## 4 Verification Between Numerical and Analytical Results  
### Analysis of the Ultimate Bearing Capacity of Shallow Foundations in Cohesive Soils  
To analyze the soil failure mechanism induced by shallow foundation loading, this study investigates the relationship between soil cohesion and the resulting failure mechanism and calculates the ultimate bearing capacity of cohesive soil. In the finite element analysis, the assumed failure mechanism is a composite sliding surface consisting of triangular and fan-shaped zones, as illustrated in Figure 1. Based on this assumed mechanism, a simulation of soil failure under a shallow foundation load is conducted, and the ultimate bearing capacity is determined accordingly. For cohesive soils, the failure mechanism is characterized by a sliding surface comprising two triangular zones and a central fan-shaped zone, referred to as the active zone (I), the transition zone (II), and the passive zone (III), as depicted in Figure 6a. The corresponding finite element mesh for this configuration is shown in Figure 6b. The ultimate bearing capacity of the shallow foundation under these conditions can be analytically determined using Equation (8). To better understand the sliding failure behavior within the fan-shaped zone and to evaluate the accuracy of different minimum upper-bound estimates, this study subdivided the sector into multiple slices and employed a multi-segment, zero-thickness interface element for simulation. Convergence analysis showed that the error between the numerical results and the upper-bound solution from the limit analysis was minimized when the sector was divided into 30 slices, corresponding to an internal angle of 3\({}^{\circ}\) for each slice within the 90\({}^{\circ}\) fan-shaped zone. To streamline the mesh generation process while maintaining sufficient accuracy, this study adopts a simplified configuration of 18 slices, where each  
Figure 6: Schematic diagram of introducing zero-thickness interface elements into the shallow foundation in cohesive soils: (**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. internal angle is \(5^{\circ}\). The analysis employs a multi-segment, zero-thickness interface element formulation, and the corresponding finite element mesh diagram is illustrated in Figure 6b. In the finite element analysis, the shallow foundation is assumed to be placed in a semi-infinite space, with two triangular zones and a single fan-shaped transition zone forming the failure mechanism The failure is modeled as five sliding surfaces, two triangles, and one fan-shaped group. For each section of the zero-thickness interface, the Mohr-Coulomb parameters are set with a friction angle (\(\varphi\)) of 0 and undrained shear strength (\(c_{u}\)), with soil cohesion normalized for the simulation. The simulation results, shown in Table 1, indicate that the error between the numerical analysis and the upper-bound value of the ultimate bearing capacity proposed by [PERSON] (1981) [PERSON] (1981) is only 0.55% (using 18 sector slices for analysis). Based on Equation (8), the ratio of the ultimate bearing capacity to the undrained shear strength of the shallow foundation is calculated as \(q_{u}/c_{u}=5.14\). At this load level, all points on the sliding surface are predicted to reach complete sliding failure, as illustrated in Figure 7. The failure process follows a progressive pattern from left to right. Initially, the triangular active zone (I) undergoes sliding failure. As the stress redistribution progresses, failure spreads into the fan-shaped transition zone (II) and eventually to the triangular passive zone (III). When the incremental accumulation reaches completion, the entire sliding surface has undergone total failure. Figure 7 illustrates the progressive failure diagrams corresponding to different stages of loading: (a) 10%, (b) 40%, (c) 70%, and (d) 100% of the applied load.