non linear fem analisi

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POLITECNICO DIMILANODEPARTMENTof Civil & Environmental Engineering

DOCTORALPROGRAMMEINStructural, Earthquake and Geotechnical Engineering

Non-Linear Soil-Foundation Interaction under Cyclic Loading

Doctoral Dissertation of:Iman Farshchi

Supervisor:

Prof. Claudio di Prisco

Co-Supervisors:Dr. Andrea GalliProf. Francesco Calvetti

XXVICycle (2011 2014)


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DOCTORAL PROGRAMME IN Structural, Earthquake and Geotechnical Engineering

DEPARTMENT of Civil & Environmental Engineering

POLITECNICO DI MILANO

XXVI Cycle (20112014)

Faculty Members

Prof. Roberto Paolucci (Co-ordinator)

Prof. Raffaele Ardito

Prof. Fabio Biondini

Prof. Gabriella Bolzon

Prof. Claudia Comi

Prof. Alberto Corigliano

Prof. Dario Coronelli

Prof. Claudio DI Prisco

Prof. Marco DI Prisco

Prof. Liberato Ferrara

Prof. Attilio Frangi

Prof. Elsa Garavaglia

Prof. Cristina Jommi

Prof. Giorgio Malerba Pier

Prof. Anna Pandolfi

Prof. Umberto Perego

Prof. Federico Perotti

Prof. Lorenza Petrini

Prof. Gianpaolo Rosati

Prof. Luigi Zanzi


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To My Mother

Shahdokht Rahimi

To My Father

Hossein Farshchi


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Acknowledgment

First and foremost I would like to express my sincere gratitude to my senior

supervisor Professor Claudio di Prisco for providing great ideas that shaped my research

and for his continuous help and support in all stages of my thesis. He always managed to find

time to meet me whenever I needed his help and provided invaluable advice to me all through

my Ph.D. I would also like to thank him for welcoming me to his office with his enthusiasm

and warm smile, being patient with me, and making me feel more motivated after each

meeting.

My heartfelt appreciation goes to Dr. Andrea Galli whose great energy and immense

knowledge guided me through my Ph.D. He was the person whose advice I sought before

starting any stage of my research and was always there to answer my questions. I learned a

lot from him particularly I learned how to become an organized researcher. He invested a lot

of time and effort not only in the process of conducting my research but also in the writing of

it as well. Frankly, without his untiring and meticulous proofreading, I would have never

succeeded in handing in my thesis.

I would also like to thank Professor Francesco Calvetti from whom I learned a lot

specifically in Distinct Element Method. It has been an honor for me to work with such a

great person whose tremendous expertise was the guiding light in the numerical part of my

thesis. In addition, I must wholeheartedly thank Dr. Marco Caruso who helped me to renew

the lab facilities, improve LABview software so that I could employ it in my research and

taught me how to enhance my lab skills.

It would not have been possible to do this doctoral thesis without the help and support

of the kind people around me, to only some of whom it is possible to give particular mention

here. Last but by no means least, I must thank my parents, brother and my sisters who have

given me their unequivocal love and support throughout, as always, for which my mere

expression of thanks does not suffice.


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[i]

Abstract

One of the important challenge and important factors that must always be considered

in the design of structures is to correctly predict all the structure movement and find out the

movement limitation that structure can resist. To get a correct prediction and safe design all

the displacements and forces induced by the nature or even by human factors on the structure

must be analyzed and taken into account carefully. Many cyclic loads of different nature may

affect civil and environmental structures, such as wind effect, sea-wave actions and

earthquake. From the geotechnical point of view and by considering the soil-structure

interaction these extreme and complex loading paths can cause large irrecoverable and plastic

deformation in the soil. Sometimes these loads become the dominant factor in the design and

may cause significant changes in the structure of the soil, even causing a shear rupture, heave

and important compaction. Hence the analytical and experimental modeling under such a

complex loading paths requires to analyze a post-yielding behavior for both soil and

foundation response. It is then evident the importance of studying effects of cyclic loads on

the foundation, but empirical data are still far from sufficient, both for shallow and deep

foundation.

This thesis deals with soil-foundation interaction by considering shallow and deep

foundation under cyclic loads. The work is divided into three parts:

1- Experimental works:This part has been performed by means of the small scale

experimental set up available at the geotechnical laboratory of the department of

the civil and environmental engineering at Politecnico di Milano. This machine is

capable to apply cyclic or monotonic, horizontal and vertical loads. The

experimental works is sub-divided into three parts, referred to shallow foundation,

shallow foundation plus piles and deep foundation. In this study loose Ticino

River sand was used, with the relative density about 40%. Several types of

monotonic and cyclic tests were performed. The tests presented hereafter were

performed with the aims of (i) defining the failure condition of the system (i.e. the


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Abstract

[ii]

interaction domain of the foundation in the generalised stress space VH), (ii)

describing the coupling between horizontal and vertical directions during

monotonic tests, and (iii) analysing the cyclic behaviour of the system.

2- Mechanical I nterpretation by Using the Macro-Element: A Soil-foundation

interaction modeling approach is presented, with an emphasis on the macro-

element model. The present part is aimed at giving a contribution to the

description of the mechanical response of the system by presenting the results of a

small scale experimental campaign on different kinds of foundations. By studying

initially monotonic tests results, by considering the generalized stress path,

regarding each monotonic test, it is possible to analytically define the interaction

domain by means of macro-element approach for the different geometries of the

foundation. In the monotonic part, the coupling between vertical and horizontal

direction is presented by studying the kinematic of the system, and also it is

confirmed that the system is non-associated. After defining the interaction domain,

a deeper investigation on the response to several cyclic loading paths, combining

vertical and horizontal loads, will be presented. The experimental results will be

interpreted in particular in terms of the average stiffness and of the damped energy

of each cycle, as well as in terms of the accumulation of permanent displacements

during cycling. A clear increase in stiffness and decay in dissipated energy will be

observed after applying number of cycles, and influence of the loading path and

the type of the foundation has been studied.

3-The numerical analysis: For this purpose, the Distinct Element Method (DEM)

has found to be a helpful numerical model for simulating the microstructure of the

geo-materials.DEMintroduced as a powerful numerical tool for discrete materials.

DEM is using as a micromechanical model and by adopting such a discrete

approach, the need of a constitutive model for the equivalent continuum is by-

passed. This method provides a synthetic material that can be used to understand

how the microstructure affects the macroscopic behavior. In Particular, the

numerical code PFC3D, adopted for all the numerical results illustrated in the

present work.


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Abstract

[iii]


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Abstract

[iv]

Abstract

Uno degli aspetti principali che devono sempre essere considerati nella progettazione

delle strutture la corretta determinazione dei limiti di spostamento ai quali la struttura stessa

in grado di resistere. Per una corretta determinazione di tali limiti e quindi per una

progettazione sicura necessario che tutti gli spostamenti e le forze agenti sulla struttura

indotte da effetti naturali o anche dal fattore umano siano presi in considerazione. I carichi

ciclici di diversa natura possono influenzare le strutture civili, ad esempio leffetto del vento,

le azioni indotte dal moto ondoso marino oppure le azioni sismiche. Dal punto di vista

geotecnico e considerando linterazione suolo-struttura queste complesse e a volte estreme

storie di carico possono provocare deformazioni plastiche irreversibili nel terreno. Talvolta

questi carichi diventano il fattore dominante nella progettazione e possono causare

cambiamenti significativi nella struttura del terreno, a tal punto da causare rotture a taglio, per

eccessiva espansione o schiacciamento. Per queste ragioni per modellare analiticamente e

sperimentalmente il comportamento di strutture soggette a queste azioni necessario

considerare il comportamento post-snervamento sia per il terreno che per la fondaz ione. E

evidente come rivesta un ruolo fondamentale la valutazione degli effetti dei carichi ciclici

sulle fondazioni; purtroppo, ad oggi, i dati sperimentali non sono sufficienti sia per quanto

riguarda le fondazioni superficiali sia per le fondazioni profonde.

