\\ \beta A\sinh\beta a+\beta B\cosh\beta a-\beta C\sin\beta a+\beta D\cos\beta a =0\end{array} \right. \tag{4}\]  
The functions \(A\), \(B\), \(C\), and \(D\) are undetermined coefficients of the deep mode shape function. Due to the symmetry of the heavy layer in both \(x\) and \(y\) directions, the form of \(Y_{1}\) is identical to \(X_{1}\), with only the variable substituted from \(x\) to \(y\). The deep beam mode shape functions \(X_{1}\) and \(Y_{1}\) are as shown in Equation (5):  
\[\left\{ \begin{array}{l}X_{1}=(\text{ch}\alpha_{1}x-\cos\alpha_{1}x)-\alpha_{1 }(\text{sh}\alpha_{1}x-\sin\alpha_{1}x)\\ Y_{1}=(\text{ch}\alpha_{1}y-\cos\alpha_{1}y)-\alpha_{1}(\text{sh}\alpha_{1}x- \sin\alpha_{1}x)\end{array} \right. \tag{5}\]  
Substituting Equation (5) into Equation (3) yields the following results: \(L_{1}=\text{sh}(\alpha_{1}a)\), \(E_{1}=\text{ch}(\alpha_{1}a)\), \(G_{1}=\sin(\alpha_{1}a)\), \(H_{1}=\cos(\alpha_{1}a)\), \(R_{1}=\text{sh}(2\alpha_{1}a)\), \(M_{1}=\text{ch}(2\alpha_{1}a)\). The expression for the bending moment of the thick plate can then be further expressed as shown in Equation (6):  
\[ \begin{array}{l}M_{y}=-\frac{0.96q\mu^{4}}{\lambda_{c1}^{2}}\left\{\mu \left(X_{1}{}^{\prime\prime}Y_{1}+\frac{b^{2}}{a^{2}}X_{1}Y_{1}{}^{\prime \prime}\right)\right.\\ \left.+I_{2}h^{2}[X_{1}Y_{1}{}^{\prime\prime\prime}+(1+\mu)X_{1}{}^{\prime \prime}Y_{1}{}^{\prime\prime\prime}+\mu X_{1}{}^{\prime\prime\prime}Y_{1}{}^{ \prime\prime}]\right\}+qh^{2}I_{4}\end{array}  \tag{6}\]  
Based on the boundary characteristics of the Jurassic heavy conglomerate layer [28], the maximum bending moment \(M_{\text{max}}\) of the heavy conglomerate layer occurs at \(x=\frac{a}{2}\) and \(y=\frac{b}{2}\), as shown in Equation (7). \[M_{\text{max}}=\frac{0.96q\mu^{4}}{\lambda_{c1}^{2}}\left(7.84\mu+2.55I_{2}h^{ 2}+3.49\mu I_{2}h^{2}+1.43\frac{\mu b^{2}}{a^{2}}\right)+qh^{2}I_{4} \tag{7}\]  
The maximum tensile stress \(\sigma_{\text{max}}\) that the lower surface of the heavy conglomerate layer can withstand, as shown in Equation (8) [29]:  
\[\sigma_{\text{max}}=\frac{12M_{\text{max}}}{h^{2}}Z=\frac{12M_{\text{max}}}{h ^{2}}\frac{h}{2}=\frac{6M_{\text{max}}}{h^{2}} \tag{8}\]  
When the maximum tensile stress in the Jurassic heavy conglomerate layer equals the ultimate tensile strength, expressed as \(\sigma_{max}=\sigma_{\text{s}}=\frac{6M_{\text{max}}}{h^{2}}\), the heavy layer fails. Combining this relationship with the above equation yields Equation (9):  
\[\sigma_{\text{s}}=\frac{4.14q\mu^{4}}{\lambda_{c1}^{2}}\left(\frac{7.84\mu}{h ^{2}}+2.55I_{2}+3.49\mu I_{2}+1.43\mu\frac{b^{2}}{a^{2}h^{2}}\right)+6 qI_{4} \tag{9}\]  
The initial fracture of the Jurassic heavy conglomerate layer above the 7202 working face is influenced by several factors, including the overlying load (\(q\)), the tensile strength of the rock layer (\(\sigma_{\text{s}}\)), the fracture thickness of the overlying rock (\(h\)), and the length of the working face (\(b\)). Based on the actual mining conditions and ground separation detection results in the area, the heavy conglomerate layer did not fracture entirely. Calculations show that the fracture thickness (\(h\)) is approximately 35 m, the working face length (\(b\)) is 200 m, the tensile strength \(\sigma_{\text{s}}\) is 4.62 MPa, and the overlying load (\(q\)) is 1.2949 MPa. Substituting these parameters into Formula (9), the initial fracture step distance of the Jurassic heavy conglomerate layer is calculated to be approximately 466.5 m. At this point, the working face had advanced 445.4 m from the starting point. The theoretical results align well with the actual mining environment and detection data. This indicates that the largest recorded strong mining-induced earthquake at the 7202 working face on 24 May 2022 was caused by the initial fracture of the Jurassic heavy conglomerate layer.