## 5 Discussion
### C-Phi Soils  
This section examines the influence of the block number within the C-phi soil transition zone on the bearing capacity factors \(N_{c}\) and \(N_{q}\). For the C-phi soil conditions described in Section 4.2 and illustrated in Figure 2a, the shallow foundation failure mechanism is assumed to consist of two triangular zones, namely, the active zone (zone I) and the passive zone (zone III), along with a transition zone (zone II) characterized by multiple slip surfaces forming a spiral mechanism. The block number refers to the discretization used to model the spiral slip surface within the transition zone, and it plays a significant role in determining the accuracy of the computed bearing capacity factors. In this study, the spiral zone is subdivided into \(n\) blocks (including \(n=1\), \(2\), \(3\), \(5\), \(6\), \(9\), \(10\), \(15\), \(18\), and \(30\)). Figure 19 especially shows the case study with block number \(n=1\). The Mohr-Coulomb interface parameters at each stage are \(c=c_{u}\) and \(\varphi=30^{\circ}\), and the numerical analysis results are shown in Tables 3 and 4. According to the analytical solutions of \(N_{q}\) and \(N_{c}\) proposed by [PERSON] (1973) [4], it can be given as follows  
\[N_{q}=tan^{2}(\frac{\pi}{4}+\frac{\varphi}{2})e^{\pi tan\varphi} \tag{41}\]  
\[N_{c}=\big{(}N_{q}-1\big{)}cot\varphi \tag{42}\]  
Figure 19: Schematic diagram of introducing zero-thickness interface elements into the shallow foundation: (**a**) failure mechanism in finite element analysis and (**b**) finite element mesh. Where I, II, and III are the active zone, transition zone, and passive zone, respectively. Figure 17: Mesh deformation diagram at different stages of loading: (**a**) 10%, (**b**) 40%, (**c**) 70%, and (**d**) 100%. Tables 3 and 4 present the ultimate bearing capacity values calculated using \(c=cu\) and \(\varphi=30^{\circ}\). Since the analysis is conducted in terms of the undrained shear strength \(c_{u}\), the resulting ultimate bearing capacity is dimensionless. Under these conditions, the calculated bearing capacity factors by the equations above are \(N_{q}=18.40\) and \(N_{c}=30.14\). Figures 10-13 illustrate the simulation and analysis results for each loading stage when the transition zone is discretized into 18 blocks. For comparison, Figures 19-22 show the results when the transition zone is modeled as a single block, highlighting the differences in the failure mechanism and corresponding bearing capacity response. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Block Number (\(n\))**} & **Ultimate Bearing Capacity** & **Error Percentage * (\%)** \\  & **(E.E.M.), \(q_{u}\)/\(c_{u}\)** & **Error Percentage * (\%)** \\ \hline
1 & 36.53 & 21.20 \\
2 & 35.75 & 18.61 \\
3 & 34.92 & 15.86 \\
5 & 34.26 & 13.67 \\
6 & 33.35 & 10.65 \\
8 & 31.84 & 5.64 \\
9 & 30.50 & 1.19 \\
10 & 30.27 & 0.43 \\
15 & 30.21 & 0.23 \\
18 & 30.17 & 0.1 \\
30 & 30.15 & 0.03 \\ \hline \hline \end{tabular}  
* \(Error percentage(\%)=\frac{\text{E.E.M.}-\text{Equation}(42)}{\text{Equation}(4 2)}100\%\).  
\end{table} 
Table 4: Influence of block number in the transition zone of the failure mechanism of C-phi soils. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Block Number (\(n\))**} & **Ultimate Bearing Capacity** & **Error Percentage * (\%)** & **Calculation Time ** \\ \hline
1 & 22.13 & 20.27 & 1 h 23 min \\
2 & 21.64 & 17.61 & 1 h 17 min \\
3 & 21.16 & 15 & 1 h 20 min \\
5 & 20.64 & 12.17 & 1 h 15 min \\
6 & 20.23 & 9.95 & 1 h 25 min \\
8 & 19.35 & 5.16 & 1 h 14 min \\
9 & 18.56 & 0.87 & 1 h 18 min \\
10 & 18.4 As shown in Figure 23, the 18-block model provides a more refined and realistic representation of soil behavior under shallow foundation loading, leading to more accurate bearing capacity estimates. The 1-block model, while simpler, sacrifices fidelity and may misrepresent key aspects of the failure mechanism.