# Modified Pseudo-Dynamic Bearing Capacity of Shallow Strip Foundation near Slope on Non-Uniform Soil
## 3 Upper Bound Analysis of Seismic Bearing Capacity
### Failure Mechanism  
Based on the definition and formulation of the upper bound theorem, it can be found that the most critical step is to determine an accurate velocity-permissive motion field (failure mechanism), and the failure mechanism can reflect the real damage of the foundation more accurately. Therefore, based on the conclusions of previous studies ([PERSON], 1999 [1]; [PERSON] et al., 2024 [22]), the failure mechanism shown in Figure 3 is established in this paper. Since the effect of seismic action is considered in this paper, the damage pattern of the foundation is unilateral.  
Figure 3: The geometry and force diagram of unilateral failure mechanism.  
Figure 2: The depth profile of cohesion.  
The failure mechanism is a multi-block wedge structure with a total of \(n\) blocks, including \(n-1\) triangular blocks and one quadrilateral block [23; 24; 25]. From Figure 3, it can be found that the block ABC\({}_{1}\) at the bottom of the foundation moves in the direction of velocity at a certain angle to the rupture surface AC\({}_{1}\), and the other blocks move in a similar manner as ABC\({}_{1}\), all at a certain angle to the respective bottom rupture surface ([PERSON] et al. (2025) [26]). In addition, the strip footing is at a certain distance \(L\) from the apex \(D\) of the slope surface, which is called the setback distance.  
The external forces acting on the above mechanism are foundation load, uniform load, soil self-weight, and seismic force, while the internal energy dissipation exists on the velocity discontinuity line \(l_{i}\) as well as on the rupture surface \(d_{i}\). It should be noted that, based on previous experience in the literature ([PERSON] (1999) [1]), the effect of vertical seismic forces on foundation bearing capacity is small. This study exclusively focuses on the impact of horizontal seismic components. In addition to this, it can be found that there are two angles in each block that are variable, including \(\alpha_{i}\) and \(\beta_{i}\). Therefore, by geometrical derivation, the side length of each block can be obtained as follows:  
\[l_{i}=B_{0}\prod_{j=1}^{i-1}\frac{\sin\beta_{j}}{\sin\left(\alpha_{j}+\beta_{ j}\right)} \tag{14}\]  
\[d_{i}=B_{0}\frac{\sin\alpha_{i}}{\sin\beta_{i}}\prod_{j=1}^{i-1}\frac{\sin \beta_{j}}{\sin\left(\alpha_{j}+\beta_{j}\right)}\ \ \ \ \ i\leq(n-1) \tag{15}\]  
\[S_{i}{=0.5l_{i}d_{i}\sin\beta_{i}\ \ \ \ \ i\leq(n-1)} \tag{16}\]  
where \(S_{i}\) means the area of each block. Unlike the case of surface level, when the foundation is close to the vicinity of the slope, as shown in Figure 3, the \(n\)th block is no longer a traditional triangle, but a quadrilateral, and therefore its geometric relationship needs to be derived separately and can be expressed as follows:  
\[d_{n}=\frac{l_{n}\sin(\alpha_{n}-\beta)+L\sin\beta}{\sin(\alpha_{n}+\beta_{n}- \beta)} \tag{17}\]  
Furthermore, Figure 4 illustrates the relationship between the velocity vectors of the different blocks, including the overall movement velocity of the blocks as well as the velocity on the interrupted line, which can be expressed as follows:  
\[v_{i}=v_{i-1}\frac{\sin(\alpha_{i-1}+\beta_{i-1}-2\varphi)}{\sin(\beta_{i}-2 \varphi)} \tag{18}\]  
\[v_{i-1,i}=v_{i-1}\frac{\sin(\beta_{i-1}-\beta_{i}+\alpha_{i-1})}{\sin(\beta_{ i}-2\varphi)} \tag{19}\]  
Figure 4: The velocity diagram.