Questa tesi tratta linterazione suolo-struttura sia per fondazioni superficiali che

profonde soggette a carichi ciclici. Il lavoro divisa in tre parti;

1- Parte sperimentale:In questa prima parte vengono descritte le prove su piccola scala

che sono state eseguite presso il laboratorio geotecnico del dipartimento di ingegneria

civile ed ambientale del Politecnico di Milano. La macchina utilizzata in grado di

applicare carici monotoni, ciclici, orizzontali e verticali. Il lavoro sperimentale stato

suddiviso in tre parti: per fondazioni superficiali con e senza pali e per fondazioni

profonde. In questo studio stata utilizzata la sabbia sciolta del fiume Ticino,

caratterizzata da una densit relativa di circa il 40%. Sono state eseguite varie prove


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Abstract

[v]

con diverse condizioni di carico (carico monotoni e ciclico). Gli obiettivi di tali test

sono stati molteplici: (i) definire le condizioni di collasso del sistema (il dominio di

interazione della fondazione nello spazio degli sforzi generalizzati V-H), (ii)

descrivere laccoppiamento tra direzione orizzontale e verticale durante lapplicazione

di un carico monotono, (iii) analizzare il comportamento ciclico del sistema.

2- I nterpretazione meccanica uti l izzando il M acro-Elemento: In questa seconda parte

viene illustarato il modello di interazione suolo-struttura, con particolare attenzione al

modello del macro-elemento. Questa parte ha lobiettivo di fornire un contributo nella

descrizione della risposta meccanica del sistema presentando i risultati della campagna

di prove eseguite su piccola scala sulle differenti tipologie di fondazioni. Inizialmente

vengono considerati i risultati sperimentali monotoni: prendendo in esame gli sforzi

generalizzati, stato possibile definire analiticamente il dominio di interazione

utilizzando la teoria del macro elemento per le differenti geometrie delle fondazioni.

Nella parte monotona, laccoppiamento tra la direzione orizzontale e verticale viene

eseguita studiando la cinematica del sistema, a conferma del fatto che il sistema non

associato. Dopo la definizione del dominio di interazione, viene presentato uno studio

completo della risposta per vari percorsi ciclici di carico, combinando azioni verticali

ed orizzontali. I risultati sperimentali sono stati interpretati in termini di rigidezza

media, energia dissipata in ogni ciclo e spostamenti permanenti accumulati durante il

percorso ciclico. Dopo un determinato numero di cicli, si osserva un aumento di

rigidezza e una diminuzione dellenergia dissipata. Infine stato studiato come il

percorso di carico e la tipologia di fondazione influenzano questo fenomeno.

3- Anal isi numerica:per simulare la microstruttura del geomateriale stato utilizzato il

Distict Element Method (DEM). DEMviene usato come modello micromeccanico e

adottando un approccio discreto non pi necessario definire un legame costitutivo

per il continuo equivalente. Questo metodo fornisce un materiale sintetico che pu

essere usato per comprendere come la micro-struttura influenza il comportamento

macroscopico. Il codice numerico PFC3D stato usato per tutti i risultati numerici

presentati in questa tesi.


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Abstract

[vi]


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[vii]

Index

ABSTRACT

Chapter 1: I ntroduction

1.1 Engineering Problem 1

1.1.1 Soil-Foundation Interaction 2

1.1.2 Methodologies for Soil-Foundation Interaction Analysis 4

1.2 Scope and Outline of the Thesis 5

Part 1: Experimental Resul ts

Chapter 2: Macro-Element Approach (Review of Exper imental Resul ts and

Analytical Modell ing)

2.1 Introduction 10

2.2 Defining the Interaction Domain 11

2.3 Elasto-Plastic Strain Hardening Macro-Element Models 23

2.4 Cyclic Loading 27

2.5 Response of Soil-Foundation Interaction under Cyclic Loading 28

2.6 Ratcheting Phenomena 34

2.7 Stiffness and Damping Factor 37


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Index

[viii]

Chapter 3: Description of the Experimental Device

3.1 Experimental Set-up 43

3.1.1 Main Box 43

3.1.2 Sand Reservoir 44

3.1.3 Spreader Caisson 45

3.1.4 Loading System 46

3.1.5 Displacement Transducer 46

3.1.6 Load Cell 47

3.1.7. Air Pressure Transducer 48

3.1.8 LabVIEW Environment 48

3.2 Calibration 49

3.2.1 Calibration the Displacement Transducer 49

3.2.2 Calibration the Load Cells 50

3.2.3 Calibration the Air Pressure Cell 51

3.3 Type of Foundations 52

3.3.1 Rigid Piles with Fix Head 52

3.3.1.1 Installing Procedure for Piles 53

3.3.1.2 Hanging System of the Pile 54

3.3.2 Shallow Foundation 55

3.3.2.1 Installation of the Shallow Foundation 56

3.3.3 Shallow Foundation Plus Piles 57

3.3.3.1 Validation of the Piles Rigidity 60

3.4 Sand Used 60


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Index

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3.4.1 Relative Density of the Used Sand 61

3.4.1.1 Analysis Procedure of the Relative Density 61

Chapter 4: Exper imental Resul ts from Monotonic and Cyclic Tests on a

Shallow foundation and Shallow Foundation plus Piles

4.1 Introduction 65

4.2 Experimental Programs 70

4.2.1 Conceptual Framework 70

4.2.2 Foundation Configurations 72

4.2.3 Type of Tests 73

4.3 Results of the Experimental Tests from Monotonic Load (test type a) 80

4.3.1 Definition the Interaction Domain for Shallow Foundation 81

4.3.2 Interaction Domain for Shallow Foundation plus Piles 83

4.3.3 Coupling Effect 88

4.3.4 Elasto-Plastic Model with Non-Associated Flow Rule 91

4.3.5 Observing Failure Mechanism for Shallow Foundation 92

4.3.6 Observing Failure Mechanism for Shallow Foundation plus Piles 93

4.3.7 Effect of the Different Foundation Configurations on the Monotonic

Results 96

4.3.7.1 Effect of the Different Foundation Configuration on the Interaction

Domain 96

4.3.7.2 Effect of the Different Foundation Configurations on the Group

Efficiency Factor 97


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Index

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4.3.7.3 Effect of the Different Foundation Configurations on the Initial

Stiffness for the Monotonic Vertical Loading Tests 98

4.3.7.4 Effect of the Different Foundation Configurations on the Coupling

Effect 100

4.4 Cyclic Tests 101

4.4.1 Cyclic Asymmetric Tests for Shallow Foundation 102

4.4.1.1 Horizontal Asymmetric Cyclic Tests for Shallow Foundation

(test type b) 102

4.4.1.2 Cyclic Vertical Loading for Shallow foundation (test type d) 106

4.4.1.3 Cyclic Inclined Loading for Shallow Foundation (test type e) 108

4.4.1.4 Cyclic Complex Two Stages Loading for Shallow Foundation

(test type f) 111

4.4.1.4 Effects of Different Loading Paths on Results of Cyclic Asymmetric

Tests for Shallow Foundation 113

4.4.2 Horizontal Asymmetric Cyclic Tests for Shallow Foundation plus Piles

(test type b) 115

4.4.2.1 Horizontal Asymmetric Cyclic Test for Shallow Foundation plus

1 Pile 115

4.4.2.2 Horizontal Asymmetric Cyclic Test for Shallow Foundation plus

3 Piles in a Row 119

4.4.2.3 Horizontal Asymmetric Cyclic Tests for Shallow Foundation plus

3 Piles in a Column 122


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4.4.2.4 Horizontal Asymmetric Cyclic Tests for Shallow Foundation plus

9 Piles 125

4.4.2.5 Effect of the Different Foundation Configurations on the Results for the

Cyclic Asymmetric Loading Tests 128

4.4.2.5.1 Effect of the Different Foundation Configurations on the Average

Stiffness for the Cyclic Asymmetric tests 128

4.4.2.5.2 Effect of the Different Foundation Configurations on the Dissipated

Energy for the Cyclic Asymmetric Tests 129

4.4.2.5.3 Effect of the Different Foundation Configurations on the Cyclic

Pseudo-dilatancy (cyc) and cyc 1304.4.3 Horizontal Symmetric Cyclic Test for Shallow Foundation (test type c) 132

4.4.3.1 Effects of Different Loading Paths on the Cyclic Symmetric

Results for Shallow Foundation 135

4.4.4 Horizontal Symmetric Cyclic Test for Shallow Foundation plus Piles

(test type c) 136

4.4.4.1 Horizontal Symmetric Cyclic Test for Shallow Foundation plus

1 Pile 136


3 Piles in a Row 139

4.4.4.3 Horizontal Symmetric cyclic Test for Shallow Foundation plus

3 Piles in a Column 142


9 Piles 145


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Index

[xii]

