Behaviour of Single Piles under Axial Loading Hành vi của cọc đơn dưới tác động của tải dọc trục

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 theorybased methods, the Poulos and Davis method and the Randolph and Wroth method, as well as axisymmetric elastic modelling using the finite elementbased program Plaxis. The results given by each method are organized in dimensionless charts of loadsettlement 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 MohrCoulomb 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 CESARLCPC 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 loadsettlement 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.

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

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

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

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

A CKNOWLEDGEMENTS

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

C ONTENTS

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

5.3.1 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

L IST OF F IGURES

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 4Figure 2.2: Pile under axial loading – relevant parameters 5Figure 2.3: Settlement-influence factor for a rigid pile in a semi-infinite incompressible soil, I0, 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 8Figure 2.4: Correction factor for the pile compressibility, Rk: 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 9Figure 2.5: Correction factor for the finite depth of the layer on a rigid base, Rh, 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 9Figure 2.6: Correction factor for the Poisson’s ratio of the soil, Rν, 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 10Figure 2.7: Tip-load proportion for incompressible pile in uniform half-space, β0, 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 10Figure 2.8: Correction factor for pile compressibility, Ck, 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 11Figure 2.9: Correction factor for pile compressibility, Cν, 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 11Figure 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 14Figure 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 15Figure 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 16Figure 2.13: Hypothetical variation of the radius of influence of the pile, rm Adapted from (Randolph & Wroth, 1978), p 1471 18Figure 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 25Figure 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

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 27Figure 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 28Figure 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 29Figure 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 30Figure 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 31Figure 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 32Figure 3.1: Geometry of the model 35Figure 3.2: Geometry of the model, L=20m 35Figure 3.3: (a) Point load at the centre; (b) Point load on the side; (c) Distributed load; (d) Prescribed displacements 36Figure 3.4: Vertical displacement field, for 1m prescribed displacement at the pile top, L=20m 38Figure 3.5: Vertical stress field, for 1m prescribed displacement at the pile top, L=20m 39Figure 3.6: Vertical stress field near the pile base, for 1m prescribed displacement at the pile top, L=20m 39Figure 3.7: Normal stress diagram at the pile top, for 1m prescribed displacement at the pile top, L=20m 40Figure 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 40Figure 3.9: Load settlement ratio in terms of the pile slenderness ratio for homogeneous soils (ρ=1) and λ=975 41Figure 3.10: Load settlement ratio in terms of the pile slenderness ratio for inhomogeneity factor ρ=0,75 and λ=975 42Figure 3.11: Load settlement ratio in terms of the pile slenderness ratio for inhomogeneity factor ρ=0,5 and λ=975 43Figure 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 44Figure 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 45Figure 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

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

Figure 5.19: Load at the pile top in terms of the total settlement at the pile top, given by Plaxis 83Figure 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 85Figure 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 86Figure 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 86Figure 5.23: Normal stress along the pile shaft at the load peaks (steps 4 and 19), given by the SLT and Plaxis 87Figure 5.24: Shaft load between different levels along the pile and total applied load, obtained by Plaxis 88Figure 5.25 Shaft load by layer of soil and total applied load, obtained by Plaxis 88Figure 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

L IST OF T ABLES

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, Rb 77

Table 5.9: Analytical shaft resistance, Rs 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

S YMBOLS

Latin alphabet

A: area of the cross section of the pile

Ab: area of the pile base

As: area of the pile shaft

CK: correction factor for pile compressibility

Cν: correction factor for Poisson’s ratio of soil

c: cohesion of the soil

cu: undrained strength of the soil

̅ : average undrained resistance along the pile shaft

d: diameter of the pile shaft

db: diameter of the pile base

E: Young’s modulus of the soil

EL/2: Young’s modulus of the soil at the middle of the pile

EL: Young’s modulus of the soil at the pile base

Ep: Young’s modulus of the pile

Eoed: Young’s modulus of the soil for oedometer loading conditions

G: shear modulus of the soil

GL/2: shear modulus of the soil at the middle of the pile

GL: 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

I0: settlement-influence factor for a rigid pile in a semi-infinite incompressible soil (ν=0.5)

Ib: vertical displacement factor for a pile element due to the normal stress at the pile base

Is: 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

Ks: earth pressure coefficient

L: length of the pile

Nc: end-bearing capacity factor

Pb: load transferred to the pile base

Ps: load transferred to the pile shaft

Pt: total (applied) load

r0: radius of the pile shaft

rm: radius of influence of the pile, i.e maximum distance past which shear stress is negligible

