Coating Breakdown Analysis of Steel Plates in … · Coating Breakdown Analysis of Steel Plates in Marine Structures ... elo de elementos ﬁnitos em ANSYS. É estudado ... Coating

Coating Breakdown Analysis of Steel Plates in MarineStructures

Sílvia Cristina Guerreiro de Sousa

Thesis to obtain the Master Degree in

Naval Architecture and Marine Engineering

Supervisor: Professor Doutor Yordan Garbatov

Examination Committee

Chairperson: Professor Doutor Carlos Guedes SoaresSupervisor: Professor Doutor Yordan GarbatovMember of the Committee: Professor Doutor Ângelo Teixeira

July 2015

"Learn from yesterday, live for today, hope for tomorrow.

The important thing is not to stop questioning."

Albert Einstein

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Acknowledgments

There are many to whom I owe the deepest gratitude to the completion of this dissertation.

First and foremost, I wish to express my deepest gratitude to my advisor, Prof. Doutor Yordan Garbatov,

for his motivation, availability, knowledge and exceptional guidance along the development of this study.

It was a pleasure learning from him and working under his supervision.

To all my friends and colleagues throughout all the Naval Engineering journey, a heartfelt thanks, for

their friendship, companionship, help, mutual support and for sharing such remarkable moments with

me, during this years. All of them, certainly, helped me to reach this moment and to be who I am today.

To Mauro, a kindly thank you for the unconditional support, motivation and for his availability to help me

and to discuss my work with me. Thank you for being my solid anchor and for always believing in me,

even when I didn’t.

Last but not least, a very special thanks to my family, especially my parents, Cidália and José, and my

sister Sandra for all the support, patience, trust and sacrifices during this years. Without them, none of

this would be possible.

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Resumo

O objetivo desta dissertação é estudar o colapso do revestimento de chapas de aço em estruturas marí-

timas sujeitas a compressão uniaxial no plano. Este trabalho é motivado pela necessidade de melhorar

os revestimentos de modo a prevenir a corrosão marítima do aço. É desenvolvido e validado um mod-

elo de elementos finitos em ANSYS. É estudado o comportamento de encurvadura da chapa de aço

revestida, com imperfeições iniciais e uma macro-delaminação na interface, baseado no problema de

valores próprios da encurvadura e na análise não-linear. A análise não-linear permite grandes deslo-

camentos e impor o contato entre o revestimento e a placa. A compressão na placa e as imperfeições

causadas pela má preparação da superfície e/ou má aplicação do revestimento leva à criação de ten-

sões residuais e consequentemente à encurvadura local da camada de revestimento. Com o aumento

do carregamento, a encurvadura aumenta atingindo o tamanho crítico onde o revestimento começa a

delaminar. Baseado na forma exponencial do modelo de zona coesiva é implementado um elemento de

interface com espessura zero simulando a adesão entre placa e revestimento. Quando a tensão nestes

elementos excede o valor crítico, a tensão é redistribuída, resultando na deformação dos elementos e

consequente separação/delaminação ao longo da interface. É estudada a influência do comprimento de

delaminação, da espessura do revestimento, das propriedades mecânicas e de interface no comporta-

mento de encurvadura e pós-encurvadura do revestimento da placa de aço. Baseado nestes resultados,

é criado um diagrama de avaliação de falha do revestimento para diâmetros de macro-delaminações.

Palavras-chave: Colapso do Revestimento, Método dos Elementos Finitos, Encurvadura, De-

laminação, Zona Coesiva, Diagrama de Avaliação de Falha

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Abstract

The objective of this dissertation is to study the coating breakdown of steel plates in marine structures,

subjected to uniaxial in-plane compression. This work is motivated by the necessity of improving the

coating protection systems, in order to prevent the marine steel corrosion. A finite element model is

developed in ANSYS and its validation is performed. The buckling behaviour of the coated steel plate,

with an initial imperfection and a macro-delamination in the interface, is studied based on the eigenvalue

buckling and nonlinear strength analysis. Nonlinear analyses allow to take into account large displace-

ments and impose contact constraints between coating and plate. The compressive loads acting on

the plate, in addition to the imperfections caused by the inadequate steel surface preparation and/or

poor coating application, leads to the residual stresses creation and consequently the local buckling of

the coating layer occurs. As the load increases the buckling region increases reaching the critical size,

when the coating layer starts to delaminate. Based on the exponential form of the cohesive zone model,

a zero thickness interface element, to model the adhesion between plate and coating layer, is employed.

When the stress in these elements exceeds the critical value, the stress field is redistributed, resulting

in the elements deformation and separation/delamination across the interface. The influence of delami-

nation length, coating thickness, mechanical and interface properties on the buckling and post-buckling

behaviour of the steel plate coating are studied. Based on these results, a coating failure assessment

diagram for the macro-delamination diameters is created.

Keywords: Coating Breakdown, Finite Element Method, Buckling, Delamination, Cohesive Zone,

Failure Assessment Diagram

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aim and structure of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 State of the Art 5

2.1 Maritime Corrosion and Coating Protection . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Initial Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Adhesion Failure of the Coating Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Delamination Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Cohesive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Theoretical Background 21

3.1 Coating Failure and Corrosion Degradation . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 One-dimensional Coating Thin Film Buckling . . . . . . . . . . . . . . . . . . . . . 27

3.2.2 Coating Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Cohesive Zone Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Interface Rupture Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.2 Exponential Model of the Cohesive Zone . . . . . . . . . . . . . . . . . . . . . . . 33

4 Finite Element Modelling and Verification 37

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Linear Eigenvalue Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 38

xi

4.2.2 Nonlinear Strength Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Post-buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4 The Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4.1 Geometry, Loading and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 40

4.4.2 Element Type and Mesh Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.3 Interface Coating-Steel Using the Cohesive Zone Model . . . . . . . . . . . . . . . 42

4.4.4 Initial Imperfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5 Validation of the Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5.1 Buckling and Post-buckling Response of a Laminated Beam . . . . . . . . . . . . 46

4.5.2 Verification of the Buckling Shapes With the Variation of the Parameter β . . . . . 48

4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Result Analysis 51

5.1 Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Effect of the Delamination Length, l, on the Coating Buckling Behaviour . . . . . . 52

5.1.2 Effect of the Coating Thickness, h, on the Coating Buckling Behaviour . . . . . . . 53

5.1.3 Effect of the Coating Properties, Ec and νc, on the Coating Buckling Behaviour . . 55

5.2 Post-buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2.1 Effect of the Delamination Length, l, on the Coating Post-Buckling Behaviour . . . 58

5.2.2 Effect of the Coating Thickness, h, on the Coating Post-Buckling Behaviour . . . . 62

5.2.3 Effect of the Coating Properties, Ec and νc, on the Coating Post-Buckling Behaviour 66

5.2.4 Effect of the Interface CZM Constants . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Failure Assessment Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Final Discussion and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Final Remarks and Future Work 81

6.1 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography 91

A FEA Flowchart 93

B Interface Parametric Analysis 94

B.1 Auxiliary Parametric Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B.2 Coating Von Mises Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.3 Normal Interface Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

B.4 Tangential Interface Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

B.5 Normal Interface Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.6 Tangential Interface Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

C Scattered coating failures assessment scale 101

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List of Tables

4.1 Interface material constants in ANSYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Main dimensions of the composite beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Main dimensions of the laminated beam for the two cases verified . . . . . . . . . . . . . 49

5.1 Delamination length considered in the analysis . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Coating thickness considered in the analysis . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Coating properties considered in the analysis . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Variation of the coating properties in relation to the Ec = 3000 MPa and νc = 0.37 results . 56

5.5 Comparison of the buckling load, breakdown load and respective coating deflection as a

function of the delamination length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


function of the coating thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65


function of the coating properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.8 Analysed interface parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.9 Maximum normal separation (∆n), tangential separation (∆t), normal stress (Tn) and

tangential stress (Tt) values achieved in the interface, considering σmax = 25, 15, 10

MPa, as a function of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

B.1 Maximum normal separation (∆n), tangential separation (∆t), normal stress (Tn) and

tangential stress (Tn) values achieved on the interface, considering σmax = 25 MPa and

δn/δt = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

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List of Figures

1.1 Thickness of corrosion wastage as a function of time (Guedes Soares and Garbatov, 1999) 2

1.2 Coating breakdown of steel plates in marine structures . . . . . . . . . . . . . . . . . . . . 4

2.1 Mechanical, thermal and chemical bond failure (Schweitzer, 2005) . . . . . . . . . . . . . 12

2.2 Idealized sketch of delamination and blistering (Sørensen et al., 2009a) . . . . . . . . . . 13

2.3 Buckling modes of thin films on substrates: (a) coherent wrinkling of the film-substrate

system; (b) film buckling with delamination along the film-substrate interface (Tarasovs

and Andersons, 2012) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Buckling mode shapes under compressive loading conditions . . . . . . . . . . . . . . . . 16

2.5 The three fracture modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Theoretical tractions in the cohesive zone ahead of the crack tip (Travesa, 2006) . . . . . 18

3.1 (a) Coating blistering; (b) Coating delamination . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 (a) Coating degradation during the ship ballast tank life; (b) Ballast tank coating break-

down along the service life time (Contraros, 2004) . . . . . . . . . . . . . . . . . . . . . . 23

3.3 (a) Stable equilibrium; (b) Neutral equilibrium; (c) Unstable equilibrium . . . . . . . . . . . 24

3.4 Typical post-buckling equilibrium path of a plate (Kubiak, 2013) . . . . . . . . . . . . . . . 25

3.5 Typical ductile material stress-strain curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6 (a) Typical Steel load-deflection relation; (b) Coating load-deflection relation . . . . . . . . 26

3.7 Geometry of the one-dimensional blister (Hutchinson and Suo, 1992) . . . . . . . . . . . 27

3.8 Buckling of a plate under uniaxial compression . . . . . . . . . . . . . . . . . . . . . . . . 28

3.9 Cohesive model: representation of the physical damage process by separation function

within numerical interfaces of zero height - the cohesive elements (Cornec et al., 2003) . 31

3.10 Four classes of cohesive zone laws (Bosch et al., 2006) . . . . . . . . . . . . . . . . . . . 32

3.11 (a) Variation of normal traction, Tn, with ∆n for ∆t = 0; Variation of shear traction, Tt, with

∆t for ∆n = 0 (Chandra et al., 2000) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1 Incremental Newton-Raphson procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Geometry of the structure modelled: (a) Intact state; (b) Buckled state . . . . . . . . . . . 40

4.3 Coated plate’s boundary conditions (Top View) . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 SOLID45 geometry in ANSYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.5 Example of the meshed coated plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xv

4.6 Schematic of interface elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.7 Interface modelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.8 Modelled initial imperfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.9 Plate model, Bohoeva (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.10 Deformed laminated plate with a single delamination . . . . . . . . . . . . . . . . . . . . . 47

4.11 Comparison of the theoretical results obtained from Bohoeva (2007) and the current FEM 48

4.12 Buckling shapes, Bohoeva (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.13 Buckling shapes obtained from the FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1 Euler and FEM critical bulking load as a function of the delamination length, l . . . . . . . 53

5.2 Euler and FEM critical bulking load as a function of the coating thickness, h . . . . . . . . 54

5.3 Euler and FEM critical bulking load for different values of Young’s modulus, E, and Pois-

son’s ratio, ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Euler and FEM critical buckling load for different Ec (a) and νc (b) . . . . . . . . . . . . . . 56

5.5 Coated plate behaviour until corrosion initiation . . . . . . . . . . . . . . . . . . . . . . . . 57

5.6 Compressive load-deflection relations of the coating layer as a function of the delamina-

tion length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.7 Coating shapes and vertical displacement distribution for each variation of the delamina-

tion length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.8 Coating deflection shape for each variation of the delamination length . . . . . . . . . . . 60

5.9 Coating Von Mises stress distribution for each variation of the delamination length . . . . 60

5.10 Compressive load-Von Mises stress relations of the coating layer as a function of the

delamination length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.11 Compressive load-deflection relations of the coating layer as a function of the coating

thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.12 Coating shapes and vertical displacement distribution for each variation of the coating

thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.13 Coating deflection shape for each variation of the coating thickness . . . . . . . . . . . . . 63

5.14 Coating maximum deflection for each variation of the coating thickness . . . . . . . . . . 63

5.15 Coating Von Mises stress distribution for each variation of the coating thickness . . . . . . 64

5.16 Compressive load-Von Mises stress curves of the coating layer as a function of the coating

thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.17 (a) Over thick coating detaches easily. Poor preparation between coats of paints causes

early failure ; (b) Where the coating is too thin, early failure occurs in service (ABS, 2007) 65


properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.19 Coating deflection shape for each variation of the coating properties . . . . . . . . . . . . 67

5.20 Coating maximum deflection for each variation of the coating properties . . . . . . . . . . 67

5.21 Coating Von Mises stress distribution for each variation of the coating properties . . . . . 67

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properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.23 Schematic representation of the interface parametric analysis . . . . . . . . . . . . . . . . 69

5.24 Compressive load-deflection relations and respective deflection shapes of the coating

layer, considering σmax = 25 MPa, as a function of δn/δt . . . . . . . . . . . . . . . . . . . 70





5.27 Compressive load-Von Mises stress relations of the coating layer considering σmax = 25,

15, 10 MPa, as a function of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.28 Considered values for the coating delamination size/diameter . . . . . . . . . . . . . . . . 74

5.29 Compressive load-deflection behaviours including the Von Mises failure criteria . . . . . . 75

5.30 Coating failure assessment diagram for macro-delamination diameter, considering 85%

of the breakdown load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.31 Failure assessment diagram for macro-delamination diameter values . . . . . . . . . . . . 76

5.32 Poor, fair and good coating conditions for the delamination diameters . . . . . . . . . . . . 77

5.33 Zoom of the limit conditions as function of the deflection for the lower values of l . . . . . 78

A.1 Flowchart of the developed FEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

B.1 Coating load-deflection and load-Von Mises stress relations, considering σmax = 25 MPa

and δn/δt = 1, as a function of δn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B.2 Coating deflection shapes, considering σmax = 25 MPa and δn/δt = 1, for each variation

of δn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.3 Coating maximum deflection, considering σmax = 25 MPa and δn/δt = 1, for each variation

of δn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.4 Coating Von Mises stress distribution, considering σmax = 25 MPa, for each variation of

δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.5 Coating Von Mises stresses distribution, considering σmax = 15 MPa, for each variation of

δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.6 Coating Von Mises stresses distribution, considering σmax = 10 MPa, for each variation of

δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.7 Normal interface separation (∆n) distribution, considering σmax = 25 MPa, for each varia-

tion of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


tion of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


tion of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

xvii

B.10 Tangential interface separation (∆t) distribution, considering σmax = 25 MPa, for each

variation of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98





B.13 Normal interface stresses (Tn) distribution, considering σmax = 25 MPa, for each variation

of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99


of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99


of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.16 Tangential interface stresses (Tt) distribution, considering σmax = 25 MPa, for each varia-

tion of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100


tion of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100


tion of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

C.1 Original scatter diagrams for corrosion and coating breakdown assessment (ABS, 2007) . 101

xviii

Nomenclature

Greek symbols

β Ratio between the normalized length and normalized thickness

βpl,0 Intact plate slenderness

∆ Opening displacement

δ Imposed displacement

∆n Normal opening displacement

δn Normal separation across the interface

∆∗n Value of ∆n after complete shear separation

∆t Tangential opening displacement

δt Shear separation across the interface

νc Coating Poisson’s ratio

νs Substrate Poisson’s ratio

φ Surface potential function

φn Work of separation in normal direction

φt Work of separation in tangential direction

σ1 First principal stress

σ2 Second principal stress

σ3 Third principal stress

σcr Critical compressive stress

σmax Maximum normal traction at the interface

σvm Von Mises equivalent stress

σx Compressive stress applied

xix

σY P Material yield stress

τmax Maximum shear stresses

Roman symbols

h Normalized thickness ratio

l Normalized length ratio[KTi

]Tangent stiffness matrix

{∆ui} Displacement increment vector

{F a} External load vector

{Fni r} Internal load vector

a Plate length parameter in the plate’s initial imperfection equation

b Plate width

D Bending stiffness

E Young’s modulus of a plate

Ec Coating Young’s modulus

Es Substrate Young’s modulus

H Steel plate thickness

h Coating thickness

h0 Intact plate thickness

L Total plate length

l Delamination length

m Number of half-waves in longitudinal direction

n Number of half-waves in transverse direction

Nxy Plate’s membrane shear force per unit length

Nx, Ny Plate’s membrane forces, per unit length, in x and y -direction respectively

p(x, y) Applied load per unit length normal to plate’s

Pcr Critical buckling load

q Coupling parameter

r Coupling parameter

xx

T Traction

Tn,max Maximum normal traction without tangential separation

Tn Normal traction

Tt,max Tangential traction without normal separation

Tt Tangential traction

Ux, Uy, Uz Displacement in x, y and z-direction respectively, referent to finite element method

w(x) Deflection in z-direction along the delamination length

w0 Plate’s initial out-of-plane deflection/initial imperfection

wmiddle Deflection at the middle of the delamination

Subscripts

c Coating

n Normal component

s Substrate/steel

t Tangential component

x, y, z Cartesian components

xxi


Glossary

1-D One-Dimensional.

2-D Two-Dimensional.

3-D Three-Dimensional.

ABS American Bureau of Shipping is a Classification Society.

ANSYS ANSYS is an engineering simulation software (computer-aided engineering).

APDL ANSYS Parametric Design Language is a scripting language that can be used to automate com-

mon tasks or even build a model in terms of parameters.

CPS Coating Protection System.

CZM Cohesive Zone Model consists of a constitutive relation between the traction acting on the interface

and the corresponding interfacial separation.

FAD Failure Assessment Diagram.

FEA Finite Element Analysis is the practical application of the FEM.

FEM Finite Element Method is a numerical technique for finding approximate solutions to boundary

value problems.

FPSO Floating Production Storage and Offloading unit is a floating vessel used by the offshore oil and

gas industry.

HGSM Hull Girder Section Modulus.

IACS International Association of Classification Societies.

NDFT The Nominal Dry Film Thickness is the thickness of the paint coat after it has cured.

PSPC Performance Standard for Protective Coatings for water ballast tanks is a IMO standard.

Python Python is a widely used general-purpose, high-level programming language.

xxiii


Chapter 1

Introduction

1.1 Motivation

For many years, steel has been used in marine structures industry as a low-cost material with excellent

mechanical properties for welding. Marine structures such as ships, since the increasing request for

lighter, cheaper and more reliable structures, can be considered as a thin-walled structures, i.e., can be

modelled basically as a box girder consisting of a number of stiffened plates (Khedmati et al., 2007).

Due to the flexing of the ship beam in seaway conditions, these stiffened plates experience significant

compressive stresses, which make these structures susceptible to failure by instability. Thus, the com-

pressive strength of steel plates is of primary concern to the designer.

The marine environment is particularly aggressive for steel-made structures due to its high corrosion

susceptibility. Many marine structural components have failed at sea because of excessive degradation

caused by corrosion, even when all these structures have met the requirements of design defined by

the Classification Societies. Corrosion is a major cause of marine structural failures. Melchers (1999)

reported that 90% of ship structural failures are attributed to corrosion. The penalty of this type of failure,

including safety hazards and interruptions in ship operations, have became more costly and specifically

recognized in the last years, which resulted to an increase of the attention given to the control and pre-

vention of corrosion.

Corrosion wastage may take the form of general corrosion, pitting corrosion, stress corrosion cracking,

corrosion fatigue, bacterial corrosion, etc. The pitting corrosion is a form of extremely localized corrosion

that leads to the creation of small holes in the metal surface. Once it has been initiated its continuation

is determined by reactions within the pit itself, which at the point of attack is anodic, and with the outer

surface being cathodic. This type of corrosion is among the major types of physical defects found largely

in ship structures. The areas of the ship, more susceptible to local corrosion, are the ballast tanks owing

to the intense contact with seawater on both sides, humidity and the chloride-rich environment, even

when empty. Because of the double hull configuration, the access to the ballast tanks is limited and so

1

the maintenance are very difficult and expensive. Double hull ballast tanks act as the achilles hull of the

ship (Baere et al., 2013).

As corrosion protection, the ship steel structures in marine environments are provided with protective

surface coatings and in addition precaution, cathodic protection in the form of sacrificial anodes. This

additional measure is required because coating defects and discontinuities will inevitably be present in

protective coatings. A good paint system is the first line of defence against the corrosive marine environ-

ment, principally in the ballast tanks. The ability of coatings to resist corrosion over extended periods is

an important contributor in safeguarding the capital investment in the structure of a vessel (ABS, 2007).

Unfortunately, coatings do not last forever and its very difficult to predict its durability. They age, weaken,

deteriorate and eventually their useful life ends. The most common cause for premature coating failure

is insufficient care during the mixing, application and curing processes. Poor application technique and

inadequate surface preparation result in imperfections and consequently poor adhesion to the substrate.

These adhesive coating defects, in addition to the compressive forces experienced by the plates leads

to local blistering, peeling, delamination and ultimately, coating breakdown. When the surface coating

deteriorates locally, the salt water penetrates the coating and the steel starts to corrode by pitting.

Many authors have investigated the effects of corrosion in the fatigue life of ship and offshore struc-

tures. Some mathematical corrosion wastage or corrosion growth rate as a function of time models have

been developed. A few of them expressing this dependence between corrosion growth and time as lin-

ear. More recent models presented nonlinear formulations, staged relationships between the corrosion

growth and time considering different characteristics for corrosion degradation such as a protective coat-

ing or protective systems and their failure in time, and corrosion arrest due to lack of oxygen. Guedes

Soares and Garbatov (1999) presented a model of nonlinear corrosion that was adopted by many au-

thors. This time-dependent corrosion model may be separated into three phases (see Figure 1.1).

d

τc tτd(t)

tO' BO

A

∞

Figure 1.1: Thickness of corrosion wastage as a function of time (Guedes Soares and Garbatov, 1999)

In Figure 1.1, the period of time t ∈ dO′, Oe represents the first stage of degradation where the protec-

tion of the metal surface works properly and for that reason it is associated with zero depth corrosion

2

wastage. The point t = O, identify a random point in time when the coating fails and pitting corrosion

starts to increase at a rate described by the slope OA. The second phase is initiated when the corrosion

protection is damaged and it corresponds to the existence of corrosion, which decreases the thickness

of the plate (t ∈ dO,Be). This process was observed to last a period around 4-5 years in typical ship

plating. Finally, the third phase corresponds to t > B and the corrosion process stops and corrosion rate

becomes zero. The corroded material stays on the plate surface, protecting it from the contact with the

corrosive environment. Cleaning the surface or any involuntary action that removes that surface material

originates the new start of the non-linear corrosion growth process. This model is flexible and can be

fitted to any specific situation, once the long-term corrosion wastage and the duration of the corrosion

process is known.