4.4.4.5 Cyclic Symmetric Test with Zero Vertical Loads for Shallow

Foundation plus Piles 148

4.4.4.5.1 Gap Effect 151

4.4.4.6 Effect of the Different Foundation Configurations on the Results for

the cyclic Symmetric Loading Tests 153

4.4.4.6.1 Effect of the Different Foundation Configurations on the

Average Stiffness for the Cyclic Symmetric Tests 153


Dissipated Energy for the Cyclic Symmetric Tests 155


cycand Horizontal Displacement for the Cyclic Symmetric Tests 156

Chapter 5: Exper imental Resul ts from Monotonic and Cycl ic Tests on a Deep

Foundation

5.1 Introduction 159

5.2 Experimental Programs for Deep Foundation under Horizontal and

Vertical Loads [case a] 162


5.2.2 Results of the Experimental Tests for Monotonic Load 167

5.2.2.1 Comparison of Interaction Domains for Different Piles Configurations 172

5.2.2.2 Coupling Effect 174

5.2.2.3 Elasto-Plastic Model with Non-Associated Flow Rule 176


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Index

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5.2.2.4 Effect of the Different Foundation Configurations on the

Group Efficiency Factor 178

5.2.3 Cyclic Tests 178

5.2.3.1 Horizontal Asymmetric Cyclic Tests 179

5.2.3.2 Horizontal Symmetric Cyclic Tests 184

5.2.3.3 Effect of Different Loading Paths 188

5.2.3.4 Effect of Different Foundation Configurations 189

5.3 Experimental Programs for a Single Rigid Pile with Fix Head [case b] 190


5.3.2 Results of the Experimental Tests for Monotonic Load 199

5.3.2.1 Calibration of the Parameters 200

5.3.2.2 Comparison between Fix Head and Free Head Pile 203

5.3.3 Cyclic Tests 205

5.3.3.1 Some Definitions 205

5.3.3.2 Cyclic Asymmetric Tests 206

5.3.3.3 Cyclic Symmetric Tests 210

5.3.4 Effect of the Geometrical Dimensions of the Pile, Relative Density

of the Soil and Loading Path on the Stiffness and Dissipated Energy 217

5.3.5 Effect of Number of Cycles on the Horizontal Displacement 219

5.3.6 Effect of the Number of Cycles on the Stiffness 222

5.3.6 Effect of the Number of Cycles on the Dissipated Energy 224


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Index

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Part 2: Numeri cal Resul ts

Chapter 6: Numerical Analysis

6.1 Introduction 229

6.1.1 Particle Contact Model 230

6.1.2 Force-Displacement Law 231

6.1.3 The Calculation Cycle 233

6.1.4 PFC3D DEM Code 234

6.1.4.1 The Geometry of the System 234

6.1.4.2 Modeling in PFC3D 235

6.1.4.3 Mechanical Parameters 235

6.1.4.4 Balls Scale-up 236

6.2 PFC3D Simulation 237

6.2.1 Calibration of Numerical Material, Specimen Preparation 237

6.2.1.1 Calibration Parameters for Ticino River Sand 237

6.2.1.2 Calibration and Scaling Up the Size of the Balls in PFC3D 237

6.2.1.3 Size of the Box in PFC3D 239

6.2.1.4 Generated Box in PFC3D 240

6.2.2 PFC3D Results 241

6.2.2.1 Monotonic Loading 241

6.2.2.1.1 Coupling Effect 256

6.2.2.1.2 Elasto-Plastic Model with Non-Associated Flow Rule 259

6.2.2.1.3 Observation from Micromechanics 261


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Index

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6.2.2.2 Cyclic Loading 265

6.2.2.2.1 Cyclic Asymmetric 265

6.2.2.2.2 Cyclic Symmetric 268

6.2.2.2.3 Cyclic Vertical 272

Chapter 7: Conclusions 276

References 283


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Index

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[1]

Chapter

1

Introduction

1.1 Engineeri ng Problem

One of the important challenge and important factors which must always beconsidered in engineering design of the structures is to correctly predict all the structure

movement and find out all the movement limitation that structure can resist. To get a correct

prediction and safe design all the displacements and forces induce by the nature or even by

human factors on the structure must be analyzed and taken into account carefully.

Many cyclic loads of different nature may affect the structure, such as wind effect and

earthquake. From the geotechnical point of view and by considering the soil-structure

interaction these extreme and complex loading paths can cause large irrecoverable and plastic

deformation in the soil.

Sometimes these loads become more effective and may cause significant changes in

mechanics of the soil, such as shear rupture, heave and void deformation, which may produce

large interaction on embedded structures such as shallow or deep foundations.

Hence the analytical and experimental modeling under such a complex loading paths

requires analyze a post-yielding behavior for both soil and foundation response.


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Chapter 1 Introduction

[2]

Non-linarites can develop in case of soil-foundation interaction, such as:

- Geometric nonlinearity such as the separation of the foundation from the soil or

uplift phenomena in case of shallow foundation. This happens when the seismic overturning

moment induces tensile stresses at the edge of the foundation. The uplift phenomena can

cause rocking motion, which can be considered as the geometric nonlinearity. Since the

ductility will be reduced while the rocking motion will take place, many authors such as

Hounser (1963), Meek (1978) and Chopra and Yim (1985), reported the benefit of the

uplifting on the performance of the supported structure.

- Interface inelasticity such as sliding at soil-foundation interface. This can

happen when the lateral loading exceeds the frictional resistance of the foundation. As it is

mentioned by Newmark (1965) the sliding usually does not induce any failure but permanent

deformation can caused by this phenomena.

- Mechanical nonlinearity such as mobilization of bearing capacity failure

mechanism in supporting soil. In statics large factor of safety will be applied to be far enough

from the bearing capacity failure. In seismic analysis sort of plastic hinge will be introduced

which can reduce the transmitted load by limiting capacity of the foundation and make the

superstructure separated by the ground motion. The new design code for foundation system

allows accumulating this non-linearity through the soil-foundation system rather than in the

superstructure.

1.1.1 Soil -Foundation I nteraction

Some factors mainly affecting the response of the structure under dynamic loading for

instance during an earthquake: the source of seismic loading (bed rock), traveling path, local

site effects and soil-structure interaction (SSI). At the primary stage the motion is coming

from the bedrock and propagates through the supporting soil stratum, the propagation

characterization such as frequency and shear wave velocity can be affected by the soil stratum

and the local topography. The propagated motion will induce the dynamic structural response

and in this part the modified velocity and frequency by the structure will be transmitted to the

footing and the soil system. In critical conditions the energy can be dissipated in the soil

domain and the failure and crush for the structure may happen because of the failure of thesoil layers. All the described stages will happen at the same time, so the coupling between the


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[3]

soil and the structure response is obvious and their reaction are linked together. Results of

these stages are free field seismic ground motion at the ground surface, where here free

field is referred to the area which the structural vibration does not have any effects on the

motion, and SSI effects, which is determined the actual loading experienced by the structure-

foundation-soil system resulting from the free-fieldseismic ground motion.

Figure 1.1: schematic of SSI in engineering assessment of seismic loading of a structure (Stewart et al. 1998)

Two kinds of interaction between the structure, foundation and soil can be occurred:

1- Kinematic interaction: Even if the mass of structure and the foundation is

considered to be zero, the presence of stiff foundation elements has an effect on

the seismic wave field and as a consequence on the formation of the underlying

soil.

2- Inertial interaction: This mechanism refers to the response of the completestructure-foundation-soil system to excitation by D Alembert forces associated

with acceleration of the super structure due to kinematic interaction.

According to Varun (2010) three mechanisms can have an effect on such a deviation:

a- Base-slab averaging: Free-field motions associated with inclined and/or

incoherent wave fields are averaged within the footprint area of the base slab

due to the kinematic constraint of essentially rigid-body motion of the slab.


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[4]

b- Embedment effects: the reduction of the seismic ground motion with depth for

embedment foundation

c- Wave Scattering: Scattering of seismic waves off of corners and asperities of

the foundation.

1.1.2 Methodologies for Soil -Foundation I nteraction Analysis

The general methods to quantify soil structure interaction are:

a- Direct approach: In a direct approach, the soil and the foundation are accounted in one

mathematical step calculation all in the same time. Typically, the soil is discretized

with solid finite element method and the structure with finite beam element. This

method provides more generality as it is capable of incorporating all nonlinear

behavior of the structure, the soil and also the interface between those two (sliding,

uplift). Among the direct methods the finite element method is the most famous one.

The advantage of the finite element method is that foundations of arbitrary complexity

can be dealt with, and soil heterogeneities and non-linarites can be treated extensively.

On the other hand this method needs wide discretization of the soil massive, and the

size of elements must be chosen in a correct manner and they must be appropriate in

order to be able to transfer the high wave frequency. So this method needs high

number of degrees of freedom and it makes the finite element method very expensive

in term of computational time.

b- Hybrid methods: The main consideration of hybrid methods is to divide the entire

system into three different sub-structures, far field, near field and the superstructure.

