behaviour of single piles under axial loading (mestrado)

8/20/2019 Behaviour of Single Piles Under Axial Loading (Mestrado)

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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


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To my parents

Aos meus pais


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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


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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 previousfinite 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


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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. Oprimeiro 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


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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 en gineer 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 andGuilherme. 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.


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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


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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


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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, 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. .................................................................................................................. 8

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.

................................................................................................................................................................. 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, 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. ................................................................................................................................ 10

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. ................................................................................................................................ 10

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. ........ 11

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.

............................................................................................................................................................... 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, fordifferent 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


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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 factorρ=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


homogeneous soils (ρ=1) and λ=975. ................................................................................................... 44


inhomogeneity factor ρ=0,75 and λ=975. .............................................................................................. 45


inhomogeneity factor ρ=0,5 and λ=975. ................................................................................................ 46


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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


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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


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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, 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


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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 shaftd: diameter of the pile shaft

db: diameter of the pile base

E: Young’s modulus of the soil

Einterface: Young’s modulus of the soil-pile interface

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

Eoed, interface: Young’s modulus of the soil-pile interface for oedometer loading conditions

G: shear modulus of the soil

Ginterface: shear modulus of the soil-pile interface

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

r: horizontal distance to the pile axis


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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

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 baseRk: correction factor for pile compressibility

Rs: resistance of the pile shaft

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


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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


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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 maximumallowable 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.


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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 ofthe 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 andsubsequent 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.


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2 ELASTIC

THEORY

-BASED

METHODS FOR

ANALYSIS OF SINGLE AXIALLY LOADED

PILES

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 methods usually 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, P t. Equilibrium andcompatibility 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).


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(a) (b) (c)

Figure 2.1: (a) Stresses acting in the soil adjacent to the pile; (b) Stresses acting on the pile; (c) Stresses actingon 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 shearstress 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, d b,

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 r 0, 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 ef fect 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

of this thesis.

There is an applied axial load Pt in the pile head. The settlement and load results refer to the applied

load only: the difference between specific weights, γ, of the pile and of the soil is not considered.

Since the conditions of this analysis are purely elastic, the interface between the soil and the pile is

considered to be rigid – there is no relative movement between them.


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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):

(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/r 0, the proportion of load transferred to the

pile base, Pb/Pt, and the load settlement ratio, Pt/(wtr 0GL).

Figure 2.2: Pile under axial loading – relevant parameters


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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):

(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, w t, is provided by eq.

(2.4):

∑ (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.


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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):

(2.5)

Where:

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


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

(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):

(2.7) The displacement compatibility condition is satisfied by imposing the same displacement for the pile

and the soil in each element (rigid interface).

The load settlement ratio is expressed in terms of a coefficient, I, as shown in eq. (2.8):

(2.8)

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)


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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/(wtr 0GL).

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

(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


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Figure 2.4: Correction factor for the pile compressibility, Rk: in terms of the relation between the pile’s and thesoil’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):

(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 betweenthe 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


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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 R ν 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


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Figure 2.8: Correction factor for pile compressibility, Ck, in terms of relation between the pile and the soil’sYoung’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 therelation 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


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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. Theresult 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):

∑

(2.13)

Where:

Ei: Young’s modulus of layer i

hi: thickness of layer i

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):

(2.14)

In the extreme case of ρ=0.5, the shear modulus at the surface must be null – this is called a “Gibson

soil”.

The factor of inhomogeneity enables a comparison between different types of soil, with more or less

vertical inhomogeneity. Since eq. (2.13) is also applicable in this case, it is used in the calculation of ρ,

through eq. (2.14), using the relation described in eq. (2.1).


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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, r 0, 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 soilaccording 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):

(2.15)

Since both the Young’s modulus of the pile, Ep, and the shear modulus of the soil at the pile base, GL,

are the same for every case, the soil-pile stiffness ratio is constant and λ=975.

Below the pile, the Young’s modulus is constant and has the same value as at the pile base. The

Young’s modulus of the soil, E, used in its calculation is the one at the middle of the pile. The Young’s

modulus of the pile is 30×106 kPa.

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.


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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 distancebetween the curves increases slightly with the slenderness ratio, L/r 0.

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 withthese charts is considerable. The non-linearity of these functions is the cause of irregularity of the

resulting curves.


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.

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80 90 100

P t /

( w t × r 0 × G L

)

L/r0

ρ=1

ρ=0,75

ρ=0,5


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Figure 2.11: Proportion of the load taken by the pile base in terms of the pile slenderness ratio, for differentinhomogeneity 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/r 0, 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 C k (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.

