## 3 Results and Analysis
### Theoretical Analysis
#### 3.3.3 Analysis of Stresses on CPB Roof  
The stress on the CPB roof is a key indicator for determining the strength of the filling body. This stress is affected by complex factors such as the overlying load of the CPB roof, the physical and mechanical parameters of the filling body and the dimensions of the stope, which makes it difficult to express accurately by a simple expression [52; 53]. Currently, the simple beam theory and elastic thin plate theory are commonly used in engineering to calculate the stresses on CPB roofs. However, because the ratio of stope height to width for the underhand drift cut-and-fill stopping mining method is higher than 1/5, the results obtained by the above theoretical calculation deviate from the actual results. Based on this, the thick plate theory is introduced in this study to analyze the stresses on the CPB roof [54]. The CPB roof is simplified as a rectangular thick plate with four sides simply supported, and its mechanical model is shown in Figure 6.  
The four sides of the thick plate are simply supported boundaries, so the deflection, bending moment and rotation angle on the boundaries are all zero, which can be expressed as follow (1) at \(x=0\) and \(x=a\):  
\(w=0\), \(M_{\rm x}=0\), \(\psi_{\rm y}=0\)  
(2) at \(y=0\) and \(y=b\):  
\(w=0\), \(M_{\rm y}=0\), \(\psi_{\rm x}=0\)  
where b represents the length of the thick plate, a is the width of the thick plate, w is the deflection of the thick plate, \(M\rm{x}\) and \(M\rm{y}\) are the bending moments of the thick plate around the x-axis and y-axis, respectively, and \(\psi_{\rm x}\) and \(\psi_{\rm y}\) are the rotation angles of the thick plate around the x-axis and y-axis, respectively. Figure 6: Mechanical model of CPB roof. According to [PERSON]'s theory, the equilibrium differential equations for a simply supported rectangular thick plate are  
\[\frac{D}{5}\Big{[}(1-u)\
abla^{2}\psi_{x}+(1+u)\frac{\partial\Phi}{\partial x}+ \frac{1}{2}\frac{\partial}{\partial x}\big{(}\
abla^{2}w\big{)}\Big{]}+\frac{ Gh}{3}\Big{(}\frac{\partial w}{\partial x}-\psi_{x}\Big{)}=0 \tag{6}\]  
\[\frac{D}{5}\Big{[}(1-u)\
abla^{2}\psi_{y}+(1+u)\frac{\partial\Phi}{\partial y} +\frac{1}{2}\frac{\partial}{\partial y}\Big{(}\
abla^{2}w\Big{)}\Big{]}+\frac{ Gh}{3}\Big{(}\frac{\partial w}{\partial y}-\psi_{y}\Big{)}=0 \tag{7}\]  
where \(h\) is the thickness of the thick plate; \(\mu\) is the Poisson's ratio of the thick plate; \(E\) is the modulus of elasticity of the thick plate and \(\Phi=\frac{\partial\psi_{y}}{\partial x}+\frac{\partial\psi_{y}}{\partial y}\). \(D\) is the bending stiffness of the thick plate and \(G\) is the shear deformation modulus of the thick plate, which can be expressed as  
\[D=\frac{Eh^{3}}{12(1-\mu^{2})} \tag{8}\]  
\[G=\frac{E}{2(1+\mu)} \tag{9}\]  
Then, the internal forces and moments of the CTB roof are expressed as follows:  
\[M_{x}=-\frac{D}{5}\Big{[}4\Big{(}\frac{\partial\psi_{x}}{\partial x}+\mu\frac{ \partial\psi_{y}}{\partial y}\Big{)}+\left(\frac{\partial^{2}w}{\partial x^ {2}}+\mu\frac{\partial^{2}w}{\partial y^{2}}\right)\Big{]} \tag{10}\]  
\[M_{y}=-\frac{D}{5}\Big{[}4\Big{(}\frac{\partial\psi_{y}}{\partial y}+\mu\frac{ \partial\psi_{x}}{\partial x}\Big{)}+\left(\frac{\partial^{2}w}{\partial y^ {2}}+\mu\frac{\partial^{2}w}{\partial x^{2}}\right)\Big{]} \tag{11}\]  
\[M_{xy}=-\frac{D(1-\mu)}{5}\Big{[}2\Big{(}\frac{\partial\psi_{x}}{\partial y}+ \frac{\partial\psi_{y}}{\partial x}\Big{)}+\frac{\partial^{2}w}{\partial x \partial y}\Big{]} \tag{12}\]  
\[Q_{x}=\frac{2}{3}Gh\Big{(}\frac{\partial w}{\partial x}-\psi_{x}\Big{)} \tag{13}\]  
Figure 6:0  
where \(M_{xy}\) is the moment of the thick plate and \(Q_{x}\) and \(Q_{y}\) are the shear forces along the x and y directions, respectively.