Dynamic stiffness of pile in a layered elastic continuum

A set of formulas for the dynamic stiffness of a pile (spring and dashpot coefficients) to use in inertial interaction analysis is proposed, utilising elastodynamic solutions. The method is based on solving a Lagrangian system of coupled equations for the pile and the soil motions for a range of vibration frequencies and also by considering the vertical, radial and angular stresses on the pile–soil interface. The solution extensively uses Bessel functions of the second kind and results are compared with finite-element models and field pile load tests. A dimensionless frequency related to the well-known active length of pile is proposed to separate inertial and kinematic interactions. A formula is also proposed for estimation of the active length of a pile in a two-layered soil. A specific depth is introduced beyond which soil layering does not have any appreciable effects on dynamic stiffness. It is commonly (rather arbitrarily) assumed that the first natural frequency of soil strata differentiates radiation (geometric) damping from hysteretic (material) damping for both types of interactions of the pile–soil system. In contrast, this paper proposes a new formulation based on relative pile–soil stiffness and frequency of the pile head loading to differentiate these two classes of damping behaviour. The application of the formulation is shown through an example.