Xintong WangAnalysis of Iceberg-Structure Interaction During Impacts NTNU Norwegian University of Science and Technology Faculty of Engineering Department of Marine Technology
Master ’s thesis
Xintong Wang
Analysis of Iceberg-Structure Interaction During Impacts
Master’s thesis in Marine Technology
Supervisor: Prof. Jørgen Amdahl/Prof. Ekaterina Kim/Postdoc.
Zhaolong Yu June 2020
Xintong Wang
Analysis of Iceberg-Structure Interaction During Impacts
Master’s thesis in Marine Technology
Supervisor: Prof. Jørgen Amdahl/Prof. Ekaterina Kim/Postdoc.
Zhaolong Yu June 2020
Norwegian University of Science and Technology Faculty of Engineering
Department of Marine Technology
’Two roads diverged in a wood, and I—
I took the one less traveled by, And that has made all the difference.’
-Robert Frost
Summary
With the increase of the maritime production and transportation in the arctic area, more offshore structures and ships are designed and built for operation in the ice infested water.
The potential impact with the ice features is focused to maintain the structural integrity for design purpose. In this thesis, the interaction between the iceberg and the structure during the impact are analyzed to increase the understanding of the ice load on the structure and the response. The thesis mainly includes the literature review of related topics, the analysis of the design ice load in the rules, the modification of a numerical ice material model and the simulation of the slide impact between the iceberg and ship structure.
Through the review of the existed ice class rules and ice mechanics theories, the knowl- edge of the design ice load determination and ice material properties during the impact is developed to generate theoretical basis for analysis and discussion.
A new design ice load model is derived based on the first principles of the ice mechanic.
With the same impact scenario and the hull geometry, the new design ice load from the ice mechanics model is lower than the rules. Form the point view of the ice, such large loads on the structure as the rules suggest are too conservative, especially for the high polar classes suchP C1andP C2. The ship speed is considered continuously through the strain rate effect in the new model, while a discrete consideration of polar class dependent speed is taken in the rules due to lack of operating experience. The increasing trend of the design ice load with the ship displacement in the rules is not obvious in the ice mechanics model, which is believed also out of conservative consideration. Some topics are summarized into a prepared paper.
To apply the integrated analysis with both explicitly modelled objects, a consistent constitutive material model for the iceberg ice is modified based on a developed model.
A damage stage is introduced to improve the erosion procedure of failing elements. The stress degradation is implemented by damage variables, a damage evolution law and a final failure criterion, which are developed based on the continuum material mechanics.
With the damage stage, more plastic deformation is allowed before erosion for the ice material, so the non-contact space between the iceberg and the structure is eliminated.
The oscillation of the force during the application of the original material model can be reduced effectively.
The improper behavior of the original model during the unloading is improved by ad- justing Bulk Viscosity in the ABAQUS. The uniaxial loading, uniaxial loading-unloading and rigid wall crushing simulations are applied to verify the behavior of the modified material model. Acceptable results are generated and mesh-insensitivity is proved in the verification simulations. Through the calibration, the parameters in the material model are adjusted to generate the same crushing force curve as theP C1in the rules.
With the modified material model, the nonlinear simulations of the impact between the iceberg and a ship side structure are applied in the ABAQUS. Both the predefined path impact and the more realistic impact with the rigid body motion are simulated. Critical impact cases are applied in the simulations, resulting in substantial energy dissipation. The
influence to the interaction process from the impact parameters are obvious and revealed simply through the parametric study.
Based on an analytical impact model, the equivalent friction factor could be used to estimate the the tangential contact force from the normal contact force during the slide impact. From the simulation results, a value between0.5to1is suggested for the slide impact between the iceberg and stiffened panel used in the marine structure widely. The factor is found rather sensitive to the material properties, involved hydro-forces and im- pact parameters. For simplicity, it could be determined based on the lateral penetration because the factor partly depends on the deformation and damage. The more deformation or damage is generated in the structure and the iceberg, the larger is the equivalent friction factor. Thus, the influence of the other conditions could be reflected by a larger or smaller penetration distance in general.
Due to the time and personal capacity limit, there are still defects in the thesis. It is found that the derived new design ice load model is sensitive to the impact geometry, which is not discussed in detail. Several theoretical assumptions are made in the development of the damage stage for the ice material model, which will be improved if an analytical material model for the damage ice could be referred. Thus, the recommendations for the further work are addressed.
NTNU
Norges teknisk-naturvitenskapelige universitet Institutt for marin teknikk
MASTER THESIS 2020 For
Stud. Xintong Wang
Analysis of iceberg-structure interaction during impacts Analyse av interaksjon ved støt mellom isfjell og konstruksjoner
Background
With the increased activities in the artic area, operators and designers are challenged to design structures which can safely operate in iceberg infested waters. On the other hand, the aim is to optimize as much as possible the steel work reducing the new building cost without compromising on the overall safety. As part of this process, the modelling of iceberg interaction with floating structures has an important role. An industrial consensus on how the design iceberg load and response are determined for ice-structure interaction scenarios shall be determined and applied for design purposes.
According to NORSOK, an offshore installation is designed to sustain iceberg impact loads according to three different strategies. These are Ductility design, Strength design and Shared-energy design.
In the first case, all the available energy is dissipated by the deformation of the structure; the iceberg is modelled as rigid and do not contribute to the energy dissipation. This simplify the analysis as the analyst does not have to deal with the uncertainties related to the strength of the iceberg. However, the penetration model for the iceberg (i.e. its shape) needs to be assumed.
In a Strength design the structure is assumed rigid and can crush the ice, i.e. can resist the maximum pressure that the ice can deliver. The ability of structure to sustain iceberg load is evaluated using pressure-area (ISO 19906) model to describe the iceberg resistance, preferably with an associated penetration-area model.
In the third case, both the iceberg and the structure are explicitly modelled and contributing to the energy dissipation. In a simple manner, this can be achieved by combining the cases above, bearing in mind that severe limitations may appear, especially with the onset of large deformations.
Rigorously speaking, Shared-energy design approach is the most challenging as the iceberg deformation as well as the structural deformation of installation depend on each other. With the developing deformation, their stiffness will also change, thus changing the pressure that the ice can deliver to the structure and that the structure can sustain.
An integrated analysis with both objects explicitly modelled, is capable of accounting for all relevant effects for assessing the structural damages with the highest precision. This includes the progressive changes in the contact surface as well as the relative strength changes during the impact. The main challenge is however to have a consistent constitutive model for the iceberg with suitable material and crushing failure properties.
Objective
In 2015, DNV GL (Maritime Advisory) implemented in cooperation with the master student Nicole Ferrari (University of Genoa) a material model for glacial ice material response in the Abaqus FE software, the constitutive model was presented by Liu et al. (2010). However, due to time and
NTNU Fakultet for marin teknikk Norges teknisk-naturvitenskapelige universitet Institutt for marin teknikk
resources, the performance of the developed continuum model was not fully investigated. In a Master thesis by Vebjrøn Kjerstad 2019, the same material model as the one adopted by Liu et al. (2010) was implemented in Abaqus. The model worked reasonably well, but there is still a need to further validate the model.