The far field is sufficiently far from the structure, so it can be assumed that the

displacement in this zone is known without inducing non-linear effects. The near field

is located near the structure and consists of all the soil-structure non-linearity where

the behavior of the system is irreversible (Figure 1.2). The near field can be modeled

by means of different approach, one of the most powerful one is macro-element

method.


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[5]

Figure 1.2: Domain considered by hybrid methods

1.2 Scope and Outl ine of the Thesis

This thesis deals with the experimental investigation description of the behavior of

soil-foundation system under monotonic and cyclic loading, and it is consists of two parts, the

first one experimental results and the second one numerical analyses. Chapters 2, 3, 4 and 5

belong to the first part. Chapter 2 is review of the experimental and analytical results obtained

by applying the macro-element concept, since this approach is the one which has been chosen

to describe soil-foundation interaction under the monotonic and cyclic loading. This chapter

introduced the various modeling possibilities and expanded the macro-element theory for

different for different foundation types and soil characterizations.

In the chapter 3, the small scale laboratory apparatus employed to obtain the

experimental results for soil-foundation interaction, is described in details. Chapter 4

represents the experimental results obtained by performing on a shallow foundation and

shallow foundation plus piles on dry loose Ticino River sand. They are very useful to

characterize the system behavior under the monotonic and cyclic behavior. Large vertical

displacement, high influence of coupling between the horizontal and vertical load and

displacement as well as complex failure condition in loose sand compare to the dense sand are

the reasons to perfume tests on the loose sand. Initially by applying the monotonic loads for

the shallow foundation the failure condition of the system in the generalized stress path V-H,


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[6]

by taking into account the macro-element model will be defined and this equation for the case

of shallow foundation plus pile will be modified according to the best fitting of the analytical

equation to the obtained interaction domain by the experimental works. In the monotonic part

the coupling between horizontal and vertical directions also will be described by studying the

kinematic of the system, and also it is confirmed that the system is non-associated. After

defining the appropriate interaction domain for each configuration, the cyclic loads will be

applied. In the cyclic parts, in addition to analyzing the cyclic behavior of the system, the

attention will be focused on the average stiffness, dissipated energy and the cumulated

displacement at the end of each cycle. From practical point of view, simple interpolating

formulas and simple parameters fir the stiffness and the dissipated energy values as well as

vertical displacement were proposed and their dependence on the vertical and horizontal load

and on the type of test was discussed for the different type of foundation. Chapter 5 is divided

to two parts, at the first part monotonic and cyclic loading in a horizontal and vertical

direction will applied on the deep foundation with different number of piles, which already

described in chapter 3 and chapter 4 on loose Ticino River sand. In the second part of chapter

5, different tests by applying different loading paths in horizontal direction were performed,

on a single rigid pile with fix head. This part has been done by using rigid fix head pile with

two different diameters and two different lengths by using loose and dense sand. Effects of

the different piles and soils characterization on the initial stiffness of the load-displacement

curves in the monotonic tests were taking into account. While in case of cyclic tests,

stabilization in accumulation rate of the irreversible displacement as well as the gap effect is

highlighted.

The numerical analysis is presented in the second part of thesis. For this purpose, the

Distinct Element Method (DEM) has found to be a helpful numerical model for simulating the

microstructure of many materials, in particular geo-materials. In the chapter 6, The Distinct

Element Method (DEM) introduced as a powerful numerical tool for discrete materials. DEM

is using a micromechanical model and by adopting such a discrete approach, the need of

constitutive model for the equivalent continuum is by-passed. This method provides a

synthetic material that can be used to understand how the microstructure affects the

macroscopic behavior.In Particular, the numerical code PFC3D, adopted for all the numerical

results illustrated in the present work.


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PART 1: Exper imental Results


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[10]

Chapter

2

Macro-Element Approach

(Review of Exper imental Results and

Analytical Modell ing)

2.1 I ntroduction

Many authors already represent macro-element approach as a useful tool for soil-

foundation interaction by analyzing the foundation response without disregarding the soil

non-linearity and geotechnical problems.

In fact most part of the design procedure takes place by considering separately the

structure and geotechnical problem, which means either they neglect the geotechnical

problem or once the structural problem is solved, the geotechnical issue will be considered,

but this uncoupling may be not correct or even safe (di Prisco et al. 2004).

Not many studies consider these two factors all together. One of the possibility to

study in a coupled manner can be small or large scale and centrifuge experimental tests or

either applying finite element analysis but since already discussed, these numerical analysis

are very time consuming. On the other hand this interaction can be successfully considered by

macro-element theory (Butterfiel and Ticof (1979), Georgiadis and Butterfield (1988), Nova


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Chapter 2 Macro-Element Approach

[11]

and Montrasio (1991), Paolucci, (1997), Gottardi, Houlsby and Butterfield (1999), Martin and

Houlsby (2001), Cremer et al. (2001 and 2002) and di prisco et al. (2003a and b)) and it is

useful because the system can be subject to the different loading path and in case of cyclic

loading situation many cycles can be applied.

The aim of the macro-element is to model the near field soil-foundation behavior. In

this concept the entire soil-foundation system is considered as a one single element located at

the near the foundation area, which is introduced to analyzing the non-linear and irreversible

behavior of soil-foundation interaction that can takes place at the near field zone.

The basic idea of the macro-element is to follow the analysis the non-linear behavior

of shallow foundation with the plasticity theory of the Roscoe and Schofield (1956 and 1957).

In fact this theory is expanded by Nova and Montrasio (1991) in a case of shallow strip

footing on sand under monotonic loading with an isotropic hardening elasto-plastic law to

define the bearing capacity of the foundation in a vertical, horizontal and overturning moment

plane. This bearing capacity is defined as a yield surface in a plasticity model. And a

kinematic of the system has been introduced by a plastic flow rule, non-associated flow rule

in sand. Defining the spatial domain where irreversible strains will be developed is essential.

So many factors can have an effect on this capacity, for instance, different loading path,

different foundation shape and different soil conditions. This aspect is proposed by many

researchers, for example Houlsby and Martin (1993) modeled the behavior of a foundation on

clay and Gottardi and Butterfield (1995) used dense sand instead. Furthermore Gottardi et al.

(1999), Martin and Houlsby (2000 and 2001), Houlsby and Cassidy (2002), Cassidy et al.

(2005) and Bienen et al. (2006) have been proposed the macro-element theory in a case of

circular footing. This theory has been employed even in case of buried pipe line by Calvetti et

al. (2001), Zhang et al. (2002), di Prisco et al. (2004), Galli (2005), Calvetti et al. (2008),

Coccheti et al. (2009) and Galli et al. (2008).

2.2 Defining the I nteraction Domain

In macro-element theory, proposed by Nova and Montrasio (1991) for a rigid strip

footing with width B under the generalized stress variables, the vertical force (V), the

horizontal force (H) and the overturning moment (M) with the generalized displacement

represented by the vertical (v) and horizontal (u) displacements and by the rotational


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[12]

settlement (), by performing several experimental tests the shape of failure locus as one of

the basic features as foundation behavior is defined.

Figure 2.1: Footing under the generalized stress variables, the vertical force (V), the horizontal force (H)

and the overturning moment (M) and generalized displacement represented by the vertical (v) and

horizontal (u) displacements and by the rotational settlement ()

The failure locus proposed by Nova and Montrasio (1991) is obtained in two steps.

Initially pure vertical load applied to the foundation in order to define the limit load by means

of two different techniques. Unlevelled technique which is the vertical load increase in stepsuntil the system reaches to the failure load (VM),and in the second technique, after each step

of increasing vertical load, sand surface was levelled so that the free surface was always kept

at the same level as the base of footing. This step allows the limit load to be defined and also

it is possible to model the vertical load-vertical displacement curve by an analytical approach

proposed by Butterfield (1980):

Equation 2.1Where, Ro,is the initial slope of the vertical load-vertical displacement curve (Figure

2.2).


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[13]

Figure 2.2: Vertical load versus vertical displacement experimentally obtained by Nova and Montrasio (1991), theoretical

curve obtained by means of equation 2.1 (Butterfield, 1980)

For the second step to find the failure locus, a series of the tests have been performed

with vertical and horizontal loads. In this case the failure point is conducted where initially

the vertical load applied until a certain value and then the horizontal load applied at a constant

vertical load until the failure load or in the other case both horizontal and vertical load

increased in a proportion to each other (Figure 2.3).

