# Modified Pseudo-Dynamic Bearing Capacity of Shallow Strip Foundation near Slope on Non-Uniform Soil
## 3 Upper Bound Analysis of Seismic Bearing Capacity
### Upper Bound Calculation  
In this paper, it is assumed that the soils follow the linear Mohr-Coulomb criterion. According to the previous section, for the whole foundation failure mechanism, the foundation forces, uniform forces, seismic forces, and inertial forces of the soil are the external forces, while the cohesion at the rupture surfaces as well as on the velocity discontinuity lines is the internal force. To make the power of external force doing work equal to the power of internal energy dissipation, the following equation can be obtained. \[W=D \tag{20}\]  
where \(W\) means the work done by the external forces, \(D\) means the work done by the internal forces. Power of external forces. The calculation of the work done by the load \(p\) on the upper part of the foundation is as follows:  
\[W_{1}=pv_{1}\sin(\beta_{1}-\varphi)+k_{h}(0,t)\cos(\beta_{1}-\varphi) \tag{21}\]  
\[P=pB_{0} \tag{22}\]  
where \(v_{1}\) is shown in Figure 3, \(f_{1}\) is defined in Appendix A, and \(B_{0}\) means the foundation width. The power of uniform load \(q\) from self-weight of overlying soils is shown as follows:  
\[W_{2}=qLv_{1}v_{n}\left[\cos\left(\frac{\pi}{2}-\beta_{n}+\varphi+\sum\limits _{j=1}^{n-1}\alpha_{j}\right)+k_{h}(0,t)v_{n}\sin\left(\frac{\pi}{2}-\beta_{n} +\varphi+\sum\limits_{j=1}^{n-1}\alpha_{j}\right)\right] \tag{23}\]  
where \(L\) means the setback distance, which is illustrated in Figure 3, \(f_{2}\) is defined in Appendix A. The power of soil inertial force is written as follows:  
\[W_{3}=\gamma\sum\limits_{i=1}^{n}S_{i}v_{i}\cos\left(\frac{\pi}{2}-\beta_{i} +\varphi+\sum\limits_{j=1}^{i-1}\alpha_{j}\right) \tag{24}\]  
where \(\gamma\) means the soil unit weight, \(f_{3}\) is defined in Appendix A. According to Equation (4), it can be found that the acceleration coefficient \(k_{h}(z,t)\) of the seismic force varies in depth. Based on previous experience, the horizontal seismic force is characterized by the horizontal inertia force of the soil mass. Therefore, the distribution of the horizontal force inside each individual block is not uniform, and the power of the seismic force in each block should be calculated by using an integral method [27; 28]. However, due to the large number of blocks, the calculation using integration is time-consuming. Therefore, to improve the efficiency of the calculation, this paper adopts the simplified method of hierarchical calculation, and the accuracy of this method can be guaranteed. The formula is as follows:  
\[W_{4}=\gamma\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}k_{h}(z,t)\cdot S_{i,j  
Dissipation of internal energy. Internal energy dissipation occurs at the rupture surface \(d_{i}\) as well as at the velocity discontinuity surface \(l_{i}\), but it is important to note that soil cohesion is no longer constant but increases linearly with depth. Therefore, the dissipation of internal energy on the velocity discontinuity surface \(l_{i}\) can be represented by a segmentation method similar to the seismic force calculation, which can be expressed as follows:  
\[D_{1}=c_{0}\sum_{i=2}^{n}l_{i}v_{i-1,i}\cos\varphi+\lambda\sum_{i=2}^{n}\sum_{j =1}^{m}z_{ij}\cdot t_{ij}\cdot v_{i-1,i}\cos\varphi \tag{26}\]  
\[z_{i,j}=\frac{l_{i}\sin(\sum\limits_{j=1}^{i-1}\alpha_{j})}{m}\cdot(0.5+j) \tag{27}\]  
\[t_{i,j}=l_{i}/m \tag{28}\]  
The first term in Equation (26) is the part of the initial cohesion, while the second term is the part that grows linearly. The dissipation of internal energy on the rupture surface \(d_{i}\) is shown as follows:  
\[D_{2}=c_{0}\sum_{i=1}^{n}d_{i}v_{i}\cos\varphi+\lambda\sum_{i=1}^{n}z_{i}^{ \prime}v_{i}\cos\varphi \tag{29}\]  
\[z_{i}^{\prime}=0.5\left(l_{i}\sin(\sum_{j=1}^{i-1}\alpha_{j})+l_{i+1}\sin( \sum_{j=1}^{i}\alpha_{j})\right) \tag{30}\]  
The above is the power formula for the internal and external work, which is brought into Equation (20) by the definition of the upper bound theorem, then the display function on the foundation bearing capacity \(p\) is obtained, as shown in Equation (31), which includes \(2n+1\) variables.