# Bearing Capacity of Karst Cave Roof under File Foundation Load Using Limit Analysis
## 5 Validation and Application
### Discussion  
Through the analysis of the horizontal stress present in the roof, it has been determined that the roof experiences both tensile and compressive stress when supporting the load of the pile foundation. In order to minimize the destabilizing effects of this stress on the roof, increasing the \(T/D\) has been found to be effective. However, it is important to consider the \(W/D\) to ensure the roof's stability.  
The stability factor of the roof displays a significant increase during the initial stages (\(T/D<3\)), ultimately approaching a fixed value during the later stages (\(T/D>3\)). This pattern can be attributed to the different failure modes exhibited by the roof when \(T/D\) is less or greater than 3 [29]. [PERSON], [PERSON] et al. [24] conducted a study utilizing the strength reduction method to examine the safety factor of underground caverns. Their research revealed that as the rock mass deformation in the vicinity of the cavern reaches a certain threshold, the reduction factor gradually increases until it reaches a steady state. These findings bear similarity to those presented in this article.  
It is important to highlight that in this paper, the stability factor of the roof has been considerably increased in both cases (\(T/D>3\) and \(W/D>5\); \(T/D<2\) and \(W/D<1\)) through the calculation of 224 distinct working scenarios. The stress diffusion range of the roof precludes load-induced cave damage, thus increasing the cave's stability.  
The paper utilizes a shear strength reduction parameter (stability factor) to convey the bearing capacity of karst foundations, a method that is not widely employed in the stability analysis of karst foundations. To ensure the accuracy of this method, the method  
Figure 9: Contour cloud map of the roof’s stability factor \(k\) under different \(T/D\) and \(W/D\): (**a**) \(D=0.8\) m, (**b**) \(D=1.0\) m, (**c**) \(D=1.5\) m, (**d**) \(D=2.0\) m.  
is compared to the three-dimensional FEM and the two-dimensional limit analysis [22]. The safety factor calculation outcomes of the three methods are presented in Table 6 (with \(D\) = 0.8 m, \(W\) = 3.2 m, and the same physical and mechanical parameters as in Table 1). The calculation results of this method are in alignment with the other two methods for \(T/D\) values of less than 3, signifying the accuracy of this method. However, for \(T/D\) values greater than 3, the calculation errors of the three methods begin to increase, potentially resulting from four reasons: (1) the similarity calculation of the conservative model and display function; (2) the linearization of nonlinear constraints in nonlinear mathematical programming models using the UBM; (3) the division of the mesh; (4) the influence of three-dimensional spatial effect. The two-dimensional and three-dimensional limit analyses have the largest calculation errors that continue to remain within 10% (when \(T/D\) is equal to 4), validating the effectiveness of the proposed method in this paper for the stability analysis of three-dimensional karst foundations.