## 5 Conclusions  
This study investigates the horizontal transient dynamic response of pipe piles embedded in saturated soil, under flexible support at the pile head. Assuming a two-stage equivalent linear relationship between the bending moment and the rotation angle at the pile head, a time-domain solution was derived, and numerical analyses provided the following key insights:  
1. The peak displacement at the pile head increases as the duration of the applied load becomes longer. Core soil inside the tubular pile reduces the peak displacement at the pile head. For piles with small inner diameters, the effect of core soil is minimal and can be ignored, but for larger inner diameters, considering core soil improves the accuracy of the analysis. 2. When the pile length exceeds a critical value, known as the effective pile length, the displacement at the pile head stabilizes, and the pile can be treated as infinitely long. Pile displacement decreases rapidly with depth and is significantly reduced by increasing the rotational stiffness at the pile head. Strengthening the pile head constraint effectively enhances the horizontal resistance of the pile. 3. The pile head displacement peaks noticeably later than the impact load's peak force. This lag becomes increasingly significant as the impact duration shortens. Furthermore, the influence of the soil inside the pile on horizontal impact effects is minimal compared to the surrounding soil. 4. In both high and low kdi\({}^{\prime}\) regions, the pile head displacement remains nearly constant, forming plateau-like trends. Within the intermediate region, however, displacement increases with kdi\({}^{\prime}\), and the rate of this increase intensifies as soil porosity grows. 5. The maximum displacement ratio \(\chi\) and constraint degree \(\lambda\) are largely invariant in regions of extreme rotational stiffness \(\kappa\) (both high and low). However, within the intermediate flexible constraint range, \(\chi\) declines steadily with increasing \(\kappa\). Our computational model simplifies the analysis of pile-soil interactions but comes with certain limitations. Neglecting soil heterogeneity and anisotropy may reduce the model's accuracy when applied to layered or stratified soils. Additionally, omitting pile-side friction could underestimate the soil's restraining effect on the pile, particularly in soft clay or sandy soils. In future work, we aim to incorporate soil nonlinearity, slippage effects, and pile-side friction into the model to improve its accuracy and broaden its applicability to more complex working conditions. Figure 17: Effect of yield moment on pile head bending moment. **Author Contributions:** A.S.: writing-original draft, conceptualization, data curation, software, formal analysis. M.Z. : writing-review and editing, resources, funding acquisition, supervision, methodology. W.S. : data curation, resources. Q.W. : supervision, resources. All authors have read and agreed to the published version of the manuscript. **Funding:** Natural Science Foundation for Young Scientists of Shanxi Province (No.20210302123168, 20210302124652) and Central Government Guides Local Science and Technology Development Fund Project (No. YDZJSX2024B005, YDZJSX20231A021). The research project was also supported by the Shanxi Scholarship Council of China (No. 2021-61). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Data are contained within the article. **Conflicts of Interest:** The authors declare no conflicts of interest.