FCG Final MSC Thesis 02

Prediction of Fatigue Crack Propagation in
Orthotropic Steel Decks using XFEM based on

LEFM and VCCT
Ravi Shankar Gupta

4751310
Cover picture: Suurhoff Bridge ©. https://www.ad.nl/voorne-putten/suurhoffbrug-raakt-aan-einde-van-zijn-latijn-brug-
ernaast~a864ea600/64387466/
Prediction of fatigue crack propagation
in orthotropic steel decks using XFEM
based on LEFM and VCCT
by
Ravi Shankar Gupta

in partial fulfilment of the requirements for the degree of
Master of Science
at the Delft University of Technology,
to be defended publicly on Friday July 19, 2019 at 11:00 AM.
Student number: 4751310

Supervisor: Prof. dr. M. Veljkovic, TU Delft
Thesis committee: MsEng. O. Joostensz, Rijkswaterstaat
Dr. Ir. F.P. van der Meer, TU Delft
Dr. H. Xin TU Delft
Ir. L.J.M Houben TU Delft
An electronic version of this thesis is available at http://repository.tudelft.nl/.

Acknowledgement
This report has been written as a final work in partial fulfilment for the degree of Master of Science at
the Delft University of Technology, Delft, Netherlands. It was a great experience for learning and
professional development. I am grateful for having a chance to work and meet with wonderful
professionals who led me through this task.
Firstly, I would like to express my gratitude to Prof. dr. M. Veljkovic, Chair of the thesis committee, who
despite being extraordinarily busy with his duties, took out the time to guide me on the right path
throughout this project. His critical reviews have enormously assisted this research.
Secondly, it is radiant sentiment to place on record my best regards to Dr. Haohui Xin, my daily
supervisor for his guidance and approach which was valuable for my research.
Thirdly, my deepest sense of gratitude to MsEng. Ostar Joostensz, for his constant professional support
in providing me valuable feedback that greatly assisted the research and allowed me to execute a part
of the research at Rijkswaterstaat.
Fourthly, I would like to thank Dr. Ir. F.P. van der Meer, who was always open to my questions and
provided me important advice on technical aspects of this research.
A great thanks to all my friends for their constant encouragement and motivation and for making my
masters journey a memorable one. Finally, my special gratitude to my father Mr. Surendra Prasad, my
mother Mrs. Manorama Devi and my two lovely sisters Mrs. Shashi and Mrs. Jyoti for their unconditional
love and support throughout my career.
Ravi Shankar Gupta

Delft, July 2019
Summary
Orthotropic Steel Decks (OSDs) are widely used in various types of steel bridges due to their benefits
of light weight, high load bearing capacity and speedy construction. Although many improvements in
aspects of design, fabrication, inspection, and maintenance have been achieved over the years for such
bridge decks, fatigue remains a predominant problem, mostly because of the complexity of prediction
methods. Many researchers have tried to investigate this component through experiments. However,
performing only experiments may not lead to a cost-effective solution. Therefore, it is necessary to
combine the experimental data with the numerical approaches.
Particularly Linear Elastic Fracture Mechanics (LEFM) allows to model and analyse the crack propagation
until subsequent failure, and significantly reduces the requirement of experiment. ABAQUS® provides
an enriched feature, commonly referred to as the Extended Finite Element Method (XFEM) which
incorporates two enrichment function namely the discontinuity function which represents the gap
between the crack surface and asymptotic function which captures the singularity and thus can be used
to model discontinuity independent to the finite element mesh. To evaluate the modelling efficiency
and validate the simulation methodology, two XFEM-model based on LEFM and Virtual Crack Closure
Technique (VCCT) are developed and the simulated results are compared with the experimental data.
The first phase of the thesis deals with the numerical simulation to investigate the crack propagation
rate in Compact-Tension (CT) specimen for different stress ratios. The results of two-dimensional (2D)
model are found to be in good agreement (within 1.48%) with the fatigue coupon test results. As most
of the work concentrates on 2D shell model, the extension to three-dimensional (3D) solid requires the
investigation of related parameters to consider through-thickness effects. Nevertheless, the mechanism
of 3D model is studied, and the simulated results match with the 2D results for fatigue crack growth
(a, N). Moreover, a reliable technique of computing Stress Intensity Factor (SIF) is obtained by
comparing with the ISO 12108 standard formulation. However, when the SIF and fatigue crack growth
are combined, the crack propagation rate in 3D is overestimated (about 26%) when compared to the
experimental data possibly because of the imperfection in the application of boundary conditions.
The second phase deals with the numerical simulation in welded connection of OSD to determine the
Paris law constants (C, m) by correlating the numerical result of fatigue crack growth with the beach
mark measurements obtained in the fatigue experiments. Prior to automated XFEM simulation, a set of
finite element analyses are performed to determine the vertical deformation, longitudinal stain
distribution and hotspot stresses to validate the numerical model as per the test setup. The results of
numerical analyses showed a good correlation (within 18%) with test data and Paris law constant C is
predicted to be lower than the recommended value by IIW standard.
The validated methodology is then applied on large scale to an existing bridge (Suurhoff bridge)
structure which was built in 1971. In this case study, a crack length of 230 mm was detected in the
deck plate originating from the root of the stiffener-to-deck plate welded connection between the cross-
beams using TOFD measurements. To verify the problem, a numerical model is developed based on
the dimension of the bridge to evaluate the crack initiation period and the crack propagation period.
The crack initiation period is predicted using hotspot stress method and the crack propagation period
is evaluated using automated XFEM simulation. Overall, the total fatigue load cycles are predicted to
be 7.86 million which is equivalent to 48 years. A similar crack length was however detected after a
service life of 44 years. This overestimation can be possibly explained as the model did not take residual
stresses and other welding defects into account. The numerical model showed a good correlation with
the real scenario and is therefore used to predict the permissible limit of deck plate crack length of 500
mm. The model predicted 8.02 million load cycles for a crack length of 500mm, which is equivalent to
34 years after the crack initiation period. Nevertheless, the fracture mechanics approach showed
improvements in the assessment of fatigue life.
List of Abbreviations
2D Two-dimensional
3D Three-dimensional
ASTM American Society for Testing and Materials
CCRB Circumferentially Cracked Round Bars
CT Compact-Tension
FE Finite Element
FEM Finite Element Method
HAZ Heat Affected Zone
IIW International Institute of Welding
ISO International Organization for Standardization
LEFM Linear Elastic Fracture Mechanics
OSD Orthotropic Steel Deck
PHILSM Signed function to describe the crack surface
PSILSM Signed function to describe the initial crack front
RP Reference Point (ABAQUS®)
RWS Rijkswaterstaat
SEM Scanning electron microscope
SIF Stress Intensity Factor
STATUSXFEM Status of the enriched element
TOFD Time-of-flight-diffraction
VCCT Virtual Crack Closure Technique
XFEM eXtended Finite Element Model
List of Symbols
2c Crack length
a Crack size
C, m Material dependent parameters of the Paris Law
C1, C2, C3, C4 Material constants based on fracture energy release rate
E Elastic modulus
Gpl Energy release rate upper limit
Gthres Energy release rate threshold
N Number of load cycles
R Stress ratio
ΔF Applied load range
ΔK Stress Intensity Factor range
ΔKeff Effective Stress Intensity Factor range
υ Poisson’s ratio
Contents
1 Introduction .......................................................................................................................... 1
1.1 Background information .................................................................................................. 3
1.1.1 Motivation .............................................................................................................. 3
1.1.2 Methodology ........................................................................................................... 3
1.2 Research Objective ......................................................................................................... 4
1.3 Thesis Structure ............................................................................................................. 5
2 Literature Overview ............................................................................................................... 7
2.1 Linear Elastic Fracture Mechanics (LEFM) ......................................................................... 9
2.1.1 Crack characterization ............................................................................................. 9
2.1.2 Stress intensity factor ............................................................................................ 11
2.2 Fatigue of welded connections ...................................................................................... 13
2.3 XFEM (eXtended Finite Element Method) ....................................................................... 16
3 Compact-Tension Specimen ................................................................................................. 21
3.1 General........................................................................................................................ 23
3.2 XFEM model ................................................................................................................. 24
3.2.1 Geometry ............................................................................................................. 24
3.2.2 Boundary conditions .............................................................................................. 25
3.2.3 Material property .................................................................................................. 26
3.2.4 Mesh quality ......................................................................................................... 28
3.3 Output ......................................................................................................................... 29
3.3.1 2D-XFEM model .................................................................................................... 29
3.3.2 3D-XFEM model .................................................................................................... 30
3.4 Result and Discussion ................................................................................................... 31
3.4.1 2D shell XFEM-model ............................................................................................ 31
3.4.2 Effect of stress ratio .............................................................................................. 32
3.4.3 3D solid XFEM-model ............................................................................................ 34
3.4.4 Stress Intensity Factor........................................................................................... 38
3.4.5 Effect of LEFM parameters (2D-XFEM) .................................................................... 40
4 Orthotropic Steel Deck Specimen .......................................................................................... 45
4.1 General........................................................................................................................ 47
4.2 Numerical simulation of fatigue- FEM ............................................................................. 48
4.2.1 Experimental setup [25] ........................................................................................ 48
4.2.2 Development of FE model ...................................................................................... 50
4.3 Output ......................................................................................................................... 52
4.3.1 Results and Discussion .......................................................................................... 53
4.4 Numerical simulation of crack propagation- XFEM........................................................... 56
4.4.1 Fatigue Test [25] .................................................................................................. 56
4.4.2 Development of XFEM model ................................................................................. 56
4.4.3 Output ................................................................................................................. 59
4.4.4 Results and Discussions ......................................................................................... 60
5 Fatigue life Assessment: Suurhoff Bridge............................................................................... 61
5.1 General........................................................................................................................ 63
5.1.1 Motivation ............................................................................................................ 63
5.1.2 Problem description .............................................................................................. 63
5.1.3 Suurhoff bridge description .................................................................................... 64
5.2 Outline: Fatigue life Assessment .................................................................................... 67
5.3 Literature Data ............................................................................................................. 68
5.3.1 Material Parameters .............................................................................................. 68
5.3.2 Loading Parameters .............................................................................................. 70
5.4 Numerical model .......................................................................................................... 73
5.4.1 Material properties ................................................................................................ 74
5.4.2 Loading conditions ................................................................................................ 74
5.4.3 Boundary conditions .............................................................................................. 75
5.4.4 Mesh .................................................................................................................... 76
5.4.5 Output: ................................................................................................................ 77
5.5 Fatigue crack initiation period ....................................................................................... 79
5.5.1 Fatigue detail category .......................................................................................... 79
5.5.2 Hot spot stress method ......................................................................................... 80
5.5.3 Fatigue Life prediction ........................................................................................... 83
5.6 Fatigue crack propagation period .................................................................................. 84
5.6.1 Development of XFEM model ................................................................................. 84
5.6.2 LEFM implementation ............................................................................................ 85
5.6.3 Time-of-flight-diffraction (TOFD) measurement....................................................... 86
5.6.4 Stationary model ................................................................................................... 86
5.6.5 Propagating model ................................................................................................ 87
5.6.6 Output ................................................................................................................. 88
5.6.7 Result and Discussion ............................................................................................ 89
5.7 Combined fatigue assessment ....................................................................................... 93
6 Conclusions and Recommendations ...................................................................................... 95
6.1 Conclusions .................................................................................................................. 97
6.2 Recommendations for future studies ........................................................................... 100
6.2.1 CT-Specimen ...................................................................................................... 100
6.2.2 OSD- Specimen ................................................................................................... 100
6.2.3 Suurhoff Bridge................................................................................................... 100
7 References ........................................................................................................................ 103
8 Appendix ........................................................................................................................... 105
A. Paris law formulation ...................................................................................................... 105
B. Beach mark measurement .............................................................................................. 106
C. Traffic distribution .......................................................................................................... 107
D. Fatigue detail category ................................................................................................... 110
E. TOFD result ................................................................................................................... 111
List of Figures
Figure 2.1 Different crack types [3] ................................................................................................ 9
Figure 2.2 Crack tip region- 2D [4] ................................................................................................. 9
Figure 2.3 Crack tip region-3D [4] .................................................................................................. 9
Figure 2.4 Crack closure effect at the material surface on crack front shapes [8]: (a) Through crack-
Curved crack front (b) Deviations of semi-elliptical crack front at material surface.......................... 10
Figure 2.5 Fracture modes a) Mode I b) Mode II c) Mode III [9].................................................... 10
Figure 2.6 Three regime of the crack propagation rate as a function of 𝛥𝐾 [8] ............................... 12
Figure 2.7 Length scales of the life cycle of a component subjected to cyclic loading [12] ............... 13
Figure 2.8 Illustration of the deformation and cracks of the bridge deck [13]: (a) Representative
loading scenarios and the corresponding deformations (b) Typical fatigue crack patterns ................ 14
Figure 2.9 (a) Representation of normal and tangential coordinates for a smooth crack. [4] (b)
Representation of enriched nodes and enrichment radius in an arbitrary 2D mesh .......................... 17
Figure 2.10 Illustration of a non-planar crack in the 3D by two signed distance function ∅ and 𝜔 [4] 17
Figure 2.11 Fatigue crack growth [4] ........................................................................................... 18
Figure 3.1 Geometry of CT-Specimen [22] .................................................................................... 24
Figure 3.2 (a) Boundary condition of CT-Specimen (b) XFEM-model of CT-Specimen ....................... 25
Figure 3.3 Illustration of reference -point coupled to a (a) Shell edge in 2D (b) Solid surface in 3D .. 25
Figure 3.4 Graphical representation of different stress ratios.......................................................... 26
Figure 3.5 Mesh quality (a) Two-dimensional XFEM-model (b) Three-dimensional XFEM-model ........ 28
Figure 3.6 XFEM output (i) STATUSXFEM (ii) PHILSM variable output (iii) PSILSM variable output .... 29
Figure 3.7 Representation of different crack propagation stages at (a) N= 1.23x10 4 load cycles (b) N=
13.16x104 load cycles (c) N= 24.42x104 load cycles ...................................................................... 30
Figure 3.8 (a) 3D-Model STATUSXFEM output indicating the crack shape (b) Representation of crack
propagation path and change in mechanism (c) Regular crack propagation mechanism (d) Irregular
crack propagation mechanism...................................................................................................... 30
Figure 3.9 Fatigue crack propagation rates obtained from the numerical simulation (2D-XFEM)
compared with the test results ..................................................................................................... 31
Figure 3.10 Fatigue crack growth for different stress ratios (b) Crack propagation rate (Similarity
principle) (c) Effective stress intensity factor (Elber's crack closure mechanism) .............................. 32
Figure 3.11 Non-uniform stress distribution over the thickness at holes .......................................... 34
Figure 3.12 Comparison of fatigue crack growth between 2D-XFEM model and 3D-XFEM model ...... 35
Figure 3.13 Stress Intensity factor distribution along the crack front in through-thickness direction
compared to ISO 12108 standard for mesh size and crack propagation mechanism (STATUSXFEM
output) (a) 0.6 mm (b) 0.3 mm ................................................................................................... 36
Figure 3.14 Fatigue crack propagation rates obtained from the numerical simulation (3D-XFEM)
compared with the test results ..................................................................................................... 37
Figure 3.15 2D-shell FE model for contour integral calculation ....................................................... 38
Figure 3.16 Stress intensity values for various crack size using different approaches ....................... 39
Figure 3.17 Fatigue crack propagation rates obtained from the numerical simulation for different
mesh sizes .................................................................................................................................. 40
Figure 3.18 Fatigue crack propagation rates obtained from the numerical simulation for different Gpl
ratios ......................................................................................................................................... 41
Figure 3.19 Fatigue crack propagation rates obtained from the numerical simulation for different (a)
GIc (b) am ................................................................................................................................... 42
Figure 4.1 (a) Dimension of the OSD specimen [25] (b) Strain gauge location at the stiffener-to-deck
plate connection [25] .................................................................................................................. 48
Figure 4.2 Detail of the loading and support conditions [25] .......................................................... 49
Figure 4.3 FE model: Interactions and boundary conditions ........................................................... 50
Figure 4.4 XFEM-model: Mesh quality ........................................................................................... 51
Figure 4.5 Deformation (a) Magnitude U (b) Vertical U2 ................................................................ 52
Figure 4.6 Strain Distribution (a) Max. Principal strain distribution (b) Strain distribution in x-direction
(c) Strain distribution in y-direction .............................................................................................. 52
Figure 4.7 (a) Von-Mises stress distribution (b) Stress distribution along x-direction (c) Stress
distribution along y-direction ....................................................................................................... 52
Figure 4.8 Comparison of the vertical deformation of FEM with test results ..................................... 53
Figure 4.9 Comparison of longitudinal strain distribution between FEM and test results (a) Deck plate
(b) Stiffener ................................................................................................................................ 54
Figure 4.10 Comparison of hot-spot stress derived from FEM with the test results at weld toe (a) Deck
plate (b) Stiffener ....................................................................................................................... 55
Figure 4.11 Longitudinal crack geometry during crack propagation for test specimen [25] ............... 56
Figure 4.12 XFEM model: Location and definition of the initial elliptical shape ................................. 57
Figure 4.13 (a) Stages of crack propagation displayed as STATUSXFEM output (b) XFEM crack
simulation including the initial semi-elliptical crack (c) Side view corresponding with the weld toe
crack simulation (d) Crack front dimension ................................................................................... 59
Figure 4.14 Fatigue crack growth from XFEM simulation is fitted with test results in two sequence .. 60
Figure 5.1 An overview of the location of the Suurhoff bridge ....................................................... 64
Figure 5.2 An overview of the global dimensions of the bridge (Top View)...................................... 65
Figure 5.3 Cross-section of the fixed part of Suurhoff bridge (SV01 and SV02) ............................... 65
Figure 5.4 Details of the investigated bridge (units: millimetres) - Cross-section of the detail ........... 66
Figure 5.5 Location of the fatigue crack (units: mm) ..................................................................... 66
Figure 5.6 Outline of the research ................................................................................................ 67
Figure 5.7 Crack propagation curves based on literature ................................................................ 69
Figure 5.8 Influence lines of the bending moments in the deck plate and stiffener for an axle load of
1N using wheel type A and B. [39] ............................................................................................... 70
Figure 5.9 Recorded traffic numbers on the slow lane in one direction traffic [42] ........................... 71
Figure 5.10 Illustration of numerical model ................................................................................... 73
Figure 5.11 Loading sequence for fatigue simulation ..................................................................... 74
Figure 5.12 Numerical model: Boundary conditions ....................................................................... 75
Figure 5.13 Numerical model: Mesh quality .................................................................................. 76
Figure 5.14 (a) Stress distribution along the thickness of the deck plate (b) Stress distribution at the
weld root along the longitudinal direction (c) Stress distribution in the bottom part of the deck plate
at the weld root (For wheel load = 45KN)..................................................................................... 77
Figure 5.15 (a) Numerical model consisting of a moving load (b) Comparison of influence line
between the cyclic load and moving load based on Max. Principal stresses ..................................... 78
Figure 5.16 Fatigue strength curves for stiffener-to-deck plate connections .................................... 79
Figure 5.17 (a) Overview of the RWS numerical model (Fixed bridge) [42] (b) RWS Fatigue
verification model under single-axle load C [42] ............................................................................ 80
Figure 5.18 Comparison of max. and min. principal stresses of different model ............................... 81
Figure 5.19 Hotspot stresses derived at the root of the wheel for various wheel loads .................... 82
Figure 5.20 Wheel load frequency for 1972-2010 derived from the standard NEN 8701 ................... 83
Figure 5.21 Location and definition of the initial elliptical shape ..................................................... 85
Figure 5.22 Representation of TOFD measurement [45] ................................................................ 86
Figure 5.23 (a) Stages of crack propagation displayed as STATUSXFEM output variable (b) Crack
propagation mechanism (c) Side view corresponds with the weld root (d) Isometric view of the crack
on the surface of the deck plate (e) Deck plate crack length for wheel load 45 KN .......................... 88
Figure 5.24 Fatigue crack growth for several wheel loads in through-thickness direction ................. 89
Figure 5.25 Fatigue crack propagation in the surface of the deck plate ........................................... 90
Figure 5.26 Fatigue crack propagation in a longitudinal direction on the deck plate surface ............. 91
Figure 5.27 Wheel load frequency for 2011-2040 derived from the standard NEN 8701 ................... 91
Figure 5.28 Summary of fatigue life estimation ............................................................................. 93
Figure B.8.1 Beach mark measurement by microscope [25] ......................................................... 106
List of Tables
Table 3.1 Direct cyclic parameters................................................................................................ 26
Table 3.2 Constants of Paris' Law and XFEM Abaqus ..................................................................... 27
Table 3.3 Critical energy release rate Gc ....................................................................................... 27
Table 3.4 Meshing details of XFE model ........................................................................................ 28
Table 3.5 Representation of modelling detail and computational time ............................................. 40
Table 3.6 Illustration of output results obtained from different mesh size ....................................... 41
Table 3.7 Representation of fracture toughness and final crack size for different G Ic values ............. 42
Table 3.8 Representation of fracture toughness and final crack size for different am values .............. 42
Table 4.1 Meshing details of XFE model ........................................................................................ 51
Table 4.3 Critical energy release rate Gc and Paris constant .......................................................... 58
Table 5.1 Paris constants proposed by different authors for structural steel .................................... 69
Table 5.2 Summary of the recorded traffic number based on NEN 8701 ......................................... 72
Table 5.3 Meshing details of the numerical model ......................................................................... 76
Table 5.4 Comparison of hotspot stresses (MPa) based on different models ................................... 81
Table 5.5 Comparison of hotspot stresses (N/mm 2) based on different approach ............................ 82
Table 5.7 Crack dimension obtained using TOFD method .............................................................. 86
Table 5.8 Paris constants C3 and C4 .............................................................................................. 87
Table 5.9 Critical energy release rate Gc ....................................................................................... 87
Table A.8.1 Keyword: Paris law formulation implemented for CT-specimen in XFEM-model ........... 105
Table A.8.2 Keyword: Paris law formulation implemented for OSD-specimen in XFEM-model ........ 105
Table A.8.3 Keyword: Paris law formulation implemented for Suurhoff bridge in numerical model . 105
Table C.8.4 NEN 8701: Period 1972-1990 ................................................................................... 107
Table D.8.7 Fatigue detail category ............................................................................................ 110
Introduction Background information
Chapter 1
INTRODUCTION
1 Introduction
1
2
1.1 Background information
1.1.1 Motivation
Structural components are often found to fail under stresses below the ultimate or even below the yield
stresses in the presence of fatigue loading. The fatigue phenomenon is due to micro-cracks initiation,
nucleation and gradually forms macrocracks [1]. The macrocracks will propagate under cyclic loading.
The conventional static strength analysis is not enough to predict the service behaviour of steel
structures. Therefore, over the past few decades many scientists and structural engineers have
focussed their attention to fatigue fracture problems while designing and analysing structural
components. Orthotropic steel decks (OSDs) are one of such typical structural components which has
suffered from fatigue problems over the past decades. Although many improvements in aspects of
design, fabrication, inspection, and maintenance have been achieved for such bridge decks, fatigue
remains its predominant problem, mostly because of the complexity of prediction methods. One of the
critical fatigue details is the welded connection between the deck plate and the longitudinal stiffener
due to direct contact of wheel load and its corresponding high-stress ranges. The closed stiffeners
restrict the transversal movement of the deck plate making this critical detail prone to fatigue failure.
Many researchers have tried to investigate this detail through experiments [2]. However, performing
only experiments may not lead to a cost-effective solution. Therefore, it is necessary to combine the
experimental data with the numerical approaches and preferably assuming basic material properties to
predict behaviour of critical details.
1.1.2 Methodology
Methods based on fracture mechanics could be used to model and analyse the fatigue crack propagation
and subsequent failure of the structure. These methods have already shown their reliability in the
aerospace and automobile industry. The use of Linear Elastic Fracture Mechanics (LEFM) model has
several advantages as it significantly reduces the requirement of experiments. Furthermore, this
method can predict the crack propagation until subsequent failure, which implies that the total fatigue
life of the structure can be predicted for a certain crack length. Therefore, remaining lifetime predictions
could be made for existing bridges [3].
ABAQUS® provides an enriched feature, commonly referred to as the Extended Finite Element Method
(XFEM) to model discontinuity independent to the finite element mesh. This removes the requirement
of the modelling domain and mesh to correspond to each other explicitly. Using XFEM, it is possible to
evaluate automated crack propagation by arbitrarily inserting the crack into the existing model. The
mesh around the crack tip should be sufficiently small to have to get an accurate prediction which leads
to the high computational effort. Two options are available to model crack propagation, either by
cohesive segment method or the linear elastic fracture mechanics (LEFM) approach in conjunction by
phantom nodes [4].
Therefore, to understand the fatigue crack propagation mechanism is essential to study the mechanics
of the material involved such development microcracks, defects, etc. This information together with the
understanding of the basis of the XFEM is essential to better understand and analyse the simulation
results.
3
Introduction Research Objective
1.2 Research Objective
The main objective of this research is to numerically model and verify the problem of fatigue crack
propagation using XFEM-model based on LEFM which can be summed up in two main questions.
1. How to model the fatigue crack propagation in Compact Tension (CT) -specimen
and in welded connection (rib-to-deck plate) of an Orthotropic Steel Deck (OSD)
using XFEM based on LEFM?
How to implement the material parameters and formulate Paris law in the XFEM model for numerical
simulation of fatigue crack propagation?
2. What is the accuracy of XFEM model developed for this research when compared
with the experimental/inspection data to predict the fatigue crack propagation?
2.1. Compact- 2.2. Orthotropic

