Behaviour of Single Piles under Axial Loading
Analysis of Settlement and Load Distribution


Joana Gonçalves Sumares Betencourt Ribeiro


Thesis to obtain the Master of Science Degree in
Civil Engineering


Examination Committee
Chairperson:
Prof. José Manuel Matos Noronha da Câmara
Supervisor:
Prof. Jaime Alberto dos Santos
Member of the Committee:
Prof. Peter John Bourne-Webb


May 2013

To my parents
Aos meus pais




























May the God of hope fill you with all joy and peace in believing,
so that by the power of the Holy Spirit you may abound in hope.
Romans 15:13


Que o Deus da esperança vos mantenha felizes e cheios da paz que nasce pela fé,
para que abundeis na esperança pelo poder do Espírito Santo.
Romanos 15:13









ABSTRACT
The purpose of this thesis is to analyse the behaviour of single piles under axial loading, as far as
settlement and load transfer mechanisms are concerned.
It includes a literary review of two elastic theory-based methods, the Poulos and Davis method and the
Randolph and Wroth method, as well as axisymmetric elastic modelling using the finite element-based
program Plaxis. The results given by each method are organized in dimensionless charts of load-
settlement ratio and proportion of load transferred to the pile base in terms of the pile slenderness
ratio and the soil inhomogeneity and compared amongst each other.
This thesis also includes two comparison studies which involve axisymmetric elastoplastic modelling in
Plaxis, considering the Mohr-Coulomb failure criterion. The first one is a comparison with two previous
finite element simulations subject to very similar conditions: a GEFdyn 3D simulation and a CESAR-
LCPC 2D axisymmetric simulation. It showed that in the simulation performed by Plaxis the value of
load transferred to the pile base was lower than the others, although the total load-settlement curve
was very similar in all three cases. The second one is a case study of the simulation of a static load
test performed on a test pile. It includes the geological description of the site and the justification of the
choice of parameters introduced in the model. Although limited information is available regarding the
geological and geotechnical conditions of the site, the overall results were quite satisfactory.

Key words: axially loaded pile, numerical modelling, soil-pile interaction, Poulos and Davis, Randolph
and Wroth







RESUMO
O objectivo desta tese é analisar o comportamento de estacas isoladas sob carga axial, em termos de
assentamento e mecanismos de transferência de carga.
Inclui uma revisão literária de dois métodos baseados na teoria da elasticidade, o método de Poulos e
Davis e o método de Randolph e Wroth, bem como a modelação elástica em estado axissimétrico
utilizando o programa de elementos finitos Plaxis. Os resultados obtidos a partir destes métodos
estão organizados em ábacos da relação carga-assentamento e carga da ponta-carga total em
termos do factor de esbelteza da estaca e da variação de rigidez do solo, e são comparados entre si.
Esta tese inclui também dois estudos de comparação que incluem modelação elastoplástica em
estado axissimétrico utilizando Plaxis, considerando o critério de cedência de Mohr-Coulomb. O
primeiro estudo trata-se de uma comparação com duas simulações numéricas realizadas
anteriormente em condições idênticas: uma simulação 3D utilizando GEDdyn e uma simulação em
estado axissimétrico utilizando CESAR-LCPC. Este estudo mostra que na simulação em Plaxis o
valor da carga transferida para a base era inferior ao das outras simulações, apesar de a curva de
carga no topo-assentamento no topo ser semelhante nos três casos. O segundo é um caso de estudo
que consiste na simulação de um ensaio de carga estático executado numa estaca experimental.
Inclui uma descrição geológica do local e a justificação da escolha de parâmetros introduzidos no
modelo. Apesar da pouca informação disponível relativa às condições geológicas e geotécnicas do
local, os resultados foram bastante satisfatórios.