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0 10 20 30 40 50 60 70 80 90 100

P b

/ P t

L/r0

ρ=1

ρ=0,75

ρ=0,5


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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 amongstthem, 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.

(a) (b)

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):

(2.16)

Where:

r: horizontal distance to the pile axis

τ: shear stress

x: horizontal coordinate

σz: vertical stress

z: depth


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Since the shear stress caused by pile loading is much greater than the vertical stress, the second

summand is insignificant. Thus, the integration of the previous equation gives the shear stress in the

soil surrounding the pile, according to eq. (2.17):

(2.17)Where:


τ0: shear stress at the pile shaft

The shear strain of the soil is calculated through eq. (2.18):

(2.18)

As mentioned before, the displacement is considered to be mainly vertical (the radial component is

negligible), so eqs. (2.16) to (2.18) may be rewritten as eq. (2.19):

∫

∫

(2.19)

When including r m in the equation, an upper boundary of the radius of influence of the place, i.e. the

distance past which shear stress becomes negligible, eq. (2.19) may be rewritten as eq. (2.20):

(2.20)

The factor ζ is as a relation between the radius of influence of the pile and the radius of the pile shaft,

as shown in eq. (2.21):

(2.21)

The pile acts on the layer below as a rigid punch; its effect is more significant, i.e. the lower layer

suffers more deformation, nearer the pile. The lower layer restrains the deformation of the upper layer.

The result is the generation of positive vertical stresses (σ z>0, compression), which indicates that, by

the equation of soil equilibrium, shear stresses will decrease more rapidly than linearly, contrary to

what eq. (2.17) implies.


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The consequence is that the magnitude of ∂/∂r(rτ) (the variation of shear stress with distance do the

pile) decreases with depth along the pile and, consequently, so does the value of r m, as represented in

Figure 2.13.

.

Figure 2.13: Hypothetical variation of the radius of influence of the pile, r m. Adapted from (Randolph & Wroth,1978), p. 1471.

Rigid Piles in Homogeneous Soil

For rigid piles, the shaft settlement, ws, is independent of the depth, since there is no shortening (the

pile is considered as non-compressible). Thus, the shear stress at the pile shaft, τ0, must also vary,

increasing with depth, so that the shaft displacement given by eq. (2.20) is constant. Thus, the higher

the shear stress, the smaller the distance at which its value is significant.

The variation of r m with depth is generally disregarded, and its value is taken, in the case of

homogeneous soil, as an average, given by eq. (2.22):

(2.22) Generally, ζ varies between 3 and 5.

The shear stress at the pile shaft may be considered as described in eq. (2.23):

(2.23)

Assuming both this and the radius of influence of the pile, r m, constant with depth, the settlementrelative to the pile shaft is given by eq. (2.24):

(2.24)


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Since the behaviour of the pile base resembles a rigid punch, the resulting displacement is obtained

by eq. (2.25):

(2.25)

η is the factor of interaction between the upper and lower layers of soil.

The factor η refers to the stiffening effect of the soil above the loaded area. The value to attribute to

this factor is subject to discussion, but it is generally accepted that to adopt the unity is adequate,

according to (Randolph & Wroth, 1978).

In (Fleming, 1992), this factor is considered as the relation between the base and shaft diameters, as

shown in eq. (2.26):

(2.26)

This factor is included in the following equations, but is considered equal to the unity whenever

calculations are performed.

In a rigid pile, there is by definition no relative displacement within, and so eq. (2.27) is applicable:

(2.27)

Therefore, every point in the pile has the same settlement.

Besides, the total load is given by eq. (2.28):

(2.28) For rigid piles in a homogeneous soil, the load settlement ratio may be calculated through eq. (2.29):

(2.29)

Also, the fraction of the load that is taken by the base may be calculated by eq. (2.30):

(2.30)


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Compressible Piles in Homogeneous Soil

Not always may the pile be considered as rigid. For piles with certain properties, relative displacement

within the pile must be taken into consideration. Thus, for compressible piles, the shaft settlement

varies with depth, since shear stress can no longer be assumed as constant along the pile, as in eq.

(2.31):

(2.31)

The pile is considered as elastic, and its compressive strain in depth may be expressed by terms of

the transmitted load, as in eq. (2.32):

(2.32)

The soil-pile stiffness ratio, λ, is calculated by eq. (2.15).

It is relevant to point out that the Young’s modulus of the pile, Ep, is taken into account in the

calculations relative to compressible piles only, since for rigid piles it is considered to be infinite.