The corrosion phenomena, is still not a fully quantifiable due to its dependence of many variables, such

as the effective duration of the coating protection. In the most studies about marine corrosion developed

in the last years, the bigger concern is to predict the behaviour of the structures after the corrosion initia-

tion. The period without corrosion, which is equal to the time interval between the painting of the surface

and the time when its effectiveness is lost, has been considered in the these works, but in terms of time

only based on statistics. The reasons that lead to the coating failure are still poorly studied. At this point,

it is necessary to turn the attention to the prevention of corrosion and to study what happens before the

coating breakdown occurs (point O in Figure 1.1). Advances in coating technology can offer significant

cost saving if developed and successfully demonstrated. In the recent years, the Classification Societies

have been concerned not only with the effect of corrosion on the structural behaviour, but also with the

prevention of the onset of corrosion, promoting the improvement of the coating properties and different

application techniques. However, in accordance with its rules, they only require repair of the coating

systems when already exists corrosion. To complete the existing requirements, it is extremely important

to study the behaviour of the coating layer on top of steel plates and to define criteria that show when a

certain coating defect requires to be repaired in order to prevent corrosion initiation at these local. This

represents a pertinent objective for this thesis and also for future works.

In laminated composite materials due to the intelaminar stresses created by impacts, eccentricities in

the structural load or from discontinuities within the structure itself, a similar failure problem occurs. How-

ever, the problem of the coating delamination in steel plates presents some differences from the studies

already made for composites, being the material properties of the layers the biggest difference.

Recent advances in computations and software technology have made possible to analyse complex

structures. The Finite Element Method (FEM) in the last years has become the most powerful tool

in terms of structural analysis, allowing to analyse the strength of simple and complex structures in a

better way than other existing numerical methods. This tool has been widely used to study the fracture

mechanism that occurs in delamination phenomena, using the cohesive interface elements.

3

Thus, the proposed scope of this thesis is to model and analyse the delamination of the coating of ship

hull structures using the FEM to better understand the breakdown of corrosion protection systems.

L

hH

Disp(δ)

b

w

l

b

a

LDisp(δ)

BOTTOM

a

L

SHIP HULL

PLATE

Figure 1.2: Coating breakdown of steel plates in marine structures

1.2 Aim and structure of the dissertation

The aim of the present dissertation is to perform a finite element analysis of the initial phase of the

coating breakdown. It will be concerned with a particular failure mode of the thin film coating layer, the

buckling delamination of macro diameters. For that, a nonlinear finite element model of a coated steel

plate localized in ballast tanks of ship hull structures is developed. In order to simulate the glue between

the steel plate and the coating layer, a cohesive finite element with a zero thicknesses is used. Based

on some important parametric variations, the delamination of the coated plate subjected to a uniaxial

compressive load and specific boundary conditions of support with initial imperfections is studied. A

failure assessment diagram, only applied for macro-delamination diameters, is developed to help in the

coating failure prevention and to complete the work performed until now by the Classification Societies.

To achieve the aim, the dissertation is developed in the following structure: Chapter 2 presents a brief

review of the state of the art; Chapter 3 is dedicated to the theoretical background used in the thesis;

Chapter 4 describes the finite element model used in the analysis; Chapter 5 shows the results and their

discussion; Chapter 6 presents the final conclusions and the future work.

4

Chapter 2

State of the Art

In this chapter, a historical review of what has been done related to the topic of this thesis will be

presented. Starting by a brief review about the corrosion problem in maritime structures and the coating

protection used, passing to the presentation of some works where imperfections were modelled, after to

the description of some studies about the adhesion of the thin films, after that the delamination problem

in composites is also reviewed and finally the cohesive zone model used many times for modelation of

fracture in interfaces is presented.

2.1 Maritime Corrosion and Coating Protection

Ship structures operate in a very aggressive environment, during the lifetime. They are subjected to sea

water salinity, oxygen content, temperature, chemistry, pH level and pollution, which causes obviously

the degradation. The phenomena of generation and progress of corrosion is a result of three sequential

processes: degradation of paint coatings, generation of pitting point and their subsequent growth (Ya-

mamoto, 1998).

Several authors have analysed the effect of corrosion on the structural behaviour of steel marine struc-

tures. Melchers (2005) described the structural reliability theory necessary for the assessment of the

risks associated with corroding infrastructure, indicating that the appropriate probabilistic models to de-

scribe the loss of material due to corrosion. With a pipeline example, he showed a dramatic increase in

the probability of failure as the pipeline ages and pitting corrosion grow.

A few years earlier, Guedes Soares and Garbatov (1996) presented a time-variant formulation to model

the degrading effect that corrosion has on the reliability of ship hulls. The effect of general corrosion

was represented as a time dependent decrease of the plate thickness that affects the midship section

modulus. The results of this work showed that the effect of the plate replacement when its thickness

reached 75% of the original thickness and how the limit value of the thickness in the repair criteria

influences the reliability and the decision about repair actions.

5

In order to establish a more rational criteria for corrosion margins and permissible corrosion levels, Ya-

mamoto (1998) applied a probabilistic corrosion model. The effects of these established criteria to the

reliability for thickness diminution due to corrosion were analysed. Numerical analyses were made for

the lower part of Bulk Carrier hold frames to evaluate the effect of the standards on decisions. In this

work, it is assumed that the life of paint coatings, which is the period before active pitting points are

generated, follows a log-normal distribution.

For reasons of economy, the mild steel and low alloy steel are preferred materials for offshore struc-

tures, ship hulls and other structures. Melchers (1999) reviewed the various important factors in marine

corrosion, outlining previous models and described an ongoing work aimed at developing a probabilistic

phenomenological model for the time-dependent material loss of mild and low alloy steels in immersion

conditions.

Floating production, storage, and offloading (FPSO) systems have been used for a development of off-

shore oil and gas fields, exposing themselves to aggressive environmental conditions with regard to

corrosion. Sun and Bai (2003) presented a methodology for the time-variant reliability assessment of

FPSO hull girders subjected to degradations due to corrosion and fatigue. The corrosion defect was

modelled as an exponential time function with a random corrosion rate and when corrosion occurs, the

plate thickness is uniformly reduces. Paik et al. (2003) also present a mathematical model for predicting

time-variant corrosion wastage of the structures of single and double-hull tankers and FPSOs and FSOs

based on a statistical analysis of a corrosion measurement database.

By studying some of the available corrosion models, Qin and Cui (2003) concluded that these models

may not fully reflect the reality. In order to improve this situation, a new model was proposed. By using

an assumed corrosion database, the flexibility and accuracy of this model was briefly demonstrated and

the influence of the corrosion models on reliability is confirmed. Melchers (2003a) showed that statisti-

cal models using pooled data are of poor quality with a very wide variety. Also argued that there is an

urgent need for better-quality models to represent adequately the deterioration mechanism of corrosion.

In the second part of his work, Melchers (2003b) considered probabilistic corrosion modelling based on

corrosion mechanical principles including the effect of environmental and others factors.

Damages to ships due to corrosion are very likely, and the likelihood increases with the ageing of ships.

For that reason, Wang et al. (2003b) explored the assessment of corrosion risks to ageing ships using

an experience database of corrosion wastage in oil tankers. This study can be used to develop de-

sign requirements for corrosion additions and wastage allowance, define design limits to the hull girder

strength, develop a time reliability approach and also schemes for risk based inspection. Later, a statis-

tical study of the hull girder section modulus (HGSM) of ageing tankers was presented by Wang et al.

(2008). An attempt was also made to quantify coating life or coating longevity. Preliminary results of

this study show that the coating life is about 6.5 years on average with a standard deviation of about 1.5

6

years. These values appear to be reasonably in line with the understanding of the industry that coating

in ballast tanks starts to breakdown when the ships have between 2 to 10 years old. In this study, the

assumption of coating life helps to measure the overall influence of coating breakdown anywhere in the

entire transverse section. However, such assumption does not precisely track when coating breakdown

takes place, because coating breakdown can take place in localized areas without having a sensitive

impact on HGSM.

The rate of corrosion depends on the rate of each partial reaction, and for simple cases it is possible

to quantify this rate by use of electrochemistry theory. An accurate estimation of corrosion rates plays

an important role in determining corrosion allowances for structural designs, planning for inspections,

and scheduling for the maintenance. Wang et al. (2003a) present an estimation of corrosion rates of

structural members in oil tankers based on a corrosion wastage database. The aim of this study was to

update the knowledge on corrosion rates in steel ships and to contribute to the efforts of mitigating the

risks of corrosion.

Corrosion varies in its forms. It is convenient to classify corrosion by the forms in which it manifests it-

self. Many studies have been presented on mathematical formulations for corrosion modelation. Guedes

Soares and Garbatov (1999) presented a model of nonlinear corrosion that was adopted by many au-

thors. Later in 2007 this model was fitted to real plate corrosion data of deck plates of ballast and cargo

tanks by Garbatov et al. with data collected by the American Bureau of Shipping (ABS). As described in

the Section 1.1, the model suggested by Guedes Soares and Garbatov (1999) has three main phases.

In the first phase, it is assumed that there is no corrosion because the coating protection is effective.

Failure of the protection will occur at a random point of time and the corrosion wastage will start a non-

linear process of growth with time. In this transition period the corrosion grows asymptotically to a point

of depth, evolving from the transition stage for a period where the growth rate decreases to zero. The

influence of adopting this model instead of a linear one is demonstrated by studying the reliability of a

corrosion-protected plate subjected to compressive loads, and maintenance actions.

The effects of different marine environmental factors on the corrosion behaviour of steel plates totally

immersed in salt water were studied by Guedes Soares et al. (2005). A new corrosion wastage model

was proposed, based on the one of Guedes Soares and Garbatov (1999). Sea water properties such as

salinity, temperature, dissolved oxygen concentration, pH and flow velocity are considered in this new

study. Also a numerical example was illustrated for ships trading in different routes in the Pacific Ocean.

Later, Guedes Soares et al. (2011) proposed a new non-linear time dependent corrosion model to as-

sess the short term and long term corrosion degradation under marine immersion conditions. In this

new model only sea water temperature, dissolved oxygen concentration and flow velocity are taken into

account. It is said that the coating effectiveness is not independent of the surrounding environment and

can vary with it. The corrosion degradation rate was shown to be linearly proportional to seawater tem-

perature and dissolved oxygen concentration. It was also shown that development of corrosion models is

7

not a straightforward process based only on theory due to the many variables and uncertainties involved.

In 2008, Guedes Soares et al. introduced a corrosion model that extends one of the existing models

by adding three variables that reflect the relative level of temperature, carbon dioxide and hydrogen

sulphide concentrations which is relevant to the rates of corrosion to be expected in ship tanks. They

proposed an equation that can serve as a guide to shipowners and Classification Societies about which

variables need to be monitored to allow more accurate predictions of corrosion wastage in ship tanks.

Two main corrosion mechanisms are generally present in steel plates. One is a general wastage that

is reflected in a generalised decrease of the plate thickness. Another mechanism is pitting which con-

sists of much localised corrosion with very deep holes appearing in the steel plate (Guedes Soares and

Garbatov, 1999). Pitting may be initiated by a small surface defect, being a scratch or a local change

in composition, or damage to the protective coating. This phenomenon is more commonly found in the

bottom plating, the aft bays of tank bottoms, welds of seams, stiffeners, horizontal surfaces or side shell

plating where the way of water flow is bigger. If the pitting corrosion is left unchecked, it may cause

severe problems such as loss of structural strength, integrity and resulting in hull penetration, which

leads to leakage and eventually serious potentially pollution incident. For that reason, strategies for the

inspection, maintenance and repair of the parts that can corrode and their protection systems, should

be planned and implemented.

Many maritime accidents have been caused by corrosion and this has led to stringent regulations con-

cerning protective coatings for ballast tanks. The Coating Performance Standard for Ballast Tank Coat-

ings (PSPC) became effective in 2008. According to the International Association of Classification So-

cieties (IACS) standard, a tank condition can be divided into three categories: "GOOD", "FAIR" and

"POOR". When it comes to general ship assessment, this division offers a practical evaluation method.

The large and complex structure of the ballast tanks is one of the most affected areas by the corrosion

because are frequently wetting and drying of sea water. Under conditions of high temperatures, inappro-

priate ventilation, high stress concentration, high stress cycling, high rates of corrosion can be achieved

in ballast tanks. The effective corrosion control in segregated water ballast spaces is probably the single

most important feature, next to the integrity of the initial design, in determining the ship’s effective life

span and structural reliability. Guedes Soares et al. (2008) wrote that the corrosion in ballast tanks is

much different from that in cargo spaces and both of them are different from the corrosion behaviour in

the void spaces of the double bottom, double hull and machinery spaces. Even in the same tank bottom,

the corrosion through the void space above the liquid level is different from the immersed part.

The primary form of corrosion protection for steel is the application of coatings. When such coatings

represent a physical barrier to the environment, cathodic protection is usually applied as an additional

precaution. This additional measure is required because coating defects and discontinuities will in-

8

evitably be present in protective coatings (Tezdogan and Demirel, 2014). Qin and Cui (2003) assumed

that in the reality the coating protection system (CPS) deteriorates gradually so corrosion may start as

pitting corrosion before the CPS loses its effectiveness completely. The corrosion rate was defined by

equating the volume of pitting corrosion to uniform corrosion. Structural integrity as well as economical

and environmentally safe operation of ships depends to a great extent on effective and durable corrosion

protection. Taking into consideration the corrosion rates applicable to the different areas of ship struc-

tures, special attention should be given to the fact that they are dependent on the existence of corrosion

protection (Emi et al., 1991).

Ballast tanks need careful attention since the useful life of a ship is often dependent on the condition

of these big structural components. For that reason, high performance anti-corrosive and anti-fouling

epoxy coatings are used in these structures to protect the steel. The IMO Standards mandates a target

useful ballast tank coating life of 15 years, which is considered to be the time period, from initial appli-

cation, over which the coating system is intended to remain in good condition (Hoppe, 2005). However,

the coating effective life depends largely on the steel surface and the edge preparation and application

conditions. With the time, a significant contributing factor in coating degradation is increasing brittleness

and loss of flexibility, causing cracking and disbanding at structural “hot spots”. The coating may be

flexible enough when newly applied and a few years afterwards (Schweitzer, 2005). Then, due to cyclic

temperature variations, the more volatile, low molecular weight coating constituents are lost by evapora-

tion or washed away by ballast water. Oxidation and other chemical changes of the coating constituents

further contribute to the gradual loss of flexibility. Also the presence of soluble salts of the metal/paint

interface have a detrimental effect on the integrity of most paint systems, as Fuente et al. (2006) studied.

So, the behaviour of coatings under simulated ballast tank conditions is of primary interest.

Abdel-samad et al. (2014) investigated the effect of recent coatings used in marine ship surfaces for pre-

venting corrosion. Experiments were performed according to standard tests to evaluate and measure

the coating adhesion to steel and to measure the corrosion wear rate if any for three types of coatings.

The results of this work indicated that all tested types of paint have resulted in a reduction in the corro-

sion rate compared with the uncoated steel.

Heyer (2013) studied the influence of the microbiology in the ship ballast tanks. A perfect coating appli-

cation should be aspired to decrease the possible attachment sites of microorganisms. She concludes

that the biodegradation of ballast tank coatings should be considered in the future in order to develop

new strategies to overcome microbial deterioration processes of the applied coatings. Another study

about the degradation process for ballast tank coatings in the marine environment was developed by

Heyer et al. (2014). A commercially available ballast tank coating was exposed to a bacterial commu-

nity. As a conclusion, the bacterial leave the coating system highly vulnerable to deterioration in real

environmental conditions.

9

An economic modelling approach in order to reducing the cost of ballast tanks corrosion was made by

Baere et al. (2013). They conclude that the best way to protect ballast tanks is by applying a standard

PSPC15 coating on a perfectly prepared substrate and under good application conditions. Lifetime last-

ing aluminium anodes could then be used as a backup system, if they are well distributed across the

ballast tank and properly maintained.

The ability of a coating to protect the metal surface against atmospheric corrosion is decided by water

and oxygen permeability apart from other factors like wet adhesion of the film, pigment volume concen-

tration, presence of other additives, etc. Sangaj and Malshe (2004) reviewed the relationship between

the structure of polymers and its permeability to oxygen and water.

Kobayashi (2007) studied the deterioration of the surface coating on steel structure, exposing steel ma-

terials with tar-epoxy coatings in a marine environment for one and nineteen years. Some specimens

were provided with artificial defects to investigate the effect of a coating defect on the progress of de-

terioration. The results demonstrated that after nineteen years of exposure, the coated steel corroded

more in a tidal zone and submerged zone than in a splash zone; After one year of exposure, the steel

with artificial defects deteriorated most severely in the splash zone and the surface coating starts to

deteriorate in the tidal and submerged zone.

2.2 Initial Imperfections

The behaviour of ship plating is influenced by various factors, namely material and geometrical proper-

ties, boundary and loading conditions, initial distortions, residual stresses and the degree of use (Ro-

drigues, 2011). Assessment of the reliability, safety and stability of structures with initial imperfections

belong to the most complex problems in applied mechanics.

Along the years, many authors, such as Faulkner (1975) and Smith et al. (1988), have tried to estab-

lish design methods to predict the ultimate strength. These authors have developed design methods

to predict the collapse strength that have implicitly a level of imperfections on the equations. Faulkner

(1975) showed that the compressive strength of plates depends mainly on their slenderness, despite

other effects also influencing it, such as residual stresses and weld induced initial imperfections.

Initial distortions in steel structures are a consequence of fabrication procedures and processes and

shipyard workshop operations (e.g. welding, manoeuvring, assembling, cutting, etc.). The welding and

cutting processes also induce residual stresses in the structures. Although nowadays the cutting process

can be done without variations of temperature. The dependency between the initial distortions and these

processes makes the initial distortions a highly variable factor (Rodrigues, 2011). The residual stresses

and the geometrical imperfections influence considerably the behaviour of ship plating.

10

Smith et al. (1988) proposed a expression that approximates the initial imperfections by a Fourier series.

These sine series have been used by many authors in the last years in studies of the plate behaviour.

w(x, y)

= w0sin(mπx

a

)sin(nπyb

)(2.1)

where x,y,z are the plate’s coordinate system, a is the length of the plate, b is its width, m is the number

of half waves in x-direction, n is the number of half waves in y -direction and w0 is the maximum out-of-

plane deflection.

The amplitude of the imperfection is given by the following expression proposed by Faulkner (1975):

w0 = 0.1h0β2pl,0 (2.2)

where, h0 is the intact plate thickness and βpl,0 denotes the intact plate slenderness given by:

βpl,0 =( bh0

)√σY PE

(2.3)

where, E represents the Young modulus and σY P the yield stress.

Using the previous equations for the modelation of initial geometrical imperfections, Guedes Soares and

Kmiecik (1993), Sadovský et al. (2005), Teixeira and Guedes Soares (2008), Rodrigues (2011), Silva

(2011) simulated the initial imperfections in order to predict its effect on the ultimate strength of plates.

The behaviour of the studied imperfect plates under compressive loads were analysed by using a non-

linear finite element analysis.

To define the geometrical imperfections, one possibility is to have available measured data. Another con-

sists in assuming an empirical or design initial imperfection. In some of the previous refereed works, the

initial geometrical imperfections shapes are defined based on the measurements of the real distortions

that are present in ship plates after all the construction procedure.

2.3 Adhesion Failure of the Coating Film

Regardless of what excellent properties a coating might possess, it is useless unless it also has a good

adhesion. The coating’s resistance to weather, chemicals, scratches, impact, or stress is only of a value

while the coating remains on the substrate. Consequently, the knowledge of the adhesion of coatings is

of importance. As Schweitzer (2005) described, the adhesion is a complex phenomenon related to the

physical effects and chemical reactions at the "interface". Some theories were proposed to explain the

phenomenon of the adhesion, including the mechanical attachment, electrostatic attraction, true chemi-

11

cal bonding, and true paint diffusion. The adhesive strength is affected by the coating thickness and the

solvent retention when solvents containing coatings are used (Schweitzer, 2005).

A paint coating is, in essence, a polymer. The substrate materials can inhibit a rigidity higher than that

of the coating. Under such conditions, fracture will occur within the coating if the system experiences an

external force of sufficient intensity. Several external factors can induce stress between the bond and

the coating, causing eventual failure (see Figure 2.1).

Figure 2.1: Mechanical, thermal and chemical bond failure (Schweitzer, 2005)

The factors presented in Figure 2.1 can act individually or in combination and were described by

Schweitzer (2005). The mechanical interface failure occurs due to a combination of tensile and shear

stress. The termal failure is due to changes in the temperature that causes differences in the contraction

and expansion coefficients. And finally, the chemical bond failure that is described by the penetration of

a media and absorption at the interface.

Some important topics related to the use of marine and protective coatings for the anti-corrosive purpose

were studied by Sørensen et al. (2009a). A description of the different environments and anti-corrosive

coating systems are made. Some of the mechanisms leading to a degradation and failure of coating sys-

tems are described, and the reported types of adhesion loss are discussed. The existence of internal

stresses in the coating, which are developed due to an inability of the coating to shrink, may be added

further to the complexity of the coating system. Internal stresses in coatings can significantly affect the

durability of anti-corrosive coatings by resulting in loss of adhesion, cracking, or cohesive failure (Hare,

1996a). Hare (1996b) states that the internal stresses are originated during the film formation and curing

processes as a result of a solvent evaporation in all films and the cross-linking of thermosetting films. In

most of the coating systems, the internal stress is also produced by the paint film’s ageing processes.