Palavras-chave: estacas sob carga axial, modelação numérica, interacção solo-estaca, Poulos e
Davis, Randolph e Wroth





ACKNOWLEDGEMENTS

Firstly, I want to thank Professor Jaime Alberto dos Santos, for proposing this theme and guiding me
through my research. I appreciate the confidence placed in me by permitting that I develop each
chapter according to my will, allowing me to learn for myself. I also acknowledge the transmission of
the value of discipline, organization and patience at work – I will not forget the importance of finishing
a task before beginning the next one.
Secondly, I want to thank João Camões, for answering my every question so promptly and thoroughly.
His help with working with finite elements was priceless, teaching me how to find solutions
methodically and observing details critically. He explained to me how useful a tool like Plaxis can be,
and also how easily it can become a “black hole”. I will never forget that “no model works well the first
time you run it”. I wish him all the luck with his career, and I know he will excel at whatever he
attempts.
The gratitude I feel towards my parents cannot be expressed. I thank my father for encouraging me to
pursue this career and setting up a fine professional example; he taught me that “to be an engineer is
not a job, it is a way of life”. I thank my mother for listening with infinite patience and being an endless
source of care and support. I thank both for providing me with the best conditions to study and learn
and never pressing or demanding anything in return. There is nothing more a daughter can ask.
I also thank my grandparents, my brothers and the rest of my family, for the good environment I grew
up in, and for allowing me to discharge whenever I go home. I particularly thank my uncle António
Carlos, for clarifying my doubts whenever I needed; I hope to have inherited at least some of his talent
for engineering.
I thank Marta for being the sister I never had, my most trustworthy friend. I thank her for always being
there, and for so often disregarding her own work to help me with mine. We shared almost everything
the last few years, the very best and the very worst moments. There is no one I would rather have
spent this time with.
I thank all my friends and colleagues who have in one way or another helped during university years,
especially Francisco, Caldinhas, António, Miguel Melo, Ana Bento, Tiago Schiappa, Margarida and
Guilherme. Because of them, Lisbon became a home to me; they made studying and working in
projects much easier and more agreeable. I am proud to belong to such a group of civil engineers, as I
am sure they will all be excellent professionals.



i

CONTENTS
1 Introduction 1
1.1 Context and Motivation 1
1.2 Objectives and Methodology 2
2 Elastic Theory-Based Methods for Analysis of Single Axially Loaded Piles 3
2.1 Introduction 3
2.2 Poulos and Davis Method 6
2.3 Randolph and Wroth Method 16
3 Comparison between Elastic Theory-Based Methods and the Finite Element Method in the
Analysis of Single Axially Loaded Piles 33
3.1 Introduction 33
3.2 The Model 34
3.2.1 Geometry 34
3.2.2 Loading 36
3.2.3 Materials and Interfaces 37
3.2.4 The Mesh 38
3.3 Initial Conditions 38
3.4 Calculations and Results 38
4 Numerical Validation of Elastoplastic Modelling of a Single Axially Loaded Pile 47
4.1 Introduction 47
4.2 The Model 48
4.2.1 Geometry 48
4.2.2 Loading 48
4.2.3 Materials and Interfaces 49
4.2.4 The Mesh 52
4.3 Initial Conditions 52
4.3.1 Water pressure generation 52

4.3.2 Initial stress field generation 53
4.3.3 Introduction of the Pile 55
4.4 Calculation and Results 56
4.4.1 Sensitivity Analysis of the Interface 56
4.4.2 Plaxis Results Compared to CESAR-LCPC and GEFdyn Results 57
4.4.3 Analysis Including Soil Dilatancy 59
5 Case Study of a Static Load Test Performed on a Test Pile 61
5.1 Introduction 61
5.2 Geological Characterization 62
5.3 The Static Load Test (SLT) 63
ii

5.3.1 Preparation 63
5.3.2 Procedure 64
5.3.3 Instrumentation 67
5.3.4 Results 68
5.4 Numerical Analysis by Plaxis 72
5.4.1 The Model 72
5.4.2 Initial Conditions 79
5.4.3 Calculations and Results 83
6 Concluding Remarks 91
6.1 Conclusions 91
6.2 Further Research 93
Bibliography 95
References 95
Consulted Bibliography 96
Appendixes 97
Appendix A – Dimension of the Model Used in the Elastic Simulations 97
Appendix B1 – Load Settlement Ratio in Terms of the Pile Slenderness Ratio 98
Appendix B2 – Proportion of Load Transferred to the Pile Base in Terms of the Pile Slenderness