Figure 2.3: Procedure of applying loads to obtain failure locus

Figure 2.4 shows the interaction diagram at failure between the vertical load (V) and

horizontal load (H) normalized with respect to the maximum vertical load (VM) obtained by

Nova and Motrasio (1991).


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[14]

Figure 2.4: Failure Locus for inclined load (Nova and Montrasio, 1991)

According to Nova and Montrasio (1991) the best way to fit eventually the obtained

failure point in the V-H plain is:

Equation 2.2

, and are the constitutive parameters and. In particular is the traditional soil-

foundation friction coefficient, and it is linked to foundation roughness. From experience in

laboratory experiments, it can be evaluated as (Nova and Montrasio 1997):

Equation 2.3

Where is the soil friction angle and B is the foundation width. For a small verticalloads failure occurs when (Nova and Montrasio 1991):

Equation 2.4


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[15]

Parameter controls the shape of the interaction domain and maximum horizontal

load, if is equal to 2.1 the domain described by a parabolic shape and H is maximum in

. Experimental results are better fitted if is chosen as 0.95 (Nova and Montrasio,1991).

Another series of tests have been performed by Nova and Montrasio (1991) by an

eccentric vertical loads and the new interaction diagram was defined between the overturning

moment (M) and the vertical load. The shape of the failure is quite similar to the interaction

domain in V-H plane (Figure 2.5)

The analytical description for this new failure locus also can be defined in a similar

way of the equation 2.2 as follow:

Equation 2.5

is non dimensional constitutive parameter which can be selected according to the

failure value under eccentric loading; e.g., =0.33 according to the Meyerhof theory (1953),or =0.48 according to Vesic (1970). Typical values of lie within the previous range.

Figure 2.5: Cross-sections of bearing capacity surface (Failure locus) at M=0 and H=0

From a mathematical point of view the generalized non dimensional stress vector (Q)

and the corresponding strain vector (q) are:


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[16]

[

] Equation 2.6

Equation 2.7

The overall yield surface in 3D space will be described by the equation 2.8,

(Butterfield and Ticof 1979, Georgiadis and Butterfield 1988, Nova and Montrasio 1991,

Butterfield and Gottardi 1994, Nova and Montrasio 1997), this ultimate space has the shape ofrugby ball (Figure 2.6).

Equation 2.8Where in this equation,

,

and

. Equation 2.7 will be

reduced to equation 2.2 and 2.5 when m = 0 and h = 0 respectively.

Figure 2.6: Yield Surface in 3D space


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[17]

From the large and small scale experimental tests it found in the literature, some main

facts can be observed. For example nonlinearity can be seen even from the beginning of the

monotonic loading tests.

Different generalized stress and strain variables work as sort of system and this

coupling between them is obvious until the failure is approached. This coupling means you

can observe vertical displacement, even if just the horizontal loading will be applied.

The well-known shear failure will occur when the soil is sufficiently dense but on the

other hand when the relative density of the soil is low, sort of punching mechanism is more

likely to take place (di Prisco 2003) and in this case, second order effect must be considered

and the behavior of the corresponding stress and strain curve instead of reaching the peak and

the plateau, it will be increased continuously.

The bearing capacity that already shown in figure 2.6 and defined in equation 2.8, the

so called interaction domain, is so affected by the inclination and eccentricity of the loads.

This is function of relative density of the soil and the foundation characterization, such as

shape, roughness and the embedment depth.

As it is suggested by figure 2.7, if the penetration mechanism is uncoupled from the

sliding mechanism or either from the toppling mechanism, since it is assumed to be used the

rigid system, meanwhile the interaction domain in the two planes H V or MV, also must

be considered as uncoupled too. In fact the sliding mechanism just concern the interface zone

and toppling would be affected by the uplift and detachment of the footing from the soil layer.

As already cited before, it cannot be uncoupled and just defined by the four straight lines as in

figure 2.7.

(a) (b)

Figure 2.7: Uncoupled interaction domain for a shallow rigid strip footing (a) H-V plane (b) M-V plane


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[18]

As already mentioned the failure mechanism and the shape of failure locus in the soil

completely depend on the applied loading path, relative density of the soil and the foundation

type, such as shape, roughness and the embedment depth.

For instance the difference in failure mechanism is obvious for the shallow foundation

under vertical loading and shallow foundation under the inclined loading path, as it shown in

the figures 2.8a and 2.8b from the research proposed by Nova and Montrasio (1988) on a

small scale experimental work.

(a) (b)

Figure 2.8:Small scale experimental results (Nova and Montrasio 1988) concerning the rigid shallow foundation (a): undervertical loading (b) under inclined loading path

Montrasio and Nova (1997) studied about the effects of the different foundation shape,

foundation embedment and foundation roughness on the shape of the interaction domain. Theconclusion from this study is that the foundation shape can have an effect on V M, Foundation

embedment effects on the VM, and and foundation roughness can have an influence onthe . For instance in the figure 2.9 different shape of interaction domain can be observed,

which experimentally and analytically obtained by applying different depth ratio (d). Here the

depth ratio d, is equal to the embedment depth of the foundation (D) divided by the width of

the foundation (B).

Figure 2.9: Comparison of experimental data and computed failure loci for different depth ratio (Montrazio and Nova,1997)


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[19]

The shape of both failure mechanism and the interaction domain is affected by soil

inhomogeneity. This is confirmed by many authors such as di Prisco et al. (2003b). As it

shown in the figure 2.10 the failure domain is changed due to the presence of the

georeinforcement under the shallow foundation. In this study, the failure mechanism and the

shape of failure locus has been experimentally investigated. For example in the case of loose

sand with fastened and unfastened georeinforcement, it was observed, on evident increase in

the bearing capacity. In particular this increasing was very important when the fastened

geogrids were used. In the considered cases not only the failure locus increases markedly its

size, but also its shape changes abruptly (Figure 2.11).

(a) (b)

Figure 2.10: Failure mechanism from laboratory small scale experiment test result (di Prisco 2003b)

(a): unreinforced loose sand (b): reinforced unfastened loose sand

(a) (b) (c)

Figure 2.11: Sketches Of the three different loose sand failure loci (di Prisco et al. 2003b): (a) Unreinforced shallow

foundation (b) Reinforced shallow foundation (Unfastened geogrids) (c) Reinforced shallow foundation (fastened

geogrids)

Calvetti et al. (2001), Zhang et al. (2002), di Prisco et al. (2004), Galli (2005), Calvetti

et al. (2008), Cocchetti et al. (2009) and Galli et al. (2008), modified and developed macro-

element theory by investigating the behavior of buried pipe line and took into consideration


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[20]

the shape of interaction domain in a V-H plane with respect to the pipe line cross section. In

this situation the new V-H-N space is defined for the pipe line (Figure 2.12).

(a) (b)

(c)

Figure 2.12: Soil-pipe interaction model (a) V-H plane (b) V-H-N space (c) Definition of static and cinematic quantities

It can be cited even the work proposed by Calvetti et al. (2001 and 2004), which

investigated the shape of interaction domain of the soil-pipe interaction by means of distinct

element method code. In this work, the coupling effect at failure between V and H plane is

evident. The interaction domain defined by considering the different relative depth of the

buried pipe line and velocity vectors at failure is shown in Figure 2.13.

(a) (b)

Figure 2.13: (a) Exploded view of the distinct element model (b) effect of relative depth = Z/D on the interaction domain

and velocity vectors at failure (Calvetti et al. 2004)


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[21]

In order to describe the numerical data regarding to the interaction domain, inspired by

the formula proposed by Nova and Montrasio for the shallow foundation, the following

equation is used (di Prisco et al. 2004):

Equation 2.9

Where Vmax and Vmin, are maximum vertical load in a downward and upward

directions respectively. And more over another authors such as di Prico et al. 2004 applied

such an equation not for different relative depth of the pipe line. Figure 2.14 shows the

different shape of interaction domain in V-H plane calculated by means of limit analysis by

using equation 2.9 for a different relative depth and different slope surface.

(a)

(b)

Figure 2.14: (a) Failure locus in V-H plane at varying relative depth (i = 0) (b) Failure loci in normalized V-H plane at

varying inclination i ( Z/D = 2 ) , (di Prisco et al. 2004)


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[22]

More recently Pisano et al. (2014) improved Nova and Montrasio macro-element

techniques by inspiring the soil-pipe interaction model, to define the interaction domain in V-

H-M plain of shallow foundation with emphasis on mechanical aspects characterizing

laterally loaded historical towers. So a non-dimensional representation has been introduced

for the footing failure in H-V plane (M=0) and for the M-V plane (H=0) in introduced by

equation 2.10 and 2.11 respectively.

