## 2 Quantitative Relationship of Roof-Cutting Parameters for the Retaining Roadway Roof
### Quantitative Relationship Analysis of Roof Cutting Parameters for Retaining Roadway
\begin{array}{l}l_{4}=l_{5}-l_{1}-l_{2}-l_{3}\\ h_{1}=m_{1}+m_{2}-h\end{array} \right. \tag{4}\]  
where \(m_{1}\) is the immediate roof thickness, m; \(h\) is the cutting top height, m; \(l_{1}\) is the width of the limit equilibrium zone, m; \(l_{2}\) is the width of the roadway, m; \(l_{3}\) is the horizontal projection distance of the cutting top line, m; and \(l_{5}\) is the distance from the limit equilibrium zone to the end of the cantilever beam, m.  
Determining \(l_{5}\) from Equation (5), the distance from the limit equilibrium point to the end of the cantilever beam is:  
\[l_{5}=l\left(\sqrt{\frac{l^{2}}{S^{2}}+\frac{3}{2}}-\frac{l}{S}\right) \tag{5}\]  
where \(l\) is the main roof cycle pressure step distance, m; and \(S\) is the length of the working face, m.  
The tensile stress of the uncut part obtained by substituting Equation (3) into Equation (2) is:  
\[\sigma_{\text{nt}}=\frac{3q_{1}l_{4}^{2}+3\rho_{2}gm_{2}l_{4}^{2}}{{h_{1}}^{2}} \tag{6}\]  
According to the on-site geological conditions, the selected parameters are as follows: \(l_{1}\) = 4.0 m, \(l_{2}\) = 5.0 m, \(l_{3}\) = \(h\tan\theta\), \(l\) = 17 m, \(S\) = 180 m, \(q\) = 0.15 MPa, and \(\rho_{2}\) = 2500 kg\(/\)m\({}^{3}\). By substituting the above equation, the relationship between the tensile stress of the uncut part and the height and angle of the cut top can be obtained:  
\[\sigma_{\text{nt}}=\frac{(0.45+0.075m_{2})(10.3-h\tan\theta)^{2}}{\left(m_{1}+ m_{2}-h\right)^{2}} \tag{7}\]  
According to Equation (7), after inputting the relevant parameters, the relationship between different main roof thicknesses, top-cutting heights and angles, and tensile stress can be obtained as shown in Figure 3.  
As shown in Figure 3, (1) there was a negative correlation and then a positive correlation between the tensile stress and the cutting angle of the non-cut main roof with different main roof thicknesses and all top depths. Within a range of 30\({}^{\circ}\) angles of roof cutting, as the top-cutting angle decreased, the tensile stress on the main roof of the uncut joint increased. When the roof cutting angle was 0, the tensile stress on the main roof of the uncut joint reached its maximum. However, in order to facilitate on-site construction and reduce the disturbance of the goal side roof on the roadway roof, a certain angle needed to be deflected towards the cutting joint surface. The most common angle on site is 10-20\({}^{\circ}\)[24]. As the cutting height increased, the tensile stress of the uncut main roof was greater than the allowable range of the ultimate tensile strength of the main roof (4.2 MPa). The range of cutting angles increased with the increase in cutting height; that is, as the cutting height increased, the limitation of the cutting angle on the cutting height became less. (2) Under different main roof thicknesses and the same top angle conditions, the greater the cutting height, the greater the tensile stress generated on the main roof of the uncut surface, and the better the cutting effect. As the thickness of the main roof increased and changed, the tensile stress on the main roof of the uncut surface was greater than the allowable cutting depth of the main roof's ultimate tensile strength (4.2 MPa), which increased accordingly. This means that there was a minimum cutting height required to ensure that the main roof of the uncut surface fractured along the cutting line.