## 3 Law of Strata Fracture Movement
### Theoretical Analysis of Strata Fracture Movement  
In order to further analyze the fracture motion of super-thick layers and its temporal and spatial correlation with the occurrence of SMTs, the characteristics of the fracture motion of super-thick layers are here theoretically analyzed. At present, the \"plate\" and \"beam\" theories are used to study the stability of the roof. For the Shilawusu Coal Mine with its super-thick strata structure, it is of more significance to analyze the fracture motion law of super-thick strata by the \"plate\" theory. With the \"plate\" theory, according to the score of plate width and thickness, thin plates and thick plates can be distinguished by the following formula [23]:  
\[ \begin{array}{ll}\left(\frac{1}{100}\sim\frac{1}{80}\right)\leq\frac{h}{b} \leq\left(\frac{1}{8}\sim\frac{1}{5}\right)&\text{thin plate}\\ \left(\frac{1}{8}\sim\frac{1}{5}\right)<\frac{h}{b}&\text{thick plate}\end{array}  \tag{1}\]  
where \(h\) is the thickness of the super-thick strata and \(b\) is the width of LW 1208 and LW 1206A. The incline length of LW 1208 increases to 585 m after the goat mining stage, where total thickness of the super-thick strata is about 300 m. The width-thickness ratio of the super-thick strata is \(300/585\approx 1/2\), which belongs to the thick category. Therefore, the thick plate correlation theory was used to solve the fracturing and instability conditions of the super-thick roof in LW 1208. In summary, the theoretical mechanical model of the super-thick strata at the goat mining stage is shown in Figure 8[24]. [PERSON]'s theory of thick plates [25] was applied to analyze the fracture movement characteristics of super-thick strata. Figure 7: Surface subsidence of strike H line during mining of LW 1208. (a) 29 January 2021–26 April 2021; (b) 7 May 2021–8 September 2021; (c) 17 October 2021–30 December 2021. According to Figure 8b, based on the assumption of plane geometric similarity, the basic equation of the mechanical model of thick plates with super-thick strata can be obtained:  
\[\left\{ \begin{array}{l}D\Delta\Delta\omega{=q-\frac{2-\mu}{10(1-\mu)}h^{2} \Delta q}\\ \frac{h^{2}}{10}\Delta Q_{x}-Q_{x}{=D\frac{\partial}{\partial x}\Delta\omega+ \frac{h^{2}}{10(1-\mu)}-\frac{\partial q}{\partial x}}\\ \frac{h^{2}}{10}\Delta Q_{y}-Q_{y}{=D\frac{\partial}{\partial y}\Delta\omega+ \frac{h^{2}}{10(1-\mu)}-\frac{\partial q}{\partial y}}\end{array} \right. \tag{2}\]  
where \(\omega\) is vertical deflection, \(q\) is overburden load, \(\mu\) is [PERSON]'s ratio, \(Q_{x}\) and \(Q_{y}\) are shear forces, \(D\) is bending stiffness, \(D=\frac{Eh^{3}}{12(1-\mu^{2})}\), \(E\) is [PERSON]'s modulus, and \(h\) is the thickness of the super-thick strata. As shown in Figure 8c, based on the boundary conditions before the initial fracture of the super-thick strata, the bending moment of the super-thick strata at \(x=a/2\) and \(y=b/2\) reaches the maximum value \(M_{max}\)[24]:  
\[M_{max}=-\frac{0.96 qb^{4}}{\lambda_{c1}^{2}}\bigg{(}7.84\mu+2.55I_{2}h^{2}+3.49 \mu_{2}h^{2}+1.43\frac{\mu b^{2}}{a^{2}}\bigg{)}+qh^{2}I_{4} \tag{3}\]  
where \(\lambda_{c1}\) is the function determined by the deep beam function, and \(I_{1}\) and \(I_{2}\) are parameters related to \(\mu\), \(I_{1}=\frac{1}{5(1-\mu)}\), \(I_{4}=\frac{\mu}{10(1-\mu)}\). The maximum tensile stress \(\sigma_{max}\) exists on the lower surface of the super-thick strata [26]:  
\[\sigma_{max}=\frac{12M_{max}}{h^{3}}Z=\frac{12M_{max}}{h^{3}}\frac{h}{2}=\frac {6M_{max}}{h^{2}} \tag{4}\]  
When the ultimate tensile strength \(\alpha\)s and maximum tensile stress \(\sigma\)max of the super-thick strata are \(\sigma_{s}=\sigma_{max}=6M_{max}/h^{2}\), the initial failure of super-thick strata is achieved.