This stress is a result of the long-term, environmentally induced changes in the molecular morphology

and structure.

Among the most severe and common forms of visible failure in immersed organic coating systems are

those of blistering and delamination (Figure 2.2). Sørensen et al. (2009a) established that the difference

12

between cathodic blistering and cathodic delamination is addressed by the events that occur after hy-

droxyl ions have interacted with the metallic substrate. Blistering is the result of an osmotic pressure,

which is developed due to the high water solubility of the reaction products from the cathodic reaction.

Delamination from damage is the result of bonds breaking at the coating-metal interface, resulting from

the alkalinity of the cathodic reaction products (Nguyen et al., 1991).

Figure 2.2: Idealized sketch of delamination and blistering (Sørensen et al., 2009a)

The delamination of both defect-free and artificially damaged barrier coatings has been reported to be

significantly reduced when the thickness of the coating is increased because coatings behave as semi-

permeable membranes (Sørensen et al., 2009b).

The process of elastic deformation of thin films on substrates, under thermal and mechanical loadings

were investigated by Panin and Shugurov (2009). In their work, mechanisms of a formation of wrinkle

and buckle patterns on the surfaces of metal and oxide films were studied. There are two main buckling

modes of thin films on substrates: coherent buckling of the film and the substrate called "wrinkling", and

buckle delamination of the film called "buckling" (see Figure 2.3).

Figure 2.3: Buckling modes of thin films on substrates: (a) coherent wrinkling of the film-substratesystem; (b) film buckling with delamination along the film-substrate interface (Tarasovs and Andersons,2012)

The competition between the two common failure modes as presented in Figure 2.3 were analysed by

Tarasovs and Andersons (2012) in order to assess the critical strain when the buckling may occur, at

given geometric and material parameters. An approximate scaling relation is derived for the energy re-

lease rate of buckling-driven delamination of a coating deposited on a compliant substrate. An overview

13

about the origins and details of the stresses, which are developed in the thin films and multilayers and

the failure modes stemming from these stresses was provided by Hutchinson (1996).

Some authors have studied the buckling and post-buckling delamination of thin films on top of substrates

subjected to varying conditions. Gioia and Ortiz (1997), Hutchinson (2001), Chiu and Erdogan (2003),

Ruffini et al. (2012), Zhuo et al. (2015) investigated the delamination of thin films subjected to com-

pressive loads. The film layer buckles away from the substrate forming a blister. After that, the buckled

film may grow by interfacial fracture, a process which, under the appropriate conditions, may result in

catastrophic failure of components. The energy release rate and mode mixity induced by buckling were

evaluated, and the growth and arrest of the interface crack under mixed loading induced by buckling

were discussed based on the energy criterion and fracture mechanisms. Xue et al. (2012) went further,

studying buckling of thin compressed films under mechanical and thermal loads. External uniaxial com-

pressive load and thermal load were applied to the specimen in order to produce the buckling distortion

of thin films. Thermal stresses, caused by the mismatch of thermal expansion between the film and

substrate, induced large delamination and buckling of the film during cooling. The interfacial toughness

of the film buckling has been discussed through an elastic buckling model under mechanical and thermal

strain.

In the most part of the reported studies about the thin film behaviour, the influence of the substrate

properties was neglected. In the works of Suo and Hutchinson (1990) and Yu and Hutchinson (2002),

buckling and the consequent interface delamination crack were calculated as a function of the elastic

mismatch between the film and substrate using the Dundurs’ parameters. Also, Djokovic et al. (2014)

studied the influence of the elastic characteristics of the substrate, concluding that the elastic character-

istics of the substrate have significant influence on the buckling delamination of the coating in the form

of a straight-sided blister when the ratio of the Young’s moduli of the coating and substrate is bigger than

3.

To induce buckling in the compressed thin film from the substrate is commonly induced an initial small

imperfection. The influence of prototypical imperfections on the nucleation and propagation stages of

the delamination of compressed thin films was analysed by Hutchinson et al. (2000). The energy release

rates for separations that develop from imperfections were calculated.

The ability of the interface to sustain a certain loading without fracturing is called fracture toughness.

This quantity has been measured in experiments for a variety of interfaces. Audoly (2000) studied the

mode dependent toughness in a buckle-driven delamination of compressed thin films. For a wide class

of patterns of delamination, it was shown that the loading on the delamination front progressively goes

from mode I to mode II during the growth of the blister. He also studied a model of the interfacial fracture

with friction. An overview about the existing approaches to investigate the interfacial toughness in coated

systems was performed by Chen and Bull (2011).

14

A kinematically nonlinear finite element analysis of stability and finite growth of buckling driven delam-

ination was developed by Nilsson and Giannakopoulos (1995). A method to account automatically for

the redistribution of the stress field as the shape of the advancing delamination is used. Also a load

perturbation was employed as well as a front perturbation. Later, Erdogan and Chiu (2000) developed

an extensive work whose aim was to study a solution to the buckling instability problem by using the

continuum elasticity rather than a structural mechanics approach.

2.4 Delamination Analysis

Laminated composites are becoming the preferable material system in a variety of industrial applica-

tions, such as aerospace structures, ship hulls in naval engineering, automotive structural parts, micro-

electromechanical systems and also civil structures for strengthening concrete members. The increased

strength and stiffness for a given weight, increased toughness, increased chemical and corrosion resis-

tance in comparison to conventional metallic materials are some factors that contributed to the advance-

ment of laminated composites (Raju and O’Brien, 2008).

The behaviour of the coating layer on top of the steel plate subjected to compressive axial load can

be compared to the behaviour of the laminate composites, which are particularly prone to delamination

type of failures. Such delamination can occur as a result of interlaminar stresses created by impacts,

eccentricities in the structural load paths or from discontinuities within the structure itself. These types

of failure are particularly dangerous because: they generally reduce the overall laminate strength due

to material discontinuity; they act as imperfections when located eccentrically, and thus substantially

reduce the overall buckling strength of the laminate; and they grow under in-plane compressive loads,

since the delamination often buckles locally much earlier than global structural buckling, resulting in a

progressive reduction in laminate strength, finally leading to fatal failure. By means of the Finite Element

Analysis (FEA), experimental works and/or using theoretical models and formulations, investigators have

studied the buckling of delaminated plates. Chai et al. (1981), Rajendran and Song (1998), Bruno and

Greco (2000), Shan and Pelegri (2003), Obdrzálek (2010), Wang et al. (2011) studied the buckling be-

haviour of symmetric delaminated beams subjected to compressive loading in one-dimensional (1-D) or

two-dimensional (2-D) modelling. The models were also studied in post-buckling range which gives an

indication of the residual capacity of the plate after the instability. It was shown that the buckling load vary

with the position, length and number of delaminations, boundary conditions, and fibre orientation angles.

Multiple delaminations cause severe degradation of the stiffness and strength of composites. In the

work of Liu and Zheng (2013), the interactions between multiple delaminations of symmetric and un-

symmetrical composite laminates were studied. As conclusion, the asymmetry affects the delamination

and buckling behaviours of composite laminates largely when the initial multiple delamination sizes are

relatively small.

15

In the case of thin delaminated structures under compressive loading conditions, buckling can occur at

the global level, locally or in mixed level (Figure 2.4).

Global Buckling

Mixed Buckling

Unbuckled

Local Buckling

Figure 2.4: Buckling mode shapes under compressive loading conditions

Naganarayana and Atluri (1995) presented numerical methods for the evaluation of the energy release

rate along a delamination periphery under conditions of local buckling of the delaminate, as well as

global buckling of the entire laminate. It is observed that the energy release rate is a function of the

stress resultants, displacement gradients and strain energy density in the vicinity of the delamination

front. It is also observed that the computation of energy release rates, with any of the approaches pro-

posed, can be used in conjunction with any efficient analytical/computational post-buckling solutions for

the delaminated plate under given loads and boundary conditions.

The problems of buckling delamination can be both linear and nonlinear (Kachanov, 1988). Normally,

the post-buckling behaviour of thin delamination is modelled through a nonlinear procedure, which allow

to take into account of large displacements. Kardomateas (1989) studied the influence of large deflec-

tions of the energy release rate that characterizes delamination growth.

Fracture of structures or their load-carrying components is one of the primary causes of potentially

dangerous failures in such vital engineering systems as aircraft, ships, bridges, pipelines and offshore

platforms. These failures are, in fact, caused by a loss of stability, or a sudden unexpected transfer

from one state to another. Bolotin (1996) studied deeply the stability problems in fracture mechanics.

In his book, both the linear and nonlinear elements of fracture mechanics were presented, along with

the simplest approaches to the fatigue crack growth prediction. A generalized approach of the analytical

fracture mechanics was presented. Fracture of bodies with both single-parameter and multi-parameter

cracks were also studied in his work. The problems of local and global instabilities, such as buckling and

its interactions with fracture, debonding and delamination, were also analysed.

Based on the fracture mechanics concepts, there are three possible ways of subjecting a force to enable

a crack to propagate: Mode I, Mode II and Mode III as shown in Figure 2.5. Mode I represents the

16

opening mode of the crack faces, Mode II represents the sliding mode or transverse shear mode, and

Mode III represents the tearing mode deformation or longitudinal shear mode (Bolotin, 1996; Raju and

O’Brien, 2008). The combination of these three fracture modes is possible, being the mixed mode

cracking the most common in layered materials (Hutchinson and Suo, 1992).

Mode IIIMode I Mode II

Figure 2.5: The three fracture modes

Bohoeva (2007) developed a complete study about the stability of composite plates with defects such

as delamination subjected to compressive loads. In this study, the problem is solved as a nonlinear

formulation based on the energy approach and the method of perturbations. She performed numerical,

analytical and experimental analyses of a delaminated rectangular plate for the possible types of buckling

(see Figure 2.4). The results obtained from the analytical solutions are comparable with the numerical

and experimental data.

2.5 Cohesive Model

In recent years, the Cohesive Zone Model (CZM) approach has emerged as a popular tool for investigat-

ing fracture processes in materials and structures. It offers an alternative way to view failure in materials

or along material interfaces. This model was originally suggested by Needleman (1987), in order to

simulate the process inclusion debonding from a metal matrix. CZM is based on the cohesive zone con-

cepts of Dugdale (1960) and Barenblatt (1962) and is a purely continuum formulation. This approach

has been successfully employed in various numerical investigations including the crack growth analysis

in homogeneous ductile materials, interface debonding, impact damage in brittle materials, analysis of

sandwiched structures, etc. (Chowdhury and Narasimhan, 2000).

The cohesive damage zone models relate tractions to displacement jumps at an interface where a

crack may occur (see Figure 2.6). The damage initiation is related to the interfacial strength, i.e., the

maximum τo in the traction-displacement jump relation. When the area under the traction-displacement

jump relation is equal to the fracture toughness Gc, i.e., the energy needed for separation is achieved,

the traction is reduced to zero and new crack surfaces are formed (Travesa, 2006). When a damage

growth occurs, these cohesive zone elements open in order to simulate a crack initiation or crack growth.

17

Figure 2.6: Theoretical tractions in the cohesive zone ahead of the crack tip (Travesa, 2006)

The cohesive model has been formulated such that it can be used for practical applications, as was

refereed before. Cornec et al. (2003) developed a procedure for the application to the assessment of

engineering structures. This procedure consists of a specific traction-separation law, which is mainly

given by the cohesive stresses and the cohesive energy and also methods for determining the material

parameters.

One method to gain insight to the failure mechanism is the use of the finite element analysis. The

cohesive zone formulations are well-suited to be implemented in finite element codes. The cohesive

elements are normally designed to represent the separation at the zero-thickness interface between lay-

ers of three-dimensional (3-D) elements.

CZM are suitable to simulate the delamination because they can be used for both damage-tolerance

and strength analyses. For this reason, they have been successfully employed in numerical simulations

of delamination in laminated composite materials. Travesa (2006), Bohoeva (2007), Gozluklu (2009),

Waseem and Kumar (2014) performed a simulation of delamination in composites using a finite element

formulation of a cohesive interface elements by means of the CZM approach. Also a cohesive finite

element formulation was defined by Chowdhury and Narasimhan (2000) for modelling fracture and de-

laminations in solids. This formulation was developed for incorporating the CZM within the framework

of a large deformation finite element procedure. A special Ritz-finite element technique was employed

to control nodal instabilities and a few tests are presented in order to validate the developed work. In

addition, a quasi-static crack growth along the interface in an adhesively bonded system is simulated,

employing the cohesive zone model.

Nekkanty et al. (2007) created a two-dimensional finite element model to simulate the response of a

coating layer on top of a substrate subjected to an in-plane uniaxial tension. Coating cracking was

simulated with cohesive zone elements that followed the bilinear cohesive law. The effects of different

coating modulus, tangent modulus for the interlayer hardening, and critical stress values (σc) of the

cohesive zone elements were studied. The distribution of the crack spacing for different parameter

changes were quantified and compared to an experimental crack spacing distribution.

18

As refereed before, the cohesive zone models are typically expressed as a function of normal and tan-

gential tractions in terms of separation distances. The forms of the functions and parameters vary from

model to model (Chandra et al., 2000; Bosch et al., 2006). Chandra et al. (2000) used two different

forms of CZM (exponential and bilinear) to evaluate the response of the interfaces in titanium matrix

composites reinforced by silicon carbide fibres. The computational results were compared with experi-

mental data. Also, Bosch et al. (2006) wrote about all the classes of the cohesive zone laws that exists in

literature. Special attention was given to an improvement description of the exponential Xu and Needle-

man cohesive zone law for the Mixed Mode decohesion. All developed work were having in mind their

applicability to analyse the delamination of a polymer coating on a metal substrate.

Cohesive parameters can sometimes be determined by fitting the numerical simulations of fracture tests

to experimental data. The hope is that the extracted parameters represent material properties and can

be used to model the fracture of the same material under different loading conditions. However, this

(usually implicit) assumption is often not satisfied and, as a consequence, the cohesive parameters are

not unique, their values depend on the specimen geometry and loading history (Hui et al., 2011).

The adhesion loss is one of the most frequent defects between two attached surfaces. Gaspar (2011)

studied the factors that might have an impact in the adhesion property of renders. For that, she built

a numerical model in ANSYS, using the CZM, where it was possible to introduce those factors and to

analyse their impact in the model interface. A parametric analysis was performed by varying the inter-

face constants σmax, δn and δt in order to understand the general behaviour of the interface under these

variations.

The advantages, limitations and challenges of the cohesive zone model were studied by Elices et al.

(2002). A review of the cohesive model and some examples were performed in order to predict the

capability of the CZM when applied to different materials: concrete, Polymethyl-methacrylate (PMMA)

and steel. The advantages of cohesive zone models are their simplicity and the unification of the crack

initiation and growth within one model. The Cohesive Zone Model formulations are more powerful than

the Fracture Mechanics approaches because they allow the prediction of both initiation and propagation

stages, and thus, the damage tolerance and strength analyses can be done with the same design tool

(Travesa, 2006).

19


Chapter 3

Theoretical Background

The objective of this chapter is to describe the important concepts and formulations in the development

of this study. First a brief section about the coating failure and consequent corrosion degradation is

presented. After that, the concept of stability is presented, describing the mathematical formulations

used for the buckling analysis and also the failure method used for predicting the coating breakdown.

Finally, the cohesive zone model, used for simulating the interface between the coating layer and the

steel plate, is discussed, including the exponential cohesive law employed and the interface ruptures

modes that the CZM can model in ANSYS, using the interface elements.

3.1 Coating Failure and Corrosion Degradation

Steel structures are a common design solution because of their mechanical properties and machinability

at a low price, while at the same time they should be corrosion resistant. Very seldom these properties

can be met in one type of material. This is where coatings are necessary. By applying an appropriate

coating, a steel construction can resists to corrosion during years.

Coatings can be applied to large surface areas and they are capable to protect, even though a relatively

thin layer. An understanding of the basic principles that describe the interfacial interactions between the

coating layer and the substrate is necessary for an effective formulation and its efficient application.

The majority of coatings are applied on the external surfaces to protect the metal from natural atmo-

spheric corrosion and pollution. In some cases, coatings are also applied internally in vessels for cor-

rosion protection. Basically, there are four different classes of coatings: Organic, Inorganic, Conversion

and Metallic (Schweitzer, 2005). Organic coatings have been widely used to protect metals, as the case

of steel ships, in corrosive environment due to their good physical and chemical performance and low

cost. The epoxy coating systems, an example of an organic coating, is widely used to protect water

ballast tanks of ship from corrosion.

21

The adhesion is a complex phenomenon related to the physical effect and chemical reactions at the

interface between the coating and steel. The actual mechanism of adhesion is not fully understood.

Several theories have been proposed to explain the phenomenon of adhesion, including the mechanical

attachment, when a substrate surface contains pores, roughness, holes, voids, etc.; electrostatic attrac-

tion, when both the coating and surface contain electrical charges; true chemical bonding, when reactive

chemical groups exist in the substrate surface and coating; and true paint diffusion, when segments of

the macromolecules will diffuse across the interface (Schweitzer, 2005). Based on the coating used and

the chemistry and physics of the substrate surface, one or a combination of these mechanisms may be

involved.

Bond failure and coating breakdown can result from any one or a combination of the following causes

(Schweitzer, 2005):

• Poor or inadequate surface preparation and/or application of the paint to the substrate (that may

cause peeling, delamination, cracking, wrinkling, etc.);

• Atmospheric effects (such as humidity, oxygen content, temperature, chemistry, pH level, pollu-

tants, etc.);

• Structural defects in a paint film (e.g. hardness, flexibility, brittleness resistance, abrasion resis-

tance, mar resistance, etc.);

• Stresses between the bond and the substrate (that may cause mechanical, thermal and chemical

bond failure (see Figure 2.1));

• Corrosion (caused by wet adhesion, osmosis, blistering, cathodic delamination, etc.).

The surface preparation, which includes cleaning and pretreatment, is the most important step in any

coating operation. For coatings to adhere, surfaces must be free from oily soils, corrosion products, and

loose particulates. In ballast tanks, the surface preparation, both for new build structures and during

the maintenance is difficult, time consuming and expensive. It is necessary to remove all impurities

and old coating, and anything else that can cause imperfections which may lead to adhesion loss and

consequently blistering and delamination of the coating layer (Figure 3.1), causing corrosion on steel.

Figure 3.1: (a) Coating blistering; (b) Coating delamination

22

The coating breakdown in the ballast tankspaces of ships is a well-known phenomenon, but it is very

difficult to predict. IMO and the leading Classification Societies have introduced limited requirements

that have been applied through their respective instruments of regulations to address this issue.

As Contraros (2004) presented, in order to avoid the “Domino Effect” by which the coating breakdown

leads to corrosion, the corrosion causes scantling diminution, which then leads to structural failure, it is

necessary to recognise each and every variable affecting the coating behaviour on the structure. The

designer and shipowner need to consider these variables at the earliest stage so that the specification

and design of structural components can take into account the necessity for contributing to the coating

breakdown and incorporate appropriate remedies in the design and construction stages.

According to Contraros (2004), the coating breakdown process has three basic stages, each of which

can occupy a variable time frame, depending on the type of coating and the environmental conditions.

The first stage is the initiation phase, where the coating is still relatively plastic (extended over the first

six months of the coating life); followed by the stabilisation phase, where small areas of coating become

detached from steel by one or more blister or delamination (during a period of 6-20 months); and finally

the breakdown phase, which allows the corrosion to occur at a rate that may not easily be arrested by

regular maintenance (starts after about 6 years of the coating’s service life) (Figure 3.2).

(a)

Coating Breakdown (%)

Stabilization

Failure Point

Breakdown

Service Life

Initiation

Breakdown

Failure Point

Stabilization

Initiation

Service Life

Coating Properties

(b)

Figure 3.2: (a) Coating degradation during the ship ballast tank life; (b) Ballast tank coating breakdownalong the service life time (Contraros, 2004)

The corrosion degradation of ballast tanks varies in its forms and degree and is often the reason for se-

vere damage of ships. It is convenient to classify corrosion by the forms in which it manifests itself. Thus,

the basis for this classification is the appearance of the corroded metal. Each form can be identified by

a simple visual observation (Silva, 2011). The most widespread forms of corrosion in marine structures

are uniform corrosion and pitting corrosion due to their direct attack to the net thickness of the structural

components. Pitting is the type of corrosion more related to the coating breakdown because it occurs at

the points when the coating fails (called pitting points) (Yamamoto, 1998).

23

3.2 Buckling

The stability of a structure can be analysed by estimating its critical load, i.e., the load corresponding to

the situation in which a perturbation of the deformation state does not disturb the equilibrium between

the external and internal forces (Novoselac et al., 2012). The behaviour of a mechanical system, and of

structures in particular, can be tested (experimentally or numerically) by evaluating how it reacts when

external disturbances are applied.

Stability qualifies the state of equilibrium of a structure, i.e., whether it is in stable or unstable equilib-

rium. In the state of stable equilibrium, if the structure (e.g. column, plates, etc.) is given any small

displacement by some external load, which is then removed, it will return back to the undeflected shape.

Here, the value of the applied load P is smaller than the value of the critical load Pcr. By definition, the

neutral equilibrium state is the one at which the limit of elastic stability is reached. In this state, if the

column is given any small displacement by some external load, which is then removed, it will maintain

that deflected shape. Otherwise, the column is in a state of unstable equilibrium (See figure 3.3). The

instability is a strength-related limit state.

(b)(a)

δ

PPP

P = Pcr P > PcrP < Pcr

(c)

Figure 3.3: (a) Stable equilibrium; (b) Neutral equilibrium; (c) Unstable equilibrium

The equilibrium state becomes unstable essentially due to large deformations of the structure and in-

elasticity of the structural materials. The decreasing of the structural stiffness lead to instability failure.

The behaviour of the structure subjected to load higher than the critical one can be described by a

stable (the growth of displacement is caused by increased load) or unstable (displacements grow with

a decrease in load) post-buckling equilibrium path. The post-buckling behaviour of structures depends

on their type (Kubiak, 2013). Thin plates/beams supported on all edges lose their stability having a local

buckling mode and the stable post-buckling equilibrium path (Figure 3.4). The stability or instability of a

thin-walled plate/beam in terms of displacement (δ) and applied load P is presented in Figure 3.4.

