## 4 Computational Analysis and Discussion
### Factor of Safety  
From the analysis in Section 4.1, it is understood that the morphology of the fracture surface is mainly influenced by the horizontal geostress coefficient and the internal friction angle. When the depth is sufficiently deep, the volume of the overlying rock mass uplift will rise significantly with the increase in burial depth, leading to an increase in the factor of safety with depth. At higher geostress coefficients, the fracture surface tends to extend in a nearly horizontal direction, in which case the fracture surface is unlikely to extend to the surface, and the volume of the overlying rock mass is relatively large, making the construction of LRCs more appropriate for sites with geostress coefficients greater than one. When the geostress coefficient is low, the starting point of the fracture surface is closer to the top of the cavern, leading to a smaller area of the potential uplifted rock mass affected by internal pressure, which may result in an overestimation of the safety factor. Given the analysis above, this study focuses on calculating and analyzing the safety factor in situations where the horizontal geostress coefficient is equal to 1. Tunnel-type LRCs can be designed with longer horizontal lengths depending on the required volume; however, silo-type LRCs cannot have larger lengths in the vertical direction, so it is not practical to directly use the same radius when comparing safety factors. In this paper, assuming that the required volume of the air storage chamber is 40,000 m\({}^{3}\), the tunnel radius is the common 5 m, and assuming that the waist length of the silo type is equal to the radius, the radius is about 18 m.  
Figure 8: Potential fracture surfaces in different rock grades of tunnel-type LRCs: (**a**) Class I rock mass; (**b**) Class II rock mass; (**c**) Class III rock mass; (**d**) comparison of different rock grades (burial depth of 80 m). Figure 9: Potential fracture surfaces in different rock grades of silo-type LRCs: (**a**) Class I rock mass; (**b**) Class II rock mass; (**c**) Class III rock mass; (**d**) comparison of different rock grades (burial depth of 80 m). Figure 11: Potential fracture surfaces for different types of LRCs in the Class II rock mass (burial depth of 80 m). Figure 12: Potential fracture surfaces for different types of LRCs in the Class III rock mass (burial depth of 80 m). Figure 10: Potential fracture surfaces for different types of LRCs in the Class I rock mass (burial depth of 80 m). Figures 14-16 illustrate the factor of safety for the two types of LRC under Class I, II, and III rock, respectively. Additionally, the figures also show the factor of safety calculated using [PERSON]'s method [12] at a burial depth of 80 m. [PERSON]'s method is also a limit equilibrium approach, which assumes the slip surface extends vertically from the cavern side wall to the surface. However, this paper indicates that the slip surface extends obliquely to the surface, which aligns more closely with reality. This suggests that the volume of the overlying block poised to uplift and the area of the slip surface in this method will be significantly larger than the rock mass assumed by [PERSON]'s method, implying that the component of uplift resistance due to the rock mass's own weight and the shear resistance along the slip surface will be greater, especially in silo-type caverns. In terms of the definition of the factor of safety, [PERSON]'s method defines it as the ratio of resistance forces to acting forces, whereas the factor of safety in this paper is defined based on the concept of strength reserve on the slip surface, without any reduction in the rock mass's own weight. Thus, [PERSON]'s method may be overly conservative. Regarding applicability, this method offers better flexibility. For example, if other researchers determine more precise but geometrically irregular potential slip surfaces using different methods, this method can still be applied to calculate the factor of safety.