Tension (CT) Steel Deck (OSD) 2.3. Suurhoff Bridge
Specimen Specimen
2.1. What is the accuracy of XFEM model developed in this research to predict the fatigue crack
propagation rate in CT-specimen for different stress ratios?
2.2. How to predict the Paris law constants (C and m) using XFEM-model based on the beach mark
measurement?
2.3. What is the total fatigue life (crack initiation period and crack propagation period) of the Suurhoff
bridge (existing bridge) based on numerical analyses in welded connection (rib-to-deck plate) of
an Orthotropic Steel Deck (OSD)?
4
Introduction Thesis Structure
1.3 Thesis Structure
In order to approach the main objectives of this research project, it is essential to divide the workflow
into distinctive parts. Hence, the chapters are categorised based on the numerical model and the
methodology is discussed below.
Chapter 2 illustrates some fundamental information in understanding the linear elastic fracture
mechanics. Some valuable insight on the evolution of fatigue crack is discussed, mainly focussing in
the welded joint between the deck plate and the longitudinal stiffener. In addition to that, some basic
of numerical XFEM feature in ABAQUS® is covered.
Chapter 3 deals with the prediction of fatigue crack propagation rate in CT-specimen using 2D-XFEM
and 3D-XFEM model for several stress ratios. The simulated results are compared with the experimental
data, to determine the efficiency of assumed parameters. The mechanism of 3D-XFEM is studied and
the results are compared with 2D-XFEM. In addition to that, the effect of LEFM parameters on crack
propagation is discussed.
Chapter 4 deals with the prediction of fatigue crack growth in OSD using XFEM-model. The simulated
results are correlated with the beach marks measurements, to estimate the Paris law constant C and
m. Prior to crack simulation, some static analyses such as deformation, strain measurement, and hot-
spot stress were performed, and the results are compared with the experimental data to ensure actual
behaviour of the test specimen.
Chapter 5 involves in developing a numerical model to predict the crack initiation period and the crack
propagation period of Suurhoff bridge (existing bridge). The simulated results are compared with the
inspection data (TOFD measurement) and/or existing numerical model to determine the accuracy of
the numerical model and to verify the problem.
Finally, chapter 6 contains a set of conclusions drawn from this research and corresponding answers to
the research questions. In addition, some recommendations are made for further research in improving
the XFEM model on this topic.
5
Introduction Thesis Structure
6
Literature Overview Thesis Structure
Chapter 2
THE STATE OF THE ART (LITERATURE OVERVIEW)
2 Literature Overview
7
Literature Overview Thesis Structure
8
Literature Overview Linear Elastic Fracture Mechanics (LEFM)
2.1 Linear Elastic Fracture Mechanics (LEFM)
Linear elastic fracture mechanics was first introduced by A. A. Griffith [5], to explain the behaviour of
flaw in materials [3]. He proposed a relation that the product of the square root of the flaw length 𝑎
and the stress at fracture 𝜎𝑓 is nearly constant (equation 2.1). He tried to illustrate this relation in
terms of linear elastic theory. As this approach showed some problems as the stress (or strain) at the
tip of a sharp flaw in linear elastic material was infinite. Later, this theory was therefore explained
taking energy considerations through thermodynamic approach, which was later modified by Irwin [6].
f a C (2.1)
2.1.1 Crack characterization
Crack geometry
The extreme ends of the crack can be considered as crack tip. If the crack is considered in two-
dimensional (line), the crack tip is a single point. Whereas, if the crack is analysed in three-dimensional,
then the crack tip can be complex crack front. This complexity depends upon the crack propagation in
through-thickness direction. Such type of fatigue which propagates through the entire thickness can be
referred as through cracks (Figure 2.1). Moreover, the crack front is generally curved in thick materials.
Such cracks are referred as part through cracks (Figure 2.1-corner cracks and surface crack). The first
conditions can be treated as the 2D-crack tip region can be used, while the later will lead to more
complex 3D crack tip region [7].
Both the types of crack are characterized by a cartesian coordinate system (𝑥, 𝑦, 𝑧) and polar reference
system (𝑟. 𝜃) with x lying on the uncracked region and z being the tangent to the crack line.
Figure 2.1 Different crack types [3]
Figure 2.2 Crack tip region- 2D [4] Figure 2.3 Crack tip region-3D [4]
9
Crack propagation
It is true that direction of crack growth derives from the stress intensity distribution along the crack
front and the rate of propagation depends of the magnitude of the stress intensity factor however, 3D
problems are more complex and are associated with the mechanistic phenomenon known as crack
closure effect. A typical example of crack propagation direction is shown in
Figure 2.4 for curved crack front in through thickness direction and an elliptical crack front situated at
the surface.
(a) (b)
Figure 2.4 Crack closure effect at the material surface on crack front shapes [8]: (a) Through crack- Curved crack front
(b) Deviations of semi-elliptical crack front at material surface
Fracture modes
There are three ways that a crack can extend namely Mode I, Mode II and Mode III. From
Figure 2.5, Mode I is referred as opening mode where the crack surface moves apart; Mode II is
referred as sliding mode (in-plane shear mode) where the crack surface slide apart perpendicular to
the crack front; and Mode II is referred as tearing mode (anti-plane shear mode) where the crack
surface slides apart parallel to the crack front. Mode I is considered as most common and important in
crack growth analysis because the crack developed under pure shear loading quickly transit to a tensile
mode [9].
(a) (c)
(b)
Figure 2.5 Fracture modes a) Mode I b) Mode II c) Mode III [9]
10
2.1.2 Stress intensity factor
The Griffith criterion
Griffith criterion states that the crack extension 𝛥𝑎 in a plate of thickness 𝑡 is only possible “if the work
done by the applied force is more than or equal to the summation of the change in the elastic energy
and the energy absorbed at the crack tip, then it will lead to unstable fracture” [10]. This statement
can be represented as
dWe  dU el + Gct a (2.2)
where 𝑑𝑊𝑒 implies the work done to form a crack extension 𝛥𝑎 , 𝑑𝑈 𝑒𝑙 is the change in elastic strain
energy and 𝐺𝑐 is the critical energy release rate which signifies the toughness of the material or
extension of crack. For a simple case, the failure in an infinite plate with central crack can be expressed
as (equation 2.3) in terms of critical energy release rate in relation with critical stress (σc ).
 c2 a
Gc = (2.3)
E
which can be further related in
 c  a = Gc E = K IC (2.4)
where 𝐾𝐼𝐶 denotes the fracture toughness of the specimen. The stress intensity factor can be expressed
3
in 𝑁. 𝑚𝑚 ⁄2 or 𝑀𝑃𝑎 √𝑚. Through the fundamental definition, the potential energy P is related to the
crack growth 𝑑𝑎 proposed by Irwin [8]
dP
G=− (2.5)
da
From equation 2.5 and equation 2.3, the most important relation can be derived as the change in
potential energy to close a small section of crack da, equating with the quantity to the work required
to close that section of crack without any external effort.
The following equation (2.6 and 2.7) [10] is valid for different crack mode, when plane stress situation
is assumed.
(1 + v) 2
G=
E
(
1 2
K I + K II2 ) +
E
K III (2.6)
Similarly, when the plane-strain situation is assumed:
1 − v2 (1 + v) 2
G=
E
( K I2 + K II2 ) +
E
K III (2.7)
11
Fatigue crack propagation regime
Paris and Erdogan [11] studied crack growth behaviour through experiments and found a relation
between the crack growth rate 𝑑𝑎/𝑑𝑁 and the stress intensity factor range 𝛥𝐾. Additional test indicated
two vertical asymptotes occur when 𝑑𝑎/𝑑𝑁 is plotted against the 𝛥𝐾 in log-log scale. The two extreme
asymptotes indicate the start and the end of the crack life. More precisely, the left asymptote at 𝛥𝐾 =
𝛥𝐾𝑡ℎ signifies that 𝐾-values below this threshold level are two low to cause crack growth. On the other
side, the right asymptote at 𝐾𝑚𝑎𝑥 = 𝐾𝑐 signifies for a 𝛥𝐾 cycle with 𝐾𝑚𝑎𝑥 = 𝐾𝑐 reaches a critical value
which leads to complete failure of the structure. With these two vertical asymptotes the function can
be divided into three different regimes as I, II and III illustrated in Figure 2.6 as (i) the threshold fatigue
crack propagation regime (ii) the fatigue crack propagation regime and (iii) the near unstable fatigue
crack propagation regime respectively. Therefore, the fatigue crack propagation regime can be
formulated as:
da
= C K m (2.8)
dN
Figure 2.6 Three regime of the crack propagation rate as a function of 𝛥𝐾 [8]
12
Literature Overview Fatigue of welded connections
2.2 Fatigue of welded connections
Fatigue crack development
In general, fatigue can be attributed to the crack development until complete fracture after ample
amount of stress fluctuation. Thereby in context of fatigue, the lifetime of structure can be divided into
three stages: fatigue crack initiation, fatigue crack propagation and final failure.
The crack nucleation stage can be stated as the period from an initial defect to the detectable crack.
According to [12], this stage can be further subdivided into two phases: microstructural and mechanical
(Figure 2.7).
Figure 2.7 Length scales of the life cycle of a component subjected to cyclic loading [12]
In the first phase, the crack develops due accumulation of irreversible plastic deformation at pre-existing
defects such as pores, inclusions, etc. which acts as the micro-notches. However, the crack is still small
which can be comparable to the dimensions of the grain size and therefore it is attributed as
microstructurally short crack. These cracks extend and tries to overcome the microstructural barriers
(twin or grain boundaries) when the applied stress range is high enough. Once the crack encloses the
number of grains the influence of local microstructural characteristics diminishes. As a result, the crack
propagation in the next phase becomes steadier and these cracks can be attributed as mechanically
short cracks. However, these crack still does not behave by the linear elastic K factor due to crack
closure effects.
For non-welded components, the crack initiation stage covers the majority portion of total fatigue life.
In contrast, the crack initiation stage is comparatively small for welded component due to welding
defects and residual stresses. Lack of penetration, lack of fusion, slag inclusion, linear porosity (gas),
sagging, undercut, overlap, excess of weld metal, incompletely filled groove are common examples of
welding defect. Furthermore, poor workmanship can significantly decrease the fatigue strength.
Although, these defects are sometimes unavoidable but with the help weld improvement techniques
fatigue strength can be significantly enhanced. Grinding, re-melting, peening, coining, overstressing
are some common weld improvement techniques. These techniques are used to reduce the stress
concentration, to remove crack like defects at the weld toe and harmful tensile residual stress.
13
Deck-to-Rib welded joint
Orthotropic steel decks (OSDs) are one of such typical structural components which suffered from
fatigue problems. One of the critical fatigue details is the welded connection between the deck plate
and the longitudinal stiffener due to direct wheel loading and local high stress ranges
For this type of welded connections, there are four types of possible crack paths.
Crack I: initiates at the weld toe in the deck plate and propagates through the deck plate.
Crack II: initiates at the weld root and propagates through the deck plate.
Crack III: initiates at the weld toe in the trough web and propagates through the trough web.
Crack IV: initiates at the weld root and propagates through the weld throat.
(a) (b)
Figure 2.8 Illustration of the deformation and cracks of the bridge deck [13]: (a) Representative loading scenarios and the
corresponding deformations (b) Typical fatigue crack patterns
Fatigue crack originating from the weld root propagate simultaneously in both vertical (through-
thickness of the deck plate) and longitudinal directions, thus becoming large invisible cracks which are
not detected until it damages the wearing surface. The significant effect due to out-of-plane bending
moment between the deck plate and the trough induced by vehicles can pose serious threat to the
structure’s integrity and service life when the length becomes large [14].
Furthermore, several researchers have tried to investigate this detail through experiments and
numerical approaches which can be summarised as follows:
✓ (Nagy, Backer, & Bogaert) [3] studied crack propagation behaviour in deck-to-rib detail of
Temse bridge in Belgium. Two approaches namely the traditional approach (Palmgren-
Miner hypothesis and SN-curves) and the fracture mechanics approach were performed
and compared. To investigate the problem, they used an FEM-model to predict the number
of cycles needed for a crack length of 600 mm. Approximately 38.5 x 106 load cycles was
evaluated which is more than the traditional calculations (21 x 106 load cycles). However,
this difference was justified as they did not take residual stresses and other welding defects
into account in their model. Nevertheless, this paper showed a sign of improvement in
14
calculation of fatigue by use of fracture mechanics. As a continuation of work [15], a crack

length of 461.703 mm was evaluated based on XFEM-simulation followed by exponential
extrapolation, while a crack length of 600 mm was detected in reality. However, their model
did not consider any residual stresses, which is most likely to be present in weld especially
because multiple welds intersect at the crack location and the welds were not chamfered.
Even though the residual stresses were not implemented, such results showed that fracture
mechanics can be used for improved fatigue life assessment. Further, an improved
analysing tool using LFEM and XFEM [16] evaluated the thickness effects for both, deck
plate and longitudinal stiffener incorporating the residual stresses into their model. Based
on the result, the authors concluded that the fatigue life increases with increase in thickness
of deck plate while this is not the case with longitudinal stiffener thickness.
✓ In the analytical study (Xiaochen et. al) [17] on crack propagation behaviour based on FEA
(Finite Element Analysis) and fracture mechanics. In analysing the crack direction, SIF
under the mixed modes I, II and III were considered which indicated the complexity of
stress field around the fatigue crack. Their numerical results showed that the crack direction
change was due to Mode II and Mode III deformations near the crack tip caused by the
out-of-plane bending of the rib wall along the crack. In addition to this, they also claimed
that the crack direction could be evaluated by investigating the equivalent stress intensity
factor.
15
Literature Overview XFEM (eXtended Finite Element Method)
2.3 XFEM (eXtended Finite Element Method)
In modelling a crack with traditional finite element method, it is necessary to refine the mesh around
the crack in order to capture the singular asymptotic field accurately. Furthermore, modelling a growing
crack can be even more cumbersome as the mesh should be at every crack propagation step to match
the discontinuity. On the other hand, XFEM alleviates the drawbacks associated with meshing crack
surfaces.
The XFEM was first developed by Belyschko and Black (1999) [18], an extension of the traditional finite
element method based on the concept of partition of unity by Melenk and Babuska (1996) [19]. Two
enrichment functions are incorporated in XFEM namely the discontinuity function 𝐻(𝑥) which represent
the gap between the crack surfaces and asymptotic function 𝐹𝑎 (𝑥) which capture the singularity around
the crack tip. These functions are enriched with nodal degree of freedom (𝑎⃗𝐼 , 𝑏⃗⃗𝐼𝑎 ) and the vector
function 𝑢
⃗⃗ with the partition of unity enrichment can be defined by the following equation:
N
u =  N I ( x) uI + H ( x)aI  (2.9)
I =1
where 𝑁𝐼 (𝑥) is the nodal shape function; 𝑢⃗⃗𝐼 represent the nodal displacement vector associated with
the continuous part of the finite element solution.
However, the terms (equation 2.9) are associate with different domains. For instance, the 2nd term
𝐻(𝑥) in only applicable to the nodes with the discontinues function (crack interior), while the last term
is applicable to the crack tip enrichment function (crack tip) and the 1st term 𝑢 ⃗⃗⃗⃗⃗1 is associated to all
nodes in the model. H(x) is the discontinuous jump function across the crack surface shown in
Figure 2.9(a) is given in equation (2.10)
1 𝑖𝑓(𝑥⃗ − 𝑥⃗ ∗ ). 𝑛⃗⃗ ≥ 0 ;
𝐻(𝑥) = { } (2.10)
−1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
where 𝑥⃗ is a Gauss point, 𝑥⃗ ∗ is the point on the crack closest to 𝑥⃗ and 𝑛⃗⃗ is the unit outward normal to
the crack at 𝑥⃗ ∗ .
16
(a) (b)
□ Crack tip function

○ Jump function
— Enrichment radius
Figure 2.9 (a) Representation of normal and tangential coordinates for a smooth crack. [4] (b) Representation of enriched
nodes and enrichment radius in an arbitrary 2D mesh
An arbitrary crack in 2D mesh is illustrated in Figure 2.9(b). The nodes highlighted with square shape
are associated with both the enrichment functions and responsible for capturing the singularity around
the crack tip. Moreover, ABAQUS® provides the user the freedom to define the enrichment radius
highlighted with circle (red) indicating the crack domain. Besides that, the nodes highlighted with circle
(black) are only associated with discontinuous function. However, a more complex three-dimensional
crack is represented using the level set method which is being used for analysing and computing
interface motion. A description of such discontinuous geometry is illustrated in Figure 2.10.
Figure 2.10 Illustration of a non-planar crack in the 3D by two signed distance function ∅ and 𝜔 [4]
17
The behaviour of the XFEM-based LEFM for automated crack propagation analysis is determined based
on modified VCCT. XFEM enriched feature in ABAQUS® can be used to simulate crack propagation by
using the direct cyclic approach in combination with Paris law formulation. The former setting provides
the user to define a cyclic loading sequence (periodic, decay, user-define etc) while the latter setting
defines the crack propagation, which adds more degree of freedom through XFEM. The direct cyclic
simulates the cyclic load and requires the definition of a periodic function (for example-equation 2.12),
which will be used by the software to apply the amount of load at its corresponding time.
𝑥 = 𝐴0 + 𝐴1 cos(𝑡 − 𝑡0 ) + 𝐵1 sin(ω(𝑡 − 𝑡0 )) (2.12)
The crack growth is described using Paris law, which is based on relative fracture energy release rates
(Figure 2.11).
Figure 2.11 Fatigue crack growth [4]
The onset of the crack propagation indicates the starting of fatigue crack growth can be defined as
(equation 2.13):
N
f = ≥ 1.0 (2.13)
C1G C2
where ΔG is the relative fracture energy rate between its maximum and minimum values. Furthermore,
the crack growth rate using Paris law can be formulated as (equation 2.14) only if 𝐺𝑡ℎ𝑟𝑒𝑠ℎ < 𝐺𝑚𝑎𝑥 <
𝐺𝑝𝑙 :
da
= C3 G C4 (2.14)
dN
Abaqus® provides three common mixed mode model for evaluating the equivalent fracture energy
release rate GequivC : the BK law, the Power law and the Reeder law [4] . Although the choice of model
18
is not well defined for a given analysis, the most commonly used model is power law which can be
illustrated as (equation 2.15):
am an ao
Gequiv G  G  G 
=  I  +  II  +  III  (2.15)
GequivC  GIC   GIIC   GIIIC 
19
20
Compact-Tension Specimen XFEM (eXtended Finite Element Method)
Chapter 3
COMPACT-TENSION SPECIMEN
3 Compact-Tension Specimen
21
Compact-Tension Specimen XFEM (eXtended Finite Element Method)
22
Compact-Tension Specimen General
3.1 General
Compact-Tension (CT) specimen is a notched sample which is used to generate a fatigue crack through
cyclic loading. Such specimens are standardised in accordance to ISO [20] and ASTM [21] , are
extensively used in the field of fracture mechanics, to evaluate the material parameters through
experiments. Fatigue being the predominant problem in steel infrastructure, such notched samples are
good representation of real-life inconsistencies (discontinuities) introduced during manufacturing of
steel products. Many researchers [22] have tried to investigate the fatigue behaviour of CT-specimen
through experiments. However, performing only experiments may not be a cost-effective solution.
Therefore, it is necessary to combine the experimental data with numerical approaches and preferably
assuming basic material properties to predict behaviour of critical detail.
Numerical model based on fracture mechanics approach can be used to analyse fatigue crack
propagation and has already shown its reliability. Commercial software such as ABAQUS ® incorporates
XFEM techniques to model discontinuities as an enriched feature. Using XFEM, it is possible to simulate
automated crack propagation by inserting the crack into the model.
The main objective of this chapter is to predict the crack propagation rate for different stress ratios of
CT-specimen using 2D and 3D XFEM-model based on LEFM and VCCT. To evaluate the efficiency of the
assumed parameters, the simulated results are compared with the results of fatigue coupon tests [22].
The simulated results of crack propagation rate with different stress ratios were also studied and
correlated with Elber’s equation to study the crack closure mechanism. As the majority of works in LEFM
concentrates on 2D shell model, the extension to three-dimensional (3D) solid requires the investigation
of some parameters mainly through-thickness effect. Firstly, the mechanism of crack propagation will
be studied and results of fatigue crack growth (a,N) will be compared with the 2D results. Secondly, an
investigation is performed in determining a reliable technique to compute Stress Intensity Factor (SIF).
Finally, the crack propagation rate in 3D is determined and will be compared with the test result.
23
Compact-Tension Specimen XFEM model
3.2 XFEM model
3.2.1 Geometry
Figure 3.1 Geometry of CT-Specimen [22]
The set of simulation carried out in an adequate manner according to the experiment [22].
A full scale XFEM-model is developed based on the dimensions shown in the
Figure 3.1. Moreover, the thickness of specimen is different for different stress ratios (see Table 3.2).
The crack propagation behaviour is analysed using both 2D-XFEM and 3D-XFEM model.
24
3.2.2 Boundary conditions
(a) (b)
Initial crack flaw

(5mm)
Enrichment region
Figure 3.2 (a) Boundary condition of CT-Specimen (b) XFEM-model of CT-Specimen
In modelling relatively realistic boundary conditions of CT specimen, two reference points namely RP-
1 and RP-2 were incorporated at the centre of the holes which were coupled (kinematically constraint
in all the direction for translation and rotation) with the two-interior half holes of the CT specimen. The
shell edge in 2D-XFEM model and the solid surface in 3D-XFEM model (see Figure 3.3). The boundary
conditions were applied on these reference points as specified in Figure 3.2. RP-1 is translationally
restrained in x-direction whereas RP-2 was restrained in both x- and y- direction. Moreover, the tensile
cyclic load is applied at RP-1. Since XFEM simulation require a crack tip to be predefined. Therefore, a
straight crack flaw of 5 mm length is assumed, and it is positioned at the notch of the model indicated
in Figure 3.2 (b). Hence, the initial crack size was assumed to be of 15 mm. Crack domain represents
the enrichment region contains a crack tip placed at the notch of the specimen illustrated in Figure 3.2
(b).
(a) (b)
Figure 3.3 Illustration of reference -point coupled to a (a) Shell edge in 2D (b) Solid surface in 3D
25
3.2.3 Material property
The elastic material properties were assigned to the XFEM model as: Young’s modulus E=210500 MPa
and Poisson’s ratio υ=0.3. Furthermore, the fracture contact properties and Paris law constants are
discussed in the following section.
LEFM implementation
Virtual Crack Closure Technique (VCCT) was used in the XFEM-based linear elastic fracture mechanics
for crack propagation analysis using the direct cyclic approach with a time increment size of 0.05 per
cycle. The direct cyclic load simulation is based on the periodic function (equation 2.12) and the
parameter used for different stress ratios (Figure 3.4) are tabulated in Table 3.1.
Table 3.1 Direct cyclic parameters
Rσ Load (N) A0 A1 B1 to 𝛚
0.0 6118.6 0.5 0 0.5 0 2𝜋
0.25 7246.2 0.625 0 0.375 0 2𝜋
0.50 9345.9 0.75 0 0.25 0 2𝜋
Figure 3.4 Graphical representation of different stress ratios
The direct cyclic approach is combined with the Paris law formulation to simulate the crack propagation.
These fatigue crack growth rates are evaluated based on assigned VCCT parameters (Table 3.3). The
crack propagation appears when the energy available for the crack is high enough to overcome the
fracture resistance of the material. Since ABAQUS® analyses the fracture by the strain energy criterion
approach, the Paris law parameters C3 and C4 were calculated assuming plane stress situation from
equation (2.8) and (2.14) and are listed in Table 3.2 for several stress ratios. To ensure the start of
crack growth process, material constants C1 and C2 were kept negligible as 0.001 and 0 respectively.
Once the onset of the fatigue crack growth is satisfied (equation 2.13), the crack propagation rate can
be computed based on the fracture energy release rate (equation 2.14).
26
Table 3.2 Constants of Paris' Law and XFEM Abaqus
Rσ B (mm) Fmax (N) Fmin (N) Experimental data XFEM Abaqus

Ca m C3 C4
0.0 7.81 6118.6 61.8 2.5893E-15 3.5622 7.8419E-06 1.7811
0.25 7.47 7246.2 1811.5 2.5491E-15 3.7159 1.9790E-05 1.8579
0.50 7.41 9345.9 4672.9 8.2764E-16 3.8907 1.8768E-05 1.9453
In this study, Power law mix-mode model is selected for evaluating the equivalent fracture energy
release rate represented in the equation (2.15) because of its simplicity in the relation of different
modes of fracture.
The test results for the fracture toughness KIC of the S355 steel grade obtained in the experiment [23]
using circumferentially cracked round bars (CCRB) ranges from 35.78 MPa√m to 40.4 MPa√m. This
scatter can be possibly used in Compact-Tension (CT) specimen, a single edge notched bend or three-
point loaded bend specimen, which is standardized by a different institution. Therefore, this data was
taken as the base of this study and applied to the XFEM-model tabulated in Table 3.3.
Table 3.3 Critical energy release rate Gc
XFEM model Critical energy release rate 𝑮𝒄 (Nmm-1) Exponent
Mode I Mode II Mode III αm αn αo

CT- Specimen 6.5 6.5 6.5 1 1 1
27
3.2.4 Mesh quality
The 2D XFEM-model consisted of shell elements and was modelled using a 4-node plane stress
quadrilateral with linear geometric order. The mesh size should be small to capture accurate stresses
near the crack tip. However, a numerical model with fine mesh can be time-consuming, therefore a
variable mesh was used. In the enrichment area (XFEM region) 0.5 mm mesh size was used and 2 mm
in the non-enrichment area was used, as shown in Figure 3.5. On other hand, the 3D XFEM-model
consisted of solid elements and was modelled using 8-node brick elements with linear geometric order.
Similar to the shell model, a variable mesh was used but with 0.33 mm mesh size for the enrichment
elements. All the meshing details are tabulated in Table 3.4.
(a) (b)
Figure 3.5 Mesh quality (a) Two-dimensional XFEM-model (b) Three-dimensional XFEM-model
Table 3.4 Meshing details of XFE model
Model Region Element type Mesh size

2D-XFEM Enriched shell 4-node plane stress quadrilateral (CPS4) 0.50 mm
2D-XFEM Non- Enriched shell 4-node plane stress quadrilateral (CPS4) 2.00 mm
3D-XFEM Enriched solids 8-noded linear brick (C3D8) 0.66 mm
3D-XFEM Non-Enriched solids 8-noded linear brick with reduced integration 2.00 mm
(C3D8R)
28
Compact-Tension Specimen Output
3.3 Output
An output result of crack propagation using direct cyclic loading is presented in this section. In the
case of 2D-XFEM, it has been observed the crack propagated without changing the direction by
fracturing the element (critical) at the end of the stabilised cycle ahead of the crack tip with zero
stiffness. As the element (enriched) is cracked, the load is redistributed for the next cycle, and the
stress intensity factor is captured based on fracture energy release. The fracture energy release rate
was accounted for the enriched element ahead of the crack tip followed by the next enriched element
when the previous enriched element is completely fractured. Simultaneously, the number of load cycles
was precisely noted as the fatigue crack propagated over the element length. An example of the 2D-
XFEM output is illustrated in Figure 3.6 representing the status of enriched element, the crack tip and
crack surface from the crack tip opening as STATUSXFEM, PHILSM and PSILSM output variable
respectively at 1.87x105 load cycles for zero stress ratio. In addition to that, different propagation
stages are represented in Figure 3.7. In the case of 3D-XFEM, the mechanism is bit complex explained
in section 3.4.3. An example of the 3D-XFEM output is presented in Figure 3.8 illustrating the crack
shape and crack propagation mechanism along through-thickness direction at 2.15x104 load cycles for
zero cycles.
3.3.1 2D-XFEM model
(i)
(ii)
(iii)
Figure 3.6 XFEM output (i) STATUSXFEM (ii) PHILSM variable output (iii) PSILSM variable output
(a (b (c)
) )
29
Compact-Tension Specimen Output
Figure 3.7 Representation of different crack propagation stages at (a) N= 1.23x104 load cycles (b) N= 13.16x104 load cycles
(c) N= 24.42x104 load cycles
3.3.2 3D-XFEM model
(a)
(b)
Point of change in mechanism
Regular crack propagation mechanism
(c) (d)
Figure 3.8 (a) 3D-Model STATUSXFEM output indicating the crack shape (b) Representation of crack propagation path and
change in mechanism (c) Regular crack propagation mechanism (d) Irregular crack propagation mechanism
30
Compact-Tension Specimen Result and Discussion
3.4 Result and Discussion
The results of the fatigue crack propagation obtained from numerical simulations are presented in this
section. It should be noted that the stress intensity factor values were computed in the experiment
using the formulation proposed in ASTM E647 for the CT specimens [21].
F  (2 +  ) 
K =
B W

 (1 +  ) 3 2
( 0.886 + 4.64 − 13.32 2
+ 14.72 3
− 5.6 4
)  (3.6)
 
where α= a/W, a is the crack size; B is the thickness of the specimen, W is the width of the specimen
and ΔF is the applied load range. In automated crack simulation, the stress intensity factor range is
computed based on fracture energy release rate. As the software (ABAQUS®) fails to compute Kmin for
corresponding applied Fmin (cyclic load) load, the lower bound of SIF (Kmin) was then calculated based
on the following relation (equation 3.7).
K min
R= (3.7)
K max
3.4.1 2D shell XFEM-model
Figure 3.9 Fatigue crack propagation rates obtained from the numerical simulation (2D-XFEM) compared with the test
results
In Figure 3.9, the numerical prediction of the fatigue crack growth rate da/dN is plotted as a function
of the stress intensity factor range ΔK in a log-log graph. The stress intensity factor values are computed
in numerical simulation using the fracture energy-based criterion and the crack propagation rate is
evaluated as the crack propagated along the length of the element starting from 15 mm as the initial
crack size as shown in the Figure 3.6.
After comparing with different stress ratios, it is observed that the rate of fatigue crack propagation is
significantly increased as the stress ratio changes from 0 to higher positive values. This can be evaluated
31
comparing the slopes of the curve. For stress ratios R=0.0, R=0.25 and R=0.50, the slopes were 3.56,
3.71 and 3.89 respectively. The increase in crack propagation rate can be due to crack closure effects.
Comparing with the test result, the simulated crack propagation provided good agreement with a
maximum difference of 0.03% in the slope (m) and 1.48% in the intercept (C) of the power law
equation.
3.4.2 Effect of stress ratio
Based on the above simulation for 2D-XFEM model, a fatigue crack growth is represented in a graph,
with the crack size plotted as a function of number of load cycles for different level of stress amplitude
(Figure 3.10 (a)). It is clear from the graph, the fatigue crack resulted in an exponential growth for all
the stress ratios. However, the steepness of curve is significant as the stress ratio increase from zero
to a positive value. With the decrease in the load range (ΔF) and corresponding increase to positive
stress ratio (R), the cycles to failure is reduced drastically. It is to be noted that with a drop of 622.1 N
in the force range (ΔF) from R=0.00 to R=0.25, the cycles to failure falls by 6 times. Moreover, the
final crack size is diminished by 3.5 mm. However, the difference is more prominent, when the stress
ratio R=0.25 is compared with R=0.50. With the decrease of load range ΔF of 761.7 N, the cycles to
failure is decreased only by 4 times but the final crack length is diminished by 5 mm. This increase in
the fatigue crack propagation rate can be explained by crack closure mechanism through similarity
principle.
(a)
(b) (c)
Figure 3.10 Fatigue crack growth for different stress ratios (b) Crack propagation rate (Similarity principle) (c) Effective
stress intensity factor (Elber's crack closure mechanism)
32
According to similarity principle based on stress intensity factor, the extension of crack Δa is constant
for constant ΔK cycles, which implies the crack growth rate is the function of K min and Kmax of the load
cycles. In other words, the crack propagation rate is not only depending upon the stress intensity factor
range (ΔK= Kmax - Kmin) but also depends upon the stress ratio (R= Kmin/ Kmax).
da
= f (K , R) (3.8)
dN
This effect can be observed with the simulated results, when the crack propagation rate is plotted
against the crack size (Figure 3.10 (b)). The crack growth rates for different stress ratio R are partly
overlapping, which indicates the similar crack growth rates occurs, although at different values of the
crack size. This difference can be eliminated by taking in account the concept of crack closure.
The important aspect to be noted is that the stress singularity at the tip of the crack. The stress intensity
factor is present as long as the crack tip is open, which can be termed as the effective stress intensity
factor ΔKeff (equation 3.8).
K eff K max − K op
U= = (3.9)
K K max − K min
Following the concept of Elber [24], the fatigue crack growth rate is defined (equation 3.10) and
adapting the relation between the U and R which is valid for R-values in the range of -0.1 to 0.7
(equation 3.11):
da
= f (K eff ) (3.10)
dN
U = 0.5 + 0.4 R (3.11)
Using equation 3.9 and equation 3.11, the effective stress intensity factor is evaluated and plotted
against the crack propagation rate for all the three stress ratios (Figure 3.10 (c)). The factor U is 0.5,
0.6 and 0.7 for stress ratios R=0.00, R=0.25 and R=0.50 respectively. The dependency of different
crack size is eliminated by taking different stress ratio into account. Hence, the simulated results hold
the crack closure mechanism and showed a good correlation with Elber’s equation.
33
3.4.3 3D solid XFEM-model
Similar to 2D-XFEM model, a predefined straight crack front of 5 mm in 3D-XFEM model is assumed
and positioned in the mesh in the middle of the crack element ensuring similar crack propagation path.
When the crack simulation is performed, the crack propagates perpendicular to the load applied by
fracturing the element from the middle of the element. Moreover, a regular crack propagation
mechanism in through-thickness direction is observed. However, after a certain crack length, the crack
propagation direction changes and intersect the boundary of the elements (see Figure 3.8 (b)). When
the crack front encounters the top interface of the elements, the crack starts to propagate in an irregular
fashion in through-thickness direction due to which the crack size calculation becomes more complex
(see Figure 3.8 (d)). As a result, the crack growth is evaluated until the point of change in mechanism
for several stress ratios. The change in direction of crack propagation is due to the imperfection in the
method of applying boundary conditions. The reference point coupled to the surface of the hole
generate an additional rotation due to eccentricity leading to the change in direction of propagation.
Moreover, a non-uniform stress distribution is observed over the thickness of the respective holes. An
example of principal stress distribution is presented in Figure 3.11 at crack size 15.1 mm.
Figure 3.11 Non-uniform stress distribution over the thickness at holes
34
Comparison of fatigue crack growth
The results of 3D-XFEM for different crack size until the point of change in mechanism and
corresponding number of cycles is plotted against each other and compared with the results of 2D-
XFEM in Figure 3.12.
Figure 3.12 Comparison of fatigue crack growth between 2D-XFEM model and 3D-XFEM model
It was found that the simulated results of 3D-XFEM model is well correlated with the results of 2D-
XFEM model up to a crack size of 17.8 mm. Although a small difference, both the models showed a
similar crack growth. The difference between the models is higher, as the stress ratio increases from
zero to a positive value. For stress ratio R=0.00, R=0.25 and R=0.50, the difference in load cycles goes
up to 2.1%, 11.9% and 13.8% respectively. This is mainly due to crack propagation mechanism in
through-thickness direction which is governed by non-uniform stress intensity factor distribution along
the crack front.
35
Fatigue crack mechanism- 3D XFEM
The determination of stress intensity factor in 3D-XFEM model is more complex mainly due to its
through-thickness effect. It is observed that the crack propagation starts from the centre of the
thickness and propagates towards the edge of thickness for every crack length increment (mesh size
of 0.3 mm) in the longitudinal direction (see Figure 3.8(c)). This crack mechanism holds for different
mesh sizes up to 1 mm and is mainly due to the distribution of stress intensity factor along the crack
front (see Figure 3.13). However, it is to be noted that the mechanism is depended on Paris law
constant, which defines the rate of crack propagation. Figure 3.13 illustrates the stress intensity factor
distribution in through-thickness direction along the crack front for different mesh sizes. This implies
that the direction of crack propagation derives from the stress intensity factor distribution.
(a)
(b)
Figure 3.13 Stress Intensity factor distribution along the crack front in through-thickness direction compared to ISO 12108
standard for mesh size and crack propagation mechanism (STATUSXFEM output) (a) 0.6 mm (b) 0.3 mm
The through-thickness effect can also be explained by the crack closure mechanism. At the material
surface, a greater crack closure mechanism is experienced due to which the crack opens first in the
centre of the thickness followed by either side at the material surface. As a result, the crack front lag
where the crack interests the surface leading to a curved crack front.
However, this crack growth mechanism does not provide a clear indication to compute the stress
intensity factor. Therefore, in order to find a reliable way to determine the stress intensity factor an
investigation is performed with different techniques available in ABAQUS® using 2D model and 3D
model illustrated in section 3.4.4.
36
Fatigue crack propagation- 3D solid XFEM
Figure 3.14 Fatigue crack propagation rates obtained from the numerical simulation (3D-XFEM) compared with the test
results
In Figure 3.14, the numerical prediction of the fatigue crack growth rate da/dN is plotted as a function
of the stress intensity factor range ΔK in a log-log graph. The stress intensity factor values are computed
in numerical simulation using the fracture energy-based criterion from the edge (surface) element and
the crack propagation rate is evaluated as the crack propagated along the length of the element starting
from 15 mm as the initial crack size up to 17.8 mm where the regular crack propagation is valid.
It is observed that the rate of fatigue crack propagation is significantly increased as the stress ratio
changes from 0 to a higher positive value. This can be evaluated comparing the slopes of the curve.
The slopes for stress ratio R=0.00 and R=0.25 were 4.22 and 4.65 respectively for a crack length of
17.8 mm, which sets the boundary of the model as indicated in Figure 3.14. The increase can be
explained through the crack closure effect. Furthermore, comparing the slopes with the test result, the
simulated crack propagation provided a rough correlation with a maximum difference of 25.23 % in the
slope (m) of the power law equation. The difference in the slope can be explained by the assumed
crack flaw. A straight crack flaw resulted in variation of stress intensity factor along the crack front.
Moreover, the effect the boundary conditions becomes predominant as the crack propagates over the
length. Nevertheless, for comparison there is clear requirement of more evidence (value) as the crack
only propagated in a regular fashion until the crack size 17.8 mm before changing the direction.
Nevertheless, the range of applicability of 3D-XFEM model developed in this research is indicated in the
figure as red and black for stress ratio R=0.0 and R=0.25 respectively. Therefore, a further investigation
is recommended mainly to access the through-thickness effect and implementation of boundary
conditions in the numerical model.
37
3.4.4 Stress Intensity Factor
ABAQUS® provide different techniques to capture the stress intensity factor around the crack tip.
However, the investigation is limited to three techniques which will be performed using 2D-shell XFEM
and 3D-solids XFEM. To evaluate the accuracy, the simulated results of all the techniques are compared
with the stress intensity factor range formulation by ISO 12108 standard (equation 3.6).
Energy based -2D XFEM
Firstly, the automated simulated results of 2D-shell XFEM model (section 3.3.1) is taken as one the
techniques to compute SIF which is evaluated based on fracture energy criterion is plotted against
different crack sizes in Figure 3.16.
Contour Integral -2D
Beside automated XFEM crack simulation, it is also possible to determine the stress intensity factor
range using stationary crack analysis. One of the most significant aspect of stationary crack analysis is
contour integral calculation as it gives the measure to access critical crack size. Thereby, a stationary
crack simulation is performed using 2D-shell model to determine the SIF values based on contour
integral calculation. This 2D-FE model is similar to the previous 2D-XFEM model. Only, the mesh is
modified with five concentric contour rings at the crack tip as illustrated in Figure 3.15. The size of the
total contour domain is determined by the number of element rings which are included in the calculation
of the integral. Theoretically, this calculation is independent of the size of the contour domain. However,
the computed SIF varied for different element rings. This can be due to the approximation with the
finite element solution. It was observed that the SIF values were converging as the domain is increased.
Thereby, the SIF was calculated by taking the average value of last three contours.
Figure 3.15 2D-shell FE model for contour integral calculation
38
XFEM- Stationary crack analysis
Similar to the contour integral method, it is also possible to compute the SIF value using 3D-XFEM
stationary crack analysis. However, the only difference is the way of computation. XFEM stationary
analysis requires the user to specify the enrichment radius, which by default is three-times the element
characteristic size. It should be noted that the crack tip should be surrounded by a sufficient number
of elements to obtain path independent contours. Using the default setting, SIF values are computed
along the crack front for a finite number of positions. These points are chosen automatically by the
software where the crack front intersects the element boundaries. This way the stationary crack
simulation is performed by inserting crack of different sizes. An example of SIF values along the crack
front is illustrated in Figure 3.13 for a crack size of 26.5mm. It is observed that the SIF values is not
constant along the crack front, therefore two extreme values (minimum: at the edges and maximum:
at the centre) were taken into consideration and plotted against the crack size as shown in Figure 3.16.
Thus, providing a bandwidth (range) of SIF value.
Figure 3.16 Stress intensity values for various crack size using different approaches
Figure 3.16 represents the variation of stress intensity factor value when plotted against different crack
sizes using different techniques. Although difference in magnitude, the SIF obtained from different
technique showed similar trend when compared to ISO 12108 standard. The difference increases as
the crack size increase. This is because the crack size approaches towards the critical crack size 𝑎𝑐 .
Moreover, it is observed that 2D-energy based is underestimated throughout with a maximum variation
of 5.81% compared to ISO 12018 standard. Furthermore, contour integral technique showed its
reliability throughout with a maximum difference of 1.92%. However, in the case 3D-XFEM, the
variation itself is quite significant between the max. SIF and min. SIF. Although XFEM-max showed an
overestimation of SIF, XFEM-min showed its promising sign of being reliable with a maximum difference
of 1.16%. Thus, in case of 3D-XFEM, XFEM-min (at the edges) can be the reliable technique in
predicting SIF.
39
3.4.5 Effect of LEFM parameters (2D-XFEM)
One of the major complexities of XFEM is that there are too many constants required to set-up a
problem. Since XFEM calculation is highly sensitive to LEFM parameters, it is important to study the
effect on the crack propagation. Therefore, an investigation is performed using different parameters.
For simplicity the simulation is carried out in 2D shell XFEM model and is limited to mode-I fracture (𝐾𝐼 )
using power law mix-mode behaviour for zero stress ratio (Rσ=0). Firstly, the dependency of mesh on
crack propagation rate and corresponding stress intensity factor range. Secondly, two parameters 1 ,
 2 used in defining the boundaries of Paris law regime. Lastly, the parameters GIC , am used in power
law mix-mode behaviour to compute the equivalent strain energy release rate.
Mesh dependency
Table 3.5 Representation of modelling detail and computational time
Mesh size Nodes Elements CPU time

(mm) (sec)
1.00 2949 2222 157.1
0.50 7064 4778 215.9
0.25 22477 14106 1318.9
Figure 3.17 Fatigue crack propagation rates obtained from the numerical simulation for different mesh sizes
A finer mesh will certainly help in capturing the singularity around the crack tip simultaneously
demanding high computational power. Therefore, it is important to find an equilibrium between the
accuracy of result and the computational effort.
An initial crack size of 16.8 mm is kept constant and an automated crack simulation is performed for
different mesh sizes to compute the stress intensity factor range and crack propagation rate. It was
observed that the final crack size varied significantly especially for 1 mm mesh size. The final crack size
for mesh size 1 mm, 0.5 mm and 0.25 mm were 25.8 mm, 26.8 mm and 26.5 mm respectively. As a
result, the unstable crack asymptote varied noticeably. The inconsistency is observed mainly for 1 mm
mesh size. Taking the computational time and accuracy of results into consideration, it is recommended
to use 0.5 mm mesh size to adequately simulate crack propagation.
40
Table 3.6 Illustration of output results obtained from different mesh size
Mesh Size Kthreshold Kc Ninitial Nfinal Crack size [mm]

[mm] [N.mm-1.5] [N.mm-1.5] [cycles] [cycles] Initial Final
1.00 643.786 1040.67 45641.3 240179 16.80 25.80
0.50 633.647 1087.79 66661.0 243414 16.80 26.80
0.25 645.311 1091.49 75698.3 244154 16.75 26.50
Paris Law regime
Gthresh G pl
1 = and  2 =
Gc Gc
The crack simulation is performed keeping constant Gc = 6.50 N/mm and correspondingly varying 1
and  2 .The Gthresh depends on various factors, one of which is elastic modulus E. When E=210500 MPa
was assigned, the strain energy released to fracture the first element was found out to be 1.729 N/mm,
which implies that the 1 ratio should be lower than 0.266 to initiate the crack propagation. Many
researchers have suggested a threshold value (see Figure 5.7) based on their investigation. If the
threshold value (𝐾𝑡ℎ𝑟𝑒𝑠ℎ ) and the fracture toughness value (𝐾𝐼𝐶 ) are known, 1 ratio can be determined.
Figure 3.18 Fatigue crack propagation rates obtained from the numerical simulation for different Gpl ratios
On the other side, four different 2 ratios were used ranging from 1.00 to 0.85. It was observed that
the final strain energy release (N/mm) before entering unstable crack propagation region comes out to
be 6.42, 6.00, 5.62 and 5.27 which is lower than 6.50, 6.175, 5.85 and 5.525 for  2 ratio 1.00, 0.95,
0.90 and 0.85 respectively. This illustrates the usage of  2 ratio which can be assigned as the fracture
toughness of the specimen ensuring unstable crack propagation beyond the limit.
41
Power law equation
am
Gequiv G 
= I  → GIC and am
GequivC  GIC 
The crack simulation is performed keeping constant 1 ratio = 0,  2 ratio = 0.85 and varying two
parameters critical strain energy release GIC and exponent am. These two parameters are used to
calculate the equivalent fracture energy release rate.
(a) (b)
Figure 3.19 Fatigue crack propagation rates obtained from the numerical simulation for different (a) GIc (b) am
Table 3.7 Representation of fracture toughness and final crack size for different GIc values
GIC [N.mm-1] Final crack size Final Strain energy release Fracture toughness
[mm] [N.mm-1] [N.mm-1.5]
5.5 25.8 4.65 990.20
6.5 26.8 5.27 1053.45
7.5 27.8 6.00 1124.15
Figure 3.19(a) depicts the range in crack propagation by varying 𝐺𝐼𝐶 . It is observed as the crack
propagation progresses the SIF range increase with increase in crack size. This parameter helps to
define the fracture toughness of the material ensuring unstable crack propagation beyond the limit.
Furthermore, the final failure crack size is changed correspondingly. Table 3.7 illustrates the fracture
toughness (plane stress situation) and respective final failure crack size.
Table 3.8 Representation of fracture toughness and final crack size for different am values
am exponent Final crack size Final Strain energy release Fracture toughness
[mm] [N.mm-1] [N.mm-1.5]
1 26.8 5.62 1087.79
2 27.3 6.00 1124.15
3 27.8 6.42 1162.68
42
Figure 3.19(b) illustrates the range in crack propagation when 𝑎𝑚 exponent is varied for constant 𝐺𝐼𝐶
6.5 N/mm. The 𝑎𝑚 exponent has a significant influence in computing GequivC. As a result, final crack size
is varied by 0.5 mm for every increase in the exponent. Table 3.8 illustrates the fracture toughness
(plane stress situation) and respective final failure crack size.
With mesh size 0.5 mm, the difference is realised. Moreover, it is recommended to use finer mesh to
precisely evaluate the above effect. Nevertheless, this investigation performed on various LEFM
parameters allowed to explore the capabilities and understand the limitations of XFEM.
43
44
Orthotropic Steel Deck Specimen Result and Discussion
Chapter 4
ORTHOTROPIC STEEL DECK (OSD) SPECIMEN
4 Orthotropic Steel Deck Specimen
45
Orthotropic Steel Deck Specimen Result and Discussion
46
Orthotropic Steel Deck Specimen General
4.1 General
Orthotropic Steel Decks (OSDs) are widely used in various types of steel bridges due to its benefits of
light weight, high load bearing capacity and speedy construction. Over the past decades, many
improvements have been achieved in various aspects of design, manufacturing, inspection and
maintenance. Thereby, the structural behaviour of such bridge decks has been significantly enhanced.
However, fatigue still remains its predominant problem. This is mainly due to numerous welding
operations and its complexity involved in OSD. As a result, such bridge decks suffer from many sensitive
crack locations. Moreover, various fatigue cracks were detected in recently built OSD [3], which proves
the lack of understanding of the fatigue behaviour. Therefore, many researchers [2] have tried to
investigate the fatigue behaviour of OSD through experiments. However, performing only experiments
may not be a cost-effective solution. Therefore, it is necessary to combine the experimental data with
numerical approaches and preferably assuming basic material properties to predict behaviour of critical
detail.
Numerical model based on fracture mechanics approach can be used to analyse fatigue crack
propagation and has already shown its reliability. Particularly, the use of LEFM models have several
advantages as it significantly reduces the requirement of experiments. Commercial software such as
ABAQUS® incorporates XFEM techniques to model discontinuities as an enriched feature. Using XFEM,
it is possible to simulate automated crack propagation by inserting the crack into the model.
The main objective of this chapter is to predict the fatigue crack growth in OSD using XFEM-model. The
set of analyses aims to simulate in an adequate manner according to the experiment [25]. Prior to
automated XFEM simulation, a set of FE analyses are performed to validate the numerical model as per
the test setup by evaluating the vertical deformation, longitudinal strain distribution and hotspot stress
based on static cyclic loading sequence. Moreover, the simulated results will be correlated with the
beach marks measurement derived from the fatigue experiment to determine the Paris law constants
C and m.
47
Orthotropic Steel Deck Specimen Numerical simulation of fatigue- FEM
4.2 Numerical simulation of fatigue- FEM
4.2.1 Experimental setup [25]
Wim Nagy [25] described a series of a fatigue test on orthotropic steel deck focusing on stiffener-to-
deck plate detail. The main aim of his experiment was to adequately represent existing OSDs through
a small specimen with possible realistic boundary conditions. As a result, the following geometry and
boundary conditions were adopted illustrated in Figure 4.1
(a) (b)
Figure 4.1 (a) Dimension of the OSD specimen [25] (b) Strain gauge location at the stiffener-to-deck plate connection [25]
The test specimen consists of one closed stiffener and a deck plate extended to the right by 300 mm
and to the left by 150 mm. A closed stiffener of 275 mm high and 6 mm thick is welded to a 15 mm
deck plate. The upper width of the closed stiffener is 300 mm while the lower soffit is 150 mm with 24
mm of radius. The deck plate is further extended to 40 mm and 50 mm from the left and right support
respectively. The left support is fixed whereas the right support is roller (pinned). The load is situated
at 70 mm from the right welded connection between the deck plate and the stiffeners.
Figure 4.1(b) describes the strain gauge pattern used in the fatigue test. Since the hot-spot stresses
are computed from the linear extrapolation from the stress measured at a distance of 25 mm and 50
mm from the stress location, the strain gauges were placed at that location on either side of the
corresponding plates.
Figure 4.2 illustrates the loading and support conditions in detail. The left support consists of fixed
bolted connection whereas the right support is a free roller bearing with rectangular steel bars of 60
mm high and 40 mm wide. For loading, hydraulic jack system INOVA actuator AH200-200 is used.
48
Figure 4.2 Detail of the loading and support conditions [25]
The cyclic loading sequence used in the static load test was different from the loading sequence used
in the fatigue load test. In case of static load test, the load is gradually varied from 40KN to -40KN with
an increment of 10KN held for 10 seconds before the next increment. However, a periodic cyclic loading
sequence was used in fatigue test which varied from 0KN to -31KN with frequency 𝑓 = 1 𝐻𝑧 . A negative
load corresponds in pushing the hydraulic jack system or a tensile load cycle at the weld toe.
49
4.2.2 Development of FE model
Geometry
A full-scale FE model is built based on the dimensions and boundary conditions specified in Figure
4.1(a) for a length of 400 mm. FEM calculations can sometime be time-consuming and can utilize a
huge amount of computational power for a large model, the FE model was simplified. The OSD
specimen model was developed in a combination of shell and solid elements. The solid elements were
used in the welded connection between the deck plate and the stiffener where the crack investigation
has to be carried out and the shell element were used in the remaining part. To ensure a rigid
connection between these two parts, the edge surface of the shell was constrained to face region of
the solid using shell-to-solid coupling. Since it is not possible to incorporate line-load in three-
dimensional geometry in Abaqus®, a reference point (RP-1) was incorporated which is kinematically
coupled in all the directions to a straight line on the surface and cyclic load is applied on it.
Fixed
Pinned roller (Uy=0)
400 mm
■ Shell part ■ Solid part
Figure 4.3 FE model: Interactions and boundary conditions
The displacement and rotation of the left support are restrained in all the three orthogonal x-, y- and
z-direction. Whereas, the displacement is fixed in y- and z-direction allowing rotation in all the
directions. In both cases, these conditions are applied to the edges of the top and bottom surface
(Figure 4.3).
Material property
The elastic material properties were assigned to the FEM model as: Young’s modulus E=210000 MPa
and Poisson’s ratio υ=0.3. Furthermore, the fracture contact properties and Paris law formulation will
be discussed under LEFM implementation (section 4.4.2).
50
Mesh
3D tetrahedron elements are easily applicable to almost every structure and can be an ideal choice for
complex structures. One of the major disadvantages of using tetrahedron elements is the locking
problem. In fact, in case of bending, the shear should be zero or negligible but the inconsistent terms
in the interpolation functions of linear elements make the shear strain much different from zero.
Thereby, the non-zero artificial shear strains absorb more energy leading to a stiffer element
consequently. To alleviate the shear locking issue standard high-order (for instance quadratic) elements
can be used. Therefore, the solid part (enrichment region) was modelled using a 10-node tetrahedron
(C3D10) with quadratic geometrical order of mesh size 2.5 mm. A numerical model with finer mesh can
be time-consuming, therefore a variable mesh is used for non-enrichment region. Solid part (non-
enrichment) was modelled using an 8-node linear brick with reduced integration (C3D8R) of average
mesh size 5 mm whereas a 4-noded shell element of 10 mm mesh size is adapted in the shell part
(Part-3). Meanwhile, the incompatibility mesh (interfaces between a tetrahedron and hexahedral) was
automatically generated using tie-constraints.
Figure 4.4 XFEM-model: Mesh quality
There are three types of element adapted in meshing the XFE model depending upon the shape and
function in different regions is summarised in Table 4.1.
Table 4.1 Meshing details of XFE model
Part Region Element type Mesh size

Part-1 Enriched solid 10-node quadratic tetrahedron (C3D10) 1.00 mm
Part-2 Non-enriched 8-node linear brick with reduced integration (C3D8R) 5.00 mm
solid
Part-3 Non-enriched 4-node shell with reduced integration (S4R) 10.0 mm
shell
51
Orthotropic Steel Deck Specimen Output
4.3 Output
(a) (b)
Figure 4.5 Deformation (a) Magnitude U (b) Vertical U2
(a) (b) (c)
Figure 4.6 Strain Distribution (a) Max. Principal strain distribution (b) Strain distribution in x-direction (c) Strain distribution
in y-direction
(a) (b) (c)
Figure 4.7 (a) Von-Mises stress distribution (b) Stress distribution along x-direction (c) Stress distribution along y-direction
Note: All the output results presented in this section are obtained from the static load simulation at -
40 KN.
52
4.3.1 Results and Discussion
The reliability of FE model of defining fatigue behaviour depends upon the quality of the numerical
model. Thus, a set of FE analyses are performed to validate the numerical model as per the test setup
by evaluating the vertical deformation, longitudinal strain distribution and hotspot stress based on static
cyclic loading sequence.
Comparison: Vertical deformation
Figure 4.8 Comparison of the vertical deformation of FEM with test results
Figure 4.8 compares the results of vertical deformation of FEM simulation with the test data [25]. The
simulated results showed a similar trend when compared with the test result. However, it can be noticed
that the numerical vertical deformation is underestimated throughout. The difference increases with
the increase in load (compression or tension) and goes up to a maximum difference of 9.2% at 40KN.
The reason can be due to its configuration. In case of FEM, the results are those in vertical direction
while those of the hydraulic jack system is according to the axle of the hydraulic jack itself which implies
that if the hydraulic jack in setup was slightly inclined, this will result in large deflection due to its
length. Another concern can be due to the imperfection in distribution of line load in the longitudinal
direction. It is possible that more pressure is transferred to the edges or in the middle of the test
specimen. Nevertheless, the simulated results were in roughly good agreement with the test data.
53
Comparison: Strain measurement
(a) (b)
Figure 4.9 Comparison of longitudinal strain distribution between FEM and test results (a) Deck plate (b) Stiffener
The comparison of strain distribution along the longitudinal direction between the FEM simulation and
the test result at -40KN of the applied cyclic load is illustrated in Figure 4.9. The simulated strain values
(z-direction) are obtained at a distance of 25 mm away from the weld toe in both the plates while in
the fatigue test, the strain was continuously recorded at specific locations in the longitudinal direction.
When the simulated results are compared with test results, it is observed that the peak appears at one-
quarter of the specimen in the both cases which can be due to the distribution of line load. However,
there is a difference in the value which can be explained through the contact property. Since the FE
model does not take friction (contact property) into account. Therefore, all the results from the strain
gauges are arranged on an independent y-axis to able to shift the data points to the curve of the FE
model for comparison. The difference in the magnitude is approximately about 7.8% in the case of
deck plate whereas in case of the stiffener, the difference is much higher. Nevertheless, the simulated
strain results showed a good correlation with the test results along the longitudinal direction for the
respective plates.
54
Comparison: Hotspot stresses
(a) (b)
Figure 4.10 Comparison of hot-spot stress derived from FEM with the test results at weld toe (a) Deck plate (b) Stiffener
The hotspot stresses are derived at the weld toe in accordance with [26] by linearly extrapolating the
stresses obtained at a distance of 25 mm and 50 mm away from the stress location. This methodology
is used in FEM as it may lead to inaccurate stress peak result due to singularity (discontinuity). The
results of the hot-spot stresses obtained at two critical locations from FEM are compared with test
results illustrated in Figure 4.10. The left graph (Figure 4.10 (a)) represents the stresses in the deck
plate towards the weld toe location. It is observed that the line of extrapolation almost coincides with
the test result. However, the difference in the hotspot stresses can be due to the use of linear
tetrahedral elements. These elements have one constant strain which can lead to the discretization of
error. Moreover, an incompatibility of mesh is realised at the interface between the tetrahedral elements
and the brick elements. On the other side, the right graph (Figure 4.10 (b)) represents the stresses in
the stiffener towards the weld root location. In this case, the simulated results along with linear
extrapolation line perfectly matches with the test result but slight difference in the magnitude. The
possible explanation of this inaccuracy can be due to the imperfection in the geometry of the weld. As
per the recommendation [27] , there should be a gap of maximum 2 mm in rib-to-deck welded
connection at the root of the weld. However, this imperfection will hardly influence the result, if
compressive stress exists at the location. Nevertheless, the simulated result showed a good agreement
with the test data and this method of determining the hot-spot stress showed its reliability.
55
Orthotropic Steel Deck Specimen Numerical simulation of crack propagation- XFEM
4.4 Numerical simulation of crack propagation- XFEM
4.4.1 Fatigue Test [25]
In the fatigue test, crack evolution was determined through beach mark measurement. These beach
marks were measured at the location with the highest crack propagation in depth. Furthermore, the
global crack shape resembles an elliptical shape. Furthermore, the crack not always originated from the
middle of the specimen but sometimes at a quarter distance from the edge.
Figure 4.11 Longitudinal crack geometry during crack propagation for test specimen [25]
Based on the results carried out from the fractographic analysis, the initial crack length was estimated
using SEM (Scanning electron microscope) while the final length was determined through the ductile
tensile fracture. The initial and final crack length was found out to be 288.25 𝜇𝑚 and 2173.21 𝜇𝑚
respectively.
4.4.2 Development of XFEM model
While developing the XFEM-model for automated crack propagation, some assumptions were made
based on the fatigue experiment (section 4.4.1). Firstly, a semi-elliptical initial flaw was assumed shape
with half-length 𝑎 of 0.3 mm along the minor axis and a half-length 𝑐 of 0.6 mm along the major axis.
Based on the fractographic results from the fatigue experiment, the size of the crack varies around 0.3
mm. The choice of the initial crack size is extremely sensitive to the fatigue life predictions. Often, an
initial crack length is chosen between 0.1 mm and 1 mm [28]. Since the longitudinal stiffener is welded
from only one side to the weld, and even the level of penetration is questionable, the initial elliptical
crack length can go up to 1 mm and 0.5 mm in the longitudinal and transversal direction respectively
[28]. Secondly, the crack is assumed to be originating from the centre of the specimen. Thereby, the
initial flaw was positioned at the weld toe perpendicular to the deck plate (Figure 4.12
Figure 4.12).
56
` 0.3
mm
1.2 mm
Figure 4.12 XFEM model: Location and definition of the initial elliptical shape
LEFM implementation
VCCT was used in the XFEM-based linear elastic fracture mechanics for crack propagation analysis using
the direct cyclic approach with a time increment size of 0.05 per cycle. The direct cyclic load simulation
is based on the periodic function (equation 2.12) and the parameter used is tabulated in Table 4.2.
Rσ Load (N) A0 A1 B1 to 𝛚
0.0 31000 0.5 0 0.5 0 2𝜋
The direct cyclic approach is combined with the Paris law crack growth to simulate the crack
propagation. These fatigue crack growth rates are evaluated based on assigned VCCT parameters
(Table 4.3). The crack propagation appears when the energy available for the crack is high enough to
overcome the fracture resistance of the material. Since ABAQUS® analyses the fracture by the strain
energy criterion approach, the Paris law parameters C3 and C4 were calculated assuming plane stress
situation from equation (2.8) and (2.14) and are listed in Table 4.3.To ensure the start of crack growth
process, material constants C1 and C2 were kept negligible as 0.001 and 0 respectively. Once the onset
of the fatigue crack growth is satisfied (equation 2.13), the crack propagation rate can be computed
based on the fracture energy release rate (equation 2.14).
Power law mix-mode model is selected for evaluating the equivalent fracture energy release rate
represented in the equation (2.15) because of its simplicity in the relation of different modes of fracture.
It should be noted that the fracture toughness depends upon the temperature, steel quality and loading
frequency. The temperature plays a vital role in determining fracture toughness, because the steel
57
becomes more brittle at low temperatures, resulting in a lower fracture toughness. The test results [29]
for the fracture-toughness KIC of A588 structural steel grade varied from 30 𝑀𝑃𝑎 √𝑚 and 67𝑀𝑃𝑎 √𝑚.
This scatter can be possibly used, as the S355 steel grade was used in fatigue test. Therefore, this data
was taken as the base of this study and applied to the XFEM-model assuming equal fracture modes
tabulated in Table 4.3.
Table 4.3 Critical energy release rate Gc and Paris constant
Paris law constant (XFEM) Critical energy release rate 𝑮𝒄 (Nmm-1) Exponent
C3 C4 Mode I Mode II Mode III αm αn αo
12.99E-06 1.5 11.9 11.9 11.9 1 1 1
58
4.4.3 Output
Figure 4.13 display the output of automated XFEM simulation and crack growth mechanism. The crack
initiating from the weld toe propagated in both longitudinal and vertical direction. It should be
remembered that the direction of crack growth is governed by the distribution of SIF at the crack front.
As the shape of the initial flaw was assumed to be semi-elliptical, the growth followed almost in elliptical
fashion. Figure 4.13 (a) displays the crack growth at different stage in propagation. As the crack
propagated in depth, three parameters namely crack size 𝑎, crack length 2𝑐 and corresponding number
of cycles 𝑁 were precisely noted. Figure 4.13 (d) displays the simulated crack front dimension after
2.30 x 105 cycles.
(a) 𝑁𝑓 = 1.2 x 105 cycles 𝑁𝑓 = 2.0 x 105 cycles 𝑁𝑓 = 2.15 x 105 cycles
cycles
(b) (c)
7.5 mm
15 mm
7.5 mm
Initial crack flaw
(d)
Figure 4.13 (a) Stages of crack propagation displayed as STATUSXFEM output (b) XFEM crack simulation including the initial
semi-elliptical crack (c) Side view corresponding with the weld toe crack simulation (d) Crack front dimension
59
4.4.4 Results and Discussions
Figure 4.14 displays the results of fatigue crack growth obtained from XFEM simulation are compared
with the test results [25]. The test results were derived from the beach marks measurement in 11
cycles intervals (Appendix B). Assuming the final failure when a crack size is one half of the deck plate’s
thickness [26], the results of XFEM simulation were calculated until the crack propagated to 7.5 mm in
depth (Figure 4.14). The shape of fatigue crack propagated keeping the elliptical shape due to the use
of tetrahedron elements the corresponding SIF distribution along the crack front of the assumed crack
shape.
Figure 4.14 Fatigue crack growth from XFEM simulation is fitted with test results in two sequence
There are two different sequences considered to define the curve fitting. The first sequence is based
on all the simulated crack results including the initial crack size whereas the second sequences is based
on all the simulated crack results excluding the initial crack size. After a crack depth of 5 mm the crack
growth seems to be arrested. The possible reason can be due to higher stress redistribution at the
crack tip due to which the crack growth shifted in the longitudinal direction. Later, the crack growth
becomes unstable after 7.5 mm resulting in irregular crack propagation.
From Figure 4.14 , it has been observed that the fitted curve of sequence 1 (includes initial crack size)
overestimates the number of cycles to failure. Moreover, the final length is conservative and can be
explained by the assumed size of the initial crack. Since the accuracy of the crack propagation analyse
is extremely sensitive to the initial crack size, the conservative result in sequence 1 can be explained.
Thereby, it is not desirable that the Paris constants depend on the size of the initial flaw. As a result,
the initial crack size is not considered in the sequence 2. It is noted that sequence 2 resulted in the
good correlation with the beach mark measurement and the calibrated Paris law constants C comes out
to be 55 % lower when compared with recommended IIW standards [30]. Hence, sequence 2 can be
used in estimating the number of cycles required the crack to reach the deck surface. Furthermore, this
curve fitting (sequence 2) can be used to compute the initial crack size by extrapolation.
60
Fatigue life Assessment: Suurhoff Bridge Numerical simulation of crack propagation- XFEM
Chapter 5
FATIGUE LIFE ASSESSMENT: SUURHOFF BRIDGE
5 Fatigue life Assessment: Suurhoff Bridge
61
Fatigue life Assessment: Suurhoff Bridge Numerical simulation of crack propagation- XFEM
62
Fatigue life Assessment: Suurhoff Bridge General
5.1 General
5.1.1 Motivation
Post world-war, many orthotropic steel bridges have been built between 1960 and 1980 across Europe,
especially in the Netherlands. Since these bridges were not designed to withstand the current traffic
intensities and are therefore prone to fatigue damage. Many localised failures have developed in welded
steel bridge components due to fatigue crack propagation which eventually lead to brittle fracture. The
amalgamation of low steel quality, high traffic intensities and suboptimal weld detailing can create
fatigue issues in existing bridges that could compromise the structural integrity of the bridge. Numerous
fatigue cracks are found in the welded connection which are susceptible to fatigue crack propagation
only after they came into function. As the infrastructure ages, the costs of renovation and maintenance
escalates and are becoming significant to the continued service. As such, no set of norms can
adequately ensure the safety and reliability of all the existing structures [31]. As a result, periodic
inspection is being executed at regular intervals to accumulate the damage.
5.1.2 Problem description
At this moment, Rijkswaterstaat is busy with the detection, reparation and renovation of the existing
steel bridges in Netherlands. Several fatigue crack problems are being regularly inspected in Suurhoff
Bridge, Rotterdam. One such type of fatigue crack is the welded connection of the deck plate to the
trapezoidal closed longitudinal closed ribs between the cross-beams. This detail is often decisive for the
fatigue behaviour of the OSD because of high stress ranges and direct wheel loading. The closed ribs
constrain the transverse deformations of the deck plate making this detail prone to fatigue failure [32].
Although these cracks do not endanger the overall structure, but the in-situ repairs are difficult and
expensive. In 2016, a crack length of 230 mm was detected in the stiffener-to-deck plate connection
at mid-span between two cross-beams using TOFD method. The crack propagated through the deck
plate originating from the weld root. These types of cracks are generally considered dangerous as they
are not visible when the crack starts to penetrate through the deck plate because of the location of the
crack initiation. Moreover, these cracks already cause significant damage before the crack front reaches
the surface. At this stage, the crack is not stable and propagates proportionally. Moreover,
Rijkswaterstaat has set a permissible limit for deck plate crack length of 500 mm [33]. At this moment,
these cracks must be repaired immediately.
To verify this issue, fracture mechanics can be an ideal choice since it can be used to model and analyse
fatigue crack propagation till subsequent failure and correlate the results with inspection data. This
approach is a reliable alternative, especially when the S-N curve-based calculation procedure does not
predict enough structural capacity, consequently avoiding unnecessary strengthening of the detail.
However, analyses based on fracture mechanics requires high computational effort and a detail insight
of geometry, material condition. This method allows the concerned authority to choose between the
renovation interval and the preventative strengthening.
63
5.1.3 Suurhoff bridge description
The Suurhoff bridge is a beam-girder bridge with an orthotropic steel deck, situated over the
Hartelkannal in Rotterdam, Netherlands. It spans 232.75 [m] and is the last bridge of A15 national
highway before reaching the Maasvlakte. The bridge is a combination of a steel girder bridge and
bascule bridge. The movable part is located on the north side of the Hartelkannal. There is a connection
of slow traffic on the eastern part of the bridge and a double-track railway line over a separate steel
cable-stayed bridge is situated adjacently on the western side. Figure 5.1 depicts an overview of the
location of the Suurhoff bridge.
In early 1970s, Suurhoff bridge was built at the time when the Hartelkanaal was dug. The bridge with
2x2 lanes opened for traffic in 1972. Next to the traffic bridge, a railway bridge was built in 1973. Both
the bridges were named after Minister Suurhoff (1905-1967).
Figure 5.1 An overview of the location of the Suurhoff bridge
The steel road bridge consists of two parts: fixed and movable part which can be sub-divided into three
segments as SV01, SV02, SB01. The movable part (SB01) is in between the fixed parts (SV01 and
SV02). Thereby, the total length of 232.75 [m] is divided into 150.80 [m], 33.90 [m] and 48.05 [m] for
SV01, SB01 and SV02 segment respectively (see Figure 5.2).
64
Figure 5.2 An overview of the global dimensions of the bridge (Top View)
The width of the bridge is 24.12 [m] consisting of 2x2 traffic lanes with a pedestrian lane on the eastern
side as shown in Figure 5.3. The lane configuration has been used since the operation of the bridge in
1972. For current use, lane no. 1 and 4 are the slow lanes and lane no. 2 and 3 are fast lanes.
Figure 5.3 Cross-section of the fixed part of Suurhoff bridge (SV01 and SV02)
The OSD of the fixed part of bridge (SV01 and SV02) consist of a 10 mm deck plate with 50 mm asphalt
surfacing is welded to 5 mm thick trapezoidal stiffeners. The longitudinal stiffeners are 320 mm high
and 300 mm wide on top and have a width of approximately 200 mm with radius of 17 at the lower
soffit. The spacing between the two consecutive longitudinal stiffeners is equal to 300mm. An overview
of these dimensions is given in Figure 5.4. The spacing between the cross-beams is 4540 mm.
65
Figure 5.4 Details of the investigated bridge (units: millimetres) - Cross-section of the detail
The inspection was executed between trough number 24 and trough number 27 to accumulate the
fatigue crack in the deck plate originating from the weld root using TOFD method in the fixed part
(SV01) of the bridge. Therefore, this research focusses on the fatigue cracks originating from the root
of the welded connection between the deck plate and longitudinal stiffener in the span between the
cross-beam of SV01 segment of the bridge.
Fatigue Crack
10
Figure 5.5 Location of the fatigue crack (units: mm)
66
Fatigue life Assessment: Suurhoff Bridge Outline: Fatigue life Assessment
5.2 Outline: Fatigue life Assessment
The main objective of the research is to develop a numerical model and correlate its simulated results
with the inspection data and/or existing numerical model provided by Rijkswaterstaat in fatigue
assessment. Later, the model will be used in predicting the permissible limit of deck plate i.e. 500 mm
crack length. In developing the numerical model, some parameters were studied from the literature
and implemented in the model. Further, the analysis is performed on fatigue assessment aiming to
predict the crack initiation and crack propagation period. Thereby, this report is divided into two
segments namely fatigue crack initiation period and fatigue crack propagation period. An outline of this
research is illustrated in Figure 5.6.
Suurhoff bridge Literature data

