## 5 Conclusions  
Based on the modified Vlasov foundation model, the displacement and internal force of laterally loaded long piles in layered foundations are obtained by using the separation  
Figure 6: Effect of the contact surface separation on the mechanical characteristics of a laterally loaded long pile. (**a**) The displacement. (**b**) The rotating angle. (**c**) The bending moment. (**d**) The shearing force.  
variable method. Through numerical calculation, the influence of the slenderness ratio of the pile and the detachment of the pile-soil contact surface on the lateral loading performance of the pile foundation is analyzed. The main conclusions are as follows:  
1. As the slenderness ratio of the pile increases, the flexibility of the pile also increases, resulting in significant growth in lateral deformation, while the maximum bending moment and maximum shear force decrease. In this study, quantitative calculations were performed to obtain the variation patterns of pile head displacement, rotation angle, and internal forces at different slenderness ratios. The results can provide a reference for pile foundation design under similar conditions.
2. When detachment occurs at the interface between the upper soil layer and the pile, the pile head displacement and rotation angle increase by approximately 2 times, and the maximum bending moment of the pile significantly increases, with the increase in negative shear force reaching approximately 3 times. Therefore, the support provided by the upper soil layer is crucial for the rigidity of the pile foundation. Special attention should be paid to this factor during design, and measures such as improving the soil layer can be taken to mitigate the adverse effects caused by detachment.
3. When detachment occurs at the interface between the middle soil layer and the pile, the increase in the pile head displacement and maximum bending moment is less than 25 %, while the maximum negative shear force decreases significantly. Compared to detachment at the upper soil layer, detachment at the middle soil layer has a smaller impact on the performance of the pile foundation.  
This study conducted a precise numerical analysis to quantify the variation patterns of pile foundation stress and deformation under different contact conditions. It emphasized the significant impact of local detachment at the pile-soil interface on the pile head displacement, bending moment, and shear force, offering accurate results and theoretical support for engineering practice. However, only two ideal conditions, complete contact and full detachment, were considered, without accounting for the gradual weakening of the pile-soil interface or its correlation with deformation and soil parameters. Further research is needed to explore the lateral loading performance of long piles in layered foundations, considering interface weakening effects.  
Methodology, F.L. and J.M.; software, D.L.; validation, F.L. and D.L.; formal analysis, J.M.; writing--original draft preparation, F.L. and D.L.; supervision, J.M. and D.L. All authors have read and agreed to the published version of the manuscript.  
This work was funded by the National Natural Science Foundation of China (Grant No. 11502072), and the Henan Province Higher Education Young Backbone Teachers Training Program (Grant No. 2019 GGJS076), and the Research and Development of Decision-Making Model and Software for TBM Excavation and Support Parameter Optimization in High Geostress Soft Rock Tunnels (Grant No. 242102220037).  
Not applicable.  
Data Availability Statement: Data is contained within the article.  
We sincerely thank all the reviewers and editors for their professional comments and suggestions regarding this manuscript.  
The authors declare no conflicts of interest.