## 4 Results and Discussion
### Influence of the Soil Strengths on the BC/BC\({}_{\rm NV}\) of the Footing-Cavity System  
#### 4.2.1 Normalization  
The analysis of the bearing capacity under several conditions considering different geometrical parameters and soil types (cohesive soils, intermediate soils) is investigated. This study elucidates the correlation of the influence zone by introducing a novel normalization concept based on the boundaries of the footing's failure surface. These boundaries are established in both horizontal and vertical directions within the ground without cavity, as depicted in Figure 4. The failure surface beneath the footing is conceptualized with three zones, as documented in the literature [25; 27], specifically applicable to homogeneous soil subjected to vertical loading. The lateral expansion of the failure domain primarily depends on the shear resistance angle. Figure 3: Effects of different geometrical parameters on the normalized bearing capacity BC/BC\({}_{\rm NV}\). (**a**) Footing width (for cases \(B=1\), \(2\) m; \(D=1\) m); (**b**) cavity shape (for cases \(B=2\) m; \(D=1\) m); (**c**) cavity size (for cases \(B=1\) m, \(D=1\), \(2\) m); (**d**) cavity depth (in case \(B=1\)m). The newly introduced normalization parameters are the dimensions \(R\) and \(H\). Table 3 shows the parameters \(R\) and \(H\) for the various types of ground without the cavity obtained by the RPFEM results. It is worth noting that these values are empirically obtained from the results of simulation with RPFEM on deformation analysis of footing on the soil without voids (Figure 5). As a reference, empirical determination of the plastic mechanism or failure surface has been used to determine failure domains in the case of total and effective stress analyses [23; 26]. \begin{table}
\begin{tabular}{c c c c c} \hline
**Cohesion \(c\) (kPa)** & \(\phi\) (deg) & \(B\) (m) & \(R\) (m) & \(H\) (m) \\ \hline \multicolumn{5}{c}{(a)} \\ \hline \multirow{5}{*}{50} & 0 & 1 & 2.5 & 1.5 \\  & 0 & 2 & 4.5 & 2.25 \\ \hline \multicolumn{5}{c}{(b)} \\ \hline \multirow{5}{*}{10} & 10 & 1 & 3.5 & 1.5 \\  & 2 & 5.5 & 3 \\ \cline{2-5}  & 20 & 1 & 4 & 1.5 \\  & 2 & 6 & 3 \\ \cline{2-5}  & 30 & 1 & 4.5 & 1.75 \\  & 2 & 7.5 & 2.75 \\ \cline{2-5}  & 40 & 1 & 7.5 & 2.25 \\  & 2 & 7.5 & 2.5 \\ \hline \multicolumn{5}{c}{(c)} \\ \hline \multirow{5}{*}{50} & 10 & 1 & 3.5 & 1.5 \\  & 2 & 5.5 & 2.75 \\ \cline{2-5}  & 20 & 1 & 4 & 1.75 \\  & 2 & 7 & 3 \\ \cline{2-5}  & 30 & 1 & 4.5 & 2.25 \\  & 2 & 8.5 & 3.25 \\ \cline{2-5}  & 40 & 1 & 7.5 & 2.5 \\ \cline{2-5}  & 2 & 12 & 4.5 \\ \hline \end{tabular}
\end{table} 
Table 3: (a) Normalization parameters \(R\) and \(H\) obtained with the RPFEM results. Cohesive soils. (b) Normalization parameters \(R\) and \(H\) obtained with the RPFEM results. Intermediate soils with small cohesion soils. (c) Normalization parameters \(R\) and \(H\) obtained with the RPFEM results. Intermediate soils with large cohesion soils. Figure 4: General bearing failure pattern with the normalization parameters. These parameters exert significant influence by governing the development of the failure mode in the footing-cavity system. When the cavity lies outside the failure domain, i.e., when (\(r\)-\(D/2\)) \(>\) \(R\) and (\(h\)-\(D/2\)) \(>\) \(H\), its impact on the footing's performance diminishes with increasing distances of (\(r\)-\(D/2\)) and (\(h\)-\(D/2\)). Conversely, critical failure modes emerge when the cavity is situated within the footing's failure domain. Consequently, the results are presented as the distribution of BC/BC\({}_{\rm NV}\) within the plane of normalized distances, with (\(r\)-\(D/2\))/\(R\) representing the \(X\)-axis and (\(h\)-\(D/2\))/\(H\) for the \(Y\)-axis, respectively. The space (\(X\), \(Y\)) is divided into two primary regions, distinguished by BC/BC\({}_{\rm NV}\) values below 1 and equal to 1, with the critical line serving as the demarcation between these areas.