Dynamic Winkler modulus for axially loaded piles

The problem of axial dynamic pile soil interaction and its analytical representation using the concept of a dynamic Winkler support are revisited. It is shown that depth- and frequency-dependent Winkler springs and dashpots, obtained by dividing the complex-valued side friction and the corresponding displacements along the pile, may faithfully describe the interaction effect, contrary to the common perception that the Winkler concept is always approximate. An axisymmetric wave solution, based on linear elastodynamic theory, is then derived for the harmonic steady-state response of finite and infinitely long piles in a homogeneous viscoelastic soil stratum, with the former type of pile resting on rigid rock. The pile is modelled as a continuum, without the restrictions associated with strength-of-materials approximations. Closed-form solutions are obtained for: (a) the displacement field in the soil and the pile; (b) the stiffness and damping ('impedance') coefficients at the pile head; (c) the actual, depth-dependent, dynamic Winkler moduli; and (d) a set of fictitious, depth-independent Winkler moduli to match the dynamic response at the pile head. Results are presented in terms of dimensionless graphs, tables and simple equations that provide insight into the complex physics of the problem. The predictions of the model compare favourably with existing solutions, while new results and simple design-oriented formulae are presented.