The effect of head wear on rail underhead radius stresses and ......Sagheer Abbas Ranjha B.Eng. (Hons), M. Eng. A thesis submitted in fulfilment of the requirements for the degree

Effect of head wear on rail underhead radius stresses

and fracture under high axle load conditions

by

Sagheer Abbas Ranjha

B.Eng. (Hons), M. Eng.

A thesis submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

Centre for Sustainable Infrastructure

Faculty of Engineering and Industrial Sciences

Swinburne University of Technology

2013

Abstract

i

Abstract

Since the Hatfield incident in the UK in 2000, the possibility of catastrophic rail failure

as a result of the propagation of surface-initiated rolling contact fatigue (RCF) cracks

that go on to form transverse defects (TDs) is of on-going concern to the rail industry.

Generally, a rail can fail by either wear (loss of profile) or by RCF (surface cracks

growing by fatigue to cause rail break), and requires replacing. The heavy haul sector is

not immune to such concerns, although the combination of improved rail steels and

optimisation of the wheel-rail interface has reduced the extent to which rail wear

influences rail life. Field measurement results show a short duration tensile stress spike

at the underhead radius (UHR) of the rail as a heavy wheel passes over. This observed

stress is due to the lateral bending of the whole rail profile and the vertical and lateral

bending of the head-on-web. It is induced by the complex wheel-rail contact conditions

that are associated with, but are not limited to, a combination of wheel-rail contact

eccentricity from the rail centreline, lateral (transverse) forces and track support

conditions. In the presence of heavily worn rail, the magnitude of these stresses could

be even higher. Since the stress state is multi-axial with out-of-phase stress components

and varying principal stress directions, the cyclic wheel-rail contact, bending, residual

and thermally induced stresses interact with geometric features to produce a mixed

mode non-proportional stress history. This directly influences the location of fatigue

crack initiation, propagation and the severity of damage in rail with or without

macroscopic defects. Defect types that can be considered include, but are not limited to

head wear, reverse detail fracture, and long surface initiated rolling contact fatigue

cracks.

This study was conducted to parametrically evaluate the possibility of the failure risk

associated with rapid fracture behaviour of pre-existing RCF cracks and fatigue damage

at the rail underhead radius, and in particular, the qualitative assessment of the transition

Abstract

ii

to Mode I crack growth in the presence of a tension spike at the underhead radius. The

mechanical responses at the underhead radius were explored by modelling a single rail

on discrete elastic foundations using the finite element method (FEM). The wheel

contact load was modelled as a Hertzian contact pressure applied to an elliptical patch

on the rail head, assuming fully slipping conditions. The results revealed that the

longitudinal tensile stress at the rail underhead radius was highly dependent on several

operational parameters. The magnitude of tension spike increases and the region over

which tensile stresses occur moves closer to the top of the rail as a result of an increase

in the contact patch offset (CPO), the L/V ratio (ratio of lateral (L) to vertical (V) load),

head wear and the foundation stiffness. This stress is further increased by residual and

thermally induced stresses, and may result in a transition to Mode I crack growth for

pre-existing RCF cracks on the gauge corner of the rail that turn down to form

transverse defects (TD), as has been observed in practice [1-4]. These tensile stresses

may also result in fatigue crack initiation at the underhead radius [3-10].

The potential fatigue mechanism of crack initiation at the underhead radius was

implemented in a FORTRAN-code embedded in ABAQUS, as the damage parameter

defined by the Dang Van criterion. The Palmgren - Miner damage accumulation law

was used as an approach embedded in the FORTRAN-code to quantify damage

accumulation and cycles to failure. Hardness testing was conducted at the rail

underhead radius to estimate and quantify the fatigue properties. An examination was

conducted for different high strength rail steels, including the heat-treated low alloy,

eutectoid and hypereutectoid rail grades that are used under heavy haul conditions. The

fatigue behaviour of high strength rail steels was compared to predict the rail wear

limits and these results were correlated with field observations. As was reported in the

field [6-10], a reverse detail fracture is initiated in poorly lubricated, heavily worn

curved rail on stiff track carrying traffic under high axle load conditions.

A further study was conducted to investigate the unstable growth behaviour of single

and multiple pre-existing RCF cracks, especially the tendency for a rail break due to

rapid fracture under the high axle load conditions typical of those that exist in

Australian heavy haul operations. The occurrence of tension spikes as a result of the

Abstract

iii

lateral bending of the whole rail profile and localised vertical and lateral bending of the

head-on-web is exacerbated with increasing rail head wear and the propensity of rapid

fracture associated behaviour correlates with the extent of rail head wear. This

behaviour was examined using the extended finite element method (X-FEM), to

parametrically study the unstable growth behaviour of RCF cracks. The extended finite

element method results revealed that existing RCF cracks, when subjected to high

tensile stresses at the underhead radius, could contribute to the development of a rapid

(unstable) fracture. The results of this thesis can be used to examine the influence of

wheel-rail interaction behaviour and rail head wear on the possibility of a catastrophic

rail failure developing from RCF damage.

This page is left intentionally blank

Declaration

v

Declaration

I declare that no portion of the work referred to in the thesis has been submitted in

support of an application for another degree or qualification of this or any other

university or other institution of learning. I also declare that to the best of my

knowledge it contains no material previously published or written by another person

except where due reference is made in the text.

Sagheer Abbas Ranjha

June 2013

© Copyright by Sagheer Abbas Ranjha


Publications

vii

Publications

1. Ranjha SA, Ding K, Mutton PJ, Kapoor A. Finite element modelling of the rail

gauge corner and underhead radius stresses under heavy axle load conditions.

Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and

Rapid Transit 2012; 226 (3):318-330.

2. Ranjha SA, Ding K, Mutton PJ, Kapoor A. Mechanical state of the rail underhead

region under heavy haul operations. In Proceedings of the Conference on Railway

Engineering (CORE 2012), Global perspective; 10-12 September 2012; Brisbane,

Australia 2012.

3. Ranjha SA, Ding K, Mutton PJ, Kapoor A. Fatigue life prediction of the rail

underhead region influenced by wear in heavy haul operations. Extended abstract

published in the proceedings of 9th International Conference on Contact Mechanics

and Wear of Rail/Wheel Systems (CM2012), Chengdu, China, pp. 675-676, August

27-30, 2012

4. Ranjha SA, Ding K, Mutton PJ, Kapoor A. Rapid fracture behaviour of rolling

contact fatigue cracks under high axle load conditions. In Proceedings of the

Conference of International Heavy Haul Associations (IHHA 2013); New Delhi,

India, pp. 382-390, February 4-6, 2013.

5. Ranjha SA, Mutton PJ, Kapoor A. Rapid fracture behaviour of rolling contact

fatigue cracks under high axle load conditions. Submitted to the proceedings of the

Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 2013.

Publications

viii

6. Ranjha SA, Mutton PJ, Kapoor A. Effect of wear on fatigue behaviour of rail under

high axle load conditions. Submitted to Advanced Materials Research, ISSN: 1022-

6680, Trans Tech Publications, Switzerland.

ix

Dedicated to:

My beloved parents and wife


Acknowledgement

xi

Acknowledgement

My heartiest gratitude goes to Almighty Allah, who gives me strength, good health and

wisdom to complete this thesis. I would like to acknowledge the assistance provided by

the following in the completion of this thesis.

Prof. Ajay Kapoor (Principal Supervisor) for his consistent support, patient guidance,

continuous encouragement, constructive comments and valuable insight on this thesis.

Dr Kan Ding for his valuable discussions throughout the work on the model

development and also for keeping me motivated and efficient.

Peter Mutton of the Institute of Railway Technology (IRT) Monash University, for his

industrial support in providing field measurements and rail samples. The data on

measured rail stresses under heavy haul conditions was based on research activities

undertaken by the Institute of Railway Technology, Monash University for Rio Tinto

Iron Ore and is provided in Appendix B. The author acknowledges these organisations

in provision of this support.

A/Prof. Palaneeswaran Ekambaram for his valuable discussion during the course of my

research project for the last three and half years.

I really thank Swinburne University of Technology for the provision of funding through

a SUPRA scholarship. I also wish to thank my colleagues, Anna Maria Sri Asih, Iman

Salehi and all the other staff at Swinburne University of Technology, for making this

study a positive experience.

I also acknowledge the professional copy-editing undertaken by Christine De Boos

(BA, B.Ed., CELTA) from De Boos Editing and Education Services, 23 Nantes Street

Newtown, VIC 3220, who ensured that this was undertaken in line with the guidelines

Acknowledgement

xii

set out by the DDOGS Committee & the Australian Standards for Editing Practices

(ASEP).

Finally, I thank all my family members: My father and mother (Muhammad Abbas

Ranjha and Azra Parveen), my wife (Zunaira Sagheer), my sisters (Rozeela Abbas and

Samina Abbas) and my brothers (Zaheer Abbas and Sibghat Abbas). They all have

provided me with the moral support required to complete this research work, as well as

irreplaceable encouragement that made my research a reality.

Table of Contents

xiii

Table of Contents

Abstract…. ........................................................................................................................ i

Declaration ....................................................................................................................... v

Publications .................................................................................................................... vii

Acknowledgement .......................................................................................................... xi

Table of Contents ......................................................................................................... xiii

List of Figures ............................................................................................................... xix

List of Tables ............................................................................................................. xxvii

List of Notations and Acronyms ............................................................................... xxix

Introduction .................................................................................................. 1 Chapter 1

1.1 Research background and motivation ................................................................... 1

Introduction .................................................................................................... 1 1.1.1

Application of a ―Whole Life Rail Model” to high axle load conditions ....... 4 1.1.2

1.2 Aims and objectives ............................................................................................ 12

1.3 Methodology ....................................................................................................... 13

1.4 Thesis outline ...................................................................................................... 14

Literature review ........................................................................................ 17 Chapter 2

2.1 Introduction ......................................................................................................... 17

2.2 Basics of wheel-rail interface .............................................................................. 17

Rail cant........................................................................................................ 18 2.2.1

Hertz contact theory ..................................................................................... 19 2.2.2

Table of Contents

xiv

2.2.2.1 Line contact ........................................................................................ 20

2.2.2.2 Circular contact .................................................................................. 21

Wheel-rail contact pressure distribution ...................................................... 22 2.2.3

2.2.3.1 Wheel conicity in the steering............................................................ 24

Stress field under a Hertzian contact ............................................................ 27 2.2.4

2.3 Rail failure mechanism........................................................................................ 29

Plastic flow and shakedown ......................................................................... 30 2.3.1

Low cycle fatigue and ratchetting failure ..................................................... 32 2.3.2

2.4 Wear and rolling contact fatigue ......................................................................... 34

Wear ............................................................................................................. 34 2.4.1

Rolling contact fatigue ................................................................................. 35 2.4.2

Surface and subsurface rolling contact fatigue ............................................ 36 2.4.3

Rail underhead radius failure ....................................................................... 38 2.4.4

2.5 Crack initiation .................................................................................................... 42

Multi-axial fatigue criteria ........................................................................... 44 2.5.1

Dang Van criterion ....................................................................................... 45 2.5.2

Critical shear plane ....................................................................................... 46 2.5.3

Damage Accumulation ................................................................................. 47 2.5.4

2.6 Linear elastic fracture mechanics ........................................................................ 48

Crack propagation and rapid fracture ........................................................... 49 2.6.1

2.7 Wear fatigue interaction ...................................................................................... 54

2.8 Allowable wear limits ......................................................................................... 56

2.9 Material grades in high axle load rail operations ................................................ 58

Residual and thermal stresses ....................................................................... 61 2.9.1

2.10 Summary ............................................................................................................. 61

Table of Contents

xv

Finite element model development ........................................................... 63 Chapter 3

3.1 Introduction ......................................................................................................... 63

3.2 Finite element model development ..................................................................... 64

Geometric model .......................................................................................... 64 3.2.1

Flow diagram of the FE model ..................................................................... 65 3.2.2

Track modelling ........................................................................................... 66 3.2.3

Finite element model .................................................................................... 66 3.2.4

Boundary conditions .................................................................................... 68 3.2.5

Loading......................................................................................................... 68 3.2.6

Vehicle curving and hunting ........................................................................ 69 3.2.7

Mesh development ....................................................................................... 73 3.2.8

3.3 Model Validation ................................................................................................. 74

Field data ...................................................................................................... 74 3.3.1

Hertzian contact stresses .............................................................................. 76 3.3.2

Octahedral shear stresses .............................................................................. 77 3.3.3

Effect of elastic foundation .......................................................................... 78 3.3.4

Sensitivity analysis ....................................................................................... 80 3.3.5

3.4 Thermal stresses .................................................................................................. 81

3.5 Residual stresses distribution .............................................................................. 81

3.6 Summary ............................................................................................................. 83

Underhead radius stresses ......................................................................... 85 Chapter 4

4.1 Introduction ......................................................................................................... 85

4.2 Finite element model ........................................................................................... 89

4.3 Effect of different contact patch offset and lateral tractions ............................... 89

4.4 Stress state at the underhead radius and gauge corner ........................................ 92

Table of Contents

xvi

4.5 Effect of lateral traction direction ....................................................................... 96

4.6 Effect of track foundation stiffness ..................................................................... 98

4.7 Effect of seasonal temperature variation ........................................................... 100

4.8 Fatigue damage ................................................................................................. 102

4.9 Summary ........................................................................................................... 103

Effect of head wear on underhead radius stresses ................................ 105 Chapter 5

5.1 Introduction ....................................................................................................... 105

5.2 Finite element analysis model ........................................................................... 106

5.3 Stress state at the rail underhead radius ............................................................ 109

Effect of residual stress .............................................................................. 109 5.3.1

Effect of head wear .................................................................................... 110 5.3.2

Effect of contact patch offset ..................................................................... 112 5.3.3

Effect of the L/V ratio ................................................................................ 113 5.3.4

Effect of track foundation stiffness ............................................................ 114 5.3.5

Evaluation of depth of tensile longitudinal stress ...................................... 116 5.3.6

5.4 Summary ........................................................................................................... 120

Fatigue damage prediction ...................................................................... 123 Chapter 6

6.1 Introduction ....................................................................................................... 123

6.2 Model development ........................................................................................... 127

Dang Van fatigue criterion ......................................................................... 129 6.2.1

Implementation of the critical plane approach ........................................... 130 6.2.2

Estimated fatigue properties of different rail grades .................................. 131 6.2.3

Experimental results for hardness distribution ........................................... 132 6.2.4

Damage Accumulation ............................................................................... 134 6.2.5

6.3 Residual stresses ................................................................................................ 136

Table of Contents

xvii

6.4 Plasticity of the underhead radius ..................................................................... 137

6.5 Dang Van (DV) damage parameter ................................................................... 138

6.6 Damage accumulation and fatigue life .............................................................. 143

6.7 Summary ........................................................................................................... 145

Rapid fracture modelling using X-FEM ................................................ 147 Chapter 7

7.1 Introduction ....................................................................................................... 147

7.2 Crack model development ................................................................................. 149

Extended finite element method (X-FEM)................................................. 149 7.2.1

Discontinuous geometry using the level set method .................................. 150 7.2.2

X-FEM crack model ................................................................................... 151 7.2.3

7.3 Variations of KI max during loading cycle .......................................................... 155

Effect of contact patch offset (CPO) and L/V ratio ................................... 155 7.3.1

Effect of crack length on SIFs .................................................................... 157 7.3.2

Effect of residual stresses on SIFs.............................................................. 158 7.3.3

7.4 Stress intensity factors (SIFs) on the crack front .............................................. 160

Effect of crack orientation on SIFs ............................................................ 160 7.4.1

Effect of head wear on SIFs ....................................................................... 162 7.4.2

7.5 Summary ........................................................................................................... 164

RCF cracks under mixed-mode loading................................................. 165 Chapter 8

8.1 Introduction ....................................................................................................... 165

8.2 Multiple cracks model development ................................................................. 169

Mixed-mode fracture criteria ..................................................................... 173 8.2.1

8.3 Variations of SIFs (K I, K II, K III) during loading cycle .................................... 174

8.4 Variations of Keq along crack front during loading cycle .................................. 176

Effect of contact patch offset (CPO) .......................................................... 176 8.4.1

Table of Contents

xviii

Effect of L/V ratio ...................................................................................... 181 8.4.2

Comparison of head wear (HW) ................................................................ 184 8.4.3

Crack shielding by multiple cracks ............................................................ 186 8.4.4

8.5 Summary ........................................................................................................... 189

Conclusions and future work .................................................................. 191 Chapter 9

9.1 Conclusion ......................................................................................................... 191

9.2 Suggestions for further work ............................................................................. 197

References ................................................................................................................... 201

Appendices .................................................................................................................. 221

Appendix A. Equations for rail bending stresses ....................................... 221

Appendix B. Rail underhead radius (UHR) stresses data .......................... 227

Appendix C. FORTRAN-code based on the Dang Van fatigue criterion .. 261

List of Figures

xix

List of Figures

Chapter 1

Figure 1.1 Appearance of RCF damage on high rail [2] 2

Figure 1.2 Development of transverse defect [1] 3

Figure 1.3 Transverse defect (TD) developed from RCF damage [1, 2, 16, 17]

3

Figure 1.4 ―Whole Life Rail Model‖ ‗WLRM‘ (a): Crack propagation rate v/s length (adopted from [1, 20, 22-24, 30, 31]); (b): Sequence of events in the development of RCF [22, 25, 26]

5

Figure 1.5 Assumed longitudinal bending response of rail in ―Whole Life Rail Model‖ [27]

7

Figure 1.6 Measured longitudinal stress response under in service loading; example shown for gauge side underhead radius of high rail in 609 m radius curve [1, 32, 33]

7

Figure 1.7 Representation of localised head-on-web bending and lateral bending of the whole rail profile as a beam model on a continuous elastic foundation with eccentric vertical and outward lateral loading (adopted from Orringer et al [46, 47], Jeong et al [7], and Salehi [49])

9

Figure 1.8 RDF development [16] 11

Figure 1.9 RDF (Courtesy of TSBC [3, 9]) 11

Figure 1.10 RDF development [16] 11

Figure 1.11 RDF (Sperry rail services [6-8]) 11 Figure 1.12 RDF development in aluminothermic welded rail [50] 11

Chapter 2

Figure 2.1 Wheel-rail contact (a) side view, (b) cross-section view [55] 18

Figure 2.2 Track gauge [56] 19

Figure 2.3 Hertzian contacts: (a) rectangular contact; (b) circular contact [25] 21

Figure 2.4 Rail-wheel Hertzian contact [60] 23

List of Figures

xx

Figure 2.5 Hertzian pressure distribution at the contact patch [the picture to the left is from , whereas the picture to the left is from

24

Figure 2.6 (a) Vehicle on a curve; (b) Rolling radius difference [64] 25

Figure 2.7 Lateral shift of wheel set leads to contact patch offset from the center of railhead cross-section [66]

26

Figure 2.8 Representation of surface traction on elastic half space [58] 28

Figure 2.9 (a) Stress distribution versus depth caused by Hertz pressure acting on a circular area of radius ‗a‘ [58] (b) subsurface stresses; (c) contours of principal shear stress [58]

29

Figure 2.10 Material response to repeated loading [73] 30

Figure 2.11 Shakedown map for repeated sliding of a rigid cylinder over an elastic-plastic half space [74, 75, 77]

31

Figure 2.12 Shakedown diagram and RCF predictions [79] (high rail on a moderate curved track with axle load of 30 tonne)

33

Figure 2.13 Photomicrograph of an etched sectioned test sample from twin disc test [88]

35

Figure 2.14 Schematic of the mechanism of RCF crack formation and branching [91]

36

Figure 2.15 Broken rail showing shell origin of detail fracture [Courtesy of Transportation Technology Centre, Inc. (TTCI)] [8, 10]

37

Figure 2.16 Predicted head stresses in 60 kg/m and 68 kg/m rails [36] 40

Figure 2.17 Effect of contact patch offset (CPO) on longitudinal stresses at underhead radius (UHR) for 68Kg/m rail at nominal axle load of 30 and 35 tonnes, dynamic factor of 1.5 and L/V ratio of 0.2 [37]

41

Figure 2.18 Predicted rail head stresses for different head designs [36] 41

Figure 2.19 Fatigue prediction loading paths at different points of the rail surface [52, 53, 99]

46

Figure 2.20 Representation of critical plane with shear stress at a material point in a shear plane defined by normal vector ‗n‘, coordinate system xyz (adopted from Ekberg et al [62])

47

Figure 2.21 Modes of crack growth (a) Mode I, opening (b) Mode II, shearing (c) Mode III, tearing [20]

48

Figure 2.22 Three phases of the life of a RCF crack initiated at the surface [95, 114]

49

Figure 2.23 Fluid assisted mechanisms of crack growth: (a) shear mode crack growth, accelerated by fluid reduction of friction between the crack faces, (b) hydraulic transmission of contact pressure, (c) entrapment and pressurization of fluid inside the crack and (d) squeeze film pressure generation [117, 118].

50

List of Figures

xxi

Figure 2.24 Effect of wear on crack growth [22, 23] 54

Figure 2.25 Life cycle of a crack in a rail [125] 55

Figure 2.26 Strategies for rail life management [22, 23, 25] 56

Figure 2.27 Hardness distribution along vertical transverse at rail centerline [2]

59

Figure 2.28 Relationship between yield strength and hardness in eutectoid and hypereutectoid heat treated rail steel [2]

60

Figure 2.29 Relationship between tensile strength and fatigue limit [130] 60

Chapter 3

Figure 3.1 68 kg/m rail [80] 64

Figure 3.2 Flow diagram for FE model solution technique development based on rail geometry, material properties, loading and boundary conditions

65

Figure 3.3 The model description: (a) Top view of single rail supported with discrete elastic foundations; (b) Cross-section view of the rail, the measurement location and the elastic foundations

67

Figure 3.4 Variations of contact patch sizes and shapes 70

Figure 3.5 Representation of a contact patch offset (CPO) from the center of rail head cross-section for modelling eccentric loading

71

Figure 3.6 Representation of inward and outward lateral shear tractions by leading and trailing wheelsets respectively during curving on high rail of a 600 m radius curve (adopted from Xiao et al [134])

72

Figure 3.7 The rail-wheel contact loads in detail (a) wheel loads [the picture to the left is the original work]; (b) Hertzian pressure distribution at the contact patch [picture to the right is adopted from Ekberg et al [62]]

73

Figure 3.8 Mesh development with dense mesh at the contact patch 74

Figure 3.9 The strain gauges at the measured points [33] 75

Figure 3.10 Measured longitudinal stress response under in service loading; example shown for high rail in 609 m radius curve [1, 32]

76

Figure 3.11 Stress distribution in the rail head versus depth caused by Hertz pressure acting on a circular area (a=6.8 mm, Fz =171.7 KN)

77

Figure 3.12 Octahedral shear stress ( ) of a rail subjected to a non-uniform (Hertzian) contact pressure (a = 6.8mm, Fz = 171.7 KN)

78

Figure 3.13 Longitudinal stress at the underhead offset of measurement point 20 mm

80

Figure 3.14 Rail section for residual stress distribution 82

List of Figures

xxii

Chapter 4

Figure 4.1 Longitudinal stress (S11) contour; (a) centric loading; (b) eccentric loading on gauge side (contact patch 30 mm offset); (c) eccentric loading on gauge side (contact patch offset of 30 mm with L/V=0.4 towards gauge side)

90

Figure 4.2 Longitudinal stress at the underhead offset of measurement point 34 mm; (a) at different contact patch offsets (0, 15 and 30 mm) with L/V=0; (b) at contact patch 30 mm offset with different L/V= (0, 0.2 and 0.4)

91

Figure 4.3 Longitudinal stress at different underhead offset of measurement points (20, 34 and 37 mm from longitudinal centerline of rail) at the contact patch 30 mm offset and L/V=0.4

92

Figure 4.4 Longitudinal stress distribution with depth on a vertical plane at 20 mm offset from the rail vertical centreline towards the gauge corner for the different contact patch offsets (0, 15, and 30) mm with L/V ratios (0, 0.4)

93

Figure 4.5 Longitudinal stress distribution with depth on different vertical planes at 16.5, 20, 25 and 35 mm offset from the rail vertical centreline towards the gauge corner at a contact path 30 mm offset and different L/V ratios (0, 0.2, and 0.4)

95

Figure 4.6 Longitudinal stress (S11) contours on the cross-section of rail with the contact patch offset of 30 mm at midpoint of rail span ( in longitudinal direction at x=2100 mm), during different lateral loads with the L/V ratio of 0, 0.2 and 0.4. The affected region at which the stress becomes tensile shifts up from the rail underhead radius to surface

96

Figure 4.7 Mechanical response at the underhead and base fillet with lateral shear tractions (inward and outward); (a) Longitudinal stress variations with different magnitude and direction of shear traction at the underhead offset of measurement point 34 mm along the midpoint (x = 2100 mm) of rail with contact patch 30 mm offset; (b) Inward lateral deformation (U2) contour with L/V = 0.4; (c) Outward lateral deformation (U2) contour with L/V = 0.4

97

Figure 4.8 Longitudinal stress variations at the underhead radius with vertical foundation stiffness at the contact patch 30 mm offset and L/V = 0.4: (a) along longitudinal position of the rail at the underhead offset of measurement point 34 mm; (b) at the underhead offset of measurement point 34 mm and at base fillet at midpoint of rail span (in longitudinal direction x=2100 mm) in between two middle sleepers

99

Figure 4.9 Longitudinal stress variations at the underhead radius with 101

List of Figures

xxiii

seasonal temperature at contact patch offset of 30 mm and L/V = 0.4 (a) along longitudinal position of the rail at the underhead offset of measurement point 34 mm; (b) at the underhead offset of measurement point 34 mm at midpoint of rail span (in longitudinal direction at x = 2100 mm) in between two middle sleepers

Chapter 5

Figure 5.1 Worn rail profiles [143] 107

Figure 5.2 The model description for the cross-section view of the worn rail (Head wear, HW = 25 mm) profile, the strain gauge measurement location and the elastic foundations with variations of contact patch size and shape

108

Figure 5.3 Variation of maximum longitudinal stress (S11) at the underhead region vs rail head wear for an eccentric load located at 20 mm offset from the rail centreline with L/V = 0.4

111

Figure 5.4 Longitudinal stresses (S11) distribution at the rail head vs the different head wear (HW) profiles for an eccentric load located at 20 mm offset from the rail centreline with L/V = 0.4

111

Figure 5.5 Longitudinal stress distribution at the underhead for the different contact locations vs the HW of 0, 22 mm and 25 mm

112

Figure 5.6 Variation of longitudinal stress at the gauge side underhead radius for eccentric loading from rail centreline towards the gauge side for different rail worn profiles versus the lateral traction coefficient

114

Figure 5.7 Longitudinal stress variations at the underhead radius with vertical foundation stiffness at the contact patch 20 mm offset and L/V = 0.4: at the underhead offset of measurement point 34 mm and at the base fillet at the midpoint of the rail span (in longitudinal direction x=2100 mm) in between two middle sleepers

115

Figure 5.8 Longitudinal stress distribution with depth on different vertical planes offset from the rail vertical centerline towards the gauge corner, at a contact patch offset of 20 mm L/V=0.4: (a) HW = 25 mm, (b) HW = 15 mm

117

Figure 5.9 Longitudinal stress distribution with depth on different vertical planes at 15, 20, 25 and 30 mm offset from the rail vertical centreline towards the gauge corner at a contact patch offset of 25 mm L/V=0.4 HW 20mm

118

Figure 5.10 Longitudinal stress distribution (S11) with depth on a vertical plane at a 25 mm offset from the rail vertical centerline towards the gauge corner for contact patch offsets of 20 mm with L/V 0.4

119

List of Figures

xxiv

Chapter 6

Figure 6.1 Stress analysis at underhead offset of 25mm for head wear of 22 mm, contact patch offset = 15mm, L/V=0.4

125

Figure 6.2 Representation of model setup for DV fatigue damage analysis 128

Figure 6.3 Description of UVARs 2 to 5 on a critical plane 131

Figure 6.4 INDENTEC-Hardness measurement apparatus (Courtesy of Swinburne University of Technology)

133

Figure 6.5 Rail sample specimen (section and the shape) for hardness testing 133

Figure 6.6 Hardness distribution along transverse plane (starting from underhead radius measurement point offset 20 mm) of high strength rail steels

134

Figure 6.7 The fatigue threshold stress is in a range of t-1 corresponding to failure at the t-1 for 106 stress cycles (the highest cycle) and at the at the DV magnitude of the fracture stress for 101 stress cycles (the lowest cycle)

135

Figure 6.8 Von Mises stress distribution at the rail underhead radius vs different head wear profiles: for eccentric loads located at different offset from the rail centerline with L/V = 0, 0.2, 0.4

137

Figure 6.9 Variation of fatigue damage with rail head wear at the underhead radius for an eccentric load located at (a) 15 mm offset from the rail centerline with L/V=0; (a) 20 mm offset from the rail centerline with L/V=0.2, for a plain C-Mn Head Hardened (HH) grade [80], with t-1 = 205 MPa

139

Figure 6.10 Variation of DV damage parameter versus rail head wear HW at the underhead radius for an eccentric load located at different offset from the rail centreline with L/V=0, 0.2, 0.4, for three rail grades: a plain C-Mn Head Hardened (HH) grade [80], a Low Alloy Heat Treated (LAHT) grade [127], and Hypereutectoid (HE) Heat Treated grade [81]

141

Figure 6.11 DV damage parameter vs different contact patch offset with a load of L/V=0.2 for the HW of 18 and 22 mm

142

Figure 6.12 The damage accumulation via the head wear at the underhead radius for an eccentric load of L/V=0.2 at the contact patch offset of 20mm

143

Figure 6.13 Prediction of fatigue life under the extreme loading conditions 144

Chapter 7

Figure 7.1 Illustration of normal and tangential coordinates for a smooth crack [138]

150

List of Figures

xxv

Figure 7.2 Representation of a non-planar crack in three dimensions by two signed distance functions υ and ψ [138]

151

Figure 7.3 A single crack at the gauge corner: (a) crack shape (crack length = R and surface length = 2R); (b) crack orientation ( = 70o); (c) fine mesh at highlighted enriched region including crack and contact patch potions; (d) Contact patch offset (CPO), L/V ratio and position of the crack from the centre of rail head cross-section

153

Figure 7.4 Loading steps at different positions relative to the crack 154

Figure 7.5 The maximum KI (KI max) at the crack front (R = 17 mm, = 90o) under the different loading positions

157

Figure 7.6 The maximum KI (KI max) at the crack front: as a function of crack length R (5, 17, 30) under the loads of L/V =0.2 at CPO = 20 mm

158

Figure 7.7 The maximum KI (KI max) at the crack front ( = 90o) under the loads of L/V =0.2 at CPO = 20 mm (a) no RS; (b) with RS

159

Figure 7.8 The SIFs at the crack front (R = 10 mm) under the local bending with L/V =0.2 at CPO = 20 mm for rail HW = 22 mm.

161

Figure 7.9 The SIFs at the crack front ( = 90o) under the local bending with L/V = 0.2 at CPO = 20 mm

163

Chapter 8

Figure 8.1 Appearance of RCF damage [33]: (a) longitudinal section with gauge corner cracking; (b) Transverse defect

166

Figure 8.2 The model description for multiple RCF cracks at the gauge corner

170

Figure 8.3 Development of fracture surface for Mode I, Mode II, Mode III and mixed-mode-loading of cracks [189, 190]

173

Figure 8.4 Variation of maximum (over the entire crack front position) SIFs during one wheel passage (loading cycle) with different contact patch offsets (CPO)

175

Figure 8.5 SIFs and Keq at the crack front when a wheel passes with CPO = 0 mm, L/V = 0 for crack (R = 10 mm, = 90o) to the rail running surface with HW 22 mm

177


178

Figure 8.7 Figure 8.7 SIFs and Keq at the crack front when a wheel passes with CPO = 15 mm, L/V = 0 for crack (R = 10 mm, = 90o) to the rail running surface with HW 22 mm, for three rail grades: a plain C-Mn Head Hardened (HH) grade [80], a Low Alloy Heat

180

List of Figures

xxvi

Treated (LAHT) grade [127], and Hypereutectoid (HE) Heat Treated grade [80]

Figure 8.8 SIF at the crack front when a wheel passes with CPO = 15 mm, L/V = 0.2 for crack (R = 17 mm, = 90o) to the rail running surface with HW 15 mm

182


183

Figure 8.10 SIF at the crack front when a wheel passes for crack angle = 90o to the rail running surface with different HW, crack sizes R, CPO and L/V, for three rail grades: a plain C-Mn Head Hardened (HH) grade [80], a Low Alloy Heat Treated (LAHT) grade [127], and Hypereutectoid (HE) Heat Treated grade [81]

185

Figure 8.11 The effect of modelling seven multiple cracks on SIF predicted for the central crack with SIF on crack front when a wheel passes with CPO = 15 mm, L/V = 0 for crack angle 90o to the rail running surface with HW 22 mm

187

Figure 8.12 The effect of modeling multiple cracks on Keq SIF predicted for the central crack when a wheel passes with CPO = 15 mm, for crack angle 90o to the rail running surface with HW 22 mm; L/V 0, 0.2.

188

Appendices

Figure A.1 Dimension of generic rail section with a reference point set near the lower gauge corner (underhead radius position) [7]

222

Figure A.2 Eccentric vertical loading and lateral loading of rail [7, 15] 224

Figure B.1 Longitudinal stresses at UHR, 26 km, 41 km, 74 km - parent rail (gauge and field side) [32, 148]

227

Figure B.2 Longitudinal stresses at UHR, 26 km-parent rail (high gauge tension / compression) [32, 148]

228


229


230

List of Tables

xxvii

List of Tables

Chapter 3

Table 3.1 Model parameter values 68

Table 3.2 Results of simulation of rail deformation with respect to the support of elastic foundation

79

Table 3.3 Residual stress scatter in the worn rail section given above [136] 82

Chapter 4

Table 4.1 Effect of track support stiffness 100

Chapter 5

Table 5.1 Effect of stress state at the underhead region due to residual stress 110

Table 5.2 Effect of rail support stiffness on rail longitudinal UHR stresses (S11)

116

Chapter 6

Table 6.1 Material properties and constants of rail 129

Table 6.2 Definition of UVARs for the output variables 130

Table 6.3 Chemical composition [43, 130] 132

Table 6.4 The estimated fatigue limits at the UHR for different high strength rail grades

134

Table 6.5 Approximate comparison of predicted head wear limits for head hardened rail based on fatigue limits with extreme loading cases for different track conditions compared to current head wear limits [124, 148]

140

Chapter 7

Table 7.1 Crack model parameter values 155

List of Tables

xxviii

Chapter 8

Table 8.1 Crack model configurations simulated 172

Table 8.2 Predicted head wear limits based on fracture strength with extreme loading cases representative of different track conditions [valid for studied crack sizes (radius) R =10mm, 17 mm and 22 mm as shown in Fugures 8.2 and 8.10]

186

Appendices

Table B-1 Rail stress data - Longitudinal stresses at UHR, 26 km, 41 km and 74 km - parent rail (high gauge tension / compression) [32, 148]

231

List of Notations and Acronyms

xxix


a and b Major and minor semi-axes BEM Boundary element method C3D4 4 node 3D tetrahedral solid elements C3D8R 8 node 3D solid elements with reduced integration COD Crack opening displacement CPO Contact patch offset CWR Continuously welded rail C Constant (comparable to the fracture strain in monotonic loading) Crack growth rate DB Deutche Bahn DHH Deep Head Hardened rail grade Di Damage corresponding to the ith equivalent stress cycles DV Dang Van E Modulus of elasticity FEM Finite element method FZ, FY Normal and tangential forces Shear modulus GC Gauge corner GCC Gauge corner cracking HDPE High density polyethylene HE3 Hypereutectoid rail grade HH a plain C-Mn Head Hardened rail grade (AS 1085.1) HR High rail HW Head wear IHHA International heavy haul association IRT Institute of Railway Technology k Yield limit in cyclic shear K Stress intensity factor Keq Equivalent stress intensity Keq

max Maximum (over the entire crack front) equivalent stress intensity factor KI Mode I stress intensity factor on the crack front KII Mode II stress intensity factor on the crack front KIII Mode III stress intensity factor on the crack front KI

max Maximum (over the entire crack front) Mode I stress intensity factor KL Lateral (elastic foundation) stiffness/area


xxx

KV Vertical (elastic foundation) stiffness/area kφ Torsional stiffness L Contact patch length (in section 2.2.2.1) L Lateral load L/V Ratio of lateral to vertical wheel load LAHT3 Low Alloy Heat Treated rail grade LEFM Linear Elastic Fracture Mechanics Ls Support length LSM Level set method M Multiple crack run number m+ Positive normal to the crack front MGT Million gross tonnes Mode 1 In-plane tensile opening mode Mode II In-plane shearing mode Mode III Out of plane tearing mode N Vertical applied load n+ Positive normal to the crack surface P Total load P(x) Normal pressure distribution P(x,y) Contact pressure PIRD Pilbra Iron Ore Railway Division Pm Mean pressure Po Maximum contact pressure q(x) Tangential pressure distribution R Crack length or crack radius RCF Rolling contact fatigue RDF Reverse detail fracture rn Rolling radius of the wheel Rrx Transverse radius of the rail RS Residual stresses Rwx Transverse radius of the wheel TSBC Transportation safety board of Canada s Crack separation S Rail span SMises Von Mises stress S11 Longitudinal stress SC Standard carbon SIF Stress intensity factor SWT Smith-Watson-Topper TDs Transverse defects u1 Longitudinal translation U2 Lateral deformation


xxxi

UHR Underhead radius uR1 Longitudinal rotation UVAR1 Equivalent stress EQ, (MPa) (The inequality‘s Equation 6.1 left side) UVAR2 Angle between the x axis and the external normal, n, of the critical plane UVAR3 Angle β between the y axis and the external normal, n, of the critical plane UVAR4 Angle between the y axis and the external normal, n, of the critical plane UVAR5 The shear stress at the critical plane, (MPa) UVAR6 The shear stress amplitude, a(t), (MPa) UVAR7 The damage parameter, Di, at the ith number of the equivalent stress cycle UVAR8 The predicted number of cycles to failure, Nf UVAR-M User output variable V Vertical load Poisson ratio VCCT Virtual crack closure technique WLRM Whole life rail model Ws Support width X Distance from the contact pressure peak to the crack mouth X-FEM Extended finite element method Crack orientation or inclination T Ultimate tensile stress Y Yield stress , Material parameters in Dang Van criterion Fatigue limit in alternate bending , Vertical and lateral bending inertia of the entire rail , Vertical and lateral bending inertia of the rail head only Shear yield stress Fracture toughness Threshold stress intensity factor Number of cycles to failure or crack initiation Number of cycles to failure by ratcheting Peak Hertzian pressure Fatigue limit in reversed pure torsion Strain to failure in a monotonic test , Axial fatigue ductility and axial fatigue strength coefficients /2 Maximum principal strain amplitude Longitudinal stress due to lateral head-on-web bending Longitudinal stress due to vertical head-on-web bending stress ( ) Time dependent hydrostatic stress Longitudinal stress due to lateral bending Thermal stresses Longitudinal stress due to vertical bending Warping stress


xxxii

Longitudinal bending stresses Maximum stress on the maximum principal strain plane The ith equivalent stress ( ) Time dependent shear stress amplitude on the shear plane Equivalent stress , Shear fatigue strength and shear fatigue ductility coefficients Octahedral shear stress Shear stress in a specified shear plane Closed cycle plastic strain range Ratchetting strain per cycle , Average ratchetting axial and shear strains per cycle Wear rate 2R Crack surface length Thermal expansion coefficient Traction coefficient Wheel point φ Crack surface, representative of crack kinking ψ Orthogonal surface representative of crack twisting

Chapter 1. Introduction

1

Chapter 1

Introduction

1.1 Research background and motivation

Introduction 1.1.1

In order for rail operations to be safe and without interruptions, it is vital that the

railway industry maintains rails in the best possible condition. However, rail

maintenance is costly. For example, significant rail maintenance that includes the

reprofiling and repairing of rail defects costs the European Union several hundred

million Euros per annum. Recently, treatment of head checks alone was said to cost the

Netherlands around € 5 million per annum [11]. In the United States, the estimated

annual cost of replacing and repairing worn out and degraded rails is approximately US

$2 billion [12]. The railway companies in other countries spend similarly high amounts.

