## 4 Parametric Analysis
Figure 14 depicts the radial surface displacement distribution around the pile along the loading direction under various pile head constraints, where \(r\) denotes the distance from the observation point to the pile center. The results reveal that radial surface displacement decreased approximately exponentially with horizontal distance. The displacement was maximal under free-head conditions, minimal under fixed-head conditions, and decreased significantly with increasing rotational stiffness. Notably, at a distance of \(6\tau_{1}\) from the pile center, the radial surface displacement was approximately 20% of the pile head displacement. Figure 15 illustrates the effect of pile head constraints on pile body displacement at the conclusion of loading (t = 20 ms). The pile body displacement decreased sharply with depth and diminished significantly as the rotational stiffness of the pile head increased. Strengthened pile head constraints effectively enhanced the horizontal resistance of the pile, leading to reduced pile body displacement. These findings underscore the critical role of pile head constraints in modulating the deformation behavior of the pile-soil system under lateral loading. Figure 2:1 Effect of pile head boundary conditions on the radial displacement of surface soil. Figure 2:2 Influence of the soil–pile elastic modulus ratio on pile body deformation. The rotational stiffness \(\kappa\) of the pile head under flexible constraints significantly affected the maximum displacement ratio \(\chi\) and the degree of constraint \(\lambda\), with a particularly pronounced impact on \(\lambda\). Figure 16 illustrates the relationship among \(\chi\), \(\lambda\), and \(\kappa\), where \(\kappa_{1}=\kappa_{2}\) = \(\kappa\). The results indicate that \(\chi\) and \(\lambda\) remained constant in regions of very low or very high \(\kappa\), signifying minimal sensitivity to stiffness under these conditions. When \(\kappa<10^{8}\) N\(\cdot\)m/rad, the pile head behaved as approximately free; conversely, for \(\kappa>10^{10}\) N\(\cdot\)m/rad, it approached a fixed condition. In the intermediate range \(\kappa=10^{8}\)\(\sim\)\(10^{10}\) N\(\cdot\)m/rad, it decreased from 100% to 50% with increasing \(\kappa\), while \(\lambda\) increased from 0 to 100%, reflecting a transition from near-free to fixed-head behavior. Figure 17 further examines the influence of rotational stiffness on pile head bending moments by analyzing the time history of the moment under various yield moments \(M_{c}\). The yield moment was set to 0.2, 0.6, 0.8, and 1.0 times the fixed-head moment \(M_{fixed}\)(0,\(t\)), maintaining these ratios throughout the analysis. The rotational stiffness was fixed at \(\kappa_{1}=10^{8}\) kN\(\cdot\)m/rad and \(\kappa_{2}=10^{4}\) kN\(\cdot\)m/rad. When \(M_{c}\)(0,\(t\)) = \(M_{fixed}\)(0,\(t\)), the moment-rotation relationship followed a single-stage linear \(M\)-\(\theta\) model. The results demonstrate that the temporal evolution of the pile head moment closely mirrored the displacement pattern, with an initial rise to a peak followed by a gradual decline to zero. Larger yield moments \(M_{c}\), which correspond to a greater initial stiffness relative to the rotation range \(\theta\), resulted in higher peak bending moments at the pile head. These findings highlight the interplay between rotational stiffness and yield moment in shaping the mechanical  
Figure 2:3 Influence of rotational stiffness (\(\kappa\)) on the maximum displacement ratio (\(\chi\)) and constraint degree (\(\lambda\)). Figure 2:4 Effect of pile head boundary conditions on pile body displacement. response of piles under lateral loading, providing valuable insights into their deformation and stability behavior.