24

Pcr

Load, P

Stable Stable

UnstableUnstable

Deflection, δ

Figure 3.4: Typical post-buckling equilibrium path of a plate (Kubiak, 2013)

The change in geometry of the structure subjected to compressive load, which results in its ability to

resist loads, is called buckling. The load producing buckling is called the critical buckling load, and it is

usually calculated using the eigenvalue linear buckling analysis for perfect structures. The importance

of buckling is the initiation of a deflection pattern, which if the loads are further increased above their

critical values, rapidly leads to very large lateral deflections. Consequently, it leads to large bending

stresses, and eventually to complete failure of the structural component.

The linear buckling analysis of plates makes it possible to determine accurately the critical loads, which

are of practical importance in the stability analysis of thin plates. However, this analysis gives no way

of describing the behaviour of plates after buckling, which is also of considerable interest. The post-

buckling analysis of plates is usually complex because it requires a nonlinear solution. The eigenvalue

buckling problems of plates can be formulated using the equilibrium method, the energy method and the

dynamic method. In this thesis, only the equilibrium method will be used in order to analyse the buckling

behaviour of the coated layer.

Buckling failure is usually caused by the elastic instability. Due to thin-walled configurations, and plate/shell

structures are more likely to buckle under compressive loads. External load usually represents a non-

linear relationship with the structural deformation in a buckling phase, and the gradient of the load-

deflection curve might be significantly decreased, representing a decrease of the structural stiffness.

In a complex structure, various buckling modes might occur according to the matching relationship be-

tween structural layout, stiffness parameters, dimensions of components, etc. For an isotropic plate, the

buckling wave shape is mainly related to its aspect ratio.

The deformation of materials is characterized by stress-strain relations. These relations are an extremely

important graphical measure of a material’s mechanical properties. In this work, both the steel and the

coating, are assumed to be homogeneous, isotropic and elastic materials. As shown in Figure 3.5, for

elastic-behaviour of materials, the strain is proportional to the stress. The maximum or ultimate tensile

strength is referred to as load carrying capacity after which the failure phase begins (until the fracture be

achieved).

25

Necking

Strain

Stress

Strain Hardening

Ultimate TensileStrength

Fracture Point

E

Yield Point

Proportional Limit

Figure 3.5: Typical ductile material stress-strain curve

Even assuming, in this study, that the coating layer and the steel plate have the same material behaviour

input, its load-deflection responses are very different. Figure 3.6 shows the compressive behaviour of

a typical steel plate and a thin coating layer, both with a stable post-buckling equilibrium. The relation

between the applied load and deflection is presented.

Pos

t-buc

klin

gSt

age

(b)

Buck

ling

Stag

e

(a)

Load

Deflection

Load

Deflection

E

Pcr

UltimateLoad P

ost-b

uckl

ing

Stag

eBu

cklin

gSt

age

RuptureLoad

CoatingBreakdown

Load

Buckling Load

Buckling Load

Pcr

Figure 3.6: (a) Typical Steel load-deflection relation; (b) Coating load-deflection relation

The left part of Figure 3.6 represents a typical diagram of a steel structure under compressive load.

The first part of this curve is linear and elastic and represents the pre-buckling state, where if the load

is removed the structure returns to the original shape/size (as also happens in the material tensile

behaviour (Figure 3.5)). When the structure is subjected to compressive load, and after buckling occurs

(somewhere near the proportional limit), the structure is in the post-buckling phase and also in the elasto-

plastic region. Following a nonlinear relationship between load and deflection, the structure will achieve

the ultimate load after which the failure phase begins. Figure 3.6 (right), shows an example of the load-

deflection curve of a thin film coating layer. An approximately exponential behaviour is described by

the coating (as the applied load increases, the deformation also is increased). The buckling occurs at

a determined point that is obtained by a linear buckling analysis (Section 3.2.1). After that point, the

coating is in the post-buckling stage and shows a nonlinear behaviour. With the increasing of the load,

the coating stresses achieve its limits and the breakdown occurs (see Section 3.2.2). Depending on the

properties of the coating and the conditions that is subjected, the failure of the film can also happen at

the same time that the buckling occurs.

26

When the compressive material is relatively ductile, the Poisson effect causes the cross-sectional area

to increase under load (the inverse of necking found in ductile tensile loading). In the coating layer

behaviour, this phenomena normally doesn’t happen.

3.2.1 One-dimensional Coating Thin Film Buckling

In order to analyse the buckling behaviour of the compressed coating film layer, 1-D blister model is con-

sidered. As Kachanov (1988), Hutchinson and Suo (1992) and Hutchinson (1996) did in their studies,

the delaminated part of the film that is conducive to buckling is treated separately from the remaining

film/substrate system. The thin film layer with a thickness h and length l is treated as a wide column and

characterized by the Von Karman nonlinear plate theory, with fully clamped conditions at its edges (x =

± l2 ).

In Figure 3.7, the coating layer on a steel plate has an initial debonded zone in the interface of a size of

l. The substrate is assumed to be infinitely deep, compared to the film thickness ( hH � 1).

y

Unbuckledl

l

- σx

w

h

# 2

# 1M

Δ N

Local loading ofinterface cracking

Buckled

h

Figure 3.7: Geometry of the one-dimensional blister (Hutchinson and Suo, 1992)

Both the thin film and substrate are taken to be isotropic and homogeneous. The subscript ’c’ and ’s’ is

attached to the reference of the properties of the coating layer and the steel plate (Figure 3.7) such that

Ec and νc are the Young’s modulus and Poisson’s ratio of the coating and Es and νs are for the substrate.

Compressive stresses are applied laterally in the x-direction, leading to buckling of the coating in the

y -direction (Figure 3.7). The deflection along the delamination length is denoted by w (x).

The differential equation that governs the plate’s deflection, normal in xy plane, accounting with the

membrane stresses may be derived as (Timoshenko and Woinowsky-Krieger, 1987):

∇4w =1

D

(p(x, y) +Nx

∂2w

∂x2+Ny

∂2w

∂y2+ 2Nxy

∂2w

∂x∂y

)(3.1)

where p (x, y) is the load normal to plate’s plane and Nx, Ny, Nxy are the external membrane forces

27

per a unit length acting on the middle plane of the plate. D denotes the so-called flexural plate rigidity

(bending stiffness) that is given by:

D =Ech

3

12 (1− ν2c )

(3.2)

l/2

l/2z

xNx

Nx b

y

Figure 3.8: Buckling of a plate under uniaxial compression

Since the plate is only subjected to an in-plane compressive load, Nx (Figure 3.8), and considering

that the deflection w is not a function of the y -coordinate for the one-dimensional buckling model, the

Equation 3.1 is simplified to:∂4w

∂x4− Nx

D

∂2w

∂x2= 0 (3.3)

Since Nx is independent of x (Frank, 2011), the force per a unit of length is substituted by Nx = σxh:

∂4w

∂x4− σxh

D

∂2w

∂x2= 0 (3.4)

For a plate with both ends clamped, the boundary conditions to be satisfied are w (x) = w′ (x) = 0 for

x = ± l2 . So, the deflection w (x), when the load reaches the critical buckling force, that is taken to be

symmetrical about x = 0, is defined by:

w(x) =wmiddle

2

(1 + cos

2πx

l

)(3.5)

where wmiddle is the buckling deflection in the middle of the deflection, i.e., at x = 0.

28

The critical compressive stress σcr, which is required to induce the film buckling, is calculated by deter-

mining the nontrivial solution of the equation 3.4, which yields:

σcr =4π2D

l2h=

π2Ec3 (1− ν2

c )

(h

l

)2

(3.6)

And the critical Euler buckling load, Pcr, is given by:

Pcr =π2h3Ec

3l2 (1− ν2c )

(3.7)

For σx < σcr the film is unbuckled, but for σx > σcr the thin film is already buckled. In that case, the

deflection amplitude wmiddle is given by:

w2middle =

4l2(1− ν2

c

)π2Ec

(σx − σcr) (3.8)

3.2.2 Coating Breakdown

As shown in the coating load-deflection curve, represented in Figure 3.6 (right), after the buckling oc-

curs, the applied compressive load continues to increase until the point when the stresses exceed the

film breaking stresses (failure point). During the buckling and post-buckling analysis, and depending on

the loading conditions and the interface properties, the delamination growth of the film on top of the steel

plate can occur.

A failure criterion is used to check if the structure has failed (brittle failure (fracture) or ductile fail-

ure (yielding)) due to the applied loads. It is more commonly used to evaluate the fracture of or-

thotropic/anisotropic materials (e.g. composite materials), but also can analyse the failure of structures

with isotropic behaviour. The anisotropic materials will turn out to require many more failure type proper-

ties, compared to isotropic materials, because the values of the properties are different in all directions

(ANSYS, 2009). Failure theories to characterize the mechanical behaviour of isotropic conventional

materials are well established (e.g. Maximum stress failure criterion, maximum strain failure criterion,

Von Mises failure criteria, Tresca criteria, etc.). Many and more sophisticated criteria are used to model

anisotropic materials (e.g. Tsai-Wu failure criterion, the Hill yield criterion, etc.). The criteria must take

account for the fact that the material is stronger in some directions than others (Bower, 2009).

In ANSYS, during a composite material analysis, there is a need to introduce a failure criteria, other-

wise the program doesn’t know if the material has failed or not and continues its calculations until the

nonlinear equilibrium be achieved. The same happens with the coating behaviour, i.e., it is necessary to

define a criterion for estimating the point where the coating breakdown occur, because from the stress-

strain definition it is impossible to determine that point (see Figure 3.6 (right)). So, in order to predict the

29

coating failure and, since that the coating layer is considered as an isotropic and ductile material, the

well-known Von Mises criterion is employed.

The maximum Von Mises stress criterion, also known as the maximum distortion energy criterion, is

mathematically expressed as (Andriyana, 2008):

σvm =

√(σ1 − σ2

)2+(σ2 − σ3

)2+(σ3 − σ1

)22

(3.9)

where σvm is the Von Mises equivalent stresses and σ1, σ2 and σ3 are the three principal stresses at

the particular point of interests. This particular failure criterion is considered because the entire state of

stress is utilized in determining the limit stress.

The theory states that a ductile and isotropic material starts to fail at a location where the Von Mises

stress becomes equal to the stress limit. Thus, the corresponding failure criteria is written as:

σvm ≥ σlimit (3.10)

The coated plate is subjected to uniaxial compressive loads so, when the Von Mises stress on the

coating layer achieves the value of the coating compressive strength, the coating breakdown is initiated.

30

3.3 Cohesive Zone Model

Cohesive zone technology models the interface, delamination and progressive failure where two mate-

rials are joined together. This approach introduces a failure mechanism by a gradually degradation the

material elasticity between the surfaces. The basic idea behind this method, as was explained by Cornec

et al. (2003), is showed in Figure 3.9. The ductile tearing process, consisting of an initiation, growth and

coalescence of voids is represented by a traction-separation law, simulating the deformation and finally

the decohesion of the material in the immediate vicinity of the crack tip (see left-hand of Figure 3.9). The

centre of Figure 3.9 shows schematically the implementation of the CZM in a finite element model. In-

terface elements representing the damage implemented between the continuum elements representing

the elastic–plastic properties of the material.

Figure 3.9: Cohesive model: representation of the physical damage process by separation functionwithin numerical interfaces of zero height - the cohesive elements (Cornec et al., 2003)

The area under the traction separation curves corresponds to the energy needed for separation (right-

hand of Figure 3.9). The initial stiffness of the cohesive zone model has a big influence on the overall

elastic deformation and should be very high in order to obtain realistic results. Chandra et al. (2000)

showed that the form of the traction-separation relations plays an important role in the macroscopic me-

chanical response of the system.

According to Bosch et al. (2006), there are a large variety of cohesive zone laws. Most of them can be

categorized as polynomial cohesive zone laws (a), piece-wise linear cohesive zone laws (b), exponential

cohesive zone laws (c) and rigid-linear cohesive zone laws (d), as is shown in Figure 3.10. The main

31

difference lies in the shape and the constants that describe that shape.

Figure 3.10: Four classes of cohesive zone laws (Bosch et al., 2006)

As represented in Figure 3.10, in the upper row, the normal traction is given as a function of the normal

opening Tn(∆n

)and on the lower row, the tangential traction as a function of the tangential opening

Tt(∆t

). The maximum normal traction and the maximum tangential traction are indicated by Tn,max and

Tt,max, respectively. δn and δt are characteristic opening lengths for the normal and tangential direction,

respectively.

In ANSYS, for characterizing the behaviour of the interface elements is only possible to use the expo-

nential traction separation law as defined by Xu and Needleman in 1994 (ANSYS, 2011).

3.3.1 Interface Rupture Modes

As refereed before, the interface elements are specifically designed to represent the cohesive zone be-

tween the components and to account for the separation across the interface.

The interface rupture is a phenomena that imply a relative movement between the two surfaces. De-

pending on the type of movement, which differs depending on the direction of the significant actuating

force, two rupture modes are distinguished (Bosch et al., 2006):

• Mode I - opening or tensile mode where the loadings are normal to the crack. In this case, the

shear stresses are negligible (see left-hand of Figure 2.5);

• Mode II - sliding or in-plane shear mode where the crack surfaces slide over one another in a

direction perpendicular to the leading edge of the crack. This is typically the mode for which

the adhesive exhibits the highest resistance to fracture. In this mode, the normal stresses are

negligible (see the centre-hand of Figure 2.5).

32

In fact, most of the interfaces are subject to a significant normal and tangential stresses, i.e., both the

Mode I and Mode II exist in the interface, which is referred to as the Mixed Mode.

3.3.2 Exponential Model of the Cohesive Zone

As explained by Chandra et al. (2000) and Bosch et al. (2006), the exponential model presented by Xu

and Needleman in 1993, uses a surface potential (φ) in order to define the stresses (T ) that are acting

on the interface in a function of the relative separation (∆):

T =∂φ(∆)

∂∆(3.11)

The normal (n) and tangential (t) components are given in Equation 3.12 and 3.13:

Tn =∂φ(∆)

∂∆n(3.12)

Tt =∂φ(∆)

∂∆t(3.13)

The potential represents the dissipated energy necessary to produce a displacement (∆) between the

two adjacent surfaces. As described by Chandra et al. (2000) and Bosch et al. (2006) the interface

potential is given by:

φ (∆n,∆t) = φn + φnexp

(−∆n

δn

)[{1− r +

∆n

δn

}(1− qr − 1

)−{q +

r − qr − 1

∆n

δn

}exp

(−∆2

t

δ2t

)](3.14)

where q = φt

φn, r = ∆∗

n

δn, where φn and φt are the work of the normal and shear separation, respectively;

∆n and ∆t are the normal and tangential displacement jumps, respectively; δn is the characteristic

length of the interface in the normal direction and corresponds to the value of the normal separation in

the corresponding interface to the maximum normal stress when δt = 0; δt is the characteristic length

of the interface in the normal direction and corresponds to the value of the tangential separation for

which with√

22 δt the maximum tangential stresses is obtained in the interface; and ∆∗n is the value of

∆n after complete shear separation takes place under the condition of normal tension being zero, Tn = 0.

The work of separation in the normal direction (φn) represents the energy necessary to dissipate in or-

der to cause the complete interface separation in the normal direction when δt = 0. On the other hand,

the work of separation in the tangential direction (φt) is the energy necessary to dissipate to cause the

complete interface separation in the tangential direction when δn = 0. Chandra et al. (2000) pointed out

that the works previously described are related to σmax and τmax according to the next expressions:

φn = exp (1)σmaxδn (3.15)

33

φt =

√exp (1)

2τmaxδt (3.16)

The parameters q and r introduced in Equation 3.14 are defined as coupling parameters. Their function

is to combine the behaviour of the interface behaviour in both directions and to consider the mixed mode

interface request. With the coupling, the normal stress developed at the interface are as a function of the

normal separation, the tangential separation and tangential stresses. In practice with the introduction

of these parameters it is intended that if the interface breaked by shear, its ability to withstand loads in

the normal direction is zero, and vice versa (Bosch et al., 2006). According to Bosch et al. (2006), it is

commonly considered that q = 1, i.e., that φn = φt. When q = 1 is assumed, Equation 3.14 is simplified to:

φ (δ) = φn

[1−

(1 +

∆n

δn

)exp

(−∆n

δn

)exp

(−∆2

t

δ2t

)](3.17)

In ANSYS, the calculations of the tensions already accounts for the supposition of q = 1. Considering

Equation 3.12 and 3.13, the normal (Tn) and tangential stresses (Tt) that are acting on the interface are

given by:

Tn = exp (1)σmax∆n

δnexp

(−∆n

δn

)exp

(−∆2

t

δ2t

)(3.18)

Tt = 2exp (1)σmaxδnδt

∆t

δt

(1 +

∆n

δn

)exp

(−∆n

δn

)exp

(−∆2

t

δ2t

)(3.19)

Equation 3.18 and 3.19 represents the traction separation relation at the modelled interface in ANSYS.

Since that both the normal and tangential stresses depend on both the normal and tangential separa-

tions, the graphical representation of the constitutive relations is more complex than a representation

in the plan, because it requires three axes: (Tn,∆n,∆t) for the relation presented in Equation 3.18 and

(Tt,∆n,∆t) for the constructive relation presented in Equation 3.19.

Chandra et al. (2000) presented graphically the constitutive relations described in Equation 3.18 and

3.19 for the simplest case that corresponds to the shear and normal separation null. Thus, Equation

3.18 represents the relation between normal stress (Tn) and the normal separation (∆n) and Equation

3.19 shows a relation between the shear stress (Tt) and the tangential separation (∆t). The graphical

representations are normalized (see Figure 3.11): in the plot (a), the normal traction is normalized by

the maximum normal stress (σmax) and the normal separation by the characteristic length in the normal

direction (δn); in plot (b), the shear traction is normalized by the maximum shear stress (τmax) and the

tangential separation by the characteristic length in the tangential direction (δt).

34

As can be seen from Figure 3.11, in the exponential model, initially with an increasing separation of the

interface surfaces, the stresses along the interface increases up to the maximum, after which it begins to

decrease until eventually be reduced to zero, thereby allowing the complete separation (which in theory

occurs when ∆→ 0) (Chandra et al., 2000).

(a) (b)

Figure 3.11: (a) Variation of normal traction, Tn, with ∆n for ∆t = 0; Variation of shear traction, Tt, with∆t for ∆n = 0 (Chandra et al., 2000)

The maximum shear stress that the interface can support is a function of σmax, δn and δt and is given by:

τmax =√

2exp (1)δnδtσmax (3.20)

The value of the parameters σmax, δn and δt must be specified in ANSYS Multiphysics 11.0 so that

the interface can be modelled, and represent the interface element inputs. As for the interface element

outputs, they consist in the normal and shear interface traction and separation.

The exponential cohesive zone law is one of the most popular cohesive zone laws. It has some advan-

tages compared to other cohesive zone laws. First of all, a phenomenological description of the contact

is automatically achieved in normal compression. Secondly, the tractions and their derivatives are con-

tinuous, which is attractive from an implementation and computational point of view (Bosch et al., 2006).

Because of its shape (smooth traction-separation curves), also helps with the nonlinear convergence

(ANSYS, 2011).

The great advantage of modelling the interface, between the coating layer and the steel plate, by ANSYS

is that it provides in its library interface elements and the model of the cohesive zone (ANSYS, 2011).

35


Chapter 4

Finite Element Modelling and

Verification

The main goal of this chapter is to present the finite element modelling in ANSYS and analyses. All the

important considerations and steps of the 3-D finite element modelling are described herein, giving a

special emphasis to the cohesive zone modelling between the coating layer and the steel plate. At the

end of this chapter, a validation of the finite element model is also presented.

4.1 Introduction

The Finite Element Method is a very powerful numerical engineering tool, computer assisted, that

presents approximate solutions to a wide range of problems. The theory behind this method is already

very well developed and studied by many authors (Ochoa and Reddy, 1992; Krishnamoorthy, 1994;

Wriggers, 2008; Madenci and Guven, 2015). The basic principle is to divide the analysed structure into

a number of finite elements, so-called discretization. The advantage of this method is that each element

can have much simpler geometry than the whole structure and is therefore much easier to analyse. FEM

requires from the designer enough experience to guarantee that the results are trustable and that every

intermediate step of the modelling process are performed adequately. Therefore, it is important to do

a validation of the model developed, comparing the results with other numerical and/or experimental

results.

The problem of the coating breakdown in steel plates subjected to compressive uniaxial load is solved

by employing a linear and also a nonlinear buckling analysis and using the FEM. Deformation shapes,

critical loads, stresses and other parameters are calculated during the finite element analysis. These

calculations are used to understand the buckling and post-buckling behaviour of the coating layer.

The software used to perform FEM calculations is the commercial software ANSYS Mechanical APDL

11.0.

37

4.2 Buckling Analysis

Buckling analysis is a technique used to determine buckling loads and buckled mode shapes, which are,

respectively, the critical loads at which a structure becomes unstable and the characteristic shape asso-

ciated with a structural buckling response. A structure subjected to compressive axial loads will buckle

when the load reaches the critical value. In the problem of a layered structure such as a coated steel

plate, compressive residual stresses can occur. In the case of an internal crack in the layered structure,

these stresses cause a buckling shape along the crack face.

ANSYS provides two techniques for predicting the buckling load and buckling shape of a structure:

eigenvalue (or linear) buckling analysis. and nonlinear buckling analysis.

4.2.1 Linear Eigenvalue Buckling Analysis

The eigenvalue buckling analysis predicts the theoretical buckling strength of an ideal elastic structure.

It computes the structural eigenvalues for the given loading and constraints. This method corresponds

to the textbook approach to elastic buckling analysis: for instance, an eigenvalue buckling analysis of a

column will match the classical Euler solution. However, imperfections and nonlinearities prevent most

real-world structures from achieving their theoretical elastic buckling strength. Thus, eigenvalue buck-

ling analysis often yields unconservative results, and should generally not be used in actual day-to-day

engineering analyses.