Rb: resistance of the pile base

Rc: total resistance of a pile under compression

Rh: correction factor for finite depth of layer on a rigid base

Rk: correction factor for pile compressibility

Rs: resistance of the pile shaft

Rν: correction factor for the Poisson’s ratio of the soil (when ν<0.5)

u: radial displacement of the soil

w: vertical displacement of the soil

wb: vertical displacement of the pile base

ws: settlement of the pile shaft

wt: total vertical displacement (settlement) of the pile

x: horizontal coordinate

z: vertical coordinate (depth)

Greek alphabet

α: adhesion factor of the pile shaft

β0: 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

τ0: shear stress at the pile shaft

φ’: effective friction angle of the soil

A CRONYMS

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

1 I NTRODUCTION

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

1.2 Objectives and Methodology

This thesis has three main objectives:

1) To describe two elastic theory-based methods of analysis of single piles under axial loading; 2) To compare solutions given by finite element 2D elastic modelling of piles with the results of the elastic theory-based methods;

3) To perform finite element 2D elastoplastic modelling of single piles under axial loading, validating its results with former numerical simulations and finally analysing a real case study

This thesis consists of six chapters, the first being the introduction

Chapter 2 describes two elastic theory-based methods that analyse the behaviour of single piles under axial loading: the Poulos and Davis method and the Randolph and Wroth method This review focuses

in settlement and load transfer mechanisms Results are organizes in dimensionless charts

Chapter 3 presents the results of 2D axisymmetric elastic modelling of single piles under axial loading, also organized in dimensionless charts These are compared to the ones obtained from the analytical methods

Chapter 4 compares the results of 2D axisymmetric elastoplastic modelling of a single pile under vertical loading with the ones obtained by other authors under similar conditions The objective is to validate the used finite element method-based program with other similar ones

Chapter 5 presents a 2D axisymmetric elastoplastic modelling of a single pile in a real case study, simulating a static load test performed on a test pile It includes a geological description of the site and subsequent choice of the attributed soil parameters that calibrate the numerical model

Chapter 6 list the concluding remarks derived from this thesis, as well as indications on further research

2 E LASTIC T HEORY -B ASED M ETHODS FOR

A NALYSIS OF S INGLE A XIALLY L OADED

2.1 Introduction

In this thesis, two elastic theory-based methods are analysed: the Poulos and Davis method, introduced by (Poulos & Davis, 1968) and the Randolph and Wroth method, firstly described in (Randolph & Wroth, 1978)

Elastic theory-based methodsusually consist of dividing the pile into uniformly-loaded elements, as shown in Figure 2.1(a) Shear stress, τ, acts along the shaft, whereas normal stress, σ, acts on the base of the pile, as represented in Figure 2.1(b) These are assumed to be uniform in each division (as shown in Figure 2.1(c)), and the resultant is equal to the total applied load, Pt Equilibrium and compatibility between the displacements of the pile and of the soil adjacent to it are imposed for each element Usually, there is no imposition of radial compatibility of displacements between pile and soil, since it is assumed that there is no horizontal movement (du/dz=0)

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

The methods are distinct because of the different assumptions made on the distribution of shear stress along the pile It may be represented as a single point load acting on the axis of each element

or as a uniformly-loaded circular area at the centre of each element, for instance In the Poulos and Davis method, shear stress is considered to be uniformly distributed around the circumference of the pile, which has proved to be an acceptable assumption, especially for shorter piles, according to (Poulos & Davis, 1980)

For both elastic theory-based methods, a cylindrical pile is considered, of length L and diameter of the shaft d Although the possibility of the shaft diameter, d, and the diameter at the pile base, db, assuming different values is considered in both methods, that case is not analysed in this thesis Therefore, in the rest of this document, the diameter is considered to be constant along the entire pile The pile radius is represented by r0, and the cross section area by A As a general rule, the index “s” refers to the pile shaft, and the index “b” to the pile base

The soil is considered to be an ideal isotropic elastic mass, being the Young’s modulus, E, and the Poisson’s ratio, ν, its linear elastic parameters that are not influenced by the presence of the pile The total depth of the soil layer, i.e the distance between the soil surface and the rigid layer, is represented by h The Young’s modulus of the pile, Ep, is assumed to be constant The Poisson’s ratio

of the pile is generally not taken into consideration, as it has negligible effect in the overall behaviour

In fact, even the Poisson’s ratio of the soil, ν, has little effect in the end results

