## 4 Computational Analysis and Discussion
### Potential Failure Surfaces  
From the previous section, to solve the potential rupture surface, it is necessary to obtain the fracture initiation point first. In this paper, the fracture initiation point refers to the research results of [PERSON] et al. [13; 15]. The research results show that the location of the fracture initiation point is strongly affected by the horizontal geostress, so the selection of the crack initiation point is referred to in Figure 7.  
Figure 7: Influence of the coefficient of horizontal geostress on fracture initiation point [13; 15].  
Figure 8 shows the potential fracture surfaces of tunnel-type LRCs at depths of 80 m and 100 m under geostress coefficients of \(K=0.5\), \(K=1.0\), and \(K=3.0\). Figure 8a-c correspond to rock mass conditions of Grades I to III, respectively, while Figure 8d presents a comparison of different rock mass grades at a depth of 80 m. The results show that the extension direction of the potential fracture surface is strongly influenced by the horizontal geostress coefficient and has little relation to the burial depth. As the geostress coefficient increases, the direction of fracture surface extension tends to be horizontal. However, this effect varies depending on the rock mass class, as shown in Figure 8c, the fracture surfaces are more concentrated in Grade III rock masses. Interestingly, it can be observed from Figure 8d that the fracture surface tends to be more horizontal in the case of poorer rock mass when the horizontal geostress coefficient \(K=0.5\), while the fracture surface also tends to be horizontal in the case of better rock mass when \(K=3.0\), and the direction of expansion of fracture surfaces is almost the same in the different rock mass classes for the case of \(K=1.0\). This suggests that the extension direction of the fracture surface of the rock mass is not only affected by the geostress coefficient but may also be related to the internal friction angle, which can be explained in the Mohr-Coulomb criterion: the inclination angle of the fracture surface is related to the internal friction angle. When the geostress coefficient is less than 1, the direction of maximum principal stress is vertical; on the contrary, when the geostress coefficient is greater than 1, the direction of maximum principal stress is horizontal. Figure 9 shows the potential fracture surfaces of silo-type LRCs at depths of 60 m and 80 m under geostress coefficients of \(K=0.5\), \(K=1.0\), and \(K=3.0\). Figure 9 demonstrates a similar regularity to that of Figure 8 and is not repeated here. In order to visually compare the potential fracture surfaces of different types of LRCs (tunnel vs. silo), a burial depth of 80 m was selected as an example in this study, and the fracture surfaces of tunnel-type and silo-type LRCs are shown in Figures 10-12, respectively, under different rock quality classes. From the results shown in the figure, it is evident that when the horizontal ground stress coefficient \(K=0.5\), the two types of fracture surfaces differ significantly, with the tunnel-type fracture surface being more inclined to be horizontal. However, at \(K=1.0\) and \(K=3.0\), there is almost no difference between them. It is noteworthy that in Class III rock masses, the potential fracture surface of the silo type exhibits local abrupt changes, which may be related to the number of mesh elements, calculation steps, or other factors. Figure 13 illustrates the potential fracture surfaces for the silo-type cavern overlying rock masses at different numbers of mesh elements and computational steps. As can be seen from the figure, although increasing the number of mesh elements and decreasing the computational step length caused a tendency for the fracture surface to be inwardly inclined, the overall effect was not significant, and thus, these two factors can be ruled out. Considering that this occurs only in the Class III rock mass, the current speculation is that the cause of this phenomenon is a complex internal stress field due to the poor strength of the rock mass.