The transmitted load is related to the variable shear stress on the shaft surface, as in eq. (2.33):

(2.33)

Differentiating and combining the last equations will result in the governing differential equation, as in

eq. (2.34):

(2.34)

And its solution is described in eq. (2.35):

(2.35) Taking the following measure of pile compressibility, described in eq. (2.36):

(2.36)


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The constants A and B are found by using the boundary conditions at the base of the pile, and

equation (2.35) may be rewritten as eq. (2.37):

[ ] [ ] (2.37)

The term (πr 0λμ)-1 is very small (inferior to 0.02 in normal cases) and, through eq. (2.25), eq. (2.37)

may be simplified to eq. (2.38):

( ) ( ) (2.38)

It is possible to particularize eq. (2.38), expressing the total settlement of the pile in terms of the

settlement of the base, as in eq. (2.39):

( ) (2.39)

The mentioned simplification may also be used, as in eq. (2.40):

(2.40)

The integration of eq. (2.33) and substitution of the latest equations provides the expression of the

total load supported by the pile in terms of depth, as in eq. (2.41):

( ) ( ) (2.41)

Thus, the expression of the load settlement ratio is provided by eq. (2.42):

(2.42)

This allows for a comparison between the pile compressibility and the load deformation behaviour.


(2.43)


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It is also useful to adapt the former equations to the case of a non-homogenous soil. Only vertical non-

homogeneity is considered – the concept of radial inhomogeneity not being developed, given its lack

of relevance for the present study.

As in the chapter referent to the Poulos and Davis method, two types of non-homogeneous soils may

be considered: a layered soil (in which every layer has a constant shear modulus) and a soil in whichthe shear modulus varies linearly.

For the case of the layered soil, the shear strain γ dist ribution is unaltered (see eq. (2.18)), the shear

stress τ0 distribution being obtained by multiplying by an appropriate shear modulus.

Once again, the second type of soil inhomogeneity, in which the shear modulus varies linearly, is more

carefully analysed.

Rigid Piles in Non-Homogeneous Soil

The behaviour of a rigid pile in this type of soil is considered. The shear stress, which would be

assumed constant in a homogenous soil, increases approximately linearly with depth in this case. The

distance at which the shear stresses become negligible, r m, will also decrease.

For a pile in an infinite half space, r m may be calculated by eq. (2.44):

(2.44) And, for a pile in a space where there is a rigid layer at the depth of 2.5L, which is more commonly

used since it allows a comparison with finite element method programs’ results, r m may be calculated

by eq. (2.45):

(2.45) The inhomogeneity factor, ρ, is given by eq. (2.14).

Below the pile base, the shear modulus is considered to be constant. However, analyses using the

finite element method have proved that the difference between this and the case where the shear

modulus continues to increase is negligible (inferior to 5% in the total settlement value).

The value of shear stress may be written as in eq. (2.46):

(2.46) And thus, the shaft settlement, ws, may be calculated through eq. (2.47):

(2.47)


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The total load taken by the pile shaft may be written as in eq. (2.48):

∫ (2.48)

Eq. (2.48) may be rewritten as eq. (2.49):

(2.49)

And the load settlement ratio may be written as in eq. (2.50):

(2.50) The fraction of the load that is taken by the base may be calculated by eq. (2.51):

(2.51)

Compressible Piles in Non-Homogeneous SoilFor compressible piles, eq. (2.42), used for homogeneous soils, may be used for non-homogeneous

soils when modified by the introduction of the inhomogeneity factor, as in eq. (2.52):

(2.52)


(2.53)


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A relevant statement is that a pile with L/r 0=100 could hardly be considered “rigid” – this would be the

case of a pile with L=100m and d=2m, or L=8m and d=16cm. The authors of this method have not

provided any means of distinction between rigid and compressible piles. According to (Fleming, 1992),

eq. (2.54) can be used as a general rule to determine if a pile is may be considered as rigid:

√ (2.54)

In this case, for the conditions described, eq. (2.54) is calculated as shown in eq. (2.55):

(2.55)

In the charts that display results from the Randolph and Wroth method presented from now on, a black

line represents this limit, and the charts are divided in “Rigid” and “Compressible” areas.


factor , ρ. The factor of interaction between layers, η, is given the value of the unit y, and so is the pile

radius, r 0, in every case. The value of Poisson’s coefficient, ν, is 0.3. The relation between the radius

of influence of the pile and the radius of the pile shaft, ζ, is given by equation (2.21), using values of

the radius of influence of the pile, r m, given by equations (2.22) and (2.45), for homogeneous (ρ=1)

and inhomogeneous (ρ≠1) soils respectively; thus, the rigid layer is assumed to be at the distance of

2.5L from the surface. 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. Below the pile, the Young’s modulus is

constant and has the same value as at the pile base.