It is assumed that both the pile and the soil are initially stress-free, and there is no residual stress, i.e effects of installation are not taken into consideration in any way It may be argued that this is an oversimplification; actually, there are quick ways of simulating the installation, such as employing adjusted shear factor values, as stated in (Poulos & Davis, 1980) However, that is beyond the scope

The shear modulus of the soil, G, is used instead of the Young’s modulus, E, in (Randolph & Wroth, 1978) because the soil deforms primarily in shear and also because G is assumed to be unaffected by whether the load is drained or undrained The shear modulus of the soil may be obtained from the Young’s Modulus, through eq (2.1), which results from Hooke’s law of isotropic linear elasticity:

(2.1)

The relevant parameters that influence the vertical displacement, w, of a floating pile under axial loading are stated in eq (2.2):

These parameters are illustrated in Figure 2.2

It is useful to have dimensionless solutions for the pile behaviour, so as to simplify and quicken their employment Results from the methods subsequently presented are therefore arranged in dimensionless units, such as the pile slenderness ratio, L/r0, the proportion of load transferred to the pile base, Pb/Pt, and the load settlement ratio, Pt/(wtr0GL)

Figure 2.2: Pile under axial loading – relevant parameters

2.2 Poulos and Davis Method

This method, firstly presented in (Poulos & Davis, 1968), allows a quick estimation of both the proportion of load which reaches the pile base and the total settlement of a pile in the conditions described in the last section

It is necessary to determine the values of the stress acting on the pile (see Figure 2.1) that satisfy the condition of displacement compatibility As previously stated, only vertical displacement are considered In order to obtain the values of shear stress, τ, normal stress, σ, and displacement of the pile top, i.e total displacement of the pile, wt, expressions that relate vertical displacement with unknown stresses must be determined, imposing the compatibility conditions and solving the resulting equations

The vertical displacement of the soil adjacent to a pile element due to shear stress at the pile shaft is given by eq (2.3):

Where:

d: diameter of the pile shaft

Is: vertical displacement factor for the pile element due to the shear stress at the pile shaft

τ0: shear stress acting at the pile shaft

E: Young’s modulus of the soil

Considering all n pile elements, the resulting vertical displacement of the soil, wt, is provided by eq (2.4):

Where:

Ib: vertical displacement factor for the pile element due to the normal stress at the pile base

σ0: normal stress acting at the pile base

This expression is valid for piles with constant diameter The mentioned factors are determined using the integration of the (Mindlin, 1936) equations for the displacement caused by a point load within a semi-infinite mass If the presence of a rigid layer at a certain depth is to be accounted for, then the factor Is is to be altered accordingly

For calculating the displacement of the pile elements, only axial compression of the pile is considered The vertical equilibrium of a cylindrical pile is provided by eq (2.5):

Where:

σ: normal stress acting on the pile (average over the cross section)

r0: radius of the pile shaft

Eq (2.5) may be applied to the pile top, resulting in eq (2.6):

Where:

Pt: total applied load

A: area of the pile cross section

Eq (2.5) may also be applied to the pile base, resulting in eq (2.7):

However, it has been mentioned that the shear modulus, G, is used instead of the Young’s modulus,

E Thus, eq (2.8) may be rewritten as eq (2.9):

(2.9)

The difference between the general shear modulus, G, and the shear modulus at the pile base, GL, is clarified as the soil inhomogeneity is taken into consideration, further in this chapter The load settlement ratio is from now on represented as in eq (2.9): Pt/(wtr0GL)

This coefficient I is obtained by multiplying other coefficients, as shown in eq (2.10):

Where:

I0: settlement-influence factor for a rigid pile in a semi-infinite incompressible soil (ν=0.5)

Rk: correction factor for pile compressibility

Rh: correction factor for finite depth of layer on a rigid base

Rν: correction factor for the Poisson’s ratio of the soil (when ν<0.5)

The values of I0, Rk, Rh and Rν are plotted in Figure 2.3, Figure 2.4, Figure 2.5 and Figure 2.6, respectively

Figure 2.3: Settlement-influence factor for a rigid pile in a semi-infinite incompressible soil, I0, 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

= 1

L

Figure 2.4: Correction factor for the pile compressibility, Rk: 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

The correction factor for the pile compressibility, Rk, is function of the relation between the pile’s Young’s modulus, Ep, and the soil’s, E This relation is represented by K, as shown in eq (2.11):

The more relatively compressible the pile, the smaller the value of K

Figure 2.5: Correction factor for the finite depth of the layer on a rigid base, Rh, 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

