2855
Semi-Analytical Solutions for Laterally Loaded Piles in Multilayered Soils
Solutions Semi-analytiques pour des pieux soumis à des charges latérales dans les sols
multicouches
Salgado R.
1
, Basu D.
2
, Prezzi M.
1
, Tehran F.S.
1
1
Purdue University, West Lafayette, Indiana, USA
2
University of Waterloo, Waterloo, Ontario, Canada
ABSTRACT: Piles subjected to lateral forces and moments at the head are often analyzed in practice with the p-y method. However,
the p-y method is not capable of capturing the complex three-dimensional interaction between the pile and the soil. The continuum
approach is conceptually more appealing but it requires the use of numerical techniques, such as the three-dimensional (3D) finite
element (FE) method. In order to save computational time, researchers have explored the development of closed-form solutions based
on linear elasticity that can be used to obtain lateral pile deflection with depth. In this paper, we present semi-analytical methods
developed to calculate the response of laterally loaded piles with general-shape cross sections embedded in multilayered elastic soil.
The displacement field of the pile-soil system is taken to be the product of independent functions that vary in the vertical and
horizontal directions. The differential equations governing the displacements of the pile-soil system are obtained using the principle of
minimum total potential energy and calculus of variations. The input parameters needed for the analysis are the pile geometry, the soil
profile, and the elastic constants of the soil and pile. The method produces results with accuracy comparable with that of a 3D FE
analysis but requires much less computational time.
RÉSUMÉ : Les pieux soumis à des charges latérales et des moments à la tête sont souvent analysés dans la pratique par la méthode p-
y. Toutefois, cette méthode n'est pas capable de prendre en compte les interactions complexes trois dimensionnelles entre le pieu et le
sol. L'approche continue est conceptuellement plus attrayante, mais elle nécessite l'utilisation de techniques numériques, tels que les
des éléments finis (EF) en trois dimensions (3D). Afin de gagner du temps de calcul, les chercheurs ont étudié le développement des
solutions analytiques basées sur l'élasticité linéaire qui peuvent être utilisé pour évaluer les déplacements latérales du pieu avec la
profondeur. Dans cet article, nous présentons des méthodes semi-analytiques développés pour calculer la réponse des pieux chargés
latéralement avec des sections transversales en forme générale incorporés dans le sol élastique multicouche. Le champ de déplacement
du système pieu-sol est considéré comme le produit de fonctions indépendantes qui varient dans les directions verticale et horizontale.
Les équations différentielles qui régissent les déplacements du système pieu-sol sont obtenues en utilisant les principes variationnels
et le principe de l'énergie potentielle totale minimum. Les paramètres d'entrée nécessaires pour l'analyse sont la géométrie du pieu, le
profil du sol, et les constantes élastiques du sol et du pieu. Cette méthode donne des résultats d'une précision comparable à celle d'une
analyse par éléments finis 3D mais nécessite beaucoup moins de temps de calcul.
KEYWORDS: laterally loaded single piles, elastic soil, continuum method, energy principles.
1 INTRODUCTION
Piles subjected to lateral forces and moments at the head are
analyzed in practice with the p–y method (e.g., Reese and Cox,
1969). According to the p–y method, the pile is assumed to
behave as an Euler–Bernoulli beam with the soil modeled as a
series of discretely spaced springs, each connected to one of the
pile segments into which the pile is discretized. The springs
model the soil response to loading through p–y curves (p is the
unit resistance per unit pile length offered by the springs, and y
is the pile deflection), which are developed empirically by
adjusting the curves until they match actual load–displacement
results (e.g. Cox et al., 1974; Ashour & Norris, 2000). However,
the p–y method often fails to predict pile response (Anderson et
al., 2003; Kim et al., 2004), for it is not capable of capturing the
complex three-dimensional interaction between the pile and the
soil.
The continuum approach is conceptually more appealing;
however, in order to model the soil as a continuum, the use of
numerical techniques such as the three-dimensional (3D) finite
element (FE) method, finite elements with Fourier analysis, the
boundary element (BE) method or the finite difference (FD)
method is often required (Poulos, 1971a, 1971b; Banerjee &
Davis, 1978; Randolph, 1981). The 3D FE or FD methods can
capture the most important features of the complex pile–soil
interaction, but three-dimensional analyses are computationally
expensive for routine practice.
In this paper, an analysis is developed, based on variational
principles, by which the deflection, slope of the deflected curve,
bending moment and shear force of laterally loaded piles with
rectangular and circular cross section can be obtained. A multi-
layered, elastic soil deposit is considered. The analysis captures
the 3D pile–soil interaction and produces pile response using
closed–form solutions. As a consequence of the analysis, the
lateral response of piles can be obtained with a degree of rigor
and speed not previously possible. The method can be extended
to capture the non-linear pile response due to soil non-linearity.
2 PROBLEM DEFINITION
A circular cross-section pile with radius
r
p
and a rectangular
cross-section pile with cross sectional dimensions 2
a
× 2
b
and
both with length
L
p
embedded in a multilayered soil profile are
considered (Figure 1). Each soil layer extends to infinity in all
horizontal directions, and the bottom layer extends to infinity in
the downward direction. The pile is subjected to a horizontal
force
F
a
and a moment
M
a
at the pile head. The goal of the
analysis is to obtain pile deflection as a function of depth caused
by the action of
F
a
and/or
M
a
at the pile head.