Two inevitable damage modes, head wear and rolling contact fatigue, are therefore of

major concern to rail maintenance and safety. Rails in high axle load conditions can

withstand severe head wear without increased safety risks. However, the effect of severe

head wear on RCF is not well understood and permitting heavy head wear either

through natural wear or after profile grinding to increase rail life may compromise rail

safety by increasing the risk of failure [1]. RCF is a primary damage mode in high

strength (Figure 1.1) and standard carbon (Figure 1.2) rail steels, occurring

predominantly at the top of the rail surface and in the gauge corner region. Additionally,

damage occurs at the lower gauge corner [6, 7] (Underhead radius, UHR).


2

Figure 1.1 Appearance of RCF damage on high rail [2]

RCF accounts for approximately 90% of all rail defects [13] such as head checks, gauge

corner cracking (Figure 1.1) and transverse defects (Figures 1.2 and 1.3). The

development of transverse defects (TDs) from RCF damage is considered a major

source of rail failure in parent rail under high axle load conditions [8, 13, 14], as shown

in Figure 1.2. Transverse defects, classified as detail fractures (Figure 1.3a) in North

American railroad tracks, account for around 75% of total rail failure in continuously

welded rails [8, 10, 15]. Transverse defects are considered to represent a potential risk

for rapid fracture, which can lead to catastrophic rail failure, as shown in Figure 1.3b [1,

2, 16, 17].

Factors contributing to the rapid fracture of long pre-existing RCF cracks, fatigue

damage initiation at the underhead radius (lower gauge corner) and, in particular, failure

risk associated with transverse defect (TD) development due to crack turning behaviour

(transition from Mode II/III to Mode I), is generally not fully understood for high axle

load conditions. Field investigations indicate that the occurrence of tension spikes at the

Eutectoid rail Hypereutectoid rail


3

rail underhead radius as a result of the lateral head bending of the whole rail profile, and

localised vertical and lateral bending of the head-on-web are significant if this

behaviour is to be understood.

Figure 1.2 Development of transverse defect [1]

Figure 1.3 Transverse defect (TD) developed from RCF damage [picture to the right is from [1, 2, 17], picture to the left is from [16]]

Standard carbon rail

(a) (b)


4

Application of a “Whole Life Rail Model” to high axle load conditions 1.1.2

The most comprehensive methodology dealing with RCF cracks is related to the

Hatfield incident in the UK in the year 2000 [18-21]. The research work arising from

this event includes the development of a ―Whole Life Rail Model‖ ‗WLRM‘ (see Figure

1.4), which is used when designing maintenance strategies for a rail system (Kapoor et

al [18-20, 22-24], Burstow et al [30] and Dutton et al [21]). All phases of crack

development, from initiation to the growth created by contact stresses and from rail

bending stresses, were modelled in this approach. Figure 1.4a shows crack initiation and

early propagation by ratchetting (curve A). As the length increases the crack

propagation rate increases (curve B). However, relatively long cracks move away from

the contact stress field and the rate of propagation drops (curve C) until, finally, the

crack is driven by bending (curve D). The ―Whole Life Rail Model‖ also describes the

sequence of events in the development of RCF that leads to a rail break, as shown in

Figure 1.4b [22, 25, 26]. A rapid fracture resulting from a transverse defect (TD), such

as that shown in Figure 1.4b, may occur if an RCF crack turns down at a certain depth

from the rail head, also driven by tensile bending stress.

The longitudinal bending stresses included in the ―Whole Life Rail Model‖ considered

vertical wheel load, and as a result, the rail bends and a bending moment can be

generated [27]. With rail uplift ahead and behind the rail/wheel contact, longitudinal

tensile stresses develop due to rail vertical bending (reverse bending), as shown in

Figure 1.5. It is these stresses that cause the rolling contact fatigue cracks to turn down

into transverse defects (TD‘s) and make them grow until a final fracture takes place, as

has been reported in references [22, 23, 25-29].

The approach adopted in the ―Whole Life Rail Model‖ for the prediction of crack growth

behaviour appears relevant under high axle load conditions. By comparing the assumed

longitudinal bending response of a rail in the “Whole Life Rail Model‖ with the in-track

measurements under high axle load conditions (Figure 1.6), it has been revealed that the

actual response of the rail section is more complex than that assumed in the “Whole Life

Rail Model‖. In the latter, there is a large and more variable tensile component when the


5

Figure 1.4 ―Whole Life Rail Model‖ ‗WLRM‘ (a):Crack propagation rate v/s length (adopted from [1, 20, 22-24, 30, 31]); (b): Sequence of events in the development of RCF [22, 25, 26]

(b)

(a)


6

wheel is directly on the top of the reference (measurement) position, as shown in Figure

1.6 [1]. This indicates that the effect of global rail bending stresses was considered in

the “Whole Life Rail Model‖ but that the issue of local rail bending (lateral head

bending of the whole rail profile and localised vertical and lateral bending of the head-

on-web) was ignored. WLRM included the effect of discretised sleepers but the

dynamic and thermal effects were not considered

In addition to a whole rail cross-section vertically bending on an elastic foundation, the

rail head also undergoes lateral bending of the whole rail profile and vertical and lateral

bending of the head-on-web. This effect corresponds to an additional local bending

stress being superimposed due to the lateral bending of the whole rail profile and the

localised bending of the rail head-on-web. This results in a much greater magnitude of

tensile stresses (tension spikes, see Figure 1.6) at the underhead radius when the wheel

is directly on the top of the rail head than those associated with rail uplift ahead and

behind the contact patch (as was identified in the post-Hatfield study, see Figure 1.5).

The stress state generated near the underhead radius rapidly changes the stress

components where the geometry of the rail changes due to a sharp radius. It is in this

region that the longitudinal stress shows tension spikes when measured under high axle

loading conditions, as shown in Figure 1.6 [1, 32, 33].

The work presented here is the first step towards modelling the effect of lateral head

bending and localised head-on-web bending, in order to understand the possibility of the

failure risk associated with the rapid fracture behaviour of pre-existing RCF cracks and

fatigue damage initiation at the underhead radius. In particular, a qualitative assessment

of the tendency to form transverse defects due to Mode I transition in the presence of

tension spike at the underhead radius was conducted. In this thesis, underhead radius

stresses due to local bending were calculated for use as an input to fatigue life and rapid

fracture models.

Various researchers [1, 3, 4, 6, 7, 34-45] have analysed rail bending stresses with a

special focus on the additional local bending stresses superimposed due to the localised

bending of the rail head-on-web and lateral bending of the whole rail profile and will be

described next.


7

Figure 1.5 Assumed longitudinal bending response of rail in “Whole Life Rail Model‖ [27]

Figure 1.6 Measured longitudinal stress response under in service loading; example shown for gauge side underhead radius of high rail in 609 m radius curve [1, 32, 33]

Tension spikes

Compressive stresses


8

Eisenmann [34] measured the tensile stresses at the underhead location in the field (at

lower gauge corner), and also calculated theoretically the tensile stresses at the gauge

and field side underhead location for eccentric and inclined loading. The local tensile

stresses on the field side underhead location due to lateral wheel loading were measured

under laboratory conditions and in the field by Sugiyama et al [35]. Marich [36-38] has

also reported that the presence of tensile longitudinal stresses at the underhead radius of

the rail are influenced by an increase in head wear.

Jeong [39], Jeong et al [7, 40], Orringer et al [46, 47] and Lyon et al [48] reported that

the rail head is supposed to be a beam resting on the rail web. The web is considered to

behave as an elastic foundation. Therefore, the local bending stresses are the result of

lateral bending of the whole rail profile and localised bending of the head-on-web.

These stresses are additional to the vertical bending of the whole rail profile (the so

called global bending). Orringer et al [47] analysed the longitudinal bending stresses in

the rail head in detail. They found it to consist of five possible components: (1) vertical

bending, (2) lateral bending, (3) warping, (4) vertical head-on-web bending, and (5)

lateral head-on-web bending. The field measurements made by Orringer et al [47]

showed that, for tangent track, the longitudinal stress caused by rail head-on-web

bending is much smaller than that caused by rail vertical bending. The warping term of

the longitudinal stress is similarly small. On curved track, the additional components of

lateral head bending and lateral head-on-web bending significantly increase the

longitudinal stress.

Jeong [39], Jeong et al [7, 40] and Orringer et al [46, 47] also reported that bending

stresses can be calculated by assuming the rail to be a beam on a continuous elastic

foundation with additional localised bending response of the head-on-web, under

applied contact load and foundations, as shown in detail in Figure 1.7. Their model

includes the vertical (kv), the lateral (kL), and the torsional (kφ) stiffness of the elastic

foundation of the whole rail profile, along with the rail head as a separate beam bending

on an elastic foundation formed by the web. Where, ‗e‘ and ‗f‘ are the locations of

vertical and lateral loads with reference to the shear centre of a rail, as shown in Figure

1.7. Orringer et al [47] also reported the complete live-load stress sequence of five


9

possible components that contributing to longitudinal bending stresses (see Figure

1.7), namely:

(1.1)

where

= Longitudinal bending stress component due to vertical bending (dominant

component due to the vertical wheel loading)

= Longitudinal bending stress component due to lateral bending

= Warping stress

= Longitudinal bending stress component due to vertical head-on-web bending

= Longitudinal bending stress component due to lateral head-on-web bending

Figure 1.7 Representation of localised head-on-web bending and lateral bending of the whole rail profile as a beam model on a continuous elastic foundation with eccentric vertical and outward lateral loading (adopted from Orringer et al [46, 47], Jeong et al [7], and Salehi [49])

𝜎𝑥𝑥𝐵

Head

Web

Base


10

The equations for rail bending stresses are given in Appendix A.

Due to the complete live-load stress sequence of five possible components contributing

to longitudinal bending stresses, , the underhead radius stresses become high for

the high axle loads that are common in heavy haul operations. This produces reverse

detail fractures (i.e. transverse defects that are initiated at the lower gauge corner of high

rail), as shown in Figure 1.8–1.11. This has occurred in North American rail systems [3,

5-9] because the poorly lubricated, heavily worn curved rails on stiff tracks are

subjected to high axle loads. Failures due to fatigue cracking in the underhead radius of

aluminothermic welds have also been found in Australia on rails subjected to high axle

load conditions, as shown in Figure 1.12. In practice, fatigue failure at the lower gauge

corner is generally associated with the presence of pre-existing defects or stress

concentrations in the form of a sharp radius, which may include a cold rolled flow /

shear lip (as was evident in the reverse detail fracture, Figure 1.8-1.11 and complex

local geometry in rail welds.

Jeong [8] specifically mentions flow lips as initiators, in addition to the possible

contribution of higher residual stresses at rail welds (which has been confirmed in

subsequent research on aluminothermic welds by Salehi et al [42-45], and also Mutton

[50] in connection with aluminothermic welds). The reverse detail fracture failure mode

is influenced by longitudinal bending stresses at the underhead radius, as reported by

Jeong et al [6]. In the presence of heavily worn rail, the magnitude of these stresses

could be even higher. Therefore, a pre-existing defect when combined with severe head

wear should be considered as increased risk for rail safety and integrity. The local

bending stresses also superimpose on axial stresses. These include, but are not limited

to, the effect of track bed support stiffness, and residual and continuously welded rail

stresses. Additional stresses are associated with dynamic axle loads and cant deficiency,

and other stresses are produced due to the influence of several wheels.

This research adds to the current understanding of the contributing factors in the rapid

fracture of pre-existing RCF cracks and the initiation of fatigue damage at the rail

underhead radius, especially the qualitative assessment of the tendency to form

transverse defects due to Mode I transition. These investigations have been conducted


11

Figure 1.8 RDF development [16] Figure 1.9 RDF (Courtesy of TSBC [3, 9])

Figure 1.10 RDF development [16] Figure 1.11 RDF (Sperry rail services [6-8])

Figure 1.12 RDF development in aluminothermic welded rail [50]


12

under the high axle load conditions typical of those, which exist in Australian heavy

haul operations. The Mode I transition and rapid fracture analysis results are correlated

with Australia heavy haul operation. The underhead radius fatigue damage modelling

results are correlated with North American railroad systems.

1.2 Aims and objectives

The aim of this research has been to investigate the parameters that affect underhead

radius stresses, in particular, fatigue damage prediction of rail underhead radius (lower

gauge corner) and rapid fractures of pre-existing RCF cracks. A further aim was to

qualitatively analyse the transition to Mode I crack growth of RCF cracks. The research

has been based on field measurements conducted for rails in heavy haul, iron ore

operations. The measurement results show a short duration tension spike at the

underhead radius of a rail as a heavy wheel passes over. This observed stress is due to

the lateral head bending of the whole rail profile and localised bending of the rail head-

on-web and is influenced by the wheel-rail contact load and its location. The

preliminary results revealed that the magnitude of the tension spike increases, and the

region over which tensile stresses occur moves closer to the top of the rail as a result of

an increase in the contact patch offset (CPO), the L/V (lateral (L) to vertical (V) load)

ratio, the head wear and the foundation stiffness. This stress is further enhanced by

residual and thermally induced stresses and may result in fatigue cracking at the

underhead radius, and / or rolling contact fatigue cracks on the gauge corner of the rail

that turn down to form transverse defects (TD). A 3D finite element model of the rail on

discrete elastic foundations was used to undertake the analyses under high axle load

conditions.

In particular, the main objectives of this doctoral study have been:

To demonstrate that the underhead radius stresses are influenced by the wheel-

rail contact eccentricity and the displacement of the contact running band away

from the rail centreline, the magnitude and direction of the lateral (transverse)


13

loading / L/V ratio, the track support and foundation stiffness, and the

continuously welded rail (CWR) stresses.

To address the effects of head wear on rail underhead radius stress conditions

considering: wheel-rail contact eccentricity and / or movement of the contact

running band away from the rail centreline, the L/V ratio (representative of rail

curvature), the foundation stiffness and the residual stresses.

To develop a fatigue life model to investigate the nucleation of fatigue failures

and to examine the fatigue damage life for crack initiation at the rail underhead

radius. To conduct hardness testing to estimate the fatigue limits of the

underhead radius of high strength rail grades used under high axle load

conditions

To develop a rapid crack propagation model to investigate the governing

parameters that affect the unstable propagation and rapid fracture of pre-existing

single and multiple RCF cracks by comparing them with the fracture toughness

values of the high strength rail steels used under high axle load conditions.

1.3 Methodology

The methodology has allowed a systematic analysis of the effects of variables such as

the head wear, the wheel rail contact conditions in combination with track support and

foundation stiffness on the fatigue behaviour of the rail underhead radius, and the rapid

fracture behaviour of pre-existing RCF cracks.

To achieve these objectives the following approaches were followed:

Finite element analysis of in-track bending behaviour, which uses a single rail

on a discrete elastic foundation considering rail local bending at the underhead

radius in the context of heavy haul (high axle load) conditions.

Fatigue life modelling with high cycle Dang Van criterion [28, 51-53] in

FORTRAN-code embedded in ABAQUS 6.11-2 was used to examine the

fatigue behaviour for a range of wheel-rail contact conditions and varying

amounts of rail head wear. The Pamgren - Miner damage accumulation law [54]


14

was used to determine the extent of material damage at the underhead radius and

to predict the number of cycles to failure at a specified reference point at the

underhead radius.

Rapid fracture analysis based on Linear Elastic Fracture Mechanics (LEFM)

using the Extended Finite Element Method (X-FEM), to parametrically study the

unstable growth behaviour of single and multiple RCF crack.

1.4 Thesis outline

This thesis consists of nine chapters.

Chapter 2, Literature review: This outlines studies on the basics of wheel-rail contact

interface, in the context of rolling contact fatigue and wear. The definitions and

terminology of rail defects are broadly reviewed. The chapter also briefly outlines the

method of fatigue and fracture analysis. Separate sections are devoted to a review of the

wear fatigue interaction, allowable wear limits and rail grades under high axle load

conditions.

Chapter 3, Finite element model development: The methodology and background of

finite element modelling is presented in this chapter, which was used to study the failure

mode described in chapter 1.

Chapter 4, Underhead radius stresses: This addresses the longitudinal bending stresses

in the underhead radius positions of the rail that were investigated as a function of the

contact patch position, and the magnitude and direction of lateral loading. The effect of

foundation stiffness and seasonal temperature variations were also investigated.

Chapter 5, Effect of head wear on rail underhead radius stresses: This examines the

effect of head wear and operating conditions (the L/V ratio, the contact locations,

foundation stiffness, thermal and residual stress) on underhead radius stresses. This

study focused especially on the qualitative evaluation of the effect of local rail bending

and the stress state at the rail underhead radius for heavily worn rail. The results enable

an understanding of the mechanism of crack initiation and propagation qualitatively.


15

Chapter 6, Fatigue damage prediction: The quantitative effect of the magnitude of a

tension spike on the probability of fatigue crack initiation in the rail underhead region

was determined using the Dang Van criterion, based on a critical plane approach. The

effects of operational conditions on fatigue behaviour have been quantified. A

comparison of fatigue behaviours was made, based on the characteristics of a number of

heat-treated low alloy and hypereutectoid rail grades used under heavy haul operations,

in order to predict rail wear limits.

Chapter 7, Rapid fracture modelling using extended finite element method: Studies

were undertaken of the tendency to rail fracture due to the change of contact patch

offset from the rail centreline, the (L/V) ratio of lateral (L) to vertical (V) loads, and the

head wear. Different crack sizes, orientations and loading positions have been analysed

to study the rapid fracture behaviour of a pre-existing defect. The extended finite

element method (X-FEM) results revealed that existing RCF cracks, when subjected to

high tensile stresses at the gauge corner region, could contribute to the development of a

rapid (unstable) fracture. The results of this work can be used to examine the influence

of wheel-rail interaction behaviour and rail head wear on the probability of a

catastrophic rail failure developing from RCF cracks.

Chapter 8, RCF cracks under mixed-mode loading: The initiation of RCF cracks

normally occurs in the length intervals along the rail track running surface and / or the

gauge corner of the outer rail head with a typical interspacing of a few millimetres. The

equivalent stress intensity factor was considered in order to investigate the fracture

behaviour under the influence of lateral head bending and localised bending of the head-

on-web. The extended finite element method (X-FEM) results revealed that the existing

gauge corner cracking, under high tensile stresses at the underhead radius, could

contribute to a rapid (unstable) fracture along the crack front of a single crack and

multiple cracks. This chapter examines the influence of wheel-rail interaction and rail

head wear on the probability of rail failure from existing gauge corner cracking and also

demonstrates the differences in the rapid fracture behaviour between a single and

multiple cracks.


16

Finally the conclusions and recommendations for future work are presented in chapter

9.

Chapter 2. Literature review

17

Chapter 2

Literature review

2.1 Introduction

Rails in heavy haul iron ore operations are subjected to very high mechanical loads and

perform in harsh environmental conditions. In the Pilbara region of Western Australia,

high strength heat-treated pearlitic rail steels are widely used [2]. Several service related

conditions under which rail defects develop include: cyclic loading impacted from

rolling stock, wheel-rail profiles, track curvature, lubrication and grinding practices but

wear and rolling contact fatigue are the major concerns in relation to rail maintenance

costs in a railway system. Field experience has shown that rail head wear can affect the

RCF behaviour [1].

This chapter highlights important points concerning basic aspects of the wheel-rail

interface, in particular focusing on previous investigations into rail RCF and wear, with

special reference to the basics of wheel-rail interface, fatigue and fracture analysis as

these aspects are interrelated.

2.2 Basics of wheel-rail interface

In a vehicle / track dynamic system, the wheel-rail interface plays a significant role in

the overall performance of the railway systems. The salient features of the wheel-rail

interface include but are not limited to the shape and size of the contact patch, material

characteristics and mechanical loading conditions. The shape and size of the contact

patch are related to the wheel-rail profile. A schematic describes the wheel-rail contact


18

profile from side and cross-section view, as given in Figures 2.1 (a) and (b)

respectively. As shown in Figure 2.1b, the rail has a non-uniform curvature across the

head and gauge corner region. The rail serves two purposes: to support the weight of the

vehicle and to provide guidance to the wheel. The wheel has conical cross-section rolls

on the head (Figure 2.1b) and makes contact over the railhead. This conical cross-

section also helps to prevent slipping by providing steering ability on a curved track

[55]. The concept of rail cant is discussed further and wheel-rail contact and wheel

conicity in the steering is explained in the subsequent sections.

Figure 2.1 Wheel-rail contact (a) side view, (b) cross-section view [55]

Rail cant 2.2.1

The rail is installed in track at the relevant cant (or inclination) as shown in Figure 2.2.

The rail inclination is 1:40. The rail cant (inclination) of 1:20 would be considered a

normal value, for example, in combination of a conical wheel tread profile of 1:20 (eg

ANZR profile). However the heavy haul rail systems on which this research is based

utilise a number of modified (wear-adapted) wheel and rail profiles; in addition, the rail

cant may differ from the nominal 1:20 value; both 1:30 and 1:40 are also present. 1:40

cant is also used on other countries.

(b) (a)


19

The track comprises two rails laid on sleepers at a particular gauge, as Figure 2.2. The

rails are laid at an angle β, to the sleeper to generally match the angle γ, of the wheelset

profile. This assists in stabilizing the rail against rollover as the normal reaction to the

contact with the wheel passes through the foot of the rail.

Figure 2.2 Track Gauge [56]

Cant deficiency occurs when the super-elevation (or cant) is less than that required for

the curve radius; i.e only occurs if the outer (high) rail is not raised enough and is also

affected by speed. The wheel-rail contact can be idealized through Hertz contact theory,

which does not describe the entire wheel-rail contact, as described in the next section.

Hertz contact theory 2.2.2

Pioneer researcher, Henrich Hertz [57], famously formulated a theory of contact

mechanics between two elastic solids many years earlier than 1982. When two elastic

solids are brought into contact and are subjected to normal load, they deform elastically

to form an elliptical contact area. A classic description of the Hertz contact theory

(HCT) is given in [58]. HCT is based on some important assumptions, as given below

[58, 59].

1. The contacting surfaces are smooth, frictionless and only normal pressure is

transmitted between them.


20

2. The contacting surfaces are continuous and non-conforming [Johnson, 1985,

p 91].

3. Each solid can be considered as an elastic half space [Johnson, 1985, p 92]. The

contact area is small compared to the radii of curvature of the contacting surface.

4. The contacting surfaces are linear elastic, and the strains are small and within

elastic limits.

2.2.2.1 Line contact

The line contact is formed between two elastic parallel cylinders and results in a

rectangular contact area as shown in Figure 2.3a. The normal pressure distribution (x)

is given by HCT as follows [58]

( ) √.

/ (2.1)

Where is the peak pressure at the centre of the contact and ‗a‘ is the half width of the

contact. The peak pressure is

(2.2)

P is the total load on the contact patch and L is its length.

Equivalent radius R and equivalent modulus of elasticity are given by

(2.3)

(2.4)

Where E is young modulus, v is poisson ratio and R is the radius of the cylinder.

Subscripts refer to body 1 and body 2.

The contact size is given by


21

√( )

(2.5)

By combining Equations 2.2 and 2.5

√( )

(2.6)

2.2.2.2 Circular contact

Circular contact is produced by the contact of two spheres, as shown in Figure 2.3b. The

pressure distribution is a semi-elliptical form with a circular contact area. The normal

pressure distribution P(r) given by HCT is as follows [58]:

Figure 2.3 Hertzian contacts: (a) rectangular contact; (b) circular contact [25]

( ) √.

/ (2.7)

Where and a is the radius of the contact circle

(a) (b)


22

The equivalent modulus of elasticity and equivalent radius are calculated from

Equations 2.3 and 2.4 respectively. The equations for contact size, deformation and

maximum pressure as calculated by Johnson [58], are as follows

√

(2.8)

√

(2.9)

√

(2.10)

More recent details of HCT, which include the subsequent relaxation in limitation and,

in particular, the extension of its application to practical situations, are given in [60, 61].

Wheel-rail contact pressure distribution 2.2.3

In a 3D wheel-rail contact problem, the more generic shape of a contact area on the

centre of the railhead surface is elliptical, having major and minor-semi axes of a and b

respectively. The wheel in contact with rail is shown in Figure 2.4. Iwincki [60]

reported governing equations to calculate the dimensions of the elliptical patch from the

curvature of the rail and wheel surface in the contact point, as given below:

.

/

| |

(2.11)

.

/

(2.12)

.

/

(2.13)

Where, N is the vertical load, Rwx and rn are the wheel transverse and rolling radii

respectively and Rrx is the transverse radius of the rail running surface, as shown in

Figure 2.4. The values m and n as a relation of are given in [60].


23

Figure 2.4 Rail-wheel Hertzian contact [60]

The ellipsoidal Hertzian pressure distribution is given as

( ) √.

/ (2.14)

(2.15)

0

(

)

1

0

(

)

1

(2.16)

Where is the wheel load perpendicular to the contact patch, is the maximum

contact pressure. The stresses are presented in a cartesian coordinate system for the

wheel-rail contact case and will be discussed in section 2.2.4. Accordingly, x is along

‗a‘ and y is along ‗b‘ respectively as shown in Figure 2.5.

Rwx

Rrx

rn


24

Figure 2.5 Hertzian pressure distribution at the contact patch [the picture to the left is from [62], whereas the picture to the left is from [63]]

2.2.3.1 Wheel conicity in the steering

In a curve, the leading wheelset tracks towards the high rail, outside of the curve,

following Newton's third law and the trailing wheelset will tend to roll towards the

inside, as shown in Figure 2.6a-b. As the wheels are conical the rolling radius of the

outer wheel is increased while on the inner wheel it is decreased, as in Figure 2.6b. Both

wheels are rotating at the same speed; the larger radius wheel tries to roll further than

the smaller radius wheel, thus steering the wheelset towards a radial alignment, when it

will roll smoothly around the curve. The opposite process happens on the trailing

wheelset as it moves inwards on the curve (Figure 2.6a). This provides creep forces to

yaw the wheelset self-steer relative to the rail. These forces are generated by the leading

wheelset moving out beyond the equilibrium rolling line to give an excess of rolling

radius difference. Similarly, the required steering forces at the trailing wheelset are

generated by moving inwards from the equilibrium rolling line [64].


25

Figure 2.6 (a) Vehicle on a curve; (b) Rolling radius difference [64]

On tangent track the contact normally occurs at the central region of the railhead cross-

section. However, even on tangent track flange contact can occur if the vehicle is

unstable or has hunting behaviour. Due to the profile design we also see the high rail

and outer wheel contact patch move toward the gauge face of the rail. And conversely,

the contact stays central or may move outward on the low rail. The contact patch

dimensions at the gauge corner would depend on wheel-rail profile match. The contact

patch dimensions in the lateral directions are greater as the contact patch moves towards

the gauge corner if wheel and rail profiles are reasonably conformal, which they tend to

be in heavy haul applications. The field reports suggest that conformal wear adapted

(nonstandard) profiles are commonly used in Australian heavy haul railway systems

[50]. The literature also suggest the dimensions of the major semi-axis along the

longitudinal direction are around 8-12 mm [65]. The wheel flange makes contact with

the gauge face and subjects the sides of the railhead to sliding contact which

corresponds to rather severe curving and in this case wear would most likely be the

main mode of deterioration.

(a)

(b)


26

Figure 2.7 Lateral shift of wheelset leads to contact patch offset from the center of railhead cross-section [66]

In all wheel-rail contact cases, especially on curved rail tracks, the contact patch

location changes as the vehicle tends to exhibit lateral shift. Additional lateral forces


27

may arise from cross-level errors or incorrect super-elevation (cant) in curved track. In

real life, the contact patch shape and size vary as the contact moves towards the gauge

corner. A detailed analysis of the lateral shift of wheel rail contact was presented by

Piotrowski et al [66], as shown in Figure 2.7. The lateral shift of a wheel set leads to a

contact patch offset from the centre of the railhead cross-section. The size of the contact

area is very small, typically 80-120 mm2, as stated by Marshall et al [67], approximately

the size of a thumbnail [29]. The shape of the contact area may be circular, elliptical or

even two ellipses (sometimes even joined together, as in Figure 2.7). The contact patch

size and location depends on the rail and wheel profiles and wheel-rail forces. Any

change in rail or wheel profiles (due to wear or due to profile grinding) will change the

contact patch size, shape and location (and thus the running band) [37]. Additionally,

the wheel-rail profile changes continuously due to on-going operational and

maintenance activities that ultimately affect the performance of the system.

Stress field under a Hertzian contact 2.2.4

In order to find Hertzian contact stresses a general representation of contact pressure

with a normal pressure profile p(x) on elastic half space is given in Figure 2.8. The

corresponding Hertzian stresses are represented in both the cartesian and polar

coordinate systems. The contact stresses due to Hertz pressure acting on a circular

contact area can be calculated as follows [58]. The calculations are for normal Hertz

pressure distribution without traction.


28

Figure 2.8 Representation of surface traction on elastic half space [58]

By using the equations 2.17-19, the analytical solution is calculated based on a circular

contact area = 145.3 mm2 for radius a = 6.8 mm, with Pm = 1182 MPa. The contact

stresses distribution versus depth caused by Hertz pressure acting on a circular area of

radius ‗a‘ is plotted in Figure 2.9a.

Similarly the contact stresses as a result of Hertzian pressure distribution of contact

between two cylinders is given by Johnson [58]. The following equations are used for

the calculations of subsurface stresses along the axis of symmetry and are plotted with

ratio of stresses to maximum contact pressure as shown in Figure 2.9b. The

maximum shear is plotted in Figure 2.9c.

2 .

/ 3 (2.20)

.

/ (2.21)


29

[*( ) + ( )] (2.22)

The signs of m and n are the same as the signs of z and x respectively.

Figure 2.9: (a) Stress distribution versus depth caused by Hertz pressure acting on a circular area of radius ‗a‘ [58] (b) subsurface stresses for line contact; (c) contours of principal shear stress for line contact [58]

Stresses for more general cases of contact pressure distribution are provided in ref [58]

2.3 Rail failure mechanism

Different regions of the rail are affected differently due to induced stresses and

deformations as a result of rail operations. In the region close to the contact zone, the

plastic deformation due to the repeated rolling / sliding contact between the wheels and

the rail is of great significance. This plastic deformation causes wear, crack initiation

and propagation leading to rail defects such as head checks, gauge corner cracking and

squats if the contact loads are sufficiently high compared to the strength of the rail

material(s). Several models exist to predict the initiation and propagation phases of

crack development. The shakedown limit approach and a more scientifically advanced

0

0.5

1

1.5

2

2.5

3

-1.5-1-0.50

Dim

ensi

onle

ss d

epth

(z/

a)

Ratio of stresses to mean pressure ( )

(a) (c) (b)


30

methodology, ‗Whole Life Rail Model‘ (WLRM), utilised in the post-Hatfield study, are

the most extensively used approaches. The WLRM was developed through extensive

scientific research activities combined with both numerical modeling and field

correlations. A description of the application of WLRM will be presented in section 2.7.

Several recent studies suggest the use of the shakedown limit approach [68-71] for

describing crack initiation phenomena.

Plastic flow and shakedown 2.3.1

In wheel-rail contact the material is subjected to repeated loading. How a material

responds to repeated loading depends on the magnitude of applied load and this can be

used to explain the concept of a shakedown limit. If the maximum stress is below the

elastic limit, the material will behave in a perfectly elastic manner, as shown in Figure

2.10. However, when the maximum stress is greater than the elastic limit, plastic

deformation occurs. The deformation results in residual stresses when the material is

unloaded. Strain hardening increases the (apparent) yield limit and the steady state

behaviour is perfectly elastic. This is called the elastic shakedown limit, below which

failure is due to high cycle fatigue. This leads to the long life of a component [72, 73].

Figure 2.10 Material response to repeated loading [73]


31

Johnson [74, 75] and Bower et al [76] developed a shakedown map, as shown in Figure

2.11. The traction coefficient ( ) levels are plotted against the x-axis, and the ratio of

the maximum contact pressure, Po to yield limit in cyclic shear, k, of the material is

plotted along the y-axis. The corresponding elastic, elastic-perfectly plastic, and

kinematic hardening shakedown limits are plotted in Figure 2.11.

Bower and Johnson [76, 77] also presented shakedown limits for a line and point

contact with longitudinal and lateral tractions. The relationship for the shakedown limit

of line contact is given below [76]:

{

.

/

}

(2.23)

Where

and .

/

Figure 2.11 Shakedown map for repeated sliding of a rigid cylinder over an elastic-plastic half space [74, 75, 77]


32

Low cycle fatigue and ratchetting failure 2.3.2

When the contact pressure is above the elastic shakedown limit and below the plastic shakedown limit, no accumulation of plastic strain occurs and the phenomenon is called cyclic plasticity with a closed cycle, as shown in Figure 2.10, the cyclic plastic shakedown limit or ratchetting threshold. The failure of material is generally by low cycle fatigue (LCF). Failure by low cycle fatigue life was studied by Coffin-Manson, and the equation given, as reported by Kapoor [72], is:

(

)

(2.24)

Where is the number of cycles to failure by low cycle fatigue, is the closed cycle

plastic strain range, and is constant with a magnitude comparable to the fracture strain

in monotonic loading, and n is equal to 0.5.

If the load exceeds over the plastic shakedown limit, ratchetting occurs, as shown in

Figure 2.10. There is an open elastic-plastic loop, and the material accumulated uni-

directional plastic strain in each cycle though the ratchetting process. For example, the

cross-section of material subjected to rolling contact reveals large unidirectional shear

plastic strain accumulation (>10) near the surface due to ratcheting. This is possible

because of high hydrostatic compression of about 1.5 GPa. The ratchetting failure

occurs if the accumulated plastic strain reaches the critical strain to failure, as the

ductility of the material is exhausted. This failure of surface material leads to cracking

and wear [72, 78].

Kapoor [72] reported that the number of cycles to failure due to ratchetting is as follows

(2.25)

√( ) ( √ ) (2.26)

Where is the number of cycles to failure by ratchetting, is the ratchetting strain

per cycle expressed in terms of and , which are the average ratcheting axial and


33

shear strains per cycle as given in equation 2.26. Kapoor [72] hypothesized that low

cycle fatigue and ratcheting failure are competitive and the criterion in Equation 2.26

should be compared with low cycle fatigue crack initiation criteria. The one that has the

lower number of cycles to failure is activated.

For high axle load conditions, Welsby et al [79] used the shakedown diagram to

illustrate the relative behaviour of standard carbon and high strength rail steels in heavy

haul applications (Figure 2.12).

Figure 2.12 Shakedown diagram and RCF predictions [79] (high rail on a moderate curved track with axle load of 30 tonne)

The latter included plain C-Mn head hardened (HH) grade [80] and hypereutectoid (HE)

heat treated grade [81]. The shakedown map suggests ratchetting is expected in standard

carbon as the kinematic hardening shakedown limit is exceeded for a high rail on a

moderate curved track under an axle load of 30 tonne. Repeated plastic deformation will

lead to surface and subsurface plastic flow and damage, as was discussed in the studies

mentioned above. In the case of a plain C-Mn head hardened (HH) grade, the


34

shakedown ratio decreases within the elastic shakedown region. After some initial

yielding, the material stabilises and hence behaves more-or-less elastically under

subsequent similar loading. Some initial mild deformation can be expected under these

conditions. No accumulation of surface and subsurface damage is expected. For

hypereutectoid (HE) heat treated grade, the shakedown ratio is below the elastic limit

and the material will behave in a perfectly elastic manner and no deformation is

expected under these loading conditions [79].

2.4 Wear and rolling contact fatigue

Many researchers have extensively studied wear and RCF formation in wheel - rail

contact [48, 82-85]. Rail can fail by either wear (loss of profile) or by RCF (long

surface cracks or complete rail break) and requires replacing. This behaviour is of great

significance to the rail industry due to the increased risk of rail failure. A detailed

description of wear and RCF is provided next.

Wear 2.4.1

Wear is defined as damage to one or both surfaces as a result of the progressive loss of

material due to relative movement [60, 86]. Williams [87] has conventionally quantified

wear rate as the volume lost per unit sliding distance i.e. m3 / m = m2. The loss is tiny

but leads to a loss of functionality of the component. Economic consequences of wear

are widespread and pervasive. The wear rate can be calculated according to Archard

wear equation as:

(2.27)

Where is the wear rate, which is directly proportional to the load on the contact

patch P and dimensionless coefficient K, is inversely proportional to material

hardness H. The literature survey on wear fatigue interaction and allowable wear limits

is given in sections 2.7 and 2.8.


