## 3 Pseudo-Static Analysis of Ultimate Bearing Capacity  
The implementation of the upper-bound limit analysis entails the first establishment of a kinematically admissible velocity field, thus, a shallow strip footing that is placed on the homogeneous geomaterial governed by the P-L criterion is depicted in Figure 2. The buried depth from the footing level to the ground surface is \(D\) and the width is \(B\). The bottom of the foundation is presumed to be rough. The lateral soil pressure is equivalent to uniformly distributed loads denoted as \(p=\gamma D\). In the presence of an earthquake, a non-symmetrical failure is prone to happen due to the existence of a horizontal seismic load, whereby a non-symmetrical multi-block translational mechanism is necessarily established. The inertia forces acting on the lateral covering layer are considered as two uniformly distributed loads that are perpendicular to the surface and parallel to the surface, respectively. It is common sense to the researchers that decreasing the soil strength and/or raising the failure loads are two primary forms by which the earthquake action influences the soil-foundation system. Herein, only the latter is taken into consideration.  
Figure 2 shows a simple sketch map of a shallow foundation in a broken state. The whole failure region is decomposed into a multi-block collapse mechanism that is comprised of \(n\) rigid wedges with point \(B\) as the same vertex. The first rigid block lying at the bottom of the strip footing translates in the meantime with footing without relative sliding, and the last block terminates at the footing level \(BC\). The relative sliding between the successive blocks along the contacted face is permitted. The geometry shape of each block is determined by total \(2n\) independent angular parameters \(\delta_{i}\) and \(\kappa_{i}\).  
Then, the kinematic analysis of the rigid blocks is developed herein to formulate the energy equilibrium equation. The associative flow rule requires that materials in the plastic flow state should possess the property that the rate of plastic strain is perpendicular to the yield surface. This requests a normal detaching velocity along the contacted face, which results in an inclined angle \(\varphi_{i}^{*}\) between the relative velocity and slip surface. In this view, given the translational velocity of block 1 denoted as \(v_{1}\), combining the aforementioned requirements, one can readily derive the relative velocity \(v_{1,2}\) from the velocity vector triangle, as sketched in the left-hand of Figure 3. For simplicity, the generic recurrence relationships between \(v_{i}\) and \(v_{i,i+1}\) of the \(i\)th block are presented as:  
\[v_{i+1}=\frac{\sin(\delta_{i}+\kappa_{i}-2\varphi_{t}^{*})}{\sin(\kappa_{i+1} -2\varphi_{t}^{*})}\cdot v_{i} \tag{11}\]  
\[v_{i,i+1}=\frac{\sin(\delta_{i}+\kappa_{i}-\kappa_{i+1})}{\sin(\kappa_{i+1}-2 \varphi_{t}^{*})}\cdot v_{i} \tag{12}\]  
In analogy, the lengths of the slip surface \(a_{i}\) and \(b_{i}\) of the \(i\)th wedge can be readily derived:  
\[a_{i}=\frac{\sin\delta_{i}}{\sin(\delta_{i}+\kappa_{i})}b_{i-1} \tag{13}\]  
\[b_{i}=\frac{\sin\kappa_{i}}{\sin(\delta_{i}+\kappa_{i})}b_{i-1} \tag{14}\]  
Figure 3: Velocity vector diagram of failure blocks.  
Figure 2: Sketch map of shallow foundation failure mechanism under the seismic condition.  
The initial values of \(a_{1}\) and \(b_{1}\) in the first block attribute to width \(B\) are depicted as:  
\[a_{1}=\frac{\sin\delta_{1}}{\sin(\delta_{1}+\kappa_{1})}\,B \tag{15}\]  
\[b_{1}=\frac{\sin\kappa_{1}}{\sin(\delta_{1}+\kappa_{1})}\,B \tag{16}\]