# Experimental and Theoretical Study of One-Way Cyclic Lateral Responses of Piles in Sloping Ground
## 3 Results and Discussion
### Lateral Static Loading Tests  
The developments of the pile head deflection \(y_{0}\) and bending moment at the ground surface \(M_{0}\) with the lateral load atop the pile \(Q_{0}\) in lateral static loading tests are illustrated in Figure 6. The pile head deflections \(y_{0}\) in either level ground or sloping ground increase with increasing lateral load at a decreasing rate, which is similar to the experimental observations by [PERSON] and [PERSON] [25]. The \(Q_{0}\)-\(y_{0}\) curves for the case in level ground and the case of \(\theta=45^{\circ}\) can be approximately fitted by power functions \(Q_{0}=19.45\)\(y_{0}\)\({}^{0.63}\) and \(Q_{0}=14.82\)\(y_{0}\)\({}^{0.59}\), respectively. The \(M_{0}\)-\(Q_{0}\) curves for the case in level and sloping ground (\(\theta=45^{\circ}\)) can be approximately fitted by elementary functions \(M_{0}=0.01\)\(Q_{0}+0.0044\)\(Q_{0}\)\({}^{1.54}\) and \(M_{0}=0.01\)\(Q_{0}+0.1832\)\(Q_{0}\)\({}^{0.70}\), respectively; they are composed of two terms, \(0.01\)\(Q_{0}\) and \(0.0044\)\(Q_{0}\)\({}^{1.54}\) (\(0.1832\)\(Q_{0}\)\({}^{0.70}\)), and the former is induced by lateral load eccentricity (\(e=0.01\) m), whereas the latter is caused by the second-order effect. As in published studies [28; 40; 41], the lateral load producing a pile head deflection of \(0.2\) times the pile diameter (\(6.4\) mm) has been frequently taken as the allowable bearing capacity of piles (\(62\) N and \(43\) N for cases of level and sloping ground, respectively). In this regard, the load amplitudes in the one-way cyclic lateral loading tests should be less than the pile bearing capacity (\(43\) N) in the sloping ground and tentatively chosen as \(20\), \(30\), and \(40\) N, respectively. Compared to the pile head deflection and bending moment profile, the pile deflection profile is more difficult to measure directly due to the restriction of experimental conditions, and thus a supplementary theoretical analysis is required. Consequently, the finite beam element method (FBEM) is introduced and detailed in Appendix A [6; 7; 8]. To validate the FBEM (Figures 7 and 8), the pile responses (pile head deflections and bending moment  
Figure 5: Comparisons between subgrade reaction profiles indirectly inferred from bending moment profiles and those directly measured by earth pressure cells. (**a**) \(Q_{0}=20\) N; (**b**) \(Q_{0}=30\) N; (**c**) \(Q_{0}=40\) N.  
Figure 6: Developments of pile responses with lateral load in lateral static loading tests. (**a**) \(Q_{0}\) versus \(y_{0}\); (**b**) \(M_{0}\) versus \(Q_{0}\). profiles) predicted by FBEM are compared with observations and those from the power series method (PSM) detailed in Appendix B (B) [38]. Note that the coefficients for the case of sloping ground required by PSM are unavailable from the specification JTG 3363-2019 [38], that is, the slope effect cannot be considered by PSM in the specification. The slope effect, however, can be considered in FBEM by halving the proportional coefficient of subgrade reaction coefficient \(m\) when \(z=0\) to \(3b\)[6; 38]. These comparisons demonstrate that the pile head deflections predicted by FBEM are consistent with measured results and particularly close to those estimated by PSM. However, there are some errors between the predictions (FBEM and PSM) and observations in terms of the profile of dimensionless bending moment. This is mainly because the moment at the ground surface used in the theoretical methods only includes the component induced by lateral load eccentricity (0.01 \(Q_{0}\)) while neglecting the one induced by the second-order effect (0.0044 \(Q_{0}\)[1.54]). The second-order effect cannot be ignored and thus would be considered in the subsequent analysis for cyclic lateral loading tests when using the FBEM. For further validation in terms of \(p\)-\(y\) curves, the pile deflection profiles predicted by FBEM and measured subgrade reaction profiles are shown in Figures 9 and 10, respectively. Note that the subgrade reaction opposite to the lateral load is denoted as positive.