35

Rolling contact fatigue 2.4.2

Rolling contact fatigue is the process of crack initiation and propagation due to the

stresses caused by rolling/sliding contact. RCF occurs in ball bearings, gears, wheel -

rail, and many other components where components roll or slide. Wheel - rail material

develops small crack-like flaws as a result of many passes of the wheel. These crack-

like flaws grow into cracks due to repeated loading by a rolling/sliding contact and may

develop into rail breaks [29]. RCF damage occurs, predominantly at the top of the rail

surface and gauge corner regions (see Figures 1.1-1.3). Additionally, damage occurs at

the lower gauge corner (Underhead radius, UHR), as has already been shown in Figures

1.8-1.12.

Figure 2.13 Photomicrograph of an etched sectioned test sample from twin disc test [88]

Fletcher el al [88] presented a computer simulation of wear and rolling contact fatigue

considering surface roughness and additional failure mechanisms, and includes the

integration of crack initiation with the wear simulation as shown in Figure 2.13. They


36

investigated RCF behaviour in terms of cyclic ratchetting material response and wear

simulation, based on a twin disc test.

Surface and subsurface rolling contact fatigue 2.4.3

RCF cracks are generally classified as surface and sub-surface initiated. Typical surface

initiated cracks are found in modern rails as head checks [2, 25, 29, 63, 89], which are

clusters of fine surface cracks with an interspacing of approximately 0.5-10 mm at the

gauge side of high rails [90], as shown in Figure 1.1 [1, 2]. These surface initiated

rolling contact fatigue (RCF) cracks may turn downwards vertically through the rail

head as transverse defects (TD) (Figure 1.2), as examined by Mutton et al [1, 2] under

heavy haul conditions. It was reported that the presence of a single or multiple TDs in

heavily worn rail increases the derailment risk [1].

Figure 2.14 Schematic of the mechanism of RCF crack formation and branching [91]

Datsyshyn et al [91] has also reported the mechanism of RCF crack formation and

branching. A squat type defect was investigated in these studies. Rail bending, the

friction force, the thermal and residual stresses and hydraulic effects were identified as

major contributing factors for RCF crack branching, leading to transverse defects, as

shown in Figure 2.14 in addition to material anisotropy and texture. It was not


37

confirmed which of these factors or combination leads to turning down as a transverse

defect.

Jeong [8, 10] contended that a transverse defect originating close to the running surface

of the rail head, is classified as a detail fracture (DF) by the Federal Railroad

Administration in North America, as given in Figure 2.15 [8]. Jeong [8, 10] defined the

detail fracture as, ― a progressive fracture starting from a longitudinal separation close

to the running surface of the rail head, then turning downward to form a transverse

separation substantially at right angle to the running surface”. A detail fracture is

associated with RCF damage and is classified as a transverse defect (TD) in Australian

railway systems. These are two different names for the same phenomenon in two

different systems. Jeong [8, 10, 15] reported that, based on railroad safety statistics from

the US Department of Transportation, Federal Railroad Administration (FRA), detail

fractures account for approximately 75% of the rail defect population in continuous

welded rail (CWR) tracks in North America. Transverse defects and detail fractures

arise as a result of both surface and subsurface initiated cracking.

Figure 2.15 Broken rail showing shell origin of detail fracture [Courtesy of Transportation Technology Centre, Inc. (TTCI)] [8, 10]


38

Shells are subsurface initiated cracks, particularly the initiation of rolling contact fatigue

cracks at the inclusion bands that are normally present in rail steels that were

manufactured prior to the introduction of improved steelmaking technologies such as

vacuum degassing and continuous casting. Shells can generally be more shallowly and

deeply initiated and different load components are dominant causes for the different

phenomena of surface and subsurface initiation. Shells normally turn down to give rise

to detail facture. Figure 2.15 reveals a typical broken rail showing the shell origin of a

detail fracture [8, 10].

Bower et al [76] reported that the rail surface layer undergoes plastic deformation to a

depth of around 15 mm. Clayton [92] presented a review of experimental research based

on tribological aspects of wheel-rail contact. It was reported that the nucleation of

fatigue cracks is deeper, at around 3-15 mm from the rail head running surface. The

crack nucleation at these positions was attributed to the presence of large tensile stresses

[93, 94]. Sugino et al [93] suggested that shells may occur in the absence of inclusions.

Farris [94] argued that subsurface shells are RCF cracks initiated at inclusion in older

rail steels. The material defects associated with geometric conditions such as flow or

shear lip formation in the case of RDF in parent rail and stress concentration at the

collar edge of the rail weld which give rise to stress concentration. These are the most

common reasons for the initiation of subsurface RCF cracks in high rails under high

axle load conditions.

Rail underhead radius failure 2.4.4

The underhead (UHR) region is away from the contact surface but is influenced by local

bending behaviour due to wheel-rail contact conditions. When the head wear increases,

under the same wheel-rail contact conditions, stresses in the underhead region become

critical. Damage investigation conducted by Canadian Pacific Railway [9] concluded

that the response of the underhead region may also be relevant to the propagation

behaviour of defects, which are initiated at, or close to, the primary wheel-rail contact

region. It is within this region, which is subjected to high contact stress and creepage

levels, that gauge corner collapse, plastic flow, shelling, and the development of


39

transverse defects may occur [9]. In sharper curves, as the curving forces increase, the

rail material potentially results in a flow lip at the lower gauge corner (underhead

radius). With the continued passage of rail traffic, a fatigue defect may be initiated and

propagated into the rail head [9].

Jeong et al [7] studied the influence of longitudinal bending stresses at the underhead

radius (lower gauge corner of rail) on the behaviour of reverse detail fractures (i.e.

transverse defects which initiate at the lower gauge corner of heavily-worn rail). This

defect type, observed in poorly lubricated, heavily worn curved rails on stiff track

subjected to high axle load conditions in the North American rail systems [6, 7, 9] was

shown in Figures 1.8-1.11. A similar failure mode due to the initiation of fatigue

damage at the underhead radius at aluminothermic welds in heavy axle load railway

operation in Australia is shown in Figure 1.12. However, such welds normally exhibit

lower material strength and higher residual stress levels than in normal rail, in addition

to the complex geometry associated with the presence of a weld collar, leading to stress

concentration. A reverse detail fracture is occasionally seen in aluminothermic welds

but generally occurs in the rail underhead [8]. Jeong specifically mentions flow lips as

initiators, in addition to the possible contribution of higher residual stresses at rail welds

(which has been confirmed in subsequent research on aluminothermic welds by Salehi

et al [42-45], and also by Mutton [50] in connection with aluminothermic welds).

Marich [36-38] reported the presence of high tensile longitudinal stresses at the

underhead radius (lower gauge corner) of the rail due to localised head-on-web bending.

The tensile stress was a function of vertical and lateral load eccentricity and was

influenced by changes in the head wear. A comparison of different worn profiles of 60

kg/m and 68 kg/m rail sections was conducted under eccentric vertical and lateral

loading. The FE results revealed that a smaller rail section would be preferable in terms

of allowable rail head wear due to the presence of lower stress values [36], as shown in

Figure 2.16. This did not take into consideration the increased beam strength with the

larger rail, which may be necessary for the axle loads considered. This work also did not

take into consideration the possible impact of RCF damage. Subsequent research by


40

Mutton et al [1] demonstrated that the approach taken by Marich was therefore non-

conservative in the presence of RCF damage.

Figure 2.16 Predicted head stresses in 60 kg/m and 68 kg/m rails [36]

The results of the analysis reported by Marich [37] demonstrated that the longitudinal

stresses at the underhead radius are increased by moving the contact patch away from

the centre of the rail head cross-section on a heavily worn rail, as shown in Figure 2.17.

An increase in axle load from 30 to 35 tonnes, along with a dynamic load factor of 1.5,

results in a corresponding increase in longitudinal stress, as shown in Figure 2.17.

Marich [36] also proposed modifications in geometry at both the gauge and field side

underhead radius positions. The results of these modifications are presented in Figure

2.18. The plots suggest that the current rail sections are far from optimum in terms of

assessment for critical rails stresses. The literature suggests that very little work has

been conducted regarding the modifications proposed at the underhead radius.

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30 35 40 45 50 55 60

Long

itudi

nal s

tres

s (M

Pa)

Percentage rail head loss

68 kg/m, @ 22.5 mm eccentricity60 kg/m, @ 22.5 mm eccentricity68 kg/m, @ 15 mm eccentricity60 kg/m, @ 15 mm eccentricity


41

Figure 2.17 Effect of contact patch offset (CPO) on longitudinal stresses at underhead radius (UHR) for 68 Kg/m rail at nominal axle load of 30 and 35 tonnes, dynamic factor of 1.5 and L/V ratio of 0.2 [37]

Figure 2.18 Predicted rail head stresses for different head designs [36]

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25 30 35 40 45 50 55 60

Long

itudi

nal s

tres

s (M

Pa)

Percentage rail head loss

Dynamic wheel load 220.50 kN, @ 15 mm eccentricity

Dynamic wheel load 220.50 kN, @ 22.5 mm eccentricity

Dynamic wheel load 257.25 kN, @ 15 mm eccentricity

Dynamic wheel load 257.25 kN, @ 22.5 mm eccentricity

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25

Long

itudi

nal s

tres

ses (

MPa

)

Eccentricity of vertical load (mm)

Modification 2Modification 1Standard

347 MPa, Stress limit for SC rails


42

Crack initiation and propagation is of great significance to the rail industry due to the

increased risk of rail failure, as described above. Predictions associated with this

behaviour are based on analytical, finite element analysis and experimental techniques.

However, analytical and experimental techniques are beyond the scope of this thesis.

Finite element analysis techniques are reviewed below. Several models exist to predict

the initiation and propagation phases of crack development. The literature suggests

many approaches and in the following sections the important models for crack initiation

and propagation are described.

2.5 Crack initiation

As described above, in a rail section, the contact and underhead radius regions are two deformation zones that can be simply classified in terms of the mechanical responses caused by the wheel-rail contact conditions. The former region, which is close to the contact surface, is the region of plastic deformation due to repeated rolling/sliding contacts between the wheels and the rail. As a consequence, if the contact loads are high, a crack initiates and propagates in the contact region leading to rail defects such as rolling contact fatigue. This behaviour is of great significance to the rail industry due to the increased risk of rail failure. The possibility of catastrophic rail failure and consequent fatalities has resulted in a great deal of research [48, 82-85, 95-98]. The underhead radius (UHR), which is away from the contact surface but is influenced by local bending behaviour due to wheel-rail contact conditions, results in rail defects such as reverse detail fractures (RDF), as was described in section 2.4.4. As stated above, the underhead radius (UHR) becomes critical when the wear increases in the rail head under the same wheel-rail contact conditions.

A study of the cyclic ratcheting material response and the RCF of a pearlitic rail steel was conducted by Ringsberg et al [68], based on Dang Van criterion, Coffin-Manson and SWT criterion and ratchetting failure. Dang Van criterion will be described in the next section 2.5.2. In Coffin-Manson, the failure of material is due to low cycle fatigue. For shear dominated failure, this is based on maximum shear strain amplitude, as given in Equation 2.28.


43

( )

( ) (2.28)

Where and are the shear fatigue strength and the shear fatigue ductility coefficient, b and c are the fatigue strength and the fatigue ductility exponents, G is the shear modulus and is the number of cycles for crack initiation.

The SWT is an energy based criterion and is based on mean stress, as given in Equation 2.29

. /

( )

( )

(2.29)

In Equation 2.29,

is the maximum principal strain amplitude and is the

maximum stress on the maximum principal strain plane. In addition and are the

axial fatigue ductility and axial fatigue strength coefficients, b and c are the fatigue

strength and the fatigue ductility exponents. The approach given in Equation 2.28 by

Coffin-Manson underestimated the crack initiation, whereas the SWT approach

(Equation 2.29) overestimated it but later predicted better results

For cyclic ratchetting material responses, Ringsberg et al used finite element (FE) simulations for different magnitudes of contact pressure and traction forces in the rolling contact. A large accumulation of shear deformation of material was found at the sub-surface of the running band. The ratchetting material model calculates the accumulation of shear plastic strains on the contact surface under different contact pressures and traction forces. RCF damage develops when the accumulated strain reaches the critical value for the rail material [68]. Two rail materials have been considered in order to prevent the ratchetting related failure described by Ringsberg et al [70]. They used functionally-graded material, i.e. the application of a surface coating. This is an option, but it is unclear if this would prevent ratchetting failure under all conditions.


44

Multi-axial fatigue criteria 2.5.1

A number of methods have been proposed for the prediction of crack initiation based on multi-axial fatigue criteria. The hypothesis for these studies is generally based on the theory of plasticity with a suitable fatigue criterion for rail life prediction. Equivalent stress criterion proposed by Dang Van [28, 53, 99-103] and a more complex energy-density based model proposed by Jiang and Sehitoglu [104] are the most extensively used approaches.

To predict fatigue crack initiation, a comparison of multi-axial fatigue criteria as applied to rolling contact fatigue was conducted by Conrado et al [105] and Ciavarella et al [106]. Ciavarella et al [106] compared the Dang Van, the Crossland and the Popadopoulos criteria and results of the comparison were obtained. Conrado et al [105] investigated the comparison between the Dang Van and the Liu-Zenner criterion for an assessment of fatigue limits of wheel-rail contact. Significant differences of predicted contact fatigue limits were obtained. It was suggested that careful correlation with experimental data is needed for the validity of predicted results using these criteria.

Ekberg et al [107, 108] developed an engineering model for fatigue damage prediction

in railway wheels. A comparison of sub-surface and surface initiated fatigue damage

was conducted based on the Dang Van and the shakedown model respectively, to

determine the dominating mechanism. The strengths and weaknesses of each model for

wheel fatigue damage analysis were elaborated. In another study, Ekberg et al [109]

reviewed multi-axial fatigue criteria as applied to rolling contact fatigue for the Sines,

the Crossland and the Dang Van criteria and a comparison of these criteria was

presented based on the strength and weaknesses of each model. As the wheel-rail

contact loads cause a multi-axial state of stress with out-of-phase stress components and

varying principal stress directions, it was suggested that the Dang Van criterion would

be suitable under these circumstances, as compared to the Sines and the Crossland

criteria and was thus required for predicting fatigue crack initiation and damage life in a

rail for this study. Ekberg et al [62] also used a high cycle fatigue model based on the

Dang Van theory for fatigue life and damage prediction in wheel – rail rolling contact.

The Dang Van fatigue initiation criterion has previously been used in many industrial


45

applications, particularly in the automotive industry [28, 52]. The fatigue criterion is

based on a multi-scale approach and on a shakedown limit hypothesis.

Dang Van criterion 2.5.2

The Dang Van (DV) [28, 51, 53, 99-103] criterion is a shear stress based criterion,

which is applicable for stress levels below the elastic shakedown limit of the material. If

the following inequality is satisfied on a shear plane passing through each material point

at least once in the whole stress cycle, damage occurs. This inequality is expressed as:

( ) ( ) (2.30)

The value of the inequality‘s left side represents a numerical index for fatigue damage.

τa(t) is the time dependent value of shear stress on the specified shear plane at the

specified material point and is defined as the difference between the instantaneous and

mean shear stress of the loading cycle; σh is the time dependent hydrostatic stress at the

material point. The constants (aDV and bDV) are the functions of material fatigue limits

[28]. The fatigue limits, f-1 and t-1 can be obtained from the classic experimental bending

and twisting tests, respectively. The constants are calculated as:

( )

(2.31)

As reported by Dang Van et al [52, 53, 99], if the rail material fatigue limits are, f-1 =

470 MPa and t-1 = 270 MPa, then the fatigue prediction loading path at different points

of the rail surface suggests fatigue failure may be initiated in the central zone of the rail

head surface due to the presence of shear traction as shown in Figure 2.19.and that

ought to depend heavily on contact geometry etc,


46

Figure 2.19 Fatigue prediction loading paths at different points of the rail surface [52, 53, 99]

Critical shear plane 2.5.3

The plane on which the mentioned inequality (Equation 2.30) is satisfied is called the

critical plane. However, the critical plane is not obvious at the beginning of the analysis,

so the inequality should be assessed in all shear planes passing through each material

point being investigated for potential fatigue crack initiation. A description of critical

plane with shear stress, at a material point ‗O‘ in a shear plane defined by normal

vector ‗n‘, with respect to coordinate system ‗xyz‘ (adopted from Ekberg et al [62]) is

given in Figure 2.20. The stress history during a stress cycle can be evaluated for

comparison to the fatigue threshold limits of the material. The Dang Van critical plane

approach is advantageous, compared to the Sines or Crossland criteria [109]. Ekberg

[103] also employed another approach based on the amplitude of mean shear stress,

defined as the centre of the minimum circle circumscribing the tip of the shear stress

vector of the loading cycle. The aforementioned approach was adapted in this thesis.


47

Figure 2.20 Representation of critical plane with shear stress at a material point in a shear plane defined by normal vector ‗n‘, coordinate system xyz (adopted from Ekberg et al [62])

Damage Accumulation 2.5.4

To quantify the damage, the Palmgren-Miner linear damage accumulation rule [103], in

conjunction with the Wöhler curve [62], was also used. In terms of the Palmgren-Miner

linear damage accumulation rule, the damage degradation can be determined through

the following expression:

Di = ( ) ( ) (2.32)

Where, Di: is the damage corresponding to the ith equivalent stress cycles, i: is the

number of the shear stress cycles, amd τEQi: is the ith equivalent shear stress cycles

calculated by Eq. (2.30). The damage degradation at each material point via the ith stress

cycles to failure, Nf, can be defined as:

Di = 1/Nf. (2.33)

O


48

2.6 Linear elastic fracture mechanics

Linear elastic fracture mechanics (LEFM) are based on the stress intensity factor (SIF).

The stress intensity factor is used to quantify the stress state (―stress intensity‖) at the

crack tip caused by multi-axial loading that is either remotely applied or is residual

stress. The stress intensity factor (K) is defined as the product of the nominal stresses in

a body, and the square root of the half-length of the crack (a), as given in Equation 2.34

√ (2.34)

Figure 2.21 Modes of crack growth (a) Mode I, opening (b) Mode II, shearing (c) Mode III, tearing [20]

Where Y is a factor depending on the geometry and location of the crack and the

loading conditions. K can be used to predict fracture, but to predict crack growth , ΔK is

needed. The crack will grow when a crack stress intensity factor range (as a result of

repeated loading of rail) exceeds an experimentally determined threshold. There are

three modes of growth, categorised as Mode I (opening), Mode II (shearing) and Mode

III (tearing), as shown in Figure 2.21. The calculated stress intensities are converted into

a crack growth rate by using a crack growth law [20, 110-113].


49

Crack propagation and rapid fracture 2.6.1

The possibility of crack propagation and the rapid fracture of pre-existing RCF under

Mode I loading has been investigated by a number of researchers. Most of the attempted

crack growth studies use finite element analysis procedures. In a study to develop a

strategy for the life prediction of RCF crack initiation and propagation, Ringsberg [114]

conducted multi-axial fatigue analysis to predict the fatigue crack initiation and

propagation life of RCF cracks, as shown in Figure 2.22. Elastic plastic finite element

modeling was conducted, along with multi-axial fatigue crack initiation analysis, using

a critical plane approach. These investigations were correlated by laboratory and field

measurements.

Figure 2.22 Three phases of the life of a RCF crack initiated at the surface [95, 114]

In another study, Ringsberg et al [95] described the crack propagation stages in which

small cracks with a length of around 0.1 mm initially grew at a shallow angle of 10-25o

until they reached a critical length of 1-2 mm. At this length the fatigue parameter as a

weighted sum of products of stress and strain at the crack tip will govern the crack

propagation either upward or downward to the rail surface, as shown in Figure 2.22.

The same behaviour of crack branching was reported by Wong et al [115]. Fischer et al

[116] investigated crack propagation behaviour influenced by the loading and material


50

characteristics of rail steel. Initiation and propagation of cracks in rail steel are

unavoidable due to rail operations. The qualitative result suggested that the fatigue

resistance of a rail against crack propagation could be improved by variations in

strength of the rail materials. Fischer et al [116] also reported that a crack branches

downward leading to a break in the rail if high tensile stresses due to low temperature

are present and the Mode I stress intensity factor exceeds the threshold.

Figure 2.23 Fluid assisted mechanisms of crack growth: (a) shear mode crack growth, accelerated by fluid reduction of friction between the crack faces, (b) hydraulic transmission of contact pressure, (c) entrapment and pressurization of fluid inside the crack and (d) squeeze film pressure generation [117, 118].

Fletcher et al [117, 118] investigated the effect of fluid penetration in RCF cracks that

leads to modification of the crack face friction and ultimately affects and accelerates the

Mode II growth of RCF cracks. Figure 2.23 presents shear mode growth influenced by

friction between crack faces, and shear mode crack growth that is accelerated by fluid

due to a reduction of friction between the crack faces. Other fluid assisted mechanisms


51

of crack growth, such as hydraulic transmission of contact pressure, entrapment and

pressurization of fluid inside the crack and squeeze film pressure generation are also

shown in Figure 2.23.

Fletcher et al [119] presented semi-elliptical crack modelling using a boundary element

approach. The contact load was included along with variable contact patch shapes

passing on different crack shapes and orientations. Green‘s function was used to

quantify the SIF at the crack faces. Kapoor et al [18] conducted an examination of the

influence of residual stress distribution on the growth of RCF cracks. During crack

closure, the residual stresses can increase the stress intensity factor and change the

loading cycle. It was found that the compressive residual stresses in the vertical

direction were more significant than the longitudinal direction, as they suppress the

tendency for RCF cracks to develop Mode I growth [18].

Liu et al [120] conducted crack propagation analysis based on a critical plane approach

for a semi elliptical crack in a wheel representative of subsurface initiated cracking. As

the wheel is subjected to the same contact pressure, a subsurface crack in the wheel is

therefore comparable with a RCF crack in the rail. Additionally, wheels may contain

different initial residual stress distributions, due to differences in the manufacturing

processes between wheels and rails. In this study both the crack propagation direction

and the growth rate were investigated. The results suggested that a crack propagates

faster in a semi major axis direction as compared to semi minor axis direction.

Bogdanski et al [121] studied crack propagation by considering the stress intensity

factor influenced by crack pressurization due to water entrapment. A semi elliptical

crack was considered, located on the surface of a block with fixed boundary conditions.

The results suggested that the entrapped water increased the opening mode stress

intensity factor resulting in an increase in the crack growth rate.

Farjoo et al [122] conducted 2D crack modelling on the surface of a rail laid on 9

sleepers, using FEM and X-FEM modelling approaches that incorporate the elastic

foundation effects. The sleepers and ballast were modelled as springs. It was found that

stresses due to the elastic foundation increased the crack growth rate. Farjoo et al [123]


52

also conducted a 3D squat model and incorporated the effect of water entrapment,

lateral traction and elastic foundation on the stress intensity factor by using the

displacement at the crack tip, using an extended finite element modelling (X-FEM)

technique. In all of the analyses, the effect of loading eccentricity and head wear were

not considered.

The growth rate of multiple rolling contact fatigue cracks under the influence of tensile

longitudinal stresses in the rail was investigated by Fletcher et al [27], who found that

multiple cracks shield each other, thus reducing the crack growth rate. The crack growth

rate was found to increase as the bending stress increased. They studied the influence of

crack spacing for longer cracks using a boundary element method to investigate SIF (KI)

at the crack tip. It was found that KI is a function of crack spacing and that wider crack

spacing is directly proportional to KI, which leads to an increase in the cracking growth

of RCF cracks.

Dutton et al [21] presented a bending model to investigate the effect of an elastic

foundation on the stress intensity factor. The results suggested that for smaller cracks

vertical bending does not contribute significantly to crack growth, and that cracks larger

than a 30 mm radius are driven by vertical bending [19]. The modelling approach did

not consider crack closure and this resulted in negative values of KI.

Sandström et al [111, 112] investigated the risk of rail breaks from a mechanical and

statistical point of view. The influence of impact loads from flat wheels was particularly

considered. The flats increased the bending moments (and resulting stresses). However

depending on the impact position it did not have to influence the growth of an existing

crack. Hence, the influence on fatigue crack growth was minor. In contrast, bending

stress was a major concern in regards to the final fracture.

In an important study, Zerbst et al [124] conducted a damage tolerance investigation of

the rail for a typical squat defect approximating a semi-elliptical shape with different

sizes and an orientation of 70o on a rail running surface with a head wear of 10 mm. The

effects of the contact patch offset due to hunting, and the wear of wheel / rail profiles

were considered. The out-of-plane tearing mode was not investigated. Furthermore,


53

bending of the rail head-on-web due to the eccentricity of the contact load was not

considered in the crack growth investigations conducted by Fletcher et al [27, 119] and

Dang Van et al [28]. However, Jeong et al [6] included this behaviour in an

examination of the growth behaviour of reverse detail fractures that were initiated at the

lower gauge corner of heavily-worn rail. Kapoor et al [23] stated that relatively little

work had been done on large cracks that approach a critical crack length and may

rapidly propagate and result in rail failure. This effect is of increased importance under

high axle load conditions and for increased head wear [1].

More recent studies by Ranjha et al [3, 4] reported the occurrence of tensile stresses

resulting from lateral bending of the rail head and a localised response of the rail head-

on-web bending directly under wheel loading, resulting in tensile bending stresses at the

underhead radius. This effect is highly localized and is additional to the stresses

generated due to bending of the whole rail profile (so called global bending). These

stresses can initiate a crack at the underhead radius. In addition to head wear the

magnitude and direction of the L/V ratio (lateral (L) to vertical (V) load), the effect of

the contact patch offset (CPO), the foundation stiffness, and the residual and thermal

stresses were considered. The magnitude of these stresses may increase with increasing

rail head wear. In practice, as a result of variations in wheel-rail contact conditions, the

peak tensile stress at the underhead radius in both the gauge and field side of rail may

fluctuate considerably. A typical tension spike at the rail underhead radius occurs in a

narrow band corresponding to approximately 100 mm of wheel travel. Increases in

wheel-rail contact offset from the rail centreline and lateral forces, i.e. the ratio of lateral

to vertical loads (defined as the L/V ratio), that occur as a result of vehicle steering,

curving and hunting, may result in an increase in the longitudinal tensile stress. In

addition, the depth below the running surface at which the stress becomes tensile

decreases, which means that the region subjected to tensile stresses extends further from

the rail underhead radius towards the rail surface, and could result in an increased

tendency for Mode I crack growth behaviour in any pre-existing rolling contact fatigue

cracks [3, 4].


54

2.7 Wear fatigue interaction

The ‗Whole Life Rail Model‘ developed by Kapoor et al [18, 23] and Dutton et al [21]

describes the crack growth rate during the life of the RCF crack from initiation to final

fracture, as shown in Figures 1.4 (a) and (b) and was described in detail in introductory

chapter [20, 22, 23, 30, 31]. A rapid fracture resulting from a transverse defect (TD)

such as that shown in Figure 1.4b [22, 25, 26], may occur if a RCF crack turns down at

a certain depth from the rail head, additionally driven by the tensile bending stress. The

extensions of ―Whole Life Rail Model‖ application to practical situations are described

next.

Fletcher et al [118] reported that wear affects crack propagation. Figure 2.24 describes

the effect of wear, in which a thin layer of material removed by wear truncates the

crack. Crack growth is the difference between fatigue led crack propagation minus

crack mouth truncation due to wear, as was represented by the expression given below

[22, 23]. Further, wear will also change the contact geometry and thus influence both

wear rate and RCF damage magnitudes.

Crack growth = crack tip propagation – crack mouth truncation

Figure 2.24 Effect of wear on crack growth [22, 23]

The modification to the Paris equation is

( )

.

/ (2.35)


55

High wear rate can lead to the removal of cracks. Grinding is a standard practice to

control rail degradation.

Franklin et al [78] presented the modeling of wear and crack initiation in rails. Different

wear rate levels were selected to analyze the effect of wear on crack growth rate, as

shown in Figure 2.25.

Figure 2.25 Life cycle of a crack in a rail [125]

If the wear rate is high, as per level 1, no crack formation is evident. At level 2, the

crack is truncated by wear and the crack will reduce in length or disappear. At level 3,

the crack growth rate is positive and the crack will initiate and propagate leading to rail

fracture [78].

Generally rail can fail by either wear (loss of profile) or by RCF (long surface cracks or

a complete rail break). The length of life affected by wear (line B) decreases with an

increase in the wear rate, whereas the RCF life (line A) increases as the net crack

growth rate drops. The superimposing of these two lives gives the replacement life, as

shown in Figure 2.26. The peak on the curve represents the point of maximum life.

Historical experience reveals that it is difficult to maintain operation at the point of

maximum life, as variable operating and loading conditions reduce the optimum life


56

point. Both A and B have the same life, but B is inherently safer. These rail life

management strategies are valuable for the railway industry in terms of both rail safety

and economic benefits [22, 23, 25].

Figure 2.26 Strategies for rail life management [22, 23, 25]

2.8 Allowable wear limits

Marich [38] measured rail stress levels under both laboratory and field conditions to

define acceptable rail wear limits for high axle load conditions, based on the fatigue

behaviour of the rail material. Marich found an acceptable head wear limit of 27 mm for

600-800 m radius curve at an L/V ratio of 0.3, deemed to be outward for rail material

with a fatigue strength of 240 MPa. The head wear limit could be reduced to 20 mm as

an additional tensile stress of 80 MPa was added, due to the differential thermal

stresses. It was found that the rail wear limits proposed by Marich could be considerably

greater than those currently accepted in practice [33]. Marich [36, 37] identified the

underhead radius (UHR) as one of the most critical locations in determining the head

wear limit in the presence of a heavily worn rail profile. This work did not take into

consideration the possible impact of RCF damage. Subsequent research by Mutton et al

A B


57

[1] demonstrated that the approach taken by Marich was therefore non-conservative in

the presence of RCF damage.

Jeong et al [126] undertook rail strength investigations in relation to rail wear limits

based on the fracture strength of the internal transverse defects, classified as detail

fractures in North American Railroad systems. This study shows that for safe operation

on railroad track, allowable head wear limits should be estimated on the basis of

fracture strength. The research work conducted by Jeong et al [126] predicted head wear

limits for the lightest rail section. The head wear limits were estimated as 1.27 cm head

height wear or 1.52 cm gauge face wear, based on an assumption that a rail is inspected

for internal defects every 20 million gross tonnes (MGT). The limitations of this

research, to be used for the estimation of wear limits, include the accumulated MGT, the

axle load for different rail sizes, materials and also the above rail parameters.

Lyon et al [48] reported on a method to estimate rail wear limits based on a fracture

mechanics approach. A limitation of this study was that contact stresses were not

considered, based on an assumption that compressive stresses do not contribute to crack

growth. Also the axial (longitudinal) component of Hertzian stresses was calculated to

be sufficiently low that it would be neglected at depths where detail fractures (DFs) are

initiated.

Ranjha et al [17] examined unstable crack growth for long cracks at the gauge corner of

rail resulting from multiple GCC. In that investigation, a large surface crack was

introduced at the gauge corner. Crack growth behaviour with different contact loads

applied to the rail head at varied locations was evaluated in respect to different rail worn

profiles. The influence of tensile bending stresses at the underhead radius, associated

with contact loads, along with a combination of the local response of the head-on-web

bending and the lateral bending of the whole rail profile [3, 4], were included. As the

crack grows, the growth rate is dependent on whether it is driven by bending stresses or

contact stresses. The bending stresses associated with this behaviour are significant in

phase 3. It was predicted that the presence of high tensile underhead radius stresses

could lead to catastrophic rail failure as a result of the rapid fracture of pre-existing RCF

cracks extending to the underhead radius (UHR) region.


58

Estimation of head wear limits is critical for rail performance and plays a vital role in

terms of maintenance activities throughout the life cycle of a rail. In practice, a head

wear limit of 68 kg/m rail was stated by Duvel et al [13] to be in the range of 20 to 15

mm in curves of decreasing radii. The current head wear limit is set at 22 mm for

tangent track and 15 mm for tight rail radius curves, as reported by heavy haul railways

in Australia [33]. Typically, head wear (HW) of 20 mm is allowed for Deutche Bahn

(DB), as reported by Zerbst et al [124]. Therefore, head wear limit estimation and

predictions of rail life based on wear rate are very important research topics requiring

careful consideration. Investigation into rail parameters, including the different section

curve radii and geometric features offer huge potential for future research.

2.9 Material grades in high axle load rail operations

Rail material grades play a significant role in performance in terms of wear and rolling

contact fatigue. Selecting a high strength rail grade can resist wear and increase the

service life of rails, but it may be uneconomic for some railway operations. For high

axle load rail operations, high strength, heat-treated rail grades such as a plain C-Mn

Head Hardened (HH) grade [80], a Low Alloy Heat Treated (LAHT) grade [127], and a

Hypereutectoid (HE) heat treated grade [81] are commonly used, as these provide

increased resistance to damage in the form of wear and rolling contact fatigue [2, 128-

130]. Results for hardness distributions along the vertical transverse at the rail centreline

are plotted for high strength rail steels, as shown in Figure 2.27. Hardness throughout

the head of these rail types decreases with an increase in depth below the outer surface.

For Eutectoid and 1% C HE rail grade, near surface hardness varies from 375-430HV.

These grades show a trend in reduction of hardness with depth, particularly in head

hardened rail [2].

The proof stress (yield stress) for Head Hardened rail grades is around 600 MPa,

increasing to 1100 MPa for the Hypereutectoid (HE) heat treated grade, as shown in

Figure 2.28. The relationship between proof stress and hardness for low alloy heat

treated (LAHT3) rail grade is also shown in Figure 2.28. Fatigue limits are calculated


59

through rotating bending tests. Figure 2.29 presents the relationship between the tensile

strength (fracture stress) and fatigue limit [130].

Figure 2.27 Hardness distribution along vertical transverse at rail centerline [2]

This shows that with an increase in tensile strength the fatigue limit increases. Although

the ductility and the surface roughness of the material are the contributing factors that

influence the fatigue resistance generally the Hypereutectoid, (HE) rails (0.9%- 1% C)

have high fatigue limit, therefore fatigue defects are less likely to originate. These

calculations are for material on the rail surface. With depth, the hardness decreases, as

does the tensile strength and, ultimately, the fatigue limit. Therefore a thorough analysis

of hardness testing should be conducted for rail grades under investigation for an

estimation of fatigue life.

Alwahdi et al [128, 129] conducted an investigation on the effect of rail material grades

(pearlitic and bainitic) and surface roughness on the wear of rail steels. The preliminary

results showed that the wear rate of pearlitic rail steel was found to be lower in

comparison to bainitic rail steel at the similar hardness level.

250

270

290

310

330

350

370

390

410

430

450

0 5 10 15 20 25 30 35 40 45 50

Har

dnes

s (H

V20

)

Distance from surface (mm)

HE3LAHT3HH (AS1085.1)


60

Figure 2.28 Relationship between yield strength and hardness in eutectoid and hypereutectoid heat treated rail steel [2]

Figure 2.29 Relationship between tensile strength and fatigue limit [130]

400

500

600

700

800

900

1000

1100

1200

250 300 350 400 450

0.2%

Pro

of st

ress

(M

Pa)

Hardness (HV)

HE3LAHT3HH (AS1085.1)

400

420

440

460

480

500

520

540

560

580

600

1000 1100 1200 1300 1400 1500 1600

Fatig

ue li

mit

(MPa

)

Tensile strength (MPa)

HE3 (1.0%C)HE (0.9%C)DHH (0.8%C)


61

Residual and thermal stresses 2.9.1

Generally, there are two methods of rail straightening which affect the residual stresses

during rail manufacturing, namely roller [80] and stretch straightening [131]. In

addition, repeated rolling contact between wheel and rail is another important parameter

that can lead to a complex distribution of residual stresses in the rail. Tensile residual

stress at the rail underhead region could be detrimental to fatigue behaviour. The

differential temperature due to seasonal effect is another factor that influences the stress

state at the UHR in continuously welded rail (CWR) due to buildup of thermal residual

stresses, which are uniformly distributed in the cross section.

2.10 Summary

This chapter reviewed the basic concepts of contact mechanics, in particular the Hertz

contact theory, Hertzian contact stresses, plastic flow, shakedown, wear and rolling

contact fatigue, from the point of view of their definitions and their relationship with

their application in wheel-rail contact mechanics. It provided definitions and

terminology for rail defects that are found widely and described a method of fatigue and

fracture analysis. It also briefly outlined theories related to crack initiation based on

fatigue analysis. Furthermore crack propagation and fracture analyses based on linear

elastic fracture mechanics were carefully reviewed. Separate sections were devoted to

the review of wear fatigue interaction, allowable wear limits and rail material grades

under high axle load conditions.


Chapter 3. Finite element model development

63

Chapter 3

Finite element model development

3.1 Introduction

This chapter presents the basics of finite element modelling strategy, which was used to

study the failure mode, as described in chapter 1. The proposed numerical models are

based on the Finite Element Method (FEM), Multi-axial Fatigue Analysis into

ABAQUS through UVAR-M (User output variable) and an extended finite element

method (X-FEM).

Firstly, a model of a rail section was constructed using the Finite Element Method

(FEM), to calculate the stress state around the underhead radius reference point by

considering the loading applied to an elliptical Hertzian contact patch on a rail surface

with a fully slipping condition, to give a worst case.

Secondly, several new finite element (FE) models were developed to study the effect of

worn profiles. These will be explained in chapter 5. The FE models developed to

calculate stress state were extended to evaluate fatigue damage around the underhead

radius (UHR) reference point. The Dang Van criterion was programmed into

FORTRAN-code and implemented into ABAQUS through UVAR-M (User output

variable), in order to identify any potential fatigue damage. This model development

will be explained in detail in chapter 6.

Thirdly, a numerical model based on the extended finite element method (X-FEM), was

used to examine the unstable RCF crack growth behaviour of long cracks in rail. The

multiple cracks introduced at the gauge corner of the model in the transverse orientation


64

had the approximate shape of typical long and turned down head checks that occur

under heavy haul conditions (Figure 1.2). These modelling details will be explained in

chapters 7 and 8.

