# Experimental and Theoretical Study of One-Way Cyclic Lateral Responses of Piles in Sloping Ground
## 3 Results and Discussion
### Cyclic Lateral Loading Tests  
To generalize the trends of measured pile head deflection, it is fairly common to develop the relationships between the dimensionless pile head deflection \(y_{0,\alpha}=y_{0}/b_{0}\) and cycle number \(n\), as illustrated in Figure 12. As anticipated, the measured dimensionless pile head deflection increases with cycle number at a decreasing rate and is positively related to both the load amplitude and slope angle. In addition, the dimensionless pile head deflection in the case of \(n=100\) accounts for almost 60% of that in the case of \(n=2500\), indicating the fundamental role of the first 100 loading cycles for the one-way cyclic lateral responses of piles. Interestingly, the dimensionless pile head deflection versus cycle number curves can be well fitted by power functions [10]\(y_{0,\alpha}=An^{0.11}\), and the fitting coefficient \(A\) is related to the load amplitude \(Q_{0}\) and slope angle \(\theta\) as in Figure 12.  
Figure 13 demonstrates the dimensionless bending moment and subgrade reaction profiles under different cycle numbers in test C2 (\(Q_{0}=30\) N and \(\theta=45^{\circ}\)). Both the dimensionless bending moment and subgrade reaction increase with cycle number as  
Figure 11: Comparisons of subgrade reaction modulus. Figure 12: Developments of dimensionless pile head deflection with cycle number in cyclic lateral loading tests. expected, indicating that the soil at the shallow depth is progressively softened; however, the increase slows down when \(n>100\). The locations of the inflection point (\(M=0\)) and critical point (\(p=0\)) move downward with increasing cycles, indicating that the increase in cycle number promotes the mobilization of subgrade reaction at the deeper depth to maintain the lateral equilibrium of piles. The dimensionless bending moment is observed to increase with cycle number when \(z_{\alpha}<4.58\) but remains almost zero when \(z_{\alpha}>4.58\), which is consistent with the suggestion that the pile deflections and bending moments below \(z_{\alpha}=4\) are almost zero and can even be neglected [38]. In reality, the soil near the ground surface would yield when the subgrade reaction reaches the lateral limiting force; hence, the subgrade reaction close to the ground surface would be small. Obviously, the continuous distributions of measured subgrade reactions are reasonable, similar to the typical distribution of subgrade reactions [30]. To further investigate the effects of slope angle and load amplitude on the cyclic lateral responses of piles, the dimensionless bending moment and subgrade reaction profiles (\(n=100\)) in different cases of load amplitude and slope angle are compared in Figure 14. The negative bending moments and obvious inflection points occur at the lower parts of these bending moment profiles, proving that the model pile can be classified as a long flexible pile. The pattern of these bending moment profiles with inflection points is similar to the observations for long flexible piles in level ground [9; 11; 16] and sloping ground [25; 28]. As the load amplitude varies from \(Q_{0}=20\) to \(30\) to \(40\) N, the maximum of \(M_{\alpha}\) increases from \(0.010\) to \(0.019\) to \(0.029\) and the maximum of \(p_{\alpha}\) increases from \(0.094\) to \(0.290\) to \(0.441\); however, the locations of maximums of \(M_{\alpha}\) and \(p_{\alpha}\) remain unchanged at \(z_{\alpha}=1.7\) and \(2\), respectively. In addition, as the slope angle ranges from \(\theta=30^{\circ}\) to \(45^{\circ}\) to \(60^{\circ}\), the maximum of \(M_{\alpha}\) varies from \(0.011\) to \(0.019\) to \(0.025\) and appears approximately at \(z_{\alpha}=1\), \(1.7\), and \(2.5\), respectively; as the slope angle ranges from \(\theta=30^{\circ}\) to \(45^{\circ}\) to \(60^{\circ}\), the dimensionless maximum subgrade reactions above the critical point vary from \(0.246\) to \(0.290\) to \(0.322\) and appear approximately at the depth \(z_{\alpha}=1.25\), \(2\) and \(2.91\), respectively.