Ratio 107
Appendix C – Load Settlement Curves Determined in the Elastoplastic Modelling for Numerical
Validation 116
Appendix D1 – Load Settlement Curve Determined in the Elastoplastic Modelling for Comparison
with the Static Load Test 120
Appendix D2 – Normal Stress along the Pile in the Elastoplastic Modelling for Comparison with the
Static Load Test 121
Appendix D3 – Shaft Load Curves Determined in the Elastoplastic Modelling for Comparison with
the Static Load Test 122
Appendix D4 – Load Settlement Curves Determined in the Elastoplastic Modelling for Comparison
with the Static Load Test 123

iii

LIST OF FIGURES
Figure 2.1: (a) Stresses acting in the soil adjacent to the pile; (b) Stresses acting on the pile; (c)
Stresses acting on a division of the pile. Adapted from (Poulos & Davis, 1980), p. 75. 4
Figure 2.2: Pile under axial loading – relevant parameters 5
Figure 2.3: Settlement-influence factor for a rigid pile in a semi-infinite incompressible soil, I
0
, in terms
of the pile slenderness ratio, L/d, and of the relation between the base and the shaft diameters, db/d.
(Poulos & Davis, 1980), p.89. 8
Figure 2.4: Correction factor for the pile compressibility, R
k
: in terms of the relation between the pile’s
and the soil’s Young’s modulus, K, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.89.
9
Figure 2.5: Correction factor for the finite depth of the layer on a rigid base, R
h

, in terms of the relation
between the total depth of the soil layer and the length of the pile, h/L, and of the pile slenderness
ratio, L/d. (Poulos & Davis, 1980), p.89. 9
Figure 2.6: Correction factor for the Poisson’s ratio of the soil, R
ν
, in terms of the soil’s Poisson’s
coefficient, ν, and of the relation between the pile’s and the soil’s Young’s modulus, K. (Poulos &
Davis, 1980), p.89. 10
Figure 2.7: Tip-load proportion for incompressible pile in uniform half-space, β
0
, in terms of the pile
slenderness ratio, L/d, and of the relation between the base and the shaft diameters, db/d. (Poulos &
Davis, 1980), p.86. 10
Figure 2.8: Correction factor for pile compressibility, C
k
, in terms of relation between the pile and the
soil’s Young’s modulus, K, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.86. 11
Figure 2.9: Correction factor for pile compressibility, C
ν
, in terms of the soil’s Poisson’s coefficient, ν,
and of the relation between the pile’s and the soil’s Young’s modulus, K. (Poulos & Davis, 1980), p.86.
11
Figure 2.10: Load settlement ratio in terms of the pile slenderness ratio, for different inhomogeneity
factors and λ=975, according to the Poulos and Davis method. 14
Figure 2.11: Proportion of the load taken by the pile base in terms of the pile slenderness ratio, for
different inhomogeneity factors and λ=975, according to the Poulos and Davis method. 15
Figure 2.12: (a) Upper and lower soil layers; (b) Separate deformation patters of the upper and lower
soil layers. Adapted from (Randolph & Wroth, 1978), p.1469. 16
Figure 2.13: Hypothetical variation of the radius of influence of the pile, r
m

. Adapted from (Randolph &
Wroth, 1978), p. 1471. 18
Figure 2.14: Load settlement ratio in terms of the pile slenderness ratio for rigid piles, for different
inhomogeneity factors, according to the Randolph and Wroth method. 25
Figure 2.15: Load settlement ratio in terms of the pile slenderness ratio for compressible piles, for
different inhomogeneity factors and λ=975, according to the Randolph and Wroth Method. 26
iv

