## 5 Conclusions  
In this work, a model for obtaining the semi-analytical solution to the lateral responses of a pile shaft in a high-steep slope including deformation and internal force was proposed based on the transfer-matrix method. Precisely, based on the tri-parameters pile-soil interaction model, the solution for pile lateral responses induced by combined loads in which the influence of vertical friction and initial inclination angle was considered can be formulated by utilizing the Laplace transformation. Conclusions can be drawn as follows:  
(1) In the proposed model, by considering the influences of vertical friction, shear deformation and initial inclination angle for the pile shaft, the flexural differential governing equations for the piles in a high-steep slope were firstly established based on the tri-parameters pile-soil interaction model, and the semi-analytical solution for the lateral response of the pile shaft was obtained by applying the transfer-matrix method. The predictions are in close agreement with field observations and the results generated by previous methods. (2) Through sensitivity analysis of each parameter in the proposed model, the lateral responses of the pile shaft were relatively sensitive to the vertical friction in the passive segment, initial inclination angle and \(P\)-\(\Delta\) effect of vertical load; specifically, it was found that a reduction effect occurred for the horizontal deformation and bending moment of the pile shaft when the vertical friction of the passive segment is increasing. Subsequently, as the value of the vertical friction was increasing, the location of the maximum bending moment was gradually closing toward the shallow stratum, the relative bearing ratio of the load in passive segment of the pile shaft was enlarged, which improved its horizontal bearing characteristics. (3) The lateral deformation and the internal force of the pile shaft were both increasing linearly due to the growth in the initial inclination angle with a decreasing amplitude. Figure 16: Influence of passive load on lateral behavior of pile shaft under different load ratios. (**a**) Maximum lateral displacement. (**b**) Maximum bending moment. There were significant influences yielded by both the vertical and passive load on the lateral deformation and internal force of the pile shaft: when the ratio of vertical load N and passive load \(q\) was less than 10, the lateral deformation and internal force were linearly increasing as the \(q\) was largening, while a non-linearly increasing trend was found when the ratio is larger than 10. In addition, greater influence was found during the N was rising up when the \(q\) is constant. T.L. : writing--original draft, software, methodology. J.L. : conceptualization, supervision. X.L. : data curation, writing--review and editing. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (funding number:42074165); Hunan Provincial Natural Resources Science and Technology Projects (funding number:2020-03). The authors declare no conflict of interest. The following symbols are used in this paper: