Coating Breakdown Analysis of Steel Plates in … · Coating Breakdown Analysis of Steel Plates in...
Transcript of Coating Breakdown Analysis of Steel Plates in … · Coating Breakdown Analysis of Steel Plates in...
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
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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
5.6 Comparison of the buckling load, breakdown load and respective coating deflection as a
function of the coating thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.7 Comparison of the buckling load, breakdown load and respective coating deflection as a
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
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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
5.18 Compressive load-deflection relations of the coating layer as a function of the coating
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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5.22 Compressive load-deflection relations of the coating layer as a function of the coating
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
5.25 Compressive load-deflection relations and respective deflection shapes of the coating
layer, considering σmax = 15 MPa, as a function of δn/δt . . . . . . . . . . . . . . . . . . . 70
5.26 Compressive load-deflection relations and respective deflection shapes of the coating
layer, considering σmax = 10 MPa, as a function of δn/δt . . . . . . . . . . . . . . . . . . . 71
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
B.8 Normal interface separation (∆n) distribution, considering σmax = 15 MPa, for each varia-
tion of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
B.9 Normal interface separation (∆n) distribution, considering σmax = 10 MPa, for each varia-
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.11 Tangential interface separation (∆t) distribution, considering σmax = 15 MPa, for each
variation of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
B.12 Tangential interface separation (∆t) distribution, considering σmax = 10 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
B.14 Normal interface stresses (Tn) distribution, considering σmax = 15 MPa, for each variation
of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
B.15 Normal interface stresses (Tn) distribution, considering σmax = 10 MPa, for each variation
of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
B.16 Tangential interface stresses (Tt) distribution, considering σmax = 25 MPa, for each varia-
tion of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
B.17 Tangential interface stresses (Tt) distribution, considering σmax = 15 MPa, for each varia-
tion of δn/δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
B.18 Tangential interface stresses (Tt) distribution, considering σmax = 10 MPa, for each varia-
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
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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.
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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
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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
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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 ,
Analytical Eigenvalue Critical Buckling Load
Numerical Eigenvalue Critical Buckling Load
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
]
Analytical Eigenvalue Critical Buckling Load
Numerical Eigenvalue Critical Buckling Load
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
]
Length of the beam, L [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]
Von Mises Stresses [MPa]
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
Buckling Load Breakdown Load Breakdown Deflection
[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]
Length of the beam, L [mm]
0.7Ec,νc
0.85Ec,νc
Ec=3000Mpa and νc=0.37
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]
Von Mises Stresses [MPa]
0.7Ec,νc
0.85Ec,νc
Ec=3000Mpa and νc=0.37
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
Buckling Load Breakdown Load Breakdown Deflection
[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]
Length of the beam, L [mm]
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
(a) Coating load-deflection curves
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]
Length of the beam, L [mm]
δn/δt=1
δn/δt=0.1
δn/δt=0.01
(b) Coating deflection shapes
Figure 5.25: Compressive load-deflection relations and respective deflection shapes of the coating layer,considering σmax = 15 MPa, as a function of δn/δt
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
(a) Coating load-deflection curves
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]
Length of the beam, L [mm]
δn/δt=1
δn/δt=0.1
δn/δt=0.01
(b) Coating deflection shapes
Figure 5.26: Compressive load-deflection relations and respective deflection shapes of the coating layer,considering σmax = 10 MPa, as a function of δn/δt
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]
Von Mises Stresses [MPa]
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
]
Von Mises Stresses [MPa]
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]
Von Mises Stresses [MPa]
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]
Delamination diameter, 𝑙 [mm]
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]
Delamination diameter, 𝑙 [mm]
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]
Delamination diameter, 𝑙 [mm]
Von Mises Failure Curve
Poor Condition Curve
Fair Condition Curve
Good Condition Curve
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]
Von Mises Failure Curve
Poor Condition Curve
Fair Condition Curve
Good Condition Curve
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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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
(a) Coating load-deflection curves
0
20
40
60
80
100
120
0 10 20 30 40 50
Axia
l L
oa
d,
P[k
N]
Von Mises Stresses [MPa]
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]
Length of the beam, L [mm]
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
Figure B.8: Normal interface separation (∆n) distribution, considering σmax = 15 MPa, for each variationof δn/δt
(a) δn/δt = 1 (b) δn/δt = 0.1
(c) δn/δt = 0.01
Figure B.9: Normal interface separation (∆n) distribution, considering σmax = 10 MPa, for each variationof δn/δt
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
Figure B.11: Tangential interface separation (∆t) distribution, considering σmax = 15 MPa, for eachvariation of δn/δt
(a) δn/δt = 1 (b) δn/δt = 0.1
(c) δn/δt = 0.01
Figure B.12: Tangential interface separation (∆t) distribution, considering σmax = 10 MPa, for eachvariation of δn/δt
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
Figure B.14: Normal interface stresses (Tn) distribution, considering σmax = 15 MPa, for each variationof δn/δt
(a) δn/δt = 1 (b) δn/δt = 0.1
(c) δn/δt = 0.01
Figure B.15: Normal interface stresses (Tn) distribution, considering σmax = 10 MPa, for each variationof δn/δt
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
Figure B.17: Tangential interface stresses (Tt) distribution, considering σmax = 15 MPa, for each variationof δn/δt
(a) δn/δt = 1 (b) δn/δt = 0.1
(c) δn/δt = 0.01
Figure B.18: Tangential interface stresses (Tt) distribution, considering σmax = 10 MPa, for each variationof δn/δt
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