Dynamic stiffness of pile in a layered elastic continuum

Abstract

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

A numerical model for the dynamic analysis of inclined pile groups

“…Discrepancies may be consequence of the adopted damping model [30], which neglects 3D effects in the definition of a plane-strain dynamic soil-pile stiffness. However, this drawback can be overcome by implementing more sophisticated damping models, which are able to account for the different classes of damping behaviour (hysteretic and geometric), depending on the type of interaction (kinematic or inertial) and the frequency [34]. Figure 9 refers the non-dimensional stiffness and damping coefficients of the vertical impedance obtained from the same analysis cases.…”

“…However, it should be mentioned that Green's functions obtained also by other formulations may be implemented, accounting for new developments in the research. As an example, the readers may refer to the more recent solution of dynamic stiffness of flexible piles obtained by Shadlou and Bhattacharya , who calibrate spring and dashpot coefficients obtained from a rigorous two‐dimensional elastodynamic solution starting from results of finite element analyses where 3D soil–pile interaction phenomena are naturally included.…”

An analytical continuum model for axially loaded end‐bearing piles in inhomogeneous soil

“…The analysis of pile settlement is a classical problem in geotechnical and piling engineering, which has attracted research interest for a long time. Relevant numerical approaches include finite elements, [1][2][3][4][5] Green's functions and boundary element solutions, [6][7][8][9][10][11] simplified t-z models, [12][13][14][15] various hybrid formulations, [16][17][18][19][20][21] and various reviews. [22][23][24][25][26][27][28][29][30] Despite the research and the publications, important aspects of the problem such as the effect of soil inhomogeneity, pile slenderness, and pile-soil stiffness contrast on load transfer remain poorly understood, mainly because of the lack of pertinent analytical tools, which can provide insight into the physics of pile-soil interaction.…”

Comparison Between Models for the Evaluation of the Seismic Response of Offshore Wind Turbines on Deep Foundations

“…For this reason, many models with different degrees of simplifying assumptions have been developed over the last years, see e.g. [6,7,8]. Particularly attractive are one-dimensional models, which are typically cheap to run.…”