# Stability and Control of Surrounding Rock of a Trapezoidal Roadway Retained with Hard Roof Cutting
## 2 Stability Analysis of the Roof with Top-Cutting
### Factors Affecting Roof Stability  
According to Equation (3) and previous research [25], it can be seen that the tensile stress of the uncut roof was closely related to the roof thickness, coal seam dip angle, top-cutting height and top-cutting angle. The control variable method was used to study the influence of the above factors on the stability of the top-cutting roof.  
(1) Influence of the top-cutting height and top-cutting angle.  
The relationship between the tensile stress of the uncut roof and the top-cutting height and angle is shown in Figure 4, when the coal seam dip angle is set to 6\({}^{\circ}\) and roof thickness is set to 5.0 m.  
Figure 4 illustrated that when the coal seam dip angle is 6\({}^{\circ}\) and the roof thickness is 5.0 m, the tensile stress of the uncut roof was inversely proportional to the top-cutting height, while it was directly proportional to the top-cutting angle. Furthermore, when the top-cutting angle is \(\geq\)9\({}^{\circ}\), the tensile stress of the uncut roof increased sharply with an increase in the top-cutting angle, which facilitated roof fracturing. Conversely, when the top-cutting angle was <9\({}^{\circ}\), the tensile stress of the uncut roof remains at a relatively low level, potentially falling below its tensile strength, which made fracturing difficult.  
(2) Influence of the coal seam dip angle and roof thickness.  
The relationship between the tensile stress of the uncut roof and the coal seam dip angle, as well as the roof thickness, is shown in Figure 4, when the cutting angle was set to 10\({}^{\circ}\).  
Figure 5 shows that when the top-cutting angle is 10\({}^{\circ}\), the tensile stress of the uncut roof was inversely proportional to both the coal seam dip angle and the roof thickness. Moreover, the sensitivity of the tensile stress to the roof thickness was greater than that to the coal seam dip angle.  
The tensile strength of the hard limestone roof in the 21,100 panel was measured at 3.3 MPa. Based on the present analysis, it was evident from Equation (4) that an increase in top-cutting height resulted in a corresponding increase in tensile stress within the uncut roof, thereby heightening the likelihood of exceeding its tensile strength and making the roof more susceptible to fracture. Given that the hard limestone roof of the 21,100 panel had a thickness of only 5.0 m, full thickness top-cutting was implemented to ensure effective top-cutting, which established the top-cutting height at 5.0 m. Taking into account the roof thickness and the dip angle of the coal seam in the 21,100 panel, it was essential for the top-cutting angle to be at least 9\({}^{\circ}\) to facilitate the effective fracturing of the roof. The top-cutting effect became significant at the top-cutting angle of 10\({}^{\circ}\). Furthermore, as the  
Figure 4: The relationship between tensile stress and the top-cutting height and angle.  
top-cutting angle decreased, both the top-cutting length and engineering volume associated with top-cutting also decreased. Therefore, the reasonable top-cutting angle for the roof was comprehensively determined to be 10\({}^{\circ}\).