## 1 Introduction  
The determination of the ultimate bearing capacity of foundations is an important topic in classical soil mechanics and foundation engineering that has received widespread attention and research from the academic and engineering communities [1, 2]. The ultimate bearing capacity of homogeneous strip foundations calculated by various classic theoretical analysis methods varies greatly, and even theoretical formulas derived from the same theory will have significant differences in their calculation results. Moreover, layered soil of upper soft clay and lower sandy soil in coastal areas are relatively common [3]. For the ultimate bearing capacity of layered foundations, scholars mainly apply the formula for calculating the bearing capacity of homogeneous foundations to layered foundations in the form of empirical coefficients, correction coefficients, etc., without fully considering the actual situation of layered soils, and the calculation results lack reliability [4]. Regarding the ultimate bearing capacity of homogeneous foundations, many international scholars have conducted extensive theoretical formula research. Due to different analysis methods and basic assumptions, the bearing capacity coefficients they derived have significant differences. [PERSON] derived the formula for the ultimate bearing capacity of a rigid foundation in weightless soil based on plastic theory [5]. Due to [PERSON]'s failure to consider the weight of the soil in the foundation bearing layer, the formula has limitations. Scholars represented by [PERSON] have proposed a widely used formula for calculating the ultimate bearing capacity of foundations by considering the weight of the soil in the foundation bearing layer based on [PERSON]'s basic principle [6]. However, [PERSON]'s theory assumes that the sliding surface extends to the bottom of the foundation, without considering the influence of burial depth. Continuous uniformly distributed loads on both sides of the foundation bottom are used to replace the shear strength of soil within the buried depth range. When the foundation is buried deep, this simplified method will underestimate its bearing capacity, making the foundation design biased conservatively. [PERSON] [7; 8] made modifications to this by assuming that the sliding surface extends to the surface of the foundation and equivalently converting the interaction between the foundation side and the soil, as well as the weight of the soil on both sides, into stresses on equivalent free surfaces. This is done to reasonably consider the shear strength of the soil on both sides, which is more in line with the actual state of foundation failure [9]. [PERSON] and [PERSON] [10] assume that the ultimate bearing capacity of a double-layer foundation is obtained by placing the foundation separately on soft and hard foundations, and then using the difference analysis method to derive a semiempirical formula for the ultimate bearing capacity of the double-layer foundation. [PERSON] [11] used the weighted average method to represent the strength parameters of a layered foundation using the average index, thus simplifying the layered foundation into a homogeneous foundation for the ultimate bearing capacity calculation. In recent years, international scholars have conducted in-depth research on the bearing characteristics of foundations from theoretical derivation, model testing, numerical simulation, and other aspects. The related research has guiding significance for the optimization design of foundations and engineering construction. [PERSON] [12] introduced reduction coefficients and depth coefficients to derive the theoretical formula for the ultimate bearing capacity of skirt foundations on soil slopes under undrained conditions and established a numerical model to study the influencing factors of bearing capacity.