In this thesis, and in addition to the complex nonlinear analysis that is carried out, also a simpler linear

buckling analysis is developed in ANSYS in order to compare the estimated results with the Euler critical

buckling loads obtained from the equation described in Section 3.2.1.

4.2.2 Nonlinear Strength Analysis

Nonlinear buckling analysis is the more accurate approach and is therefore recommended for design

or evaluation of actual structures. This technique employs a nonlinear static analysis with gradually

increasing loads to seek the load level at which the structure becomes unstable. Utilizing nonlinear

analysis the structural model can include initial imperfections, residual stresses and the solution is ac-

companied by large deflection responses. For that reason, the behaviour of the coating on the top of a

steel plate under uniaxial load will be estimated fundamentally based on the nonlinear technique.

Nonlinear structural behaviour arises from a number of causes, which can be grouped into three principal

categories: changing status (contact forms, etc.); geometric nonlinearities (large displacements and/or

rotations); material nonlinearities (elasto-plastic response, environmental effects, creep response, etc.).

38

To solve nonlinear problems, ANSYS employs the Newton-Raphson or Arc-Length method. In Newton-

Raphson approach, the load is subdivided into a series of load increments. The load increments can be

applied over several load steps (see Figure 4.1). The iteration is done according to Equation 4.1.

[KTi

]{∆ui} = {F a} − {Fni r} (4.1)

Where[KTi

]is the Jacobian matrix (Tangent Stiffness Matrix), {∆ui} the displacement increment vec-

tor, {F a} the external load vector and {Fni r} the internal load vector.

At each solution step the tangent stiffness matrix and the difference between external load and internal

load is updated. The solution converges when the difference between two loads, represented as R

in Figure 4.1, is in an admissible tolerance. Figure 4.1 shows the iterations done by the incremental

Newton-Rapson method.

Figure 4.1: Incremental Newton-Raphson procedure

In order to use nonlinear analysis, a disturbance of the structure must be initially introduced. There are

several methods to introduce disturbances in a finite element model. In this thesis an initial imperfection

is implemented in order to leave the geometry more prone to buckling (Section 4.4.4).

4.3 Post-buckling Analysis

A post-buckling analysis is the continuation of a nonlinear buckling analysis. After a load reaches its

critical buckling value, the load value may remain uncharged or it may decrease, while the deformation

continues to increase. For some problems, after a certain amount of deformation, the structure may

start to take more loading to keep deformation increasing, and a second buckling may occur. The cycle

may even repeat several times.

The post-buckling stage is unstable. Theoretically the structures may still bear extra loads in this phase.

However, due to the forces and moments acting on the buckling shape, laminated parts might debond

from each other. As mentioned in Chapter 2, some studies have been done in order to predict the

bucking and post-buckling behaviour of laminated plates with delamination (Chai et al., 1981; Rajendran

39

and Song, 1998; Bruno and Greco, 2000; Shan and Pelegri, 2003; Obdrzálek, 2010; Wang et al., 2011).

These researchers found that the extremely large deformations might cause delamination grow between

layers in the post-buckling phase.

4.4 The Finite Element Model

In this Section is described the 3-D FEM developed in ANSYS. Since that the objective is to perform a

parametric analysis and for this purpose there is a need to run a lot of simulations, became necessary

to develop a program in order to automate the process. The program was made in Python language.

Appendix A presents a diagram that describes how the parametric analysis is performed in ANSYS with

the aid of the program developed in Python. Some of the most important considerations are described

in the next Subsections.

4.4.1 Geometry, Loading and Boundary Conditions

A thin coating layer bonded to a steel plate with an initial debonded zone of size l and with an initial

imperfection, as will be explained in Section 4.8, is modelled in ANSYS (Figure 4.2). In reality, a typical

coating system includes a primer, an intermediate coat and a top coat. However, in order to simplify

the study developed herein is considered a single layer for the coating system. As showed in Figure 1.2

in Chapter 1, is only considered for this analysis a plate strip from the middle of the ballast tank plate,

where is expected the maximum deformation of the coating layer. The main dimensions of the steel

plate are considered constants and equals to 100x2x11 mm. These dimensions are chosen based on a

the standard dimensions of a specimen so that in the future work the results obtained in this thesis can

be verified with an experimental analysis. The plate dimensions could have been larger, especially the

width b, but an increase in size would greatly increase the simulation time due to the complexity of the

problem. The thickness of the coating layer, h, and the size of the initial debonded zone l are kept as

variables during the analysis. The effect of the variation of the ratio of the normalized length (l = lL ) and

the ratio of the normalized thickness (h = hH ) on the behaviour of the coated plate will be analysed.

x

y(b)

δ > δcrit

Steel Plate

wCoating

l

w

L

h

H

L

lb

δ

Steel Plate

Coating

(a)

z

Figure 4.2: Geometry of the structure modelled: (a) Intact state; (b) Buckled state

40

The coating and the steel plate are assumed to be homogeneous, isotropic and elastic. The Young’s

modulus of the steel is considered to be Es = 210 GPa and the Poisson’s ratio νs = 0.3. For the coating

layer, the properties are difficult to predict because it is a complex combination of materials. Since that’s

one of the bigger component of the protective coatings is epoxy (Sørensen et al., 2009a), the material

properties of the epoxy resin are used as reference for the coating properties. The Young’s modulus

and the Poisson’s ratio of the coating layer are considered as variables during the analysis, taking as

reference the material properties of the epoxy polymer: Ec = 3 GPa and νc = 0.37. With that variation,

the effect of the coating material properties on the behaviour of the structure is also analysed.

In order to simulate the real loading conditions of the coated steel plate of a ship hull, an in-plane

compression with the x-coordinate direction is applied. The compression of the plate is simulated by

imposing a displacement in the left boundary (x = 0), as shown in Figure 4.2.

The boundary conditions of the beam modelled are summarized in the Figure 4.3. At x = L the beam is

restrained in all degrees of freedom (Ux, Uy and Uz) and at x = 0 is free in Ux and restrained at the Uy

and Uz. In the sides z = 0 and z = b the plate is free in all degrees of freedom.

z

xUz=0Uy=0Ux≠0

Free Condition

Uz=0

Ux=0Uy=0 b

L

y

Figure 4.3: Coated plate’s boundary conditions (Top View)

4.4.2 Element Type and Mesh Density

The finite element model of the coated steel plate is developed using 3-D solid elements, since that they

provide more accurate results than those coming from corresponding models, which incorporate 2-D

elements. The element used for the modelling of the coating layer and the steel plate is SOLID45. It’s

an eight node element with three degrees of freedom at each node: Ux,Uy and Uz. Plasticity, creep,

swelling, stress stiffening, large deflections and large strain are capacities of this 3-D element. The

geometry and node locations are shown in Figure 4.4.

Figure 4.4: SOLID45 geometry in ANSYS

41

In order to generate the finite element elements, a 3-D solid mesh is implemented for the steel plate

and the coating layer. The generated mesh is dense enough in the coating layer, where the maximum

displacements should normally be developed and could be coarser in the steel plate, in order to combine

adequate accuracy with acceptable solution times. Figure 4.5 shows an example of the meshed plate

and coating layer.

ImposedDisplacement

(δ)

L b

y

xz

Figure 4.5: Example of the meshed coated plate

4.4.3 Interface Coating-Steel Using the Cohesive Zone Model

An interface exists anywhere when two materials are joined together. The interface between the two

layers is of a special interest, because when subjected to certain types of external loading, fracture or

delamination may occur. The interface delamination can be simulated by traditional fractures mechan-

ics methods or by using the cohesive zone model technique. In ANSYS, the interface surfaces of the

materials can be represented by a special set of interface elements or contact elements, and a cohesive

zone model can be used to characterize the constitutive behaviour of that interface (ANSYS, 2011). In

this work, the glue between the steel plate and the coating layer is simulated using interface elements

meshed with zero thickness.

Interface traction-separation behaviour is highly nonlinear. The full Newton-Raphson solution proce-

dure, which is the standard ANSYS nonlinear method, is the default method for performing this type of

analysis.

4.4.3.1 Interface Element Selection and Meshing

ANSYS offers a set of four interface elements designed specifically to represent the cohesive zone

between the interface and to account for the separation across the interface. For the interface elements,

the interface separation is defined as the displacement jump, δ, i.e., the difference of the displacements

of the adjacent interface surfaces (see Figure 4.6).

42

Figure 4.6: Schematic of interface elements

The simulation of an entire assembly, consisting of the cohesive zone and the structural elements on

either side of the cohesive zone, requires that the interface elements and structural elements have the

same characteristics. So, for the selected solid element (SOLID45) only the element INTER205 could

be used. INTER205 is a 3-D 8-node linear interface element. When used in a conjunction with 3-

D structural elements, as SOLID45, INTER205 simulates an interface between two surfaces and the

subsequent delamination process, where the separation is represented by an increasing displacement

between node, within the interface element itself. The interface has zero thickness and the nodes are

initially coincident. It is defined by eight nodes having also three degrees of freedom at each node:

Ux,Uy and Uz. Figure 4.7 shows the geometry of this element and the location of the interface layer in

the structure modelled.

(a) INTER205 geometry in ANSYS

Coating

Zero ThicknessInterface Elements

Coating

Steel Plate

(b) Interface location in the coated plate

Figure 4.7: Interface modelation

With the forces acting on the model, the nodes deviate from its original position and the element gain

thickness, which occurs because the upper face of the element has a different displacement from the

bottom part. The difference between these displacements is equal to the separation (∆) that occurs at

the interface. This separation of the interface have units of length and can be decomposed in its normal

component (∆n) and tangential (∆t).

The meshing of the interface elements, between the coating layer and the steel plate, is performed using

the CZMESH command in ANSYS. This command creates both the mesh and the interface area with

the interface elements selected. The meshing of the interface could be made by hand but an additional

refining of the mesh would not bring difference in results and would increase the simulation time (Gaspar,

2011).

43

4.4.3.2 CZM Material Definition

The cohesive zone material that defines the separation at the interface has an exponential behaviour,

that is defined using the ANSYS command TB, CZM. In order to characterize the interface between the

coating and the steel plate is necessary input in ANSYS the material constants defined in Table 4.1.

Table 4.1: Interface material constants in ANSYS

Constant in ANSYS Symbol Meaning

C1 σmax Maximum normal traction at the interface

C2 δn Normal separation across the interface

C3 δt Shear separation across the interface

The determination of the three parameters that define the exponential law of the cohesive zone is the

biggest difficulty that many authors were experienced in the modelation of the interface between two

surfaces (Gaspar, 2011), principally when there is no experimental work capable to his determination.

In this thesis, no experimental work will be performed in order to predict the properties of the adhesion

between the coating layer and the steel plate. So, the initial values used for these three parameters are

from a study developed by Waseem and Kumar (2014) about the delamination in composites. Based

on these values, some variations in C1, C2 and C3 will be performed in order to predict the influence of

these parameters in the coating behaviour.

4.4.4 Initial Imperfection

Structural components may have an initial imperfection due to the limitations of manufacture or assem-

bling. The initial imperfections include geometrical imperfections, local ply-gaps, non-uniformly applied

end loads, variations of boundary conditions, etc., from which the geometrical imperfection is of most

concern. The modelled structure in FEM are perfect, what doesn’t correspond to the reality. Without the

perturbation caused by small initial imperfections, a finite element model with an ideal shape could only

represent the transmission of the stress wave, and no ultimate strength will be predicted in nonlinear

analysis.

Imperfections can be added into an analysis by applying small displacements, perturbation loads or

making changes in the initial geometry.

As shown in the Chapter 2, many authors already have used the geometric initial imperfections for sim-

ulating the real distortions that are present in ship plates after the construction procedure. The initial

geometric imperfection shape in x and y directions can be modelled by a sine series expansion as is

presented in Equation 2.1.

44

In this work, the shape of the coating layer could be perturbed by introducing an initial geometric imper-

fection. Since the problem of the coating failure is studied in 1-D, the vertical displacement of the coating

is only a function of the x coordinate and so the Equation 2.1 is simplified to:

w(x) = w0sin(mπx

l

)(4.2)

The factors that lead to small imperfections in the steel plates and coating layer are slightly different.

As was demonstrated in the previous works, the initial geometric imperfection is a good approximation

for the limitations of manufacture or assembling in the steel plates fabrication. However, for the case of

the coating layer, the initial geometric imperfection isn’t the best way to simulate the real imperfections.

These small imperfections in the coated layer are presented in the real world due to the inadequate

steel surface preparation or due to poor application technique of the coating. The compressive loads

that the beam is subjected, in addition to the imperfections, lead to creation of residual stresses and

consequently buckling may occurs and the delamination initiates until the coating breakdown occurs.

In the model developed here, and in order to perturb the shape of the coating film, as an initial imperfec-

tion is modelled a debonded zone of size l and it is induced a small destabilization load in the nodes at

the centre of the beam as represented in Figure 4.8. These nodal forces are applied at x = L2 and along

the width of the plate (0 ≤ z ≤ 2).

x

z

Steel Plate

Coating

Destabilization Load y

Figure 4.8: Modelled initial imperfection

To model of the initial debonded zone between the coating and the plate needs to use the surface-

to-surface contact elements from the CZM to bond two surface constituents of the coating layer. The

element CONTA174, defined in the "contact" surface, and the TARGE170 established in the "target"

surface are used to define the permanent contact between two boundaries. The ANSYS command

KEYOPTs and REAL are used to control the contact behaviour between the contact elements. Different

from the interface elements used to model the interface delamination between the steel and the coat-

ing layer, these contact elements are used to connect two surfaces permanently, making a joint of the

surfaces.

45

4.5 Validation of the Finite Element Model

This section is developed for the presentation of a verification study used to test the applicability of the

proposed computational model. In order to make the validation of the finite element model described

in the previous section, the ideal scenario would be to make an experimental analysis for comparing

the results obtained with those obtained by ANSYS. However, and since the experimental analysis is

beyond the scope of this thesis (Section 6.2), the model is checked by comparing the results with others

available in the literature. As is known, the coating breakdown and delamination on top of steel plates

is an understudy problem. So, in the literature there is no experimental or theoretical results to compare

and perform the verification of the model developed. To make sure that the results obtained in this work

are correct, a study about delamination in composite materials developed by Bohoeva (2007) is used.

For the same parameters, the results obtained from the analysis of the delamination in composites are

compared with the results of the developed finite element model.

As described in the Section 2.4, Bohoeva (2007) performed a numerical and experimental analysis of

delaminated rectangular plates subjected to axial compressive loading. The initial delamination in this

work was also positioned symmetrically about the centre of the beam. In the Figure 4.9 is presented the

beam-plate studied in this study with the main dimensions considered.

Figure 4.9: Plate model, Bohoeva (2007)

Although it is a composite behaviour study, the material is assumed homogeneous and isotropic with

Young’s modulus Es = 200 GPa and Poisson’s ratio νs = 0.3.

4.5.1 Buckling and Post-buckling Response of a Laminated Beam

The objective of this verification is to show the capability of the current finite element model to capture

the behaviour of a plate with a single delamination on top, subjected to compressive axial loads. As

previously mentioned, the work of Bohoeva (2007) is used as reference.

In their study, the stability of the composite beam in the presence of defects such as delamination was

predicted using the nonlinear formulation based on the energy approach. This approach allowed Bo-

hoeva (2007) to obtain explicit analytical expressions for characterizing the behaviour of the laminated

46

beam. Those expressions are used in this section to compare with the results obtained from the FEM

results.

First, the finite element program results from the study of Bohoeva (2007) are analysed, checking the be-

haviour of the beam when subjected to compressive load and drawing the respective plot load-deflection.

These results are compared with the theoretical ones obtained through the expressions developed by

Bohoeva (2007).

The main dimensions of the plate used for this first study are presented in the Table 4.2.

Table 4.2: Main dimensions of the composite beam

L 100 [mm]

l 40 [mm]

H 4 [mm]

h 0.4 [mm]

Figure 4.10 shows the deformed shape of the laminated plate with the dimensions presented in Table

4.2 obtained in the developed FEM. The variation of the displacements in the y direction, Uy, is also

presented along the structure. As expected, the maximum value of the displacement in y occurs in the

centre of the delaminated part.

Figure 4.10: Deformed laminated plate with a single delamination

47

The type of instability that occurs in the beam is the local buckling, i.e., only occurs the delamination of

the top, when the lower and the main part of the plate remains flat. This type of local loss of stability

occur due to the high concentration of the interlayer stresses in the front of the defect. The composite

layer buckles when the stress reaches its critical value. After the buckling and increasing the external

load applied on the structure, the delaminated region increases to a critical value and others types of

buckling may occur.

A study of the buckling and post-buckling behaviour of the composite plate is performed, analysing the

relationship between the load and the deflection at the centre of the delamination. This analysis is

performed using the results provided by the FEM developed. A comparison of the results obtained from

current FEA and the results obtained by using the analytical expression obtained by Bohoeva (2007) is

performed, as is possible to see in Figure 4.11.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Norm

aliz

ed L

oad

Normalized Deflection

Theoretical Results

FEA Results

Figure 4.11: Comparison of the theoretical results obtained from Bohoeva (2007) and the current FEM

Figure 4.11 presents the normalized axial load-normalized deflection curves for the composite plate in

this study. Comparing the results of the analysis conducted by Bohoeva (2007) with the results from

the nonlinear FEA, as can be seen, the corresponding load-displacement curves are similar, which

proves the validation of the program developed in ANSYS. The discrepancy of the results, principally

for lower load values, can be justified by the fact that in the FEA the initial imperfections are taken into

consideration and in the analytical expressions the imperfections aren’t considered.

4.5.2 Verification of the Buckling Shapes With the Variation of the Parameter β

To predict the effect of the length (l) and the thickness (h) in the post-critical behaviour of the defect in

the plate, Bohoeva (2007) introduce the parameter β that is defined by the ratio of the normalized length

(l) and thickness (h) of the defect (Equation 4.3).

48

When the critical load is achieved, there are three possible types of buckling of composite material with

delamination: Global Buckling, Local Buckling or Mixed Buckling (see Figure 2.4).

β =l

h=l.H

L.h(4.3)

Bohoeva (2007) concluded that if β < 1 the delaminated part is small, so then it will exist only a global

form of the defect. On the other hand, if β > 1 the length of the delamination is bigger and will exist

local or "mixed" form of buckling. These conclusions were performed using the theoretical and the finite

element model calculations.

Considering two specific cases very well studied in the work of Bohoeva (2007), one for β < 1 and an-

other for β > 1 (See table 4.3), it is expected that after running the FEM developed, for β < 1 is obtained

a global loss of stability in the beam and for β > 1 a local or mixed form.

Table 4.3: Main dimensions of the laminated beam for the two cases verified

β < 1 β > 1

L 600 600 [mm]

l 120 210 [mm]

H 30 30 [mm]

h 9 6 [mm]

l=120mm

l=210mm

Global Buckling Shape β<1

Local Buckling Shape β>1

h=9mm

H=30mm

h=6mm

H=30mm

L=600mm

Figure 4.12: Buckling shapes, Bohoeva (2007)

The buckling shapes obtained after running the finite element program developed with the main dimen-

sions presented in Table 4.2 are presented in the next Figure. Observing Figure 4.13, it is possible to

conclude that the shapes obtained from the FEM developed in this thesis are similar to the expected

buckling shapes (Figure 4.12). For β > 1 with l = 0.35 is obtained a local buckling and for β < 1 with l =

0.2 a global standard buckling of the entire beam is occurring. Again the results of the current FEM are

very similar to the results obtained in the work developed by Bohoeva (2007).

49

(a) β > 1

(b) β < 1

Figure 4.13: Buckling shapes obtained from the FEM

4.6 Concluding Remarks

In this Chapter, the description of the finite element program is presented. All the steps and assump-

tions made for FEM are explained. Some of the potentialities of the finite element program ANSYS

are identified, such as the capability of to model the interface between two surfaces, using the CZM.

An interface between the steel plate and the coating layer is modelled in order to imitate the glue and

predict the delamination between the two surfaces. This interface zone is defined by three parameters

that characterizes the adhesion of the coating layer on the plate.

After the description, also a validation of the FEM is performed. The validation is realized based on the

results from literature about the delamination in composite plates. After comparing the theoretical results

with the results of the current FEM, is concluded that the developed computational model is suitable for

prediction of the buckling and post-buckling behaviour of the coated steel plate.

50

Chapter 5

Result Analysis

In this chapter, the linear and nonlinear finite element analyses are presented and discussed. Also some

theoretical results obtained from the expressions presented in Chapter 3 are showed. The influence

of the variation of the delamination length, coating thickness, coating and interface properties on the

buckling and post-buckling behaviour of the coating are verified. A failure assessment diagram applied

for the macro-delamination sizes is developed in order to be used in the coating breakdown prevention.

5.1 Buckling Analysis

When a perfectly straight beam is compressed, its straight form is initially in a stable equilibrium, mean-

ing that the beam remains straight even if it is perturbed. As the load gradually increases, a critical point

is reached when a slight perturbation produces a stable lateral deflection. This phenomena is referred

to as buckling.

As was explained in Section 3.2.1, the problem of the local buckling of a coating layer can be considered

as a classical linear buckling problem of a column with fully clamped edges, and so the critical buckling

load, which is the lowest load that causes buckling, is given by the formula derived by Euler for columns

(Equation 3.7).

In this Section the buckling analysis of the coating layer is performed, determining the first critical buck-

ling load and respective buckling shape for each parametric variation. For that and, as refereed in

Section 4.2.1, a simpler linear buckling analysis of the coating layer is performed in ANSYS in order to

compare with the Euler critical buckling loads, estimated as explained in the previous paragraph.

The influence of the variation of the delamination length l, coating thickness h and mechanical prop-

erties of the coating Ec and νc in the buckling behaviour of the coating layer are presented herein. As

refereed in Section 4.4, the plate length (L), plate thickness (H) and plate width (b) are considered fixed

parameters during the performed analyses. Since that only the buckling of the coating layer is analysed,

51

assuming that the film is independent of the substrate and of the interface properties so, variations in

the constants σmax, δn and δt, that define the interface between the steel plate and the paint, aren’t

considered.