3.2 Finite element model development

Geometric model 3.2.1

The geometric model used a 68 kg/m rail section corresponding to AS 1085.1 [80], a

rail section commonly used in Australian heavy haul railways, as shown in Figure 3.1.

The rail height is 185.7 mm and the minimum thickness of the rail web is 17.5 mm. The

width of the rail head and the base of the rail are 74.6 mm and 152.4 mm respectively.

The vertical and lateral bending inertia for a typical unworn or new rail are 39.4 x 106

mm4 and 6.02 x 106 mm4 respectively as reported in AS1085.1 [80]

Figure 3.1 68 kg/m rail [80]


65

The rail is installed in track at the relevant cant (or inclination) as was shown in Figure

2.2. The rail inclination is 1:40. In the current model the rail inclination was not

considered as the Hertzian contact patch was modelled instead of the actual wheel. It is

customary to do FE analysis relative to rail axis.

Flow diagram of the FE model 3.2.2

The detailed schematic for FE model development based on rail geometry, material

properties, loading and boundary conditions is shown in Figure 3.2.

Figure 3.2 Flow diagram for FE model solution technique development based on rail geometry, material properties, loading and boundary conditions

-nodes 134399


66

Track modelling 3.2.3

The IHHA (International Heavy Haul Association) publication [132] provides more up-

to-date guidelines on track designs for heavy haul operations. Types indicated in the

Australian Standards for the heavy haul systems for which the current work is

applicable suggest concrete sleepers at 600 mm spacing, elastic fasteners (resilient

fastenings generally Pandrol "e-clip"), resilient high density polyethylene (HDPE)

rail/sleeper pads, 68 kg/m continuously-welded rail, an axle load of 35 tonnes and

ballast with a nominal depth of 300 mm [80, 89, 132, 133]. Ballast depths are typically

300 mm at track construction but sometimes get deeper, due to the lifting/tamping

methods that are used for line and top re-alignment. However, current ballast depths and

the condition of the ballast will vary as a result of track degradation and ongoing track

maintenance activities over the lifetime of the infrastructure. For this reason, one aspect

of interest in this thesis has been to investigate the effect of ballast pumping and

additional effects such as degradation and looseness in sleepers, fasteners, rail pads and

subsequent variations in track stiffness levels on the response of the rail. This was

achieved by applying varying stiffness values to the elastic foundation in the lateral and

vertical directions of the rail, as will be described in the next section on FE modelling.

Finite element model 3.2.4

A finite element (FE) model for this analysis was developed using the commercial finite

element package ABAQUS 6.11-2. Only one rail was modelled to reduce computation

time. An elastic stress analysis was used to calculate the stress state around the

reference point by considering the contact loading applied at the midpoint of the rail

span (in longitudinal direction at x = 2100 mm) in between two middle sleepers at a

contact patch offset as shown in Figure 3.3 (a) and (b).

Elastic foundations were used to simulate the effect of rail pad, sleepers and ballast.

These were acting directly under the foot (wide bottom portion) of the rail and on both

sides of the rail foot area at the location of the sleepers, which were spaced at 600 mm

intervals along the length of the rail, as shown in Figures 3.3(a) and (b). In the vertical


67

direction, a stiffness / area (Kv) was applied to a contact area of 230 mm x 152.4 mm

underneath the rail, while in the lateral direction the stiffness / area (KL) was applied to

an area of 230 mm x 7 mm on both sides of the rail foot. Some key model parameter

values are given in Table 3.1. The values of all the parameters used in the model were

based on typical track designs used in Australian heavy haul rail systems.

Figure 3.3 The model description: (a) Top view of single rail supported with discrete elastic foundations; (b) Cross-section view of the rail, the measurement location and the elastic foundations

(b)

(a)

S


68

Table 3.1 Model parameter values

Model Parameters Values

Rail span, S (mm) 4200

Support length, Ls (mm) 230

Support width, Ws (mm) 152.4

Axle load (kN) 343.4

Contact patch offset (mm) 0, 15, 30

L/V ratio 0, 0.2, 0.4

Thermal expansion coefficient, α (/oC) 1.2e-5

Young‘s modulus, E (MPa) 209,000

Poisson‘s ratio, 0.3

Boundary conditions 3.2.5

The rail was constrained against the longitudinal translation (u1 =0) and rotation (uR1 =

0) at both ends. The vertical and lateral translations were constrained by the vertical and

lateral stiffness applied at the sleeper locations. For an S = 4200 mm span of rail, as

shown in Figure 3.3(a), the resultant stresses considered at the centre location of the

cross-section were almost unchanged as a result of boundary condition changes at both

ends of the rail.

Loading 3.2.6

The wheel load was applied on the rail assuming a fully slipping Hertzian contact patch.

The wheel vertical load (V) was applied through a Hertzian pressure distribution over

an elliptical contact area of 125.6 mm2 (semi major axis, a = 10 mm, semi minor axis, b

= 4 mm as was reported by Mutton et al [1]). The pressure distribution is for heavy haul

operations in Auatralia for an axle load of 35 tones; the current analysis has used the

same axle load as mentioned in Table 3.1. The Hertzian pressure distribution is given by

the following expression:


69

( ) √.

/ (3.1)

(3.2)

Vehicle curving and hunting 3.2.7

Vehicle curving and hunting behaviour was of interest in this doctoral study due to its

effect on contact patch position and lateral forces. This is exacerbated if contact moves

away from the centre of the rail head cross-section. In real life the contact patch size and

shape vary as the contact moves towards the gauge corner. Figure 3.4a shows a

simplified representation of the contact patch shape versus different offsets (0, 15, and

30 mm). Depending on the curvature, double contacts are possible as well as significant

departures from the ideal Hertz assumption.

The focus of this study was the stress state of the underhead radius, as a result of

localized section bending behaviour (lateral bending of whole rail profile and localized

vertical and lateral bending of head-on-web) on sleepers rather than the contact stresses

close to the running surface. Therefore, to minimize the effects of the contact patch size,

it was kept unchanged at the different offsets on the rail running surface, as shown in

Figure 3.4b. Likewise, contact pressure distribution varies as the contact patch moves

towards the gauge corner. In order to minimize this effect, ellipsoidal Hertzian contact

pressure distribution was used at different contact patch offsets. However the effect of

changes in contact patch size and shape are only small. The underhead radius stresses

showed a maximum difference of 5%, as a result of 20% variations in the values of

major and minor semi axes due to subsequent changes in contact geometry and

maximum contact pressure.

Salehi et al [43, 45] have also made such a simplification in contact patch offset and

contact pressure distribution. They reported that the underhead radius position is far

away from the contact patch and that the difference in stress distribution at the

underhead radius between uniform and ellipsoidal contact pressure is less than 1%. The


70

underhead radius stresses are associated with a combination of lateral head bending of

the whole rail profile and localised bending of the head-on-web on the elastic

foundation due to contact patch offset. Mutton et al [1] and Marich [36-38] also

reported the contact patch offset from the centre of rail head cross-section with similar

loading conditions to those stated above. A maximum contact patch offset of 30 mm

was considered in this thesis, as shown in Figure 3.5. The contact loads were located at

the mid-point of the rail span (in longitudinal direction at x = 2100 mm) in between the

two middle sleepers (Figure 3.3a), at different contact patch offsets. The major semi

axis, a, was in the longitudinal direction and the minor semi axis, b, was in the lateral

direction at the midpoint of the rail span (x = 2100 mm), as shown in Figure 3.5.

Figure 3.4 Variations of contact patch sizes and shapes

In real life

Current analysis

Same size and shape

(a)

(b)


71

Figure 3.5 Representation of a contact patch offset (CPO) from the center of rail head cross-section for modelling eccentric loading

Vehicle dynamics studies [64, 134] have described the direction of the lateral shear

traction, which can be either inward or outward depending upon the curving, steering

and hunting mechanism of the vehicle. The high rail will experience outward lateral

shear traction (towards the field side) from the leading wheelset during curving but, due

to the rigid bogie side frame, the trailing wheelset will try to readjust itself and produce

an inward lateral shear traction (towards the gauge side) on the rail head surface, as

shown in Figure 3.6. On a tangent track, the hunting mechanism due to vehicle

dynamics is considered to be the main source of lateral shear traction.

The lateral forces are influenced by many factors, such as wheel-rail profiles, curve

radius, lubrication conditions and vehicle speed. The calculation of these forces was

beyond the scope of this thesis, and the simulations have been conducted for several

L/V ratios to cover all possible cases. The Hertzian pressure distribution for local

coordinate systems (XYZ) can be found in Figure 3.7(a) and (b). The contact pressure

PZ (X, Y) had contributions from both the wheel vertical load (V) and the lateral load

(L) to the contact patch and was applied through a Hertzian pressure distribution over an


72

elliptical contact area with an angle in the vertical direction, as shown in Figure 3.7

(a) and (b). The normal and tangential forces (FZ and FY) at the contact point in the YZ

plane for the local coordinate system (XYZ) can be expressed as

FZ = Vcos – Lsin (3.3)

FY = Lcos + Vsin (3.4)

Figure 3.6 Representation of inward and outward lateral shear tractions by leading and trailing wheelsets respectively during curving on high rail of a 600 m radius curve (adopted from Xiao et al [134])

The lateral load (L) could be caused by both the steering effects on the curved track and

the hunting of the vehicle due to vehicle dynamics, and normally these are not present at

the same time. This can produce lateral shear traction in the contact area, in combination

with the elevated friction levels due to the absence of any lubrication at the wheel-rail

interface. The magnitude of the loads can be additionally impacted by the vehicle

dynamics [60, 64]. The studies conducted by Salehi et al [43, 45] revealed that the

longitudinal traction does not affect the stress state at the underhand radius, so it can be

eliminated. In order to perform the sensitivity analysis on the effect of lateral traction,

the ratio of lateral load to the vertical load (L/V ratio) can be varied, defined as the

lateral traction coefficient. Accordingly, the contact patch offset and the L/V ratio (ratio


73

of lateral to vertical load) are representative of load cases that approximate the hunting

and steering behaviour of wagon. Cases with an L/V ratio of (0, 0.2 and 0.4) and contact

patch offsets (0, 15, 20 and 30 mm) were parametrically simulated.

Figure 3.7 The rail-wheel contact loads in detail (a) wheel loads [the picture to the left is the original work]; (b) Hertzian pressure distribution at the contact patch [picture to the right is adopted from Ekberg et al [62]]

Mesh development 3.2.8

The rail 3D FE model with dense mesh at the contact patch is shown in Figure 3.8. The

model used 8 node 3D solid elements with reduced integration (C3D8R). The dense

mesh at the contact patch used 4 node 3D tetrahedral solid elements (C3D4) [135]. In

order to reduce the element numbers, the mesh refinement was updated to obtain a fine

mesh at the rail head around the contact patch area only, at all the three offset locations

and at the underhead radius location where the stresses were being calculated. Coarse

mesh was used along the length of the rail and on the web and foot areas. The model has

177,074 elements and 135,399 nodes. The maximum element size was 20 mm in the

rail, but the smallest element size was 0.007 mm in the contact patch region.

(a) (b)


74

Figure 3.8 Mesh development with dense mesh at the contact patch

3.3 Model Validation

Field data 3.3.1

Field measurements for the model validation were based on track instrumentation

conducted by Mutton et al [1, 33] and Bartle et al [32], who measured longitudinal

stress at the underhead radius on both the field and gauge side, as well as vertical

stresses in the rail web and longitudinal stresses at the top of the rail foot at several track

locations subject to high axle load rail traffic. Vertical wheel loads were also measured

using conventional shear bridges. For this activity, the detailed positions of the strain

gauges are shown in Figure 3.9 (a) and (b). These were applied to the relevant position

on the welds (Figure 3.9b), and on the corresponding position on the rail (Figure 3.9a)

at a distance of approximately 1200 mm from the instrumented welds, in the direction

facing the approaching loaded traffic.

The field data used for the model validation are shown in Figure 3.10. Longitudinal

stress in the rail head was measured at 8 locations (4 field / 4 gauge), as given in


75

Appendix B. A typical pattern shows rail going into compression due to the normal

bending action, with a tensile spike associated with lateral bending of the whole rail

profile and localised vertical and lateral bending of the head-on-web, as shown in Figure

3.10. The variations in tension peaks for each wheel can be attributed to differences in

combinations of contact patch positions, vertical wheel loads and the lateral traction

forces for individual wheels. The magnitude of the spike is also dependent upon rail

wear conditions. It was found that the tensile bending stress at the gauge side underhead

radius of the rail reached a peak value of about 100 MPa when the wheel was directly

above the strain gauge position (underhead offset of measurement point 20 mm). This is

a 609 m radius curve as mentioned in the caption and the speed and resulting cant

deficiency is unknown. The position of the offset measurement point has been defined

in Figure 3.3 and the in-situ position is shown in Figure 3.9

Figure 3.9 The strain gauges at the measured points [33]

(a)

(b)


76

Figure 3.10 Measured longitudinal stress response under in service loading; example shown for high rail in 609 m radius curve [1, 32]

The corresponding stresses on the field side underhead radius of rail reached a peak

value of about 83 MPa. For different wheels of a single train the magnitude of peak

tensile stress varied considerably at the both gauge and field side underhead radius

positions of the rail as given in Appendix B.

Hertzian contact stresses 3.3.2

The FE model was validated by comparing the stress field below the centre of a circular

Hertz contact patch with those obtained using an analytical solution. A circular Hertzian

contact patch with a radius of 6.8 mm, supporting a vertical load of 171.7 kN, was used

in the FEA model and also for the calculation of the analytical part [58].

Tension spikes

Compressive stresses

Time (sec)

Time (sec)


77

Figure 3.11 Stress distribution in the rail head versus depth caused by Hertz pressure acting on a circular area (a=6.8 mm, Fz =171.7 KN)

The FEA results revealed that the longitudinal stress at the centre of the contact surface

had a maximum difference of 4% compared to the analytical solution, as shown in

Figure 3.11.

Octahedral shear stresses 3.3.3

The octahedral shear stress ( ) was evaluated analytically as:

√

,( ) ( ) ( ) - (3.5)

The octahedral shear stress distribution along the centreline below the contact area

showed a maximum difference of 5% between the analytical solution [58] and the

-3

-2.5

-2

-1.5

-1

-0.5

0-1.5-1-0.50

Dim

ensi

onle

ss d

epth

(z/a

)

Ratio of stresses to Pm (MPa)

σz/Pm FEMσz/Pm ANALYTICσr/Pm FEMσr/Pm ANALYTICτ/Pm FEMτ/Pm ANALYTICσθ/Pm FEM


78

current FEA prediction, as shown in Figure 3.12. This variation is within an acceptable

range when compared to the FEA solutions as reported by [1] and [69].

Figure 3.12 Octahedral shear stress ( ) of a rail subjected to a non-uniform (Hertzian) contact pressure (a = 6.8mm, Fz = 171.7 KN)

Effect of elastic foundation 3.3.4

Appropriate values of stiffness / area of elastic foundation in the vertical and lateral

directions were chosen using evaluations of the deformation of a rail head in both the

vertical and lateral directions against field measurements. Table 3.2 shows some of the

compared results. The finite element analysis (FEA) results indicate that, with the stiffness /

area of (KV) = 1.0 N/mm3 and (KL) = 1,740 kN/mm3, the vertical deformation of the rail

head is about 2-3 mm and the lateral deformation is in the order of 1 mm (for Fz = 171 kN).

These deformation values were as expected previously in field or laboratory measurements

-16

-14

-12

-10

-8

-6

-4

-2

00 100 200 300 400 500 600

Dep

th b

elow

rai

l sur

face

(mm

)

Octahedral shear stress (MPa)

FEM

ANALYTIC


79

[1]. For simplicity, stiffness was taken as uniform and was assumed to be linear, even

though, in reality it is non-linear due to different behaviour in tension and compression. KV

was applied over an area of 35,052 mm2 (230 mm x 152.4 mm) and KL over an area of

1,610 mm2 (7 mm x 230 mm). With this, the vertical stiffness applied is 35 kN/mm and the

lateral stiffness applied is 2800 MN/mm.

Table 3.2 Results of simulation of rail deformation with respect to the support of elastic foundation

Loading and support conditions with L/V ratio

Rail deformation (mm)

Vertical Lateral Longitudinal

Head Foot Head Foot Head Foot

15 mm contact patch offset 1 L/V=0.4 0.26 0.14 0.93 -0.01 -0.0015 0.00002

2 L/V=0.4 2.24 2.32 1.39 -0.013 -0.128 -0.128

3 L/V = 0 2.046 2.014 0.33 -0.018 -0.138 -0.137

L/V=0.2 2.15 2.17 0.90 -0.023 -0.180 -0.168

L/V=0.4 2.24 2.32 1.40 -0.025 -0.13 -0.125

4 L/V=0.4 11.08 11.24 2.0 -0.027 0.0002 0.0007

30 mm contact patch offset 5 L/V=0 2.55 2.69 1.66 -0.06 -0.13 -0.14

L/V=0.2 2.85 3.49 3.75 -0.11 -0.13 -0.14

L/V=0.4 3.17 4.30 5.89 -0.16 -0.144 -0.145

1. Elastic foundations: N/A. Boundary conditions: fixed at base and both sides of the rail foot at the location of sleepers. 2. Elastic foundations: Kv = 1 N/mm3. Boundary conditions: fixed in lateral direction at both sides of rail foot at the location of sleepers. 3. Elastic foundations: Kv = 1 N/mm3, and KL = 1740 kN/mm3. Boundary conditions: N/A. 4. Elastic foundations: Kv = 0.1 N/mm3, and KL = 17.4 kN/mm3. Boundary conditions: N/A. 5. Elastic foundations: Kv = 1 N/mm3, and KL = 1740 kN/mm3. Boundary conditions: N/A.

The lateral stiffness is applied to a very small area at both sides of the rail foot, as

shown in Figure 3.3b, to model the rail fastening applied on these locations. Therefore,

the stiffness values are much higher compared to the vertical stiffness value. Similarly,

the difference in deformation is a factor of about -3, whereas the difference in stiffness

is a factor of about 2000 (or 10000 if forces are considered) as shown in Table 3.2.


80

These boundary conditions, especially the vertical and lateral stiffness, provided some

constraint to the rail regarding rotation in the lateral and vertical directions.

Sensitivity analysis 3.3.5

A sensitivity analysis was performed to investigate the bending and deformation

behaviour using different support and boundary conditions, as shown in Table 3.2. Field

measurements [1] showed the lateral deformation to be of the order of 1 mm and the

vertical deformation to be of the order of 2-3 mm. Case 3 in Table 3.2 with L/V = 0.2

and 0.4 provides a sufficiently good match for both and these support conditions were

used throughout the doctoral investigation. The longitudinal stresses at the underhead

offset of measurement point 20 mm are plotted in Figure 3.13, and a tension spike can

be seen to occur directly under the wheel contact position. This behaviour has been

observed in field measurements, as shown in Figure 3.10.

Figure 3.13 Longitudinal stress at the underhead offset of measurement point 20 mm

The FE results were compared with in-situ measured stress data for loaded rail traffic

with a nominal axle load of 35 tonnes. The in-situ measured data showed variability in

-60

-40

-20

0

20

40

60

80

100

120

1800 1900 2000 2100 2200 2300 2400 2500

Lon

gitu

dina

l Str

ess (

MPa

)

Longitudinal Position (mm)

L/V=0 contact patch 15 mm offsetL/V=0.2 contact patch 15 mm offset L/V=0.4 contact patch 15 mm offsetL/V=0.2 contact patch 30 mm offsetField measurements


81

peak tensile stress values at the measurement location, which could be attributed to

variability in either one or a combination of the contact positions for individual wheels,

vertical wheel loads and the lateral traction forces that arose from the steering effects.

The variations could be due to wagon loading variations. The values of these parameters

were not determined during the in-situ measurements, and hence the current analysis

made use of peak stress data for a sample of 21 wheel passes, which represented a

portion of the total data set. The peak tensile stress at the measured location (the

underhead offset of measurement point 20 mm) from the in-situ measured data showed

100.6 MPa, which was a smaller difference of 6.8 % in the case of an L/V ratio of 0.2

and a contact patch offset of 30 mm. The other peak tensile stress values at the in-situ

measured location are also plotted against FE results and are showing sufficiently good

correlation, see Figure 3.13.

3.4 Thermal stresses

The differential temperature due to seasonal effect is another factor that influences the

stress state at the UHR in continuously welded rail (CWR). An approach to calculating

the thermal stresses due to variations of service temperature from stress free or neutral

temperatures was stated as

(3.6)

Where T is the deviation from the stress free temperature, E is the Young‘s modulus

and is the thermal expansion coefficient, as given in Table 3.1. The thermal stresses in

the longitudinal direction of the rail due to either a cold or warm temperature changes

were included in the simulation using the commercial FE package ABAQUS 6.11-2.

3.5 Residual stresses distribution

The residual stress (RS) induced by the manufacturing process and repeated rolling

contact between wheel and rail is another important parameter that can change the stress

state in rails and is often included in the specification of the rail by the manufacturer


82

[81, 127]: hence the interest in examining the influence of residual stress distribution on

the development of RCF. The Australian Standard (AS1085.1) [80] suggests a typical

distribution of longitudinal residual stresses in roller-straightened rail, which presents at

the underhead region with changes from -65 to 25 MPa.

Figure 3.14 Rail section for residual stress distribution

Worn profile

Original profile (not to scale)


83

Table 3.3 Residual stress scatter in the worn rail section given above [136]

Residual stresses, (MPa)

Rail section σx σy σz σxy σxz σyz

1 -250 -300 -300 0 0 -40

2 -150 -100 -200 0 0 -80

3 -150 -100 -100 0 0 0

4 -50 -200 -100 0 0 0

5 -87.5 -87.5 -87.5 0 0 0

6 -162.5 -80 -80 0 0 0

7 -12.5 -12.5 -12.5 0 0 0

8 100 0 0 0 0 0

For this thesis, the residual stress distribution was based on the measurements

conducted by Magiera [136]. As rail geometry is quite complex, the residual stresses

were considered in terms of the six stress components for each of the regions shown in

Figure 3.14. The input residual stresses are summarized in Table 3.3. The rail grades

used in the field are both roller and stretch straightened, therefore RS values used may

not be representative of actual rail grades operated in the field. They were applied by

defining them as initial conditions in an input file created in the FE code ABAQUS

6.11-2. For the first step in the rail FE model calculation, residual stresses became the

initial conditions and no contact loads were applied for this step, in order to prevent any

contraction and bending.

3.6 Summary

In this chapter, the author has presented the methodology and validation of the finite

element model, which was used to study the failure mode as described in chapter 1. The

proposed numerical model is based on the Finite element method and is validated with

in-track field data. The effect of the contact patch offset and L/V ratio, representative of


84

curving and hunting, was included and directly correlates with a tension spike due to

localised bending behaviour of rail. In the current model, allowance was made for the

effect of foundation stiffness, thermal stresses arising from variations around the

nominal stress-free or neutral temperature and the residual stresses, in order to

determine the applicable stress limits.

Chapter 4. Underhead radius stresses

85

Chapter 4

Underhead radius stresses

4.1 Introduction

The work presented in this chapter describes the first step in modeling the effect of local

rail bending stresses. The development of a finite element model of the rail, as described

in chapter 3, is applied to demonstrate the rail local bending behaviour at the underhead

radius, influenced by contact, bending and seasonally dependent thermal stresses. The

finite element model validation in chapter 3 demonstrated that the longitudinal stresses

in the rail underhead radius position are of special interest, as tension spikes have been

identified at this location during in-track measurements under high axle load conditions

[1]. The analysis revealed that the magnitude of the tension spike was highly dependent

on several service conditions: the contact patch offset from the rail centreline, the ratio

of lateral (L) to vertical (V) loads, the direction of lateral shear traction, foundation

stiffness and seasonal temperature variations.

In addition to the vertical bending of the whole rail cross-section on the elastic

foundation, the rail also undergoes lateral bending as well as vertical and lateral bending

of the head-on-web that was reported by Jeong [39], Jeong et al [40] and Orringer et al

[41]. The tension spike is a result of this effect, and corresponds to an additional local

bending stress due to the lateral bending of the whole rail profile and the localised

bending of the rail head-on-web. This effect is highly localised and is additional to the

stress generated due to vertical bending of the whole rail profile (the so called global

bending). Orringer et al [47] analysed the longitudinal bending stresses in the rail head.

They found the stress state to consist of five possible components: (1) vertical bending,

(2) lateral bending, (3) warping, (4) vertical head-on-web bending, and (5) lateral head-


86

on-web bending, as was discussed in section 1.1.2, chapter 1. The field measurement

from Orringer et al [47] showed that, for tangent track, the longitudinal stress caused by

rail head-on-web bending is much smaller than that caused by rail vertical bending. The

warping term of the longitudinal stress is similarly small. On curved track, the

additional components of lateral bending and lateral head-on-web bending significantly

increase the longitudinal stress.

Eisenmann [34] measured the tensile stresses at the underhead location (at the lower

gauge corner) under field conditions within the German rail system, and also calculated

the theoretical tensile stresses at the gauge and field side underhead location for

eccentric and inclined loading. The local tensile stresses on the field side underhead

location under laboratory conditions and in the field due to lateral wheel loading were

measured by Sugiyama et al [35]. Marich [37] presented finite element analysis results

that demonstrated that the longitudinal stresses at the underhead radius are increased

when the contact patch moves away from the centre of the rail head cross-section on a

heavily worn rail. The increased stress level leads to fatigue failure and is a potential

contributor to rail defects. It was suggested that rail life could be improved if the contact

patch offset is controlled by adopting appropriate rail grinding strategies. Sugino et al

and Farris et al [93, 94] also reported the presence of large tensile stresses at the

subsurface (about 3-15 mm below the running surface inside of the rail head) of the rail

head. These stresses can become high and produce reverse detail fractures (i.e.

transverse defects which initiate at the lower corner of the gauge face of heavily worn

high rail), as was shown in Figure 1.7-1.12. It would be of interest to compare this

earlier work with what has been observed under high axle load conditions.

Jeong et al [6, 7] investigated reverse detail fractures by providing a close form solution

to calculate the longitudinal bending stresses, using simplified wear geometry on the top

and side of a rail head. A sensitivity analysis was undertaken to examine the influence

of rail head wear, residual stresses, thermal stresses and track stiffness on crack growth

rates. The scope of this research, which estimated crack growth rates for reverse detail

factures, included: the accumulated MGT, axle load for simplified wear geometry for

different rail sizes, materials and above rail parameters. Jeong [8] specifically

mentioned flow lips as an initiator, in addition to a possible contribution by higher


87

longitudinal bending stresses at the lower gauge corner of the rail. The reverse detail

fracture failure mode is influenced by longitudinal bending stresses at the underhead

radius, as reported by Jeong et al [6].

Failures that initiate at the underhead radius of aluminothermic welds have been found

in Australia under high axle load conditions. Salehi et al [42-45] conducted a study

using multi-axial fatigue analysis to examine fatigue crack initiation in the vicinity of

the weld collar of aluminothermic welds. The results obtained permitted systematic

investigation of factors influencing fatigue occurrence based on careful FEM elastic

calculations, including, but not limited to, the contact zzonal local bending stress

superimposed due to the bending of the rail head on the web. The localisation of this

effect in a narrow band of 100 mm corresponded to the wavelength of this bending as

mentioned by Salehi et al [42-45]. This tension spike is very important in explaining

crack initiation at the underhead radius. The effect of head wear of weld profile and

inward lateral shear traction, which were not considered in these investigations,

exacerbate the failure at the gauge side underhead radius, and also in the area where the

tension spike generated shifts upwards from the underhead radius to the rail head

surface. This presents a potential risk for existing RCF cracks to turn down and become

transverse defects.

Mutton et al [1] modelled the elastic foundation effect in a 68kg/m worn rail section

laid on 8 discrete sleepers. Their model is capable of showing underhead radius stresses

due to lateral bending of the head and vertical and lateral bending of the head-on-web.

This effect is additional to the bending of a whole rail profile (so called global bending).

The boundary conditions and contact patch lateral movement created high longitudinal

bending stresses in the form of short duration tensile stresses at the underhead radius. It

was concluded that the tensile stress at the underhead radius increased with an increase

in head wear and L/V ratio, but the depth at which stresses became tensile reduced. This

high tensile stress on the gauge corner of rail head could cause a RCF initiated crack to

turn down to form a transverse defect. The underhead radius stresses were qualitatively

correlated with field measurements. For this activity, strain gauges were applied to the

relevant position on the welds, and on the corresponding position on the rail at a


88

distance of approximately 1200 mm away from the instrumented welds, in the direction

facing the approaching loaded traffic.

Measurements conducted by Mutton et al [1] under high axle load conditions showed

that the tensile bending stress at the underhead radius occurs as a short duration peak

when the wheel is directly above the measurement position. In the field measurements,

it was reported that the tension spikes at the gauge side underhead radius reached a peak

value of about 100 MPa when the wheel was directly over the strain gauge position

(underhead offset of measurement point 20 mm). The corresponding stress on the field

side of the rail was about 83MPa, being 17% less than that in the gauge side of the rail.

With different wheels on a single train, the magnitude of peak tensile stress varied

considerably at both the gauge and field side underhead radius positions of the rail.

Greisen [53] and Greisen et al [137] presented an estimation of rail bending stresses

from real time vertical track deflection measurements, based on studies conducted by

Lu [138]. The long range (thermal, residual and bending) stresses, especially the cyclic

axial (longitudinal) stresses, usually dominated rail stresses and lead to fatigue crack

propagation and fatigue failure. Greisen et al [137] also reported that the longitudinal

bending stresses due to vertical bending could be a potential contributor to fatigue

failure in rail. The study focused on vertical bending, and the effects of lateral bending

and localised bending of the head-on-web were not considered.

In this chapter the longitudinal bending stresses as a result of additional lateral head

bending and localised vertical and lateral bending of the head on the web are being

taken into account. Vehicle dynamics studies [134] show that some wheels develop

inward shear traction (towards the gauge side) and some outward shear traction

(towards the field side) on the rail surfaces, which lead to differences in the tracking

behaviour of individual wheelsets during curving. Both these cases were considered for

this thesis. The effect of foundation stiffness and seasonal temperature variations were

also investigated. The results generated are described and discussed next. The work

presented in this chapter is a contribution to the existing literature [3]. The qualitative

effect of these underhead radius stresses on the propagation of pre-existing RCF cracks

and fatigue crack initiation at the underhead radius position will also be discussed. For


89

this chapter, the longitudinal underhead radius stresses due to local bending were

calculated to form input to fatigue crack initiation and rapid fracture models, and the

development of these is detailed in chapters 6, 7 and 8.

4.2 Finite element model

The finite element (FE) model of nominal rail profiles was developed using the

commercial finite elements code ABAQUS (6.11-2) and validated with field

measurements. Only one rail was modelled to reduce computation time. An elastic

stress analysis was conducted for a stationary contact loading. The finite element

method was used to calculate the stress state around the reference point. The loading

was considered to be a fully slipping Hertzian contact pressure modeled on an elliptical

contact patch to give a worst case. The case of a wheel pass over a sleeper and then

between two sleepers was simulated and it was found that the case when the wheel is in

the middle of two sleepers gives the worst case. A detailed description regarding model

development and calibration was already provided in chapter 3. It should also be noted

that the details on input parameters such as loading, boundary conditions and support

characteristics will remain the same as was discussed in chapter 3 and are not repeated

here.

4.3 Effect of different contact patch offset and lateral tractions

The position of contact patch and the wheel rail forces depend on several parameters

and are not within the scope of this study. However to make these results useful, all

cases have been examined. A vehicle dynamics study would be required to determine

L/V ratios, contact patch offset and direction of lateral loading. The outputs of vehicle

dynamics studies would next be used to read the appropriate stress values.

The simulations were primarily conducted for tangent track with free rolling conditions

i.e. (L/V =0) on the centre of the rail head cross-section. In the case of a pure vertical

load (L/V = 0) on the centre of a rail head cross-section, there was no tensile stress

generated at the underhead radius in the gauge side of the rail, as shown in Figures 4.1


90

(a) and 4.2 (a). The tensile stress at the underhead position showed an increase with

movement of the contact location to the gauge corner of the rail, as shown in Figure 4.1

(b). Figure.4.2 (a) shows that the peak tensile stress was 60 MPa at the underhead offset

of measurement point 34 mm for the contact patch offset of 15 mm with L/V = 0 (wheel

load of 172 kN). The vertical load eccentricity imposed local bending and additional

torsion of the rail head. The tensile stress at the underhead radius was attributed to the

localized bending behaviour of the rail head. The narrow band (around 100 mm) of this

effect corresponds to the wave length of this bending. When the eccentric position of

loading was increased from the contact patch 15 mm offset to 30 mm, the tensile stress

rose from 60 MPa to 140 MPa consistently.

Figure 4.1 Longitudinal stress (S11) contour; (a) centric loading; (b) eccentric loading on gauge side (contact patch 30 mm offset); (c) eccentric loading on gauge side (contact patch offset of 30 mm with L/V=0.4 towards gauge side)

(a) L/V=0, CPO=0 mm

(b) L/V=0, CPO=30 mm (c) L/V=0.4, CPO=30 mm

y z x

(UHR)


91

Figure 4.2 Longitudinal stress at the underhead offset of measurement point 34 mm; (a) at different contact patch offsets (0, 15 and 30 mm) with L/V=0; (b) at contact patch 30 mm offset with different L/V= (0, 0.2 and 0.4)

When considering the hunting and steering behaviour of the rail, assuming the L/V =

0.2 and 0.4 with the contact patch 30 mm offset, the tensile stress takes place at the

gauge side underhead radius of the rail and increases consistently with an increase of

L/V ratios. As a result of the eccentric loads from L and V, the high tensile stress

imposed by the vertical and lateral bending of the head-on-web can clearly be observed

at the underhead radius, as shown in Figure 4.1 (c). The longitudinal stress at the

underhead offset of measurement point 34 mm for contact patch 30 mm offset with

-60

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1800 1900 2000 2100 2200 2300 2400 2500

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Pa)


L/V=0L/V=0.2L/V=0.4

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(a)

(b)

34mm


92

different loading ratios L/V of 0, 0.2 and 0.4, (wheel load of 172 kN) is shown in Fig.

4.2(b). The tension spike is about 324 MPa (for estimated torsional fatigue limits for

HH rail, t-1 = 205 MP) with a lateral load (L/V = 0.4), which is about two times higher

than without the lateral load (L/V = 0). It can be seen that the tension spike clearly

increases with an increase in the L/V ratio.

4.4 Stress state at the underhead radius and gauge corner

The behavior of the tension spike, in a range of positions at the underhead radius and

with depth in the rail head at gauge corner region, is considered next. Figures 4.2 a-b

show that the tension spike occurs in a narrow band (around 100 mm in length) and the

stress is highest at UHR surface. The variation of longitudinal stresses in this region is

of importance, as these high tensile stresses can cause cracks to develop and grow.

Figure 4.3 Longitudinal stress at different underhead offset of measurement points (20, 34 and 37 mm from longitudinal centerline of rail) at the contact patch 30 mm offset and L/V=0.4

Figure 4.3 shows the longitudinal stress distributions at the underhead offset of

measurement points 20, 34 and 37 mm from the longitudinal centreline of the rail. The

tension spike is highest with a peak value of 324 MPa when the underhead offset

measurement point is 34 mm, about 48% higher than that at the underhead offset of

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underhead offset of measurement point 37 mm



Offset


93

measurement point of 20 mm. Beyond the underhead offset measurement point of 34

mm, the tension spike reduces slightly by 7% at the underhead offset measurement

point of 37 mm. Figure 4.4 shows the variation of longitudinal stress with depth on a

vertical plane at 20 mm offset from the vertical rail centreline towards the gauge corner.

Figure 4.4 Longitudinal stress distribution with depth on a vertical plane at 20 mm offset from the rail vertical centreline towards the gauge corner for the different contact patch offsets (0, 15, and 30) mm with L/V ratios (0, 0.4)

The contact patches are 0, 15 and 30 mm offset and the L/V ratios are 0 and 0.4. It can

be seen that, for L/V = 0.4, the longitudinal stresses are more tensile for the 30 mm

0

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Longitudinal stress (MPa)

contact patch 30 mm offset

contact patch 15 mm offset

L/V = 0.4

0

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contact patch 30 mm offsetcontact patch 15 mm offsetcontact patch 0 mm offset

L/V = 0


94

contact patch offset. They shift from compression to tension at a depth of 14 mm below

the contact surface, whereas for the contact patch 15 mm offset, the depth is about 20

mm. Where there is no lateral loading (L/V = 0), the tensile stress occurs at a depth of

38 and 34 mm for contact patches 15 and 30 mm offset respectively, showing small

sensitivity to changes in the contact patch position.

Figure 4.5 shows the longitudinal stress variation, with depths for different vertical

plane offsets of 16.5, 20, 25 and 35 mm from the rail vertical centreline towards the

gauge corner with L/V ratios of 0, 0.2 and 0.4 at the contact patch 30 mm offset. By

increasing the vertical plane offset from the vertical centreline of the rail toward the

gauge corner (L/V = 0.4), the longitudinal tensile stress increases, but the depth at

which the stress become tensile reduces from 20 mm to 2 mm. For the case with a

lateral loading of L/V = 0.2, the longitudinal tensile stress increases, and the depth at

which stress becomes tensile reduces from 29 mm to 5 mm. If there is no lateral

loading, i.e. L/V = 0, the longitudinal tensile stress increases, and the depth at which the

stress becomes tensile reduces from 38 mm to 14 mm. This behaviour can be verified

from the deformation of the rail, as shown in case 5 of Table 3.2. The lateral

deformation on the gauge side of the rail head at the contact patch 30 mm offset

increases 3.5 times as the L/V ratio changes from 0 to 0.4. The gauge side is in tension,

which will subsequently result in tensile stresses at the gauge corner.

The contours of the longitudinal stress on the cross-section of the rail head, with respect

to the contact patch offset of 30 mm with different L/V ratios, are shown in Figure 4.6.