Equation 2.10

Equation 2.11

Where these formulas are following the soil-pipe interaction from di Prisco et al.

(2004):

Equation 2.12 Equation 2.13

Equation 2.14

Where and are the footing capacities under pure compression (downwardvertical load) and tension (upward vertical load) respectively. , and control theshape and the size of the domain in H-V cross-sections in a axis and , and control the shape and the size of the domain in M-V cross-sections in a axis.


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[23]

2.3 Elasto-Plastic Strain Hardening M acro-Element Models

The Nova and Montrasio model in a classic plasticity model is characterized by an

isotropic hardening law. As a usual in classic plasticity, it is assumed the existence of a load

function f in the generalized space Q such that:

Equation 2.15Where is an internal hardening variable assumed to be a function of permanent

strain q, stress state Q such that f(Q)>0 are not admitted. The flow rule is defined as:

and and Equation 2.16

Where is the so called plastic multiplier and g is the plastic potential. Both load

function f and plastic multiplier shall obey relations:

{

Equation 2.17

By assuming that yield locus has the same shape of the limit locus, the loading surface

fin the generalized stress pass is defined as follow:

Equation 2.18


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In the condition of:

Equation 2.19

The group of the yield loci will expand from the origin of the axis, according to the

hardening parameter and in fact representing the yield locus size. The yield surface hasthe same shape as the ultimate surface when hardening parameter =1 (Figure 2.15).

Figure 2.15: Isotropic hardening for different value of

The hardening variable, is governed by the equation:

|| || || || Equation 2.20Where , and are constitutive parameters playing the role of plastic stiffness.

Plastic potential g is define by similar expression of the load function, but modified by two

parametersand as follow:

() Equation 2.21


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[25]

Parameters and are defined as below:

Equation 2.22

Equation 2.23

and are the constitutive parameters which can be obtain experimentally. If = = 1 the plastic potential and the loading function coincide(f = g) and this is the case of when

flow rule is associated. But it cannot be accepted in case of shallow foundation, because in

this case the meaningless upward vertical displacement may be occurring (di Prisco and

Pisano 2011). However, experimental evidence suggests that and are larger than andrespectively. For shallow foundation Nova and Montrasio proposed the value of 0.4 for and 0.286 for .

Figure 2.16: Plastic potential shape in the H-V plane


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[26]

By combining equations 2.16 and 2.20 the plastic multipliers can be computed as:

Equation 2.24

The explicit formula for according to Nova and Montrasio (1991) is given as:

{ |||| ||||}

Equation 2.25

Where

Equation 2.26

Equation 2.27

If is known for a given , one may determine via the equation 2.16c. It istherefore possible, in principle, to determine the deformation history of the foundation for any

assigned loading history.


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[27]

2.4 Cycli c Loading

The basic macro-element formula has been further modified by pedretti (1998) and di

Prisco et al. (2003) by considering the foundation under cyclic loading by taking into account

the model with isotropic hardening with the bounding surface elasto plastic model. For the

first time the bounding surface plasticity was defined by Dafalias and Hermann (1982) and it

is used instead of loading surface. Here the image point P (Figure 2.17) is defined within the

surface which is associated by the specific mapping rule to the point IPon the surface. At this

time the plastic modules will be defined. As a function of the distance between the point P

and the image point IP, so the size of the plastic modules is various by changing the distance

between these two points. di Prisco et al. (2003a) define the bounding surface according to the

loading-reloading response and gives a continue variation of the plastic modules, while the

virgin loading response is defined according to the hardening rule and this model coincide

with the Nova and Montrasio model. Pedretti (1998) validated this model by experimental

cyclic tests on loose and dense sand, and di Prisco et al. (2003a) defined the purely elastic

region similar to the concept of the elastic bubble introduced by the Al Tabbaa and Wood

(1989). This model has been declared that the behavior is fully reversible limited to the sort of

ice-cream cone (figure 2.17) within the surface limited by the bounding surface, already

defined in monotonic loading as already seen also in Nova and Montrasio macro-element. The

cone is defined by the position of the center of the spherical cap, A 1, and by its radius that is a

fix quantity. The angle at the cone apex, the origin of axes, is determined by the continuity

condition between the cone and the spherical cap. Assume now that a point P1on the surface

of the ice-cream represents the current state of stress, and that the stress increment P1P2 is

such that plastic strains occur. It is assumed that these strains are given by:

Equation 2.28

In Equation 2.28, the plastic multiplier , as well as the gradient of the plastic

potential

, are calculated in the image point I1, as if the stress point were on the bounding

surface. The image point is determined by the intersection of the bounding surface with the


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[28]

straight line A1P1(mapping rule). The matrix is a diagonal matrix, the role of which is thatof a weight function. For its definition, which is rather complex, the interested reader can

refer to di Prisco et al. (2003).

As far as the evolution of the loading-unloading locus is concerned, three possibilities

exist:

(a) Irreversible strains do not occur and the current stress point is within the zone S of

Figure 2.17b. In this case the elastic domain does not evolve.

(b) Irreversible strains do not occur but the current stress point belongs to the border

between zone R and S defined in Figure 2.17b. In this case the elastic domain evolves and

shrinks. The center of the ice-cream cone, point A1, shifts along the straight line connecting itwith the axes origin, while the current stress point remains on the inner border between zones

R and S.

(c) Irreversible strains take place. Point A1shifts along the straight line connecting it

with the image point on the boundary surface previously defined, while the current stress

point belongs to the outer border of the sphere.

(a) (b)

Figure 2.17: (a) Bounding Surface and domain fully elastic behavior (b) The unloading-reloading domain: definitions of

the active zone S and of zone R (after di Prisco et al., 2003b)

2.5 Response of Soil -Foundation I nteraction under Cyclic L oading

In case the soil layer under the footing system is consists of a dense sand layer, during

the cyclic test by controlling the rotation and keeping constant the vertical load, a typical S

shape response can be observed in the M graph, and for the large the uplift phenomena


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[29]

will apparent (Figure 2.18a). On the other hand in the case of loose sand, this phenomenon is

not so obvious and sort of punching mechanism will happen (Figure 2.18b).

(a) (b)

Figure 2.18: Experimental data concerning the rigid square shallow foundation cyclically titled (a) Dense sand (b) Loose

sand (PWRI, 2005)

One of the nonlinearities and damage that may occur while the foundation is subjected

to the cyclic load is the uplift phenomena (Figure 2.19).

Figure 2.19: Defining the uplift of the foundation and parameter (Cremer et a. 2001)


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[30]

Recently many authors have been working on this issue. In this case Cremer et al.

(2001) and Cremer et al. (2002) have been proposed macro-element for dynamic soil-

foundation interaction and considering the coupled plasticity uplift model. Furthermore other

researchers such as Grange et al. (2008) modified those works for the circular footing and

Chatzigogos et al. (2009) and Chatzigogos et al. (2010) worked on the combination of

plastification, uplift and sliding. Further study on this issue can be found on the work from

Shirato et al. (2008) and Grange et al. (2009).

As already mentioned Cremer et al. (2001) and Cremer et al. (2002) have been

presented a macro-element for dynamic soil-structure interaction analyses, in which the non-

linear soil-foundation behavior is characterized by a plasticity model coupled with an uplift

model. The system displacements are written as follows (in terms of generalized kinematic

parameters):

Equation 2.29The assumption divided the total displacement (

) to the elastic displacement (

),

plastic displacement () and uplift ().Stiffness for each system is a tangent stiffness. In this phenomenon all of the

parameters are coupled although they calculated separately. Nonlinearity comes from the

plasticity and uplift subsystem.

Here the major elements according to the work proposed by Cremer will be discussed

briefly:

a) Elasticity:

For this part the dimensionless stiffness matrix will be defined as follows:

[

] Equation 2.30


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Here the is defined as the maximum bearing capacity of the shallow foundationunder the vertical load and B is the width of foundation. The diagonal parameters , and are defined as the real part of elastic impedances of foundation. It must be considered thatnon-diagonal parameters are too small to compare to the diagonal parameters and generally

they do not have too much effects on the foundation response so it is possible to neglect them

(Cremer et al. 2001).

b) Plasticity

In this part a modified shape for the failure criterion is defined, flow rule and

hardening rule suitable to define the uplift phenomenon by considering kinematic and

isotropic hardening.