The objective of this work is to further validate the material model in Abaqus and to increase the understating of ice mechanics and effect of different material data, shape, loading rate on the performance of the modelled iceberg. In order to investigate the effects on the iceberg-structure interaction for selected scenarios and iceberg geometries parametric studies should be carried out varying material properties and interaction speeds.
The following topics should be addressed:
1) Further work with the material model developed for Abaqus. This includes e.g.
implementation of the strain rate effect and the erosion procedure of failing elements. As per now, the element is eroded simultaneously when the limiting (critical) strain is reached, alternatively a stress degradation procedure based on “damage “criterion may be adopted.
2) Perform simulation of ice-impacts with the side shell of relevant ship(s). Potential candidates are a passenger vessel, an oil tanker or a gas-tanker. Existing FE models may be applied. The impact should be simulated with different procedures - a) predefined path - pushing first the ice laterally, followed by tangential motion b) more realistic
simulations, of the rigid body motion. The global shape and size of the iceberg may be modelled with a 6 DOF beam/mass model using constant added masses.
3) Compare the results from nonlinear simulations with simplified models of external- and internal mechanics. Can the tangential force be modelled with an equivalent friction factor, and can it be estimated from the resistance to lateral indentation?
4) To the extent possible parametric studies should be carried out in a systematic way. It is recommended that this could be conducted using scripting techniques.
5) A brief review of IACS unified rules for polar vessels and ISO 19906 requirements for design against ice shall be included. How are speed effects incorporated into load/resistance models?
6) Conduct analytical evaluations of pressure-area relationship and the corresponding plate thickness formulations adopted for design of Polar ships according to IACS PC-code. The starting point for the work is draft paper supplied by Ekaterina Kim. The focus should be on extending the model to include plate-thickness requirements and analysis of speed effects from an ice mechanics point of view. The aim of this work is to prepare a paper for a recognized international conference and/or technical journal.
7) Conclusions and recommendation for further work.
Literature studies of specific topics relevant to the thesis work may be included.
The work scope may prove to be larger than initially anticipated. Subject to approval from the supervisor, topics may be deleted from the list above or reduced in extent.
NTNU Fakultet for marin teknikk Norges teknisk-naturvitenskapelige universitet Institutt for marin teknikk
In the thesis the candidate shall present his personal contribution to the resolution of problems within the scope of the thesis work.
Theories and conclusions should be based on mathematical derivations and/or logic reasoning identifying the various steps in the deduction.
The candidate should utilize the existing possibilities for obtaining relevant literature.
The thesis should be organized in a rational manner to give a clear exposition of results, assessments, and conclusions. The text should be brief and to the point, with a clear language.
Telegraphic language should be avoided.
The thesis shall contain the following elements: A text defining the scope, preface, list of contents, summary, main body of thesis, conclusions with recommendations for further work, list of symbols and acronyms, references and (optional) appendices. All figures, tables and
equations shall be numerated.
The supervisor may require that the candidate, in an early stage of the work, presents a written plan for the completion of the work. The plan should include a budget for the use of computer and laboratory resources which will be charged to the department. Overruns shall be reported to the supervisor.
The original contribution of the candidate and material taken from other sources shall be clearly defined. Work from other sources shall be properly referenced using an acknowledged
referencing system.
The report shall be submitted in two copies:
- Signed by the candidate
- The text defining the scope included - In bound volume(s)
- Drawings and/or computer prints which cannot be bound should be organised in a separate folder.
Supervisors:
Prof. Jørgen Amdahl Prof. Ekaterina Kim Postdoc Zhaolong Yu
Deadline: June 10, 2020 Trondheim, January 15, 2020
Jørgen Amdahl
Preface
The presented report is the master thesis of Xintong Wang at the Norwegian University of Science and Technology (NTNU) in the spring of 2020. The work is continued based on the master project in the fall of 2019. It is the accomplishing part of the Master of Science Degree in Marine Structural Engineering at the Institute of Marine Technology, NTNU.
First of all, great thanks to my supervisor Prof. Jørgen Amdahl. He has been always available to provide guide and hint during the whole thesis work. The suggestions he gave from the methodology when I got trapped was so effective that I doubted he had predicted the whole things. I would also thank my co-supervisor Prof. Ekaterina Kim.
Her broad knowledge about ice physics and ABAQUS simulations helps me a lot to finish the thesis work. I would like to thank my co-supervisor Postdoc. Zhaolong Yu for sharing his experience of the simulation results and help to use the megacomputer. Also thanks to Vebjørn Kjerstad and Gabriele Notaro from DNV GL for their suggestions to the ice material modification.
The chosen topic is challengeable, but rather interesting. I developed a number of new knowledge during the work, especially about the continuum material mechanics. It is always exciting to step in a new area.
Readers of the thesis are preferably capable of the knowledge related to the structure mechanics and finite element analysis. It would be beneficial if knowledge of ice mate- rial mechanics is available. However, the necessary theory to comprehend the thesis is reviewed in the contents.
Xintong Wang Trondheim, June 9, 2020
Contents
Summary i
Scope iii
Preface vii
Table of Contents xi
List of Tables xiii
List of Figures xviii
Abbreviations xix
1 Introduction 1
2 Theory Review 3
2.1 Marine Structure Design Principles . . . 3
2.1.1 Limit State Criteria . . . 3
2.1.2 Relative Strength Design . . . 4
2.1.3 External and Internal Mechanics . . . 5
2.2 General Ice Physics . . . 7
2.2.1 Ice Actions and Resistance . . . 7
2.2.2 Iceberg Ice Properties . . . 9
2.3 Analytical Ice Material Model . . . 11
2.3.1 Yield Surface . . . 11
2.3.2 Return Mapping Algorithm . . . 13
2.3.3 Failure criterion . . . 14
2.3.4 Erosion Technique . . . 15
2.3.5 Flow Rule . . . 15
3 IACS Unified Requirements Review 17
3.1 General introduction . . . 17
3.1.1 Design Scenario . . . 17
3.1.2 Load Mechanics . . . 18
3.2 Criteria Division . . . 22
3.2.1 Polar Classes . . . 23
3.2.2 Hull Areas . . . 23
3.3 Load Determination . . . 24
3.3.1 General Bow Area Set . . . 24
3.3.2 Special Bow Area Set . . . 25
3.3.3 Other Hull Area Set . . . 25
3.3.4 Summary . . . 25
4 Comparison between the Ice Loads 27 4.1 New Ice load Model Derivation . . . 27
4.1.1 Ice Mechanics Model Introduction . . . 27
4.1.2 Global Ice Pressure Derivation . . . 28
4.1.3 Design Ice load Patch Determination . . . 31
4.2 Ice Load Comparison . . . 32
4.2.1 Global Ice Load comparison . . . 33
4.2.2 Local Ice Load comparison . . . 34
4.3 Summary and Discussion . . . 37
5 Numerical Modelling of the Ice Material 39 5.1 Original Subroutine Revaluation . . . 39
5.1.1 Introduction Study of the Original Subroutine . . . 39
5.1.2 Uniaxial loading Simulation of the Original Subroutine . . . 42
5.1.3 Weakness and Discussion . . . 43
5.2 Stress Degradation by the Damage Stage . . . 44
5.2.1 Plastic-Damage Model Introduction . . . 44
5.2.2 Numerical Damage Stage Realization . . . 46
5.2.3 Theoretical Summary and Discussion . . . 48
5.3 Modified Subroutine Verification . . . 49
5.3.1 Uniaxial Loading Simulation . . . 49
5.3.2 Uniaxial Loading-Unloading Simulation . . . 51
5.3.3 Mesh-insensitivity Verification . . . 61
5.3.4 Rigid Wall Crushing Simulation . . . 64
5.4 Modified Subroutine Calibration . . . 68
5.4.1 Setups of the calibration simulations . . . 68
5.4.2 Outline of the calibration procedure . . . 70
5.4.3 Results with the calibrated parameters . . . 70
6 Slide Impact Simulation 75
6.1 Simulation Setup in Abaqus . . . 75
6.1.1 Ship Side Model Description . . . 75
6.1.2 Iceberg Model Description . . . 79
6.1.3 Impact Scenarios Description . . . 84
6.1.4 Other Attributes Setup . . . 88
6.2 Result Representation and Analysis . . . 90
6.2.1 Predefined Path Impact . . . 90
6.2.2 Slide Impact . . . 99
6.3 Summary and Discussion . . . 107
6.3.1 Influence of Material Properties . . . 107
6.3.2 Influence of Hydro-forces . . . 108
6.3.3 Influence of Impact Parameters . . . 109
7 Conclusions and Recommendations 111 7.1 Conclusions . . . 111
7.2 Recommendations . . . 113
Bibliography 113 Appendix 119 A Predefined path impact 119 A.1 Deformation Process with Rigid Iceberg . . . 119
A.2 Contact Forces and Ratio with Deformed Iceberg . . . 120
B Slide impact 125
List of Tables
3.1 Polar classes division and description in IACS UR . . . 23
3.2 Summary of the sets to determine design ice load in IACS UR . . . 25
5.1 Ice material properties in the introductory study . . . 41
5.2 Illustration of the parameters setup in the uniaxial loading simulations . . 50
5.3 Illustration of the unloading setup in the uniaxial unloading-unloading simulations . . . 52
5.4 The initial parameters setup in the calibration simulations . . . 69
5.5 Properties of the ice material used in the calibration . . . 70
5.6 The final parameters setup after the calibration . . . 73
6.1 Characteristic material properties of the steel from DNV GL RP-C208 . . 78
6.2 The parameters setup after the re-calibration . . . 82
6.3 Summary of the global iceberg model in the ABAQUS simulation . . . . 84
6.4 The position of the initial contact points in the predefined path impact . . 85
6.5 Cases with the setup in the local impact scenario . . . 86
6.6 The initial position of the iceberg COG in the global slide impact . . . 87
6.7 Cases with the setup in the global impact scenario . . . 88
6.8 The boundary conditions of the ship side model in the ABAQUS . . . 88
6.9 The averaged equivalent friction factors with the applied impact angle in the slide compat simulations . . . 110
List of Figures
1.1 Illustration of the arctic transportation ship and the potential ice damage . 1 1.2 Illustration of thesis structure. Most topics addressed in the scope are
included in the three main parts in the middle. . . 2
2.1 Characterisation of design principles . . . 4
2.2 Illustration of different external mechanics . . . 6
2.3 Estimation of the force and deformation for ice-structure impacts adopted from ship collision . . . 6
2.4 Factors that can influence the ice-structure interaction scenarios . . . 7
2.5 Collision cases of a semi-submersible platform in the still water . . . 8
2.6 Illustration of the importance of the iceberg local geometry . . . 9
2.7 Illustration of potential locations for the structure-ice impact in waves in NORD ST19 . . . 9
2.8 Illustration of the strain rate dependence and hydrostatic pressure depen- dence of iceberg ice . . . 10
2.9 Illustration of "Tsai-Wu" yield surface inp−J2space . . . 12
2.10 Illustration of the stress return method corresponding to CPA . . . 13
2.11 Illustration of failure criterion of iceberg ice . . . 14
3.1 Illustration of design scenario in the IACS UR . . . 18
3.2 Definition of hull angles in the IACS UR . . . 18
3.3 Nominal contact geometry during oblique collision with an ice edge . . . 19
3.4 Ice load patch configuration in the IACS UR . . . 21
3.5 Illustration of the transformation from nominal to design rectangular load patches . . . 22
3.6 Illustration of the hull area division in the IACS UR . . . 24
4.1 Schematic illustration of the spatial pressure distribution of the ice me- chanics model . . . 28
4.2 Illustration of the combination of the impact geometry and new pressure- area relationship . . . 29
4.3 Global pressure-area curves from the ice mechanics model . . . 30 4.4 Illustration of the pressure determination for the new ice load patch . . . . 32 4.5 Comparison of the global pressure-area relationships between new ice load
model and IACS UR . . . 33 4.6 Comparison of the local pressure-area relationships for the General Bow
Area Set . . . 34 4.7 Comparison of the local pressure-area relationships for the Special Bow
Area Set . . . 35 4.8 Comparison of the local pressure-area relationships for the Other Hull
Area Set . . . 36 4.9 Comparison of the speed effect on the design pressure for non-bow areas . 37 4.10 Maximum local ice pressure with the speed of a icebreaker . . . 38 5.1 Illustration of the model and the mesh of the introductory study . . . 40 5.2 Comparison between different iceberg mesh sizes in the introduction study 41 5.3 Comparison between the ABAQUS and the LS-DYNA using the same
model in the introductory study . . . 42 5.4 Illustration of the one cube element loading simulation in the introductory
study . . . 42 5.5 Stress curves with the original subroutine in the uniaxial loading simulation 43 5.6 Illustration of the non-contact space during the impact with the original
subroutine . . . 44 5.7 Schematic illustration of the stress-strain curves with different theoretical
mechanics models . . . 45 5.8 Illustration of the damage process . . . 46 5.9 Illustration of the exponential damage evolution law . . . 47 5.10 Stress comparison between original and modified subroutine in loading test 48 5.11 Stress comparison with differentGf values . . . 50 5.12 Stress comparison with differentαvalues . . . 51 5.13 Illustration of unloading setup in the uniaxial unloading-unloading simu-
lations . . . 52 5.14 Improper material behavior in the uniaxial loading-unloading simulations
with default setup . . . 53 5.15 Strain increment in loading direction with different maximum step time
increment limitation . . . 54 5.16 Material behavior with manual maximum step time increment . . . 55 5.17 Cyclic loading behavior of a UMAT subroutine with the Drucker-Prager
yield surface . . . 56 5.18 Relationship betweenJ2andPhydrofor trial and real stress states . . . 57 5.19 Stress-strain relationship with increasing numerical damping . . . 58 5.20 Relationship betweenJ2andPhydrowith given linear bulk viscosity . . . 59 5.21 Stress-strain relationship comparison with given and default numerical
damping . . . 60 5.22 Unloading criterion change for the loading-unloading simulations . . . . 61 5.23 Illustration of the aborted linear damage evolution law . . . 62 5.24 Illustration of the mesh sensitivity of the modified subroutine . . . 63
5.25 Illustration of the central ice part . . . 64 5.26 Stress change of two neighbor elements during the crushing with the rigid
wall . . . 65 5.27 Deformation procedure of the ice hemisphere during the crushing with the
rigid wall . . . 66 5.28 Contact force curves in the ice crushing simulations . . . 66 5.29 Energy change in the ice crushing simulations . . . 67 5.30 The deformation procedure of the calibration simulation . . . 69 5.31 The contact force curves from the start cases of the calibration simulations 69 5.32 The contact force curves with changing0during the calibration . . . 71 5.33 The contact force curves with changingM during the calibration . . . 71 5.34 The contact force curves with changingN/Mduring the calibration . . . 72 5.35 The contact force curves after the calibration . . . 73 5.36 The stress-strain relationship after the calibration . . . 74 6.1 The arrangement and the modelling procedure of the ship side model . . . 76 6.2 The mesh results of the ship side structure . . . 77 6.3 Stress-strain relationship of the steel material from the DNV GL RP-C208 78 6.4 Fracture criterion plotted with respect to Fracture strain versus stress tri-
axiality . . . 79 6.5 Illustration of the iceberg model used in the impact simulation . . . 80 6.6 Illustration of the mesh of the iceberg model used in the impact simulation 81 6.7 The contact force curves of a rough re-calibration after the mesh size
changing . . . 81 6.8 Illustration of the mass-beam iceberg model used in the global impact sim-
ulation . . . 83 6.9 Illustration of the large global iceberg model with longer rigid beam . . . 84 6.10 The assembled models of the local impact scenario in the ABAQUS . . . 85 6.11 Illustration of the assumed slide impact scenario . . . 86 6.12 The assembled models of the global impact scenario in the ABAQUS . . 87 6.13 The applied boundary conditions of the impact simulation in the ABAQUS 89 6.14 Comparison of the contact forces in different impact positions of the local
impact scenario . . . 91 6.15 The stress state of the frames at the beginning of tangential move for dif-
ferent longitudinal impact locations . . . 92 6.16 Comparison of the ratio between the tangential and normal contact forces
with the rigid iceberg . . . 93 6.17 Comparison of the internal energy within the ship side model with the rigid
iceberg . . . 94 6.18 Structural deformation process of the impact positionD with the rigid
iceberg . . . 95 6.19 Contact forces and ratio of impact positionDwith different steel fracture
criteria . . . 96 6.20 Section of the structure and the iceberg at the end of the normal movement
with the characteristic steel fracture criterion . . . 97
6.21 Comparison of the ratio between the tangential and normal contact forces with the deformed iceberg . . . 97 6.22 Deformation of the structure and iceberg for for impact positionBatT =
1.13s . . . 98 6.23 Energy curves for the impact positionDwith the deformed iceberg . . . . 99 6.24 Deformation of the structure and iceberg after the slide impact with the
small iceberg model . . . 100 6.25 The interaction process of the impact simulations with the small iceberg
model . . . 101 6.26 Contact forces and equivalent friction factor curves with the small iceberg
model in the slide impact . . . 101 6.27 Internal energy curves with the small iceberg model for high and low im-
pact height . . . 102 6.28 The deformation states of the structure and the iceberg at the final step for
different impact height . . . 103 6.29 The contact forces and equivalent friction factor of curves with the large
iceberg model in the slide impact . . . 104 6.30 Illustration of the ’cut-in’ phenomenon in the slide impact simulations
with the large iceberg model . . . 105 6.31 The velocities and the rotation angles of the iceberg COG in the slide im-
pact simulations . . . 106 6.32 Energy curves with the large iceberg model for the high and low impact
height . . . 107 A.1 Structural deformation process of the impact positionAwith the rigid iceberg119 A.2 Structural deformation process of the impact position B with the rigid
iceberg . . . 120 A.3 Structural deformation process of the impact positionC with the rigid
iceberg . . . 120 A.4 Contact forces and ratio of impact positionAwith different steel fracture
criteria . . . 121 A.5 Contact forces and ratio of impact positionBwith different steel fracture
criteria . . . 122 A.6 Contact forces and ratio of impact positionCwith different steel fracture
criteria . . . 123 B.1 Bird view of the deformation pattern at the initial and final time step . . . 125 B.2 Deformation pattern of the structure and iceberg, contact forces and ratio
for impact angleα= 15◦ . . . 126 B.3 Deformation pattern of the structure and iceberg, contact forces and ratio
for impact angleα= 30◦ . . . 127 B.4 Deformation pattern of the structure and iceberg, contact forces and ratio
for impact angleα= 60◦ . . . 128 B.5 Deformation pattern of the structure and iceberg, contact forces and ratio
for impact angleα= 75◦ . . . 129
Abbreviations
AF = Hull Area Factor
AL = Abnormal Level
ALIE = Abnormal-Level Ice Event ALS = Accidental Limit State
AR = Aspect Ratio
COG = Center Of Gravity
CPA = Cutting Plane Algorithm CSA = Canadian Standards Association
DNV GL = Det Norske Veritas - Germanischer Lloyd
DOF = Degree Of Freedom
EL = Extreme Level
ELIE = Extreme-Level Ice Event
FEM = Finite Element Method
FLS = Fatigue Limit State
IACS = International Association of Classification Society IACS UR = IACS Unified Requirements for Polar ships IMO = International Maritime Organization
ISO = International Organization of Standardization
LIWL = Lower Ice Water Line
LNGC = Liquefied Natural Gas Carrier NLFEA = Nonlinear Finite Element Analysis NLFEM = Nonlinear Finite Element Method
PC = Polar Class
PPF = Peak Pressure Factor
PTIL NORD = Petroleum Safety Authority Norway SLS = Serviceable Limit State
UIWL = Upper Ice Water Line ULS = Ultimate Limit State
WMO = World Meteorological Organization
Chapter 1
Introduction
To give a general overview of the thesis, the background and the contents of the thesis are introduced simply in this chapter.
Due to the environmental change and the economical need, the maritime production and transportation in the arctic area increase. Especially for the oil and gas industry, more carriers are designed and built for the arctic area such as an arctic Liquefied Natural Gas Carrier (LNGC) shown in the Figure1.1(a). To operate such a structure safely in the ice infested water, its potential interaction with the ice features is taken into account from the beginning of the design. A typical damage case of the ship hull from an ice feature is shown in the Figure1.1(b). Besides, with the aim of optimizing the structure to reduce the cost while maintaining the integrity, the modelling of the iceberg-structure interaction process is concerned. Out of the engineering consideration, the design ice load on the structure and the response during the interaction shall be determined and applied in the design stage.
(a) The arctic LNGC ’CHRISTOPHE DE MARG- ERIE’ owned by SCF company (SCF, 2020)
(b) A damaged ship hull by the ice feature (Pekel, 2012)
Figure 1.1:Illustration of the arctic transportation ship and the potential ice damage Taken into account the complicated material properties and interaction process, the
Chapter 1. Introduction
integrated analysis is capable of accounting for all relevant effects for assessing the struc- tural damages with the highest precision. This includes the progressive changes in the contact surface as well as the relative strength changes during the impact. But, the main challenge is to have a suitable material model for the iceberg with proper material and crushing failure properties.
Focusing on the interaction between the iceberg and the ship structure, three main parts are illustrated and analyzed in the thesis. Most topics addressed in the thesis scope are included and discussed in the contents. The structure of the thesis is illustrated in the Figure1.2.
Figure 1.2: Illustration of thesis structure. Most topics addressed in the scope are included in the three main parts in the middle.
Firstly, the design ice load from the widely used rules are reviewed. A new design ice load model is derived based on the first principles of the ice mechanic. With the same impact scenario and the hull form, two design ice load models are compared qualitatively and quantitatively. The speed effects in the rules are analyzed and compared with the new model. The related conservative considerations in the rules are revealed and discussed.
Due to the limit of the time, not all rules mentioned in the scope are included.
Secondly, to apply the integrated analysis with both explicitly modelled objects, a consistent constitutive material model for the iceberg ice is further modified based on a previous developed model. As agreed by the supervisors, most time and efforts are devoted in this part to generate a reasonable numerical model with a proper erosion procedure of the failing elements, while the strain effect is not implemented due to time. Through adding a damage stage, the stress degradation of the ice is completed based on the constitutive material mechanics. The problem of a large force oscillation in the original model during the interaction process is reduced effectively. A number of simulations are applied to verify the modified numerical model. The involved parameters of the model are calibrated to result in the same ice behavior as the referred rules.
Thirdly, with the modified material model, the nonlinear simulations of the impact be- tween the iceberg and a LNGC ship side are applied in the ABAQUS. Both the predefined path impact and the more realistic impact with the rigid body motion are simulated. Based on an analytical impact model, the relationship between the tangential and normal con- tact forces is investigated. A referred value of the equivalent friction factor is suggested for the slide impact cases. The parametric study of the related factors are conducted and influences are summarized.
Chapter 2
Theory Review
To better understand the background of the thesis, some topics related to the interaction between the marine structure and the ice are addressed in this chapter. The basic design principles and methods of the marine structure are reviewed. The general ice physics are introduced simply. An analytical ice material model as the basis of the used numerical model is illustrated.
2.1 Marine Structure Design Principles
The main structure design principles and methods in limit states are described. The impor- tance of relative strength design is discussed. External mechanics and internal mechanics of the iceberg-structure interaction are described.
2.1.1 Limit State Criteria
The ships and offshore structures are designed to meet certain functions safely and eco- nomically. The term, limit state, is often referenced in the structure design check. When a certain limit state is used, it means a condition of the structure, beyond which it no longer fulfills the relevant design criterion and is considered as unsafe (Gulvanessian, 2002). Usu- ally, the limit states includes the Ultimate Limit State (ULS), the Serviceable Limit State (SLS), the Fatigue Limit State (FLS) and the Accidental Limit State (ALS):
• The ULS reflects the ultimate resistance for extreme loads, normally with a 100 year event.
• The SLS takes into account restrictions like deformations and motions during nor- mal use and occupancy.
• The FLS corresponds to the deduction of ultimate strength of the structure due to repetitive loads.
Chapter 2. Theory Review
• The ALS reflects a accidental collapse of the structure, normally with a 10000 year event.
For the ships and offshore structures operated in the ice-infested water, they are usually designed according to the ULS and checked according to the ALS.
In the ULS, mostly the linear elastic behaviour is assumed. Sometimes, for the struc- tures subjected to predominately permanent load, the plastic analysis as simplified rigid- plastic or elastic-plastic is also used. As mentioned above, the structure is designed to withstand loads with an annual exceedence probability of10−2 (with a return period of 100 years).
In the ALS, non-linear response of structure as buckling, yielding, etc, should be fo- cused. There are two conditions are considered. One is the ALS condition, in which the structure capacity to resist the accidental action effects is investigated and the structure may be damaged. These actions refers to an annual exceedance probability of10−4(with a return period of 10000 years). Another is the Post-ALS (or damaged) condition, in which the structure shall have resist progressive collapse with local damage under environmental actions. This environmental actions should correspond to the ULS criterion.
The interaction between iceberg and the structure could result in rather large pressure to penetrate or damage the structure so that the ALS criterion shall be used in this case.
Also, loads from rare iceberg impacts can be characterized as an Abnormal Level Ice Event according to ISO 19906 (ISO, 2019), which references to the ALS design criterion.
2.1.2 Relative Strength Design
Three design principles can be applied during the analysis of interaction between ice and structures. Depending on the relative strength of the structure and ice, strength design, ductile design and shared-energy design are included, which is also related to different design criteria as shown in the Figure2.1.
Figure 2.1:Characterisation of design principles depending on the distribution of energy dissipation in the ice mass and the structure. The ratio reflects the amount of the energy dissipated in the structure (Kim and Amdahl, 2013)
Strength design implies that the ice feature can be crushed and the structure is assumed
2.1 Marine Structure Design Principles to be rigid. The structure exhibits a resistance exceeding the failure strength of ice. Thus, most of the energy is absorbed by the ice and little plastic deformation is caused to the structure. This principle is identical to the conventional ULS design (NORSOK N-004), but will be conservative.
Ductile design implies that the ice feature is considered to be rigid and the structure shall have deformation. All the energy will be dissipated by the impacted structure as label ALS 1 shown in Figure2.1. In this principle, the shape of ice feature influences significantly because it is possible to choose a shape that doesn’t comply with the ALS criterion, but there is no widely accepted calculation model existing.
Shared-energy design implies that both the ice feature and the structure has consid- erable deformation. The energy dissipation relies on the detailed deformation process as label ALS 2 shown in Figure2.1. The challenge of this principle is that the mechanical properties of structure material and ice should be both modelled. The crushing or failure of the ice due to interaction with the structure will reshape the ice features locally or glob- ally. There are two kinds of analysis existing. One is the coupled analysis in which the deformation of the ice and structure at every moment shall influence each other. It is chal- lenging because of the dependence of the ice feature and structure on each other. Another is uncoupled analysis in which the pressure-deformation curve for the ice or structure is gotten assuming another one is rigid, respectively. An typical example using this method is the decoupling shared-energy analysis of an ice feature collision with a floating structure performed in NORD ST19 (Lu et al., 2018).
2.1.3 External and Internal Mechanics
In this thesis, the interaction between the iceberg and structure is mainly focused. Same as the analysis of the collision between ships in the ALS, the interaction could be split into two uncoupled processes, external mechanics and internal mechanics (Liu and Amdahl, 2010). The external mechanics solves the rigid body motions and determines the energy dissipated as strain energy. The internal mechanics deals with how the strain energy is dissipated in two collided objects.