Global structure (SV01) Fracture Toughness
Load history Paris Constant
Fatigue crack location Wheel type and Position
Numerical model
Material properties
Boundary conditions
Loading condition
Mesh
(Hot-spot stress method) (LEFM method)

𝐾𝑡 (Stress concentration factor) Analysis 𝐾 (Stress Intensity factor)
(SN-curve) (Paris Law)

Crack initiation period Evaluation Crack propagation period
(Fatigue limit) (Final failure)
RWS Numerical Model Verification TOFD measurement
Prediction
Permissible limit deck plate crack
(500 mm)
Figure 5.6 Outline of the research
67
Fatigue life Assessment: Suurhoff Bridge Literature Data
5.3 Literature Data
5.3.1 Material Parameters
Fracture toughness
The steel grade of the OSD of Suurhoff bridge is S355. However, it should be noted that the fracture
toughness depends upon the temperature, quality of steel and the loading frequency. From the
comprehensive collection of data by [29], the value of plane-strain fracture toughness (𝐾𝐼𝑐 ) varied from
27 𝑀𝑃𝑎 √𝑚 and 60 𝑀𝑃𝑎 √𝑚 for structural steel (A36) and from 30 𝑀𝑃𝑎 √𝑚 and 67𝑀𝑃𝑎 √𝑚 for structural
steel (A588).
The temperature plays a vital role in determining fracture toughness, as the steel becomes more brittle
at low temperatures, resulting in a lower fracture toughness. Particularly for bridges in the Netherlands,
the lowest possible service temperature is assumed to be -20 °𝐶. Taking in account all the above factors,
the fracture toughness is taken 𝐾𝐼𝑐 = 50 𝑀𝑃𝑎 √𝑚 as the base of this study.
However, it is advisable to either use existing test results or perform a new fracture toughness test to
obtain more accurate value of fracture toughness.
68
Paris constants
Material dependent parameters C and m determines the quality of fatigue life prediction, when using
Paris Law for describing the crack propagation. These parameters should be determined preferably
based on field measurement or material test [34]. Since material tests of the base metal (deck plate
and trough stiffener) are not available, the Paris constants used are based on the available literature.
Many researches have tried to predict the material constant through fatigue experiments (Table 5.1).
Unfortunately, these parameters vary significantly. Furthermore, a distinction should be made between
the fatigue growth behaviour in the base metal, in the heat affected zone (HAZ) and in the weld metal
for detail numerical analysis. Due to welding, different sub-layers with several microstructure changes
the behaviour of the base metal. As a result, Paris law constant can differ at such location. The
International Institute of Welding (IIW) [30] makes a distinction between the weld material and base
material and recommended a value of C equals to 3x10-13 for the base material and 5x10-13 for the weld
bead. In this research, the material constant of base-metal recommended by IIW is assumed constant
in the OSD model as the crack encounters a minute region of HAZ while originating from the weld root
and propagating in the deck plate.
Nevertheless, multiple standards/authors have proposed a safe value for Paris constants that have been
used in the past is summarised in Table 5.1 and Figure 5.7.
Table 5.1 Paris constants proposed by different authors for structural steel
Source C m ΔKth R Reference

British Standards (2007) 5.21 x10−13 3 63 0 [35]
Kuhn et al. (2008) 4.00·10−13 3 170 0 [34]
Maljaars et al. (2012) 3.00·10 −13
3 80 0 [36]
Bignonnnet et al. (1991) 4.25 x 10-13 3 50-71 -1 [37]
Hobbacher base metal 3.00·10−13 3 63 0 [38]
(2015) weld metal 5.21·10−13 3 63 0
*For dN in mm/cycle and ∆K in MPa√mm.
Figure 5.7 Crack propagation curves based on literature
69
5.3.2 Loading Parameters
Fatigue is a time-dependent phenomenon. Although the fatigue limit state is an ultimate limit state, the
approach to design or verify is different because the failure is associated with the cumulative damage
caused by repeated application of same levels of stress. Hereby, the ultimate strength is therefore
irrelevant, and the stresses must be based on the elastic stress analysis.
Wheel type-position (Transversal):
In a numerical study [39] performed on the welded connection to determine the most unfavourable
transverse position and type of the wheel load for weld root stress and the conclusion drawn are as
follows:
• The bending moments (Mx) are smaller in the case of wheel type B (FLM4 Eurocode) due to
larger dispersion of load over the surface.
• Wheel type A resulted in largest bending moment when the wheel load is in between the two
longitudinal stiffeners ( 𝑥 = −40 𝑚𝑚).
Position 𝑥 indicates the left edge of the wheel with respect to the left edge of the left stiffener-to-deck
plate connection (Figure 5.8).
Figure 5.8 Influence lines of the bending moments in the deck plate and stiffener for an axle load of 1N using wheel type A
and B. [39]
Since the geometry of the welded connection between the deck plate and stiffener resembles to the
geometry of the numerical model, above conclusions can be used as the starting point and possible
explanation of type of wheel selection. It is observed that wheel type B results in the least bending
moment whereas, wheel type 𝐴 and wheel type 𝐶 are dominant in creating higher contact stresses at
the weld location.
The usual practice for fatigue assessment is to assume high tensile residual stresses generated from
the welding and neglect the mean stress effect. This results in the simplification of the entire stress
range (Δσ = σmax – σmin) is effective in terms of fatigue crack growth and not as a function of mean
stress. However, it is to be remembered that the fatigue crack growth depends on the loading sequence
(stress ratio R = σmin/σmax) and the frequency of the load.
70
In this research, wheel type A (NEN 8701) for reverse stress ratio (R=-1) is adopted in the model
situated at the middle of two longitudinal stiffener (trough no. 25 and 26) in the transversal direction
(see
Figure 5.4) and at the centre between the two cross-beams in the longitudinal direction ensuring most
unfavourable position for the weld root stress.
Number of cycles (N)
Fatigue load models were derived from the traffic measurements on Dutch highway bridges and
implemented in NEN 8701 [40]. According to NEN 8701 Article 5.2 (3) the following fatigue load models
(FLM) should be applied in the fatigue verification of steel road bridges:
✓ FLM 4a: set of ‘standard lorries’ (for materials primarily dependent on the stress ranges).
✓ FLM 5: based on recorded road traffic data.
Moreover, NEN 8701 Annex A.1 (2) recommended to determine the number of vehicles per location
based on observation or based on the category classification of NEN -EN 1991-2 [41].
According to the standardised tables in Annex A.2 from NEN 8701, the recorded traffic data is
categorised in three time periods as Opening (1972)-1990, 1991-2010 and 2011-future (2040) with
different sets of vehicles in each period. The recorded traffic data on slow lane in one direction of the
Suurhoff bridge is graphically represented in
Figure 5.9. The detailed traffic distribution is tabulated and can be found in APPENDIX C. The dynamics
effects and development in time are here excluded here (a load increase of 20% in 100 years is
estimated).
Figure 5.9 Recorded traffic numbers on the slow lane in one direction traffic [42]
It should be noted that the traffic increase is significant once the 2nd Maasvalakte becomes fully
functional in 2020 (see Figure 5.9). Beside original traffic record, two types of traffic extrapolation were
used. One based on NEN8701 [40], which estimates the traffic to be doubled in 100 years and the
second is based on traffic study performed by Rijkswaterstaat West-Nederland Zuid (WNZ). The latter
71
seems to be the underestimated especially after 2020, therefore NEN8701 extrapolation is used in this
research.
As the Suurhoff bridge (A15) serves an important link between the Maasvlakte and the western part of
the Voorne-Putten and Botlek, Europoort and Rotterdam, heavy loaded lorries are expected and
therefore long-distance traffic type with high vehicle load is considered. Based on NEN 8701
extrapolation, the recorded traffic data per period is summarised in Table 5.2.
Table 5.2 Summary of the recorded traffic number based on NEN 8701
Period Nobs/period Nobs/year Reference Year High Traffic (15%)

1972-1990 10582163 556956 1981 83543
1991-2010 20747690 1037384 2000 155608
2011-2040 64726301 2157543 2025 323632
Nobs: Traffic observations as number of cycles
72
Fatigue life Assessment: Suurhoff Bridge Numerical model
5.4 Numerical model
To study and assess the fatigue behaviour in the welded connection between the trough and deck
plate, a full-scale FE model (Figure 5.10) is developed in an adequate manner according to the cross-
section described in the Figure 5.4 from trough no. 24 to trough no. 27. The model consists of a deck
plate welded to the longitudinal stiffener spanning between two cross-beams and extended further to
half the distance between the cross-beams on either side. The FEM calculation of such model can be
time-consuming; therefore, the model is developed in combination of shell and solid elements. At the
crack location, the part is replaced by solid elements for XFEM calculation. To ensure adequate
connection between them, the solid face is constrained to shell edge using shell-to-solid coupling.
Moreover, the nodes of the longitudinal stiffener at the crossbeam position are tied in all the degree of
freedom to the nodes of the cross-beam using tie-constraint. A gap size of 1mm is been used in rib-
to-deck welded connection at the root of the weld.
Shell-to-solid
Tie-constraint
coupling
24 25 26 27
9080 mm
■ Solid part ■ Shell part ■ Shell (cross-beams)
Figure 5.10 Illustration of numerical model
73
5.4.1 Material properties
There are three types of material parameters required for LEFM calculations. Firstly, the parameters
defined in the elastic stage of the material i.e. Youngs’ modulus and Poisons ratio which were assigned
as 210000 MPa and 0.3 respectively. Secondly, the fracture toughness of the material which determines
the ability to resist fracture. Thirdly, the materials constants (C and m) from the Paris Law equations
which defines the rate of fatigue crack propagation. The latter two parameter types were adopted from
the literature of existing steel bridge and will be discussed in LEFM implementation section.
5.4.2 Loading conditions
Figure 5.12 shows the configuration of wheel load type A (220 mm x 330 mm) used in the static
analyses without considering the dispersion of load due to asphalt. In determining the hotspot stresses
(static analyses) several loads are used ranging from 21.5 KN to 45 KN. However, a periodic cyclic
loading sequence of reverse stress ratio is used in fatigue simulation varying from negative wheel load
(KN) to positive wheel load (KN). Figure 5.11 depicts the loading sequence for a unit wheel load. Four
types of wheel loads are used for automated crack propagation i.e. 40KN, 45KN, 50KN and 52.5KN. A
positive load corresponds to a tensile load cyclic at the weld root. The direct cyclic loading parameters
can be found out in Table 5.6.
Figure 5.11 Loading sequence for fatigue simulation
74
5.4.3 Boundary conditions
In this research, a symmetric boundary condition is adopted as shown in Figure 5.12. The translation
in x-direction is constrained along the deck plate boundaries and side edges of the cross-beams.
Similarly, the translation of longitudinal troughs at either side of the longitudinal edges is restrained in
z-direction. At two locations i.e., the bottom flange of the cross-beams and edges of the longitudinal
troughs, vertical translation in y-direction is restrained. The crack flaw in the model is situated at the
centre, the influence of boundary condition will hardly generate any error to the stresses near the crack
tip.
Z-Symmetry
(Uz=0, Rx=0, Ry=0)
X-Symmetry
(Ux=0, Ry=0, Rz=0)
Uy=0
Figure 5.12 Numerical model: Boundary conditions
75
5.4.4 Mesh
Figure 5.13 Numerical model: Mesh quality
The numerical model is built using shell elements excluding the crack region as shown in the Figure
5.13. The crack part (welded connection between deck plate and the stiffeners) of 400 mm is built with
solid 8-noded brick elements as highlighted in the above figure. Particularly, at the loading region and
crack location, a fine mesh should be used to accurately capture the stresses at the joint between the
deck plate and longitudinal stiffeners. Since hotspot stress method requires to compute the stresses at
4 mm and 10 mm from the joint, the element size was therefore kept as 2mm. The size of the element
is gradually increased to 25mm towards the cross-beams and 50mm in the remaining region to reduce
the total number of elements. Due to its simplified geometry, the shape of the element is kept
hexagonal throughout. For precision, 8-noded shell element (S8) of quadratic order was adopted at
the loading region.
Table 5.3 Meshing details of the numerical model
Region Element type Mesh size

