# Modified Pseudo-Dynamic Bearing Capacity of Shallow Strip Foundation near Slope on Non-Uniform Soil
## 5 Conclusions  
This paper presents a framework for analyzing the seismic bearing capacity of foundations near slopes on non-uniform soils, considering the temporal and spatial characteristics of seismic forces. The modified pseudo-dynamic method is incorporated into the framework of the limit analysis upper bound method to obtain the formulation of the bearing capacity of the foundation near slope. Diverging from established pseudo-static and elementary pseudo-dynamic procedures, the advanced pseudo-dynamic formulation integrates the consideration of surface stress boundaries and damping ratio impacts, and more realistically reflects the spatial and temporal characteristics of seismic forces. The seismic forces are calculated using the more efficient layered superposition method. In addition, a linearly increasing cohesion model is used to characterize the non-uniformity of the soils. This paper compares and analyzes the calculated results with the existing literature and analyzes important parameters one by one. The main conclusions are as follows:  
(1) The normalized frequency has a significant periodic effect on the seismic bearing capacity of foundations. When \(\omega H/V_{s}=\pi/2+n\pi\) (\(n=0,1,2\ldots\)), the seismic bearing capacity will suddenly and abruptly decrease, and unfavorable working conditions occur, so special attention is needed in engineering practice. (2) The effect of soil damping ratio \(\xi\) on seismic capacity at \(\omega H/V_{s}=\pi/2+n\pi\) (\(n\)\(=0,1,2\ldots\)) is more significant, as the seismic bearing capacity increases with the increase in damping ratio. The effect of damping ratio at \(\omega H/V_{s}=\pi\) is minimum. (3) The relationship between seismic bearing capacity and setback distance \(L\) as well as slope angle \(\beta\) has obvious convergence. As the setback distance grows, the structure's seismic resistance improves, whereas steeper slope angles lead to reduced bearing capacity under earthquake conditions. (4) The seismic bearing capacity is positively correlated with the initial cohesion and the non-uniformity coefficient, and the non-uniformity of the soils has a significant effect on the seismic bearing capacity. This study investigates the bearing capacity of foundations adjacent to slopes under seismic conditions using a linear non-uniform soil profile, extending conventional homogeneous soil models to account for depth-dependent soil parameter variations. By integrating the upper bound theorem of limit analysis with the modified pseudo-dynamic method, the proposed framework addresses both slope proximity effects and seismic dynamics, providing practical references for engineering applications. The derived solutions demonstrate improved alignment with realistic bearing capacity behaviors compared to static or homogeneous soil assumptions. The current methodology carries inherent limitations. The linear non-uniform soil profile represents an idealized simplification, as real stratigraphy often exhibits nonlinear heterogeneity. Theoretical prerequisites of the upper bound theorem (e.g., small deformation, elastic-plastic constitutive behavior) and modified pseudo-dynamic assumptions strictly govern all calculations. Furthermore, the adoption of linear failure criteria overlooks soil nonlinearity, while unsaturated soil mechanics and seepage effects remain unaddressed. These aspects warrant future refinement. The results of this study exhibit broad applicability, with the model being validated for foundation bearing capacity analysis under seismic conditions while accounting for soil heterogeneity and slope proximity. However, the proposed framework is not applicable to unsaturated soil conditions or scenarios involving seepage effects. This paper aims to study the seismic bearing capacity of the foundation near the slope on non-uniform soil, providing reference for practical engineering. However, there are some limitations in this paper. To simplify the calculation, the linear increasing cohesion model is used to characterize the non-uniformity of the soil.