## 2 Engineering Background
### Stability Analysis of Key Blocks of Overlying Strata in the Withdrawal Space
#### 3.2.2 Stability Analysis of Key Blocks of Not Stopping Coal Caving  
The mechanical model of the stability of the key block in the case of whether the withdrawal space of the final mining stage stops coal drawing is shown in Figure 5. Within the figure, \(q_{\text{s}}\) is the support force of stopping coal pillar; \(q_{\text{z}}\) is the support force of the hydraulic support; the horizontal thrust T and shear force \(Q_{\text{g}}\) are generated by the action of A in front of the working face; and \(q_{\text{A}}\) and \(q_{\text{B}}\) are the load of the fractured rock block above A and B; G is the gravity of B itself.  
In the final mining stage, the coal is not stopped, and there is still a large gap between the goat and the main roof. The main roof is broken, the B activation and rotation space is large, and the B rotation angle becomes larger. According to the \"masonry beam\" structure \"S-R\" stability theory, it is judged whether the hinged structure of the key block is stable or not when the final mining is not stopped.  
In the final mining stage, the coal caving is not stopped, the working face is normally mined 19.0 m, and only 7.7 m is directly filled in the goat. According to the calculation formula of the rotation angle of B,  
\[\beta_{G}=\arcsin\frac{h_{\text{j}}-(\sum h_{\text{s}}(K_{\text{ps}}-1)+h_{2} (K_{\text{P2}}-1))}{L_{\text{Z}}} \tag{3}\]  
where \(\beta_{G}\) is the rotation angle of B; \(L_{G\text{Z}}\) is the fracture length of B along the strike of the working face, which is similar to 26 m; \(h_{\text{j}}\) is the mining height of the coal seam, 19.0 m; \(\sum h_{\text{s}}\) is the thickness of the rock between the main roof and the coal seam, 7.7 m; \(K_{\text{ps}}\) is the average residual bulking coefficient of direct roof strata, 1.15; \(h_{2}\) is the height of the uncaving top coal, 0 m; \(K_{\text{P2}}\) is the average residual bulking coefficient of the uncaving top coal, 1.05.  
In the case of not stopping the top coal, the rotation angle of B is 43\({}^{\circ}\).  
The key block that does not experience slip instability must satisfy Formula (4); Formula (4) further refines and derives the final judgment Formula (5) of slip instability [31]:  
\[\text{Ttg}\varphi<Qg \tag{4}\]  
\[h+h_{1}\leqslant\frac{\sigma_{\text{c}}}{30\gamma}(\text{tg}\varphi+\frac{3} {4}\sin\beta_{G})^{2} \tag{5}\]  
Figure 5: Mechanical model of key block stability under the condition of whether to stop coal caving or not. The B rock block is the key research block in this paper, and the stability of the B rock block is related to the stability of the surrounding rock in the underlying withdrawal space. Therefore, the B rock block is highlighted and changed to magenta.  
In this context, T is the horizontal thrust of B; \(\mathrm{tg}\varphi\) is the friction coefficient between rock blocks, taken as 0.6; \(Q_{g}\) is the shear force of B; \(h\) is the thickness of key block B, 14.1 m; \(h_{1}\) is the thickness of the B load rock layer, 19.4 m; \(\varphi_{c}\) is the ultimate compressive strength of B, 86 MPa; \(\gamma\) is the average bulk density of the overlying rock, 0.025 MN.  
The key block that does not experience rotary deformation instability must meet Formulas (6) and (7):  
\[\mathrm{T}\;>s\eta\sigma_{c} \tag{6}\]  
\[h+h_{1}\leqslant\frac{\sigma_{c}}{30\gamma}\left(\mathrm{tg}\varphi+\frac{3} {4}\sin\beta_{G}\right)^{2} \tag{7}\]  
In this context, \(s\) is the contact surface area of B and C\({}_{1}\); \(\eta\sigma_{c}\) is the extrusion strength of B at the corner; \(i\) is B fracture degree; \(l\) is assumed to be equal to \(L_{Z}\), 26 m.  
Under the condition of not stopping coal cave, the key block \"summary of S-R stability theory analysis parameters\" is shown in Table 1. The results show that under the condition of not stopping coal cave, there will be no sliding instability between the key blocks of the basic roof, and because the B rotation angle is too large, there may be rotational deformation instability.