## 2 Stability Analysis of Ultra-Large Span Open-Off Cutting Areas
### Theoretical Analysis of Support Parameter Design  
Ultra-large span open-off cuts differ from conventional roadways, as the critical stress-to-span ratio that the cutting area roof can withstand is only one-fourth that of the roadway. At the installation area of the underground chamber within the open-off cuts, the span reaches a maximum of 11.8 m. Numerical simulation results indicate a significant stress concentration at the expanding openings and the central roof area, which poses considerable challenges for the support of these galleries. To design and optimize the support parameters for these ultra-large span open-off cuts, [PERSON]'s theory and the natural equilibrium arch theory were employed to calculate the unstable rock layer thickness suspended in the open-off cuts [15]. The expression for the natural equilibrium arch height \(b\) can be represented as follows:  
\[b=\frac{\left[a+h\cdot cfg\left(45^{\circ}+\frac{a}{2}\right)\right]}{f} \tag{5}\]  
where \(a\) is half of the open-off cut span, taken as 4.9 m; \(f\) is the direct roof Proctor's coefficient, with a value of 3 for coarse sandstone; \(h\) is the gallery height, set at 3.8 m; \(\theta\) is the friction angle, with a value of 25\({}^{\circ}\) for the coal layer where the open-off cuts are located; \(f\) is the [PERSON]'s coefficient for coal, set at 1.13. The formula for calculating the length of the rock bolts is given by:  
\[l=l_{1}+l_{2}+l_{3} \tag{6}\]  
where \(l_{1}\) is the exposed length of the rock bolt and is taken as 0.1 m; \(l_{2}\) is the effective length of the rock bolt, ranging from 1.0 to 1.2 m; \(l_{3}\) is the length of the rock bolt deeply embedded into the basic roof, taken as 0.3-0.4 m based on field experience. The formula for calculating the rock bolt diameter is:  
\[d=\sqrt{\frac{4Q}{\pi\sigma_{t}}} \tag{7}\]  
where \(d\) is the diameter of the rock bolt in meters; \(Q\) is the anchoring force of the rock bolt, obtained from on-site tests, measured at 150 MPa; \(\sigma_{t}\) is the tensile strength of the rock bolt, which is a high-strength left-handed threaded steel bolt without longitudinal ribs. The designed length of the anchor cable should satisfy the following equation:  
\[L=L_{a}+L_{b}+L_{c}+L_{d} \tag{8}\]  
where \(L\) represents the design length of the anchor cable. \(L_{a}\) denotes the anchoring length of the anchor cable into the relatively stable rock layer. \[L_{a}\geq K\times\frac{d_{1}f_{a}}{4f_{c}} \tag{9}\]  
where \(K\) is the safety factor, commonly set at 2 based on empirical data; \(d_{1}\) is the diameter of the cable;\(f_{a}\) is the tensile strength of the cable itself;\(f_{c}\) is the anchoring strength, set at 400 MPa; \(L_{b}\) is the thickness of the unstable rock layer to be suspended, equivalent to the height of the natural equilibrium arch [18]; \(L_{c}\) is the thickness of the tray and anchors; \(L_{d}\) is the exposed lifting length. By combining Equations (4) to (8), the calculations for the length of the bolts, bolt diameter, cable length, and the spacing between bolts were obtained, as shown in Table 1.  
Based on the calculated support parameters and in conjunction with commonly used materials in mines, rational design values for the support parameters were derived. The designed support cross-sections are illustrated in Figure 5 (roadway surrounding rock support diagram). The initial design for the support of the open-off cuts in the 81309 working face, design Scheme 1 is as follows:  
Roof: \(\O\)20 mm \(\times\) 2000 mm high-strength left-handed threaded steel bolts, spaced at 1000 mm, 1400 mm, and 1500 mm, with intervals of 900 mm; \(\O\)22 mm \(\times\) 8000 mm anchor cables, spaced at 1400 mm \(\times\) 900 mm and 1600 mm \(\times\) 900 mm. Rib: \(\O\)20 mm \(\times\) 2000 mm high-strength left-handed threaded steel bolts; \(\O\)18 mm \(\times\) 2000 mm fiberglass reinforced plastic (FRP) bolts used for cut-off ribs, spaced at 900 mm \(\times\) 900 mm. Two additional rock bolts were added at the expanding openings of the open-off cuts to enhance the support strength.