# Bearing Capacity of Karst Cave Roof under File Foundation Load Using Limit Analysis
## 6 Conclusions  
This paper examines the stability of a three-dimensional karst foundation by employing the UBM of finite limit analysis. The primary conclusions are outlined below:  
1. The horizontal stress experienced by the roof due to the pile foundation load can be classified into tensile and compressive stresses. In order to mitigate the detrimental effects of these stresses on the bearing capacity of the roof, increasing the \(T/D\) proves to be a more effective approach than decreasing the \(W/D\). This is especially true when the \(T/D\) is less than 3, as a higher T/D is associated with a more remarkable reduction in the horizontal stress value of the roof. 2. When the \(W/D\) is equal, an increase in the pile diameter will necessitate a corresponding higher \(T/D\) to prevent the surpassing of horizontal stress over the tensile strength of the rock mass. 3. The stability factor of the roof demonstrates an initial linear increase with the elevation of the \(T/D\) value (\(T/D<3\)), followed by a gradual stabilization (\(T/D>3\)). This suggests that when \(T/D\) exceeds 3, the impact of roof thickness on bearing capacity becomes negligible. Furthermore, the stability factor of the roof only experiences further enhancement when \(W/D\) exceeds 5. 4. The stability factor of the three-dimensional karst foundation is more cautious compared to the outcomes obtained under two-dimensional conditions. Upon reaching \(T/D\) greater than 3, the calculation errors under two-dimensional and three-dimensional circumstances start to increase. Therefore, considering the three-dimensional spatial effects, examining the bearing capacity of the karst foundation aligns better with engineering practice. 5. This paper solely looks into the stability of regular rectangular karst caves' end-bearing pile karst foundation. However, real-life engineering projects often encounter more complex geological conditions such as karst caves with different shapes and multi-layer karst caves. Furthermore, the bearing capacity of rock socketed pile karst foundations has rarely been considered, and hence, these issues merit further investigation. Adding experimental results to scrutinize the research content will enhance its reliability and substance. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(T/D\) (D = 0.8 m) & \begin{tabular}{c} FEM \\ **(Three-Dimensional)** \\ \end{tabular} & \
\begin{tabular}{c} **Reference [22] (2021,** \\ **Two-Dimensional)** \\ \end{tabular} &
\begin{tabular}{c} **This Paper** \\ **(Three-Dimensional)** \\ \end{tabular} \\ \hline
1 & 1.02 & 1.03 & 1.00 \\
2 & 1.95 & 1.96 & 1.95 \\
3 & 2.75 & 2.76 & 2.73 \\
4 & 2.79 & 2.84 & 2.75 \\
5 & 2.83 & 2.94 & 2.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Stability factor \(k\) of karst cave roof using different methods. **Author Contributions:** Conceptualization, Z.L., K.L., W.L., H.W., P.P. and H.X. ; methodology, K.L. ; software, Z.L. and K.L. ; validation, W.L., H.W. and K.L. ; formal analysis, Z.L. ; investigation, Z.L. ; resources, W.L. ; data curation, W.L. ; writing--original draft preparation, K.L. ; writing--review and editing, Z.L. and P.P. ; visualization, H.X. ; supervision, W.L. and Z.L. ; project administration, H.X. and Z.L. ; funding acquisition, Z.L. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China, grant number 12162018 and 12262016; the Research Foundation of Science and technology plan project of Yunnan Provincial, grant number 202201 AT070283 and 202305 AD160064. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature**  
The following symbols are used in this paper: