## 1 Introduction  
Active piles are usually designed to bear the load transmitted from their superstructures (e.g., axial loads, lateral loads, eccentric loads, and inclined loads). When an active pile is affected by nearby surcharges [1] or excavation [2], the active pile also bears the additional passive load caused by the displacement of the soil generated by these surcharges or excavations at the pile side, known as the active-passive pile. The abutment piles in soft ground are usually active-passive piles. There has been considerable previous research on passive piles, which only bear passive loads, or active piles bearing only active loads. [PERSON] [3], [PERSON] [4], [PERSON] et al. [5], and [PERSON] et al. [6] have studied the horizontal displacement of soil adjacent to a surcharge load and its influence on the internal forces and the deformation of adjacent piles through tests. [PERSON] et al. [7] proposed a three-dimensional numerical model to analyze the deformation and internal force of a passive pile adjacent to a surcharge load in extensively deep soft soil. In order to study the mechanical response of passive piles in soft soil, [PERSON] et al. [8] carried out numerical simulation using a two-dimensional (2D) finite element (FE) model and two different pile-soil interaction methods (the embedded and linked-element method). The results showed that a higher loading rate will lead to a greater displacement of the pile top. Based on the deducedextended Koppejan model, [PERSON] et al. [9] established a numerical model to study the time-dependent interaction between the passive pile and soft soil. However, these studies not only fail to take into account the impact of active loads, but also lack a phased calculation of the additional load caused by an adjacent surcharge load. [PERSON] et al. [10] improved the Ito theory [11; 12] by creating a more accurate way to calculate this passive load. Based on this improvement, [PERSON] et al. [13] introduced Shen theory [14] to analyze the influence of soil flow around a pile. On the basis of analyzing the lateral deformation mode of the soil in the free field by using the three-dimensional finite element method, [PERSON] et al. [15] established the calculation model of the passive pile under the surcharge load. They only analyzed the impact of passive loads on the pile. In addition, [PERSON] et al. [16] started with the \"\(m\)\" method and proceeded to take into account factors such as the axial force, horizontal force, bending moment, and landslide thrust at the pile side, and established the deflection differential equation for an axially and laterally loaded foundation pile that bears the combined action of P-A effects and pile-soil interaction. Similarly, [PERSON] et al. [17] combined the Winkler foundation model and displacement method to put forward a simplified analytical solution for an axially loaded pile subjected to lateral soil movement. However, [PERSON] and [PERSON] both used the elastic resistance method to describe the relationship between \(p\) (the lateral soil resistance) and \(y\) (the horizontal displacement of the pile) when analyzing the active-passive piles. This method cannot reflect the nonlinearity of the pile-soil interaction, so it is only applicable when the lateral displacement of a pile is small. To deal with this, [PERSON] et al. [18] applied the \(p\)-\(y\) curve method to the analysis of piles under an axial and lateral load, and [PERSON] et al. [19] applied the \(p\)-\(y\) curve method to the analysis of piles under lateral loads. However, they only considered the effect of active loads. [PERSON] et al. [20] used the transfer matrix method to give the semi-analytical solution of the active-passive coupling load pile in layered soil. The mechanical response of an active-passive pile is closely related to the interaction between the laterally moving soft soil and the pile. Its working mechanism is complex and influenced by many factors.