These conditions are the closest possible to the ones used to analyse the Poulos and Davis method

(see Figure 2.10).


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26

In Figure 2.15, results given by eqs. (2.42) and (2.52) are plotted, for different values of the soil

inhomogeneity factor. The Young’s modulus of the pile is 30×106 kPa. The conditions are similar to the

ones described for Figure 2.14.

Figure 2.15: Load settlement ratio in terms of the pile slenderness ratio for compressible piles, for differentinhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth Method.

This chart shows that, for compressible piles, the load settlement ratio increases non-linearly with the

pile slenderness ratio. This suggests that the total settlement does not reduce infinitely as the pile

slenderness increases, but instead that it stabilizes, indicating the existence of an asymptote. This did

not happen for rigid piles, as in that case the contribution of the shaft for the load settlement ratio

increased infinitely with the pile slenderness.

For compressible piles, the contribution of the base remains the same. However, with the introduction

of the measure of the pile compressibility, µL, the proportion of the load taken by the shaft is reduced.Thus, the load settlement ratio for a given value is lower if the pile is considered compressible than if it

were considered rigid, and the difference increases with the pile slenderness (see fig. Figure 2.16).

This is an overall more realistic approach for high pile slenderness values.

Once again, the load settlement ratio also becomes higher as the inhomogeneity factor ρ approaches

the unity.

The values that originate these curves are grouped in tables in Appendix B1.

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80 90 100

P t /

( w t × r 0 × G L

)

L/r0

ρ=1

ρ=0,75

ρ=0,5


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28

Figure 2.17: Load settlement ratio in terms of the pile slenderness ratio for rigid and compressible piles, fordifferent inhomogeneity factors and soil-pile stiffness ratios, ν=0.3 and h=2.5L, according to the Randolph and

Wroth method.

This chart shows that the load settlement ratio is quite sensitive to the soil-pile stiffness factor, for anygiven value of the soil inhomogeneity factor. However, it should be borne in mind that, for a constant

Young’s modulus of the pile of 30GPa, a soil-pile stiffness factor of 300 requires the shear modulus of

the soil at the pile base to be equal to 100MPa, whereas a soil-pile stiffness factor of 3,000 requires

the shear modulus of the soil at the pile base to be equal to 10MPa, which is quite a wide range. The

curves relative to λ=3,000 are obviously closer to the rigid pile solution (see Figure 2.14).

Application of eq. (2.54) provides the boundary values between rigid and compressible piles, for the

given values of soil-pile stiffness factor, shown in Table 2.1.

Table 2.1: Limit pile slenderness ratio between rigid and compressible piles, for different values of the soil-pilestiffness factor, according to (Fleming, 1992).

λ L/r 0

300 8.66

975 15.61

3,000 27.39

This table shows that the boundary between rigid and compressible piles is also quite sensitive to the

soil-pile stiffness factor (as is, in fact, clear in eq. (2.54)). This gives an idea of the utility of the rigid

solution (up to which value of the pile slenderness ratio the rigid solution is applicable).

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80 90 100

P t /

( w t × r 0 × G L

)

L/r0

ρ=1; λ=3000

ρ=0,75; λ=3000

ρ=0,5; λ=3000

ρ=1; λ=975

ρ=0,75; λ=975

ρ=0,5; λ=975

ρ=1; λ=300

ρ=0,75; λ=300

ρ=0,5; λ=300


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29

In Figure 2.18, results given by eqs. (2.30) and (2.51) are plotted, for different values of the soil

inhomogeneity factor. Once again, the conditions are the same as the ones described for Figure 2.14.

Figure 2.18: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid piles, fordifferent inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth method.

This chart shows that, for rigid piles, the proportion of load that reaches the pile base decreases non-

linearly with the slenderness ratio: its value reduces quickly for low values of L/r 0, but tends to

stabilize.

Besides, the values increase inversely with ρ; this is expectable, since the contribution of the base to

the load settlement ratio is not affected by ρ and the contribution of the shaft is proportional to ρ (see

eq. (2.50)), as it increases with the shear modulus along the pile (see eq. (2.24)).

The values that originate these curves are grouped in tables in Appendix B2.

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0 10 20 30 40 50 60 70 80 90 100

P b

/ P t

L/r0

ρ=1

ρ=0,75

ρ=0,5


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behaviour of single piles under axial loading (mestrado)

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