The external mechanics for the iceberg-structure interaction has been developed for decades. A wildly used model is made by Popov with three-dimension to solve six de- grees of freedom (DOF) problem (Popov et al., 1969). With his model, the International Association of Classification Societies (IACS) proposed the IACS Unified Requirements for Polar Ships (IACS UR). Besides the specified model for ship-ice contact, external me- chanics adapted form ship-ship collision is also used. Liu proposed the impact mechanics that can be applied to both two-dimensional and three-dimensional problems (Liu and Amdahl, 2010). In his model, the equation of motion is solved in a local coordinate sys- tem determined by the hull shape at the collision point. Then a transformation matrix is proposed to derive the motion in global coordinate system. The dissipated energyEi is calculated as Equation2.1.
Ei= 1
2|( ¯mi4vi2)| (2.1)
,wherem¯iis an equivalent mass variable and4v2i is the change of the squared relative velocities.
Chapter 2. Theory Review
Two different external mechanics are also illustrated in Figure2.2 below. The model by Liu and Amdahl in Figure2.2(b) could better simulate the iceberg-structure interaction because the model by Popov in Figure2.2(b) limits the ice motion as level ice and neglects heave, pitch and roll motions.
(a) Impact of a ship against an ice floe (a: ship’s side; b:
ice floe; c:along A-A) (Popov et al., 1969)
(b) Collision scenarios and the main dimen- sions of the impact model (Liu and Amdahl, 2010)
Figure 2.2:Illustration of different external mechanics
As for the internal mechanics for the iceberg-structure interaction, it could be solved using the plasticity theory or nonlinear finite element analysis (NLFEA). It is also related to the relative strength design mentioned above. Thus, the structure deformation and the ice deformation could be solved separately with the rigid assumption of another one as a simplification. The coupling analysis can also be applied but good modelling of shape and material is needed.
Figure 2.3: Estimation of the force and deformation for ice-structure impacts adopted from ship collision (Ice curve to the left; Resistance curve to the right;Es,ice, Es,str: strain energy dissipation in ice and structure;ws,ice, ws,str: deformation of ice and structure;Fice: ice force;Rstr: structure resisdence;) (Amdahl, 2019)
2.2 General Ice Physics Above two processes are linked through the total dissipated strain energy. As shown in the Figure2.3, with force-deformation relationship of ice and structure from internal mechanics, the area under curves represents the energy dissipation, which should be equal to the value from the external mechanics. This simplified method is represented by the solid curve. When the "shaping effect" (The stronger one between two collision objects would change the shape of the weaker one) during the deformation is considered, the force-deformation curve is modified. For example, the structure after local deformation may "wrap around" the ice feature making it confined. A higher pressure is needed to crush the ice, which results in the dotted curve for the ice in the Figure2.3. In turn, if the ice is shaped blunter, the resistance would be larger for the structure as shown with dotted curve for the structure. This influence would repeat between ice and structure during the contact depending on the relative strength.
2.2 General Ice Physics
Ice is a special material. The focused aspects of the ice physics vary depending the inves- tigated problem. For the iceberg-structure interaction discussed in this thesis, modelling of the ice action and resistance is described. Main properties related to impact problem using NLFEM is also addressed in the section.
2.2.1 Ice Actions and Resistance
Ice actions are often described as different structure-ice interaction scenarios. The shape and size of the structure, the ice conditions and the environmental conditions can result in a number of different interaction scenarios, failure modes and resulting ice actions.
Factors influencing the scenarios are illustrated in the Figure2.4. Different combinations of factor will give different scenarios, which would largely change the problem definition and analysis. In this thesis, same scenario as NORD ST5 and NORD ST19 is chosen, which means a iceberg from glacial ice with relative velocity would collide with the structure.
Figure 2.4:Factors that can influence the ice-structure interaction scenarios (ArclSo, 2019)
Chapter 2. Theory Review
The ice resistance is defined as time averaged force during the iceberg-structure inter- action. It could be obtained by forcing the iceberg to impact with a rigid structure. Besides the ice properties mentioned later, the geometry and the relative velocity would influence this resistance mainly. These two factors need to be considered in the modelling of sim- ulation. The geometry includes global and local geometry. The former would influence the iceberg mass as well as the collision location of the structure. The latter is related to the deformation (crushing, extrusion, etc) of ice at the contact point. The relative velocity between the structure and iceberg would influence the kinetic energy when the collision happens, also the impact location.
The Figure2.5 illustrates the importance of the global geometry. The pontoon could only be hit by a large iceberg in this case. The collision between column and iceberg will be eccentric. It means there is a large arm for the force vector with respect to the center of gravity (COG) and considerable energy would be transferred to rigid motions like roll and pitch of the ice. This would result in less energy dissipation and thus less ice resistance.
For the smaller ice feature like spheroidal ice feature on the right, it is prone to larger wave induced velocity than the bigger one and the collision may be centric leading to an considerable energy dissipation. Also, it may be harder to be detected. When the global geometry is chosen, it should be representative to calculate the global response or impact energy during the impact.
Figure 2.5:Collision cases of a semi-submersible platform in the still water. The left one is with a large iceberg and the right one is with a small ice feature (Amdahl, 2019).
The Figure2.6 illustrates the importance of the local geometry. Different ice features are shown in the Figure2.6(a). A shape corner of an ice feature may penetrate the structure and cause more server damage compared to a blunt one. There is no conclusion of choos- ing local geometry, but if the ice could be crushed, a worse case may happen as shown in the Figure2.6(b). In this case, a small protrusion exists and may behave like a sharp corner to penetrate shell plating between two frames. But if it is crushed and the ice curvature is rather large, the contact would spread over several frames, which is worse than penetration (Amdahl, 2019). Thus, the local geometry of iceberg should be considered carefully.
2.2 General Ice Physics
(a) Imagined (with exaggeration) ice features with differ- ent local geometries in NORD ST19 (Lu et al., 2018)
(b) Illustration of local ice geometry after protrusion is crushed (Amdahl, 2019) Figure 2.6:Illustration of the importance of the iceberg local geometry
The existence of relative velocity is the reason why the collision happens. Both the structure and the iceberg motions in six DOF would be influenced by the wind, wave and current. With different relative velocity in different environmental conditions, the exposure and energy dissipation would vary. The Figure2.7 illustrates the importance of relative motion. The highest collision position is close to the wave crest and the lowest to the wave trough. If the relative motion changes, the possibility and exposure of hitting column or pontoon will change in this case.
Figure 2.7:Illustration of potential locations for the structure-ice impact in waves in NORD ST19 (Lu et al., 2018)
As an conclusion, when the ice action is set as impact scenario between the iceberg and structure, the ice resistance would be largely influenced by the geometry and relative velocity. Before starting the simulation, these two factors need to be chosen reasonably corresponding to the practice.
2.2.2 Iceberg Ice Properties
Unlike the sea ice, the iceberg ice belongs to the multi-year ice, which is composed of the freshwater ice falling from land-based glaciers. So, it can be reasonable considered as
Chapter 2. Theory Review
isotropic material (Sanderson, 1988). Some common properties are also listed in the Fig- ure2.4, such as the crystallography/grain size, temperate, salinity, porosity, etc. Normally, the ice becomes weaker and softer with the increasing grain size, temperature, salinity and porosity. When it is related to the impact problem, some special properties also need to be considered.