The regions subjected to tensile stresses in the rail head are clearly visible at the

underhead radius location of the gauge side. The contour plots indicate that with an

increase in the L/V ratio (lateral load), the longitudinal tensile stress increases at the

underhead and gauge corner region. The regions that are in tension can cause a crack to

grow rapidly. The depth at which stress becomes tensile decreases as the L/V ratio

increases. For L/V=0.4, this is of the order of a few mm, as evident from Figure 4.6.

This can result in a transition to Mode I crack growth and the formation of a transverse

defect for rolling contact fatigue initiated cracks that extends to this region and causes

the fatigue crack initiation at the rail underhead radius.


95

Figure 4.5 Longitudinal stress distribution with depth on different vertical planes at 16.5, 20, 25 and 35 mm offset from the rail vertical centreline towards the gauge corner at a contact path 30 mm offset and different L/V ratios (0, 0.2, and 0.4)

0

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(a) L/V = 0.4

(b) L/V = 0.2

(c) L/V = 0


96

Figure 4.6 Longitudinal stress (S11) contours on the cross-section of rail with the contact patch offset of 30 mm at midpoint of rail span ( in longitudinal direction at x=2100 mm), during different lateral loads with the L/V ratio of 0, 0.2 and 0.4. The affected region at which the stress becomes tensile shifts up from the rail underhead radius to surface

4.5 Effect of lateral traction direction

As was described in section 4.2, instead of track loading, one rail is modeled to reduce

the computational time and loading cases on high rail are considered only. The direction

of lateral forces will vary between inwards and outwards leading and trailing wheelsets

during curving of conventional three-piece bogie types commonly used in high axle

load railways. In order to understand the effect of the different directions of lateral

traction, loading cases were considered with the lateral traction pointing towards the

gauge side (inward) or towards the field side (outward). To factor these parameters in

terms of the lateral shear traction direction, simulations were conducted for a contact

patch 30 mm offset towards the gauge side from the rail centreline for both inward and

outward shear traction directions with the L/V ratios of 0, 0.2 and 0.4, as shown in

Figure 4.7. The results in Figure 4.7 (a) show high underhead radius stresses even

without inward lateral traction as compared to outward lateral traction. The contact

patch offset and the load cases with an inward / outward force of up to L/V = 0.4 are

just an assumption as a worst condition to predict the resultant stress state at the

underhead locations and this has not been quantified in track measurements. The results

show that by increasing the L/V ratios towards either the field or gauge side of the high

rail, the tensile stress at the underhead radius of the respective side increases. The peak

2

L/V=0 L/V=0.2 L/V=0.4 2

3

1


97

tensile stress is about 324 MPa for the inward traction, which is 4 times higher than that

for the outward traction.

Figure 4.7 Mechanical response at the underhead and base fillet with lateral shear tractions (inward and outward); (a) Longitudinal stress variations with different magnitude and direction of shear traction at the underhead offset of measurement point 34 mm along the midpoint (x = 2100 mm) of high rail with contact patch 30 mm offset; (b) Inward lateral deformation (U2) contour of high rail with L/V = 0.4; (c) Outward lateral deformation (U2) contour of high rail with L/V = 0.4

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)

Towards Gauge Side (Inward) Towards Field Side (Outward)

Underhead Radius Gauge SideUnderhead Radius Field SideBase Fillet Field SideBase Fillet Gauge Side

L/V ratio

Underhead Radius (UHR)

Base fillet

x z

y

High rail (L/V=0.4) High rail (L/V=0.4)

Underhead Radius (UHR)

Base fillet

(a)

(b) (c)


98

These stresses are generated by the lateral deformation of the railhead, as shown in

Figures 4.7(b) and (c). It can be seen that the lateral deformation of the railhead for the

inward traction is about 2.5 times higher than that for the outward traction. This is easy

to understand when considering the effect of L and V on the rotation and lateral

deformation of the rail head-on-web. A contact patch offset towards the gauge corner

rotates the railhead counterclockwise in Figures 4.7(b) and (c). The lateral load, L,

causes further rotation and lateral deformation of the rail head. When the lateral load is

towards the gauge corner (inwards), both rotations (due to L and V) are summative and

provide a much larger rotation and lateral deformation than when the lateral load is

outwards. In this case, the rotation due to L partly cancels out the rotation due to V,

producing significantly lower deformation and longitudinal stresses.

4.6 Effect of track foundation stiffness

An important aspect of this study was to model the support characteristics of track

system influenced by operational conditions to investigate the effect of altered stiffness

on the stress state at the rail underhead radius and base fillet. The looseness in sleepers,

deterioration due to ballast pumping and degradation of track bedding would result in

altered stiffness. The altered stiffness may also be due the effect of the looseness in

fasteners and deterioration of rail pads. The model addresses in a very simple way the

effect of altered stiffness. This was achieved by applying varying stiffness values to the

elastic foundation in the lateral and vertical directions of the rail, as described in detail

in chapter 3. This modeling evaluated the stress state at the underhead and base fillet,

when a poor track support condition is presented in a localized area.

The longitudinal tensile stress at the underhead radius and base fillet was investigated

by changing the vertical foundation stiffness underneath the two middle sleepers

adjacent to the contact area. The value of vertical stiffness of all other sleepers was kept

at KV = 1 N/mm3. The value of lateral stiffness was kept at KL = 1740 kN/mm3 for all

the sleepers to achieve a lateral deformation of 1 mm. Figures 4.8 (a) and (b) show the

results of the longitudinal stresses as the foundation stiffness value was altered. The

stress increased at the underhead radius and decreased at the base fillet as the vertical


99

Figure 4.8 Longitudinal stress variations at the underhead radius with vertical foundation stiffness at the contact patch 30 mm offset and L/V = 0.4: (a) along longitudinal position of the rail at the underhead offset of measurement point 34 mm; (b) at the underhead offset of measurement point 34 mm and at base fillet at midpoint of rail span (in longitudinal direction x=2100 mm) in between two middle sleepers

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Vertical Foundation Stiffness (MN/m)

Underhead

Base fillet

(a)

(b)


100

foundation stiffness increased from 0 to 123 MN/m, as shown in Figure 4.8 (b). Without

any support to rail at the adjacent sleepers (a vertical foundation stiffness of zero), the

maximum longitudinal tensile stress was found to be 200 MPa at the base fillet and 267

MPa at the underhead, respectively. This condition simulated ballast pumping, which

could remove support from one sleeper, as is observed in the field. Thus, the possibility

of straight breaks at the foot region driven by repeated tensile stresses was enhanced,

becoming much higher. It was noted that beyond the vertical foundation stiffness of 35

MN/m, the longitudinal tensile stress hardly changed at either of the locations. A

comparison of the longitudinal and Von-Mises underhead radius stresses resulting from

variations of support stiffness is also presented in Table 4.1.

Table 4.1 Effect of track support stiffness

Support stiffness

(Vertical)

Longitudinal stress, S11

(MPa)

Von-Mises stress, Mises

(MPa)

Loose support (0 MN/m) 267 283

Average support (35 MN/m) 322 330

Stiff support (123 MN/m)

340 350

4.7 Effect of seasonal temperature variation

The differential temperature due to seasonal effect is another factor which influences the

stress state at the UHR. An approach by the US Department of Transportation [7] for

evaluating the thermal stresses in CWR was adopted here, by taking the service

temperature difference from the neutral temperature. The neutral or stress free

temperature (assuming a temperature condition at which the rail is installed) was taken

as the reference temperature. The two other service temperatures were assumed to

represent the mean of colder months (10 oC), and the mean of the warmer months (42


101

Figure 4.9 Longitudinal stress variations at the underhead radius with seasonal temperature at contact patch offset of 30 mm and L/V = 0.4 (a) along longitudinal position of the rail at the underhead offset of measurement point 34 mm; (b) at the underhead offset of measurement point 34 mm at midpoint of rail span (in longitudinal direction at x = 2100 mm) in between two middle sleepers

-500

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oCoCoC

102642

280290300310320330340350360370

10 15 20 25 30 35 40

Lon

gitu

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MPa

)

Rail temperature (oC)

Underhead radius gauge side

(a)

(b)


102

oC) respectively. The reference temperature was taken as 26 oC and the simulation

temperature was changed for the different service temperatures of the rail. Figures 4.9

(a) and (b) show that the longitudinal tensile stress at the underhead radius increases in

the colder months and decreases in the warmer months with a difference of about 40

MPa. The longitudinal tensile stress went up to 365 MPa at the underhead radius by the

additional tensile stress generated in colder months.

4.8 Fatigue damage

The results of the analyses reveal that the tension spike is strongly dependent on the

contact patch location and the magnitude and direction of the lateral traction, and hence

on the curving and hunting behaviour of the vehicle. Additionally, stiffer tracks

operated in cold weather conditions experience increased underhead radius stresses. The

tension spike at the underhead radius of the rail can cause fatigue damage and cracking.

In addition, the presence of such high tensile stresses can cause a crack to turn

perpendicular to these tensile stresses, resulting in the formation of transverse defects.

In practice, a potential reverse detail fracture could be initiated at sharper curves in the

underhead radius where this localised stress peak has the maximum value. The axle

loads in the Australian heavy haul rail systems are continually rising [89, 139]. This

current analysis, with an axle load of 35 tonnes, shows the propensity for crack

initiation and propagation. With increased axle loads (which sometimes are up to 40

tonnes), these stresses will increase even further and more occurrences of transverse

defect development from rolling contact fatigue damage may be expected.

Vehicle dynamic effects associated with variations in track geometry will also increase

the magnitude of the vertical and lateral loads. For this thesis, the effect of dynamic

loading was assumed to be zero, as the intent was to establish a finite element analysis

approach that reproduced the stress history apparent from the in-situ measured data.

Additional (quasi-static) loads due to curving were included by means of the application

of lateral loads, as described above. Increased train speeds may also produce an increase

in dynamic loads as well as an increase in the effect of lateral forces due to lateral

irregularity on either the gauge or the field side, and subsequently an increase in the L/V


103

ratio. This will lead to elevated values of tensile stresses and ultimately an increased

risk of fatigue damage at the underhead radius. In addition, an increase in tensile

bending stresses, together with both residual and thermally induced stresses, could

cause a crack to turn perpendicular to these tensile stresses once they reach a critical

length [3]. With rail uplift ahead and behind the wheel-rail contact, longitudinal tensile

stress develop due to rail vertical bending (reverse bending). A short duration tensile

stress peak occurs due to lateral bending when the instrumented location is directly

beneath the wheel-rail contact.

The underhead region of welded rail joints could be susceptible to such a failure. Welds

have lower material strength, elevated residual stress levels and a complex geometry

leading to stress concentration [42]. The underhead region of the weld collar in

aluminothermic welds often develops fatigue cracking under high axle load conditions

[42]. The fatigue damage will be even more severe when loading is applied on the

gauge side. This is because the tensile stresses increase at the gauge side and underhead

region, as shown in Figure 4.7.

The analysis results provided in this chapter have qualitatively revealed that the

magnitude of the tension spike at the rail underhead radius is strongly dependent on the

contact patch location, the magnitude and direction of the lateral traction, and hence on

the curving and hunting behavior of the vehicle. Rail heads that are heavily worn are

expected to exhibit even higher stress levels, as was identified by Jeong et al in their

examination of reverse detail fractures [6]. The effect of head wear on underhead radius

stresses in relation to the development of transverse defects from rolling contact fatigue

at the gauge corner, is explored in chapter 5.

4.9 Summary

This chapter covered an examination of the stress state at the underhead radius and

gauge corner of the rail, using the commercial finite element package ABAQUS 6.11-2.

The tension spike at the underhead radius was found to be highly dependent on several

factors: the contact patch offset, the ratio of lateral (L) to vertical (V) loads, the


104

direction of lateral traction, the vertical foundation stiffness and the seasonal

temperature. It can be concluded from the model results that:

The tension spike at the underhead radius of a rail increases when the contact patch

moves away from the rail centreline and / or the L/V ratio increases. Inward shear

traction was found to be more damaging. The magnitude of longitudinal tensile stress

increases at the underhead radius as the vertical foundation stiffness increases. The

results suggest that the magnitude of longitudinal tensile stress at the underhead radius

increases in the colder months and decreases in the warmer months.

The depth at which stress becomes tensile decreases as the L/V ratio increases. For

L/V=0.4, this is of the order of a few mm. This can result in a transition to Mode I crack

growth and the formation of a transverse defect for both rolling contact fatigue initiated

cracks that extend to this region and the fatigue behavior of the rail underhead, as was

previously examined in connection with the reverse detail fracture in the US.

Further study is also needed to understand the fatigue mechanism and crack initiation at

the underhead location by considering factors such as worn profiles, residual stresses,

and unstable crack growth behaviour at the underhead location, which will be presented

in subsequent chapters.

Chapter 5. Effect of head wear on UHR stresses

105

Chapter 5

Effect of head wear on underhead radius

stresses

5.1 Introduction

In this chapter, the investigation into the effect of head wear on underhead radius

(UHR) stresses is described. The previous chapter qualitatively revealed that the

magnitude of the tension spike at the rail underhead radius is strongly dependent on the

contact patch location, the magnitude and direction of the lateral traction, and hence on

the curving and hunting behavior of the vehicle. The work presented in this chapter

examines the effect of head wear and operating track conditions (the L/V ratio, the

contact patch offset (CPO), foundation stiffness, thermal and residual stress) on

underhead radius stresses.

This study was especially concerned with the qualitative evaluation of the state at the

rail underhead radius for heavily worn rail profiles, which are due to the effect of the

head-on-web bending and the lateral bending of the whole rail profile. It investigated

the situation when the rail head cross-section bends on the rail web. This effect is much

more localized and could be termed local bending stress as a result of vertical and

lateral head-on-web bending. It occurs in the form of a tension spike at the underhead

radius of the rail.

Marich [36-38] has also reported the presence of high tensile longitudinal stresses at the

lower gauge corner of the rail due to localised bending of the head-on-web. The tensile

stress was found to be a function of vertical and lateral load eccentricity and was


106

influenced by changes in rail head wear (HW). A comparison of different worn profiles

of 60 kg/m and 68 kg/m rail sections under eccentric vertical and lateral loading

revealed that, in terms of allowable rail head wear, the smaller rail section would be

preferable and reported that the bigger rail section was not always better [36]. This did

not take into consideration the increased beam strength with the larger rail, which may

be necessary for the axle loads considered. This work also did not take into

consideration the possible impact of RCF damage; subsequent research by Mutton et al

[1] demonstrated that the approach taken by Marich was therefore non-conservative in

the presence of RCF damage.

Generally, wear results in a reduction of the rail cross-sectional area, and changes the

profile of the rail, which may have an influence on the wheel-rail contact conditions and

consequently impact on the magnitude of bending stresses in the rail. The peak tensile

stress at the underhead radius can be higher in terms of the head wear levels and

potentially produce reverse detail fractures (RDF) (i.e. transverse defects which are

initiated at the lower corner of the gauge face of heavily worn rail) under heavy haul

operations. This type of defect can be observed on heavily worn curved rails on stiff

tracks that are poorly lubricated and subjected to high axle loads, as reported in North

American rail systems [6, 9]. Similar failures have been found in aluminothermic welds

in the Australian heavy haul railway system. Currently, Australian heavy haul rail

systems operate with axle loads of up to 35 tonnes (some recent ones are up to 40

tonnes) [89, 139]. The previous analysis, presented in chapter 4, with an axle load of 35

tonnes considered an unworn rail head. This chapter extends the analysis to consider the

effect of a worn rail profile on longitudinal stresses and their qualitative effect on

fatigue behaviour in the underhead radius of the rail head. The effect of residual stresses

is also incorporated here.

5.2 Finite element analysis model

Several new finite element (FE) models were developed to study the effect of worn

profiles, using the commercial finite elements code ABAQUS (6.11-2). Cases with an

L/V ratio of 0, 0.2 and 0.4 and contact patch offsets 0, 15, 20 and 30 mm were


107

parametrically studied using six rail profile changes. Six worn rail profiles were adopted

for the case studies, namely: no wear (ideal profile), 5 mm, 15 mm, 20 mm, 22 mm, and

25 mm of head wear (HW), defined as the difference between the highest point on the

new rail and the worn one. The Miniprof unit provides the ability to determine these

values. The rail head profiles used for all head wear (HW) conditions were based on a

typical worn profile selected from recent assessment of rail profiles and rail grinding

activities. These rail profiles assessments were conducted by Welsby et al [140]. The

rail profiles with the different HW used for the current analysis are presented in Figure

5.1. The rail is installed in track at the relevant cant (or inclination) as was shown in

Figure 2.2. The rail inclination is 1:40. Hence the worn profile measurement is not the

vertical dimension at the rail centreline but the vertical dimension with the rail at a 1:40

inclination. However, this does not affect the results or arguments presented in this

thesis as in this work wheel on rail was not modelled, Instead a required Hertzian

contact pressure was simulated on the rail mesh.

15 600M HR - Head wear (HW) of 15 mm for high rail on a 600 m radius curve

Figure 5.1 Worn rail profiles [140]

Ideal Profile

3

4


108

Figure 5.2 shows a schematic of the modelling approach with worn rail profile. The

detailed positions of the strain gauges for the field measurements by Mutton et al [1] are

also shown in Figure 5.2. The field data used for the model validation was also

discussed in chapter 3. It should be noted that the details on input parameters such as

loading, boundary conditions and support characteristics will remain the same as was

discussed in chapter 3.

Figure 5.2 The model description for the cross-section view of the worn rail (head wear, HW = 25 mm) profile, the strain gauge measurement location and the elastic foundations with variations of contact patch size and shape

Differential temperature due to seasonal effect is another factor which influences the

stress state at the underhead region in continuously welded rail (CWR). The results

presented in chapter 4 revealed that seasonal temperature variation from ambient (26 oC) to cold (10oC) can produce an additional tensile stress of about 40 MPa at the

underhead radius. In order to simulate the stresses resulting from the effect of


109

temperature variations, the thermal stresses in the longitudinal direction of the rail due

to cold or warm climate were considered. The value of thermal expansion coefficient

was given in Table 3.1.

The input residual stress levels refer to the measurements that were used by

Magiera [136], as discussed in chapter 3. As the rail geometry is quite complex, the

input residual stresses, with six stress components, have to be simplified with uniform

distributions in the proposed rail sections, as shown in Figure 3.14. The input residual

stresses were summarized in Table 3.3 and were applied by defining them as initial

conditions in an input file created in the commercial FE package ABAQUS 6.11-2. For

the first step in the rail FE model calculation, residual stresses became the initial

conditions and no contact loads were applied for this step. It is noted that equilibrium

was reached after some iterations.

5.3 Stress state at the rail underhead radius

The stress state at the underhead radius is influenced by the contact location and the L/V

ratio during the curving and hunting of the vehicle. The work presented in chapter 4

considered in detail the variation of the tension spike and found it to depend on L/V

ratio and contact offset. The tension spike occurs in the rail underhead radius position as

a short-duration spike of about 100 mm in length. The variation in longitudinal stresses

(S11) in this region is worth special consideration as it can play an important role in the

fatigue behaviour of the underhead radius and the propagation of existing gauge corner

cracking. Figure 4.6 shows that an increase in the L/V ratio results in an increase in the

longitudinal tensile stress at the rail underhead radius, and a decrease in the depth at

which the stress becomes tensile. The region with tensile stress shifts up from the rail

underhead radius to the rail surface with an increase in the L/V ratio.

Effect of residual stress 5.3.1

Table 5.1 shows the effect of residual stresses on longitudinal (S11) and Von-Mises

(Mises) underhead radius stresses under different L/V ratios applied at the contact patch


110

offset of 20 mm for an ideal rail profile. These results reveal that the longitudinal stress

(S11) is reduced by nearly 60 % for L/V = 0, 37 % for L/V = 0.2 and 29 % for L/V =

0.4, if the residual stresses are considered, but the Von-Mises stress (Mises) is increased

at this region with an increase in the residual stress values.

Table 5.1 Effect of stress state at the underhead region due to residual stress

L/V ratio Von-Mises stress, Mises (MPa) S11 (MPa)

with RS without RS with RS without RS

L/V=0 345.2 162.1 38.40 95.5

L/V=0.2 375.4 222.25 121.2 191

L/V=0.4 405.6 282.4 204 286.5

Effect of head wear 5.3.2

Rail head wear results in an increase of the longitudinal stress at the underhead radius.

Figure 5.3 shows the variation of maximum longitudinal stress as the HW changes from

0 mm (no wear) to 25 mm. An increase in the tensile stress is seen from 130 MPa to 350

MPa as the head wear (HW) increases from no wear to a HW of 5 mm. Beyond a HW

of 15 mm, the longitudinal stress changes rapidly from 500 MPa to 940 MPa, and

reaches a maximum value of 940 MPa for a HW of 25 mm. Figure 5.4 shows the

longitudinal stress (S11) distribution in the railhead with different head wear (HW)

cases at an L/V ratio of 0.4 and a contact location of 20 mm offset. In the case of an

ideal profile (Figure 5.4a), compressive stresses are dominant in the rail head and small

values of tensile stresses are seen at the underhead radius with a small area under

tension. With an increase in the head wear, a gradual increase in the tensile stress is

evident with larger tensile values starting at the underhead radius and covering more rail

head area under tension. In addition, the region over which tensile stresses occur can be

seen to have moved closer to the top of the rail.


111

Figure 5.3 Variation of maximum longitudinal stress (S11) at the underhead region vs rail head wear for an eccentric load located at 20 mm offset from the rail centreline with L/V = 0.4

Figure 5.4 Longitudinal stresses (S11) distribution at the rail head vs the different head wear (HW) profiles for an eccentric load located at 20 mm offset from the rail centreline with L/V = 0.4

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25

Lon

gitu

dina

l Str

ess

(MPa

)

Head Wear (mm)

L/V=0.4, Patch offset=20 mm

MPa


112

The heavily worn profile with HW of 25 mm presents very high longitudinal tensile

stress (S11 > 600 MPa) at the underhead radius and near to the rail head surface, as

shown by the grey region in Figure 5.4(f). The results also indicate that these very high

longitudinal tensile stresses significantly increased up to 946 MPa when the HW = 25

mm, as compared to the other light or moderate head wear cases. The reported 0.2 %

proof stress data for HH (AS 1085.1) rail grade is 780 MPa at standard head position.

The low alloy and some of the HE (Hypereutectoid) rail grades possess the 0.2 % proof

stress ranges to 910 MPa at the standard head position.

Effect of contact patch offset 5.3.3

The position of the contact patch was a key factor under consideration in this study for

two reasons. Firstly, it is known that for curves and a tangent track, the vehicles tend to

exhibit lateral movement or hunting behaviour on the rails. Secondly, the shape and

location of the contact patch changes as the rail head wear profile is influenced by a

reduction in the cross-sectional area of the rail head.

Figure 5.5 Longitudinal stress distribution at the underhead for the different contact locations vs the HW of 0, 22 mm and 25 mm

-100

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300

400

500

600

0 5 10 15 20

Lon

gitu

dina

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ess

(MPa

)

Contact patch offset from rail centreline (mm)

HW 0 mm

HW 25 mm

HW 22 mm


113

Current simulations show that moving the contact location towards the gauge corner

away from the centre of the rail head cross-section causes the tensile stress at the

underhead radius to increase gradually, as shown in Figure.5.5. In order to study the

effect of contact patch location due to the different head wear (HW) levels, a constant

L/V ratio of 0 was taken to minimise the effect of the lateral load (L). In real life the

lateral load will be expected to increase in the presence of vehicle hunting and during

curving (especially combined with traction) and this is investigated in the next section.

Figure 5.5 shows the variation of longitudinal stress as the head wear changes from

HW of 0 mm (no wear) to HW of 25 mm with different contact patch positions. The

results show that, as a result of increasing the contact patch offset, the longitudinal

tensile stress linearly increases for the contact patch offset from 0 to 15 mm. The small

value tensile stress represents a well-maintained rail profile, but the tensile stress

increases significantly after the contact patch offset of 15 mm. The heavily worn profile

results in higher longitudinal tensile stresses at the rail underhead radius when moving

the contact location toward the gauge corner of rail. It can be seen to have peaked at 522

MPa for the case with HW of 25 mm at the contact patch location of 20 mm offset. The

tensile stress was more sensitive to head wear with a 20 mm contact patch offset, which

is representative of poorly maintained rail profiles.

Effect of the L/V ratio 5.3.4

The lateral load (L), which is influenced by vehicle steering or hunting behaviour, was

investigated to evaluate the stress state at the rail underhead radius (UHR). The values

of L were changed to obtain an L/V ratio of 0, 0.2 and 0.4 for this investigation. The

tensile stress generated at the underhead radius was due to the bending of the rail head-

on-web caused by the eccentricity of the vertical and lateral loads from the rail head

centreline. The stress analyses at the underhead radius of the gauge side of the rail

indicated that the tensile stress at the underhead radius position (measurement point

offset of 34 mm) clearly increased with an increase in L/V ratio at the contact patch

offset of 20 mm with respect to different head wear (HW) levels, as shown in Figure

5.6. It was noted that the heavily worn profile showed the worst case, presenting a very


114

high longitudinal tensile stress of 946 MPa at underhead radius for the L/V ratio of 0.4,

which potentially achieves the yield of the rail material at underhead radius, as shown in

Figure 5.4f. The graph with the no head wear (HW of 0 mm) condition for the tangent

track exhibiting free rolling (L/V = 0) showed a much lower value of tensile stress even

with an increase in L/V ratio up to 0.4, being a 19 % increase in the tensile stress.

However, in the presence of the head wear levels, the tensile stress value became high at

the same location of the underhead radius, almost doubling in value (114 % for HW of

22 mm and 81 % for the HW of 25 mm) in the tensile stress, with an increase in L/V

ratio.

Figure 5.6 Variation of longitudinal stress at the gauge side underhead radius for eccentric loading from rail centreline towards the gauge side for different rail worn profiles versus the lateral traction coefficient

Effect of track foundation stiffness 5.3.5

An investigation was undertaken into vertical foundation stiffness underneath the two

middle sleepers adjacent to the contact position on the longitudinal tensile stress at the

rail underhead radius and base fillet. A low value of vertical foundation stiffness, KV,

presents a poor track support condition. In the previous analysis of the ideal rail profile

(HW of 0 mm), presented in chapter 4, the longitudinal stress at the rail underhead

0100200300400500600700800900

1000

0 0.1 0.2 0.3 0.4

Lon

gitu

dina

l Str

ess

(MPa

)

L/V ratio

HW 25 mm, Patch offset 20 mmHW 25 mm, Patch offset 15 mmHW 22, Patch offset 20 mmHW 0 mm, Patch offset 20 mm


115

radius increased, but decreased at the base fillet as a result of increasing the vertical

foundation stiffness from 0 to 123 MN/m (see Figure 5.7). Beyond the vertical

foundation stiffness of 35 MN/m, the longitudinal tensile stress became steady for both

locations.

This thesis involved the effect of head wear on longitudinal stresses at the rail

underhead radius and the base fillet under the track support conditions. Figure 5.7

suggests the longitudinal stress increased by 115 MPa and 384 MPa at the base fillet

and underhead radius respectively as a result of increasing the HW from 0 to 22 mm

under average track support stiffness (35 MN/m) conditions, as described in chapters 3

and 4. The increase in the longitudinal tensile stress with HW of 22 mm is more

pronounced at the underhead radius as compared to the base fillet. This is because of

increased local bending stresses at the underhead radius with an increase in the head

wear. This effect has already been discussed in the previous section. The longitudinal

underhead radius stresses resulting from variations of support stiffness for different rail

head wear profiles are also presented in Table 5.2.

Figure 5.7 Longitudinal stress variations at the underhead radius with vertical foundation stiffness at the contact patch 20 mm offset and L/V = 0.4: at the underhead offset of measurement point 34 mm and at the base fillet at the midpoint of the rail span (in longitudinal direction x=2100 mm) in between two middle sleepers

0

100

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300

400

500

600

700

800

900

0 20 40 60 80 100 120 140

Lon

gitu

dina

l Str

ess (

MPa

)

Vertical Foundation Stiffness (MN/m)

UHR - HW = 22 mm

UHR - HW = 0 mm

Base fillet - HW = 22 mm

Base fillet - HW = 0 mm


116

Table 5.2 Effect of rail support stiffness on rail longitudinal UHR stresses (S11)

Support stiffness Vertical

Longitudinal stress, S11(MPa)

HW 0 mm HW 22 mm

Loose support

(0 MN/m) 267 599.2

Average support

(35 MN/m) 322 705.8

Stiff support

(123 MN/m)

340 736.6

Evaluation of depth of tensile longitudinal stress 5.3.6

The mechanical response at the underhead radius is significantly influenced when

heavily worn rail profiles are considered. Rail head wear can result in changes to the rail

profile and a reduction in the cross-sectional area of the rail head, both of which can

affect the stress state. The extent to which the loss of a cross-sectional area can be

sustained is related to the direct relationship between the resultant increase in stress

levels and the probability of rail failure. An increase in head wear results in an increase

of the longitudinal stress at the rail underhead radius, as was discussed in section 5.3.2.

As the tension spike occurs in the rail underhead radius position as a short-duration

spike, about 100 mm in length, the variation in longitudinal stresses (S11) in this region

was of special interest as it can play an important role in the fatigue behaviour of the

underhead radius and the propagation of existing gauge corner cracks.

Figure 5.8a shows the longitudinal stress variation with depth for different vertical plane

offsets of 15, 20, and 25 mm from the rail vertical centreline towards the gauge corner,

with an L/V ratio of 0.4 at the contact patch offset 20 mm for HW of 25 mm. By



117

gauge corner, the longitudinal tensile stress increases, and the depth at which the stress

becomes tensile reduces from 8 mm to 0.7 mm for HW of 25 mm

Figure 5.8 Longitudinal stress distribution with depth on different vertical planes offset from the rail vertical centerline towards the gauge corner, at a contact patch offset of 20 mm L/V=0.4: (a) HW = 25 mm, (b) HW = 15 mm

02468

101214161820

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

Dep

th (m

m)


Plane offset 15 mmPlane offset 20 mmPlane offset 25 mmPlane offset 30 mm

0

2

4

6

8

10

12

14

16-1000 -800 -600 -400 -200 0 200 400 600 800 1000

Dep

th (m

m)


Plane offset 15 mmPlane offset 20 mmPlane offset 25 mm

(a) HW = 25 mm

(b) HW = 15 mm


118

For HW of 15 mm (Figure 5.8b), the depth at which stress becomes tensile reduces from

20 mm (plane offset 15 mm) to 0 mm (plane offset 30 mm).

As the contact patch offset can reach a maximum of 25 mm, the longitudinal stress

variation with a depth for different vertical plane offsets of 15, 20, 25 and 30 mm from

the rail vertical centreline towards the gauge corner, with an L/V ratio of 0.4 at the

contact patch offset 25 mm for HW of 20 mm, was examined (Figure 5.9). By


gauge corner, the longitudinal tensile stress increases, but the depth at which the stress

became tensile reduced from 13 mm (plane offset 15 mm) to 0 mm (plane offset 30

mm) for HW of 20 mm as shown in Figure 5.9.

Figure 5.9 Longitudinal stress distribution with depth on different vertical planes at 15, 20, 25 and 30 mm offset from the rail vertical centerline towards the gauge corner at a contact patch offset of 25 mm L/V=0.4 HW 20mm

A comparison of the stress variation with realistic head wear is illustrated by contour

plots (Figure 5.10). The line shows the tension / compression boundary and a

subsequent upward shift in this boundary as a function of increasing head wear. Figure

02468

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-1000 -800 -600 -400 -200 0 200 400 600 800 1000

Dep

th (m

m)


Plane offset 15 mmPlane offset 20 mmPlane offset 25 mmPlane offset 30 mm

HW 20 mm


119

5.10 also shows the longitudinal stress distribution as a function of depth on a vertical

plane at a 25 mm offset from the rail vertical centreline towards the gauge corner for

contact patch offsets of 20 mm with L/V 0.4 with different head wear (HW). The high

head wear (HW) results in an increase of the longitudinal stress at the underhead radius

(UHR) and the region of tensile stresses moves closer to the top of the rail. These results

illustrate the severity of longitudinal tensile stresses at the underhead radius (UHR)

region with increasing head wear (HW) conditions.

Figure 5.10 Longitudinal stress distribution (S11) with depth on a vertical plane at a 25 mm offset from the rail vertical centerline towards the gauge corner for contact patch offsets of 20 mm with L/V 0.4


120

Figures 1.2, and 1.3 [2] show the development of a crack at a gauge corner in a worn

rail, its turning down in the rail head and the final fracture as a transverse defect. Such

high tensile bending stress could help progress transverse defects (TDs) from the RCF,

which generally start to turn down at about 5 mm below the rail head surface, as

examined by Mutton et al [1, 2]. Another example was given by Mutton et al [2],

indicating that the RCF cracks at the gauge corner turn down in low alloy heat treated

rail grades as well.

The crack is expected to turn down and grow rapidly when it enters the region of tensile

bending stresses. Its growth will depend on the magnitude of the tensile stress as

compared to the contact stress field, material properties such as anisotropy etc. In the

case of an HW of 25 mm, the zone of tensile bending stress was very close to the

contact surface (Figure 5.10), which could cause a rolling contact fatigue crack to turn

perpendicular to the running surface, resulting in the formation of transverse defects due

to Mode I, as has been observed in practice [1, 2]. The presence of a high underhead

radius stress peak in a heavily worn rail head is evident, as is shown in Figure 5.10. In

this case, the Mode I transition of a pre-existing RCF crack to form transverse defects

has been caused by local bending rather than global bending.

5.4 Summary

This chapter has described the stress state at the rail underhead radius, influenced by the

head wear (HW) levels in conjunction with the wheel-rail contact conditions and / or

L/V ratio. The rail underhead radius in a heavily worn rail presents the most critical case

with very high tensile stress close to the contact surface. The wheel-rail contact location

is another critical factor that influences the stress state at the rail underhead radius,

particularly when moving laterally towards the gauge side. Lateral loading generated by

curving and hunting operations can also be highly influential on the stress state at the

rail underhead radius. Stiff tracks are more prone to high underhead radius stresses.

The depth at which tensile longitudinal stresses occurred was evaluated and it was also

demonstrated that residual and CWR stresses interact with these high local bending


121

stresses, and can potentially initiate a fatigue crack or cause existing rolling contact

fatigue cracks to turn downwards and form transverse defects. In addition, the results

presented in this chapter have helped to understand the mechanism of rail damage

qualitatively.


Chapter 6. Fatigue damage prediction

123

Chapter 6

Fatigue damage prediction

6.1 Introduction

This chapter examines the possibility of fatigue damage initiation at the rail underhead

radius (UHR) due to the occurrence of a short duration tensile stress peak when a wheel

directly passes over this region. Heat-treated low alloy, euctectoid and hypereutectoid

rail grades operated under heavy haul conditions are considered. It was demonstrated in

chapters 4 and 5 that tensile stress generated at the underhead radius is mainly due to the

localised bending of the rail head-on-web induced by the complex wheel-rail contact

conditions that are associated with a combination of lateral offsets in the contact

positions and lateral (transverse) forces. In the presence of heavily worn rail, the

magnitude of these stresses could be even higher. The main focus of this chapter has

been an evaluation of the potential risk of fatigue damage at the underhead radius

(UHR) resulting from a combination of wheel-rail contact conditions and rail head wear

(HW) states.

Mechanical responses at the underhead radius have been explored using the finite

element method (FEM). The potential fatigue mechanism and crack initiation at the

underhead radius are described in this chapter as damage parameters. The Dang Van

(DV) criterion, implemented as a customised computer programme, was used to identify

any potential fatigue damage at a specified location. The Palmgren - Miner law was

used to quantify damage accumulation and potential cycles to failure. Fatigue behaviour

was compared for the different high strength rail steel grades used for heavy haul

operations in order to predict rail wear limits.


124

Rail wear can result in changes to the rail profile and a reduction in the cross-sectional

area of the railhead, both of which can affect the stress state under in-service loading

conditions. The extent to which a loss in the cross-sectional area can be sustained is

related to the direct relationship between the resultant increased stress levels and the

probability of rail failure. Wear itself may therefore be a direct threat to rail integrity,

but excessive wear combined with the presence of a defect that increases the risk of

fatigue cracking may pose an even greater risk [48]. Wear limits for rails can therefore

be determined by examining the relationship between rail wear and stress levels

throughout the rail section, and the probability of fatigue damage either in defect-free

rail, or in the presence of known defect types. Defect types that may be considered

include rolling contact fatigue (RCF) and reverse detail fractures (RDF). Additional

defect types may be associated with rail welds such as those produced using flash butt

or aluminothermic welding procedures.

Marich [36] reported that rail wear under high axle load conditions can generally be

observed on the gauge face of the high rail of curves with radii less than 500-800 m due

to the increased lateral loads and the occurrence of wheel flanging. Loss of material at

the running surface of the rail also occurs due to a combination of wear resulting from

normal wheel/rail interaction, and rail maintenance activities such as rail grinding. The

influence of increasing rail head wear levels on rail stress levels can be examined

numerically using finite element analysis. Marich [38] also measured rail stress levels

under both laboratory and field conditions to define acceptable rail wear limits for high

axle load conditions, based on the fatigue behaviour of the rail material. It was found

that rail wear limits could be considerably greater than what was currently accepted in

practice.

In a rail section, there are two deformation zones, which can be simply classified in

terms of the mechanical responses caused by the wheel-rail contact conditions. The first

zone, which is close to the contact surface, is the region of plastic deformation due to

repeated rolling/sliding contacts between the wheels and the rail. As a consequence, if

the contact loads are high, crack initiation and propagation in the contact region leading

to rail defects, such as rolling contact fatigue and squats, could be very significant. This


125

behaviour is of great significance to the rail industry due to the increased risk of rail

failure, and has therefore been the subject of much research [48, 82-85]. The hypothesis

for these studies is based on the theory of plasticity with a suitable fatigue criterion for

the rail life prediction. The second zone is the underhead region, which is away from the

contact surface but is influenced by the same wheel-rail contact conditions. To predict

the initiation of a crack in this region, the Dang Van fatigue criterion can be used, as it

has previously been used in many industrial applications, particularly in the automotive

industry [28, 52]. The fatigue criterion is based on a multi-scale approach and on a

shakedown limit hypothesis. As the wheel-rail contact loads cause a multi-axial state of

stress with out-of-phase stress components and varying principal stress directions, such

a fatigue criterion is suitable and is thus used to predict fatigue limits and crack

initiation in a rail for this thesis [51]. Figure 6.1 shows the results for in-depth analysis

of the stress components.

