Elastodynamic analysis of laterally loaded piles: Modifications to a simplified energy approach

A simple, yet powerful analytical model for determining the dynamic response of laterally loaded piles has recently been proposed in a series of papers by Karatzia and Mylonakis. The method is essentially a finite-element formulation, based on the evaluation of a set of energy integrals to establish the dynamic stiffness and damping matrices at the pile head. The key ideas/assumptions behind the method are: (1) the soil around the pile is replaced by a bed of dynamic Winkler springs and dashpots accounting for soil stiffness and energy dissipation; (2) a shape function for pile deflection is employed along the whole pile length in each vibration mode; (3) the associated integrals can be solved in closed form. Solutions have been obtained for different soil profiles which provided realistic predictions of pile response to flexural loads. An implicit assumption of the method lies in the use of real-valued shape functions. While such functions greatly simplify the analysis by separating real and imaginary parts (thus leading exclusively to real-valued integrals), they have the disadvantage of ignoring the phase differences between pile movements at different depths. This paper recognises that this simplification can lead to inaccurate results for inhomogeneous soil profiles, especially at high frequencies where phase differences among different points along the pile are significant. To overcome the problem, the possibility of using complex-valued shape functions, analogous to those employed in spectral finite-element methods is explored. It is shown, through comparison with rigorous numerical solutions (including FEM and BEM), that use of complex-valued shape functions improves the predictive power of the method.