5.1.1 Effect of the Delamination Length, l, on the Coating Buckling Behaviour

The delamination length is one of the most important parameters in the buckling delamination analysis,

as is known from the works of delamination in composite materials. Since that the coating layer, that

protects the steel surface, can fail by buckling and consequently delamination, like happen in layered of

composite materials, it is fundamental to analyse the influence of the initial delamination length in the

buckling behaviour of the coating layer.

For this part of the study, the coating mechanical properties are considered fixed and equal to those

defined in Section 4.4 (Ec = 3000 MPa and νc = 0.37). The coating thickness (h) is also considered con-

stant and is chosen according the standards of the coatings for ballast tanks, that defines a minimum

nominal dry film thickness (NDFT) equal to 0.32 mm, for epoxy-based coating systems (GL, 2010). The

delamination length used in the present analysis is given in Table 5.1:

Table 5.1: Delamination length considered in the analysis

l 10 20 30 40 50 [mm]

l 0.1 0.2 0.3 0.4 0.5 [-]

Figure 5.1 presents the comparison of the critical buckling loads obtained by the Euler expression and

from the linear finite element analysis performed by ANSYS, for each value of the delamination length

ratio l, as presented in Table 5.1.

The effect of the delamination length on the buckling response is clearly evident. Figure 5.1 shows that

for bigger values of l, i.e., bigger values of the delamination length, the compressive load necessary for

the local buckling of the coating layer occurs is increasingly smaller. The values of the critical load varies

between 0.3 - 7.5 N, which are very small values comparatively, e.g., with the buckling loads of the steel.

Between the l = 0.1 and l = 0.2, the difference in the critical load is around 75%, which is a big difference

taking into consideration that the increment of the delamination length is only 10 mm. However, for the

other delamination sizes the variation in the critical buckling loads is smaller. For 0.2 ≤ l ≤ 0.3, 0.3 ≤ l ≤

0.4 and 0.4 ≤ l ≤ 0.5 the difference in Pcr are ≈ 56%, ≈ 44% and ≈ 36%, respectively. A coating layer

with only 10 mm of an initial delamination requires more than 96% of load for its local buckling, than in

the case of a coating with a 50 mm of delamination.

52

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

Critica

l B

ucklin

g L

oa

d,

Pcr

[N

]

Delamination Length ratio,

Analytical Eigenvalue Critical Buckling Load

Numerical Eigenvalue Critical Buckling Load

𝑙

Figure 5.1: Euler and FEM critical bulking load as a function of the delamination length, l

The values obtained from the theoretical calculations and the ones from the FEA are slightly different for

the lower delamination length ratio (≈ 1 N of difference), but for the others lengths the variation is very

small, which give a good agreement of the results.

This kind of analysis is very important because it suggests that for a larger size of defects caused by

the inadequate steel surface preparation, poor coating application and/or incompatible coating systems,

is easier to occur the formation of the coating blistering (local buckling shape) and perhaps the coating

breakdown in this local. So, when the visual inspection is performed, e.g., in ballast tanks, it is important

to pay attention to the parts that have defects with a bigger size, and requires repair of those parts,

because as seen in Figure 5.1, the load necessary for a bigger defect to deform is much smaller (ap-

proximately less 96% considering the studied cases), and so the failure of these parts may occur faster,

which leads to a corrosion initiation.

As refereed before, in this preliminary study is only considered a single delamination problem in the

centre of the coated plate for simulating the coating failure. In reality, a coated plate from a ballast tanks

has much more failure points and of different sizes but, these considerations are left for the future work.

5.1.2 Effect of the Coating Thickness, h, on the Coating Buckling Behaviour

The coating thickness, h, is also a very important parameter on both adhesion and corrosion resistance.

Due to the careless application of the coating on top of the steel plate, the achieved film thickness can be

excessive or insufficient for resisting to the loads and the surrounding corrosive environment. A coating

system that has been applied an inadequate thickness often leads to premature breakdown. In order to

understand this behaviour, the effect of the coating thickness on the local buckling critical load is anal-

ysed here.

Opposed to the previous Section, the delamination length (l) is considered constant and equal to 30 mm.

The coating mechanical properties are the same as used for the previous analysis. Table 5.2 presents

53

the coating thickness, which are chosen based on the minimum film thickness presented in the coating

for the ballast tank standards, also for the epoxy-based coating systems (200 - 500 µm) (GL, 2010).

Table 5.2: Coating thickness considered in the analysis

h 0.2 0.32 0.4 0.5 [mm]

h 0.018 0.029 0.036 0.045 [-]

The theoretical and numerical results of the critical buckling load are shown in Figure 5.2. For both

FEA and theoretical results, the critical load increases with the increasing of the coating thickness. The

difference in terms of Pcr in between the two curves is also increased as the thickness increases. The

maximum load, even being a small load (≈ 3.2 N), is achieved by the maximum thickness considered

(h = 0.5 mm). It is also interesting to observe from Figure 5.2 that an increase of 0.3 mm of the coating

thickness (0.2 mm ≤ h ≤ 0.5 mm) can represent an increase of about 94% in the critical load. So, it can

be concluded that a thicker paint film is more difficult to buckle because it can support higher load.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

Critica

l B

ucklin

g L

oa

d,

Pcr

[N

]

Coating Thickness Ratio ,



ℎ

Figure 5.2: Euler and FEM critical bulking load as a function of the coating thickness, h

An adequate film thickness is necessary for a coating system to correspond to requirements and to

provide a good anti-corrosion protection or achieve the expected anti-fouling lifetime, etc. As concluded

from Figure 5.2, under thickness will result in premature failure. However, over application can also

cause problems, such as solvent entrapment and subsequent loss of adhesion, cracking of the paint or

splitting of the primer coats. So, ideally the coating thickness in the ballast tanks should be that specified

by the manufacturers and/or Classification Societies, allowing for practical application variations (GL,

2010).

54

5.1.3 Effect of the Coating Properties, Ec and νc, on the Coating Buckling Be-

haviour

The mechanical properties classify the material and also describe how it will react to physical forces. As

refereed before, it is difficult to find information about the ballast tank coating properties in the literature.

For the execution of this work, the properties found for the epoxy resin (a component of the coating)

are used, because there aren’t values for the Young modulus and Poisson ratio of the coating available.

Since that the coating material properties used in this study aren’t the properties of a real coating, com-

monly used in ballast tanks. It is important to investigate the sensitivity of the coating buckling behaviour

to the material properties variation, i.e., to the variation of the coating type.

In order to investigate the effect of the coating properties, it is assumed that the delamination length and

the coating thickness are constant and equal to 30 mm and 0.32 mm, respectively. The Young’s modulus

and Poisson’s ratio, are chosen based on the epoxy resin properties, Ec = 3000 MPa and νc = 0.37, as

already explained in Section 4.4. A variation of these values is assumed as ± 15% and ± 30% of Ec

and νc (see Table 5.3).

Table 5.3: Coating properties considered in the analysis

Ec 2100 2550 3000 3450 3900 [MPa]

νc 0.259 0.315 0.370 0.426 0.481 [-]

Variation -30% -15% - +15% +30%

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.7Ec,νc 0.85Ec,νc Ec=3000Mpa and νc=0.370

1.15Ec,νc 1.3Ec,νc

Critica

l B

ucklin

g L

oa

d,

Pcr

[N

]



Figure 5.3: Euler and FEM critical bulking load for different values of Young’s modulus, E, and Poisson’sratio, ν

The way in which buckling occurs depends on how the structure is loaded and on its geometrical and

material properties. Figure 5.3 shows a comparison between the critical loads with the variation of the

coating material properties. Both the calculated and ANSYS results revealed that the buckling critical

load is influenced positively and linearly by the coating properties values. An increase in Ec and νc re-

sults in a bigger resistance to buckle from the compressed coating and also causes a bigger difference

55

between the theoretical and numerical results. The percentage variation of the critical load in relation to

the initial assumed coating properties (Ec = 3000 MPa and νc = 0.37) is presented in Table 5.4.

Table 5.4: Variation of the coating properties in relation to the Ec = 3000 MPa and νc = 0.37 results

Variation in Euler critical buckling load Variation in buckling critical load from FEA

0.7Ec, νc - 35% -30%

0.85Ec, νc -19% -15%

1.15Ec, νc +21% +15%

1.3Ec, νc +46% +30%

It can be concluded that the variation in Pcr is relatively close to the variation in the coating properties,

e.g., a variation of -30% in the coating properties represents a diminution in the buckling critical load of

30 - 35%.

In Figure 5.3, the variation of Ec and νc is done simultaneously, in order to simulate the properties of

different coating types. However, to analyse the influence of each parameter, also a separated study

is performed. Due to the order of magnitude of both values, is foreseeable that the influence of the

Poisson’s ratio, although not negligible, is much smaller than the influence of the Young’s modulus (see

Figure 5.4).

0.05

0.15

0.25

0.35

0.45

0.55

0.65

0.75

0.85

0.95

1.05

2000 2500 3000 3500 4000

Critical B

ucklin

g L

oad, P

cr

[N

]

Young's Modulus, [MPa]

Euler Critical Buckling Load

Critical Buckling Load from FEA

𝐸𝑐

(a) Variation of Ec

0.14

0.24

0.34

0.44

0.54

0.64

0.74

0.84

0.25 0.3 0.35 0.4 0.45 0.5

Critica

l B

ucklin

g L

oa

d, P

cr

[N]

Poisson's Ratio,

Euler Critical Buckling Load

Critical Buckling Load from FEA

ν𝑐

(b) Variation of νc

Figure 5.4: Euler and FEM critical buckling load for different Ec (a) and νc (b)

For 2100 ≤ Ec ≤ 3900 MPa the difference in the critical load is about 46% and for 0.259 ≤ νc ≤ 0.481

the difference is only 17%. Figure 5.4 shows that the influence of the Ec is much more significant in

terms of critical load than νc.

56

5.2 Post-buckling Analysis

A coated plate, subjected to a uniaxial in-plane compressive load, may have two behaviours. If the

painting was well performed and the adhesion between the paint and the steel surface is strong enough

to resist to the applied compressive loads, the steel plate may fail with the coating glued to its surface

(global buckling). The steel plate behaviour under compressive loads is a structural problem that is al-

ready well studied and for that reason is out of the scope of this study.

However, if the coating was poorly applied and there are imperfections in the interface between the paint

and the steel surface, the local buckling of the coating layer can occur while the steel plate remains

flat. The critical load that is necessary for the local buckling of the coating was already studied in the

previous Sections. Considering this case, two behaviours may be experienced by the coating. One

is the incapacity of the coating to resist to extra loads, without breakdown, after the buckling shape is

formed (coating breakdown and buckling happen at the same time). Another is the resistance of the

coating to support additional loads, even after the local buckling occurs. In this case, the delamination

growth (increasing of the delamination size) may occur until the moment of the coating failure. As well

described along this work, the coating protects the steel against the ballast tank corrosive environment

and, at the moment when the coating failure occurs the steel corrosion initiates.

After buckling, the coating still resists without breakdown. Delamination growth occurs until the moment when the coating failure happen

Coating breakdown when the buckling occurs

Buckling of the coated steel plate

Buckling of the coating layer(the steel plate remains flat)

Imposed Displacement (δ)

Coated steel plateunder uniaxial compression

Steel corrosion initiation

Figure 5.5: Coated plate behaviour until corrosion initiation

The linear behaviour of the coating film until it buckles locally, determining the critical load for each

studied case is analysed in Section 5.1. Now is time to analyse what happens to the coating layer after

the local buckling occurs. A nonlinear strength analysis is performed by ANSYS for this purpose. As a

result, the displacements and stresses along the coating layer and the load-deflection curves for each

case are presented. Based on the Von Mises failure criteria, explained in Section 3.2.2, the coating

breakdown is also discussed. It is assumed that when the Von Mises stresses on the coating layer is

equal to the coating compressive strength, the coating breakdown initiates. The compressive strength of

the coating is considered to be around 50 MPa, which is associated to some epoxy coating catalogues.

57

Considering that the steel plate thickness (H) and length (L) are constant, the effect of the delamination

length, coating thickness, coating properties and interface parameters (σmax, δn and δt) on the coating

post-buckling behaviour are verified in the next sections. For all simulations, the calculations are based

on a total of 100 load substeps. A higher number of substeps aren’t considered because this would

increase the time of simulation and wouldn’t provide additional data.

5.2.1 Effect of the Delamination Length, l, on the Coating Post-Buckling Be-

haviour

The influence of the delamination size in the local buckling critical load is investigated in Section 5.1.1.

It is concluded that the longer is the delamination length, the lower is the load necessary for the local

buckling of the coating layer to occur. In this section, the sensibility of the coating layer post-buckling

behaviour to the delamination size is analysed. Since that this analysis is the continuation of the study

started in Section 5.1.1, the values used for h, Ec, νc and l are the same. The interface properties are

maintained constant and equal to the values used by Waseem and Kumar (2014), σmax = 25 MPa and

δn = δn = 0.0224 mm.

Figure 5.6 shows the load-deflection curves for l equal to 0.1, 0.2, 0.3, 0.4 and 0.5, resulted from the

nonlinear analyses performed by ANSYS.

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6

Axia

l L

oa

d,

P[k

N]

Deflection, w [mm]

𝑙 =0.1

𝑙 =0.2

𝑙 =0.3

𝑙 =0.4

𝑙 =0.5

Figure 5.6: Compressive load-deflection relations of the coating layer as a function of the delaminationlength

The described load-deflection curves of the coating layer have an approximately exponential behaviour,

as demonstrated in Figure 3.6 (right). However, the curve defined by l = 0.1 is a slightly different be-

haviour for the lower values of load (<20 kN). That initial behaviour can be explained by the fact of be a

small defect length and so for achieving the same deflection is necessary an extra load, compared to the

other cases. These results are in agreement with the results obtained for the critical buckling load, where

it is found that for l = 0.1 is required a bigger value of Pcr for the buckling to occur. The different shape

in the beginning of the l = 0.1 curve confirms the existence of a different coating behaviour for small val-

58

ues of delamination (micro-delamination) and bigger values of delamination size (macro-delamination),

being the l = 10 mm near to the boundary. For the same applied axial load, it can be verified that the

deflection achieved by each curve is different. Considering, e.g., the axial load of P = 50 kN, it is ob-

served that the deflection that this force causes in the coating layer increase as the delamination size

also increases. The difference in between l = 0.1 and l = 0.2 it is visibly more significant than in other

curves (for P = 50 kN the difference in the deflection is around 52%), which is also clearly in accordance

to the results obtained in the buckling analysis.

Figure 5.7 shows the post-buckling coating shapes and also the distribution of the vertical displacement

along the coated plate, for any l. The behaviour of each case, presented in Figure 5.7, is described by

the load-deflection curves shown in Figure 5.6 shows.

(a) l = 0.1 (b) l = 0.2

(c) l = 0.3 (d) l = 0.4

(e) l = 0.5

Figure 5.7: Coating shapes and vertical displacement distribution for each variation of the delaminationlength

A resume of the deflected shape and the correspondent vertical displacement along the length of the

coating layer, achieved for each considered case, is also presented in Figure 5.8. For all presented

cases, the coating defect simulated in the middle of the steel plate has a perfect symmetrically deflected

shape. Between each case, the delaminated curve described by the coating layer is only an expansion

of the previous case, not only in the size, but also in the amplitude. It is interesting to conclude that an

59

increase of 10 mm in the delamination size means an increase of about 1.2 mm in the final deflection.

The maximum deflection is achieved by the l = 0.5 curve and is about w = 6.064 mm, which corresponds

to an increase in amplitude of about 80%, comparatively to l = 0.1.

0

1

2

3

4

5

6

7

20 30 40 50 60 70 80

De

flection,

w

[mm

]

Length of the beam, L [mm]

𝑙 =0.1

𝑙 =0.2

𝑙 =0.3

𝑙 =0.4

𝑙 =0.5

Figure 5.8: Coating deflection shape for each variation of the delamination length

The stresses verified along the coating layer are also analysed, in order to understand when the coating

breakdown occurs. The distribution of the Von Mises stresses along the delaminated part of the coating

layer, which is the interesting part and where the stresses are bigger, are presented in Figure 5.9.

(a) l = 0.1 (b) l = 0.2

(c) l = 0.3 (d) l = 0.4

(e) l = 0.5

Figure 5.9: Coating Von Mises stress distribution for each variation of the delamination length

The distribution of stresses in all cases is quite similar, i.e., the maximum and minimum of the stresses

are located in the same place. With the increasing of the delamination length, it is visible that the

60

stresses estimated for the same load are smaller, and so the region in the middle of the delamination,

where the maximum occurs, is also smaller. As refereed before, it is considered the value of 50 MPa for

the coating compressive strength and the Von Mises criterion for defining theoretically the place where

the coating breakdown occurs. When the Von Mises stresses reach the compressive strength value, it is

considered that the coating breakdown occurs. The Von Mises stress values in the coating layer as the

compressive load increases, until reaching the compressive strength are presented in Figure 5.10. The

Table 5.5 shows the buckling and post-buckling results, i.e., the buckling critical load and the breakdown

load.

0

20

40

60

80

100

120

140

0 5 10 15 20 25 30 35 40 45 50

Axia

l L

oa

d,

P[N

]

Von Mises Stresses [MPa]

𝑙 =0.1

𝑙 =0.2

𝑙 =0.3

𝑙 =0.4

𝑙 =0.5

Figure 5.10: Compressive load-Von Mises stress relations of the coating layer as a function of thedelamination length

Table 5.5: Comparison of the buckling load, breakdown load and respective coating deflection as afunction of the delamination length

Buckling Load Breakdown Load Breakdown Deflection

[N] [kN] [mm]

l = 0.1 6.470 32.452 0.513

l = 0.2 1.617 74.845 1.765

l = 0.3 0.719 106.350 3.111

l = 0.4 0.404 123.912 4.483

l = 0.5 0.259 133.500 5.849

As concluded before, the buckling load decreases for bigger values of l. Opposed, the load necessary

to produce the coating compressive strength increases as the defect size also increases (Table 5.5). A

smaller delamination offers more resistance at the beginning, but after the buckling, due to its reduced

size, the estimated stresses for the same applied force are bigger and theoretically it will break at a lower

load than a bigger delamination (Figure 5.10).

61

Although these results are consistent accounting for the considerations assumed in this thesis, they may

not be in complete agreement with the reality. The environmental conditions, such as the temperature

and humidity, the consequent loss of flexibility and increasing brittleness are important factors, among

many others, that also contribute to the coating failure and that aren’t considered in this study. If these

factors are accounted for, probably the estimated breakdown loads are smaller or the coating breaks at

the moment when the buckling occurs. So, the influence of the refereed factors in the coating breakdown

will need to be checked in the future work.

5.2.2 Effect of the Coating Thickness, h, on the Coating Post-Buckling Behaviour

During the painting process, it is important to achieve a good and uniform coating thickness because it is

one of the most important factors that influences the longevity of the coating. The discussion presented

herein is about the influence of the parameter h in the post-buckling behaviour of the coating layer, and

it is in the sequence of the study already presented in Section 5.1.2. The same parameters (l, Ec, νc)

are used in this study and also the same variations in h are assumed.

The results of the parametric analysis, in terms of load-deflection curves, are shown in Figure 5.11. In

each studied case, the behaviour varies from the others ones, i.e., at each substep, the applied axial

load has a different influence on the coating behaviour. For the lower thickness (h = 0.2), compared with

the maximum studied value (h = 0.5), the displacement that the same applied load causes is bigger.

This is more visible in the beginning, for lower values of P , and at the end of the four curves. The results

are in agreement with the ones from the buckling analysis, where it is concluded that a smaller thickness

needs a smaller load to buckle.

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

160.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Axia

l L

oa

d,

P[k

N]

Deflection, w [mm]

h=0.018

h=0.029

h=0.036

h=0.045

Figure 5.11: Compressive load-deflection relations of the coating layer as a function of the coatingthickness

Figure 5.12 and 5.13 show the symmetrically deflected shapes corresponding to the curves presented

previously. Since that, only the coating thickness is varied in the present analysis, the delaminated part

is equal and the amplitude is very similar (3.5 mm ≤ w ≤ 4 mm). The maximum value achieved by

62

each curve shape isn’t clear. Figure 5.14 shows the maximum vertical displacement for each coating

thickness. Although the difference in the results is not very large (≈ 7%), the conclusion earlier withdrawn

is now again observed, i.e., the lower the thickness, the bigger is the deflection that is achieved.

(a) h = 0.018 (b) h = 0.029

(c) h = 0.036 (d) h = 0.045

Figure 5.12: Coating shapes and vertical displacement distribution for each variation of the coatingthickness

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

30 35 40 45 50 55 60 65 70

De

flection,

w

[mm

]


h=0.018

h=0.029

h=0.036

h=0.045

Figure 5.13: Coating deflection shape for each variation of the coating thickness

3.827

3.664

3.600 3.553

3.40

3.50

3.60

3.70

3.80

3.90

h=0.018 h=0.029 h=0.036 h=0.045

Maxim

um

D

efle

ction

, w

ma

x[m

m]

Figure 5.14: Coating maximum deflection for each variation of the coating thickness

63

After analysing the forces and deflections at each load substep, it is time to check the stresses along the

coating layer (Figure 5.15).

(a) h = 0.018 (b) h = 0.029

(c) h = 0.036 (d) h = 0.045

Figure 5.15: Coating Von Mises stress distribution for each variation of the coating thickness

It is clearly visible the difference in the stress distribution throughout each delaminated part. For the

higher coating thickness, the stresses along the delaminated coating layer are more variable. From h =

0.018 to h = 0.045 the size of the zone with the smaller stresses decreases and in the area correspond-

ing to the maximum estimated stresses increases. With the increasing of the thickness it is also possible

to see that at the ends of each blister, the stresses are bigger, which means that the coating with higher

thicknesses are more favourable to delaminate.

