## 2 Upper-Bound Limit Analysis Theorem
### C-Phi Soils  
In the upper-bound limit analysis of cohesive-frictional (C-phi) soils, it is necessary to first assume a kinematically admissible failure mechanism. The geometry of this mechanism must satisfy the condition that the internal work dissipated along the failure surface balances the external work done by applied loads. For C-phi soils, typical failure mechanisms consist of straight-line slip surfaces, spiral curves (such as logarithmic spirals), or a combination of both. To ensure the accuracy of the upper-bound estimation, the assumed failure mechanism should reflect realistic soil behavior under limit conditions. The resulting prediction of ultimate bearing capacity must closely approximate the true collapse load. Under plane strain conditions, deformation in the out-of-plane direction is neglected, and the analysis focuses on the in-plane motion along continuous and discontinuous failure surfaces. Furthermore, the direction of relative displacement along the slip surface is influenced by the dilation angle, which governs the orientation of plastic strain increments and affects the geometry of the assumed failure mechanism. When assuming a failure mechanism in cohesive-frictional (C-phi) soils, the mechanism must conform to the fundamental characteristics of such materials. Specifically, under the condition of limit equilibrium, the internal work dissipated along the sliding surface  
Figure 1: The failure mechanism of the shallow foundation in cohesive soils is composed of (**a**) two triangular zones and a single fan-shaped transition zone, and (**b**) the displacement diagram. in resisting external loads must balance the external work input. In the idealized case where the mechanism is perfectly kinematically admissible and satisfies the yield criterion everywhere, the net virtual work is zero. This relationship can be expressed as  
\[\delta W=0 \tag{9}\]  
The external work, \(\delta E\), conducted by the external force or the soil's weight can be given by  
\[\delta E=\sum\delta wF+\int_{A}\delta w(q-u)dA+\int_{V}\delta w(\gamma-\gamma_{ w})dV \tag{10}\]  
where \(F\) is the ultimate bearing load above the foundation, \(q\) is the surcharge pressure, \(u\) is the pore water pressure, \(\gamma\) is the unit weight of the soil, \(\gamma_{w}\) is the unit weight of water, \(V\) is the volume of the soil, \(A\) is the area under the surcharge, and \(\delta w\) is the displacement increment of the soil settlement caused by external force [16]. The failure mechanism of a shallow foundation on a horizontal surface is illustrated in Figure 2a. This mechanism consists of a sliding surface formed by two triangular zones and a single logarithmic spiral. As shown in Figure 2b, the horizontal displacement increments in sliding zones I and III result from external forces. According to the geometric relationship, if the angle of the sector formed by the logarithmic spiral is \(90^{\circ}\), the corresponding spiral equation can be derived as follows:  
\[r_{b}=r_{a}exp\Big{(}\frac{\pi}{2}tan\varphi\Big{)} \tag{11}\]  
Figure 2: The failure mechanism of the shallow foundation in C-phi soils is composed of (**a**) two triangular zones and a single logarithmic spiral transition zone, and (**b**) the displacement diagram.