Solid part 8-noded brick element (C3D8) 2.00 mm
Shell (Loading region) 4-node shell (S4) 10 mm - 25 mm
Shell (Remaining region) 4-node shell with reduced integration (S4R) 50 mm
76
5.4.5 Output:
A preliminary stress distribution is presented in this section mainly focussing at the weld root and aiming
to determine the unfavourable position, where the crack can be positioned.
(a)
(b)
(c)
Point of Extrapolation
for hotspot stress
Figure 5.14 (a) Stress distribution along the thickness of the deck plate (b) Stress distribution at the weld root along the
longitudinal direction (c) Stress distribution in the bottom part of the deck plate at the weld root (For wheel load = 45KN)
Note: All the output results presented in this section are obtained from the static load simulation for
wheel load 45KN. Stress distribution mainly focusses at the weld root at the centre span between the
cross-beams are obtained near the crack location.
77
Comparison with moving load
(a)
(b)
Figure 5.15 (a) Numerical model consisting of a moving load (b) Comparison of influence line between the cyclic load and
moving load based on Max. Principal stresses
Since the numerical model consist of a single cyclic load acting at one position (centre), it is important
to know the practical relevance by comparing with a moving vehicle. Therefore, the influence line based
on cyclic load 45KN (for zero stress ratio) is been compared with a similar wheel load moving at speed
of 100 km/hr. Figure 5.15 (b) depicts the comparison of principal stresses on the deck plate between
the two scenarios. Firstly, it is clear that the most unfavourable case of position of load is at the centre
(at 4540 mm) of the model. Secondly, the influence of load is localised, and it is mostly in region of
340 mm from the centre on either side for principal stress. From the comparison of peak principal
stress, it has been observed that cyclic load produces 2.12 times higher than the vehicle moving with
100km/hr speed. It recommended to model the moving load in predict the crack propagation.
Considering the computation effort require to model the moving load combined with XFEM calculation,
this research is limited to a cyclic load acting at a single position and excludes other dynamics factors.
78
Fatigue life Assessment: Suurhoff Bridge Fatigue crack initiation period
5.5 Fatigue crack initiation period
5.5.1 Fatigue detail category
The ‘FAT’ or detail category represents the fatigue stress range which gives the fatigue life at 2 million
cycles. The shape of the fatigue strength S-N curves recommended in the IIW document for structural
hotspot stress is similar to the direct nominal stress S-N curve consisting of a line with constant slope
(m) of 3 when plotted in log-log graph. The fatigue limit is defined for stress range at 5 million load
cycles below which the fatigue life is infinite for a constant amplitude loading.
This research focusses on the welded connection of the deck plate to the longitudinal stiffener at the
span between the cross-beams. The welding detail is designed based on the available standards.
According to the Eurocode 3 [41], a minimum weld penetration of 67%-75% should be achieved.
Moreover, based on Kolstein research [43], a nominal value of penetration of 80% is suggested. Despite
these recommended values, the manufacturer tries to achieve penetration as much as possible.
Thereby, when the weld is executed with care, the fatigue strength increases. Therefore, the fatigue
detail category for a single-sided fillet welded connection between the deck plate and longitudinal
stiffeners differs.
Figure 5.16 Fatigue strength curves for stiffener-to-deck plate connections
A reference value Δσc of 71 MPa is defined in the Eurocode NEN-EN 1993-1-9 [27]. However, a recent
fatigue test showed a reference value up to 140 MPa and proposed a detail category of 125 MPa for
cracking in the deck plate [43]. Furthermore, Dutch national annex NEN-EN 1993-2+C1/NB
recommended a value of 125 MPa for deck plate cracking originating from the weld root [44]. Moreover,
IIW [30] recommends a FAT class 100 for this detail if the stresses are derived using hotspot method.
For this research the fatigue life (crack initiation) prediction is carried out using hot-spot stress method.
Therefore, a reference value of 100 MPa is adopted and the corresponding fatigue limit (CAFL) is
evaluated as 73.68 MPa, ensuring no fatigue damage occurs below this stress range.
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5.5.2 Hot spot stress method
The investigation of crack initiation at weld root is evaluated based on hot spot stress method. Firstly,
the numerical model developed in this research is aimed to validate against the full-scale RWS existing
numerical model. Mainly, the hotspot stress is compared with the hotspot stress obtained by the existing
numerical model [42] developed by Rijkswaterstaat (RWS). The existing numerical model was
developed on a full-scale (fixed part of the bridge) using beams elements in combination with shell
elements (see Figure 5.17(a)) for recalculating the capacity of the existing bridges and to design
strengthening schemes for fatigue and static strength. Particularly, for fatigue verification the RWS
model consisted of one axle loading (type C axle of truck type 6) acting over 50 mm of hot asphalt
(E=0.5 GPa) in local fatigue assessment (see Figure 5.17 (b)). However, the FEM-model developed in
this research consist of a single wheel load (type C axle of truck type 6) acting over the deck plate.
(a)
(b)
Figure 5.17 (a) Overview of the RWS numerical model (Fixed bridge) [42] (b) RWS Fatigue verification model under single-
axle load C [42]
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Figure 5.18 displays the output results of principal stress of both the models. Moreover, the
comparison between the models is tabulated in Table 5.4.
FEM-model RWS- Model
Figure 5.18 Comparison of max. and min. principal stresses of different model
From RWS model, the hotspot stress derived at the root of the weld was 109.08 MPa by extrapolating
the stresses obtained at 5 mm (Point A) and 10 mm (Point B) away from the heel of the weld. On the
other hand, the hotspot stress derived using a single wheel load (wheel print C) by extrapolating the
stresses from 4mm and 10mm distance away from the weld root. The hotspot stress comes out to be
to be 113.84 MPa, which is 4.3% higher than the existing model. The difference in the magnitude can
be due to the dispersion of load from asphalt and difference in the points of extrapolation. Nevertheless,
the XFEM numerical model showed a good correlation with the existing numerical model.
Table 5.4 Comparison of hotspot stresses (MPa) based on different models
Numerical Model Load Max. Min. Max. Stress Max. Stress Hot spot
model (KN) Principal Principal Point A Point B Stress (MPa)
(MPa) (MPa) (MPa) (MPa)
FEM-model Wheel Load-52.5 123.9 -185.8 109.0 `101.8 113.84
RWS- Model Axle Load-105 160.4 -158.5 108.0 104.3 109.08
After the validation of simulated results, the numerical model is used to determine the hot-spot stresses
for different wheel loads with wheel print A throughout this research. It should be noted that the
determination of stress range is highly sensitive to fatigue life estimation. With a twice increase in the
stress range, the number of cycles can overestimate by 8 (23) times. Considering the importance of
accuracy, the stress range is evaluated using shell and solid models.
According to NEN 8701, the wheel load (type A and type C) ranges from 21.25 KN to 45 KN for high
traffic category. A comparison between their hotspot stress based on different approach for different
wheel loads is tabulated in Table 5.5. It is noted that the variation of the hotspot stress between the
solids and shell approach is constant (5.46 %) throughout from wheel load 21.25 KN to 45 KN. In
solids, the geometry of weld is well defined and variation of stresses in through-thickness of the deck
plate can be observed. This can be the possible reason between the difference of hotspot stresses in
solids and shell model.
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Table 5.5 Comparison of hotspot stresses (N/mm2) based on different approach
Model Approach Wheel load - 45KN (Max.) Wheel load - 21.25KN

(Min.)
Solids Model 108.46 50.84
Shell Model 114.40 53.63
Further, hotspot stresses are derived using the solid model for several wheel loads and plotted against
each other shown in Figure 5.19. Based on fatigue limit (see Figure 5.16), it is possible to segregate
the wheel load which are not contributing to the fatigue damage. It was found that the wheel load
more than 31 KN have an impact on fatigue life of the joint. With this analysis, the wheel load range is
reduced to four types: 35KN, 40KN, 42.5KN and 45KN for further investigation.
Figure 5.19 Hotspot stresses derived at the root of the wheel for various wheel loads
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5.5.3 Fatigue Life prediction
It would be conservative to consider only the maximum load (45KN) to evaluate the fatigue life. The
standard NEN8701 [40] provides the percentage of traffic based on the type of vehicle but does give a
clear distribution of traffic based on wheel load. Since the hotspot is computed for a wheel load,
therefore it was important to redefine the traffic distribution based on wheel load.
Figure 5.20 represents the distribution of traffic based on wheel load for period 1972-2010.
Figure 5.20 Wheel load frequency for 1972-2010 derived from the standard NEN 8701
It was found out that wheel load 45KN had maximum occurrence about 45.20 % and 42.5 KN being
the least of about 14.55% for period 1972-2010.Taking the frequency of the wheel load in account, the
fatigue life is evaluated using Palmgren-miner damage model (5.1).
k
ni
D= (5.1)
i =1 Ni
where 𝑘 different stress level, 𝑁𝑖 is the average number of cycles to failure, 𝑛𝑖 is the number of cycles
accumulated and 𝐷 is damage fraction (when reaches 1, failure occurs).
The fatigue life is predicted to be 1.91 million load cycles which is equivalent to 20 years. In other
words, it can be said that the fatigue crack initiated in the year 1992.
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Fatigue life Assessment: Suurhoff Bridge Fatigue crack propagation period
5.6 Fatigue crack propagation period
It is true that the accuracy of simulated fatigue crack propagation relies on various factors such as
LEFM parameters, weld root conditions and most importantly the used traffic load. It is often assumed
to consider the maximum axle load in predicting the fatigue crack propagation rate. However, this
assumption is safe and reliable in accessing the extreme situation. Furthermore, this assumption can
lead to underestimation of the remaining life of the structure. Thus, reducing the advantage of load-
bearing capacity of OSD. Therefore, it is important to reduce the degree of conservative assumptions
as much as possible and effectively utilise the capacity of numerical approach. Thereby, a range of
wheel load is considered to predict the crack propagation period.
To predict the fatigue crack propagation period, numerical simulation can be performed in two ways
using XFEM-model i.e. Stationary model and Propagating model. Stationary simulation refers to
inserting certain number of crack fronts in the weld location and evaluating stress intensity factor. On
the other hand, propagating simulation refers to evaluating the crack front using the full advantage of
automated XFEM simulation. The procedure to predict the crack propagation period using stationary
model is discussed briefly in this report but the final estimation to predict the crack propagation period
is performed using propagating model.
5.6.1 Development of XFEM model
Although FE model with shell elements reflects the real behaviour of the structure, a more detail model
is needed for studying the fracture crack growth behaviour. Therefore, the crack part was replaced with
solid elements and LEFM based VCCT contact property was applied to it. While developing the XFEM
model, certain assumptions were taken for numerical XFEM calculation. Firstly, the initial crack was
assumed to be semi-elliptical of 1 mm half-length along both the major and minor axes. The selection
of the initial crack flaw is extremely sensitive to the simulated results. The choice of the initial crack
flaw often depends upon the welding detail, used manufacturing technologies and the lifetime of the
structure. The initial crack length is generally selected between 0.1 to 1 mm [28]. Nowadays, the
manufacturer tries to manage a higher degree of weld penetration during welding. Taking into the
consideration of welding technology at the time of construction (1970s), the size of the initial crack is
selected. Secondly, the fatigue crack is assumed to originate from the centre of the mid-span between
the cross-beam. Therefore, the initial crack flaw was positioned at the weld root perpendicular to the
deck plate. The implementation of the initial crack flaw is well illustrated in Figure 5.21.
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1mm
2mm
Figure 5.21 Location and definition of the initial elliptical shape
5.6.2 LEFM implementation
VCCT was used in the XFEM-based LEFM for crack propagation analysis using the direct cyclic approach
with a time increment size of 0.05 per cycle. The direct cyclic load simulation is based on the periodic
function (equation 2.12) and the parameter used for different wheel load is tabulated in Table 5.6. In
this study, Power law mix-mode model is selected to determine the equivalent fracture energy release
rate represented in the equation (2.15) because of its simplicity in the relation of different modes of
fracture.
Rσ Load (KN) A0 A1 B1 to 𝛚
0.00 40 0 0 1 0 2𝜋
0.00 45 0 0 1 0 2𝜋
0.00 50 0 0 1 0 2𝜋
0.00 52.5 0 0 1 0 2𝜋
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5.6.3 Time-of-flight-diffraction (TOFD) measurement
On 8th October 2016, a semi-mechanized TOFD was conducted to inspect fatigue cracks in selected
areas on the east side, direction of the Maasvlakte of the Suurhoff bridge. TOFD equipment consisted
of two ultrasonic transducer which were positioned on either side of the weld between cross-beam no.
29 and 32. A crack length of 230 mm was measured at the west side of the trough no. 25 after the
asphalt was removed. The crack originated from the weld root and was reached the surface of the deck
plate. The placement of the ultrasonic transducer and corresponding TOFD data [45] can be found in
Appendix E.
Table 5.7 Crack dimension obtained using TOFD method
Location Start (mm) Length (mm) Depth (mm) Height (mm) Remarks
25 trough 8460 230 0.0 10.0 Surface
(West) Breaking
Figure 5.22 Representation of TOFD measurement [45]
5.6.4 Stationary model
From the TOFD measurement, the final dimension of the crack front can be obtained. Furthermore, the
crack front can be divided into certain number of equal intervals (da) including the dimension of the
initial crack flaw. In this way, the stress intensity factor at the tips of the various crack fronts can be
computed. This parameter is most important in predicting the crack growth and its lifetime using the
Paris Law equation (5.2).
da
= C.K Im (5.2)
dN
where 𝐶 and 𝑚 are the material parameters and ∆𝐾𝐼 represents the stress intensity factor range for
mode I cracks. Paris law constant 𝐶 and 𝑚 are adopted from the literature as discussed previously.
Knowing the stress intensity factor range for a different range of crack dimension (da), it is possible to
compute the required number of cycles (N) using equation (5.3) manually.
af
da
Nf =  C.K
ai
m
I
(5.3)
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Furthermore, geometrical dependent parameter f(a) can be evaluated from equation (5.4).
K I = f (a ). .  .a (5.4)
where ∆𝜎 is the stress range at the crack front and 𝑎 is the crack length. 𝑓(𝑎) is the geometrical
parameters which not only depends upon the crack length 𝑎 but also on the overall dimension of the
bridge. However, computing geometrical factor 𝑓(𝑎) is beyond the scope of this research.
5.6.5 Propagating model
It is to be noted that the stationary model gives limited information of number of cycles (N) as it is
derived only through stress intensity factor range for mode I fracture. However, in complex structures
such as OSD model, mode II and mode III can have significant effect in crack propagation. Therefore,
a second and more accurate method to evaluate the crack front including three modes of fracture using
automated crack propagation simulation. This method utilises the full advantage of XFEM possibilities
and even is able to determine the crack propagation path in a complex three-dimensional way.
Although, it requires a high computational effort, this method has the potential to evaluate the weld
geometry and its corresponding cracks. At the beginning, an initial crack is positioned in the model and
material parameters of fracture is applied to it. Later, the XFEM-model uses the Paris law to simulate
the crack propagation automatically. This simulation is carried out using fracture energy-based criterion
which uses the least energy to crack.
The crack growth is characterized by the Paris law, which relates the relative fracture energy release
rates to fatigue crack growth rate (Figure 2.11). These fatigue crack growth rates are evaluated based
on assigned VCCT technique. The crack propagation appears when the energy available for the crack
is high enough to overcome the fracture resistance of the material. Since ABAQUS® analyses the
fracture by the Griffith energy criterion approach, the Paris law parameters C3 and C4 were calculated
assuming plane stress situation see equation (2.8) and (2.14) listed in Table 5.8.
Table 5.8 Paris constants C3 and C4
Literature Data ABAQUS (XFEM)

C m C3 C4
Hobbacher (2015) 3.00x10−13
3 28.87x10−6
1.5
To ensure the start of crack growth process, material constants C1 and C2 were kept negligible as 0.001
and 0 respectively. Once the onset of the fatigue crack growth is satisfied (see equation 2.13), the
crack propagation rate can be computed based on the fracture energy release rate (equation 2.14).
The material fracture property was implemented using Power law mix-mode behaviour illustrated in
Table 5.9.
Table 5.9 Critical energy release rate Gc
Critical energy release rate 𝑮𝒄 (Nmm-1) Exponent

