# Experimental and Theoretical Study of One-Way Cyclic Lateral Responses of Piles in Sloping Ground
## 4 Conclusions  
This study investigated the effects of cycle number, load amplitude, and slope angle on the one-way cyclic lateral responses of piles in sloping ground by model experiments and the finite beam element method. Based on these experimental and theoretical observations, conclusions can be drawn as follows:  
1. The increase of cycle number, load amplitude, and slope angle can increase pile responses, including the maximums of bending moment and subgrade reaction, and pile head deflections. However, only the increases in cycle number and slope angle lead to the increased depth of inflection point and critical point, indicating that the increases promote the mobilization of subgrade reaction at the deeper depth to maintain the lateral equilibrium of piles.
2. The increase of these pile responses slows down when \(n>100\), indicating that the first 100 loading cycles are fundamental for the one-way cyclic lateral responses of piles. Interestingly, the variations of dimensionless pile head deflection with cycle number can be well fitted by power functions \(y_{0,\alpha}=An^{0.11}\).  
Figure 16: Comparisons of cyclic lateral responses predicted by FBEM and those measured by experiments in test C2. (**a**) pile head deflection versus cycle number; (**b**) bending moment profile.  
Figure 15: Predicted dimensionless pile deflection and bending moment profiles under different cycle numbers in test C2. (**a**) pile deflection profile; (**b**) bending moment profile.  
3. The finite beam element method (FBEM) is compared with experimental observations and other theoretical methods, and it is more realistic than other theoretical methods. The missing points of the pile deflection caused by incompetence of direct measurement can be alternatively obtained by FBEM in a supplementary analysis. On this basis, a full picture of the bending moment profile can be obtained and better characterize the maximum pile response and its location, compared to the one outlined by connecting the scattered points of the measured bending moment. This proves that the FBEM can well predict the one-way cyclic lateral responses and lateral static responses of piles in sloping ground, and the slope effect and second-order effect should be considered in the analysis for cyclic lateral loading tests when using the FBEM.
4. Although the FBEM is verified and demonstrates the above advantages, its use relies on the known condition of bending moment at the ground by observation when the subgrade reaction modulus is unknown. This prerequisite restrains its general application to a certain extent, and the FBEM is thus only used as a supplementary analysis method for experimental means in this paper. A more systematic theoretical method including the assessment method of subgrade reaction modulus under cyclic load will be proposed by comparative analysis of more in-depth tests and numerical studies in the future.  
Conceptualization, W.P.; Data curation, M.Z.; Formal analysis, W.P.; Funding acquisition, C.Y.; Investigation, C.Y.; Project administration, M.Z.; Resources, M.Z. and C.Y.; Software, S.Z.; Validation, W.P. and S.Z.; Writing--original draft, W.P.; Writing--review & editing, S.Z. 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 (Grant Nos. 52108317 and 51908208) and the Natural Science Foundation of Hunan Province of China (Grant No. 2022 JJ40074).  
Not applicable.  
Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request (the data in the graph and the code of calculation, etc.).  
No conflict of interest exists in the submission of this manuscript, and the manuscript is approved by all authors for publication.