apparently affected by the magnitude of the constant axial loads [9,10]. In more recent developments, the numerical modelling treated as an efficient tool for evaluating the performance of laterally loaded piles has attracted significant attention in the academic field. In detail, [PERSON] et al. [11, 12, 13] investigated the \(P\)-\(\Delta\) effect on the response of inclined loads piles in sand and clay soil by using the finite element method (FEM), and proposed a numerical model to quantify the influence of inclined-loads piles in sand [14]. [PERSON] et al. [15] successfully conducted a series of three-dimensional finite differences analysis for investigating the influence of vertical loads on the lateral performance of pile. In addition to numerical methods, some theoretical methods [16, 17, 18, 19, 20, 21, 22, 23] have also been used to facilitate designs. Although the above-mentioned studies have been applied widely, the slope effect (which refers to the effect of soil resistance on a pile shaft) has not been taken into account. In detail, the slope effect mainly consists of three aspects of resultants: (i) the reduction in the soil horizontal resistance in front of piles; (ii) the earth pressure caused by potential sliding soils behind piles; and (iii) the initial inclination of piles resulting from the thrusting of sliding soils. Reflection on reduction in the soil horizontal resistance has mainly been performed by experimental investigation [24, 25, 26, 27, 28, 29], numerical simulation [30, 31, 32, 33] and theoretical methods [34, 35, 36, 37]. In addition, the earth pressure behind piles caused by potential sliding soils can be approximatively predicted by landslide thrust [38]. In further developments, studies about the behavior of a pile with complicated loads in sloping ground have been performed by experimental investigation [39], and field test research [40]. Theoretical approaches to the combined effect of vertical and lateral loads on sloping ground are still insufficient [41, 42], especially for combinatorial loaded large diameter pile in sloping ground. In engineering practice, it is noteworthy that the diameter \(D\) for most piles in usage is greater than 0.8 m, a number of studies [43, 44, 45, 46, 47, 48] indicated that, for such piles, the moment of resistance induced by vertical friction could affect the behaviors of both the lateral deformation and internal force of a pile shaft, and this effect will be enhanced with an increasing in diameter \(D\) or the rigidity of the surrounding soils. [PERSON] et al. [45] conducted a centrifugal model test and developed relevant theoretical and numerical calculation models for estimating the moment of resistance for the vertical friction of the pile shaft. [PERSON] et al. [47] proposed a simplified theoretical calculation model for evaluating the above-mentioned moment of resistance by adopting friction enhancement theory and several vertical friction models, and further conducted lateral horizontal deformation analysis for pile shafts based on the transfer-matrix method. However, up to now, the previous studies were mainly focusing on the analysis of the lateral response for pile shafts under horizontal load; the studies on the lateral deformation of the pile shaft in a slope by considering coupling effects, including the vertical friction, are extremely limited. Therefore, based on the above assertions, the main objective of the current research is to develop an analytical model for evaluating the lateral deformation of a pile shaft in a slope under the aforementioned combined loads by considering vertical friction. With this objective, the remainder of this work is organized as follows. Firstly, the governing equations for mimicking the lateral deformation of different characteristic segments, including the passive, active and free segment of the pile shaft, are determined.