## 2 Upper-Bound Limit Analysis Theorem
### Colesive Soils  
In the upper-bound limit analysis of cohesive soils, an essential first step is to assume a kinematically admissible failure mechanism. The geometry of this mechanism must satisfy the condition that the internal work dissipated along the failure surface (\(\delta W\)) equals the external work performed by the applied loads (\(\delta E\)). For purely cohesive soils (where \(c=c_{u}\) and \(\varphi=0\)), the assumed failure mechanism can be represented by either straight-line slip surfaces or circular arc slip surfaces. Two common types of failure geometries for cohesive soils are (1) the planar slip surface mechanism, composed of straight-line segments, and (2) the circular slip surface mechanism, defined by arc-shaped failure paths. Any proposed failure mechanism must be physically reasonable and reflect the expected collapse behavior to ensure that the calculated ultimate bearing capacity closely approximates the actual value. In scenarios involving plane strain conditions, deformation is often assumed to be negligible in directions orthogonal to the primary failure mechanism. Thus, the analysis focuses on continuous deformation within the soil mass and discontinuities along the sliding surfaces [16].  
The relationship between the internal work conducted on the sliding surface to resist the external force and the displacement increment of the soil caused by the external force is:  
\[\delta W=\sum c_{u}L\delta w+\sum 2c_{u}R\theta_{f}\delta w \tag{3}\]  
where the first term \(\delta w\) on the right is the internal stress of the plane sliding surface, and the second term \(\delta w\) on the right is the internal stress of the arc sliding surface; where \(c_{u}\) is the undrained shear strength of the soil on the sliding surface, \(L\) is the length of the plane sliding surface, \(R\) is the radius of the arc sliding surface, and \(\theta_{f}\) is the angle of the arc sliding surface. The external work, \(\delta E\), generated by an external force or soil weight is expressed as  
\[\delta E=\sum\delta\omega F+\int_{A}\delta\omega qdA+\int_{V}\delta\omega\gamma dV \tag{4}\]  
where \(F\) is the ultimate bearing capacity above the foundation, \(q\) is the surface soil load, \(\gamma\) is the unit weight of the soil, \(V\) is the volume of the soil, \(A\) is the area under the surface soil load, and \(\delta w\) is the displacement increment caused by soil settlement due to external force.  
Assuming the failure mechanism of a shallow foundation occurs within a semi-infinite soil mass, the failure geometry can be represented by a composite mechanism consisting of two triangular zones and a central circular fan zone, as illustrated in Figure 1a. As shown in Figure 1b, the displacement increment resulting from the applied external load corresponds to the vertical settlement of the foundation. This displacement is directly related to the increment in the surface load, \(q\), which drives the assumed failure mechanism. Its geometric relationship is given as  
\[\int_{A}\delta\omega_{q}qdA=-\delta w_{s}qC \tag{5}\]  
where \(C=B\) and \(\delta w_{q}=-\delta w_{s}\); therefore, the work performed by the force on these five sliding surfaces is  
\[\delta E=q_{u}B\delta w_{q}-qC\delta w_{q} \tag{6}\]  
and  
\[\delta W=\pi c_{u}B\delta w_{q} \tag{7}\]where \(\delta W\) is the sum of the internal work conducted by each line segment on the sliding surface to resist stress. According to the principle of conservation of energy (\(\delta W=\delta E\)), the ultimate bearing value of the foundation can be obtained as follows.  
\[q_{u}=(2+\pi)c_{u}+q \tag{8}\]  
When the overburden pressure \(q=0\), then