- Failure criteria

Following the work by Selenon and Pecker (1995a and b) for the homogeneous

cohesive soil without tensile strength, the equation of the bounding surface, Cremer suggested

the following formula for the failure criteria:

Equation 2.31

Where V, H and M are the parameters for describing the non-dimensional variables

which already defined as , h and m by the Nova and Montrasio (1991) and a,b,c,d,e and f are

the constitutive parameters and they will define the bounding surface shape.

- Hardening rule

The evolution of hardening rule here is defined by observing the hardening from the

uplift behavior and following the Prevost (1978), multi-yield model for anisotropic undrained

clay (Figure 2.20).


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(a) (b)

Figure 2.20: Prevost's multi-yield model for anisotropic undrained clays (1978): (a) Triaxial compression (b) Extension

stress-strain curves.

For any loading, the surface, initially reduced to a straight line segment, is dragged

along by the forces point F. It simultaneously undergoes an isotropic growth and a kinematic

translation of the ellipse centers in the (H'-M') plane, with a movement of the extreme point P

along the V' axis. This point moves in such a way that, when point F reaches the failure

surface, point P simultaneously reaches the extreme failure point V'= 1 (Figure 2.21).

(a) (b)

Figure 2.21: Failure locus according to the evolution of the hardening rule (Salencon and Pecker, 1995a and b) (a) H'- V'

surface (b) H'- M' surface


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[33]

Yield locus according to this rule is written as:

Equation 2.32

, and are the hardening variables defining the former the isotropic hardening andthe latter two kinematic hardening of the yield function, respectively, whereas:

Equation 2.33

Equation 2.34

Equation 2.35

Here is the isotropic hardening parameter.

- Uplift

A new parameter will be defined here, which shows the area of the footing that

detached from the soil (Figure 2.19). The assumption which here is considered by Cremer, is

that the horizontal loading does not effect on the uplifting percentage.

This behavior of the uplift domain is divided into an elastic-uplift and a plastic-uplift

domain. The elastic domain defines the set of forces that have already been applied to the

foundation. The plastic domain covers the set of forces that have not yet been reached during

the loading history.


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2.6 Ratcheting Phenomena

For large number of cyclic loads applied to the foundation, while there is not any

damage observed, the irreversible displacement will be accumulated at a decreasing rate, and

a sort of stabilization takes place (Figure 2.22). The experimental work by di Prisco et al.

(2003a), already observed such phenomena and confirmed experimentally that the results are

affected by the amplitude of the cycles, generalized stress path and the image point which

already discussed (see Figure 2.17).

(a) (b)

Figure 2.22: Experimental result (di Prisco et al. 2003a), obtained by keeping constant the vertical load and apply

horizontal cyclic load (a) generalized strength path (b) horizontal load versus horizontal displacement

It is quite difficult to find an approach capable quantitavely to define the ratcheting

phenomenon. di Prisco (2012) in the research based on multi-mechanism viscoplasticity

assumption defines elasto plastic constitutive relationship as below:

Equation 2.36

Where will describe response of material at a very small strain rate refer to theelastic/reversible strain and is for the small strain rate of the mechanical response ofmaterial, and these two parameters control typical standard cyclic response and show that


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[35]

when the cyclic amplitude is small, irreversible strain do not develop and the typical shake

down response is reproduced (Figure 2.23).

Figure 2.23: Mechanical response of the system in case of shake down (di Prisco, 2012)

The third parameter ,is control the dissipation of energy and reduction in stiffnessdue to the change of the size of the cycles. By adding this parameter it is possible to

generalize the ideal plastic adaptation where no accumulation of irreversible displacement

will take place (Figure 2.24).

Figure 2.24: Mechanical response of the system in case of an ideal-plastic adaption (di Prisco, 2012)


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[36]

The last part will describe the ratcheting phenomena and the coupling betweenthese mechanisms cause the progressive accumulation of irreversible strain due to repeated

loading and unloading.

In fact the parameter which concerns the ratcheting phenomena can be described inthree ways (Figure 2.25):(a) First it can produce this accumulation with the same rate, in the case of no

coupling between the principal plastic mechanism and the one that introduce the ratcheting

phenomenon

(b) Second it can be stabilized and the rate of accumulation will be decreased and sort

of positive hardening can be taking place

(c) The ratcheting can produce continually increase in the accumulation rate.

In case of granular material as already discussed in experimental work with di Prisco

(2003a) the second case is suitable which employed the isotropic hardening and non-

associated flow rule.

(a) (b) (c)

Figure 2.25: Mechanical response of the system (a) Perfect ratcheting (b) Progressive stabilization (c) Increase the

accumulation (di Prisco, 2012)

Figure 2.26: Coupling between the ratcheting plastic mechanism and the principal one (di Prisco, 2012)


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2.7 Sti ff ness and Damping Factor

Many experimental results such as works done by Pedretti (1998), PWRI (2005) and

Paolucci et al. (2007), by taking into account the well-known concepts for secant stiffness

(K), and damping ratio () observed a clear decay in stiffness K and increase in after

applying cyclic load and having horizontal or even rocking displacement. Damping ratio ()

which is calculated as a ratio between dissipated energy D and the stored elastic energy

w (Equation 2.37), (Figure 2.27 d). It must be noted that these abacus are affected by the

amplitude of the cycles.

Equation 2.37

The example of the results illustrated in the figure 2.27 refers to the sand. In case of

stiffness K, it is normalized with respect to the initial small strain stiffness K0. In the same

plot, the numerical results employing the macro-element concept which is developed by di

Prisco et al. (2003a) are also reported. It is shown that even for the small foundation rocking

angle such as 0.001 rad the rotational secant stiffness of the foundation reduces by a

percentage varying from 40% to 60%, depending on the relative density of the soil.

Damping factor values vary between 5% and 10% for rotations up to 0.001 rad, while

they increase significantly for higher rotations up to about 20% for dense sands and 30% for

medium dense sands.

As already discussed when the rocking angle is sufficiently high, the uplift dominates

the response of the system and damping ratio will not be observed any more.


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(d)

Figure 2.27: (a) Normalized rocking (b) translational stiffness (c) damping factor for Dr = 90% (Paolucci et al.2007) at

increasing values of rocking angle and horizontal displacement, respectively (d) definition of rocking/translational

stiffness and damping factor

To investigate the effects of the different factors such as amplitude of loading and

different relative density of the soil on the stiffness and damping ratio di Prisco et al. (2003a)

and Paolucci et al. (2007) performed couple of experimental and numerical tests. As it is

obvious form the figure 2.28, by decreasing the valuethe decay in become more rapid

while the increase in damping become more rapid.


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(a) (b)

(c) (d)

Figure 2.28: Influence of theratio on the dependence secant rotational stiffness and damping factor for dense and

medium dense sand respect to the rocking angle (a),(b): the secant rotational stiffness K (c),(d): The damping factor

(Paolucci et al. 2007)

The numerical results by Paolucci et al. (2007) interpolated according to the following

expression:

, ( ) Equation 2.38


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The dimensionless parameters: a, m and b, which are listed in table 2.1, allowing the

best fitting of the analytical to the experimental results.

Dense san Me um san

Vmaxc/V a m a m2 458.36 1.30 37.73 686.26 1.30 39.39

3 281.95 1.11 32.76 386.24 1.11 47.61

4.5 262.90 1.00 43.93 339.87 0.98 67.79

6 292.81 0.94 62.25 352.13 0.92 90.64

7.5 324.76 0.91 66.96 398.44 0.89 104.49

9 785.05 0.89 85.08 433.12 0.86 119.20

10 415.50 0.88 95.60 452.44 0.84 130.85

15 575.36 0.83 164.42 653.02 0.79 210.42

20 1010.90 0.86 233.70 1219.47 0.83 285.15

25 2461.06 0.95 305.97 2461.06 0.89 367.70

30 5192.13 1.02 382.51 5192.13 0.96 442.47

Table 2.1: Values of parameters a, b and m for equation 2.38 (Paolucci et al. 2007)


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[42]

Chapter

3

Descr iption of the Experimental Device

In chapter 3 a specific discussion about the experimental set up which is available at

the geotechnical laboratory of the department of the civil and environmental engineering at

Politecnico di Milano will be presented. This small scale apparatus is capable of applying

cyclic or monotonic, horizontal and vertical loads to the foundation. It was originallydesigned for the shallow foundation but by modifying some parts of the system, it can be

possible to apply different foundation configurations such as single pile or shallow foundation

plus piles.


