## 5 Conclusions  
An elastodynamic continuum model to investigate the kinematic response of end-bearing piles under the excitation of vertical P-waves was proposed in this paper. This solution is rigorous in theory and considers the actual situation during the installation of the pile. To address the soil properties changed in the radial direction during pile-sinking, the disturbed soil adjacent to the pile was modeled by a series of vertical annular  
Figure 8: The effects of the disturbance degree (\(\eta\)) on the soil displacements (\(\rho_{p}=2500\) kg/m\({}^{3}\), \(E_{p}=36\) GPa, \(r_{0}=0.5\), \(v_{j}=0.4\), \(\beta_{j}=0.2\), \(G_{N+1}=36\) MPa, \(H=20\) m). (**a**) \(\eta=0.5\); (**b**) \(\eta=1\); (**c**) \(\eta=1.5\).  
Figure 6: The effects of the pile length (\(H\)) on the kinematic response and amplification factors of piles (\(\rho_{p}=2500\) kg/m\({}^{3}\), \(E_{p}=36\) GPa, \(r_{0}=0.5\), \(v_{j}=0.4\), \(\beta_{j}=0.2\), \(G_{N+1}=36\) MPa, \(H=20\) m, \(\eta=0.5\)). (**a**) Kinematic response factor; (**b**) Amplification factor.  
Figure 7: The effects of the pile length (\(H\)) on the kinematic response and amplification factors of piles (\(\rho_{p}=2500\) kg/m\({}^{3}\), \(E_{p}=36\) GPa, \(r_{0}=0.5\), \(v_{j}=0.4\), \(\beta_{j}=0.2\), \(G_{N+1}=36\) MPa, \(H=20\) m, \(\eta=1.5\)). (**a**) Kinematic response factor; (**b**) Amplification factor.  
homogeneous zones. Based on this assumption, the detailed formulation and solution of the model was conducted. The obtained solution was verified by comparison with the existing degenerated case.  
The parametric studies show that the construction effect has a significant influence on the kinematic response of piles:  
From the investigation of the effect of disturbance degree, it was found that the resistant earthquake properties of piles increased if the surrounding soil was strengthened during the process of construction. On the contrary, the resistant earthquake properties of piles decreased if the surrounding soil was weakened during pile-sinking.  
Through the investigation of the effect of the disturbance range, it was found that there was nearly no influence of the disturbance range on the dynamic response of piles.  
It was found that the construction effect mainly affected a specific region around the pile. The soil response was slightly smaller for the strengthened soil than for the weakened soil.  
The earthquake-resistant properties of longer piles were superior to those of structures composed of shorter piles.  
Conceptualization, D.D., Y.Z. ([PERSON]) and Z.L.; methodology, D.D., Y.Z. ([PERSON]) and Z.L.; software, D.D., Y.Z. ([PERSON]) and Z.W.; validation, D.D., Y.Z. ([PERSON]), Z.W. and Z.L.; formal analysis, D.D., Z.W., Y.Z. ([PERSON]) and Y.Z. ([PERSON]); investigation, Y.Z. ([PERSON]) and Z.L.; resources, D.D.; data curation, D.D.; writing--original draft preparation, D.D.; writing--review and editing, Z.W. and Z.L.; visualization, Z.W.; supervision, D.D. and Z.L.; project administration, D.D., Y.Z. ([PERSON]) and Z.L.; funding acquisition, D.D. and Y.Z. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
This research was funded by the key science and technology special program of Yunnan province, grant number 202102 AF080001; the Fundamental Research Funds for the Central Universities, grant number B210202042; the National Natural Science Foundation of China, grant number 52108314; and the China Postdoctoral Science Foundation funded project, grant number 2020M681475.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.