L

L

Figure 2.6: Correction factor for the Poisson’s ratio of the soil, Rν, 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

Figure 2.6 confirms the previous statement that the Poisson’s ratio of the soil, ν, has no great influence in the total settlement of the pile, since the correction factor Rν varies between 0.8 and 1.0, for normal cases

The proportion of load transferred to the pile base, Pb/Pt, for a floating pile may be calculated by eq (2.12), first presented in (Poulos, 1972):

(2.12)

Where:

β0: tip-load proportion for incompressible pile in uniform half-space (ν=0.5)

CK: correction factor for the pile compressibility

Cν: correction factor for the Poisson’s ratio of the soil

The values of β0, CK and Cν are plotted in Figure 2.7, Figure 2.8 and Figure 2.9, respectively

Figure 2.7: Tip-load proportion for incompressible pile in uniform half-space, β0, 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

L

ν

Figure 2.8: Correction factor for pile compressibility, Ck, 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

The pile’s compressibility has the effect of decreasing the load transferred to the tip On the other hand, the load transferred to the tip tends to increase with the relative stiffness of this stratum, and this

is more pronounced for slender piles

Figure 2.9: Correction factor for pile compressibility, Cν, 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

The distance to a rigid layer, h, is not present nor has any influence on any term of eq (2.12) In fact, the proportion of load which reaches the pile base is not greatly affected by it, when its value is higher than 2L, according to (Poulos & Davis, 1980) This must be taken into consideration when comparing results provided by different methods – if the depth h is not to be accounted for, then the limit of 2L must be respected

There are obviously other factors that may have influence on the proportion of load that reaches the pile tip, such as the presence of a pile cap resting on the soil surface, or of enlarged bulbs along the pile, but they are beyond the scope of this study

ν

L

A layered or a vertically non-homogeneous soil may also be analysed by using the equations in (Mindlin, 1936) for a uniform mass, if approximate values of Young’s modulus and Poisson’s ratio at various points along the pile are employed

Thus, the stress distribution is assumed to be unaltered, as if the soil was homogeneous, but the soil displacement at a point adjacent to the pile is function of the soil’s Young’s modulus at that point The result of the soil-displacement equation changes (see eqs (2.3) and (2.4)), but not the pile-displacement’s one

An average Young’s modulus may be calculated using eq (2.13):

n: number of layers/divisions of the soil

This may be used when the soil is divided into different layers but the Young’s modulus does not vary much In those cases, the solution may be calculated with this new value of the Young’s modulus and

is very close to the one provided by the finite element method (errors inferior to 15%), according to (Poulos & Davis, 1980) This approach is an approximation, but its solution is considered to be accurate enough for practical purposes It must not be forgotten, however, that this is an approximation and that it does not provide an accurate solution of the load or settlement distribution along the pile Only the total values are considered relevant The variations in the Poisson’s ratio along the depth may be ignored, since, as discussed before, this parameter has little influence in the total settlement of the pile

A relevant form of soil non-homogeneity is one in which the shear modulus varies linearly with depth

A measure of this variation is the inhomogeneity factor, ρ; it is calculated through eq (2.14):

In Figure 2.10, results given by eq (2.9) are plotted, for different values of the soil inhomogeneity factor, ρ The value of Poisson’s coefficient of the soil, ν, is 0.3 The radius of the pile, r0, is equal to the unity in every case, for simplification reasons The length of the pile, L, varies between 4m and 100m The rigid layer is assumed to be at a distance of 2.5L of the surface (h) The Young’s modulus

at the pile base, EL, is equal to 80×103 kPa in every case – its value varying in the rest of the soil according to ρ

This has very little influence in the overall results, since the charts are normalized for the soil rigidity; thus, it only affects the value of K K assumes the values of 375, 500 and 750 for ρ=1, ρ=0.75 and ρ=0.5, respectively The reason why it is not given a constant value is that it is not possible, if the said values of soil inhomogeneity are to be tested and simultaneously the shear modulus at the pile base,

GL, is to be the same in all cases, for K is a relation between the pile’s Young’s modulus and the average shear modulus along the shaft Nevertheless, K has not great influence either over the load settlement ratio or over the proportion of load that is transferred to the pile base for normal values of

Ep and GL Besides, in (Randolph & Wroth, 1978) a similar relation is presented, the soil-pile stiffness ratio, λ, calculated through eq (2.15):