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Chapter 3 Description of the Experimental Device

[43]

3.1 Experimental Set-up

This experimental apparatus is composed of a rigid support frame made of metal profiles of

the type H140, which is fixed and used as a structure of a main body (Figure 3.1). The main

body is consisting of the following parts:

(a) (b)

(c)

Figure 3.1: Overall view of the apparatus

3.1.1 Main Box

It consists of the wooden box with the glass walls (Figure 3.2). This is used as the

main box of sand for doing the tests. The size of the box is 200 mm width, 900 mm long and

400 mm depth. This box is laterally confined by the tempered glass walls with the thickness

of 10 mm. The walls are reinforced by means of three steel bars. Those are 8 mm tie rods

which make the walls rigid enough to avoid any out of plane movement and also the glass


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walls are sufficiently smooth to prevent any shear stress to develop at glass-sand interface.

The box is filled with dry sand by the pluviation method.

Figure 3.2: Main box (Reinforced glass galls by tie rods)

3.1.2 Sand Reservoi r

A wooden box used as a sand reservoir, with the size of 300 mm width, 470 mm depth

and 1250 mm length (Figure 3.3). The vertical position can be modified in order to match the

desired falling height. The bottom part of this reservoir is equipped with the appropriate grid

(Figure 3.4). By changing the grid size and the falling height, the desired relative density can

be obtained.

Figure 3.3: Sand reservoir Figure 3.4: Grids employed to control the density


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3.1.3 Spreader Cai sson

A spreader caisson is provided for the system, which is placed between the main box

and the sand reservoir box (Figure 3.5), so it makes the sand fall in the uniform way, and it

makes possible to have regular and homogeneous sand. The final step to prepare the sample

consist of the regulating the sand level by means of a vacuum cleaner (Figure 3.6), and it must

be considered that this action must not make any disturbance to the soil; the entire system can

be moved backward and forward with respect to the testing box so there is not any

disturbance, induced by the system.

Figure 3.5: Spreader Caisson

Figure 3.6: Use vacuum cleaner to make the uniform sand level


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3.1.4 Loading System

The loading system allows applying two independent loading components along

vertical and horizontal direction, respectively, without inducing any overturning moment on

the model foundation. The loads are applied by means of two double acting Bellofram air

cylinders connected to a pressure control panel giving an independent regulation line for each

of the two chamber of every cylinder. In both cylinders the air pressure of right chamber is

controlled by a proportional 4-20 mA control valve while air pressure in the left chamber is

regulated by a long-term stable manually driven pneumatic regulator. The manual regulator

allows to accurately apply (0.05 kPa sensitivity) a convenient backpressure between 15 and

600 kPa at the beginning of the test and then keep it constant. According to the desired

loading history, the pushing and pulling actions are then applied regulating the control signal

(0-10 V) of the proportional valve corresponding to a pressure interval between 20 and 400

kPa. The regulation is obtained by mean of a 12 bit D/A converter (PCI port interface)

allowing an overall regulation precision of 5 kPa.

Figure 3.7: Loading system

3.1.5 Di splacement Tr ansducer

The rigid displacements of the model foundation are recorded by two 050 mm

estensimetric displacement transducers with an estimated average precision (standard

deviation) of 0.05mm.


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Figure 3.8: Displacement transducer

The positioning of the transducer takes place thanks to a bar fixed to the supporting

frame by means of bolts and it makes it possible to rotate.

Figure 3.9: Positon of the displacement transducer

3.1.6 Load Cell

Each piston is equipped with two load cells placed, the same way for horizontal and

vertical directions, at the connection between cylinder rod and its extension needed to reach

the model foundation at the middle of the testing box. Horizontal load cell allows to measure

an applied force up to 1961 N with an estimated precision of 1.5 N. Vertical load cell has a

960 N capacity with an estimated precision (standard deviation) of 0.5 N.

Figure 3.10: Load cell


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3.1.7. Ai r Pressur e Transducer

Electronically controlled air pressure line is constantly monitored by a 01000 kPa

pressure transmitter with an estimated precision (standard deviation) of 2kPa. Manually

controlled air pressure lines are regulated with the support of the same pressure transmitters,

which can be connected to the different circuit acting on some ball valves, while the long term

monitoring is obtained by a couple of bourdon tube pressure gauges (class 2.5).

Figure 3.11: Air pressure Transducer Figure 3.12: Manual control and air pressure lines

3.1.8 LabVIEW Environment

The whole system is controlled by a dedicated software (developed in the LabVIEW

environment), handling the data pre-processed by a 16 bit A/D converter. A low-noise power

supply device for all transducers has been also designed to improve measurement reliability.

Figure 3.13: Outlook of LabVIEW software


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3.2 Cali bration

The calibration of the instruments is essential to enable the correct interpretation of the

data read during the testing.

The tools that allow us to collect the data of interest for our tests are:

- Displacement transducer for the measurement of foundation displacements.

- Pressure transducer for measuring the pressure exerted by the piston on the

foundation.

- Load cell to measure the load transmitted to the foundation.

The purpose of calibration is to find the factor for each instrument which translates the

variation of microvolt initially measured by the electronic devices to the appropriate

measuring unit relative to the type of instrument. For example, for the displacement

transducer, the calibration allows to determine how many microvolts correspond to the actual

displacement in millimeter. In this way, by inserting the appropriate values identified by this

calibration, the values read from the computer will be directly expressed in millimeters. This

value will be showed in kilogram for the load cell and in Kilopascal for the pressure

transducer.

3.2.1 Calibrati on the Displacement Transducer

For calibrating this device, the micrometer is used which allows to measure the

movement of the transducers with the high accuracy.

The procedure is to turn the micrometer manually until the desire displacement, then

put the corresponding number in the computer which already has microvolt data sent by the

displacement transducer. In this way we can have the function between the millimeter and

microvolt data for each step, that for the best calibration it shows the linear line relation

between these two parameters (Figure 3.15).


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Figure 3.14: Displacement transducer attached to the micrometer

Figure 3.15: relation between displacement (mm) and microvolt

3.2.2 Calibrati on the Load Cell s

The purpose of this calibration is similar, which means to find the bind between the

microvolt (real data sent to the computer) and the corresponding force value.

As it can be seen in figure 3.16, the load cell will be placed in the frame suitably made

for this purpose, to connect the load cell with the bar which is possible to put the weights on

that and also prevent of any rotation. The first reading in microvolt is relating to the load cell

without applying load, after that for each step 10 Kilogram load will put on manually.

As it discussed before, two kinds of load cell was used, the one with maximum load

100 Kilograms and another one with 200 Kilograms limit, which the linear relationship

between microvolt and corresponding kilogram data for each load cell shows below:


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Figure 3.16: Calibration the load cell

(a) (b)

Figure 3.17: (a) Relation between load (kg) and microvolt for load cell with 100 Kg limit (b) Relation

between load (kg) and microvolt for load cell with 200 Kg limit

3.2.3 Calibrati on the Ai r Pressure Cell

For calibration, a manual pump calibration will be used. The principle of operation is

by means of plastic tubing with low deformability, the two cells are connected to the pump

pressure available for quick and easy generation of hydraulic pressures (Figure 3.18). All the

tubes and connections must be saturated by means of pure oil to apply the hydraulic pressure.

After making all the connection saturated by the oil, we will increase the hydraulic pressure in

each step until it reaches the limit of the pressure cells which it is equal to 1000 kPa.


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Figure 3.18: Pressure cells connected to the pump for generating hydraulic pressures

3.3 Type of F oundations

3.3.1 Rigid Pi les with F ix Head

Rigid steel piles are used for the tests with the following characteristics:

Density [ kg/m3] Poissons ratio Shear modulus

G [N/mm2]

Elastic modulus

E[N/mm2]

7850 0.3 80769.23 210000

Table 3.1: Piles characterizations

Four kinds of piles were used. Long pile with 655 mm length (316 mm effective

length in the soil) with 22 and 40 mm diameter and shorter pile with 555 mm height (230 mm

effective length) with 22 and 40 mm diameter are considered (Table 3.2).

Pile 1 Pile 2 Pile 3 Pile 4

Diameter (mm) 22 22 40 40Total Length (mm) 655 555 655 555

Effective Length (mm) 316 230 316 230

Table 3.2: Size of the piles


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Figure 3.19: Measurement of the piles (mm)

3.3.1.1 I nstall ing Procedure for Pil es

For this purpose, the pile will be clamped to the system, so the pile can move only

horizontally with the mobile parts of the machine. Two steel parts will be used for the

clamping system.

Part A: is bolted permanently to the apparatus through a single M8 bolt passing

through a guide on the rear part of the elements and carried through two protrusions of the

width of 50 mm and 20 mm.

Part B: is a steel element of the size of 100 mm width, 32 mm height for a width of 12

mm with the purpose of fixing the pile

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