Load Transfers During Vibratory Driving

Abstract

The vibratory driving technique consists in applying a vibratory load onto a profile to reduce the ground resistance and allow penetration of the profile under its own weight. The vibratory action is produced by counter-rotating eccentric masses actuated within the exciter block. A proper definition of this mechanical action is fundamental for vibratory driving analyses. The vibratory force transferred from the vibrator onto the pile during vibratory driving is however generally neither well defined nor understood, in particular when using simplified closed form solutions for the analysis of pile driving. Few authors have pointed out the very low ratio observed between the force measured in the pile and the nominal inertial force developed by the eccentrics, but without offering a theoretical framework to explain and predict this low ratio. The objective of this paper is to develop a better understanding of the so-called ‘efficiency factor’ of the vibratory driving process. Analytical solutions are presented, along with more advanced numerical simulations. Theoretical solutions are illustrated with reference to field measurements collected at different test sites.

Appendix

In the FE model, the pile is discretized into a limited number of two noded elements, with one axial translation degree of freedom per node. The basic equations of motion, in their general form, can be written

$$ \left[ M \right]\frac{{\partial^{2} }}{{\partial t^{2} }}\left\{ U \right\} + \left[ C \right]\frac{\partial }{\partial t}\left\{ U \right\} + \left[ K \right]\left\{ U \right\} = \left\{ F \right\} $$

(24)

where [M], [C], [K] are the mass, damping and stiffness matrices of the vibrator-pile-soil system; {U} is a vector containing the degrees of freedom and {F} is the loading vector. When considering constant value linear parameters, direct solutions of Eq. (20) can be found by decomposing the displacement and force vectors as

$$ U = \hat{U}e^{i\omega t} $$

(25)

$$ F = \hat{F}e^{i\omega t} $$

(26)

where $\hat{U}$, $\hat{F}$ are the displacement and force amplitudes; and e ^iωt expresses the harmonic time dependency of the signals. The equations of motion can then be written:

$$ \left( { - \omega^{2} \left[ M \right] + i\omega \left[ C \right] + \left[ K \right]} \right)\left\{ {\hat{U}} \right\} = \left\{ {\hat{F}} \right\} $$

(27)

Shape functions are used to interpolate the movement of each element composing the pile as a function of the node movements. To describe the axial movement, first degree polynomials can be used:

$$ \hat{s}_{p} (z) = \frac{1}{2}\left[ {\hat{s}_{p} (z_{i} ) + \hat{s}_{p} (z_{i + 1} )} \right] $$

(28)

with z _i and z _i+1 the depth of the pile element ends. Once the pile movements have been calculated, the internal normal forces for each discrete pile element are given by:

$$ F^{e} = K^{e} U^{e} $$

(29)

where the superscript ‘e’ denotes the element matrices.