Figure 6.1: In-depth stress analysis at underhead offset of 25 mm for head wear of 22 mm, contact patch offset = 15mm, L/V=0.4

-100

0

100

200

300

400

500

600

1900 1950 2000 2050 2100 2150 2200 2250 2300

Valu

e of

Sre

ss (M

Pa)

Distance x-position (mm)

sxx

Txz

Syy

Szz

Txy

Tzy

Smises

25mm


126

In sharper curves, as the curving forces increase, the deformation of the rail material

potentially results in a flow lip at the lower gauge corner. With the continued passage of

rail traffic, a fatigue defect may be initiated and propagated into the rail head [9]. The

same behaviour is considered significant in reverse detail fractures (RDF), which are

found to occur in North American rail systems [6-9] due to poorly lubricated, heavily

worn rails on stiff tracks subjected to high axle loads. The potential for reverse detail

fractures is higher in sharper curves, due to the increased probability of the localised

stress concentrations associated with heavy gauge face wear and plastic flow. Reverse

detail fracture is also influenced by longitudinal bending stresses at the underhead

radius, as reported by Jeong et al [6]. Failures due to fatigue cracking at the underhead

radius of aluminothermic welds under heavy axle load conditions were examined by

Salehi et al [42]. The welds exhibited an increased sensitivity to this behaviour due to a

combination of elevated residual stress levels, a more complex local geometry leading

to increased stress concentration, and differences in material characteristics compared to

the parent rail.

Some of the above aspects were examined in chapters 4 and 5. The current chapter

examines the probability of fatigue damage in the rail underhead radius, under the

influence of localised tensile stresses and in the absence of pre-existing defects, such as

transverse defects and reverse detail fractures. These aspects were examined in the

context of heavy haul (high axle load) conditions, based on the behaviour of 68 kg/m

rail. The influence of rail material grade was considered, based on the characteristics of

a number of heat-treated low alloy and hypereutectoid grades commonly used under

these conditions [2, 13, 130]. A high cycle fatigue analysis based on Dang Van criteria

[51] was used to examine the fatigue behaviour for a range of wheel-rail contact

conditions and varying amounts of rail wear. In addition, crack propagation was

simulated using the Pamgren - Miner damage accumulation law [62, 103], which can be

used to determine the extent of material damage at the underhead region and predict the

number of cycles to failure at a specified reference point.


127

6.2 Model development

The finite element (FE) models developed in chapters 3 and 5 to calculate a stress state

were extended to evaluate fatigue damage around the reference point set at the

underhead radius. The Dang Van criterion was programmed into FORTRAN-code and

implemented into ABAQUS through UVAR-M (User output variable), to identify any

potential fatigue damage. A wheel load was applied on the rail, assuming a fully

slipping Hertzian contact pressure, at the midpoint of the rail span (in longitudinal

direction at x = 2100 mm) in between two middle sleepers. Different contact patch

offsets (0, 15 20 and 30 mm) were considered. A contact patch offset of 20 mm is

shown in Figure 6.2. It should also be noted that the details on input parameters such as

loading, boundary conditions and support characteristics will remain the same as were

presented in chapter 3.

In real life, the whole stress cycle at a point is obtained as the contact loading moves

from one end of the rail span to the other, as shown in Figure 6.2a (not to scale), with

initially a low stress which becomes maximum when the wheel contact patch is on top

of the reference point under consideration and returning to low once the wheel has

passed the reference point. As the rail geometry being consistent for the entire rail span,

this stress cycle can be simulated by keeping the contact load fixed on one point but

moving the point under consideration from one end to the other.

A separate study was conducted for fatigue analysis at the rail underhead radius

throughout the span (S = 4200 mm, Figure 6.2b, not to scale), to check the value of

maximum fatigue damage under given extreme loading conditions. The maximum

fatigue damage was found to occur at the underhead radius, when the wheel load on the

rail surface was close to the position of interest. This approach was found to be

sufficiently good to perform the study. Therefore, fatigue damage analysis was

performed for a very fine mesh of 30 mm width of rail length in a longitudinal direction

at the underhead radius, with respect to the contact patch of 20 mm width (semi major

axis a = 10 mm) along the longitudinal direction of the rail, to accurately capture the

value of the maximum fatigue damage and reduce the computational time, as shown in


128

Figure 6.2c. Coarse mesh was used for the rest of the rail span (S = 4200 mm). In the

parametric studies, rail profiles representing no wear (ideal profile), 5 mm, 10 mm, 15

mm, 20 mm and 22 mm of head wear (HW) were used. The finite element model

construction, the required parameters and the modelling validation have already been

discussed in detail in chapters 3 and 5.

Figure 6.2 Representation of model setup for DV fatigue damage analysis

(a) (b)

(c)


129

Dang Van fatigue criterion 6.2.1

The Dang Van (DV) criterion is a shear stress based criterion that is applicable for stress

levels below the elastic shakedown limit of the material. If the following inequality is

satisfied on a shear plane passing through each material point at least once in the whole

stress cycle, damage occurs. This inequality is expressed as:

( ) ( ) (6.1)

τa(t) is the time dependent value of shear stress on the specified shear plane at the

specified material point and is defined as the difference between the instantaneous and

mean shear stress of the loading cycle; σh is the time dependent hydrostatic stress at the

material point. The constants (aDV and bDV) are the functions of the material fatigue

limits [28]. The fatigue limits f-1 and t-1 can be obtained from the classical experimental

bending and twisting tests respectively, and are stated in Table 6.1 based on estimated

fatigue properties. The calculations are shown in the next section. The constants are

calculated as:

( )

(6.2)

The value of the inequality‘s left side represents a numerical index for fatigue damage,

taken as the DV (Dang Van) damage parameter applied to the dense mesh area at the

underhead radius (see Figure 6.2c) for subsequent investigations.

Table 6.1 Material properties and constants of rail

Material Parameters Value

Young‘s modulus, E (GPa) 209 Poisson‘s ratio, 0.29 Alternate bending fatigue limit, f-1, (MPa) 353 Alternate torsional fatigue limit, t-1, (MPa) 205 aDV 0.24 bDV, (MPa) 205


130

Implementation of the critical plane approach 6.2.2

The plane on which the above-mentioned inequality is satisfied is called the critical

plane, as shown in Figure 6.3. However, the critical plane was not obvious at the

beginning of the analysis so the inequality needed to be assessed in all shear planes

passing through each material point being investigated for potential fatigue crack

initiation. The critical plane and DV damage parameters were calculated using the

corresponding algorithms and equations implemented into customised FORTRAN-

code, linked to the FE package ABAQUS, UVAR-M (user output variables) as given in

Appendix C. The Dang Van criterion was programmed based on equations presented by

Ekberg et al [62]. The validity of the criterion under various stress cycles and the mid

value of shear stress have been described elsewhere in the literature [51, 103, 141]. The

implementation in ABAQUS subroutine –UVAR-M was further checked by modeling a

shaft made of ductile material. In torsional loading, it will fail along the plane of

maximum shear, that is, a plane perpendicular to the shaft axis. The list of defined

UVAR-M‘s for the output variables is given in Table 6.2. A description of UVARs 2 to

5 on a critical plane is given in Figure 6.3.

Table 6.2 Definition of UVARs for the output variables

UVARs Meaning

UVAR1 The inequality‘s (Equation 6.1) left side, EQ, (MPa) UVAR2 Angle between the x axis and the external normal, n, of the critical plane

UVAR3 Angle β between the y axis and the external normal, n, of the critical plane

UVAR4 Angle between the y axis and the external normal, n, of the critical plane

UVAR5 The shear stress at the critical plane, (MPa)

UVAR6 The shear stress amplitude, a(t), (MPa)

UVAR7 The damage parameter, Di, at the ith number of the equivalent stress cycles

UVAR8 The predicted number of cycles to failure, Nf


131

Figure 6.3 Description of UVARs 2 to 5 on a critical plane

Estimated fatigue properties of different rail grades 6.2.3

For high axle load rail operations, high strength, heat-treated rail grades are commonly

used, as these provide increased resistance to damage in the form of wear and rolling

contact fatigue [2]. During manufacture, heat treatment is carried out using either in-line

or off-line processes to increase hardness levels throughout the rail head. In addition,

steel compositions may vary from eutectoid carbon-manganese grades to low alloy

grades and more recently hypereutectoid grades with carbon contents up to 1% [130].

Hardness levels throughout the head of these rail types decrease with increasing depth

below the outer surface. The effective depth of hardening is typically greater with in-

line processes, while hardness levels are generally higher in the low alloy and

hypereutectoid grades. For this thesis, rails representing the three categories described

above were used. The variation in hardness along a traverse from the gauge corner to

the intersection of the fishing surfaces in these grades has been examined previously [2,

13]. The material grades and typical compositions are summarised in Table 6.3.

O


132

Table 6.3 Chemical composition [43, 130]

Rail Type C %

Si %

Mn %

Cr %

P %

S %

HH 0.65 0.58 1.25 0.2 0.025 0.025

LAHT 0.72 0.13 0.65 0.6 0.025 0.025

HE 1.0 0.55 0.7 0.3 0.025 0.025

The material fatigue limit, as defined by the stress at which failure occurs after a

specified number of loading cycles, is a key factor when describing rail life. The fatigue

limit under rotating bending, f-1, is proportional to the ultimate tensile stress (Y.) of the

rail, with a proportionality ratio in the range 0.25-0.6 [142]. For the present study, a

ratio of 0.38 at 106 cycles was used for the underhead region, while the twisting fatigue

limit, t-1, was set to 58% of the f-1 [142]. Generally a ratio of 0.5 is considered suitable

for steel. However there should be a reduction for surface roughness, which would give

a ratio lower than 0.38. Since the gradients are high, this value could probably be

increased. Three fatigue limits with different rail grades are provided to observe the

sensitivity of these limits in detail. The mechanical properties of relevance are for the

underhead region, and hence may differ from those measured at other locations in

accordance with the relevant specifications. The value of the ultimate tensile stress in

this region can be estimated from the hardness according to reference [13].

Experimental results for hardness distribution 6.2.4

Figure 6.4 shows the hardness testing apparatus used. Hardness measurements were

carried out in the underhead radius region of the samples (specimen in Figure 6.5)

prepared from the above mentioned rail grades, and these results are plotted in Figure

6.6 and were used to estimate the fatigue properties summarised in Table 6.4.


133

Figure 6.4 INDENTEC-Hardness measurement apparatus (Courtesy of Swinburne University of Technology)

Figure 6.5 Rail sample specimen (section and the shape) for hardness testing


134

Figure 6.6 Hardness distribution along transverse plane (underhead radius measurement point offset 20 mm) of high strength rail steels

Table 6.4 The estimated fatigue limits at the UHR for different high strength rail grades

Rail grades Hardness (HV)

Y (MPa)

f-1 (MPa)

t-1

(MPa)

HH 270 615 353 205

LAHT3 320 753 424 246

HE3 370 895 480 278

Damage Accumulation 6.2.5

The DV criterion employed in the model was used to identify the damage accumulation

that occured during a specific time portion of the analysis when the inequality (6.1) at

each material point was fulfilled. To quantify the damage, the Palmgren-Miner linear

damage accumulation rule [103, 143] in conjunction with the Wöhler curve [62, 143]

250

270

290

310

330

350

370

390

410

430

450

0 2 4 6 8 10 12 14 16 18 20

Har

dnes

s (H

V30)

Distance from centre of railhead cross-section (mm)

HE3 (HE-X)LAHT3HH (AS1085.1)


135

was also used. The Wöhler curve defines a linear relationship between fatigue stress and

number of cycles to failure in which the fatigue threshold stress is in a range of t-1

corresponding to failure at the t-1 for 106 stress cycles (the highest cycle) and at the DV

magnitude of the fracture stress for 101 stress cycles (the lowest cycle), as shown in

Figure 6.7. For this thesis, the stress cycle for each case was evaluated to ensure that

there was no plasticity, a basic condition of the DV criterion. It should also be noted

that, in the current thesis, the Wohler curve, Figure 6.7, is used only to demonstrate the

methodology. The loading in this case is non-proportional, for practical implementation,

it would be desirable to use material properties correlated with extensive experimental

and site observations that represent the in-service loading conditions.

Figure 6.7 The fatigue threshold stress is in a range of t-1 corresponding to failure at the t-1 for 106 stress cycles (the highest cycle) and at the DV magnitude of the fracture stress for 101 stress cycles (the lowest cycle)

In terms of the Palmgren - Miner linear damage accumulation rule, the damage

degradation can be determined through the following expression:

Di = ( ) ( ) (6.3)

0

100

200

300

400

500

600

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

DV

Dam

age

Para

met

er -

UVA

R1

(MPa

)

Number of Cycles to Failure (log(N))

HE3

LAHT3

HH


136

Where, Di: is the damage corresponding to the ith equivalent stress cycles,

i: is the number of the shear stress cycles

τEQi: is the ith equivalent stress calculated by Equation 6.1

In Equation 6.3, at the low stress cycles of 101, the damage parameter, D, is equal to 0.1

when the reaches the bending fatigue limit, f-1. At the high stress cycles of 106, the

damage parameter, D, is equal to 10-6 when the τEQ reaches the twisting fatigue limit, t-1.

The damage degradation at each material point via the ith stress cycles to failure, Nf, can

be defined as:

Di = 1/Nf. (6.4)

6.3 Residual stresses

The overall stress state in rails under service loading conditions can also be influenced

by the residual stresses. Therefore, residual stresses in the rail are an important factor

contributing to the overall stress state, and hence some rail specifications include upper

limits for residual stress levels [81, 127]. The distribution of residual stresses in as-

manufactured rails is influenced by straightening procedures such as roller-straightening

[144]. The measurement of residual stresses in rails is usually undertaken using a

destructive method involving strain gauges [26]. This method does not provide

information on the distribution of residual stress at depths below the outer surface,

which can be determined using neutron diffraction [145].

In previous studies of RCF crack initiation and propagation, there was rare

consideration given to residual stress scatter in the subsurface for most models used [27,

120, 146]. Since fatigue life is dependent on the weakest point of a material, the local

extreme is most relevant for the fatigue life prediction. For the current study, the

residual stress distribution was based on the measurements conducted by Magiera [147].

As the rail geometry is quite complex, the residual stresses were considered in terms of

the six stress components for each of the regions shown in Figure 3.14. The input

residual stresses are summarized in Table 3.3.


137

6.4 Plasticity of the underhead radius

The stress state in the underhead region needs to satisfy the basic conditions of the DV

criterion, which states that stress during the loading cycle remains elastic. The

maximum value of Von-Mises stress, Mises max, at the underhead region was calculated

for L/V ratio of 0, 0.2 and 0.4, contact patch offset (CPO) values of 15 and 20 mm and

different rail worn profiles with a maximum of 22 mm of head wear.

Figure 6.8 Von Mises stress distribution at the rail underhead radius vs different head wear profiles: for eccentric loads located at different offset from the rail centerline with L/V = 0, 0.2, 0.4

The yield region is defined by Mises max > Y. The yield stress ( Y) values of three high

strength rail grades were used in this analysis. The yield stress ( Y) value used were for

a plain C-Mn Head Hardened grade, HH (AS 1085.1) - Y = 615 MPa [80], Low Alloy

Heat Treated grade, LAHT3 - Y = 753 MPa [127] and Hypereutectoid heat treated

grade HE - Y = 895 MPa [81] at the underhead radius. These values were estimated

0100200300400500600700800900

1000110012001300

0 2 4 6 8 10 12 14 16 18 20 22

Von

Miss

es S

tres

s -M

ises (

MPa

)

Head Wear (mm)

L/V=0.2, Patch offset=20 mmL/V=0.4, patch offset=15 mmL/V=0, Patch offset=15 mm

HE3- σY

LAHT3- σY

HH (AS1085.1)- σY

Yield Region

Increasing the yield strength


138

based on hardness testing conducted at the underhead radius, as reported in Table 6.4.

The yield stress values at the underhead region for the HH, LAHT and HE3 rail grade

are also illustrated in Figure 6.8 by three dashed lines, which are used to examine the

yield criteria for different materials. The effect of increasing the yield strength of the

rail material for the same range of loading conditions is also shown in Figure 6.8.

Increasing the yield strength enables the limit to be increased for high strength rail

material grades.

The results (Figure 6.8) indicated that, for a heavily worn rail profile, the Von-Mises

stress at the underhead region increased. Under the severe loading case (CPO = 20 mm,

L/V = 0.2), the Von-Mises stress remained below the yield stress of the HE3 and

LAHT3 rail grades up to the maximum head wear considered. For the HH rail grade, the

Von-Mises stress exceeded the yield stress at a head wear value of 18 mm, for an L/V

ratio of 0.2 applied and CPO of 20 mm from the rail centerline; reducing the CPO to 15

mm at an L/V ratio of 0.4 increased this limit to 22 mm. For HH rail, above these head

wear limits implies (global) plastic flow and thus the DV criterion is not applicable.

6.5 Dang Van (DV) damage parameter

The model uses a wheel load of 172 kN, which is equivalent to an axle load of 35

tonnes. The L/V ratios of 0.2 and 0.4 represent loading conditions in curves of

decreasing radii. For tangent track, an L/V ratio of 0 is used, although this may increase

slightly in the presence of vehicle hunting. The curved track loading cases also include a

contact patch offset of 15 mm from the centre of the running surface for a well

maintained rail profile, and 20 mm for poorly maintained rail profiles, as was reported

by Marich [36].

Fatigue failure is predicted to occur at the rail underhead region when the DV damage

parameter exceeds the torsional fatigue limit t-1. Figure 6.9 shows the extent of the

region at the underhead radius, which satisfies this criterion under the loading cases

examined for the HH rail grade. For a contact patch offset of 15mm and an L/V ratio of

0 (Figure 6.9(a)), fatigue damage is first predicted to occur at a head wear of 20 mm; the


139

region affected increases in size for a head wear of 22mm. For a tight radius curve,

represented by a contact patch offset of 20mm and L/V ratio of 0.2, fatigue damage is

first predicted at 14 mm head wear. The critical plane on which the fatigue damage

occurs at point A in Figure 6.9 (b) is at 108O in the x-axis, at 144O in the y-axis and at

60O in the z-axis, respectively. The angles of the critical plane are defined in Table 6.2

and its directions shown in Figure 6. 2.

(a) Contact patch offset 15 mm with L/V=0

(b) Contact patch offset 20 mm with L/V=0.2

Figure 6.9 Variation of fatigue damage with rail head wear at the underhead radius for an eccentric load located at (a) 15 mm offset from the rail centerline with L/V=0; (a) 20 mm offset from the rail centerline with L/V=0.2, for a plain C-Mn Head Hardened (HH) grade [80], with t-1 = 205 MPa

HW = 22 mm HW = 20mm HW = 14 mm

HW = 18 mm HW = 14 mm

HW = 10 mm

Point A

UVARM1 (MPa) - DV Damage Parameter


140

In practice, the head wear (HW) limit of rail has been stated by Duvel et al [13] to be in

the range of 20 to 15 mm at the curves of decreasing radii. The current head wear limit

is set to the 22 mm for tangent track and 15 mm for tight rail radius curves reported for

heavy haul railways in Australia [33, 148]. Typically, HW = 20 mm is allowed for

Deutche Bahn (DB), as reported by Zerbst et al [124]. Table 6.5 shows that the

predicted head wear limits are in line with practice under the proposed extreme loading

cases, but the predicted value may be conservative. However, by considering the

loading conditions with the lateral shear traction in the outward direction, Marich [38]

found that the acceptable wear limit would be 27 mm for 600-800 m radius curves at an

L/V ratio of 0.3 for rail material with a fatigue strength of 240 MPa. The HW drops to

20 mm if an additional tensile stress of 80 MPa due to variations in rail temperature

below the stress-free or neutral temperature were considered. Both of these values are

greater than what is currently accepted within the rail industry.

Table 6.5 Approximate comparison of predicted head wear limits for head hardened rail based on fatigue limits with extreme loading cases for different track conditions compared to current head wear limits [124, 148]

Rail track curvature

Rail profile maintenance

Assumed L/V ratio

Patch offset (mm)

Predicted HW limit (mm)

Current HW limits (mm)

Tangent rail track

well 0 15 20 22

poor 0 20 15

Tight radius curve

well 0.4 15 14 15

poor 0.2 20 13

Figure 6.10 shows the variation in Dang Van damage parameter with an increase in

head wear for a range of loading conditions and the three rail grades considered. The

fatigue prone region for the three rail grades is represented as horizontal lines for t-1 of

HH, LAHT3 and HE respectively, as shown in Figure 6.10. For the HH rail grade,


141

initiation of fatigue damage is predicted at approximately 14 mm of head wear at an

L/V of 0.2 and contact patch offset (CPO) of 20 mm, or an L/V ratio of 0.4 and 15 mm

of contact patch offset. The corresponding limit for an L/V ratio of 0 and contact patch

offset of 15mm is approximately 20 mm. However, with an increased fatigue strength

there is also an increased sensitivity to surface roughness, Figure 6.10, shows the

estimated fatigue limits for three different rail grades to see the sensitivity of these

grades against fatigue crack initiation under given loading conditions. Increasing the

fatigue strength enables the head wear (HW) limit to be increased for high strength rail

material grades.

Figure 6.10 Variation of DV damage parameter versus rail head wear HW at the underhead radius for an eccentric load located at different offset from the rail centreline with L/V=0, 0.2, 0.4, for three rail grades: a plain C-Mn Head Hardened (HH) grade [80], a Low Alloy Heat Treated (LAHT) grade [127], and Hypereutectoid (HE) Heat Treated grade [81]

0255075

100125150175200225250275300325350375400

0 2 4 6 8 10 12 14 16 18 20 22

DV

Dam

age P

aram

eter

-UVA

R1

(MPa

)

Head Wear (mm)

L/V=0.2, Patch offset=20 mmL/V=0.4, Patch offset=15 mmL/V=0, Patch offset 15 mm

HH (AS1085.1) - t-1

LAHT3 - t-1

HE3 - t-1

Fatigue Region

Increasing the fatigue strength


142

Figure 6.11 shows the variation in the Dang Van damage parameter as a function of the

contact patch offset, for an L/V ratio of 0.2 and head wear values of 18 mm and 22 mm.

Fatigue damage is predicted to be initiated at a contact patch offset of 10 mm for head

wear of 22 mm, and at a contact patch offset of 15 mm for head wear of 18 mm for HH

rail grade. For materials such as LAHT3 and HE3, rail grades cause the horizontal line

to move up and the contact patch offset can be extended to 22.5 mm under extreme

loading conditions. These conditions are representative of extreme–case loading under

heavy haul operations in Australia [13, 36]

Figure 6.11 DV damage parameter vs different contact patch offset with a load of L/V=0.2 for the HW of 18 and 22 mm

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12 14 16 18 20 22

DV

Dam

age

Par

amet

er -

UV

AR

1 (M

Pa)

Contact patch offset from rail centreline (mm)

HW 22 mmHW 18 mm

HE3 - t-1LAHT3 - t-1

HH (AS1085.1) - t-1

Fatigue Region

Increasing the fatigue strength


143

6.6 Damage accumulation and fatigue life

This analysis is primarily concerned with fatigue damage as predicted using the Dang

Van criterion. The accumulation of fatigue damage is calculated based on the Palmgren

- Miner linear damage accumulation rule in conjunction with the Wöhler curve. At the

underhead region the calculated maximum damage accumulates, considering all shear

planes through each material point under the initial assumption that a value of D = 10-6

corresponds to failure.

For the HH rail grade, the weakest of rail materials considered in this thesis, the onset of

damage at the underhead region occurs when the rail head wear increases to 14 mm at

an L/V value of 0.2 and contact patch offset of 20 mm, as shown in Figure 6.12.

Figure 6.12 The damage accumulation via the head wear at the underhead radius for an eccentric load of L/V = 0.2 at the contact patch offset of 20 mm

0.0

5.0

10.0

15.0

20.0

0 5 10 15 20

Dam

age

Acc

umul

atio

n/cy

cle

-D (

UVA

R7)

Head Wear (mm)

x10-5


144

An increase in the rail head wear from 14 mm to 22 mm results in increased damage

accumulation. As a result, fatigue failure could occur at 1.2x104 stress cycles when the

rail head wear reaches 22 mm. Figure 6.13 shows the predicted number of cycles to

failure for the range of loading conditions considered. For an L/V ratio of 0.2 at a

contact patch offset of 20 mm, at 22 mm of head wear, the predicted fatigue life is

approximately 1.2x104 cycles, increasing to 3.42 x105 cycles at 14 mm of head wear. At

an L/V ratio of 0 and contact patch offset of 15 mm, the predicted fatigue life is 3.5x105

at a head wear of 22 mm, which is identical to that of a head wear of 14 mm at an L/V

ratio of 0.2 and a contact patch offset of 20 mm.

Figure 6.13 Prediction of fatigue life under the extreme loading conditions

The simulation results did not consider the effects of pre-existing defects or other

conditions, which may increase localised stresses in the vicinity of the underhead

radius. However, the results also indicated the potential formation of defects such as

reverse detail fractures, which have previously been found to develop at the lower gauge

100

150

200

250

300

350

400

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

DV

Dam

age

Para

met

er -

UVA

R 1

(MPa

)

Number of Cycles to Failure - UVAR8 (log(N))

Wohler curve used in the current analysisHW 22 mm, L/V=0.2, Patch offset=20 mmHW 22 mm, L/V=0.4, Patch offset=15 mmHW 22 mm, L/V=0, Patch offset=15 mmHW 14 mm, L/V=0.2, Patch offset=20 mmHW 14 mm, L/V=0.4, Patch offset=15 mm

Fatigue Region


145

corner on heavily worn rail. Excessive wear combined with a pre-existing transverse

defect may pose a direct threat to rail integrity. The treatment of residual stress aspects

needs to consider the fact that they may differ from those used for the previous

measurements and may not be representative of current rail manufacturing procedures.

Although the analytical results indicate under what combination of rail grade, rail head

wear and loading conditions fatigue damage is predicted to initiate, in practice fatigue

failure is generally associated with the presence of pre-existing defects or a stress

concentrator in the form of a sharp radius which may include flow lips (as was evident

in the reverse detail fracture mode) or rolling contact fatigue cracking. Material defects

are not included but different rail material grades are studied. In the case of defects, this

provided an overview assessment and is deemed sufficient.

6.7 Summary

The fatigue damage prediction analysis for the rail underhead radius was undertaken to

investigate fatigue damage behaviour as a function of the grade of rail material and

increasing head wear under heavy haul operations. Fatigue damage was predicted using

the Dang Van criterion. The analyses revealed that under the more severe loading

conditions, fatigue damage can be predicted to develop at the rail underhead region with

increasing amounts of head wear for the HH rail grade. For rail grades with high fatigue

limits, fatigue damage rarely occurred at the underhead region under the range of

loading conditions considered with an increase in head wear. Lateral offsets in the

wheel contact patch relative to the rail centreline, in combination with increased lateral

forces associated with vehicle curving, increase the probability of fatigue damage

initiation at the underhead radius. This is due to an increase in the tensile stresses at the

underhead region, which is induced by the local bending behaviour on the rail web. In

the presence of heavily worn rail profiles, the fatigue life of rail in sharp curves is less

than in tangent track. In general, the more wear that occurs on the surface of rail, the

more fatigue damage can be predicted to initiate at the rail underhead radius.


146

The approach presented in this chapter, if extended to include the residual stress aspects

outlined above and in the presence of defects, provides a basis for an assessment of

allowable rail wear limits as a function of loading conditions and the rail material grade.

Further work is required to examine these aspects of rail fatigue damage behaviour.

Chapter 7. Rapid fracture modelling using X-FEM

147

Chapter 7

Rapid fracture modelling using X-FEM

7.1 Introduction

The results presented in chapters 4 and 5 have shown that the occurrence of tension

spikes as a result of lateral head bending and localised vertical and lateral bending of the

head-on-web are significant to the understanding of the unstable propagation of pre-

existing RCF cracks. This chapter investigates the growth behaviour of long RCF

cracks, in particular, the tendency of rail to break due to rapid fracture, under the high

axle load conditions typical of those that exist in Australian heavy haul operations. This

work is also very interesting from the perspective of focusing on the impact of tension

spikes on the unstable growth behaviour of gauge corner cracking (GCC). This

behaviour is exacerbated with increasing rail head wear (HW), and the propensity of

rapid fracture associated with this behaviour correlates with the extent of rail head wear.

An investigation of fatigue crack initiation in the rail head and the fatigue crack

propagation in the web of the weld zone of a rail was conducted by Josefson et al [149],

using multi-axial fatigue analyses incorporating Dang Van theory. The effect of global

rail bending stresses was incorporated in this model by Ringberg et al [150]. A fatigue

crack growth model developed by Skyttebol et al [151] was used to examine the crack

growth behaviour of a welded rail with respect to the influence of residual stresses.

These investigations focused on the crack growth in phases 2 and 3 of ―Whole Life Rail

Model‖. It was observed that a weld is more sensitive, or less fatigue resistant, to

surface cracks than to embedded cracks. As expected, the reason found was that stress

ranges in the location of surface cracks were larger than for embedded cracks. Nielsen


148

et al [152] also found the same results in their work and suggested that surface defects

should be avoided. The numerical investigations demonstrated that crack growth rate is

influenced by impact loads and by seasonally-dependent thermal stresses. It was found

that these factors also have a great effect on the fatigue life of a rail.

The risk of rail breaks was investigated by Sandström et al [111] from a mechanical and

statistical point of view. In particular, the influence of impact loads from flatted wheels

was considered. Wheel flats were not considered to contribute to an increase in rail

bending even though they increased the contact forces. Hence, the influence on fatigue

crack growth was minor. However, bending stress was a major concern regarding any

final fracture. Kapoor et al [18, 23] and Dutton et al [21] developed ―Whole Life Rail

Model‖ (WLRM), during their investigation of the Hatfield accident, as was described

in detail in the introductory chapters. A phase 3 crack growth dominated by bending

stresses has a significantly higher crack growth rate. A rapid fracture resulting from

transverse defects (TDs) such as shown in Figure 1.3, may occur if a RCF crack turns

down, additionally driven by the tensile bending stress.

This chapter examines rapid fracture from a large crack at the gauge corner of rail. The

influence of tensile bending stresses at the underhead radius (UHR), which are

associated with contact loads along with a combination of the lateral bending of the

whole rail profile and local bending of the head-on-web are included. This effect is of

greater importance under high axle load conditions and for increased head wear, as was

demonstrated in chapter 5. Furthermore bending of the rail head-on-web due to the

eccentricity of contact load was not considered in the crack growth investigations

conducted by Fletcher et al [27, 119] and Dang Van et al [28]. Jeong et al [6] ,

however, did include this behaviour in an examination of the growth behaviour of

reverse detail fractures that were initiated at the lower gauge corner of heavily-worn

rail. Relatively little work has been done on long cracks that approach the critical crack

length for rapid fracture, as reported by Kapoor et al [23]. In the investigation

conducted for this thesis, a long crack was introduced at the gauge corner. The unstable

growth behaviour of the crack under various situations of contact load applied to the rail


149

head at the varied eccentric locations was evaluated with respect to different rail worn

profiles.

7.2 Crack model development

A detailed literature survey revealed possible methods for an analysis of stress intensity

factors that included but was not limited to finite element modelling (FEM) [114, 150,

153-156], boundary element analysis (BEM) [27, 157-159], Green‘s functions [24, 160,

161], dislocation based studies [162-164] and body force method [165-167]. Finite

element analyses include both special purpose and general purpose crack analysis

packages. Special purpose packages include ZENCRACK (FEM), BEASY (BEM)

[168] and FRANC2D, 3D (FEM) and general purpose packages include ANSYS and

ABAQUS [135] by using manipulation techniques to characterize the stress intensity

factors. These numerical approaches are crack opening displacement (COD) [169], J-

integral [170], a virtual crack closure technique (VCCT) [171] and a force method

[172].

Extended finite element method (X-FEM) 7.2.1

In addition to all these methods, the extended finite element method (XFEM) was

developed in 1999 by Belytschko et al [173]. The X-FEM provides significant

advantages over other approaches, such as the boundary element methods [168], re-

meshing methods (Carter et al [174]; Maligno et al [175]), and element deletion

methods (Henshell et al [176]), in addressing the computational burden associated with

the insertion of arbitrary cracks. Compared to the traditional FE approach, the crack

could be enhanced by additional functions that captured the singularities at the crack

front, and allowed a more effective and accurate calculation of the characteristic values.

The X-FEM elements for the crack region provided the crack enrichment, which

represents the discontinuities in the displacement field along the crack. These singular

functions allow the asymptotic displacement at the crack front to be taken into account

http://en.wikipedia.org/wiki/Ted_Belytschko


150

for the modelling of cracks whose geometry is independent of the original finite element

region [177].

Heaviside function H(x) is used to represent a displacement jump across the crack face

and is given by the following expression:

( ) { ( )

} (7.1)

Where

is a sample (Gauss) point,

is the point on the crack closest to , and

n is the unit outward normal to the crack at , as shown in Figure 7.1 [135].

Figure 7.1 Illustration of normal and tangential coordinates for a smooth crack [135]

Discontinuous geometry using the level set method 7.2.2

In addition, the displacement field approximation was divided into continuous and

discontinuous parts. The Level Set Method (LSM) was used to locate the discontinuity

by defining the crack geometry by two orthogonal signed distance functions, as given

below:


151

υ describes the crack surface and

ψ is used to construct an orthogonal surface, as shown in Figure 7.2.

The intersection of the two surfaces gives the crack front.

n+ indicates the positive normal to the crack surface, m+ indicates the positive normal

to the crack front, as shown in Figure 7.2 [135].

Figure 7.2 Representation of a non-planar crack in three dimensions by two signed distance functions υ and ψ [135]

X-FEM crack model 7.2.3

The rapid fracture behaviour was examined using several new models by inserting X-

FEM elements into the existing finite element (FE) meshes, which were previously

validated by comparison with in-track measurements, as given in chapters 3 and 5. For

this thesis, extended finite element method (X-FEM) elements [135] were applied to

model the crack. The X-FEM elements were applied to the crack region, which was

directly located above the underhead radius (UHR) reference location, as shown in

Figure 7.3a-d; the remainder of the model used FEM elements. This newly developed

numerical model utilized the commercial code ABAQUS 6.11-2 to examine the


152

unstable crack growth behaviour. The ABAQUS 6.11-2 X-FEM models solutions were

conducted on 2.99 GHz CPU.

The extended finite element method X-FEM in ABAQUS is a mesh independent

method which is based on the enriched elements and its capabilities were extensively

verified in the literature [177]. The critical crack sizes were considered with dense mesh

as shown in Figure 7.3c-d. The validation of crack modelling using track data and the

comparison of the crack sizes using the operational fractures was not possible in the

absence of field data.

The X-FEM model used a single 68 kg/m rail with 6 m length laid on discrete elastic

foundations to parametrically study the unstable growth behaviour of RCF cracks

subjected to change of the head wear (HW), the contact patch offset (CPO) from the rail

centreline, and the (L/V) ratio of lateral (L) to vertical (V) loads.

The X-FEM model considered a single crack, which was in the transverse orientation

with an approximate shape based on the typical transversally deviated head checks that

occur under heavy haul conditions (Figure 1.2). The crack face friction is not included

in the current model. The crack shape is described by the semicircular crack radius R,

and the crack surface length 2R (Figure. 7.3a). The crack has a length of R with an

angle to the top surface of rail as in Figure 7.3b and is centered at an offset of 27 mm

from the centerline of the rail head cross-section, as shown in Figure 7.3c. The change

in angle θ was also investigated using the simulations described in the next paragraph.

A single wheel load was simulated, applying a fully slipping Hertzian contact patch on

the wheel-rail contact surface. The contact loads included lateral and vertical loads (L/V

ratio of lateral to vertical load), a contact patch offset (CPO) from the rail centerline

(Figure 7.3c) and contact patch movement from the crack location along the rail

longitudinal running direction (Figure 7.4). A discrete elastic foundation directly at the

location of the sleepers was used to support the rail. The details of the model parameters

were given in Table 7.1. The finite element modeling details have already been

discussed in chapters 3 and 5. It should also be noted that the details on input


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parameters such as loading, boundary conditions and support characteristics will remain

the same as was discussed in chapter 3.

Figure 7.3 A single crack at the gauge corner: (a) crack shape (crack length = R and surface length = 2R); (b) crack orientation ( = 70o); (c) fine mesh at highlighted enriched region including crack and contact patch positions; (d) Contact patch offset (CPO), L/V ratio and position of the crack from the centre of rail head cross-section

(a)

(c) (d)

(b)

Contact patch positions

Crack

Contact patch

Crack front

Crack


154

Figure 7.4 Loading steps at different positions relative to the crack

The simulations incorporate several steps to enable the examination of unstable crack

growth behaviour for different loading positions relative to the position of the crack

(Figure 7.4). The passing wheel was simulated by applying the contact loading at step 1,

then at step 2, then step 3, and so on. The stress intensity factor (SIF) in the plane

opening Mode I (KI) was calculated along the crack front for each of the locations. For

each assessment, the size of the crack and the rail head wear (HW) were defined as two

variables in which the crack length, R was in the range of 5 mm to 30 mm, and HW was

in the range of 10 mm to 22 mm, respectively, and the crack orientation ( θ ) was 60o,

70o and 90o. The results from the parametric analyses were used to evaluate unstable

crack propagation in the rail.


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Table 7.1 Crack model parameter values

Model Parameters Value Rail span, S (mm) 6000 Axle load, V (kN) 343.4 L/V ratio 0, 0.2 Worn rail profile / head wear, HW (mm) 10, 15, 22 Crack radius, R 22, 17, 10 Crack orientation / inclination, θ (o) 60o, 70o and 90o Contact patch offset, CPO (mm) 0,15, 20 Semi major axis, a (mm) 10 Semi minor axis, b (mm) 4 Young‘s modulus, E (MPa) 209,000 Poisson‘s ratio, 0.3

7.3 Variations of KI max during loading cycle

The position of the crack front was normalized by the total length of the crack front.