The chart in Figure 2.10 was built according to non-linear functions created from the few exact points given by Figure 2.3 to Figure 2.6, since the intermediate values cannot be interpolated linearly Therefore, the load settlement ratio was calculated for each natural number of pile slenderness between 4 and 100 All those values are grouped in tables in Appendix B1

Figure 2.10: Load settlement ratio in terms of the pile slenderness ratio, for different inhomogeneity factors, ν=0.3,

h=2.5L and λ=975, according to the Poulos and Davis method

The chart shows that the load settlement ratio increases with the pile slenderness ratio This is expected, since, when subject to the same conditions, longer piles settle less

The shapes of the three curves are very similar In fact, the inhomogeneity factor, ρ, influences the pile compressibility, K, and the expression of the load settlement ratio, eq (2.9), only; the distance between the curves increases slightly with the slenderness ratio, L/r0

The pile settlement ratio increases with the inhomogeneity factor, ρ Since the same shear modulus at the pile base is considered for the three cases, the one with the smallest ρ is the one in which the shear modulus at the middle of the pile (G, in eq (2.9)) is the smallest, i.e the average Young’s modulus Eav is the lowest Thus, according to eq (2.9) and Figure 2.4 and Figure 2.6, the load settlement ratio will also be the lowest It is expected that, the lower the ρ, the worst the results, since

eq (2.13) obviously provides a very gross approximation, the grosser the less homogeneous the soil Since there were few exact points to be extracted from the original charts, the error associated with these charts is considerable The non-linearity of these functions is the cause of irregularity of the resulting curves

In Figure 2.11, results given by eq (2.12) are plotted, for different values of the soil inhomogeneity factor, ρ The conditions of the pile and the soil are identical to the ones described for Figure 2.10 Once again, it is built from non-linear functions created from the few exact points given by Figure 2.7

to Figure 2.9 Therefore, the proportion of load transferred to the pile tip was calculated for each natural number of pile slenderness between 4 and 100 All those values are grouped in tables in Appendix B2

Figure 2.11: Proportion of the load taken by the pile base in terms of the pile slenderness ratio, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Poulos and Davis method

This chart shows that the proportion of load transferred the pile base decreases non-linearly with the slenderness ratio: its value reduces quickly for low values of L/r0, but tends to stabilize In longer piles, the shaft plays a more important role in the load transfer mechanism, given its dimension Thus, less load reaches the pile base In fact, some studies consider it to be null, for values of the pile slenderness ratio higher than a certain limit

Besides, the values of the load settlement ratio increase inversely with ρ, although very slightly In fact, the difference in the shear modulus of the soil distribution only affect Ck (Figure 2.8) and Cν

(Figure 2.9), and in this last case the change is negligible However, it is natural that, in piles with a higher value of the relation K, more load is transferred to the tip

Once again, it should not be forgotten that the information used to build this chart has come from Figure 2.7 to Figure 2.9, and so there is a significant error associated with it

Although some of these parameters are taken as independent from each other (as the measure of pile compressibility, K, and the total depth of the soil layer, h, used in the calculation of the load settlement ratio), and other factors are not considered, this method is very convenient and adequate for practical purposes

The Poulos and Davis method has proved to provide relatively good solutions, considering its simplicity Its results for the load settlement ratio are usually slightly higher than the ones given by numerical methods, i.e settlement values are lower, according to (Poulos & Davis, 1980) In Chapter

3, the pertinence of this statement is tested

2.3 Randolph and Wroth Method

This method, firstly introduced in (Randolph & Wroth, 1978), has been developed in order to explain the axial load transfer process between pile and soil It is particularly useful in cases where the soil is non-homogenous, since the previously developed methods, the Poulos and Davis method amongst them, had great limitations in that aspect

Initially, the shaft and base behaviours are studied separately An imaginary horizontal plane AB at the depth of the pile base separates base and shaft, as represented in Figure 2.12(a) Thus, it is considered that above that plane the soil deforms due to the pile shaft only, and that below the plane the soil deforms due to the pile base only, as shown in Figure 2.12(b) The deformation above and below the plane is not compatible and that allows for interaction between the upper and lower layers of soil This is a simplification which will obviously not provide the exact solution, but that has proved to

be satisfactory

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

The soil is considered to be linear elastic Thus, the effects of installation (residual stresses) are ignored As explained before, it is also assumed that the parameters of the soil are not affected by the installation of the pile

The deformation of the soil surrounding the pile is similar to shearing of concentric cylinders The vertical equilibrium of an element of soil is given by eq (2.16):