## 5 Discussion
### Discussion of the Bearing Capacity Factors \(N_{q}\) and \(N_{c}\)  
A shallow foundation was considered in a semi-infinite soil medium to investigate the effects of varying internal friction angles (\(\varphi\) = 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), and 40\({}^{\circ}\)) on the bearing capacity factors \(Nc\) and \(Nq\), as well as the progressive failure mechanisms. The assumed failure mechanism initiates with the collapse of the triangular active zone located directly beneath the foundation. As loading increases, stress accumulation leads to progressive failure within the transition zone, where slip surfaces sequentially fail. Finally, complete sliding failure occurs in the triangular passive zone. Five distinct failure mechanisms, corresponding to the five different friction angles, are illustrated in Figures 24-27. In the analysis, the logarithmic spiral region of the transition zone is discretized into 18 blocks, each spanning an angle of 5\({}^{\circ}\). The sliding surfaces are modeled using multi-segment zero-thickness interface elements, enabling deformation to occur along the tangential direction of the slip surface. The numerical results for the bearing capacity factors \(N_{q}\) and \(N_{c}\), as presented in Tables 5 and 6, respectively, obtained through finite element analysis, show an agreement with the theoretical predictions proposed by [PERSON] (1973) [4]. The numerical results for the bearing capacity factors \(N_{q}\) and \(N_{c}\), shown in Tables 5 and 6, demonstrate an agreement with the theoretical values proposed by [PERSON] (1973) [4]. For \(N_{q}\), the maximum error percentage across friction angles \(\varphi=10^{\circ}\) to \(40^{\circ}\) is only 2.43%, observed at \(\phi=10^{\circ}\), while the minimum error is just 0.12% at \(\varphi=40^{\circ}\). Similarly, for \(N_{c}\), the error remains very small across all friction angles, with a maximum of 0.48% at \(\varphi=10^{\circ}\) and a minimum of 0.04% at \(\varphi=40^{\circ}\). These results confirm the accuracy and reliability of the finite element modeling approach used in this study. For the limitations and future extensions, while the proposed method demonstrates high accuracy and efficiency for two-dimensional, homogeneous soil conditions, certain limitations must be acknowledged. First, the current implementation assumes predefined failure mechanisms in a 2D plane-strain context, which may not directly extend to 3D problems where spatial failure surfaces are more complex and may involve out-of-plane effects. Adapting the framework to 3D would require either an analytical extension of the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Friction Angle, \(\varphi\)** & **[PERSON] (1973) [4], \(N_{c}\)** & **F.E.M.** &
\begin{tabular}{c} **Error Percentage \({}^{*}\)** \\ **(\%)** \\ \end{tabular} \\ \hline
0 & 5.14 & 5.15 & 0.19 \\
10 & 8.35 & 8.39 & 0.48 \\
20 & 14.83 & 14.86 & 0.20 \\
30 & 30.14 & 30.16 & 0.07 \\
40 & 75.31 & 75.34 & 0.04 \\ \hline \hline \({}^{*}Error percentage(\%)=\frac{\text{F.E.M.}-\text{Equation}(42)}{ \text{Equation}(42)}100\%\). & & & \\ \end{tabular}
\end{table} 
Table 6: Comparison between numerical and analytical solutions of bearing capacity factor \(N_{c}\). Figure 27: Displacement field diagram at different stages of loading: (**a**) 10%, (**b**) 40%, (**c**) 70%, and (**d**) 100%. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Friction Angle, \(\varphi\)** & **Vesic (1973) [4], \(N_{q}\)** & **F.E.M.** &
\begin{tabular}{c} **Error Percentage \({}^{*}\)** \\ **(\%)** \\ \end{tabular} \\ \hline
10 & 2.47 & 2.53 & 2.43 \\
20 & 6.40 & 6.44 & 0.63 \\
30 & 18.40 & 18.48 & 0.43 \\
40 & 64.20 & 64.28 & 0.12 \\ \hline \hline \({}^{*}Error percentage(\%)=\frac{\text{F.E.M.}-\text{Equation}(41)}{ \text{Equation}(41)}100\%\). & & & \\ \end{tabular}
\end{table} 
Table 5: Comparison between numerical and analytical solutions of the bearing capacity factor \(N_{q}\). failure geometry or integration with automated mechanism detection algorithms.