## 2 Mechanical Model of Surrounding Rock for GESCP
### Multi-Layer Cracking Mechanical Model of the Side Wall for GESCP  
After the development of a goaf-side entry, the original three-dimensional stress acting on the surrounding rock was transformed into two-dimensional stress. The shallow side walls of the goaf-side entry may even bear uniaxial stress. Under the action of compressive stress generated by the roof and floor, a large number of secondary cracks, which extended along the direction of compressive stress, were generated in the shallow coal of the ribs in both the solid coal side as well as the small coal pillar side, as shown in Figure 4. As the cracks increased and joined up, a multi-layer cracking structure finally formed on the coal side [26; 27]. In GESCP engineering practice, these sides fail.  
The mechanical model of the multi-layer cracking of the side coal was simplified, as shown in Figure 5. We assumed that the multi-layer cracking consisted of thin rectangular plates with a height of \(h\), a width of \(b\), and a thickness of \(a\). The stress between the thin rectangular plates and acted on by the deep coal side was neglected. A 3D planar coordinate system (\(x\), \(y\), \(z\)) was used, in which the \(x\)-axis corresponds to the height of a thin rectangular plate and the \(z\)-axis corresponds to the direction of the excavated entry. The boundary at points \(x=0\) and \(x=\) a bore the uniform load \(\sigma\), and the boundary at points \(y=0\) and \(y=\) b were free. Owing to the poor integrity of the small coal pillar and the solid coal side, the simple support at both ends and the uniform load \(\sigma\) at the bottom of the thin rectangular plates were simplified, as shown in Figure 5c.  
According to the boundary conditions, the thin rectangular plate was buckled into m half-sine wave shapes under the action of the uniform load \(\sigma\). Its deflection was expressed as:  
\[\psi=A\sin(\frac{m\pi x}{h}) \tag{5}\]  
where \(A\) is a constant and \(m\) is a positive integer greater than \(1\).  
Using an integral method to solve the differential equations representing the deflection plate, the buckling equilibrium of the thin rectangular plate was expressed as [27]:  
\[\frac{\partial^{4}\psi}{\partial x^{4}}+2\frac{\partial^{4}\psi}{\partial x^{ 2}\partial y^{2}}+\frac{\partial^{4}\psi}{\partial y^{4}}=-\frac{\sigma}{D} \frac{\partial^{2}\psi}{\partial x^{2}} \tag{6}\]  
where \(D\) is the plate stiffness, given by \(D=Ea^{3}/12(1-\mu^{2})\), where \(E\) is the elastic modulus of the plate, and \(\mu\) is the Poisson ratio of the plate.  
Figure 4: Site photos of the vertical cracks in the side wall.  
Considering the boundary conditions at points \(x=0\) and \(x=h\), substituting Equation (5) into Equation (6) yielded the following:  
\[A(\frac{m\pi}{h})^{2}\sin\frac{m\pi x}{h}-\frac{\sigma}{D}A(\frac{m\pi}{h})^{2} \sin\frac{m\pi x}{h}=h \tag{7}\]  
The boundary conditions for the thin rectangular plate at points \(y=0\) and \(y=\mathrm{b}\) were satisfied. So, by solving Equation (7), the collapsing load of the thin rectangular plate \(\sigma_{b}\) was derived as follows:  
\[\sigma_{b}=\frac{m^{2}\pi^{2}}{h^{2}}D \tag{8}\]  
If the size and elastic modulus of the thin rectangular plate are known and \(m=1\), the buckling shape of the plate is a half-sine wave. The minimum critical collapsing load of the thin rectangular plate \((\sigma_{x})_{min}\) was derived as follows:  
\[\left(\sigma_{x}\right)_{min}=\frac{\pi^{2}}{h^{2}}D=\frac{E\pi^{2}a^{3}}{12h ^{2}(1-\mu^{2})} \tag{9}\]