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.

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An analytical elastic continuum model is developed for the settlement of end-bearing piles in a two-layer soil over a rigid stratum. The model has its roots in the point-load solution of Westergaard which was later extended by Tajimi to deep foundations and lies on the assumption of a vanishing soil stress or displacement component. For piles in homogeneous soils such solutions were elaborated by Nogami and Novak. Contrary to these solutions, the proposed generalized formulation can handle layered soils using, for the first time, two sets of eigenfunctions (static "modes") which are different for the soil and the pile. Stresses and displacements are determined in the form of Fourier series with coupled coefficients obtained by solving a system of algebraic equations of rank equal to the number of modes considered. This is in contrast with existing models, where the Fourier coefficients are obtained individually. Pile head stiffnesses obtained from this model are verified against results from rigorous finite-element analyses and other solutions. Results for pile settlement, pile stresses, side friction and Winkler moduli are presented.

a b s t r a c tThe analytical representation of dynamic soil reaction to a laterally-loaded pile using 3D continuum modeling is revisited. The governing elastodynamic Navier equations are simplified by setting the dynamic vertical normal stresses in the soil equal to zero, which uncouples the equilibrium in vertical and horizontal directions and allows a closed-form solution to be obtained. This physically motivated approximation, correctly conforming to the existence of a free surface, was not exploited in earlier studies by Tajimi, Nogami and Novak and leads to a weaker dependence of soil response to Poisson's ratio which is in agreement with numerical solutions found in literature. The stress and displacement fields in the soil and the associated reaction to an arbitrary harmonic pile displacement are derived analytically using pertinent displacement potentials and eigenvalue expansions over the vertical coordinate. Both infinitely long piles and piles of finite length are considered. Results are presented in terms of dimensionless parameters and graphs that highlight salient aspects of the problem. A detailed discussion on wave propagation and cutoff frequencies based on the analytical findings is provided. A new dimensionless frequency parameter is introduced to demonstrate that the popular plane-strain model yields realistic values for soil reaction only at high frequencies and low Poisson's ratios.Published by Elsevier Ltd.

The analytical representation of dynamic soil reaction to a laterally-loaded pile using 3D continuum modelling is revisited. The governing elastodynamic Navier equations are simplified by setting the dynamic vertical normal stresses in the soil equal to zero, which uncouples the equilibrium in vertical and horizontal directions and allows a closed-form solution to be obtained. This physically motivated approximation, correctly conforming to the existence of a free surface, was not exploited in earlier studies by Tajimi, Nogami and Novak and leads to a weaker dependence of soil response to Poisson's ratio which is in agreement with numerical solutions found in literature. The stress and displacement fields in the soil and the associated reaction to an arbitrary harmonic pile displacement are derived analytically using pertinent displacement potentials and eigenvalue expansions over the vertical coordinate. Both infinitely long piles and piles of finite length are considered. Results are presented in terms of dimensionless parameters and graphs that highlight salient aspects of the problem. A detailed discussion on wave propagation and cutoff frequencies based on the analytical findings is provided. A new dimensionless frequency parameter is introduced to demonstrate that the popular plane-strain model yields realistic values for soil reaction only at high frequencies and low Poisson's ratios.