Figure 2.16: Load settlement ratio in terms of the pile slenderness ratio for rigid and compressible
piles (λ=975), for different inhomogeneity factors, according to the Randolph and Wroth method. 27
Figure 2.17: Load settlement ratio in terms of the pile slenderness ratio for rigid and compressible
piles, for different inhomogeneity factors and soil-pile stiffness ratios, according to the Randolph and
Wroth method. 28
Figure 2.18: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid
piles, for different inhomogeneity factors, according to the Randolph and Wroth method. 29
Figure 2.19: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for
compressible piles, for different inhomogeneity factors and λ=975, according to the Randolph and
Wroth method. 30
Figure 2.20: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid
and compressible piles, for different inhomogeneity factors and λ=975, according to the Randolph and
Wroth method. 31
Figure 2.21: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid
piles, for different inhomogeneity factors and soil-pile stiffness ratios, according to the Randolph and
Wroth method. 32
Figure 3.1: Geometry of the model. 35
Figure 3.2: Geometry of the model, L=20m. 35
Figure 3.3: (a) Point load at the centre; (b) Point load on the side; (c) Distributed load; (d) Prescribed
displacements. 36
Figure 3.4: Vertical displacement field, for 1m prescribed displacement at the pile top, L=20m. 38
Figure 3.5: Vertical stress field, for 1m prescribed displacement at the pile top, L=20m 39

Figure 3.6: Vertical stress field near the pile base, for 1m prescribed displacement at the pile top,
L=20m. 39
Figure 3.7: Normal stress diagram at the pile top, for 1m prescribed displacement at the pile top,
L=20m. 40
Figure 3.8: Normal stress diagram at the pile base, for 1m prescribed displacement at the pile top,
L=20m – both at the pile and at the soil side of the interface. 40
Figure 3.9: Load settlement ratio in terms of the pile slenderness ratio for homogeneous soils (ρ=1)
and λ=975. 41
Figure 3.10: Load settlement ratio in terms of the pile slenderness ratio for inhomogeneity factor
ρ=0,75 and λ=975. 42
Figure 3.11: Load settlement ratio in terms of the pile slenderness ratio for inhomogeneity factor ρ=0,5
and λ=975. 43
Figure 3.12: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for
homogeneous soils (ρ=1) and λ=975. 44
Figure 3.13: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for
inhomogeneity factor ρ=0,75 and λ=975. 45
Figure 3.14: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for
inhomogeneity factor ρ=0,5 and λ=975. 46
v

Figure 4.1: Geometry of the model. 48
Figure 4.2: Distribution of nodes in interface elements and respective connection to 15-node triangular
elements. Adapted from Plaxis Manual (Brinkgreve, 2006). 49
Figure 4.3: Mesh. 52
Figure 4.4: Diagram of pore pressure. 53
Figure 4.5: Initial vertical effective stress field, not including the pile. 54
Figure 4.6: Initial horizontal effective stress field, not including the pile. 55
Figure 4.7: Total load, load transmitted to the pile shaft and load transmitted to the pile base in terms
of the settlement at the pile top, for smooth and rough interfaces. 56
Figure 4.8: Total load, load transferred to the pile shaft and load transferred to the pile base in terms of

the settlement at the pile top, obtained through Plaxis and GEFdyn. 57
Figure 4.9: Total load, load transferred to the pile shaft and load transferred to the pile base in terms of
the settlement at the pile top, obtained through Plaxis (for smooth and rough interfaces) and GEFdyn.
58
Figure 4.10: Plastic points due to the Mohr-Coulomb criterion at the pile base for s=8mm. 58
Figure 4.11: Total load, load transferred to the pile shaft and load transferred to the pile base in terms
of the settlement at the pile top, obtained through Plaxis (including soil dilatancy at the base) and
GEFdyn 59
Figure 5.1: Geological profile where the static load test was performed. 62
Figure 5.2: Pile and soil layers. Adapted from (Santos, 2005). 63
Figure 5.3: (a) Driving of the temporary casing; (b) Welding of the casing; (c) Introduction of the
reinforcing cage. (Viaponte, 2003). 64
Figure 5.4: Reaction system (Viaponte, 2003). 64
Figure 5.5: Loading plan. Adapted from (Santos, 2005). 65
Figure 5.6: Vibrating wire extensometer welded to the reinforcing cage (Viaponte, 2003). 67
Figure 5.7: Depth of each level of extensometers. Adapted from (Santos, 2005). 68
Figure 5.8: Load at the pile top, measured by pressure gauges, in terms of time in minutes. Adapted
from (Santos, 2005). 69
Figure 5.9: Measured load at the pile top in terms of the measured settlement of the pile top. Adapted
from (Santos, 2005). 69
Figure 5.10: Normal stress at different levels along the pile (Santos, 2005). 70
Figure 5.11: Normal stress along the pile shaft measured for load steps 4 and 19. Adapted from
(Santos, 2005). 70
Figure 5.12: Lateral stress between different levels along the pile (Santos, 2005). 71
Figure 5.13: Model geometry. 72
Figure 5.14: Mesh. 79
Figure 5.15: Diagram of pore pressure. 80
Figure 5.16: Initial vertical effective stress field, including removed soil and not including the pile. 80
Figure 5.17: Initial horizontal effective stress field, including removed soil but not the pile. 81
Figure 5.18: Total displacements after the removal of soil at the top. 82

