## 4 Results and Discussion
### Discussion  
In the preceding sections, results were sequentially presented, exploring the impact of geometrical parameters on the bearing capacity behavior of the footing in the presence of a cavity. Additionally, an examination of the influence of soil strength parameters was undertaken, specifically focusing on cohesive and intermediate soil types. Furthermore, a detailed analysis of typical failure mechanisms was conducted to comprehend variations in stability levels under various conditions. These analyses have contributed to the formulation of proposed solutions aimed at determining the critical boundary of the zone of influence of the cavity on the bearing capacity. The analysis of geometrical parameters, as depicted in Figure 9, demonstrates that an increase in the footing width \(B\) leads to a reduction in the normalized bearing capacity BC/BC\({}_{\rm NV}\). This trend is attributed to the heightened footing pressure magnitude, which intensifies the plastic mechanism and deformations, as illustrated in Figure 4. Consequently, a larger footing width is susceptible to the influence of nearby cavities, as highlighted in previous research [18]. Furthermore, the cavity shape analysis indicates that square cavities yield lower BC/BC\({}_{\rm NV}\) values compared to circular cavities. The circular shape facilitates a smoother stress distribution within the soil, while square cavities result in an expanded stress distribution at the cavity edges. Additionally, the cavity's proximity to the centerlines of the footing is revealed to exert the most significant influence on bearing capacity. At greater depths or horizontal distances, this influence diminishes, and the bearing capacity tends to recover to values similar to scenarios without a cavity. These findings align with prior studies by various researchers [9; 18]. However, it is noteworthy that [PERSON] et al. [27] reported negligible influence on bearing capacity concerning cavity shape. Consequently, our solution design is based on square cavities, representing severe scenarios and providing a conservative approach to ensure safety. The exploration of soil strength parameters in cohesive and intermediate soils underscores the significance of studying this issue across a diverse range of soil types rather than in localized regions. The introduction of normalized parameters, derived from the plastic mechanism of the footing without a cavity, serves to quantify and establish a well-defined distribution of BC/BC\({}_{\rm NV}\) within the plane (\(X\), \(Y\)) based on geometrical parameters. Cohesive soil cases, within the considered range of cohesion, demonstrate that the bearing capacity is notably affected by cavities beneath the footing, particularly those close to the centerline of the footing. Conversely, in intermediate soils, the distribution of BC/BC\({}_{\rm NV}\) within the plane (\(X\), \(Y\)) is influenced by the combination of cohesion \(c\) and shear resistance \begin{table}
\begin{tabular}{c c} \hline \hline
**Coefficients** & **Values** \\ \hline \(p\) & \(a_{0}+a_{1}\times\phi+a_{2}\times\phi^{2}\) \\ \(q\) & \(b_{0}+b_{1}\times\phi+b_{2}\times\phi^{2}\) \\ \(a_{0}\) & \(-13.24+0.17\times c\) \\ \(a_{1}\) & \(0.42\) \\ \(a_{2}\) & \(-0.0061\) \\ \(b_{0}\) & \(12.17-0.19\times c\) \\ \(b_{1}\) & \(0.25\) \\ \(b_{2}\) & \(-0.003\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Coefficients \(p\) and \(q\) for intermediate soils. angle \(\phi\). Low cohesion and low shear resistance angle soil types predominantly impact the vertical direction (\(\gamma\)), with a larger influence zone compared to cases with higher shear resistance angles. For the latter, the influence decreases in the vertical direction, while in the horizontal direction (\(X\)), an impact is observed, especially for shallow cavities at large \(X\) (Figure 9a,b). In fact, the slip surface is widened due to its large \(\phi\); therefore strain rate expands more horizontally. However, the BC/BC\({}_{\rm NV}\) varies between 0.30 and 0.98 (approx. 1).