## 2 Stability Analysis of Ultra-Large Span Open-Off Cutting Areas
### Cutting Span and Stability of Working Face  
(1) Mechanical Model  
The relationship between the span of open-off cuts and the stability of the roof is a significant factor. To investigate this relationship, a theoretical model for analyzing the stress characteristics of open-off cuts was established. In this model, the open-off cut is treated as infinitely long, with no influence on the stability of the roof. Based on principles from elasticity theory, a theoretical model with fixed supports at both ends was developed, as shown in Figure 3.  
In the figure, h (m) corresponds to the height of the roadway, 1 (m) is the span, the upper portion of the roof is subjected to a uniformly distributed load q (Pa), and the horizontal support force is F (Pa). According to the mechanical model, the relationship between the span of open-off cuts and the stability of the working face was analyzed. The critical stress when both ends are fixed is:  
\[\sigma_{s}=\frac{\pi^{2}E}{3(1-\mu^{2})}\cdot\frac{1}{i^{2}} \tag{1}\]  
where \(i\) represents the ratio of the open-off cut span to the thickness of the roof rock layer, known as the span-to-thickness ratio. From the equation, it can be observed that the critical stress for the model's stability is inversely proportional to the square of the span-to-thickness ratio. It is directly related to the elastic modulus and dependent on the Poisson's ratio but unrelated to the rock strength. This suggests that, under constant roof rock layer thickness, the span of the open-off cut and the mechanical properties of the rock  
Figure 3: Theoretical model of stress on open-off cuts. layer are the determining factors for roof stability. As the span of open-off cuts in roadway excavation is twice that of conventional mining tunnels, the ratio of the critical stress-bearing capacity of the roof in open-off cuts to conventional tunnels is only one-fourth, indicating reduced roof stability in open-off cuts. Conditions for fracture instability  
Due to the lower tensile strength of the rock layer, under significant in situ stress, it is highly likely that the rock beam may crack at the upper ends of the supports and then fracture near the middle or bottom of the beam. As the rock blocks rotate, horizontal compressive forces are formed, resulting in a three-hinge arch-shaped equilibrium. However, it is important to note that rock beams do not necessarily collapse as soon as they reach the fracture limit. The arching conditions primarily depend on the original rock stress and the magnitude of horizontal compressive forces formed during rock block rotation. When the horizontal compressive forces are sufficiently high, multiple rock blocks can still remain compressed together in a suspended state. The conditions for sliding instability of the structure involve the magnitude of the frictional force at the interlocking point, which is the product of the horizontal compressive force and the coefficient of friction at that point. This force acts in the opposite direction to the direction of rock block sliding, serving to prevent the mutual sliding of rock blocks. When the shear force exceeds the frictional force, the structure experiences instability, thus the conditions of instability, as shown in Figure 4.  
\[i\geq\frac{1}{4}kg\phi \tag{2}\]  
\[\sigma_{t}=K\cdot q\cdot\frac{l^{2}}{h^{2}/6}=6 Ki^{2}q \tag{3}\]  
The conditions for deformation instability of the structure are as follows: During the rotation of rock blocks, local stress concentration occurs at compression points, leading to plastic deformation or even tensile failure at those points. This can exacerbate rock block rotation, ultimately causing structural instability. According to the theory of three-hinge arch equilibrium, the relationship between the load \(q\) and the tensile strength \(\sigma_{t}\) of the rock beam at the point of fracture is as follows:  
\[\sigma_{t}=K\cdot q\cdot\frac{l^{2}}{h^{2}/6}=6 Ki^{2}q \tag{4}\]  
The condition for structural deformation instability is that \(\sigma_{t}\) is greater than \(\sigma_{c}\).