## 4 Results and Discussion
### Influence of the Soil Strengths on the BC/BC\({}_{\rm NV}\) of the Footing-Cavity System
#### 4.2.3 Influence of the Intermediate Soil Materials  
The intermediate soil types were differentiated based on variations in cohesion \(c\) and shear resistance angle \(\phi\). These categories include soil with low cohesion and low shear resistance angle \(\phi\), soil with low \(c\) and high \(\phi\), soil with high \(c\) and low \(\phi\), and soil with high \(c\) and high \(\phi\). In Figure 8a, the BC/BC\({}_{\rm NV}\) variation is depicted within a soil characterized by low cohesion (\(c\) = 10 kPa) and a low shear resistance angle (\(\phi\) = 20 deg). The influence zone, where the ratio BC/BC\({}_{\rm NV}\) is less than 1, remarkably expands in the vertical direction. This expansion is attributed to the low shear resistance angle, causing the soil to exhibit cohesive-like behavior. Furthermore, a small area, highlighted in black (BC/BC\({}_{\rm NV}\) = 0.3), indicates a more significant influence than observed in other cases, as it falls below the allowable BC with a factor of safety of 3. Figure 8b illustrates the distribution of BC/BC\({}_{\rm NV}\) when the shear resistance of the previous \(c\)-\(\phi\) soil is increased to \(\phi\) = 30 deg. The trend in the footing's performance with the cavity's location remains consistent with the previous case. However, the expansion in the \(Y\) direction is comparatively reduced compared to the scenario in Figure 8a, and an area showing BC/BC\({}_{\rm NV}\) less than the allowable BC is not observed.  
Figure 8c,d presents the distribution of BC/BC\({}_{\rm NV}\) with distances \(X\) and \(Y\) for a \(c\)-\(\phi\) soil with high cohesion (\(c\) = 50 kPa) and low shear resistance angle (\(\phi\) = 20 deg), and high shear resistance angle (\(\phi\) = 30 deg), respectively. In comparison to the preceding cases (Figure 8a,b), the influence zone notably contracts in both directions. This contraction implies that a cavity in soil with substantial shear strength is more stable, and the potential to collapse is reduced compared to scenarios involving soil with lower shear strength.  
Overall, the influence zone due to the presence of the cavity on the footing exhibits high sensitivity in the vertical direction and low sensitivity in the horizontal direction, particularly evident in soils with smaller shear resistance angles. It is important to highlight that the influence zone in the horizontal direction does not exceed \(X\) = 1. The denominator \(R\), integral to the calculation, depends on the shear resistance angle and the dilatancy angle (equivalent to \(\phi\), given the use of the associated flow rule). A higher dilatancy angle results in a larger \(R\). Consequently, under these conditions, significant expansion of the influence zone in the horizontal direction is limited. Similar considerations apply to the vertical direction, where the variation of \(H\) is less pronounced than that of R due to the increasing confining stress with depth, thus explaining the observed sensitivity in the vertical direction.  
Figure 7: Distribution of the BC/BC\({}_{\rm NV}\) within the \(X\) and \(Y\) direction for the cohesive soils (\(c\) = 50 kPa).