The variation of the Von Mises stresses at each load substep is given in Figure 5.16. In addition, Table

5.6 contains a summary of the critical buckling load values obtained in Section 5.1.2 and also the values

of load and displacement at the coating layer failure. It can be observed that the higher the thickness

of the coating system is, the higher is the force required for the buckling to occur and lower is the

maximum deflection reached by the blister. For h = 0.018 the breakdown load is smaller, which means

that, theoretically the compressive strength is achieved first and the failure occur. The load value at

which the compressive strength is reached, increases from h = 0.018 to h = 0.029 but after it starts to

decrease. Although the relation between the coating thickness and deflection wh increases, the deflection

that the coating has, when the breakdown occurs, has the same behaviour that the breakdown load, i.e.,

increases until h = 0.32 mm but after decreases. This may be due to the fact that the thickness be

excessive taking into account the considered coating material properties and also the loads to which it

is subjected.

64

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25 30 35 40 45 50

Axia

l L

oa

d,

P[k

N]


h=0.018

h=0.029

h=0.036

h=0.045

Figure 5.16: Compressive load-Von Mises stress curves of the coating layer as a function of the coatingthickness

Table 5.6: Comparison of the buckling load, breakdown load and respective coating deflection as afunction of the coating thickness


[N] [kN] [mm]

h = 0.018 0.175 69.669 2.716

h = 0.029 0.719 84.061 2.835

h = 0.036 1.401 83.246 2.770

h = 0.045 2.742 75.326 2.598

From the performed analysis, it is important to notice that a good coating thickness is one of the most

important parameters to determine the time that the coating resists for protecting the steel. Due to the

quality of the coating application, defects can appear related to the thickness. If the coating is very thick,

it can support more load, but after it can peel away from the steel, as shown in Figure 5.17 (left). When

the coating is too thin (less than the minimum specified by the coating manufacturer), the supported load

is smaller and the failure may occur early (Figure 5.17 (right)).

Figure 5.17: (a) Over thick coating detaches easily. Poor preparation between coats of paints causesearly failure ; (b) Where the coating is too thin, early failure occurs in service (ABS, 2007)

65

5.2.3 Effect of the Coating Properties, Ec and νc, on the Coating Post-Buckling

Behaviour

The effect in the coating post-buckling behaviour of the changes in the coating properties will be studied

here. The values used for the parameters l and h are maintained the same. The interface has the same

properties used in the previous two post-buckling nonlinear analysis.

The load-deflection curves obtained for each variation of the coating properties (± 15% Ec, νc and ±

30% Ec, νc) are presented in Figure 5.18. The objective of varying the coating properties is to simulate

the behaviour of different coating types.

0

20

40

60

80

100

120

140

160

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Axia

l L

oa

d,

P[k

N]

Deflection, w [mm]

0.7Ec,νc

0.85Ec,νc

Ec=3000Mpa and νc=0.37

1.15Ec,νc

1.3Ec,νc

Figure 5.18: Compressive load-deflection relations of the coating layer as a function of the coatingproperties

In the linear buckling analysis, the coating film is considered independent of the steel plate, what doesn’t

happen in the nonlinear post-buckling study. As shown in Figure 5.18, the variation performed in the

coating properties doesn’t affect much the post-buckling behaviour of the coating. The described curves

are almost overlapping, which means that in terms of the deflection behaviour, a variation of ± 30%

in the properties, when the coating thickness and the delamination length are maintained constants,

doesn’t have a large influence. Figure 5.19 presents all the deformed shapes for each studied case. It is

possible to see exactly what is described previously. An increase in the relation between the coating and

the steel properties, Ec/Es and νc/νs, i.e., an increase in the coating properties values, represents a

very small diminution of the coating deflection. Figure 5.20 shows the maximum values of the estimated

displacement. The variation in the amplitude between the -30% and +30% cases is about 3%, which is

a small value compared with the change in the coating properties.

Figure 5.21 shows the distribution of the FEA stress results along the coating delaminated part.

66

0

0.5

1

1.5

2

2.5

3

3.5

4

30 35 40 45 50 55 60 65 70

De

fle

ction

, w

[m

m]


0.7Ec,νc

0.85Ec,νc


1.15Ec,νc

1.3Ec,νc

Figure 5.19: Coating deflection shape for each variation of the coating properties

3.781

3.748

3.7213.700

3.679

3.60

3.65

3.70

3.75

3.80

0.7Ec,νc 0.85Ec,νc Ec=3000Mpa and νc=0.37

1.15Ec,νc 1.3Ec,νc

Ma

xim

um

D

efle

ction

, w

max

[mm

]

Figure 5.20: Coating maximum deflection for each variation of the coating properties

(a) 0.7Ec,νc (b) 0.85Ec,νc

(c) Ec = 3000 MPa, νc = 0.37 (d) 1.15Ec,νc

(e) 1.3Ec,νc

Figure 5.21: Coating Von Mises stress distribution for each variation of the coating properties

67

Visually, the distribution of stresses as a function of the coating mechanical properties doesn’t vary

much. As in the previous cases, the maximum stress is achieved in the centre of the delaminated part,

which also corresponds to the point where the maximum deflection occurs. The fact of this being the

point where it is applied the force to simulate the initial imperfection explains the maximum values in this

area. Figure 5.22 shows the compressive load-Von Mises stress curves for all the studied cases up to 50

MPa. According to the typical stress-strain curve (Figure 3.5), defined in the linear part by the Hooke’s

law, for bigger values of E, the achieved stress values are also bigger (Figure5.22). It is interesting to

conclude that for bigger values of Ec/Es and νc/νs, the buckling critical load is bigger but the breakdown

load and deflection are smaller (Table 5.7). Due to its elasticity, in the beginning, is necessary a higher

load for the buckling occurs but, after that the stresses in the coating layer have a faster increase.

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35 40 45 50

Axia

l Load,

P[k

N]


0.7Ec,νc

0.85Ec,νc


1.15Ec,νc

1.3Ec,νc

Figure 5.22: Compressive load-deflection relations of the coating layer as a function of the coatingproperties

Table 5.7: Comparison of the buckling load, breakdown load and respective coating deflection as afunction of the coating properties


[N] [kN] [mm]

0.7Ec,νc 0.503 150.662 3.774

0.85Ec,νc 0.611 122.976 3.371

Ec = 3000 MPa, νc = 0.37 0.719 102.704 3.071

1.15Ec,νc 0.826 87.246 2.822

1.3Ec,νc 0.934 75.439 2.616

The analysis performed herein can be important when applied in practice to protect against corrosion.

Knowing the influence of the properties in the coating behaviour, can be a base to choose a paint that

suits better to the physical needs and that can withstand and protect the steel for a longer time.

68

5.2.4 Effect of the Interface CZM Constants

The interface between the steel surface and the coating layer modelled in ANSYS is governed by the

Equation 3.18 and 3.19. According to these relations, the interface is dependent on the parameters

σmax, δn and δt. The parameter σmax is associated with the resistance of the interface and the parame-

ters δn and δt to the rigidity (Gaspar, 2011).

The value of σmax is the maximum stress acting in the normal direction that can be installed in the in-

terface. On the other hand, the parameters δn and δt measure how the interface can deform without

affecting its strength. δn and δt are designed by characteristic lengths of the normal and tangential di-

rection, respectively.

There are no experimental and theoretical studies that evaluate the adhesion of a coating/steel plate

system and that provide directly the values of the parameters σmax, δn and δt. So, for the previous

analysed cases, are considered the interface parameters already used in a laminated composite study.

The strength of the interface depends on many factors, e.g., the type and properties of coatings, the

degree of preparation of the surface, the effectiveness of the coating application, etc. Thus, it is justified

a parametric analysis to test new parameters and to establish the influence of these parameters in

the coating and interface behaviour. The parametric analysis performed in ANSYS, consists into the

change of the interface parameters, analysing its influence on the coating behaviour and also verifying

the deformation and stresses in the interface. In each analysis, the parameter l, h, Ec and νc will be

maintained constant and equal to 30 mm, 0.32 mm, 3000 MPa and 0.37, respectively. The variation in

the interface parameters is presented in Table 5.8 and schematized in Figure 5.23.

Table 5.8: Analysed interface parameters

Parameter Analysed Values

σmax 25;15;10 [MPa]

δn 0.0224 0.01 1 [mm]

δt 0.0224 0.1 100 [mm]

δn/δt 1 0.1 0.01 [-]

σmax

δn/δt

δn/δt

δn/δt

Figure 5.23: Schematic representation of the interface parametric analysis

69

For the σmax values is decided to use only values lower than the one used in all the above mentioned

analysis (25 MPa), because this interface proved to be strong, since it never delaminated in the analysed

cases. The additional values of δn and δt are chosen based on the study reported by Gaspar (2011).

5.2.4.1 Coating Layer Deflection

The effect of the interface parameters variation in the coating layer deflection is analysed here. Figure

5.24, 5.25 and 5.26 shows the coating load-deflection curves and the respective deflection shape for

each variation in σmax and δn/δt.

0

20

40

60

80

100

120

140

160

0 2 4 6 8

Axia

l Lo

ad

, P

[kN

]

Deflection, w [mm]

δn/δt=1

δn/δt=0.1

δn/δt=0.01

(a) Coating load-deflection curves

6.552

6.163

3.597

0

1

2

3

4

5

6

7

0 20 40 60 80 100

De

fle

ction

, w

[m

m]


δn/δt=1

δn/δt=0.1

δn/δt=0.01

(b) Coating deflection shapes

Figure 5.24: Compressive load-deflection relations and respective deflection shapes of the coating layer,considering σmax = 25 MPa, as a function of δn/δt

0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12

Axia

l L

oa

d,

P[k

N]

Deflection, w [mm]

δn/δt=1

δn/δt=0.1

δn/δt=0.01


10.982

6.184

3.606

0

2

4

6

8

10

12

0 20 40 60 80 100

De

fle

ction

, w

[m

m]


δn/δt=1

δn/δt=0.1

δn/δt=0.01



70

0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12

Axia

l L

oa

d,

P[k

N]

Deflection, w [mm]

δn/δt=1

δn/δt=0.1

δn/δt=0.01


3.625

11.052

6.203

0

2

4

6

8

10

12

0 20 40 60 80 100

De

fle

ction

, w

[m

m]


δn/δt=1

δn/δt=0.1

δn/δt=0.01



From Figure 5.24, 5.25 and 5.26 can be concluded that a smaller σmax value causes an increase in the

deflection of the coating layer, i.e., an interface with a bigger maximum stress is a stronger interface and

for the same loading conditions, the coating deformation and the possibility of occurs delamination is

smaller. The deformation of the coating is inversely proportional to the value of σmax. All the deflection

curves for δn/δt = 1 and δn/δt = 0.01 have an acceptable shape, i.e., the applied load and the coating

deflection shows an approximately exponential behaviour (Figure 3.6 (right)). The shape described by

the curves for δn/δt = 0.1 is also the expected in the beginning, but after a determined point the shape

changes and, for the same applied load at each load substep, the displacement of the coating is much

bigger. These phenomena occurs because the interface between the coating layer and the steel surface

doesn’t resists to the applied loads and the coating starts to delaminate (see Figure 5.24, 5.25 and 5.26

(right)).

According to the auxiliary analysis performed in Appendix B.1, for the lower values of δn and δt, the

coating deflection is smaller. Based on this study, it is expected that the lower deflection is achieved

by δn/δnt = 0.1, since δn has the lower value, but happens exactly the opposite. This can be explained

by the fact that the value used for δt is 0.1, which is a bigger value when compared to the value of δn,

and probably has negatively influenced the effect of δn = 0.01, enhancing the failure of the interface and

initiating the delamination. If the δt value was equal to the value of δn, then the deflection would be less

than the estimated by δn = δt = 0.0224 mm, as shown in the Appendix B.1. The study of the mutual

influence of the δn and δt parameters is also part of the future work.

71

5.2.4.2 Coating Layer Von Mises Stress

The stresses acting on the coating layer at each load substep, until the considered compressive load,

estimated as 50 MPa, are presented in Figure 5.27. From these load-stress curves, it is interesting to

observe that the higher the value of δn/δt, the higher is the force required for the compressive strength.

Gaspar (2011) wrote that for small values of δn and δt, the transmission of the stresses applied to the

structure are similar to the transmission that occurs in the situation where there is no interface, e.g.,

the coating and the steel plate acting as a block. In the contrary, a larger values of δn and δt cause a

worse stress distribution between the two faces of the interface. What is described in Figure 5.27 and

in Appendix B.2 is in accordance with the cited by Gaspar (2011). In δn/δt = 0.01 curves, the interface

is less rigid and for the same applied load, the stresses that the coating layer achieved are bigger. For

δn/δt = 1 the interface is stronger and so, it is necessary an extra load for achieve the same stresses.

It is also interesting to conclude for the coating curves with δn/δt = 0.1, which are the ones that have

delaminated, that the load necessary for the delamination begins is a bigger load that the one required

for achieve the coating compressive strength, which means that, based on this failure criteria and these

interface parameters, the coating will breakdown before its delamination begins.

0

20

40

60

80

100

120

0 10 20 30 40 50

Axia

l Loa

d,

P[k

N]


δn/δt=1

δn/δt=0.1

δn/δt=0.01

(a) σmax = 25 MPa

0

20

40

60

80

100

120

0 10 20 30 40 50

Axia

l Lo

ad

, P

[kN

]


δn/δt=1

δn/δt=0.1

δn/δt=0.01

(b) σmax = 15 MPa

0

20

40

60

80

100

120

0 10 20 30 40 50

Axia

l Load,

P[k

N]


δn/δt=1

δn/δt=0.1

δn/δt=0.01

(c) σmax = 10 MPa

Figure 5.27: Compressive load-Von Mises stress relations of the coating layer considering σmax = 25,15, 10 MPa, as a function of δn/δt

72

5.2.4.3 Interface Deformation

This Section analyses the reaction of the interface itself to the variation of the initial parameters that

characterizes it. As explained in Section 3.3.1, the interface modelled in ANSYS has two rupture modes:

normal and tangential. In Appendix B.3 to B.6 are presented the results for the normal and tangential

interface separation and traction for σmax = 25, 15, 10 MPa. Table 5.9 shows a resume of the maximum

separations and tractions estimated in the interface, for each studied case.

The interface delamination and failure process involves the stiffness softening and complete loss of the

interface stiffness, which in turn will cause numerically instability of the solution. As explained before, the

δn and δt parameters quantify the separation that occurs in the interface when it is subjected to a given

stresses, namely σmax and 2/√

2τmax. When the separation values (δn and δt) are high, the interface is

highly deformable, i.e., the transmission of the stresses between the steel surface and the coating layer

doesn’t occur correctly and the interface is subjected to higher-order stresses leading to its deformation.

In all analysis, the value of the maximum normal stresses is considered constant along the interface

between the steel surface and the coating layer, which doesn’t correspond to the real behaviour. This

parameter represents the adherence between the two surfaces, that isn’t constant due to many factors,

as the case of the poor application of the coating. In the future work would be interesting to perform

coating/steel interface analysis where the values of σmax will change along the interface.

Analysing the results presented in Table 5.9, it is possible to conclude that exists a directly proportional

relationship between the values of δn and ∆n and δt and ∆t and an inverse proportionality between

the values of δn and Tn and δt and Tt. This conclusion is in accordance with the theoretical relations

governed by Equation 3.18 and 3.19 and also with the auxiliary analysis performed in Appendix B.1. It is

also interesting to observe that the values of ∆n and ∆t are inversely proportional to the value of σmax,

i.e., an interface with a lower value of σmax can loss adherence and deform more easily (see Table 5.9).

Table 5.9: Maximum normal separation (∆n), tangential separation (∆t), normal stress (Tn) and tan-gential stress (Tt) values achieved in the interface, considering σmax = 25, 15, 10 MPa, as a function ofδn/δt

σmax = 25 MPa σmax = 15 MPa σmax = 10 MPa

δn/δt = 1 δn/δt = 0.1 δn/δt = 0.01 δn/δt = 1 δn/δt = 0.1 δn/δt = 0.01 δn/δt = 1 δn/δt = 0.1 δn/δt = 0.01

∆n 0.002 1.543 0.039 0.003 7.230 0.050 0.004 7.331 0.061 [mm]

∆t 0.007 0.692 1.046 0.011 1.188 1.049 0.019 1.151 1.052 [mm]

Tn 6.278 3.943 2.572 5.109 2.572 1.940 4.208 2.202 1.556 [MPa]

Tt 34.642 5.826 0.014 29.141 3.496 0.009 22.944 3.030 0.006 [MPa]

73

5.3 Failure Assessment Diagram

Inevitably, steel structures protected by coatings will require maintenance to recover the protective sys-

tem. The extent of maintenance depends upon the condition of the initial protective treatment and the

effectiveness of any remaining coating.

As described in the Chapter 2, the condition of the coating in ballast tanks is assigned and categorised

as "GOOD", "FAIR" and "POOR" based on a visual inspection and the estimated percentage of areas

with a coating breakdown and rusty surfaces. The determination of the extension of the coating break-

down is usually carried out using the extended diagrams with the correspondent estimated percentage,

e.g., the diagram presented in Appendix C. The Classification Societies require that the coating repair

work needs to be carried out once a "POOR" condition has been reached to bring the ballast tank back

into a "FAIR" or "GOOD" condition. However, when a ballast tank has reached the "FAIR" condition, the

usable protective lifetime of the coating has probably came to an end and the steel renewal will become

inevitable if the coatings cannot be satisfactorily repaired. So, it is important to be able to detect the

coating breakdown before the point that an extensive ballast tank refurbishment becomes necessary. In

order to do so, it is necessary to assess the coating breakdown in a way that enables the breakdown

mechanism itself to be understood.

Based on the delamination length/diameter (l), which is the unique parameter that can be visually in-

spected and measured in a ship ballast tank without affecting the coating condition, can be developed a

criteria that defines when the coating almost breakdowns and the repair is needed.

In Section 5.1.1 and 5.2.1, the parametric variation of the delamination length/diameter is performed for

l equal to 10, 20, 30, 40 and 50 mm. The behaviour of the blisters with a micro diameter is different

from the ones with a macro diameter. This thesis is focused to understand the coating delamination

behaviour with macro sizes, leaving the micro category for the future work. However, and in order to

understand the delamination size that corresponds to the point when the micro category ends and the

macro one begins, three additional simulations are performed with delamination sizes lower than l = 10

mm. Figure 5.28 shows the delamination sizes already used in the parametric study and the ones used

to determine the micro and macro delamination size rage (l equal to 2.5, 5 and 7.5 mm).

Coating Coating

Coating

Del

amin

atio

ndi

amet

er

l=2.5mmSteel Plate

l=5mmSteel Plate

l=7.5mm

Coating

Steel Plate

l=20mmSteel Plate

Steel Plate l=30mm

Steel Plate l=40mm

Coating

Coating

Steel Platel=10mml=50mm

Coating

Steel Plate

Coating

Figure 5.28: Considered values for the coating delamination size/diameter

74

According to the Von Mises criteria, i.e., assuming that the coating failure occurs when the Von Mises

stresses achieve the coating compressive strength (about 50 MPa), is determined the axial load neces-

sary for the coating breakdown and also the deflection of the coating (see Table 5.5, l = 10,20,30,40,50

mm). With these values, the Von Mises failure curve is defined, as represented in Figure 5.29. Taking

into account the load values below that curve, the coating thin film has buckled, but didn’t breakdown

yet, i.e., still resists to the water intake. However, for the forces above the curve, the coating is failing

and the corrosion has begun. A bigger defect fails with a higher load and deflection that a defect with a

smaller diameter.

Fair Condition Curve (70% of the Breakdown Load)

Poor Condition Curve (85% of the Breakdown Load)

0

20

40

60

80

100

120

140

160

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Axia

l Load,

P[k

N]

Deflection, w [mm]

𝑙 =0.025 𝑙 =0.05 𝑙 =0.075 𝑙 =0.1 𝑙 =0.2 𝑙 =0.3 𝑙 =0.4 𝑙 =0.5

Von Mises Failure Curve (100% of the

Breakdown Load)

Good Condition Curve ( 50% of the Breakdown Load)

Figure 5.29: Compressive load-deflection behaviours including the Von Mises failure criteria

Based on Figure 5.29, it is possible to conclude that when a coating has a defect of diameter l, it fails

locally with a certain deflection. However, as has been explained before, it isn’t the moment when the

breakdown occurs that interests in this work, but a state just before, where it is still possible to repair

without corrosion. For knowing this point of the coating effective life, it is assumed that when the applied

force reaches 85% of the breakdown force (Figure 5.29), the coating is in a poor condition and it is

necessary to repair the affected zone, because after that, the failure can occur soon. This assumption

is made based on the knowledge that already exists for repair and preventing the coating collapse. In

order to know the condition of the coating during its lifetime, also a "fair" and a "good" curve are defined,

based on 50% and 70% of the Von Mises failure load. These curves are also presented in Figure 5.29.

Considering the assumed 85% of the breakdown load, a simplified Failure Assessment Diagram (FAD)

is defined. This diagram is presented in Figure 5.30, where in the y -axis is represented the applied axial

load, that is calculated for the analysed cases as described before (85% breakdown load), and in the

x-axis is represented the delaminated blister diameter (l). It is important to refer that the coating thick-

ness, coating properties and interface parameters are maintained constant. If any of these parameters

change, it will be seen a difference in the failure assessment diagram, because this is a function of these

parameters.

75

0

20

40

60

80

100

120

7.5 16.0 24.5 33.0 41.5 50.0

Axia

l L

oa

d,

P[k

N]

Delamination diameter, 𝑙 [mm]

f(Coating Thickness, Coating Properties, Interface Properties)

Acceptable Domain

Unacceptable Domain Critical

Figure 5.30: Coating failure assessment diagram for macro-delamination diameter, considering 85% ofthe breakdown load

The coating failure assessment diagram is a criteria that can be used to prevent corrosion. Below the

curve defined by the 85% of the breakdown load, the coating condition is still in an acceptable condition,

i.e., for now it is not required to do maintenance of the affected area. On the other hand, in the area

above the curve, the coating system is in an unacceptable condition, which means that at any time

the coating may fail or has already failed. The curve itself represents the boundary limit between the

acceptable and unacceptable state. Considering a certain compressive force acting on the coated plate,

when it intersects the limit in the coating failure diagram, a certain critical delamination size is obtained.

