## 1 Introduction  
In foundation pit engineering, a new foundation pit is frequently constructed near or even attached to an existing structure. Thus, a finite amount of soil remains between the existing structure and the new foundation pit [1]. The finite soil pressure calculated by traditional theory is often too high [2], which affects the design parameters of the bracing structures of the new foundation pit. The distribution of finite soil pressure was obtained with model tests using finite soil pressure calculation theory. The distributions of finite active soil pressure and static soil pressure under different widths were studied by centrifugal model tests [3,4]. The slip surface of a finite amount of soil under a translational mode of failure can be represented as a right-angled trapezoid through image velocimetry (PIV) [5]. The relationship between the finite soil width and the slip surface was investigated in detail [6; 7]. It is found that the shape of the sliding surface tends to become arc-shaped when the width of the fill is larger. A considerable amount of research has focused on a theoretical method to predict finite soil pressure. The slip surface was assumed to be a plane in some studies using the limit equilibrium method, and a reasonable failure angle of the finite soil slip surface was determined. The inclination of the slip surface was closely related to the calculated depth, internal friction angle of soil, soil cohesion and limited soil width [8]. Considering the interaction between existing buildings and finite soil [9] and the free-standing height of the soil [1], a formula describing active finite soil pressure was derived. This formula assumes that a slip surface is a plane and thus simplifies the problem of finite soil pressure. However, a real soil slip surface must be described using complex models such as multiple polylines and curves. Therefore, a multiplying soil failure model was assumed to obtain a finite soil pressure calculation using the limit equilibrium theory [10; 11]. Then, the seismic active soil pressure on a narrow backfill retaining wall was studied [12]. The number of slip surfaces increases with the decrease of horizontal distance between the wall and soil by the numerical analysis method, and then, corresponding theoretical formulas were proposed [13; 14]. These theoretical studies cannot describe the nonlinear distribution of finite soil pressure. However, the horizontal differential element method can be used to calculate the finite soil pressure. In this method, the finite soil is differentiated, and then the force of the differential element is analyzed to determine the static equilibrium equation to solve the finite soil pressure. The soil pressure distribution in the semi-infinite space of the retaining wall was investigated using this method [15; 16]. The trapezoidal soil slip area was divided into zones, and cohesionless finite soil pressure was obtained. The incline angle of the slip plane in the confined cohesionless soil is in close proximity to that of [PERSON]'s solution [17; 18]. The soil arching effect was introduced [19; 20]. The shear stress between thin horizontal layers was considered [21]. The horizontal distance and the inclination angle of the adjacent rock face were considered to establish four models, and the analytical solution of active soil pressure under different conditions was obtained [22]. The soil was assumed to be in a limited state in these theoretical studies. However, displacement and deformation are primarily considered in foundation pit engineering, and the soil behind the retaining structure is in a nonlimited state. To accurately explore the relationship between the finite soil pressure and displacement, the analytical formula of the active finite soil pressure for clay in a retaining wall translational mode was investigated [23]. A model test with rotation about the bottom of the retaining wall (RB mode) was performed, and the shape of the potential slip surface was observed by PIV technology [24].