2679
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
1
A Variational Approach for Analysis of Piles Subjected to Torsion
Une approche variationnelle pour l'analyse des pieux soumis à torsion
D. Basu & A. Misra
University of Waterloo
T. Chakraborty
Indian Institute of Technology Delhi
ABSTRACT: A framework is developed using the variational principles of mechanics for analyzing torsionally loaded piles in elastic
soil. The total potential energy of the pile-soil system is minimized to obtain the differential equations governing the pile and soil
displacements. Closed-form solutions are obtained for the angle of twist and torque in the pile as a function of depth. The analysis
explicitly takes into account the three-dimensional pile-soil interaction in multi-layered soil. The results match well with the existing
solutions and with those of equivalent finite element analyses.
RÉSUMÉ : Un cadre conceptuel est élaboré en utilisant les principles variationnels de la mécanique pour l'analyse de pieux chargés
en torsion dans un sol élastique. L'énergie potentielle totale du système pieu-sol est minimisée pour obtenir les équations
différentielles régissant le pieu et les déplacements du sol. Des solutions analytiques sont obtenues pour le couple et l'angle de torsion
dans le pieu en fonction de la profondeur. L'analyse prend en compte explicitement les interactions pieu-sol tridimensionnelles dans
un système multi-couches. Les résultats correspondent bien avec les solutions existantes ainsi qu' à celles obtenues par des analyses
par éléments finis.
KEYWORDS: pile, torsion, multi-layered soil, elastic analysis, variational principles.
1 INTRODUCTION
Piles loaded laterally are often subjected to torsion due to
eccentricities of applied lateral loads. The existing analysis
methods are mostly based on numerical techniques such as the
three-dimensional finite difference, finite element, discrete
element or boundary element methods (Poulos 1975, Dutt and
O’Neill 1983, Chow 1985, Basile 2010) although some
analytical methods also exist mostly based on the subgrade-
reaction approach (Randolph 1981, Hache and Valsangkar
1988, Rajapakse 1988, Budkowska and Szymcza 1993, Guo and
Randolph 1996, Guo et al. 2007).
In this paper, a new analytical method is developed for
torsionally loaded piles in multi-layered soil using the
variational principles of mechanics. Based on a continuum
approach, the analysis assumes a rational displacement field in
the soil surrounding the pile, and explicitly captures the three-
dimensional pile-soil interaction satisfying the compatibility
and equilibrium between the pile and soil. Closed form
solutions for the angle of twist and torque in the pile shaft are
obtained. The analysis produces accurate results if the
equivalent soil elastic modulus is correctly estimated.
2 ANALYSIS
A pile of radius
r
p
and length
L
p
embedded in a soil medium
containing
n
layers is considered (Figure 1). The pile base rests
in the
n
th
layer and the pile head is at the level of the ground
surface. The pile has a shear modulus of
G
p
and
is subjected to a
torque
T
a
at the head. The soil layers extend to infinity in all
horizontal directions and the bottom (
n
th
) layer extends to
infinity in the downward vertical direction. The bottom of any
layer
i
is at a depth of
H
i
from the ground surface; therefore, the
thickness of the
i
th
layer is
H
i
H
i
1
(note that
H
0
= 0). The soil
medium is assumed to be an elastic, isotropic continuum,
homogeneous within each layer, characterized by Lame’s
constants
s
and
G
s
. There is no slippage or separation between
the pile and the surrounding soil or between the soil layers. For
analysis, a polar (
r-
-z
) coordinate system is assumed with its
origin at the center of the pile head and
z
axis pointing
downward.
H
n
-2
H
i
H
i
-1
H
2
r
0
T
a
…
…
…
…
H
n
-1
H
1
Layer 1
Layer 2
Layer
i
Layer
n
1
Layer
n
z
2
r
p
Pile
L
p
T
a
r
0
Figure 1. Torsionally loaded pile in multilayered soil.
ri ti
l
r c f r
lysis f il s
j ct t rsion
ne approche variationnelle pour l'analyse des pieux sou is à torsion
Basu D., Misra A.
University of aterloo
Chakraborty T.
Indian Institute of Technology Delhi
