## 4 Verification Between Numerical and Analytical Results
### Analysis of the Ultimate Bearing Capacity of Shallow Foundations in C-Phi Soils  
The simulation of the soil failure mechanism induced by shallow foundation loading primarily involves two components: the assumed failure mechanism and the calculation of the ultimate bearing capacity. As illustrated in Figure 2a, the failure mechanism is composed of two triangular zones, the active zone (I) and the passive zone (III), connected by multiple slip surfaces that form a logarithmic spiral group, representing the transition zone (II). This composite mechanism captures the progressive nature of soil failure under loading. In addition, the corresponding velocity field associated with this assumed mechanism is depicted in Figure 2b, providing an insight into the soil movement and deformation patterns during failure. The schematic representation of the distribution of multi-segment zero-thickness interface elements used for analyzing the ultimate bearing capacity of shallow foundations is shown in Figure 10a, while the corresponding finite element mesh is depicted in Figure 10b. The results from the finite element analysis indicate that all points along the predefined slip surface eventually reach complete sliding failure. Moreover, the numerical results closely align with those obtained from upper-bound limit analysis. For soil with an internal friction angle of 30\({}^{\circ}\), the computed ratio of ultimate bearing capacity to cohesion, \(q_{u}/c_{u}\), is 1.84. A detailed comparison of these results is provided in Table 1.  
Figure 8: Mesh deformation diagram at different stages of loading: (a) 10%, (b) 40%, (c) 70%, and (d) 100%. Figure 10: 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 9: Displacement field diagram at different stages of loading: (a) 10%, (b) 40%, (c) 70%, and (d) 100%. As illustrated in Figure 11, the numerical results under varying loading percentages, 10%, 40%, 70%, and 100%, demonstrate that failure along the slip surface progresses incrementally from top to bottom and from left to right. Specifically, as stress redistributes during loading, failure initiates in the active zone (zone I), propagates through the spiral transition zone (zone II), and finally ruptures along the slip surface in the passive zone (zone III). Upon completion of the incremental loading, as shown in Figure 11d, all points on the slip surface experience full sliding failure. These numerical results closely align with the minimum upper-bound solution derived from the limit analysis, confirming that the shallow foundation induces a progressive failure mechanism in the surrounding soil under applied load. In addition, Figure 12 depicts the deformation and rupture of the finite element mesh along the slip surface, while Figure 13 illustrates the trend of the velocity field throughout the failure process. The results, as presented in Table 1, indicate that the numerical calculations are in close agreement with the upper-bound solution for the ultimate bearing capacity of shallow foundations proposed by [PERSON] (1981) [16]. Although the method is based on plastic collapse and does not explicitly simulate elastic deformation, we varied the stiffness ratio from 100 to 1000 to evaluate any indirect influence on interface behavior. The bearing capacity predictions remained stable across this range, and load-displacement convergence behavior was unaffected. To assess the numerical stability of the method, we varied the mesh density in non-transition zones by scaling the base mesh size. It indicates that the solution is largely mesh-independent outside the slip zone, due to the structured nature of the predefined failure mechanism. Figure 11: Progressive failure diagram at different stages of loading: (**a**) 10%, (**b**) 40%, (**c**) 70%, and (**d**) 100%.