Kinematic response of single piles for different boundary conditions: analytical solutions and normalization schemes

Kinematic pile–soil interaction is investigated analytically through a Beam-on-Dynamic-Winkler-Foundation model. A cylindrical vertical pile in a homogeneous stratum, excited by vertically-propagating harmonic shearwaves, is examined in the realm of linear viscoelastic material behaviour. New closed-form solutions for bending, as well as displacements and rotations atop the pile, are derived for different boundary conditions at the head (free, fixed) and tip (free, hinged, fixed). Contrary to classical elastodynamic theory where pile response is governed by six dimensionless ratios, in the realm of Winkler analysis three dimensionless parameters suffice for describing pile–soil interaction: (1) a mechanical slenderness accounting for geometry and pile–soil stiffness contrast, (2) adimensionless frequency (which is different from the classical elastodynamic parameter), and (3) soil material damping. With reference to kinematic pile bending, insight into thephysics of the problem is gained through a rigorous superposition scheme involving an infinitely-long pile excited kinematically, and a pile of finite length excited by a concentrated force and a moment at the tip. It is shown that for long piles kinematic response is governed by a single dimensionless frequency parameter, leading to a single master curve pertaining to all pile length sand pile–soil stiffness ratios