The crack front was positioned from 0 at the underhead radius (UHR) of the lower

gauge corner point to 1 at the point near the top surface of the rail (Figure 7.3a). The

stress intensity factors (SIF) (KI, KII and KIII) at the crack tip were used to examine the

rapid fracture behaviour in response to the stress distribution resulting from the

combination of contact loads and bending stresses. Three commonly used rail grades: a

plain C-Mn Head Hardened (HH) grade [80], a Low Alloy Heat Treated (LAHT) grade

[127], and a Hypereutectoid (HE) heat treated grade [81], were used to examine the

effect of material properties on the rapid fracture of the crack.

Effect of contact patch offset (CPO) and L/V ratio 7.3.1

Two representative conditions can be used to simulate the effect of vehicle curving and

hunting: the contact patch offset and L/V ratio. The contact patch lateral displacement

was considered in this study for two reasons. Firstly, the vehicle tends to exhibit lateral


156

movement due to steering and hunting behaviour and, secondly, the contact patch offset

may change because of a change in the position of running band due to a reduction in

the cross-section area because of natural wear or after profile grinding. In order to study

the effect of contact patch location, an L/V ratio of 0 was taken to minimise the effect of

the lateral load (L). Figure 7.5 shows the maximum (over the entire crack front) Mode I

stress intensity factor (KI max) under contact loads of L/V of 0, 0.2 when applied to

different loading positions relative to the position of crack length R = 17 mm, = 90o

and HW 15 mm.

The fracture region is defined by KI max > KIC. The three fracture toughness (KIC) values

mentioned for a plain C-Mn Head Hardened grade, HH - KIC = 45 MPam [80], a Low

Alloy Heat Treated grade, LAHT3 - KIC = 39.1 MPam [127], and a Hypereutectoid

heat treated grade, HE - KIC = 35.4 MPam [81] rail steels are also illustrated in Figure

7.5 by the three dashed lines, and are used to examine the effect of material properties

on rapid fracture of the crack. When the contact load is directly imposed above the long

crack (R = 17 mm), for CPO =15 mm, L/V = 0, KI max value is small, below the fracture

region defined by KI max > KIC , as shown in Figure 7.5. This indicates that rail failure is

unlikely under these loading and rail head wear conditions.

The results shown in Figure 7.5 reveal that rapid fracture could happen when the contact

load is directly imposed above the long crack (R = 17 mm) for CPO = 20 mm, L/V =

0.2, as the KI max > KIC (KIC = 35.4 MPam for HE rails, KIC = 39.1 MPam for LAHT3

rails, and KIC = 45 MPam for HH rails). This can result in the rapid fracture of rail

under these loading and head wear conditions. The crack is driven by local tensile

bending stress having a spike at the underhead radius. The loads applied near to the rail

gauge corner (CPO = 20 mm) provide higher tensile bending stress than the loading

case of L/V = 0 at a contact patch offset (CPO) of 15 mm. As a result, KI max increases

up to 71 MPam, being almost three times higher than the case with CPO 0 mm. When

the loads are positioned away from the crack, KI max reduces significantly, as the local

bending stresses are absent at the crack front and the global bending stresses are small,

as shown in Figure 7.5.


157

Figure 7.5 The maximum KI (KI max) at the crack front (R = 17 mm, = 90o) under the different loading positions

Effect of crack length on SIFs 7.3.2

Under the same conditions (CPO = 20 mm, L/V = 0.2, = 90o and HW = 15 mm), the

crack length, R, is a key factor influencing stress intensity factors at the crack front, as

KI max value is dependent on the crack length R. Figure 7.6 shows that the crack with a

length R = 17 mm for HW of 15 mm, produces the highest value of KI max compared

with a crack length R = 5 mm and 30 mm. This is because the crack front for R = 17

mm crack is located at the underhead radius where the tensile stress is much higher,

whereas the relatively smaller crack (R = 5 mm) is away from this highly stressed

region. Under the extreme loading conditions (L/V = 0.2 and CPO = 20 mm), the crack

may not extend to 30 mm, as the probability is high for a rapid fracture at R = 17 mm.

Therefore, a long crack of 30 mm may be inappropriate for use as it is away from the

underhead radius and the possibility of rapid failure occurs only at the 17 mm long

-25

-15

-5

5

15

25

35

45

55

65

75

-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500

KIM

ax (M

Pa√

m)

Wheel position relative to the position of crack (mm)

CPO = 15 mm, L/V = 0CPO = 20 mm, L/V = 0.2

HW 15 mm

HH - KICLAHT3 - KICHE3 - KIC

Fracture Region


158

crack. Another possibility exists in that if the contact load is reduced, the value of stress

intensity factors would be consequently lower. It can be seen from Figure 7.5 for the

loading case of L/V =0 at CPO =15 mm, the 17 mm long crack has a low value of KI max,

which is less than KIC. The crack will not grow by rapid fracture and it will pass through

the underhead radius region by gentle fatigue crack growth.

Figure 7.6 The maximum KI (KI max) at the crack front: as a function of crack length

R (5, 17, 30) under the loads of L/V =0.2 at CPO = 20 mm

Effect of residual stresses on SIFs 7.3.3

The residual stresses formed as a result of the rail manufacturing process and repeated

rolling contact between wheel and rail could influence the crack propagation behaviour.

-10

0

10

20

30

40

50

60

70

-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500

K Im

ax(M

Pa√m

)

Loading position relative to the position of the crack (mm)

R = 5 mmR =17 mmR= 30 mm

HW 15 mm

LAHT3- KIC

HH - KIC

HW 15 mm

LAHT3- KIC

HH - KIC

HE3 - KIC

Fracture Region

(a)


159

Figure 7.7 The maximum KI (KI max) at the crack front ( = 90o) under the loads of

L/V = 0.2 at CPO = 20 mm (a) no RS; (b) with RS

(b)

17

Fracture Region

17

Fracture Region


160

Although the residual stresses will redistribute the stress state of the rail as a result of

head wear (HW) and during crack propagation modelling, the influence of these varying

residual stresses on the final fracture becomes cumbersome [178]. Thus, in this analysis

they were assumed not to change. In this simplified analysis, crack sizes with different

R (radii) values are considered, and crack propagation was not modeled as the crack

front was subjected to tensile stresses influenced by the local response of head-on-web

bending to analyse the rapid fracture. To study the effect of residual stresses,

simulations were conducted by incorporating the pre-existing stress distribution

associated with residual stress measurements conducted by Majera [136] (Table 3.3).

The value of KI max is influenced by the residual stress levels in rail. Figure 7.7 shows

that the KI max value incorporating the residual stress distribution is approximately 33%

higher than that without residual stresses, for a crack length R = 17 mm. The results

suggest that inclusion of residual stresses is expediting the rapid fracture behaviour of

rail leading to rail failure.

7.4 Stress intensity factors (SIFs) on the crack front

Effect of crack orientation on SIFs 7.4.1

The change of crack orientation from 60o to 70o and 90o was studied to evaluate the

stress intensity factor changes at the crack front. It was found that there is a greater

reduction in KI at the crack front along the distance from 0 to 0.2 at the orientation of

= 60o than at the orientation of = 70o and 90o, as shown in Figure 7.8 (a) and (c). The

negative KI that occurred at the rest of the crack front position was due to compressive

stress, and was treated as zero.

When KI max at the crack front is equal or greater than the fracture toughness of the

material, KIC, rapid fracture is likely to occur leading to the rail break. For example, KI

max > KIC for all three rail material grades (KIC = 35.4 MPam for HE3 rails, KIC = 39.1

MPam for LAHT3 rails, and KIC = 45 MPam for HH rails). As shown in Figure 7.8, a

possible rapid fracture could occur if the crack is at an angle greater than 70o and is

positioned in the underhead radius region under specific loading conditions (lateral


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Figure 7.8 The SIFs at the crack front (R = 10 mm) under the local bending with L/V =0.2 at CPO = 20 mm for rail HW = 22 mm.

-100

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0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

SIF

(MPa

√m)

Normalised crack front position

KKK

HH - KICLAHT3- KIC HE3 - KIC

III

III

-60

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0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

SIF

(MPa

√m)


KKK


I

IIIII

-60

-40

-20

0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

SIF

(MPa

√m)


KKK


IIIII

I

Fracture Region

Fracture Region

Fracture Region

(a) θ = 60o

(c) θ = 90o

(b) θ = 70o


162

offset on the heavily worn rail profile in combination with increased inward lateral

forces). At the underhead radius, KI max is more pronounced for the crack orientations

normal to the top of the rail surface, and the rapid fracture may be driven in Mode I.

Effect of head wear on SIFs 7.4.2

SIFs (KI, KII and KIII) around the crack front under the different cases of crack length,

crack orientation, wheel-rail contact, and head wear conditions were studied to

investigate a possible rapid fracture due to the effect of local bending stresses. The

critical crack length for these occurrences could therefore be estimated. As head wear

changed the rail profile, the crack length R was reduced in order to keep the crack in the

critical underhead radius region (as shown in Figure 7.3c). There were three crack

lengths with respect to head wear ((a) R = 22 mm for HW = 10 mm; (b) R = 17 mm for

HW = 15 mm; (c) R = 10 mm for HW = 22 mm) that were considered in the analyses.

It can be seen in Figure 7.9 that less wear (HW =10 mm) lowers the SIFs, particularly

for KIII. At the UHR region, in the normalized crack front position of 0 to 0.2, KI and KII

are positive and higher than the rest of the positions, but KI is less than KIC individually

provided for the HH, HE3 and LAHT3 rail grades. Figure 7.9c shows the SIFs for a

crack with heavily worn rail profile (HW = 22 mm). KI obviously exceeds the fracture

toughness of the HH, HE3 and LAHT3 grades from the normalized crack front position

of 0 to 0.3 at the UHR region. Hence, rapid (unstable) crack propagation may develop

for these combinations of crack length and head wear (i.e. R=17 mm for HW 15 mm;

and R=10 mm for HW 22 mm see Figure 7.9 b-c).

The maximum of the crack opening corresponds to the maximum (over the entire crack

front) Mode I stress intensity factor KI max of the tensile local bending stresses. A

possible rapid fracture could occur when the cracks are positioned at the underhead

radius, having an angle of more than 70o, under a loading with an offset on the heavily

worn rail profile. It can also be seen that the less worn rail will resist the fatigue crack

propagation. Once KI max exceeds the fracture toughness of the rail material, the

unstable crack growth will lead to rapid fracture.


163

Figure 7.9 The SIFs at the crack front ( = 90o) under the local bending with L/V = 0.2 at CPO = 20 mm

-60

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0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

SIF

(MPa

√m)


KKK


I

IIIII

-60

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0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

SIF

(MPa

√m)


KKK


I

IIIII

-60

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0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

SIF

(MPa

√m)


KKK


I

IIIII

Fracture Region

Fracture Region

Fracture Region

(a) HW = 10 mm, R = 22 mm

(b) HW = 15 mm, R = 17 mm

(c) HW = 22 mm, R = 10 mm


164

It is possible to consider more advanced approaches by combining KI, KII and KIII in Keq

to determine unstable crack propagation. However, for the predicted results, KII and KIII

are much smaller compared to KI (see Figure. 7.9). With this, the simplified approach

that used only KI was sufficiently accurate. The more advanced approaches for rapid

fracture analysis using mixed-mode fracture criterion are discussed in chapter 8.

7.5 Summary

This chapter described the rapid fracture behaviour of long transversal cracks, especially

in phase 3 of the “Whole Life Rail Model”. The X-FEM model was used to investigate

the rapid fracture behavior of the RCF cracks in rail under heavy haul operations in

terms of stress intensity factors (KI, KII and KIII). The stress state near the crack tip was

obtained from the contact loads, the loading positions on the rail head surface. A single

crack, proposed in different lengths and orientations with the HW conditions, was

investigated and was subjected to this stress state.

The results revealed that the highest KI max occurs when the contact loading is directly

above the crack location. The effect of KI max is also influenced by the contact patch

offset, a case in which high tensile bending stresses are produced at the underhead

radius due to the lateral head bending and localised bending of the rail head-on-web.

The X-FEM results revealed that existing rolling contact fatigue (RCF) cracks, when

subjected to high tensile stresses at the gauge corner region, can contribute to the

development of rapid (unstable) fracture if the high tensile stresses at the underhead

radius are produced by any extreme contact loading conditions. The results of this work

can be used to examine the influence of wheel-rail interaction behaviour and rail head

wear on the probability of catastrophic rail failure from RCF damage.

Chapter 8. RCF cracks under mixed-mode loading

165

Chapter 8

RCF cracks under mixed-mode loading

8.1 Introduction

Since a stress state is multi-axial with out of phase stress components, the cyclic wheel-

rail contact, bending, residual and seasonally-dependent thermal stresses interact with

geometric features to produce a mixed-mode non-proportional stress history. The

individual stress intensity factors such as KI, considered previously in chapter 7, are

insufficient to predict the fracture behaviour of cracks in response to global and local

stress states. An equivalent stress intensity factor (Keq) should be used to combine Mode

I, Mode II and Mode III behaviour in order to predict the fracture behaviour of RCF

cracks, a process described in this chapter. Additionally, the effect of multiple cracks as

observed in real life is included here and is shown in Figure 8.1.

The FE mesh developed in chapter 7 is extended here to include multiple RCF cracks.

RCF cracks are typically found in clusters, and have been a focus of this thesis. The

work described in chapters 4 and 5 revealed that tension spikes occur at the rail

underhead radius (UHR), generated by the local bending of the rail head-on-web. The

magnitude of the spike was found to be affected by head wear. The high tensile stress at

the underhead radius could drive a crack to rapid fracture. These effects have also been

included studying the thesis.

This work investigated the unstable growth behaviour of typical long transversal cracks

caused by head checks or gauge corner cracking. The extended finite element method

(X-FEM) was used to model a single rail supported on a discrete elastic foundation. The


166

study focused on the parametrical influence of the rail head wear (HW), the contact

patch offset (CPO) from the rail centreline, the (L/V) ratio of lateral (L) to vertical (V)

loads and foundation stiffness. The FE results revealed that existing long cracks which

have already turned down (to Mode I behaviour), under high tensile stresses at the

underhead radius, can contribute to a rapid (unstable) fracture along the crack front of a

single crack and multiple cracks. This chapter examined the probability of rail failure

from existing gauge corner cracking and the differences in rapid fracture behaviour

between a single and multiple cracks.

Figure 8.1 Appearance of RCF damage [33]: (a) longitudinal section with gauge corner cracking; (b) Transverse defect

Mutton et al [1, 2, 33] reported on a detailed metallographic examination of rail samples

operated in heavy haul operations. A longitudinal and transverse section through the

(a)

(b)


167

damaged region (Figure 8.1(a) and (b)) of high rails revealed the crack growth

morphology associated with the development of transverse defects, with an origin from

surface initiated multiple gauge corner cracks. Duvel et al [13] also reported similar

crack growth morphology for other heavy haul systems.

A recent study conducted by Tillberg et al [180] investigated the multiple crack

interaction under cyclic loading on a rail surface with an assumption of plane strain

conditions. The investigations focused on a parametric study of the multiple crack

interactions associated with the effect of the crack spacing, the crack angle, the crack

geometry and the material properties. The results demonstrated that the spacing between

adjacent cracks is an important factor for crack shielding (reduction in the stress

intensity factor of the main crack) as observed by the strain energy release rate (J-

integral) that peaked before and after the wheel passed over the cracks, and the J-

integral that increased with increasing tangential traction.

Fletcher et al [27] modelled a series of identically sized, closely spaced, adjacent cracks

using a boundary element method on rails in bending. The results suggested that closely

spaced cracks can shield each other from remotely applied bending stress, which results

in a reduction in the maximum crack growth rate of adjacent multiple cracks present in

the rail. They also studied the influence of crack spacing for longer cracks to investigate

the stress intensity factor (KI, KII, KIII) at the crack tip. It was found that KI is a function

of crack spacing and that crack shielding is inversely proportional to crack spacing.

Similar results were predicted by Guo et al [181], who also considered thermal effects

due to frictional heating on the rail surface in order to study the effects on SIFs due to

spacing between two cracks.

Several researchers have undertaken studies of multiple cracks that are unrelated to

RCFs. Lam and Phua [182] used the surface integral method to determine the

distribution of the edge dislocations to represent multiple cracks. It was found that crack

shielding depends on the position and orientation of the multiple cracks. A similar crack

configuration for coplanar (i.e. aligned end to end) and parallel semi-circular multiple

cracks under contact fatigue loading was studied by Kuo et al [183]. They found that

coplanar cracks interact to increase the stress intensity factors, driving their growth, but


168

that parallel cracks have slower propagation than single cracks. Wang et al [184]

reported on the influence of the growth direction of multiple small cracks relative to

crack separation. The predicted results agreed with the field measurements, and showed

that the direction of crack growth was highly influenced by crack separation distances.

The reduction in stress intensity factors for a series of parallel multiple cracks relative to

that for a single crack was also reported by Rooke and Cartwright [185]. Several crack

width to separation ratios were considered for a series of parallel cracks in a sheet and

the maximum reduction in KI was found to be when the width to separation ratio was

0.7.

This chapter examines unstable crack growth for long cracks at the gauge corner of a

rail. The influence of tensile bending stresses at the underhead radius associated with

contact loads, along with a combination of the local response of the head bending on the

web, and the lateral bending on the head are included. Relatively little work has been

done on long RCF cracks that approach the critical crack length for rapid fracture [23].

Furthermore, bending of the rail head on the web due to the eccentricity of a contact

load was not considered in the crack growth investigations conducted by Fletcher et al

[27, 119] and Dang Van et al [28]. Jeong et al [6], however included this behaviour in

an examination of the growth behaviour of reverse detail fractures that were initiated at

the lower gauge corner of heavily-worn rail. Zerbst et al [124] conducted an

investigation into the damage tolerance of the rail. They investigated a typical squat

defect approximating a semi-elliptical shape of different sizes and an orientation of 70o

on the rail running surface with head wear of 10 mm. The stress intensity factors (KI,

KII, KIII) were calculated for contact, thermal and residual stresses. The variations in the

contact patch offset from the centre of the rail head cross-section due to the so called

Klingel movement with worn states of the rail and wheel were taken into account. The

out of phase stress intensity factor (KIII) was not considered in these investigations. This

effect was studied with high axle load conditions for this doctoral study, and single long

and multiple surface cracks were introduced at the gauge corner. An evaluation was

undertaken of unstable crack growth behaviour when the different situations of contact

load were applied to the rail head at various locations.


169

8.2 Multiple cracks model development

A numerical model was used to examine the unstable crack growth behaviour of long

cracks in rail. The model consisted of a 68 kg/m rail of 6 m length laid on a discrete

elastic foundation at the location of sleepers spaced at 600 mm, and it was executed

using the commercial finite element (FE) package ABAQUS 6.11-2 and extended finite

element method (X-FEM). Chapter 7 described several new models that were developed

by inserting X-FEM elements in the existing finite element (FE) meshes. These were

then extended to include multiple RCF cracks, and will be described in this chapter. The

details of X-FEM have already been discussed in chapter 7. The FE model was

previously validated using a comparison with in-track measurements to verify the

occurrence of tension spikes at the underhead radius, as was demonstrated in chapter 3.

Cracks in rail start at a shallow angle (almost parallel to the rail surface) and as they

grow, they become steeper and ultimately inclined at an angle of 60o - 70o [90]. Crack

shape near the mouth is not expected to influence the stress intensity at the crack tip too

much, and so these simulations have assumed straight line crack shape as to keep the

model simple, as shown in Figure 8.2 a-b. Zerbst et al also made such an approximation

[124]. The crack shape was described by the semi-circular radius R, and the crack

surface length 2R (Figure 8.2a). The crack length ‗R‘ with an angle = 90o to the top

surface of the rail (Figure 8.2a), was centred at an offset of 27 mm at the gauge corner

from the centreline of the rail head cross-section, as was shown in Figure 7.3c. The

multiple cracks introduced at the gauge corner of the model in the transverse orientation

(see Figure 8.2 (a)-(c)) had the approximate shape of transversally deviated typical head

checks that occur under heavy haul conditions (Figure 8.1a and b). The passage of a

wheel was simulated through the longitudinal movement of the associated contact patch

and pressure on the rail contact surface, assuming a fully slipping Hertzian contact

pressure.

The contact loads included lateral (L) and vertical (V) loads (L/V ratio of lateral to

vertical load), see Figure 7.3c, a contact patch offset (CPO) from the rail centreline, and

different longitudinal loading positions relative to the crack, (see Figure 8.2a.).


170

Figure 8.2 The model description for multiple RCF cracks at the gauge corner

(a)

(CPO)

(b) 3 cracks with HW 15 mm

(d) Contact load passes a crack (side view)

(c) 7 cracks with HW 22 mm

(a) Crack shape (crack length = R and surface length = 2R) with loading steps at different positions relative to the multiple RCF cracks


171

The wheel passing behaviour was simulated by applying the contact loading at different

locations with respect to the analysis steps i, i+1, i+2 and so on, as shown in Figure

8.2a. When the contact pressure is applied on patch i+1, the contact patch ‗i‘ is

unloaded.

As the rail profile changed due to head wear (HW), the crack length R was reduced in

order to keep the crack in the critical underhead radius (UHR) region, as shown in

Figure 8.2 (a)-(c). The size of the crack and the rail head wear (HW) were varied with

the crack length, R, in a range of 5 mm to 30 mm and the HW in a range of 10 mm to 22

mm, respectively. There were three critical crack lengths selected: (a) R = 10 mm for

HW 22 mm; (b) R = 17 mm for HW 15 mm; (c) R = 22 mm for HW 10 mm and

considered in the analyses given in Table 8.1. Crack orientations of 60o, 70o and 90o

were considered and these results were presented in chapter 7. The single crack run

number (A1-A4) and the introduced adjacent multiple cracks M1 (3 cracks), to M2 (7

cracks) were modelled in the range of HW (10, 15, 22 mm), loading (CPO = 0, 15, 20

mm, L/V = 0, 0.2), crack sizes (10, 17, 22 mm), crack orientation ( = 90o) and crack

separation (5mm only), as given in Table 8.1. Interaction properties were also defined

with a friction coefficient of 0.1 along the crack surfaces to simulate some fluid in the

crack. Some simulations with a crack surface friction coefficient of 0 were conducted

and results were almost identical. The results were used for the evaluation of unstable

fatigue crack propagation of long cracks in the rail.

The residual stresses resulting from the rail manufacturing process and repeated rolling

contact between wheel and rail could influence crack propagation behaviour. Although

the residual stresses will redistribute the stress state in the rail as a result of head wear

(HW), and during crack propagation, the influence of residual stresses on the final

fracture becomes cumbersome [178]. Thus, the residual stresses were not considered.

Likewise, thermally induced stresses, which are believed to influence crack

propagation, were not considered for this thesis. These influences will be studied at a

later time.


172

Table 8.1 Crack model configurations simulated

Run cracks Rail profile Loading conditions Crack size orientation separation

No. No. HW (mm) L/V ratio CPO

(mm) R

(mm)

(o) s

(mm)

A1 1 10 0.2 20 22 90 - A2 1 15 0 15 17 90 - A3 1 15 0.2 20 17 90 - A4 1 22 0.2 20 10 90

M1 3 22 0.2 20 10 90 5

M2 7 22 0.2 20 10 90 5

The model used a wheel load of 172 kN (Table 7.1), which is equivalent to an axle load

of 35 tonnes. All the other influencing parameters such as boundary conditions and

support characteristics values are same as was given in Chapter 3. It focused on the

effect of HW conditions; contact patch offset (CPO) and L/V representative of curved

track. The contact patch offset (CPO) and L/V ratio are two representative conditions

that were used to simulate the effect of vehicle curving and hunting. Further, the CPO

changes due to the offset of running bands as the rail HW profile changes because of a

reduction in the cross-sectional area, either due to natural wear or after profile grinding.

Under these conditions the stress range in the underhead radius is higher for the rail of a

curved track due to the lateral transverse bending and torsion of the head-on-web.

Additional sets of finite element analyses referring to contact patch offset

(representative of lateral wheel positions), L/V ratios (representative of rail curvature)

and head wear (HW) conditions were carried out as described in the next section. The

crack growth behaviour, especially the probability of rapid (unstable) crack growth

behaviour, was examined for three rail grades commonly used in heavy haul

applications: a plain C-Mn Head Hardened (HH) grade [80], a Low Alloy Heat Treated

(LAHT) grade [127], and Hypereutectoid (HE) Heat Treated grade [81].


173

Mixed-mode fracture criteria 8.2.1

The literature suggests a number of fracture criteria to use when investigating how a

pre-existing mixed-mode crack becomes unstable. For example, those introduced by

Erdogan and Sih [186], Schollmann et al [187], Hussain et al [188] and Richard et al

[189-191] are well-known criteria for analysing a fracture. Richard‘s criterion has

shown to be numerically accurate and also simple and practical to apply. It is based on

approximate formulae and has proven to be in good agreement with experiments and

other accurate criteria such as the Schollmann maximum principal stress criterion [187,

191, 192].

Figure 8.3 Development of fracture surface for Mode I, Mode II, Mode III and mixed-mode-loading of cracks [189, 190]

The stress intensity factors (SIFs) in Mode I, II and III (KI, KII and KIII respectively)

were calculated along the crack front for each of the contact loading locations, as shown

in Figure 8.2a. The equivalent mixed-mode SIF (Keq ) (Equation 8.1) due to the

variations of the individual stress intensity factors can be found from Richard et al [189-

191], as shown in Figure 8.3.


174

√

( ) ( ) (8.1)

According to the criterion proposed by Richard et al [189, 190], the unstable fracture

occurs if is equal to the fracture toughness KIC of the material. i.e.

= KIC (8.2)

8.3 Variations of SIFs (K I, K II, K III) during loading cycle

A single long crack formed at the gauge corner of the rail was analysed first. The crack

was in the transverse orientation of angle with a semi-circular shape approximating a

typical transversally deviated head check, such as the one shown in Figure 8.1b, with a

crack radius R, and a crack surface length of 2R (see Figure 8.2a). The position on the

crack front was normalized by the total length of the crack front from the position 0

corresponding to the underhead radius near the lower gauge corner point to the position

1 at a point of the top surface of the rail. The passage of the wheel was considered using

the longitudinal movement of the associated contact patch and pressure, as shown in

Figure 8.2a. The resultant SIFs (KI, KII , KIII) at the crack front were used to examine the

unstable growth behaviour of this crack in response to the local bending stress

distribution resulting from the mixed-mode loading conditions of the contact and

bending stresses.

Figures 8.4 (a)-(d) show the variation of stress intensity factors (maximum over the

entire crack front) at each longitudinal position of the wheel over the rail running

surface. Figures 8.4(a) and (c) show this variation for a contact patch offset of 0,

whereas Figures 8.4 (b) and (d) are for a value of 15 mm. The maximum value was

observed when the wheel passed with a CPO of 15 mm (Figure 8.4b). Figure 8.4b

shows the maximum stress intensity factors in the tensile opening (KI), and the in-plane

sliding (KII) modes are much higher than the out-of-plane tearing (KIII) mode. The

maximum stress intensity factors in the two in-plane modes correspond to the position

of the wheel close to the top of the crack, as shown by points A and B in Figure 8.4b.


175

The plots in Figures 8.4 c and d show these small changes in values in detail. The

maximum in-plane opening mode KI max (point A in Figure 8.4 d) corresponds to the

position when the wheel is over the crack, and is the result of tensile stresses due to the

lateral bending of the whole rail profile and the localised vertical and lateral bending of

the head-on-web, as shown in Figure 8.4b and d. When the loads are positioned away

from the crack in the longitudinal direction, the KI max reduces as the local bending

stresses are absent at the crack front and the global bending stress is small.

(a) CPO 0 mm (b) CPO 15 mm

(c) CPO 0 mm (d) CPO 15 mm

Figure 8.4 Variation of maximum (over the entire crack front position) SIFs during one wheel passage (loading cycle) with different contact patch offsets (CPO), L/V = 0

For a wheel passing on the rail head without eccentricity (CPO = 0 mm), as shown in

Figures 8.4 (a) and (c), these changes in values are small, due to the absence of local

bending stresses. As with the contact patch offset CPO = 0, the head is in compression

-100

-80

-60

-40

-20

0

20

-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500

SIF

(MP

a√m

)

Wheel position f rom crack at the rail (mm)

KKK HW 22 mm, R 10 mm, θ = 90o

IIIIII

-40

-30

-20

-10

0

10

20

30

40

50

60

70

-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500

SIF

(MP

a√m

)

Wheel position f rom crack at the rail (mm)

KKK HW 22 mm, R 10 mm, θ = 90o

IIIIII

A

B

-90

-70

-50

-30

-10

10

30

50

-30 -20 -10 0 10 20 30

SIF

(MPa

√m)

Wheel position from crack at the rail (mm)

KKK

HW 22 mm, R 10 mm, θ = 90o

IIIII

I

-90

-70

-50

-30

-10

10

30

50

-30 -20 -10 0 10 20 30

SIF

(MPa

√m)

Wheel position from crack at the rail (mm)

KKK

HW 22 mm, R 10 mm, θ = 90o

IIIII

IA


176

due to global bending. KI is negative and shows compressive values, as depicted in

Figure 8.4a. The changes in values due to global bending stresses are lower compared to

local bending and are almost similar in both the loading cases, as shown in Figures 8.4

(a) and (b).

8.4 Variations of Keq along crack front during loading cycle

An equivalent SIF (Keq) is used to predict the fracture behaviour of RCF cracks by using

the Equation 8.1. The equivalent (Keq) SIF along the crack front changes, due to the

changes in the values of individual stress intensity factors when a wheel passes the

crack. The wheel position relative to the position of the crack is denoted by the index

X/a and is used in the subsequent investigation. X is the distance from the contact

pressure peak to the crack mouth and ‗a‘ is the semi-major axis of the contact patch, as

shown in Figure 8.2d and the value of ‗a‘ is stated in Table 7.1.

Effect of contact patch offset (CPO) 8.4.1

In order to study the effect of CPO, a constant L/V ratio of 0 was taken to minimise the

effect of the lateral load (L). For the first case to be described, Figure 8.5 shows the

SIFs at a crack tip when a wheel is passing on the rail head without eccentricity (CPO =

0 mm). No lateral traction (L/V = 0) is included. The crack (R = 10 mm, inclined at =

90o) to the rail running surface with rail HW of 22 mm was considered. The stress

intensity factors (KI , KII , KIII, Keq) are shown in Figure 8.5 at the normalised crack front

position during a wheel passage. Due to global bending at the contact patch offset CPO

= 0, for X/a = 0, the head is seen in compression. KI was negative and showed

compressive values, as depicted in Figure 8.5a and these negative KI values are treated

as zero for Keq calculations. Figure 8.5 d shows that Keq values are very small, far below

the fracture region defined by Keq KIC.


177


Figure 8.6 depicts the second case, when a wheel with the CPO 15 mm, L/V =0, passes

the crack (R = 17 mm, = 90o) on the rail surface with HW 15 mm. According to the

SIF values, a significant variation at normalised crack front positions is seen compared

to the case with CPO = 0 mm (Figure 8.5). Figure 8.6a depicts the SIF due to tensile

opening (KI) behaviour at the crack front. When the wheel is directly on top of the crack

mouth (X/a = 0), the maximum value of SIF in Mode I (KI max) is at the 0 crack front

position corresponding to the underhead radius near the lower gauge corner point. Other

wheel positions show a lower value of KI as compared to X/a = 0. The Mode II stress

intensity factor (KII) is plotted in Figure 8.6b. When the wheel approaches the crack, the

maximum value of SIF in sliding (KII max) is positive at the 0 crack front position (X/a

=1.5) and starts decreasing and becomes almost 0 when the wheel is over the crack (X/a

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Keq

(MP

a√m

)


X/a = 1.5

X/a = 0.6

X/a = 0

X/a = -1

X/a = -1.7

HH - KIC

LAHT - KIC

HE3 - KIC

-5

0

5

10

15

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

(MP

a√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-100

-80

-60

-40

-20

0

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KI(M

Pa√

m)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-10

-5

0

5

10

15

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

I(M

Pa√

m)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

Fracture Region

(b)

(c)

(a)

(d)


178

=0). When the wheel passes the crack the KII increases, but with a negative sign, as

depicted in Figure 8.6b.

Figure 8.6c shows the values of the SIFs in the out of plane tearing mode (KIII), which

are much lower compared to the opening (KI) and the sliding (KII) modes, suggesting

that the in-plane opening and shear forces play an important role in the unstable

propagation of the crack at the underhead radius.


Figure 8.6d suggests that the maximum Keq occurs at the loading step (X/a = 0), when

the loading is imposed directly above the long crack, as shown in Figures 8.2 (a) and

(d). Equation 8.1 shows that Keq is primarily dependent on the value of KI. Figures 8.6

(a) and (d) show that as the KI value rises, so does the Keq value. This is due to the

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

K eq

(MPa

√m)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

HH - KIC

LAHT3- KIC

HE3 - KIC

-40

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

(MPa

√m)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KI(M

Pa√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

I(M

Pa√m

)


X/a = 1.5X/a = 0.6X/a = 0 X/a = -1X/a = -1.7

(a)

Fracture Region

(c)

(b)

(d)


179

bending of the rail head-on-web forced by the eccentricity of loads from the rail

centreline. As a result of changes in the stress state and this tensile stress, KI increases

and so does Keq. Figure 8.6d shows that Keq values are below the fracture region defined

by Keq KIC, which indicates that the risk of rail failure is unlikely under these loading

and rail head wear conditions.

The third case examined is when a wheel with CPO = 15 mm, L/V =0, passes the crack

(R = 10 mm, inclined at = 90o) on the rail surface with HW 22 mm, as shown in

Figure 8.7. The higher head wear leads to a higher in-plane opening SIF (KI) at 0 crack

front position and therefore a higher Keq. The in-plane opening is attributed to higher

tensile stress as a result of an increase in the head wear, when the loading is applied at a

contact patch offset from the centre of the rail head cross-section. Moving the contact

location towards the gauge corner, away from the centre of the rail head cross-section,

leads to an increase in the tensile stress at the underhead radius. The combination of

increased bending stress due to increased localised bending of the rail head-on-web and

lateral bending of the head opens the crack (positive KI ) when the wheel contact

pressure peak is directly above the crack mouth (X/a = 0) and is exacerbated by

increasing head wear. This reveals that the in-plane opening SIF (KI) will be the

dominant driver of fracture. This is due to a linear increase in the in-plane tensile

opening SIF (KI) at the underhead radius with the highest value at the 0 crack front

position, as shown in Figure 8.7. Similarly the Keqmax is highest at the same location (see

Figure 8.7d) due to the large contribution of the positive KI mode.

The fracture region is defined by Keqmax > KIC. Fracture toughness (KIC) values of three

high strength rail grades were used in this analysis. The fracture toughness (KIC) value

used for a plain C-Mn Head Hardened grade [80], HH - KIC = 45 MPam is reported by

Marich et al [179] and Skyttebol et al [151]. For a Low Alloy Heat Treated grade [127],

LAHT3 - KIC = 39.1 MPam, and Hypereutectoid heat treated grade [81], HE - KIC =

35.4 MPam; these values are estimated by Ueda et al [130]. The representation of

these values is also illustrated in the Figure 8.7d by three dashed lines, which are used

to examine the effect of material properties on the rapid fracture of the crack. The effect

of increasing the fracture toughness of the rail material for the same range of loading


180

conditions is also shown in Figure 8.7d. Increasing the fracture toughness enables the

limit to be increased for high strength rail material grades. These values are all based on

previous studies (as reported in the respective references), and are used solely to

illustrate the effect of differences due to rail grade. A more rigorous analysis would

require testing of the rail material grades under identical test conditions, to ensure that

any differences in KIC were not attributable to the test methods used.

Figure 8.7 SIFs and Keq at the crack front when a wheel passes with CPO = 15 mm, L/V = 0 for crack (R = 10 mm, = 90o) to the rail running surface with HW 22 mm, for three rail grades: a plain C-Mn Head Hardened (HH) grade [80], a Low Alloy Heat Treated (LAHT) grade [127], and Hypereutectoid (HE) Heat Treated grade [81]

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Keq

(MP

a√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

HH - KIC

LAHT3- KIC

HE3 - KIC

-40

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

(MP

a√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-40

-20

0

20

40

60

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KI(M

Pa√

m)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-40

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

I(M

Pa√

m)


X/a = 1.5X/a = 0.6X/a = 0 X/a = -1X/a = -1.7

Fracture Region Increasing the fracture strength

(a)

(c)

(b)

(d)


181

A comparison of equivalent SIF (Keq) with fracture toughness (KIC) values (i.e. Keq max >

KIC , where KIC = 35.4 MPam for HE rails, KIC = 39.1 MPam for LAHT3 rails, and

KIC = 45 MPam for HH rails) indicated that a rapid fracture is likely to occur for these

rail material and loading conditions.

Effect of L/V ratio 8.4.2

The next case to be described is when the wheel passes the crack (R = 17 mm, inclined

at = 90o) with the lateral load (L) due to the angle of attack between the wheel and the

rail. A lateral load is generally influenced by both the vehicle steering effects on the

curved track and the hunting behaviour of the vehicle due to the vehicle dynamics, and

it can be scaled by the L/V ratio. An L/V ratio of 0 represents loading conditions in

tangent track, however this may increase slightly in the presence of vehicle hunting and

a value of 0.1 or even higher may be considered. .An L/V ratio of 0.2 represents a

loading condition in a curved track, as was reported by Marich [36, 37]. The high rail

experiences outward lateral shear traction and could reach a value of 0.4. The value of

L/V = 0.2 used for this thesis is on the lower side and simulates the inwards lateral shear

traction, rather than the outwards lateral shear traction considered in the work

referenced in [38, 42-45].

The lateral load can change the stress intensity factor pattern significantly at the

transverse rail curvature, as shown in Figure 8.8, with an L/V ratio = 0.2 applied at a

contact patch offset (CPO) of 15 mm for a crack (R = 17 mm, angle = 90o) to the rail

running surface with HW of 15 mm. The in-plane opening (KI) and sliding (KII) SIF are

plotted in Figures 8.8a and 8.8b. An increase in L/V = 0.2 leads to higher KII and the

pattern of KII is symmetric but with higher values, as expected. When the wheel is

directly on the top of the crack, the expected pattern of KI also has higher values,

particularly at the 0 crack front position. This higher tensile bending stress at the

underhead radius is caused by the superposition of eccentric loading with lateral shear

traction, as was expected and is shown in Figure 8.8a. The variation of SIFs in the out-

of-plane tearing mode KIII is similar to the last case, as shown in Figure 8.8c, presenting

much lower values when compared to KI and KII, even if the lateral traction is included.