Size Dependence
The size dependence is also refereed as the size effect. It means the ice failure mode would depend on the size of ice. When the ice is in small-scale, a ductile behaviour is observed and a brittle behaviour in large-scale.
When the impact problem is considered, this property becomes important because not all ice in the whole area will fail at the same time. The ice in the high-pressure zone will have larger load and fail first, which makes the ice in the rest unbroken parts easier to fail. In practice, the size effect is taken into account by a pressure-area curve in the design codes to determine the ice strength.
Strain Rate Dependence
The strain rate is also called the loading rate. The ice would increase strength with the increasing strain rate until the brittle failure occurs after which the strength decreases. This trend is shown in the Figure2.8(a). With a strain rate of10−3, a transition from ductile to brittle exists (Schulson, 2001). The ductile material could have the plastic deformation and absorb considerable energy before it fail, while the brittle material will fail without the significant plastic deformation.
In iceberg-structure collisions, it is believed that the strain rate is greater than the tran- sition value (Liu et al., 2011). But, due to lack of experimental data, it is still uncertain the strength will decease indefinitely or there is a limit. Hence, a yield envelope representing a high strain rate is used for iceberg ice.
(a) Stress-strain curves of the ice for different strain rates (Schulson, 2001)
(b) Schematic illustration of the main process of spalling, extrusion and high-pressure zone forma- tion (Jordaan, 2001)
Figure 2.8:Illustration of the strain rate dependence and hydrostatic pressure dependence of iceberg ice
Hydrostatic Pressure Dependence
2.3 Analytical Ice Material Model The strength of ice has a nonlinear dependence on the hydrostatic pressure. For ex- ample, the uniaxial strength of freshwater ice strength in compression is 2-10 times larger than in tension. It is also suggested that hydrostatic pressure would result in a phase change (Kim, 2013).
During the impact between the iceberg and structure, large confinement and compres- sion appears within the contact area of ice. The existence of the high-pressure zone shown in the Figure2.8(b) makes ice features next to the edge easily spall and extrusion. There is also the phase change from the ice to water in these small areas. Thus, the resistance to deviatoric stresses is reduced in high hydrostatic pressure. To model this phenomenon, a shear-cap yield surface, such as "Tsai-Wu" yield surface, is normally used to simulate ice-impact behaviours.
Temperature Dependence
A comprehensive study on the temperature profile is performed by Loset. It is found that there exists a temperature gradient from the surface to the core region for the iceberg in sea because of the low thermal conductivity of ice. The submerged part of iceberg is sub- jected to a steep temperature gradient and the core temperature is reached approximately 3 meters from the surface (Loset, 1993). When the iceberg is modelled in the simula- tion tool, this gradient needs to be considered. The yield criterion should be modelled as temperature-dependent and the temperature gradient should be applied. For example, a linear interpolation of temperature gradient is used for elements between the surface and core area using the computer code LS-DYNA by Liu (Liu et al., 2011).
2.3 Analytical Ice Material Model
It is impossible to develop an universal model of the ice for every condition due to its var- ious existence in different condition. In this thesis, only the impact problem is considered.
As mentioned previous, several special properties need to be included in the modelling of iceberg ice. An elastic-perfect plastic model based on the data from triaxial experiments to describe the ice material behaviour during iceberg impacts is developed by Liu (Liu et al., 2011). In this section, the analytical model is introduced in detail with respect to the yield surface, return mapping algorithm, failure criterion, erosion technique and flow rule.
2.3.1 Yield Surface
The yield surface is formulated as an algebraic combination of the invariants of stress tensorΣij.
Due to the dependence on the hydrostatic pressure, some widely used yield-surface techniques such as Von Mises, Drucker-Pager and Mohr-Coulomb are not suitable. In the iceberg-structure impact, the ice particles in the center contact area would be confined by neighbouring particles, which means the ice is in a triaxial stress state. A suitable yield surface should be adopted according to triaxial experiment results.
An elliptical yield envelop for the iceberg is proposed by Derradji-Aouay as shown in Equation2.2.
Chapter 2. Theory Review
(τ−η
τmax)2+ (p−λ pc
2
) = 1. (2.2)
,where η,τmax andpc are constants, τ =
qSij:Sij
3 is the octahedral stress, Sij is the deviatoric stress andpis the hydrostatic pressure.
Mathematically, this yield surface is same as the "Tsai-Wu" yield surface on the con- ditionη= 0. For an isotropic material, it is usually written as:
f(p, q) =q−p
a0+a1p+a2p2 (2.3)
,where p = Σ3kk = I31 is the hydrostatic pressure,q = 32p
Sij:Sij is the Von Mises stress, anda1,a2anda3are the constant that requires fitting to triaxial experimental data.
In order to make the implementation convenient, the "Tsai-Wu" yield surface is written as:
f(p, J2) =J2−(a0+a1p+a2p2) = 0 (2.4) , whereJ2is the second invariant of deviatoric stress tensor.
This yield function is defined that there is no loading if it is a negative value. The value zero means the elastic limit and plasticity will occur when the function lies on this limit.
The positive value is inadmissible. During plasticity deformation, the yield surface should change to lay on the limit. The shapes of the yield surface inp−J2space adopted from different data sources are shown in the Figure2.9. It is clear that the difference exists due to different data sources and fitting methods. To reflect the temperature dependence, the a1,a2anda3are set as functions of the temperature in the Equation2.4.
Figure 2.9:Illustration of "Tsai-Wu" yield surface inp−J2space (Liu et al., 2011)
2.3 Analytical Ice Material Model
2.3.2 Return Mapping Algorithm
As mentioned above, when the plasticity happens, the stress should map back to the yield surface.
In the model, the Cutting Plane Algorithm (CPA) is used. First, the elastic equations are integrated with total strain increments to obtain an elastic predictor. Then, the elasti- cally predicted stresses are mapped to a suitably updated yield surface by correcting the plastic strain increments iteratively (Huang and Griffiths, 2009). The procedure is shown schematically in the Figure2.10.
Figure 2.10: Illustration of the stress return method corresponding to CPA (Huang and Griffiths, 2009).
The plastic correction phase is driven by a plastic multiplier, 4λn+1. Thus, during the plastic-predictor stage, the plastic strain remains fixed, and during the plastic-corrector stage, the total strain is fixed. Enforcing an normality rule at the beginning of the step, we have:
{4σ(k)}={σ(k+1)} − {σ(k)}=−[De]{4σp(k)} (2.5)
{4εp(k)}=4λ(k){a(k)} (2.6)
, where k is the iteration number and[De]is the constitutive matrix.
At every iteration, linearizing the yield function f around the current stress values {σ(k)}, we have:
fk+1=fk+{a(k)}T({σ(k+1)} − {σ(k)}) (2.7) By settingfk+1= 0, we have:
Chapter 2. Theory Review
4λ(k)= fk
{a(k)}T[De]{a(k)} (2.8)
2.3.3 Failure criterion
Depending on the stress state and hydrostatic pressure, different failure modes of the ice may occur.