vi

Figure 5.19: Load at the pile top in terms of the total settlement at the pile top, given by Plaxis. 83
Figure 5.20: Load at the pile top in terms of the total settlement at the pile top, given by the SLT and
Plaxis, for the first loading cycle. 85
Figure 5.21: Load at the pile top in terms of the total settlement at the pile top, given by the SLT and
Plaxis, for the second loading cycle. 86
Figure 5.22: Load at the pile top in terms of the total settlement at the pile top, given by the SLT and
Plaxis, for both loading cycles. 86
Figure 5.23: Normal stress along the pile shaft at the load peaks (steps 4 and 19), given by the SLT
and Plaxis. 87
Figure 5.24: Shaft load between different levels along the pile and total applied load, obtained by
Plaxis. 88
Figure 5.25 Shaft load by layer of soil and total applied load, obtained by Plaxis. 88
Figure 5.26: Total load, load transferred to the pile shaft and load transferred to the pile base in terms
of the settlement at the pile top for the second loading cycle, obtained through Plaxis. 90

vii

LIST OF TABLES
Table 2.1: Limit pile slenderness ratio between rigid and compressible piles, for different values of the
soil-pile stiffness factor, according to (Fleming, 1992). 28
Table 3.1: Material Properties. 37
Table 3.2: Young’s Modulus of the soil, E. 37
Table 4.1: Material properties (Neves, 2001a). 49
Table 4.2: Interface properties used in the CÉSAR-LCPC and in the GEFdyn simulations. 50
Table 4.3: Interface properties used in the Plaxis simulation. 51
Table 5.1: Loading plan. Adapted from (Santos, 2005). 66
Table 5.2: Depth of each level of extensometers. 67
Table 5.3: Mobilized shaft load for load steps 4 and 19. Adapted from (Santos, 2005). 71

Table 5.4: Pile properties 73
Table 5.5: Soil properties 73
Table 5.6: Pile properties (2). 74
Table 5.7: Soil properties (2). 77
Table 5.8: Analytical base resistance, R
b
. 77
Table 5.9: Analytical shaft resistance, R
s
. 78
Table 5.10: Shaft load for load step 4. 89
Table 5.11: Shaft load for load step 19. 89
Table 5.12: Poulos and Davis estimation. 84
Table 5.13: Randolph and Wroth (compressible piles) estimation. 84

viii


ix

SYMBOLS
Latin alphabet

A: area of the cross section of the pile
A
b
: area of the pile base
A
s
: area of the pile shaft

C
K
: correction factor for pile compressibility
C
ν
: correction factor for Poisson’s ratio of soil
c: cohesion of the soil
c
u
: undrained strength of the soil


: average undrained resistance along the pile shaft
d: diameter of the pile shaft
d
b
: diameter of the pile base
E: Young’s modulus of the soil
E
interface:
Young’s modulus of the soil-pile interface
E
L/2
: Young’s modulus of the soil at the middle of the pile
E
L
: Young’s modulus of the soil at the pile base
E
p
: Young’s modulus of the pile

E
oed
: Young’s modulus of the soil for oedometer loading conditions
E
oed, interface
: Young’s modulus of the soil-pile interface for oedometer loading conditions
G: shear modulus of the soil
G
interface:
shear modulus of the soil-pile interface
G
L/2
: shear modulus of the soil at the middle of the pile
G
L
: shear modulus of the soil at the pile base
h: total depth of the soil layer, i. e. distance between the soil surface and the rigid layer
I: coefficient used in the calculation of the total settlement of the pile
I
0
: settlement-influence factor for a rigid pile in a semi-infinite incompressible soil (ν=0.5)
I
b
: vertical displacement factor for a pile element due to the normal stress at the pile base
I
s
: vertical displacement factor for a pile element due to the shear stress at the pile shaft
K: relation between the pile’s and the soil’s Young’s modulus
K
s

