# Bearing Capacity of Karst Cave Roof under File Foundation Load Using Limit Analysis
## 5 Validation and Application
### Stability Analysis  
Based on the previous discourse, the stability factor \(k\) is derived through the solution of a nonlinear mathematical programming model that determines the karst cave roof's ultimate bearing capacity under various scenarios. Tables 2-5 illustrate that a change in \(T/D\) significantly affects the stability factor \(k\) when \(T/D\) is lesser than 3, although all other conditions remain constant. As \(T/D\) increases, the stability factor \(k\) shows a pronounced upward trend. When \(T/D\) reaches 3, any alterations in the stability factor \(k\) due to \(T/D\) become negligible, indicating that \(T/D\) barely impacts the roof's ultimate bearing capacity. Furthermore, an increase in \(W/D\) weakens the karst cave roof's ultimate bearing capacity, as evidenced by the gradual decline in the stability factor \(k\) with an increase in \(W/D\). It is noteworthy that when the \(T/D\) is 1, and the karst cave's span \(W\) exceeds 6 m, the roof's stability factor \(k\) for each of the four varying pile diameters is less than 1, suggesting a propensity for damage. Consequently, during actual engineering, special attention must be paid to karst caves with spans exceeding 6 m, consistent with the conclusion reached through numerical simulation in reference [28]. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & & & & & \(k\) & & & \\  & \(W\) = 1 m & \(W\) = 2 m & \(W\) = 3 m & \(W\) = 4 m & \(W\) = 5 m & \(W\) = 6 m & \(W\) = 7 m & \(W\) = 8 m \\ \hline \(T\) = 1.5 m (1D) & 1.94 & 1.23 & 1.01 & 1.00 & 0.99 & 0.96 & 0.92 & 0.85 \\ \(T\) = 3.0 m (2D) & 2.73 & 2.22 & 1.91 & 1.90 & 1.95 & 1.94 & 1.94 & 1.86 \\ \(T\) = 4.5 m (3D) & 2.75 & 2.75 & 2.74 & 2.66 & 2.66 & 2.66 & 2.71 & 2.69 \\ \(T\) = 6.0 m (4D) & 2.75 & 2.76 & 2.75 & 2.79 & 2.87 & 2.89 & 2.87 & 3.04 \\ \(T\) = 7.5 m (5D) & 2.75 & 2.78 & 2.75 & 2.76 & 2.79 & 2.84 & 2.90 & 3.08 \\ \(T\) = 9.0 m (6D) & 2.67 & 2.75 & 2.74 & 2.76 & 2.81 & 2.85 & 2.82 & 3.07 \\ \(T\) = 10.5 m (7D) & 2.72 & 2.76 & 2.75 & 2.73 & 2.75 & 2.80 & 2.79 & 2.85 \\ \hline \end{tabular}
\end{table} 
Table 4: Stability factor \(k\) of karst cave roof when pile diameter \(D\) = 1.5 m.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & & & & \(k\) & & & \\  & \(W\) = 1 m & \(W\) = 2 m & \(W\) = 3 m & \(W\) = 4 m & \(W\) = 5 m & \(W\) = 6 m & \(W\) = 7 m & \(W\) = 8 m \\ \hline \(T\) = 0.8 m (1D) & 1.94 & 1.23 & 1.00 & 1.00 & 1.00 & 0.97 & 0.93 & 0.86 \\ \(T\) = 1.6 m (2D) & 2.73 & 2.26 & 1.95 & 1.95 & 1.95 & 1.95 & 1.97 & 1.94 \\ \(T\) = 2.4 m (3D) & 2.76 & 2.75 & 2.73 & 2.70 & 2.73 & 2.78 & 2.73 & 2.75 \\ \(T\) = 3.2 m (4D) & 2.73 & 2.75 & 2.75 & 2.74 In order to better illustrate the impact of two dimensionless parameters (\(T/D\) and \(W/D\)) on the stability factor \(k\), a graphical representation has been created based on the calculation results provided in Table 4, as depicted in Figure 8. The findings derived from Figure 8 indicate that the stability factor \(k\) of the roof can be significantly influenced by low (\(W/D<2\)) or high (\(W/D>5\)) values of \(W/D\). When \(W/D\) is less than 2 and \(T/D\) is less than 3, the roof's stability factor \(k\) is relatively high due to the load-sharing attribute of the rock mass surrounding the karst cave with the karst cave roof, which in turn enhances the overall roof stability. On the other hand, when \(W/D\) exceeds 5 and \(T/D\) exceeds 3, the stability factor \(k\) of the roof is also higher due to the limited stress diffusion range within the karst cave roof when the span is too large. However, the high \(T/D\) value reduces the potential for pile foundation load to cause damage to the roof, resulting in a skewed calculation outcome that overemphasizes safety in the evaluation of karst foundation stability. Figure 9 displays the contour cloud map that indicates the stability factor \(k\) of the roof under varying \(T/D\) and \(W/D\). It is imperative that the stability of the karst cave roof is designed with an appropriate safety reserve.