In the iceberg impact problem, the ice is well confined and the hydrostatic pressure is an important influencing factor. It is shown taht the hydrostatic pressure and friction may trigger different failure mechanisms, namely frictional or Coulombic faults and non- frictional or plastic faults, which are shown in the Figure2.11(a) (Schulson, 2009). At a lower confinement, the friction sliding of the ice is not suppressed and the shear force results in the Coulombic failure. At a higher confinement, the friction sliding is restricted and the plastic failure occurs.
(a) Schematic sketched of two kinds of com- pressive shear fault (Coulombic faults form under lower degree of confinement and plas- tic faults form under higher degrees of triaxial confinement) (Schulson, 2009)
(b) Illustration of the U-shape strain based failure criterion curve proposed by Liu (Liu et al., 2011)
Figure 2.11:Illustration of failure criterion of iceberg ice
When it is related to iceberg impact, the Coulombic appears at the beginning of contact.
For those elements in this state, reaching a certain level of shear force would trigger its deletion from analysis using erosion technique. This is easy to satisfy because the pressure is increasing during Coulombic faults. As the interaction continues, the appearance of plastic failure makes the failure criterion hard to reach because the increasing hydrostatic pressure results in a much stiffer ice.
In Liu’s model, the usage of the "Tsai-Wu" yield surface and the CPA makes it possible to introduce more advanced failure criteria through the strain rate. Based on the hypothesis that the stiffness of ice will not change significantly in the loading process until later when the plastic limit load is approached, a U-shaped strain-based failure criterion as shown in
2.3 Analytical Ice Material Model the Figure2.11(b) is proposed (Liu et al., 2011). This failure criterion to simulate the ice- fracture mechanics is empirical and based on the effective plastic strain and hydrostatic pressure as shown below:
εpeq= r2
3εpij :εpij (2.9)
εf =ε0+ (p
p2 −0.5)2 (2.10)
,whereεpeq is the equivalent plastic strain,εpij is the plastic strain tensor,εf is the failure stain,ε0is the initial failure strain, which should be determined by fitting experimental data andp2is the larger root of the yield function. The erosion is activated if theεpeq > εf or the pressure is not greater than the cut-off pressurepcut. The cut-off pressure is described as the strength difference between the tensile and compression states.
2.3.4 Erosion Technique
The erosion technique means that the elements violating the failure criterion will be deleted from the analysis.
In this implemented model, when the erosion occurs, the deviatoric stresses on the element are set to zero, but not the hydrostatic pressure. This can be viewed as a rapid softening process due to the fact that the removal of an element is equivalent to setting stress level zero (Kjerstad, 2019).
In practice, there are two problems existing related to the "erosion". One is that it does not simulate any additional loads caused by the extrusion of the crushed ice, which may play an important role during the crushing. Another is that this technique largely depends on the meshing of the ice , thus it is necessary to check the convergence property.
2.3.5 Flow Rule
As mentioned in the part of the yield surface, the associated flow rule, a nominal flow rule, is used. In this case, the plastic strain increment is normal to the yield surface. It means that the yield surface, as the plastic potential, could used to derive the plastic strains. The plastic strain increment,dεp, is determined by:
dεp=dλ∂f
∂δ =dλ4f (2.11)
,where f is the yield function anddλ is the plastic multiplier, which means the plastic strain magnitude is normal to the yield surface.
Chapter 2. Theory Review
Chapter 3
IACS Unified Requirements Review
In this chapter, as a design reference of the ships constructed of steel and intended for inde- pendent navigation in ice-infested polar waters, the IACS UR is reviewed. The mechanics of the ice load determination is introduced in detail and the same algorithm is used in the derivation of the ice mechanics model later.
3.1 General introduction
The IACS is a technically based non-governmental organization that currently consists of twelve member marine classification societies (IACS, 2020). To address the maritime safety in the Arctic operation, the IACS UR establishes minimum technical standards and requirements. The rules are developed and modified based on principles introduced in this part.
3.1.1 Design Scenario
The ice load on the ship plate and framing from the IACS UR is based on a design scenario, a glancing collision on the shoulders of the bow as shown in the Figure3.1(a). The ship moves forward with design speed and penetrates the ice edge, such as the edge of a channel or a floe. During the impact, the normal kinematic energy from the ship is absorbed by the crushing ice so that the maximum crushing force can be found by the energy equilibrium.
In the interaction process, the ice at the contact area is crushed, while the flexural failure would also happen as shown in the Figure3.1(b). Thus, the bending failure of the ice is included and the maximum crushing force can’t exceed the failure force required to bend the ice.
Chapter 3. IACS Unified Requirements Review
(a) Glancing collision on the bow area (b) Crushing contact and flexural failure dur- ing the glancing collision
Figure 3.1:Illustration of design scenario in the IACS UR (Daley, 2000)
This impact scenario is strictly valid only for the bow area or stern area of the double- acting ships. To balance the structure design work, the ice load on the other hull area is scaled by a factor as the proportion of the bow area. Before being applied elsewhere, the ice load from the bow area is normalized by a set of ’standard’ bow angles because the loads on other hull areas are not strongly dependent on the bow shape (Daley, 2000).
The definition of hull angles is illustrated in the Figure3.2. Normally, the hull angles are measured at the upper ice waterline. The influence of the hull angles is captured through the calculation of a bow shape coefficient in the application.
Figure 3.2:Definition of hull angles in the IACS UR (Daley, 2000)
3.1.2 Load Mechanics
The Ice loads in the IACS UR are derived based on a ’Popov’ type of collision. A wedge shaped ice edge and a pressure-area ice indentation relationship are used to modify it.
3.1 General introduction Firstly, the total ice load for the design condition is found as the minimum between the crushing and flexural limit loads. Then, a patch to distribute this load is determined. At last, the distribution of load within the patch is modified due to the potential local loading peak. The results are supported by numerical models, model tests, ship trials and opera- tional experience (Daley, 2000).
The normal kinetic energy (KEn) combining the normal velocity (Vn) with the effec- tive mass (Me) at the collision point is expressed as:
KEn= 1
2MeVn2 (3.1)
The crushing energy (Ecrushing) is integrated from the normal force (Fn) over the penetration depth (δ) as:
Ecrushing = Z δ
0
Fn(δ)dδ (3.2)
Figure 3.3:Nominal contact geometry during oblique collision with an ice edge (Daley, 2000) During the interaction between the ship hull and ice, the nominal area is found for a penetrationδas shown in the Figure3.3. Both the width (W) and height (H) of the nominal contact area can be determined by the normal penetration depth (δ) along with the normal frame angle (β0) and the ice edge opening angle (φ) as:
W = 2δtan(φ/2)/cos(β0) (3.3)
H =δ/(sin(β0)cos(β0)) (3.4) Thus, the triangle normal contact area (A) is calculated as:
A=W/2·H
=σ2tan(φ/2)/(cos2(β0)sin(β0)) (3.5) The averaged pressure (P) is found from the pressure-area relationship as:
P =P0Aex (3.6)
, where theP0is the class-dependent crushing pressure of the ice and theexis the pressure- area exponent.