: earth pressure coefficient
L: length of the pile
N
c
: end-bearing capacity factor
P
b
: load transferred to the pile base
P
s
: load transferred to the pile shaft
P
t
: total (applied) load
r: horizontal distance to the pile axis
x

r
0
: radius of the pile shaft
r
m
: radius of influence of the pile, i.e. maximum distance past which shear stress is negligible
R
b
: resistance of the pile base
R
c
: total resistance of a pile under compression
R

h
: correction factor for finite depth of layer on a rigid base
R
k
: correction factor for pile compressibility
R
s
: resistance of the pile shaft
R
ν
: correction factor for the Poisson’s ratio of the soil (when ν<0.5)
R
inter
: interface resistance factor
t
interface
: thickness of the soil-pile interface
u: radial displacement of the soil
w: vertical displacement of the soil
w
b
: vertical displacement of the pile base
w
s
: settlement of the pile shaft
w
t
: total vertical displacement (settlement) of the pile
x: horizontal coordinate
z: vertical coordinate (depth)



Greek alphabet

α: adhesion factor of the pile shaft
β
0
: tip-load proportion for incompressible pile in uniform half-space (ν=0.5)
γ: shear strain
γ: specific weight
δ: friction angle of the soil-pile interface
ζ: relation between the radius of influence of the pile and the radius of the pile shaft
η: depth factor of interaction between layers
λ: soil-pile stiffness ratio
ν: Poisson’s ratio of the soil
ρ: inhomogeneity factor of the soil
σ: normal stress
σ
0
: normal stress at the pile base
σ
z
: vertical stress






: average effective vertical stress along the pile shaft

τ: shear stress
τ
0
: shear stress at the pile shaft
φ’: effective friction angle of the soil


xi

ACRONYMS
CPTU: Piezocone Test
FVT: Field Vane Test
GEFdyn : Géo Mécanique Eléments Finis Dynamique
P&D: Poulos and Davis
R&W: Randolph and Wroth
SCPTU: Seismic Cone Penetration Test
SLT: Static Load Test
SPT: Standard Penetration Test


xii



1

1 INTRODUCTION
1.1 Context and Motivation
Piles are deep foundations, necessary when the bearing capacity of shallow foundations is not enough
to ensure the support of the superstructure. This superstructure results in vertical forces, due to its

weight as well as additional loads, which are axially transferred to the pile and, through its shaft and
base, to the soil, possibly reaching a stiffer layer.
The analysis of the load transfer mechanism in single piles under axial loading is therefore an
essential basis for deep foundation design. It is very important that the physical interaction between
pile and soil is carefully studied. The settlement analysis is also fundamental, for the maximum
allowable settlement of a foundation is often the most important criterion in its design. Thus, it should
be estimated accurately.
The behaviour of single piles under axial loading, as far as load distribution and settlement along the
pile are concerned, have been analysed through numerous methods. They can be divided into three
main categories, according to (Poulos & Davis, 1980):
1) Load-transfer methods, which involve a comparison between the pile resistance and the pile
movement in several points along its length;
2) Elastic theory-based methods, which employ the equations described in (Mindlin, 1936) for
surface loading within a semi-infinite mass (such as the Poulos and Davis method), or other
analytical formulations that impose compatibility between the displacements of the pile and of
the adjacent soil for each element of the pile (such as the Randolph and Wroth method);
3) Numerical methods, such as the finite element method.

Elastic theory based methods, such as the ones presented in this work, do not explain the behaviour
of the pile near failure. In this thesis, their results are used in comparison with the results of a finite
element method program, Plaxis 2D version 8. Numerical methods are powerful and very useful tools
when used carefully and calibrated with the appropriate tests. They also constitute a valuable way of
performing a sensitivity analysis of the soil parameters.