This means that a blister of this size has reached it critical point and it will fail soon. It is therefore

required that the delaminated blisters with the obtained critical size from the FAD, are repaired so that

the coating breakdown is avoided. Using the other conditional limits defined in Figure 5.29, a more

detailed coating failure assessment diagram is constructed. This diagram not only allow to know when

the coating is in acceptable or unacceptable condition, as the simplified diagram presented in Figure

5.30, but also permits to find out in which state of the acceptable domain (good, fair and poor condition)

the coating is. It is important to refer that the limits that are delimiting the good, fair and poor condition

areas are defined by assuming 50%, 70% and 85% of the breakdown load, respectively.

0

20

40

60

80

100

120

7.5 16.0 24.5 33.0 41.5 50.0

Axia

l L

oa

d,

P[k

N]


Fair Condition

Good Condition

FailurePoor Condition

Figure 5.31: Failure assessment diagram for macro-delamination diameter values

76

Both, the simplified and the detailed coating failure assessment diagram (Figure 5.30 and 5.31) are only

valid for the macro-delamination diameter range, that is found to be beyond 7.5 mm. For a delamina-

tion diameter lower than 7.5 mm (micro-delamination cases), the coating film has a different behaviour,

decreasing the load value as the blister size increases, as can be seen in Figure 5.32 and 5.33. The

boundary value of 7.5 mm is called the delamination diameter threshold and is determined by defining

the minimum of each of the limit curves (Figure 5.32 (right)). Since that the represented curves are

related each other by a percentage of Von Mises criterion, the threshold delamination diameter is equal

for all. The intersection of the constant lthreshold with the four limits, results in a four different Pthreshold

values (Figure 5.32 (right)). Figure 5.33 presents a zoom of the lower delamination size zone of Figure

5.29. As happens for the curves defined by the load-delamination diameter values, in the load-deflection

curves the behaviour of the macro- and micro-delamination rage is different. Unlike happens in Fig-

ure 5.32 (right), where the four threshold points defined a perfect vertical straight line, the four threshold

points defined in Figure 5.33 have all different wthreshold and Pthreshold, defining a perfect crescent curve.

It is demonstrated in this part of the work that, the behaviour of the macro- and micro-delamination is

characterized differently. However, only a detailed study of the behaviour of the macro-delamination

cases is performed. A more detailed study about the micro-delamination should be developed in the

future, in order to understand better the behaviour of these small coating blisters on the top surface of

the steel plates.

0

20

40

60

80

100

120

140

0 10 20 30 40 50

Axia

l Load,

P[k

N]


Von Mises Failure Curve

Poor Condition Curve

Fair Condition Curve

Good Condition Curve

(a) Coating condition curves for l = 0 − 50mm

0

70

140

0 10 20

Axia

l Load,

P[k

N]






c

Pthreshold,Failure

Pthreshold,Good

Pthreshold,Fair

Pthreshold,Poor

𝑙threshold=7,5

(b) Zoom of the coating condition curves for the lower val-ues of l

Figure 5.32: Poor, fair and good coating conditions for the delamination diameters

77

0

70

140

0 1.5 3

Axia

l L

oad,

P[k

N]

Deflection, w [mm]





c

Pthreshold,Failure

Pthreshold,Good

Pthreshold,Fair

Pthreshold,Poor

wthreshold,Fair

wthreshold,Poor

wthreshold,Good

wthreshold,Failure

Figure 5.33: Zoom of the limit conditions as function of the deflection for the lower values of l

Corrosion is one of the most important failure mode that ships face during their lifetime. For this reason,

numerous studies have been performed to predict the behaviour and the duration of the structures

subjected to corrosion. In recent years, mainly due to ecological concerns, it began an increasing

concern for the prevention of corrosion and pollution. The Classification Societies assess the condition

of the coating systems and only require repair when already exists corrosion (see diagram in Appendix

C.1). The work developed herein focus on prevention, determining that it is necessary to maintain the

paint before it breaks. Thus, it can be concluded that the failure assessment diagram developed in

this work (Figure 5.30 and 5.31) is very important in preventing corrosion, because it completes the

criteria that already is defined by the Classification Societies to establish a good coating condition in

ballast tanks (Appendix C.1), defining the coating blister diameter that requires to be repaired in order to

prevent the corrosion initiation at these locals.

78

5.4 Final Discussion and Concluding Remarks

Along this Chapter, the linear and nonlinear finite element analyses are presented and discussed. The

final aim of this dissertation is to analyse the coating breakdown of ship ballast tanks and to establish

a criteria that helps the ship owner and the Classification Societies to anticipate the coating breakdown

and repair the coating before corrosion begins. Therefore, herein, is verified the influence of the varia-

tion of the delamination length l, coating thickness h, coating properties and constants that define the

interface between the coating layer and the steel plate on the buckling and post-buckling behaviour of

the steel plate coating. Many other factors affect the coating resistance, e.g., temperature, humidity,

erosion, porosity, etc., but those are left for future considerations.

The study of the effect of the delamination length on the coating behaviour showed that the critical buck-

ling load decreases for bigger l and, contrary, the load necessary for the coating breakdown to occur

increases, as presented in the failure assessment diagram (see Figure 5.30 and 5.31). A smaller de-

lamination offers more resistance at the beginning, but after the buckling, due to its reduced size, the

estimated stresses are bigger and theoretically it will break at a lower load than at a big delamination.

From the parametric analysis of the coating thickness (h), it is concluded that an adequate thickness is

expressly necessary to provide a good anti-corrosion protection and to achieve the expected anti-fouling

lifetime. Due to the coating application, the thickness of the coating isn’t uniform along all the area. This

analysis allowed to conclude that under thickness will result in premature failure. However, over appli-

cation can also cause problems, such as subsequent loss of adhesion, cracking of the paint or splitting

of the primer coats. Thus, ideally the coating thickness in the ballast tanks should be that specified by

the manufacturers and/or Classification Societies, allowing for small practical application variations. The

results showed that although the relation between the coating thickness and deflection wh increases, the

deflection that the coating has when the breakdown occurs has the same behaviour that the breakdown

load, i.e., increases until h = 0.32 mm, but after that decreases. This may be due to the fact that the

thickness is excessive by taking into account the considered coating material properties and also the

loads to which it is subjected.

The coating properties depend on the coating type, and have also an important influence on the durability

of the protective system. Therefore, it is also considered their influence in this study. It is concluded that

for bigger ratios of Ec/Es and νc/νs, the buckling critical load is bigger, but the breakdown load and

deflection are smaller. Due to its elasticity, in the beginning, it is necessary a higher load for the buckling

to occur, but, after the stresses in the coating layer have a faster increase. It is up to the ship owner to

decide if he will use a paint with better resistance against delamination and also more expensive, or if

he prefers a cheaper paint, which will degrade a few years later and will need repair.

79

The interface strength between the coating and steel depends mostly on the coating properties, surface

preparation and application method. From the interface analysis is deducted that the deformation of the

coating and the Von Mises stresses are inversely proportional to σmax and directly to the parameters

that define the rigidity (δn and δt). Based on the theoretical formulations and numerical results, it is also

verified a relation of a directly ratio between the values of δn and ∆n and δt and ∆t and an inverse ratio

between δn and Tn and δt and Tt. It is also found that an interface with the lower value of σmax has

higher values of ∆n and ∆t.

Based on the developed studies, especially in the analysis of the delamination length, a failure assess-

ment diagram is created. This diagram is defined based on the Von Mises breakdown load and it is

a function of the coating thickness and coating and interface properties. The FAD is only valid for the

macro-delamination diameter range, that is found to be beyond 7.5 mm, because for a delamination

diameter lower than this value, the coating layer has a different behaviour. With this diagram, it is pos-

sible to know when the coating is in an acceptable or unacceptable condition and in which state of the

acceptable domain (good, fair or poor condition) the coating is, for each delamination diameter. When

the coating condition is still in an acceptable condition, it is not required to do the maintenance of the

affected area. However, if the coating is in an unacceptable condition means that at any time the coating

may fail or already has failed. When the critical limit ("poor" condition curve) is reached by a determined

blister diameter, is required to do its repairing because sooner it will fail. Finally, it can be concluded that

the failure assessment diagram is an added value to the corrosion prevention and to the criteria already

used by the Classification Societies to assess the condition of coating systems.

80

Chapter 6

Final Remarks and Future Work

6.1 Final Remarks

This dissertation analysed the coating breakdown of steel plates in marine structures subjected to uni-

axial in-plane compressive load. For this purpose, a nonlinear finite element model of a rectangular

coated steel plate was developed and validated with the commercial software ANSYS. The plate and

the coating layer were discretized by three-dimensional finite elements. To model the adhesion between

the plate and coating layer, a zero thickness interface element was employed, based on the exponential

form of the cohesive zone model. In order to simulate the coating defect, an initial imperfection was

modelled by introducing a symmetrically debonded zone and inducing in the middle nodes of the plate

a small destabilization load. This model is a compromise between accuracy of the coated plate size,

coating layer discretization, computational capabilities and time, since that the average computational

effective time per simulation was in between 40 to 50 minutes. Taking into account that the convergence

of the results wasn’t always a successful, the overall computational time increases considerably.

The buckling behaviour of the coated steel plate, with an initial imperfection and a central delaminated

part in the interface, was studied based on the parametric eigenvalue buckling and nonlinear strength

analysis. For the linear buckling analysis, a simpler finite element model was developed, where the buck-

ling critical load of the coating layer was obtained. Using the theoretical expressions, the critical load

was calculated for different delamination sizes, coating thickness and coating properties. A comparison

between the linear FEA and the theoretical results were carry out. The nonlinear strength analysis was

performed using the nonlinear FEM. In this analysis, essentially the post-buckling behaviour was anal-

ysed for different delamination sizes, coating thicknesses, coating properties and interface parameters.

To determine the moment when the coating breakdown effectively occurs, the Von Mises failure crite-

ria was used. It was considered that when the Von Mises stresses on the coating layer achieved the

compressive strength of the coating, the coating fails totally and the corrosion begins.

81

From the parametric variation of the coating delamination size, i.e., the blister diameter, was concluded

that the critical buckling load decreases as the delamination size increases and, the breakdown load

increases. Compared to a smaller diameter, a bigger blister achieve a bigger deflection before its break-

down occurs. Taking into account the simplifications performed in the model, these results were in good

agreement with the linear theoretical expressions and with the results already obtained by many authors

in studies about the delamination in layered composite materials.

Due to the careless application of the coating on top of the steel plates, the coating thickness can be

excessive or insufficient to resist to the loads and the surrounding ambient. The parametric variation of

the coating thickness allowed to conclude that to achieve a good and uniform thickness is one of the

most important factors in determining the lifetime of the coating. A very thick coating system supports

more load until the buckling occurs but, in the post-buckling stage, due to the high internal stresses, it

can delaminate from the steel more easily. Contrary, when the coating is too thin, the supported load for

the buckling and breakdown are smaller and, consequently, the failure occurs earlier. The Classification

Societies already have this concern with ballast tanks and have defined a minimum film thickness that is

required when the manufacturers didn’t specify the minimum thickness for a specific coating. This mini-

mum thickness required by the Classification Societies was tested in this work and, taking into account

the considerations performed (delamination length value, coating properties and interface properties),

proved to be the best thickness value compared to the other ones.

The coating properties have also a big influence on the coating protection durability. Owners, builders

and coating manufacturers are fully aware that it is more efficient and cost effective to apply quality coat-

ings than to be forced to repair them at a later stage in the service life of the vessel. In order to simulate

different types of coating and check its influence on the coating protection behaviour, some variations in

the mechanical properties were performed. It was concluded that for bigger ratio values of Ec/Es and

νc/νs the load necessary for the bucking shape creation is bigger, but the breakdown load and the re-

spective deflection of the blister are smaller. It was also concluded that when the difference between the

coating and steel properties is smaller (bigger Ec/Es and νc/νs ratios), the coating behaviour is clearly

affected by the elastic characteristics of the steel.

Delamination is a phenomenon of great importance, not only for the composite industry, but also for the

coating durability. When the stress in the interface elements exceeds the critical value, the stress field

is redistributed, resulting in the element deformation and separation/delamination across the interface.

A interface parametric variation was performed and was concluded that the deformation of the coating

and the Von Mises stresses are affected inversely by the maximum normal interface traction and directly

by the normal and tangential rigidity parameters (δn and δt). A direct relation between δn and ∆n and δt

and ∆t and an inverse in the ratios of δn and Tn and of δt and Tt was also verified. These results were

in accordance with the theoretical formulations that define the interface. Finally, it was also concluded

that an interface with a smaller maximum normal traction can loose adherence and deform more easily,

82

which means that higher values of the normal and tangential opening displacements were found in the

interface.

Managing the corrosion of marine ballast tanks is an ongoing problem for all vessels. A good protec-

tive coating applied on a well-prepared surface at the new building stage and a good coating mainte-

nance/repair are the most effective means of avoiding this problem. The ability of the coating to resist

corrosion over extended periods is an important contributor in safeguarding the capital investment in

ship structures. Understanding how coatings postpone and reduce corrosion rates, how they must be

maintained and how coatings eventually breakdown are important for safe vessel operations. For that

reason, the work developed herein was focused essentially on the coating breakdown prevention.

In the final stage of the work, it was demonstrated that the behaviour of the macro- and micro-delamination

sizes is different. This work has focused more on the study of the macro-delamination length. Based

on the delamination diameter nonlinear analysis, a coating failure assessment diagram, applied only for

macro-delamination diameter values, was created. This diagram is a function of the coating thickness,

coating properties and interface parameters and it was defined based on the breakdown load obtained

by using the Von Mises failure criterion. Three curves delimited the area in the diagram where the coat-

ing is in a good, fair, poor or failure condition. The poor condition limit represents the critical boundary,

above which the coating may fail or already has failed (unacceptable domain) and, below which the coat-

ing still is in a acceptable condition, i.e., in a good, fair or poor state. When for a certain delamination

diameter, the applied load reaches the poor condition curves, i.e., the critical point, the blister requires

repairing because it will fail sooner. The developed diagram is a very important tool to prevent the cor-

rosion, because it completes the criteria already in use by the Classification Societies for establish the

coating condition in ballast tanks, defining the coating blister diameter that requires to be repaired in

order to prevent corrosion initiation at these spots.

Finally, it can be concluded that the finite element analysis results were in accordance with the results

that have been obtained in similar studies of buckling and delamination using interface cohesive zone

modelling. This proves that the nonlinear analysis, which allows to take into account large displacements

and to impose contact contains between the coating and the steel plate, it is a good way to study this type

of problems, when there is no possibility of carrying out experimental tests. Applying a small perturbation

load also demonstrated to be the best way to simulate the coating imperfections.

6.2 Future Work

Based on the conclusions and challenges faced during the development of this work, further work can

be done to improve the coating behaviour understanding. The future work comprises:

• Study of the behaviour of the micro-delamination and develop a failure assessment diagram for

the delamination diameter values lower than 7.5 mm.

83

• Compressive experimental tests of the studied coated steel plate and comparison of the results

with those obtained in this thesis.

• Adhesion experimental tests (e.g., scrape adhesion test, pull-off tests, etc.) for determining the

coating-steel surface adhesion parameters (σmax, δn and δt) for different types of coatings and

also different surface preparation and/or application methods.

• Detailed interface parametric study. In this work, the values of the interface parameters were

assumed constant along the interface between the steel and the coating, which does not fully

correspond to the reality, due to the surface status, the coating application method and coating

properties. It would be important to consider in a future study that the properties vary along the

interface. Could also be tested the mutual influence of these parameters (σmax, δn and δt).

• Influence of other parameters in the behaviour of the coating, such as, the temperature, loss of

the coating properties along the time, humidity, surrounding ambient changes, erosion, impacts,

sea water salinity, chemistry, pH level, biodegradation by bacteria and many others factors that

contribute to the coating degradation. These considerations can be made, e.g., by manipulating

the mesh elements, varying the parameters of the interface, changing the coating properties, etc.

• Numerical and experimental study in which the position of the delamination vary, since in this study

only the case of a symmetrical blister was considered, positioned exactly in the middle of the plate.

Besides being important to consider the case of different asymmetric positions it would also be

interesting to simulate several delamination and of different sizes in the same plate, which is what

often happens in reality.

• Experimental tests to identify the correct properties of the coating so that the numerical studies

are as similar as possible to the reality.

• Development of instrumented methods that can detect the decline in coating properties, coating

thickness and interface strength before it can be detected visually. These methods can provide an

early warning of future problems and allow an effective maintenance strategy to be introduced into

ballast tanks and others critical zones.

• Buckling and post-buckling analysis of a coated plate with delamination for different boundary

conditions.

• Numerical analysis of a coated plate with bigger dimensions, specially width, that in this work were

considered smaller due to computational time and computer capacity.

84

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91


Appendix A

FEA Flowchart

Start Python

Definition of the variable to be changed in the parametric analysis ( l, h, Ec, ν

c, σ

max, δ

n, δ

t )

and the respective values

Open the file .txt where the ANSYS APDL code is written

Write in the file .txt the value from the list to be used in the variable parameter

Open Ansys

Definition of the main dimensions and material constants

Creation of the coating and steel plate geometry

Attribution of the coating and steel material properties

Definition of the element type, element size and meshing of the coating layer and steel plate

Attribution of the CZM interface properties: σmax

, δn, δ

t

Selection of the interface element, element size and creation and meshing of the

interface between the coating and the steel surface

Generation of the surface-to-surface contact zone for modelling the initial

delamination, designating and defining the contact and target surfaces and

selecting the contact elements, the element KEYOPTs and real constants

Application of the boundary conditions, imposed displacement and destabilization load

Definition of the nonlinear solution options and the required load step

Solve the nonlinear problem

The analysis converge and the solution is achieved

Save the Post1 and Post26 output results

End

All simulations were performed?

Yes

No

Figure A.1: Flowchart of the developed FEA

93

Appendix B

Interface Parametric Analysis

B.1 Auxiliary Parametric Variation

As an aid to parametric analysis discussed in Section 5.2.3, it was decided to make a variation of the

parameter δn, keeping constant the value of σmax = 25 MPa and δn/δt = 1. This analysis is performed

in order to understand the influence of varying only the parameter δn or δt in the coating and interface

behaviour. The values considered for δn are the values presented in Table 5.8 and since that the δn/δt

= 1, the values assumed by δt are equal.

Figure B.1 shows the load-deflection and load-Von Mises stress curves of the coating layer as a function

of δn. The deflection shapes and maximum deflection values, are presented, respectively, in Figure B.2

and B.3. Table B.1 presents the normal and tangential separations and stresses for each studied case.

0

20

40

60

80

100

120

140

160

0 1 2 3 4

Axia

l L

oad

, P

[kN

]

Deflection, w [mm]

δn=1

δn=0.0224

δn=0.01


0

20

40

60

80

100

120

0 10 20 30 40 50

Axia

l L

oa

d,

P[k

N]


δn=1

δn=0.0224

δn=0.01

(b) Coating load-Von Mises Stress curves

Figure B.1: Coating load-deflection and load-Von Mises stress relations, considering σmax = 25 MPaand δn/δt = 1, as a function of δn

94

0

0.5

1

1.5

2

2.5

3

3.5

4

0 20 40 60 80 100

De

fle

ction

, w

[m

m]


δn=1δn=0.0224δn=0.01

Figure B.2: Coating deflection shapes, considering σmax = 25 MPa and δn/δt = 1, for each variation ofδn

3.808 3.597 3.592

0

1

2

3

4

5

δn=1 δn=0.0224 δn=0.01

Ma

xim

um

D

efle

ctio

n,

wm

ax

[mm

]

Figure B.3: Coating maximum deflection, considering σmax = 25 MPa and δn/δt = 1, for each variationof δn

Table B.1: Maximum normal separation (∆n), tangential separation (∆t), normal stress (Tn) and tangen-tial stress (Tn) values achieved on the interface, considering σmax = 25 MPa and δn/δt = 1

δn ∆n ∆t Tn Tt

[mm] [mm] [mm] [MPa] [MPa]

1 0.012 0.087 0.780 11.713

0.0224 0.002 0.006 6.278 34.642

0.01 0.001 0.004 8.458 41.388

From this auxiliary analysis, is concluded that an increasing in δn and δt causes also an increasing in the

values of ∆n and ∆t but, contrary causes a decreasing in Tn and Tt. As verified for the displacement of

the interface, an increasing in δn and δt values causes a increasing in the deflection of the coating layer

and its compressive strength is achieved more rapidly by the Von Mises stresses. In all the results for δn

= 0.0224 mm and δn = 0.01 mm, the difference is very small which is due to the fact that they are very

close values.

95

B.2 Coating Von Mises Stresses

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01

Figure B.4: Coating Von Mises stress distribution, considering σmax = 25 MPa, for each variation of δn/δt

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01

Figure B.5: Coating Von Mises stresses distribution, considering σmax = 15 MPa, for each variation ofδn/δt

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01

Figure B.6: Coating Von Mises stresses distribution, considering σmax = 10 MPa, for each variation ofδn/δt

96

B.3 Normal Interface Separation

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01

Figure B.7: Normal interface separation (∆n) distribution, considering σmax = 25 MPa, for each variationof δn/δt

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01


(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01


97

B.4 Tangential Interface Separation

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01

Figure B.10: Tangential interface separation (∆t) distribution, considering σmax = 25 MPa, for eachvariation of δn/δt

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01


(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01


98

B.5 Normal Interface Stresses

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01

Figure B.13: Normal interface stresses (Tn) distribution, considering σmax = 25 MPa, for each variationof δn/δt

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01


(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01


99

B.6 Tangential Interface Stresses

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01

Figure B.16: Tangential interface stresses (Tt) distribution, considering σmax = 25 MPa, for each variationof δn/δt

(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01


(a) δn/δt = 1 (b) δn/δt = 0.1

(c) δn/δt = 0.01


100

Appendix C

Scattered coating failures assessment

scale

Assessment of coating breakdown in ballast tanks of ships after construction is regulated by IACS and

Class Societies, based on the following diagram of Assessment Scale for Breakdown.

Figure C.1: Original scatter diagrams for corrosion and coating breakdown assessment (ABS, 2007)

101

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