# Bearing Capacity of Karst Cave Roof under File Foundation Load Using Limit Analysis
## 5 Validation and Application
### Horizontal Stress Analysis  
The capacity of a rock mass to withstand tensile deformation is commonly referred to as its tensile strength. As a result, investigating the tensile behavior of rock masses is of significant importance in the context of surrounding rock in underground engineering caverns [21]. The horizontal stress of the roof, as determined by the analysis presented above, is derived from the stress field within the calculation results. Strength analysis of the roof is then conducted with varying pile diameters of 0.8 m and 1.5 m, as illustrated in Figures 5 and 6, respectively, depicting the influence curves of different \(T/D\) and \(W/D\) on the roof's horizontal stress. It is evident that the junction of the pile and rock mass is subjected to extreme tensile stress, while the surrounding area experiences compressive stress. This local state of tension and compression is unfavorable for the roof's stability; however, the reduction of horizontal stress can be attained through the alteration of the \(T/D\) and \(W/D\) parameters. As illustrated in Figure 5, a decrease in the roof's tensile and compressive stress can be achieved by increasing \(T/D\) when \(T/D\) is less than 3. For instance, in Figure 5, an increase in T from 1 \(D\) to 2 \(D\) results in over 70% reduction in the extreme tensile stress values of the roof, decreasing by 2 MPa and 0.6 MPa, respectively. Similarly, the extreme compressive stress values decrease by 1 MPa and 0.2 MPa, respectively, both decreasing by more than 60%. However, the influence curves of different \(T/D\) values overlap, indicating a feeble impact on the roof's horizontal stress when \(T/D\) is greater than 3. Nonetheless, the extreme values of the roof's horizontal tensile stress are less than the tensile strength of the rock mass by 0.6 MPa, implying that the horizontal stress does not affect the overall stability of the roof. Figure 6 demonstrates that the reduction of the karst cave's horizontal stress can also be achieved by decreasing \(W/D\), particularly for \(W\) spans greater than 6 m. The alteration of \(W/D\) has the most substantial influence on the roof's horizontal stress value. Of significance is the fact that even when \(T/D\) is equal to 3, and the karst cave's width, \(W\) is equal to 8 m, the extreme value of the roof's horizontal stress still exceeds the tensile strength of the rock mass. Henceforth, meticulous evaluation of the \(W/D\) and \(T/D\) parameters is imperative in ensuring the durability of the karst cave ceiling. Figure 5: Influence of different \(T/D\) on the roof’s horizontal stress: (**a**) \(D\) = 0.8 m, (**b**) \(D\) = 1.5 m.  
We select the maximum horizontal stress value of the roof, and subsequently produce stress contour cloud maps under various \(T/D\) and \(W/D\) parameters, as depicted in Figure 7. In the stress contour map, the upper left region which lies under the \(-0.6\) MPa stress contour line, denotes the extreme tensile stress values of the roof. It should be noted that said values are all lesser than the roof's corresponding tensile strength. Thus, the selection of any \(T/D\) and \(W/D\) combination falling within this specific region would ensure that the roof's stability remains unaffected by horizontal stress. Notably, as the pile diameter \(D\) increases, the aforementioned \(-0.6\) MPa stress contour line gradually shifts toward the lower right of the diagram. This constitutes that given equal \(W/D\) values, a larger pile diameter \(D\) results in a larger value of \(T/D\) required to prevent the extreme horizontal tensile stress from surpassing the tensile strength value of the roof. For instance, if \(W/D\) values are both 6, the minimum \(T/D\) value necessary to prevent the roof's extreme tensile stress from surpassing its corresponding tensile strength value is 2.5 when the pile diameter is 0.8 m and 3.5 when the pile diameter is 1.5 m, as shown in Figure 7a,b, respectively.