Mode I Mode II Mode III αm αn αo
Numerical model 11.9 11.9 11.9 1 1 1
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5.6.6 Output
(a)
(b)
10 mm
(c)
(d)
(e)
50 mm
Figure 5.23 (a) Stages of crack propagation displayed as STATUSXFEM output variable (b) Crack propagation mechanism (c)
Side view corresponds with the weld root (d) Isometric view of the crack on the surface of the deck plate (e) Deck plate crack
length for wheel load 45 KN
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5.6.7 Result and Discussion
Due to reversed loading, the weld root is affected by both tensile and compressive stress. However, it
should be noted that compressive stresses have no effect on the crack propagation. As a result, all the
negative SIF values are set to zero in LEFM. In the absence of residual stresses, the crack propagation
is limited to half-thickness of the deck plate until where the tensile stress is present for zero stress ratio.
As the crack front the neutral axis of the deck plate, the crack propagation stops. This is one of the
main reasons behind the consideration of reverse load cycles. It is assumed that fatigue resistance
becomes critical when the crack has propagated through the deck plate and reaches the surface.
However, the fatigue crack growth will not immediately develop into the unstable stage. Nevertheless,
immediate action is necessary to remedy this situation.
Although the crack propagation path is not the same, the crack propagation mechanism was similar for
all the wheel loads (Table 5.6). The crack initiating from the weld root propagates in both longitudinal
and vertical direction in the deck plate. However, the propagation rate in both directions was different.
Firstly, the crack propagates in a longitudinal direction for a certain length until it starts propagating in
a vertical direction. This process continues symmetrically until the crack reaches the deck plate surface
forming an elliptical shape (see Figure 5.23 (b)). An example is presented in Figure 5.23 (a), for wheel
load 35 KN where the crack length is 33 mm followed by 18mm in the next row. Despite the use of 8-
noded brick hexagonal elements, the crack propagated in a semi-elliptical fashion. This is mainly due
to the distribution of the stress intensity factor at the crack front. This trend holds for all the range of
wheel loads considered in the simulation.
Using the XFEM model, the fatigue crack growth originating from the weld root and propagating into
the deck plate thickness is studied in this section. The simulated results of crack growth in the through-
thickness direction of the deck plate are plotted against the number of load cycles for various wheel
loads as shown in Figure 5.24. The wheel load is adapted from the NEN 8701 for the period (2011-
2040) ranging between 40 KN and 52.5 KN.
Figure 5.24 Fatigue crack growth for several wheel loads in through-thickness direction
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From Figure 5.24, exponential crack growth is observed. The rate of crack propagation in through-
thickness increases with an increase in the wheel load. Moreover, the variation in the growth is
observed after 100,000 cycles, especially for wheel load 52.5KN. For every 12.5 % increase in wheel
load from 40KN to 52.5KN, the decrease in the cycles to the critical stage (the fatigue crack reaches to
the surface of the deck) is not constant. It is clear from the graph, as the load decrease, the difference
in the cycles to a critical stage decrease. For instance, the difference in the cycles for wheel load 52.5KN
and 50KN is about 24% whereas the difference diminishes to 10% in the case between wheel load
40KN and 42.5KN. This difference is mainly governed by bending stresses caused by the wheel load
and corresponding variation of the stress intensity factor at the crack front. As a result, the crack
propagation path varies. From the Paris law, it is clear that with a slight change in the stress intensity
factor, the rate of the propagation drastically increases which can be the possible explanation for the
degree of variation.
Figure 5.25 Fatigue crack propagation in the surface of the deck plate
Once the crack touches the surface of the deck plate, the crack seems to be arrested. The possible
explanation can be the absence of SIF at the crack front in the top side. As a result, the crack
propagation shifts in the longitudinal direction. An example of crack propagation in the deck plate for
wheel load 45 KN at 1.1 million cycles is illustrated in Figure 5.23(d) for a crack length of 50 mm.
Figure 5.25 displays the curves of crack propagation in the surface of the deck plate for various wheel
loads. At this moment, it was not possible to simulate the crack until a crack length is found like the
Suurhoff bridge of 230 mm due to convergence problem in the simulation. However, if an exponential
extrapolation function is used based on the simulated crack lengths, it is possible to predict the number
of load cycles required to reach a crack length of 230 mm. Moreover, the prediction of the number of
load cycles will depend upon the wheel load considered.
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Figure 5.26 Fatigue crack propagation in a longitudinal direction on the deck plate surface
Figure 5.26 displays the exponential extrapolated curves of crack propagation in the surface of the deck
plate for various wheel loads. From the graph, the number of load cycles requires to reach a crack
length of 230 mm is predicted in the range between 0.99 and 15.03 million load cycles for wheel load
52.5 KN and 40 KN respectively. In order to predict a reliable value from the range, the traffic
distribution (period 2011-2040) based on frequency of wheel load is considered. Since the frequency
of the wheel load 45KN is 49.3% (maximum) to the total distribution of traffic (Figure 5.27) wheel load
45KN will be used for prediction.
Figure 5.27 Wheel load frequency for 2011-2040 derived from the standard NEN 8701
Therefore, if an exponential extrapolation is made for the wheel load 45KN, the crack will reach a length
of 230mm after 5.95 million load cycles. Whereas, in reality, it just requires 4.75 million load cycles
after the crack initiation. The XFEM model predicted more than the required number of cycles due to
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multiple reasons. It must be remembered that the model does not include residual stresses and initial
imperfection. The initial imperfection can significantly influence the fatigue life. However, the residual
stresses effect may either be beneficial or detrimental, depending on their magnitude of compression
or tension and distribution in the connection. The effect of residual stresses is expected to be high as
a large amount of weld is concentrated at one location. Moreover, the welds are not chamfered at the
surface to reduce the effect of residual stresses. On the other hand, the weld material is homogenous
(without any defects) in the XFEM model. This can be a possible explanation for this overestimation.
Nevertheless, this numerical model showed a good correlation with the real scenario. Furthermore, this
numerical model is used in predicting the permissible limit of deck plate cracks length for 500 mm. If
an exponential extrapolation is made to a crack length of 500 mm for the wheel load 45KN, the number
of load cycles comes out to be 8.02 million load cycles. In other words, it will take 34 years more after
the crack initiation to reach a deck plate crack length of 500mm. This information helps to determine
the renovation and inspection interval.
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Fatigue life Assessment: Suurhoff Bridge Combined fatigue assessment
5.7 Combined fatigue assessment
With the help of the numerical model, it was possible to determine the total fatigue life of the bridge.
The determination of crack initiation period and crack propagation period were derived independently.
The fatigue life was determined based on SN-curves and the hypothesis of Palmgren-Miner. Whereas
the fracture assessment was carried out using automated XFEM simulation. Since the numerical model
consisted of a single wheel load it was important to redefine the traffic distribution based on wheel
load. Thereby, the total fatigue assessment was estimated taking into account the frequency
distribution of the wheel load for given three periods (1972-1990, 1991-2010, 2011-2040) as per
prescribed in NEN8701. Moreover, based on the fatigue limit, the wheel loads were categorised.
Based on the hotspot stress method, the crack initiation period is predicted to be 1.91 million load
cycles which is equivalent to 20 years. In other words, it can be said that the fatigue crack initiated in
the year 1992. In continuation to that, automated crack propagation resulted in 5.95 million load cycles
to reach a crack length of 230 mm in the deck plate surface. So, if both the periods are combined, the
numerical model predicts the total fatigue life to be 7.86 million load cycles which is equivalent to 48
years for the crack length of 230 mm. However, in reality, a similar crack length was detected after a
service life of 44 years. Although a bit optimistic, the numerical model showed good agreement with
the TOFD measurement. Furthermore, the numerical model predicted 8.02 million load cycles for a
deck plate crack length of 500 mm which is equivalent to 34 years after the crack initiation period.
Therefore, the total fatigue life for deck plate crack length of 500 mm is predicted to be 54 years.
Nevertheless, the fracture mechanics approach showed a sign of improvement of the fatigue life
assessment.
Figure 5.28 Summary of fatigue life estimation
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Fatigue life Assessment: Suurhoff Bridge Combined fatigue assessment
94
Conclusions and Recommendations Combined fatigue assessment
Chapter 6
CONCLUSIONS AND RECOMMENDATIONS
6 Conclusions and Recommendations
95
Conclusions and Recommendations Combined fatigue assessment
96
Conclusions and Recommendations Conclusions
6.1 Conclusions
The implementation of LEFM as a fatigue assessment tool showed promising results for assessing
fatigue cracks, particularly when combined with the XFEM-model in ABAQUS®. A detailed 3D-
visualization of crack propagation could be observed. In addition to that, some explicit information such
as SIF for every crack tip/front is available, which were helpful to determine other fracture parameters
easily. The results of these parameter (SIF) is investigated in CT-Specimen and found be to in good
agreement with the standard formulation, which signifies the reliability of this tool. However, some
limitations were encountered during this research. The determination of SIF around the crack front in
the presence of tetrahedral elements was not possible based on the contour-integral method. Moreover,
the crack propagation based on LEFM does not affect the compression zone. Also, ABAQUS ® does not
provide a clear indication of the choice of mix-mode behaviour to evaluate the equivalent fracture
energy release rate. Nevertheless, the automated crack simulation based on Paris law showed good
correlation with the test result and turns out to be conservative mainly because of its simplistic nature.
One of the important assumptions which still need to investigate is the choice of initial crack size and
shape. The size of the initial crack flaw is extremely sensitive to fatigue crack simulation result in XFEM.
A slight change in the dimension can shift the result from being unsafe to being conservative. To
evaluate the modelling efficiency, the Paris constant should be preferably verified with the test result.
Since the material test result of OSD specimen was not available, the constant was thus compared with
the standard and proved to be indeed less conservative. Furthermore, threshold SIF should be
considered in the formulation to improve the accuracy of Paris Law.
The main objective of the research was to numerically model and verify the problem of fatigue crack
propagation using XFEM-model based on LEFM and VCCT. Based on the set of simulation executed, the
conclusion of each model and corresponding answers to the research questions can be stated as
follows:
1.1. How to implement the material parameters and formulate Paris law in the XFEM
model for numerical simulation of fatigue crack propagation?
The Paris law implementation in ABAQUS® for constant C3 and C4 were computed corresponding to the
material constant C and m from the following relationship between strain energy release rate and stress
intensity factor: C4=m/2 and C3=C.E*C4 where E*=E for plane stress condition and E*=E/ (1- υ2) for
plane strain condition.
2.1. What is the accuracy of XFEM model developed in this research to predict the
fatigue crack propagation rate in CT-specimen for different stress ratios?
• The fatigue crack propagation rate in 2D CT-specimen was predicted based on assumed VCCT
fracture property using XFEM-model. The simulated crack propagation rate was validated
against the test results with a maximum difference of 0.03% in the slope (m) and 1.48% in
the intercept (C) of the power law equation.
• The simulated result of 2D-XFEM model was able to explain the crack closure mechanism
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through the similarity principle for several stress ratios. Moreover, the simulated result showed
good agreement with Elber’s equation U = 0.5 + 0.4 R , the relationship between the effective
stress intensity factor ratio (U) and stress ratio (R).
• The crack propagation mechanism in 3D-XFEM is studied for various mesh sizes and it was
found that the crack propagation starts from the centre of the thickness and propagates
towards the edge of thickness for every crack length increment. This is mainly due to the stress
intensity distribution along the crack front.
• The fatigue crack growth in 3D-XFEM model showed a good correlation with 2D-XFEM until a
crack size of 17.8 mm with a maximum difference of 13.8% for stress ratio R=0.50. This
difference is mainly due to the assumed straight crack front and corresponding non-uniform
stress intensity factor distribution along the crack front in through-thickness direction.
• The stress intensity factor distribution is not constant along the crack front i.e. maximum in the
middle and minimum at the edges. Based on the investigation, it was found out that SIF at the
edges can be a reliable technique in predicting SIF when compared with the ISO 12108
formulation for SIF for a maximum difference of 1.16 %.
• From crack simulation of 3D-XFEM, a regular propagation mechanism is been observed up to

a crack size of 17.8 mm until the crack encounter the top-face of the element. Based on the
results up to 17.8 mm, 3D-XFEM- model were roughly in good agreements with the test results
in predicting the crack propagation rate for a maximum difference of 25.23% in the slope of
the power law equation.
2.2. How to predict the Paris law constants (C and m) using XFEM-model based on
the beach mark measurement?
• The fatigue crack growth was predicted based on assumed VCCT material property using XFEM-
model. The simulated results of fatigue crack originating from the weld toe in the deck and
propagating to the surface were correlated with the beach mark measurement obtained from
the fatigue test. The calibrated Paris law constants C comes out to be 55 % lower when
compared with the recommended value in IIW standards.
• In ensuring the actual behaviour of the test specimen, static analyses were performed. It was
noticed the simulated vertical deformation overestimated with a maximum difference of 18 %
at 40 KN when compared to the test measurement. This difference can be possible when the
hydraulic jack setup is slightly inclined. Furthermore, the simulated strain results showed a
good correlation with the test results along the longitudinal direction for the respective plates
(deck plate and stiffener). It was observed that the peak appears at one-quarter of the
specimen in both cases which can be due to the distribution of line load. Lastly, the method of
determining the hot-spot stress showed its reliability as the numerical results were in good
agreement with the test data.
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2.3. What is the total fatigue life (crack initiation period and crack propagation
period) of the Suurhoff bridge (existing bridge) based on numerical analyses?
• In ensuring adequate behaviour of assumed boundary conditions, the numerical model is thus
compared with a full-scale RWS existing numerical bridge model. Based on similar wheel load,
the results of numerical model showed a good correlation in determining the stresses at the
weld root using hotspot stress method with a difference of 4.3%.
• The total fatigue life of the bridge was predicted based on an assumed material property using
a numerical model (FE and XFEM). The determination of crack initiation period and crack
propagation period were derived independently. The crack initiation period was predicted to be
1.91 million load cycles which is equivalent to 20 years based on SN-curves and the hypothesis
of Palmgren-Miner. Whereas the crack propagation period was predicted to be 5.95 million load
cycles using automated XFEM simulation for a crack length of 230 mm.
• The numerical model predicted a total fatigue life to be 7.86 million load cycles which is
equivalent to 48 years for the crack length of 230 mm. However, in reality, a similar crack
length was detected after a service life of 44 years. This can be possible as residual stresses;
weld defects were not implemented in the XFEM model. Nevertheless, the fracture mechanics
approach showed a sign of improvement of the fatigue life assessment.
• Finally, the numerical model predicted a total service life of 54 years for a deck plate crack
length of 500 mm.
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Conclusions and Recommendations Recommendations for future studies
6.2 Recommendations for future studies
6.2.1 CT-Specimen
Although 2D-XFEM model were in good agreements with the test result, it is to be noted that the crack
propagation rate is depended on the thickness of the specimen. Based on three samples out of seven
given samples of different thicknesses, it is difficult to conclude. Therefore, a parametric study is
recommended to study the thickness effect on crack propagation rate.
It is advised to model the crack propagation, by placing the crack tip/front in the middle of element to
obtain regular crack growth. Furthermore, a straight crack front is been assumed in this research which
leads to non-uniform distribution of stress-intensity factor along the crack front. It would be interesting
to investigate the shape of curved crack front, which can lead to uniform distribution of SIF. Based on
the investigation on different crack profile such as elliptical, polynomial mathematical function, it was
found 4th order polynomial returns a constant energy release rate value along the thickness [46].
The effect of boundary conditions caused a detrimental effect in crack propagation path as observed in
this research for 3D-XFEM model. It is recommended to use a semi-circular section (separate part)
attached to the respectively holes of the CT-specimen in applying loading/boundary conditions.
Furthermore, the material parameters and through-thickness effect based on three-dimensional model
needs to be further investigated to obtain comparable results with the test data till the final failure of
the model.
6.2.2 OSD- Specimen
It should be remembered that a constant material property was assigned to the OSD model. However,
it is to be noted that the fatigue crack propagation rate is different for the base material, welds and
HAZ zones and thereby Paris law constants can differ at such location. Furthermore, the effect of
residual stresses and microstructure change can be implemented in the XFEM-model. This can be a
possible investigation in the future in predicting more accurate results.
Furthermore, the imperfection in implementation of line-load causes a non-uniform distribution of load

over the surface, which can be further improved. In addition to that, contact property such as friction
should be implemented. Consequently, the marginal error observed in FE analyses can be resolved. It
is possible to predict the material parameter (Paris constant C and m) through XFEM simulation.
However, to determine the modelling efficiency, the predicted material constant should be compared
with the test sample.
6.2.3 Suurhoff Bridge
It is well known that material parameters fracture toughness KIC and Paris constants (C and m)
determines the quality of the fatigue life and are used in describing the crack propagation. These
parameters should be determined based on the material test before implementing in the model.
Furthermore, the application of loading (cyclic) resulted in 2.1 times higher stresses compared to a
100
moving vehicle of similar wheel load which is quite significant. Moreover, the magnitude of load should
be calculated depending upon on location in the bridge lane preferably based on probabilistic approach.
101
102
References Recommendations for future studies
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MODÈLE de MÉCANIQUE de la RUPTURE. 1991. p. 57–96.
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control in relation to bridge design. 1983.
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32. Xiao, Z.-G., et al., Stress analyses and fatigue evaluation of rib-to-deck joints in steel
orthotropic decks. 2008. 30(8): p. 1387-1397.
33. Technische Specificatie (Reparaties orthotrope rijdekken met trogprofielen). 2017,
Rijkswaterstaat Grote Projecten en Onderhoud p. 57.
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35. Standard, B.J.G.t.M.f.A.t.A.o.F.i.M.S., ” London, UK, BS 7910: 2005,“. 2007.
36. Maljaars, J., F. van Dooren, and H.J.S.C. Kolstein, Fatigue assessment for deck plates in
orthotropic bridge decks. 2012. 5(2): p. 93-100.
37. Bignonnet, A., J. Carracilli, and B. Jacob. ÉTUDE en FATIGUE des PONTS MÉTALLIQUES par un
MODÈLE de MÉCANIQUE de la RUPTURE. in ANNALES DE L'INSTITUT TECHNIQUE DU
BATIMENT ET DES TRAVAUX PUBLICS. 1991.
38. Hobbacher, A., Recommendations for fatigue design of welded joints and components. 2015:
Springer.
39. Nagy, W., et al., Development of a fatigue experiment for the stiffener-to-deck plate
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December 2011.
41. Eurocode, C.J.B., Belgium: Comité Européen de Normalisation, 1: Actions on structures-Part 2:
Traffic loads on bridges, NEN-EN 1991-2: 2003. 2003: p. 168.
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104
Appendix Paris law formulation
8 Appendix
A. Paris law formulation
Table A.8.1 Keyword: Paris law formulation implemented for CT-specimen in XFEM-model
R=0.0 *FRACTURE CRITERION, TYPE=fatigue, MIXED MODE BEHAVIOR=POWER,

TOLERANCE=0.001
0.001,0,7.84199E-06,1.7811,0,0.85,6.5,6.5
6.5,1,1,1
TOLERANCE=0.001
0.001,0,19.80264E-06,1.85795,0,0.85,6.5,6.5
6.5,1,1,1
TOLERANCE=0.001
0.001,0,18.7685E-06,1.94535,0,0.85,6.5,6.5
6.5,1,1,1
Table A.8.2 Keyword: Paris law formulation implemented for OSD-specimen in XFEM-model

TOLERANCE=0.001
0.001,0, 12.99E-06,1.5,0,0.85,11.9, 11.9
11.9,1,1,1
Table A.8.3 Keyword: Paris law formulation implemented for Suurhoff bridge in numerical model

TOLERANCE=0.001
0.001,0, 28.87E-06,1.5,0,0.85, 11.9, 11.9
11.9,1,1,1
105
Appendix Beach mark measurement
B. Beach mark measurement
Figure B.8.1 Beach mark measurement by microscope [25]
106
Appendix Traffic distribution
C. Traffic distribution
Table C.8.4 NEN 8701: Period 1972-1990
107
108
109
Appendix Fatigue detail category
D. Fatigue detail category
Table D.8.7 Fatigue detail category
Source Detail Structural Detail Description

Category
Weld connecting deck
Eurocode 3: plate to trapezoidal ribs;
Design of steel Partially penetrated weld
structures - Part 2: 71 with a>t;
Steel bridges [44]
Nominal Stress range
assessment
Calculated as nominal
local stress on the
NEN-EN underside of the cover
1993-1-9+C2 125
plate at the crack
[44]
initiation point,
calculated with a 3D
model
Cracking in the deck plate;
M.H. Kolstein [43] 125 Assessment based on the

nominal stress range in
the deck plate
IIW Ends of the longitudinal

Recommendations FAT stiffeners;
for fatigue design 100
of welded joints Fillet welded;
and components
[30] Fatigue resistance against
hotspot stresses
110
Appendix TOFD result
E. TOFD result
(a)
(b)
(c)
Figure E.8.1 (a) Illustration of location (top view) of TOFD measurement using skectches (b) Explanation of crack detection
of TOFD method using sketches (c) TOFD scan [45]
111

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Prediction of Fatigue Crack Propagation in

Orthotropic Steel Decks using XFEM based on

Ravi Shankar Gupta

Ravi Shankar Gupta

to be defended publicly on Friday July 19, 2019 at 11:00 AM.

Student number: 4751310

An electronic version of this thesis is available at http://repository.tudelft.nl/.

Ravi Shankar Gupta

1.1 Background information

1.2 Research Objective

2.1. Compact- 2.2. Orthotropic

1.3 Thesis Structure

2.1 Linear Elastic Fracture Mechanics (LEFM)

2.1.1 Crack characterization

Figure 2.1 Different crack types [3]

Figure 2.5 Fracture modes a) Mode I b) Mode II c) Mode III [9]

2.1.2 Stress intensity factor

The Griffith criterion

dWe  dU el + Gct a (2.2)

which can be further related in

Similarly, when the plane-strain situation is assumed:

Fatigue crack propagation regime

2.2 Fatigue of welded connections

Fatigue crack development

Deck-to-Rib welded joint

calculation of fatigue by use of fracture mechanics. As a continuation of work [15], a crack

2.3 XFEM (eXtended Finite Element Method)

□ Crack tip function

𝑥 = 𝐴0 + 𝐴1 cos(𝑡 − 𝑡0 ) + 𝐵1 sin(ω(𝑡 − 𝑡0 )) (2.12)

Figure 2.11 Fatigue crack growth [4]

3.2 XFEM model

Figure 3.1 Geometry of CT-Specimen [22]

3.2.2 Boundary conditions

Initial crack flaw

Figure 3.2 (a) Boundary condition of CT-Specimen (b) XFEM-model of CT-Specimen

3.2.3 Material property

Table 3.1 Direct cyclic parameters

Figure 3.4 Graphical representation of different stress ratios

Table 3.2 Constants of Paris' Law and XFEM Abaqus

Rσ B (mm) Fmax (N) Fmin (N) Experimental data XFEM Abaqus

Table 3.3 Critical energy release rate Gc

XFEM model Critical energy release rate 𝑮𝒄 (Nmm-1) Exponent

Mode I Mode II Mode III αm αn αo

3.2.4 Mesh quality

Table 3.4 Meshing details of XFE model

Model Region Element type Mesh size

3.3.1 2D-XFEM model

3.3.2 3D-XFEM model

Regular crack propagation mechanism

3.4 Result and Discussion

3.4.1 2D shell XFEM-model

3.4.2 Effect of stress ratio

U = 0.5 + 0.4 R (3.11)

3.4.3 3D solid XFEM-model

Figure 3.11 Non-uniform stress distribution over the thickness at holes

Comparison of fatigue crack growth

Fatigue crack mechanism- 3D XFEM

Fatigue crack propagation- 3D solid XFEM

3.4.4 Stress Intensity Factor