## 3 Determination of Key Parameters
### The Control Mechanism of Roof Cutting Angle on Roof Breaking  
A reasonable pre-splitting roof cutting angle can reduce the friction between the roof of the gob and the roof of the gob-side entry, so that the roof of the gob is easy to separate along the pre-splitting cutting surface and cut off smoothly.  
With the advance of the working face, the roof rock breaks after reaching the limit span. The stability of the formed masonry beam structure is mainly controlled by the key rock block B. The force of the hinge point of rock block A and rock block B is shown in Figure 5. The horizontal thrust formed by rock block B on the pre-splitting slit section is T, and the shear force on the slit section is Q.  
Figure 4: The relationship between \(\sigma_{w}\) and \(h_{1}\).  
Figure 3: Stress analysis of non-penetrated surface during partial roof cutting.  
According to the 'S-R' stability theory of masonry beam structure, the relationship and equilibrium conditions between the forces of rock blocks are as follows:  
\[(T\cos\theta-Q\sin\theta)\tan\varphi\geq Q\cos\theta+T\sin\theta \tag{5}\]  
\[\frac{Q}{T}\leq\tan(\varphi-\theta) \tag{6}\]  
In order to ensure that the key rock block B of the main roof slides smoothly along the pre-splitting cutting surface after roof cutting, the structural instability conditions can be obtained according to the above formula:  
\[\theta\geq\varphi-\arctan\frac{Q}{T} \tag{7}\]  
Substitute into \(T=\frac{d{l_{2}}^{2}}{2(l-\Delta)}\), \(Q=ql_{2}\) can be obtained:  
\[\theta\geq\varphi-\arctan\frac{2(h-\Delta)}{l_{2}} \tag{8}\]  
In the formula: \(\theta\) is the roof cutting angle (\({}^{\circ}\)); \(\varphi\) is the internal friction angle of the rock (\({}^{\circ}\)); \(h\) is the thickness of the rock mass (m); \(\Delta\) is the subsidence of the rock mass (m); \(l_{2}\) is the lateral span (m) of rock block B.  
Among them, the rotary subsidence \(\Delta\) of rock block B is the gap between the immediate roof collapse and the basic roof. The collapse thickness of the immediate roof rock layer is \(\Sigma h\), and the height of the rock layer collapse and accumulation is \(K_{\rm P}\Sigma h\). The calculation formula of \(\Delta\) is obtained:  
\[\Delta=M-\sum h(K_{\rm P}-1) \tag{9}\]  
In the formula: \(M\) is the mining height of the coal seam (m); \(K_{\rm P}\) is the bulking coefficient of the rock.  
Taking \(M=2.2\) m, \(K_{\rm P}=1.3\) into the Equation (9), the rotary subsidence of rock block B is \(\Delta=1.15\) m.  
The lateral span \(l_{2}\) of rock block B can be determined according to the cantilever breaking length of the basic roof rock block:  
\[l_{2}=h\sqrt{\frac{R_{\rm T}}{3q}} \tag{10}\]  
In the formula, \(h\) is the thickness of the rock block (m); \(R_{\rm T}\) is the ultimate tensile strength (MPa) of the base roof; \(q\) is the load per unit area of the main roof rock mass (MPa).  
Figure 5: Stress analysis of key block of roof cutting.  
Take \(h=14\) m, \(R_{\rm T}=3.78\) MPa, \(q=0.5\) MPa. Substituting into Equation (10), the lateral span \(l_{2}\) of rock block B is 22.2 m.  
Substituting the rotary subsidence \(\Delta\) of rock block B, the lateral span l2 of rock block B and the internal friction angle \(\varphi=42^{\circ}\) into Equation (8), \(\theta\geq 11.9^{\circ}\) is obtained.