# Modified Pseudo-Dynamic Bearing Capacity of Shallow Strip Foundation near Slope on Non-Uniform Soil
## 1 Introduction  
In many cases where foundations have to be placed near slopes, slope instability under the seismic effect is one of the most common geological disasters in engineering, which is a great threat to people's lives and property safety. Therefore, the accurate assessment of seismic bearing capacity of strip foundation has an indispensable role in guiding actual engineering, which has attracted the attention of many scholars. The calculation methods of foundation bearing capacity are mainly classified into the following four types: (a) slip line method, (b) limit equilibrium method, (c) limit analysis method, and (d) numerical simulation. Because of the simplicity of the limit analysis method, it has been employed by many scholars. Compared with the later emerging numerical simulation, including the finite element method and the discrete element method, the limit analysis method can quickly obtain many calculation results to carry out parameter analysis to explore the influence of different parameters. However, it is difficult to use the limit analysis method to deal with some complex working conditions. In addition, the limit analysis approach comprises two distinct methodologies: the upper bound and lower bound techniques. The upper bound methodology employs the virtual work principle to determine the maximum possible bearing capacity through analysis of kinematically admissible velocity fields, while the lower bound method obtains the lower bound valueof bearing capacity from the static permissible stress field by using the stress equilibrium. [PERSON] (1999) [1] provided a method for the assessment of the bearing capacity of the foundations based on the upper bound method of the limit analysis based on the two different failure mechanisms and verified the validity of the method. This field has attracted the attention of many scholars due to the growth of cases where foundations are placed on adjacent slopes. Based on the method of [PERSON] (1999) [1], [PERSON] et al. (2011) [2] similarly calculated the seismic bearing capacity of strips near rock slopes for different failure mechanisms using the upper bound method. [PERSON] (2018) [3] evaluated the seismic bearing capacity of foundations near \(c\) and \(\varphi\) soil slope using finite element limit analysis upper limit method (FELA) and finite element limit analysis lower limit method, respectively. [PERSON] et al. (2021) [4] calculated the ultimate bearing capacity of strip foundations on rock slopes by using FELA. [PERSON] et al. (2023) [5] gave the seismic bearing capacity of strip foundations on rock slopes under eccentric loading conditions by using the FELA method. [PERSON] et al. (2024) [6] investigated the bearing capacity of foundations on rock slopes based on slip line method and FELA method. It is to be noted that there are currently three main methods of dealing with seismic forces. The first method is known as the pseudo-static method (PS), which is widely used because of its simplicity. This method uses a constant coefficient to characterize the magnitude of lateral or vertical seismic forces, and, therefore, most scholars believe that the pseudo-static method can only simply represent the dynamic properties of seismic forces (through the variation of \(k_{h}\)), ignoring the time factor. The second method is the conventional pseudo-dynamic method (CPD). Considering the phase difference in seismic waves in the backfill and the amplification effect, [PERSON] and [PERSON] (1990) [7] gave a means of the pseudo-dynamic analysis of the active earth thrust of a retaining wall. Subsequently, the method was adopted and further extended by many scholars [8; 9; 10]. The method assumes that both pressure and shear seismic waves are upward-propagating harmonics, which have the same frequency but different amplitudes. Although the conventional pseudo-dynamic method has been widely used in the field of seismic stability analysis of geotechnical engineering, it still has some disadvantages.