182

Figure 8.8d depicts a detailed analysis regarding the fracture behaviour of the rail

underhead radius. Increasing the L/V ratio from 0 to 0.2 with contact patch eccentricity

(CPO = 15) shows that on the lower corner of the gauge face of the rail, the

combination of tensile bending stress due to superposition of eccentric and inward

lateral loading opens the crack (positive KI ) when the wheel is directly on the top of the

crack (X/a = 0). Similarly, the Keq is highest at the same location. When the Keqmax > KIC

for all the three rail material grades is considered, as shown in Figure 8.8d, the rail will

break with a rapid fracture.


0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Keq

(MP

a√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

HH - KIC

LAHT3- KIC

HE3 - KIC

-40

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

(MP

a√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-20

0

20

40

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KI(M

Pa√

m)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

I(M

Pa√

m)


X/a = 1.5X/a = 0.6X/a = 0 X/a = -1X/a = -1.7


(a)

(c)

(b)

(d)


183

Results of extreme loading conditions (HW = 22 mm, CPO = 20 mm, and L/V =0.2) on

a heavily worn rail head are shown in Figure 8.9. As with the last case, a higher L/V

=0.2 leads to a higher Keq and is strongly correlated with the lateral load, which results

in a higher tendency to rapid fracture leading to catastrophic rail failure.


This modelling showed how an inwards lateral shear traction of L/V = 0.2 can expedite

the rapid fracture of a pre-existing crack on a heavily worn rail of HW 15, 22 mm with

CPO of 15, 20 mm, as shown in Figures 8.8 and 8.9 respectively. This investigation

revealed that the lateral load (L/V) with a contact patch offset (CPO) could play a

significant role in the rapid fracture behaviour of pre-existing RCF cracks extending to

the rail underhead radius, if a heavily worn rail exists in practice. In the field, most of

0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Keq

(MP

a√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

HH - KIC

HE3 - KIC

LAHT3- KIC

-40

-20

0

20

40

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

(MPa

√m)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KI

(MPa

√m)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-40

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

I(M

Pa√m

)


X/a = 1.5X/a = 0.6X/a = 0 X/a = -1X/a = -1.7

Fracture Region

(a)

(c)

(b)

(d)


184

the cracks occurred on the high rail in curves of decreasing radii, around 850m – 350m,

as was reported by Kerr et al [193], and these are more prone to rail fracture. The

curved rails are heavily worn out, with contact patch eccentricity due to steering effects

and cant deficiency as compared to tangent tracks. If these curves are poorly lubricated

then lateral shear traction may rise and contribute to rail failure due to rapid fracture.

Comparison of head wear (HW) 8.4.3

The value of Keq along the crack front, when a wheel with different loads passes a crack

( = 90o to the rail running surface) with different head wear, is utilized to predict the

safe head wear limit. Figures 8.10 (a)-(b) show the variation in Keq with increasing head

wear and a range of crack sizes (R) and loading conditions. It should be noted from

Figure 8.10a that the least head wear (HW =10 mm) presents the lowest Keq for the

same loading condition due to the small values of tensile stress seen at the underhead

radius. At the underhead radius from the normalized crack front position of 0 to 0.2, Keq

is positive but Keq KIC for the HH, HE3 and LAHT3 rail grades considered in this

thesis, so no rapid fracture was expected.

Figure 8.10a also shows the stress intensity factors for a crack with heavily worn rail

profiles (HW 15 and 22 mm). In these cases, Keq exceeds the fracture toughness of the

HH, HE3 and LAHT3 grades from the normalized crack front position of 0 to 0.153 at

the underhead radius. Hence, rapid (unstable) crack propagation may develop for these

combinations of the crack length and head wear.

A rapid fracture can be predicted with 15 mm of head wear at an L/V of 0.2 and contact

patch offset of 20 mm (Figure 8.10a). The corresponding HW limit for an L/V ratio of 0

and contact patch offset of 15mm is approximately 22 mm, as shown in Figure 8.10b.

The safe HW limits are also summarised in Table 8.2. The prediction of safe HW limits

was based on the three HW profiles considered. Further study is required with multiple

sets of analysis with a range of HW, crack size, orientations, and loading conditions, in

order to examine the sensitivity of these limits in detail.


185

(a) CPO 20 mm, L/V = 0.2

(b) CPO 15 mm, L/V = 0

Figure 8.10 SIF at the crack front when a wheel passes for crack angle = 90o to the rail running surface with different HW, crack sizes R, CPO and L/V, for three rail grades: a plain C-Mn Head Hardened (HH) grade [80], a Low Alloy Heat Treated (LAHT) grade [127], and Hypereutectoid (HE) Heat Treated grade [81]

0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7K

eq(M

Pa√

m)


HW = 22 mm, R = 10 mmHW = 15 mm, R = 18 mmHW = 10 mm, R = 23 mm

HE3 - KIC

HH - KICLAHT3 - KIC

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

K eq

(MPa

√m)


HW = 22 mm, R = 10 mmHW = 15 mm, R = 17 mm

HE3 - KIC

HH - KIC

LAHT3 - KIC

R = 22 mm

R = 10 mm R = 17 mm


Increasing the fracture strength

No rapid fracture

Rapid fracture (d)

No rapid fracture R = 10 mm R = 17 mm

Fracture Region


186

Table 8.2 Predicted head wear limits based on fracture strength with extreme loading cases representative of different track conditions [valid for studied crack size (radius) R =10 mm, 17 mm and 22mm as shown in Figures 8.2 and 8.10]

Rail track curvature Assumed L/V

ratio Patch offset (mm) Safe HW limit (mm)

(predicted)

Tangent (prone to hunting) / shallow radius 0 15 15

Tight radius curve 0.2 20 10

Crack shielding by multiple cracks 8.4.4

The effect of crack shielding by multiple cracks was examined. This was done by

considering clusters of both 3 and 7 cracks with a spacing s = 5 mm from the middle

crack, as shown in Figure 8.2 (a)-(c). This spacing was chosen as representative of the

separation of cracks seen on curved rails prone to RCF cracks [27]. The SIF predicted at

the crack front for the central crack of seven multiple cracks when a wheel passes with

CPO = 15 mm, L/V = 0 for crack angle = 90o to the rail running surface with HW of

22 mm is shown in Figure 8.11.

Figure 8.7 presents the results for the SIF for a single crack under a CPO 15 mm, with

an L/V = 0 and head wear of 22mm. The results for 7 cracks are shown in Figure 8.11a.

A comparison of these results suggests that there is a reduction in the maximum KI of

the middle crack of around 37% relative to the single crack. The results in Figure 8.11b

indicate that the maximum KII is reduced by 19 % of the single-crack value (Figure

8.7b). The KIII values for single and seven crack cluster models were close. The results

in Figure 8.11d indicate that the maximum Keq is reduced by 19 % from the single-crack

value (Figure 8.7d). The maximum of Keq was found to be sensitive to the presence of

multiple cracks to a similar degree to the maximum of KI and KII values, as shown in

Figure 8.11d.


187

Figure 8.11 The effect of modelling seven multiple cracks on SIF predicted for the central crack with SIF on crack front when a wheel passes with CPO = 15 mm, L/V = 0 for crack angle 90o to the rail running surface with HW 22 mm

The Keq for a single crack and clusters of 3 and 7 cracks were compared. The Keq

predicted on a normalised crack front for the central crack, as used to model 1, 3 and 7

multiple cracks, is shown in Figure 8.12. The results for the reduction in the maximum

of Keq for normalised crack front position of 0 are taken for comparison. The results for

L/V = 0 (Figure 8.12a) suggest that when compared to a single crack, there is a 27%

reduction in the Keq max of the middle crack when the model with 3 cracks is considered.

Similarly the reduction in Keq max of the middle crack is around 35% when 7 cracks are

considered in comparison to the single crack model. In the presence of these reductions

in SIFs, it can be seen that the multiple cracks shield each other and prevent the rail

fracturing through a reduction in the stress intensity factors. For the 3-crack model, the

predicted Keq max just reached the KIC for HE and LAHT3 rail grades, for a head wear of

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Keq

(MP

a√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

HH - KIC

LAHT3- KICHE3 - KIC

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

(MP

a√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-40

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KI(M

Pa√m

)


X/a = 1.5X/a = 0.6X/a = 0X/a = -1X/a = -1.7

-20

-10

0

10

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

I(M

Pa√m

)


X/a = 1.5X/a = 0.6X/a = 0 X/a = -1X/a = -1.7


(a)

(c)

(b)

(d)


188

22 mm. Nevertheless, the 7 crack model shows Keq max < KIC for both LAHT3 and HH

rail grades, which is a significant reduction when compared to the single crack case.

(a) L/V = 0 (b) L/V =0.2

Figure 8.12 The effect of modeling multiple cracks on Keq SIF predicted for the central crack when a wheel passes with CPO = 15 mm, for crack angle 90o to the rail running surface with HW 22 mm; L/V 0, 0.2.

The results for L/V = 0.2 (Figure 8.12b) suggest a subsequent increase in the Keq max due

to increased tensile stresses at the underhead radius, and as a result of the superposition

of the load eccentricity with lateral shear traction, as shown in Figure 8.12b. Compared

to a single crack, there is a 43% reduction in Keq max of the middle crack if the model

with three cracks is considered. Similarly, the reduction in Keq max of the middle crack is

around 50 % if seven cracks are considered, as compared to the single crack model.

In this doctoral study, multiple cracks of the same size and shape were modelled, which

is not what is generally observed in the field. It has been commonly observed that one

crack in the series can become longer compared to others. In this way, there is a

possibility that the shielding effect may be reduced and a single crack may become

critical, resulting in the rapid fracture of the rail, as presented in the case of single crack

models. Therefore, the single crack case is more critical and the analysis and design

against failure of a single crack should be considered accordingly.

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

K eq

(MPa

√m)


Single crack3 cracks7 cracks

HH - KIC

LAHT3- KIC

HE3 - KIC

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7K e

q(M

Pa√m

)


Single crack3 cracks7 cracks

HH - KIC

LAHT3- KICHE3 - KIC

Fracture Region Fracture Region Increasing the fracture strength


189

8.5 Summary

This chapter has discussed the prediction of the rapid fracture behaviour of rails

associated with local bending of the rail head-on-web. A method has been developed to

predict unstable growth of multiple RCF cracks under multi-axial loading at the gauge

corner of the rail. The extended finite element method (X-FEM) model was used to

investigate the stress intensity factor near the crack front that results from contact and

bending stresses. The equivalent stress intensity factor was considered in order to

investigate the fracture behaviour. Both single and multiple cracks were modelled at

different lengths and orientations, over a range of head wear conditions.

The results of the rapid fracture analysis using mixed-mode loading suggest that when a

wheel passing on the rail head without eccentricity, and approaches a position above the

crack mouth, the head is in compression as a result of global bending. In-plane tensile

opening mode, KI is negative and the Keq values are very small, far below the fracture

region defined by Keq KIC. Moving the contact patch towards the gauge corner away

from the centre of the rail head cross-section increases the tensile stress at the underhead

radius. This is due to bending of the rail head-on-web forced by the eccentricity of loads

from the rail centreline. The maximum of Mode I stress intensity factor KI max occurs

when the contact loading is applied directly above the crack location. As a result of

changes in the stress state, the KI increases and so does the Keq.

Higher head wear leads to a higher in-plane opening stress intensity factor (KI) at the

crack front position of 0, corresponding to the underhead radius near the lower gauge

corner point, and it therefore has a higher Keq. The in-plane tensile opening behaviour

can be attributed to higher tensile stress as a result of an increase in the head wear, if the

loading is applied at the contact patch offset from the centre of the rail head cross-

section. However the L/V ratio also has a significant effect on KI and KII and

consequently Keq. This is due to the higher tensile bending stress at the underhead radius

caused by the superposition of the eccentric loading with inwards lateral shear traction,

as was expected. The values of the stress intensity factors in the out-of-plane tearing KIII

mode are much lower, as compared to the opening KI and the sliding KII modes,


190

suggesting that the in-plane opening and shear forces play an important role in the

unstable propagation of pre-existing RCF cracks that extend to the rail underhead

radius. Therefore, KI max and KII max were more significant in terms of rapid fracture than

KIII max.

The results revealed that the lateral load (L/V) with a contact patch offset (CPO) could

play a significant role in the rapid fracture behaviour of pre-existing RCF cracks

extending to the rail underhead radius, if a heavily worn rail exists in practice.

Therefore, pre-existing RCF cracks at the gauge corner region with crack lengths

extending to the underhead radius region could increase the risk of rapid (unstable)

fracture if high tensile stresses at the underhead radius result from extreme contact

loading conditions in combination with rail material and head wear. It was predicted

that the presence of high underhead radius stresses could lead to catastrophic rail failure

as a result of the rapid fracture of pre-existing RCF cracks extending to the underhead

radius (UHR) region.

The work also examined the rapid fracture behaviour of multiple RCF cracks. A

significant reduction in stress intensity factors, especially for KI and KII, and

consequently in Keq was predicted for multiple cracks compared to a single crack. The

thesis examined the influence of wheel-rail interaction and rail head wear on the

probability of rail failure from the existing gauge corner cracking. It also demonstrated

the differences in rapid fracture behaviour between a single and multiple cracks. The

results were presented for the bending stresses, as they are likely to be different to the

highly localised contact stress driven phases of the “Whole Life Rail Model”. Therefore,

further investigation of multiple gauge corner cracks in ratchetting and contact stresses

phases would be valuable.

Chapter 9. Conclusions and future work

191

Chapter 9

Conclusions and future work

9.1 Conclusion

The effect of the lateral bending of a whole rail profile and the localised vertical and

lateral bending of the head-on-web has been investigated in order to understand the

phenomenon of fatigue cracking and rapid fracture. Finite element modelling and

extended finite element method (X-FEM) modelling approaches were used. In

particular, the depth at which tensile longitudinal stresses occurred was evaluated, in

relation to the development of the longitudinal stress peak associated with local bending

directly under the wheel position, but this does not need to be the depth of the mode

transition. The magnitude of tensile bending stress was much higher than that associated

with rail uplift ahead and behind the wheel contact (as was identified in the post-

Hatfield studies). The current study only features a single load. To obtain the full effect

of (relative) uplift, a bogie loading should be considered and a vehicle dynamics study

should be required to determine the L/V ratio, contact patch offset and direction of

lateral loading. The results have been described and discussed in detail in each chapter

(4 to 8). This chapter highlights the main conclusions that have been drawn in each

chapter from this research.

The finite element commercial package ABAQUS 6.11-2 was used to examine the

stress state at the underhead radius of the rail. To validate the predicted results, a

comparison with measured field data was conducted with the focus on the longitudinal

stresses at the underhead radius of the rail. The tension spike at the underhead radius

was found to be highly dependent on several factors: the contact patch offset (CPO), the

ratio of lateral (L) to vertical (V) loads, the direction of lateral traction, the vertical


192

foundation stiffness and the seasonal temperature. The wheel-rail contact location is a

critical factor that influences the stress state at the rail underhead radius, particularly

when offset laterally towards the gauge side. The lateral shear traction generated by

vehicle curving and hunting operations greatly influences the stress state at the rail

underhead radius. Foundation stiffness also has an effect: the stiffer the track the higher

the underhead radius stresses. The magnitude of longitudinal tensile stress increases at

the underhead radius as the vertical foundation stiffness increases. The results also

suggest that the magnitude of longitudinal tensile stress at the underhead radius

increases in the colder months and decreases in the warmer months. This is the result of

rail temperature variations around the nominal stress-free or neutral temperature. This

indicates that stiffer tracks operated in cold weather conditions experience increased

underhead radius stresses.

The tension spike at the underhead radius of a rail increases when the contact patch

moves away from the rail centreline and / or the L/V ratio increases. The direction of

lateral shear traction can be either outward or inward. Inward shear traction was found

to be more damaging. This is easy to understand by considering the effect of L and V on

the rotation and lateral deformation of the rail head-on-web. A contact patch offset

towards the gauge corner rotates the rail head in the same direction due to eccentric

load. The lateral load, L, causes further rotation and lateral deformation of the rail head-

on-web. When the lateral load is towards the gauge corner (inwards), both rotations

(due to L and V) are summative and provide a much larger rotation and lateral

deformation than when the lateral load is outwards. In this case, the rotation due to L

partly cancels out the rotation due to V, producing significantly lower deformation and

longitudinal stresses at the field side underhead radius position. The depth below the rail

running surface at which stress becomes tensile decreases as the L/V ratio increases. For

example, when the L/V=0.4, the depth may be of the order of a few millimeters.

Several new finite element (FE) models were developed to study the effect of worn

profiles, using the commercial finite elements code ABAQUS (6.11-2). The stress state

at the rail underhead radius is influenced by the head wear levels in conjunction with the

wheel - rail contact conditions and L/V ratio. The rail underhead radius with a heavily

worn rail presents the most critical case with very high tensile stress close to the contact


193

running surface. These high local bending stresses can potentially initiate a fatigue

crack or cause existing rolling contact fatigue cracks to turn downwards. It was found

that the residual and thermally induced stresses interact with these high local bending

stresses and alter the cracking behaviour in a complex way. This can result in a

transition to Mode I crack growth and the formation of a transverse defect for rolling

contact fatigue initiated cracks that extend to this region, as has been observed in

practice [1-4]. RCF cracks generally started to turn down at about 4-5 mm below the

rail head surface for the AS1085.1 [80] head hardened and low alloy heat treated rail

grades operated under high axle load conditions [1, 2, 4] and this depth is dependent on

the rail manufacturing and operational conditions. These tensile stresses may also result

in fatigue crack initiation at the underhead region, as was previously investigated in

connection with the reverse detail fracture in the United States [6, 7, 9].

The results also suggest that stiffer tracks are more prone to high tensile stresses at the

rail underhead radius in the presence of heavily worn rail. The predicted conditions

were in line with field observations of reverse detail fracture defects in the North

American railroad systems [3-10]. Reverse detail fracture defects have been observed

on heavily worn curved rails on stiff tracks subjected to high axle loads. With no

support to the rail from the adjacent sleepers (a vertical foundation stiffness of zero), the

maximum longitudinal tensile stress decreases at the underhead radius but increases at

the base fillet. This condition simulates the effect of ballast pumping, which can remove

support from one sleeper.

A fatigue damage prediction analysis for the rail underhead radius was undertaken to

investigate the fatigue damage behaviour as a function of wheel – rail contact

conditions, head wear and rail material grades under heavy haul operations. Fatigue

damage was predicted using the Dang Van criterion. Lateral offsets in the wheel contact

patch relative to the rail centreline, in combination with the increased lateral forces

associated with vehicle curving, increase the probability of fatigue damage initiation at

the underhead radius. This is due to an increase in the tensile stresses at the underhead

radius, which is induced by the local bending behaviour on the rail web. In the presence

of heavily worn rail profiles, the fatigue life of rail in sharp curves is less than in tangent


194

track. In general, the more wear that occurs on the surface of the rail, the more fatigue

damage is predicted to initiate at the rail underhead radius.

The analyses revealed that, under severe loading conditions, fatigue damage is predicted

to develop at the rail underhead radius with increased head wear and in a rail grade with

lower fatigue limit. It was reported in the North American railroad systems that a

reverse detail fracture is initiated in poorly lubricated, heavily worn curved rail on stiff

track carrying traffic under high axle load conditions, with flow lips as initiators [6-10],

and additionally there is a possible contribution of higher residual stresses at a rail weld.

Failures that initiate at the underhead radius of the aluminothermic welds have been

found in Australia on lines subjected to heavy axle load railway operations. However,

for high strength rail material with higher fatigue limits, it was rarely predicted that

fatigue damage would occur at the underhead radius under the range of loading

conditions considered with an increase in the extent of the head wear. Examples of the

high strength rail materials used included the low alloy or hypereutectoid heat-treated

grades used in Australian heavy haul operations. The results of the fatigue analysis

model showed sufficiently good correlation with field observations.

The prediction of rapid fracture behaviour of pre-existing RCF cracks that have already

turned down (to Mode I behaviour), associated with the lateral bending of the whole rail

profile and localised vertical and lateral bending of the rail head-on-web, was described

in chapters 7 and 8. In particular, the rapid fracture behaviour of long RCF cracks in

phase 3 of the ―Whole Life Rail Model‖ was considered. The extended finite element

method (X-FEM) was used to investigate the stress intensity factor (KI, KII and KIII) near

the crack front that results from contact and bending stresses. Several new crack models

were developed by inserting X-FEM elements in the existing finite element (FE) meshes

for the prediction of RCF cracks unstable growth behaviour under mixed-mode loading

at the gauge corner of the rail. Both single and multiple cracks were modelled at different

crack lengths and orientations, over a range of head wear conditions.

The results of rapid fracture analysis using mixed-mode loading suggest that when a

wheel passing on the rail head without eccentricity is above the crack mouth, the head is

in compression due to global bending. In the plane tensile opening mode, KI is negative

and the Keq values are very small, far below the fracture region defined by Keq KIC.


195

The underhead radius stresses are higher in heavily worn rails. An increase in the

contact patch offset increases these stresses and stress intensity factors even more. The

effect of KI max influenced by the contact patch offset is a case in which high tensile

bending stresses are produced at the underhead radius due to the localised bending of

the rail head-on-web and lateral bending of the whole rail profile. As a result of changes

in the stress state, the KI increases and so does the Keq. The results revealed that the

highest KI max occurs when the contact loading is directly above the crack location and is

exacerbated by increasing head wear.

The L/V ratio also plays a significant role in the tensile bending stresses at the

underhead radius. The highest tensile bending stress at the underhead radius is caused

by the superposition of the eccentric loading with inward lateral shear traction in

combination with an increase in head wear when the contact loading is applied directly

above the crack location. This leads to the higher in-plane opening stress intensity factor

KI at the 0 crack front position, corresponding to the underhead radius near the lower

gauge corner point. Similarly the Keqmax is highest at the same location due to the large

contribution of the positive in-plane tensile opening KI. The results revealed that the in-

plane tensile opening stress intensity factor KI is the dominant driver of a fracture.

Nevertheless, the L/V ratio also has a significant effect on in-plane sliding KII mode and

consequently Keq when the contact position is away from the crack. The values of the

stress intensity factors in the out-of-plane tearing KIII mode are much lower than the

opening KI and the sliding KII modes, suggesting that the in-plane tensile opening and

shear forces play an important role in the unstable propagation and extension of pre-

existing RCF cracks to the rail underhead radius. Therefore, KI max and KII max were more

significant in terms of rapid fracture than KIII max.

The results revealed that a contact patch offset with a L/V ratio could drive the rapid

fracture of pre-existing RCF cracks at the gauge corner region, with the crack length

extending to the rail underhead radius, if a heavily worn rail exists in practice for the

combination of rail material grades considered. In practice, most of the cracks occur on

the gauge corner of a high rail with curves of decreasing radii, around 850 m – 350 m,

as was reported by Kerr et al [193] and these are more prone to rail fracture. The curved

rails are generally heavily worn, with contact patch eccentricity due to steering effects


196

and cant deficiency compared to tangent tracks. In addition, the curves are prone to high

lateral shear traction, which may increase the risk of rail failure due to the rapid fracture

of pre-existing RCF cracks.

An examination was also made of the rapid fracture behaviour of multiple RCF cracks.

A significant reduction in stress intensity factors, in particular, for KI and KII, and

consequently in Keq is predicted for multiple cracks compared to a single crack. It has

been commonly observed that only a small number of widely spaced cracks in the series

can become longer compared to others. In this way, there is a possibility that the

shielding effect may be reduced and a single crack may become critical, resulting in the

rapid fracture of the rail, as presented in the case of single crack models. The single

crack case is more critical and the analysis and design against failure of a single crack

should be considered accordingly. The results of this work can be used to examine the

influence of wheel-rail interaction behaviour and rail head wear on the probability of

catastrophic rail failure resulting from long turned down pre-existing RCF cracks. The

thesis has also demonstrated the differences in the rapid fracture behaviour between

single and multiple cracks. It is suggested that KI and KII should be primarily reduced to

prevent rapid fracture.

The analyses results reported in this thesis have applications in the improvement of the

modelling of RCF crack propagation and fatigue damage prediction at the rail

underhead radius, leading to improved asset management and risk assessment. The

approach taken in this thesis forms a basis for more extensive studies into the growth

behaviour of RCF cracks, as outlined above, in order to be better able to provide firm

guidelines to rail management, particularly in relation to RCF damage. The following

measures are proposed in this regard:

Catastrophic rail failure due to the rapid fracture behaviour of pre-existing long

turned down (to Mode I behaviour) RCF cracks can be prevented by following safe

head wear limits through appropriate maintenance strategies. It is recommended that

the approach studied in relation to RCF cracks be used when determining safe head

wear levels.


197

Rail life can be improved by reducing the contact patch lateral displacement. Some

of the approaches include but are not limited to re-profiling the rail by appropriate

maintenance and grinding strategies.

Rail performance can be improved by controlling the L/V ratio through the

application of lubrication or other friction management strategies (friction modifier)

at the wheel - rail interface, particularly in short radius curves.

In addition to lubrication strategies, the bogie suspension design should be

optimised to control the lateral loading, especially the inward lateral shear traction,

which is the more damaging phenomenon. Nevertheless, more rigorous studies are

required to work out a strategy as to how this problem can be solved.

As far as possible, sharp curves should be avoided. There are several examples of

curve radius reductions in connection to speed increases of existing railway lines.

This could also be done when new routes are planned.

Stiffer tracks should be avoided, as the tensile bending stresses increase at the

underhead radius and leads to a higher propensity for rapid fracture. Therefore, it is

beneficial to keep the foundation stiffness (sleeper type, fasteners, rail sleeper pad

and ballast) within the design range through regular maintenance. A revision of

asset management procedures can also be considered.

Fatigue crack initiation at the underhead radius can be prevented by using rail steels

with high fatigue limit, such as low alloy or hypereutectoid heat treated grades. High

strength steels tend to be more sensitive to surface roughness and geometric defects

and the risk of a rapid fracture of these high strength rail materials should be

balanced with their fracture toughness ‗KIC‘

9.2 Suggestions for further work

The inferences drawn from analyses results are true for the limited combination of

operating conditions and rail material grades considered. Further detailed analyses and

field investigations are required to understand how the combinations of operating

conditions and rail material grades affect rolling contact fatigue cracks and head wear.

The recommendations for future work are summarized in this section.


198

The fatigue life model results indicate under which combination of loading conditions,

rail head wear and rail material grades fatigue damage is predicted to be initiated.

However, in practice, fatigue failure is generally associated with the presence of pre-

existing defects or a stress concentrator in the form of a sharp radius, which may include

flow lips (as was evident in the reverse detail fracture failure mode) or rolling contact

fatigue cracking. The simulation results for fatigue damage prediction did not consider

the effects of pre-existing defects or other conditions, which may increase localised

stresses in the vicinity of the underhead radius. However, the results do indicate the

potential formation of defects such as reverse detail fractures, which have previously

been found to develop at the underhead radius (lower gauge corner) on heavily worn

rail. Further work is required to examine these behaviours in the presence of pre-

existing defects.

The residual stresses were incorporated using results obtained by the neutron diffraction

method. These values may not be representative of current rail manufacturing

procedures, and hence the treatment of residual stress aspects needs to consider the fact

that those in the rail grades examined may differ from those used for the previous

neutron diffraction measurements. The approach presented can also be extended to

include the original residual stress distribution for both roller-straightened rail, and that

straightened by other methods, such as stretch straightening. The influence of pre-

existing residual stresses resulting from the rail manufacturing process could be

examined by undertaking a sensitivity analysis to residual stresses. In addition, the

repeated rolling contact between wheel and rail is another important parameter that can

lead to a complex distribution of the residual stresses in the rail and is of interest for

future study.

The vehicle dynamic effects associated with variations in track geometry will also

increase the magnitude of the vertical and lateral loads. The dynamic effects will change

the contact patch offset and L/V ratio in addition to the increased load. Increased train

speeds may also produce an increase in dynamic loads as well as an increase in the

effect of lateral forces due to lateral irregularity on either the gauge or the field side, and

subsequently an increase in the L/V ratio. The current analysis did not consider vehicle

dynamics. This model can be updated by using the contact load conditions (contact


199

patch size, shape and position, L/V ratio etc.) based on vehicle dynamic studies to

obtain a detailed prediction of the stress state under dynamic loading conditions and

would be of interest for future study.

The fatigue crack growth rate and the direction of the propagation of pre-existing RCF

cracks using the contact patch offset, lateral traction and head wear conditions are still

unclear. Further study is required to model the quantitative assessment of fatigue crack

growth using more realistic data, both in field and laboratory conditions. Moreover,

water entrapment and crack closure are factors that are not considered in the extended

finite element method (X-FEM) fracture modelling. Further development of the current

model could include crack closure and liquid entrapment, in addition to localized

bending effects. The behaviour of multiple RCF cracks also requires examination under

more realistic service conditions. The results of rapid fracture analysis of multiple RCF

cracks were presented for phase 3 of the “Whole Life Rail Model”, mainly driven by

bending stresses that are likely to be different to highly localised contact stress driven

phases. Therefore, further investigation of multiple RCF cracks in ratchetting and

contact stress phases would be valuable.

Rapid fracture analysis performed for pre-existing RCF cracks incorporated fracture

toughness KIC values based on previous studies (as reported in the respective

references), and was used solely to illustrate the effect of differences due to rail grade.

Newly introduced rail steels, such as bainitic grades, also have the potential to replace

the widely-used pearlitic steels. A more rigorous analysis would require testing of the

rail material grades under identical test conditions, to ensure that any differences in KIC

were not attributable to the test methods used. Therefore, a thorough review of the rail

steel grades should be considered and incorporated into a future study.

The prediction of safe head wear limits was based on the limited head wear profiles

considered. Further study requires multiple sets of analyses with a range of head wear,

crack size, orientations, separation and loading conditions, in order to see the sensitivity

of these limits in detail. The approach presented in this thesis, if extended to include the

residual stress aspects in the presence of pre-existing defects or the other conditions

outlined above, provides a basis for an assessment of allowable rail wear limits as a


200

function of loading conditions and rail material grades and may be worth investigating

in a future study.

References

201

References

1. Mutton PJ, Tan M, Bartle P, Kapoor A. The effect of severe head wear on rolling

contact fatigue in heavy haul operations. 8th International Conference on Contact

Mechanics and Wear of Rail/Wheel Systems Comitato (CM2009); September 15-

18, 2009, ISBN. 978-88-904370-0-7 Firenze, Italy, 2009.

2. Mutton PJ, Welsby D, Alvarez E. Wear and rolling contact fatigue behaviour of

heat-treated eutectoid and hypereutectoid rail steel under high axle load

conditions. 8th International Conference on Contact Mechanics and Wear of

Rail/Wheel Systems Comitato (CM2009); September 15-18, 2009 Firenze,

Italy2009. p. 569-75, ISBN. 978-88-904370-0-7.

3. Ranjha SA, Ding K, Mutton PJ, Kapoor A. Finite element modelling of the rail

gauge corner and underhead radius stresses under heavy axle load conditions.

Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail

and Rapid Transit 2012;226(3):318-30.

4. Ranjha SA, Ding K, Mutton PJ, Kapoor A. Mechanical state of the rail underhead

region under heavy haul operations. In: CORE 2012: Global Perspectives;

Conference on railway engineering, 10-12 September 2012, Brisbane, Australia

Barton, ACT: Engineers Australia, 2012: 185-194 ISBN: 9780987398901.

5. Ranjha SA, Ding K, Mutton PJ, Kapoor A. Fatigue life prediction of the rail

underhead region influenced by wear in heavy haul operations, 9th International

Conference on Contact Mechanics and Wear of Rail/Wheel Systems:

CM2012,State Key Laboratory of Traction Power (TPL), Southwest Jiaotong

University, Chengdu, China, 27-30 August 2012. 2012.

6. Jeong DY, Tang YH, Orringer O, Perlman AB. Fracture mechanics analysis of

transverse defects originating at the rail lower gauge corner. Proceedings of the

References

202

World Congress on Railway Research; Colorado, Springs, CO, USA,1996 pp.

391-397.

7. Jeong DY, Tang YH, Orringer O, Perlman AB. Propagation Analysis of Trasverse

Defects Originating at the Lower Gage Corner of Rail, Volpe National

Transportation System Center, Cambridge, Massachusetts, Report No.

DOT/FRA/ORD-98/06.1998.

8. Jeong DY. Progress in Rail Integrity Research, Volpe National Transportation

System Center, Cambridge, Massachusetts, Report No. DOT/FRA/ORD-

01/18.2001.

9. Main-Track Train Derailment: Railway Investigation Report-R06C0104 (July

2006). Canada CPR, Freight Train CP 803-111, Mile 97.4, Canadian National

Ashcroft Subdivision, Lytton, British Columbia.

<http://www.tsb.gc.ca/eng/rapports-reports/rail/2006/r06c0104/r06c0104.asp>. .

10. Jeong DY. Progress in Rail Integrity Research, in proceedings of the AREMA

Annual Conference & Exposition, September 11, 2001 Chicago, IL, Available:

http://www.arema.org/files/library/2001_Conference_Proceedings/00044.pdf2001

11. Dollevoet., Petrus R, Johannes B. Design of an anti head check profile based on

stress relief. University of Twente, Enschede, Ir RPBJ Dollevoet, Netherland2010.

ISBN:978-90-365-3073-6.

12. Francisco CRH, Demas NG, Davis DD, Polycarpou AA, Maal L. Mechanical

properties and wear performance of premium rail steels. Wear. 2007;263(1–

6):766-72.

13. Duvel J, Mutton PJ, Alvarez E, McLeod J. Rail requirement for 40 tonne axle

loads. 8th International Heavy Haul Railway Conference; Brazil2005.

14. Moynan M, Cowin A, Offereins G, Tew G. Axle Loads - The BHP Iron Ore

Experience, BHP Iron Ore, BHP Melbourne Research Laboratories, Available

online through web link: http://www.ncc.gov.au/images/uploads/DeRaFoAp-

0019.pdf.[11 March 2013]

http://www.tsb.gc.ca/eng/rapports-reports/rail/2006/r06c0104/r06c0104.asp%3e

http://www.ncc.gov.au/images/uploads/DeRaFoAp-0019.pdf

http://www.ncc.gov.au/images/uploads/DeRaFoAp-0019.pdf

References

203

15. Jeong DY. Correlations between rail defect growth data and engineering analyses,

part II: field test, Volpe National Transportation System Center, Cambridge,

Massachusetts2003.

16. Federal Railroad Adminstration, Office of Railroad Safety, Track inspector rail

defect reference manual, Aug 01, 2011, available from

http://www.fra.dot.gov/eLib/Details/L03531 (accessed on 27 February 2013).

17. Ranjha S, Ding, K., Mutton, P.J., Kapoor, A., editor. Rapid fracture behavior of

rolling contact fatgue cracks under high axle load conditions, In proceedings of

the conference of International Heavy Haul associations (IHHA 2013); 6 February

2013; New Delhi, India 2013. ; 2013.

18. Kapoor A, Fletcher DI. Post Hatfield rolling contact fatigue - The effect of

residual stress on contact stress driven crack growth in rail, : Newcastle

University School of Mechanical and System Engineering, New Rail Report no:

WR061106(Parts 1-6)2006.

19. Kapoor A FD, Franklin FJ, Beagels AE, Burstow M, Allen R, Evans G, Jaiswal J.

Management and understanding of rolling contact fatigue, WP2: Phase 2 Crack

Growth. London, UK: Railway Safety and Standards Board, 2007. Deliverable 2

Mechanism of Crack Growth.

20. Kapoor A FD, Franklin FJ, Beagels AE, Burstow M, Allen R, Evans G, Jaiswal J.

Management and understanding of rolling contact fatigue, WP1: Mechanims of

Crack Initiaton and WP2: Crack Growth. London, UK: Railway Safety and

Standards Board, 2005. Deliverable 1 Literature Review.

21. Dutton JT, Hobbs JW, Fletcher DI, Kapoor A. Post-Hatfield rolling contact

fatigue - Modelling of bending stresses in rails, Report No: MM/04/29, Office of

Rail Regulation, Health and Safety Laboratory. 2005.

22. Kapoor A, editor. Wear fatigue interaction and maintenance strategies,

Proceedings of the Railway Engineering, LondonJuly 2002.

23. Kapoor A, Schmid, F. and Fletcher, D. I. . Managing the critical wheel/rail

interface. Railway Gaz. Int., 2002, 158(1), 25-28.

http://www.fra.dot.gov/eLib/Details/L03531

References

204

24. Kapoor A, Fletcher D, Franklin F, Alwahdi F. Whole life rail model. Interim

Report, Technical Rcport MEC/AK/AEAT/September, The University of

Shieffield and AEA Technology Rail, Vol 2. 2002.

25. Lewis R, Olofsson U. Wheel-rail interface handbook: Oxford Cambridge, New

Delhi: Woodhead Publishing Limited; 2009.

26. Kapoor A, Fletcher DI, Franklin FJ. The role of wear in enhancing rail life. in D.

Dowson, M. Priest G. Dalmaz A. A. Lubrecht, Tribology Series, Elsevier,

Volume 41, 331-340, http://dx.doi.org/10.1016/S0167-8922(03)80146-3,

http://www.sciencedirect.com/science/article/pii/S0167892203801463; 2003.

27. Fletcher DI, Hyde P, Kapoor A. Growth of multiple rolling contact fatigue cracks

driven by rail bending modelled using a boundary element technique. Proceedings

of the Institution of Mechanical Engineers -- Part F -- Journal of Rail & Rapid

Transit. [Article]. 2004;218(3):243-53.

28. Dang Van K, Maitournam MH, Moumni Z, Roger F. A comprehensive approach

for modeling fatigue and fracture of rails. Engineering Fracture Mechanics. [doi:

DOI: 10.1016/j.engfracmech.2008.12.020]. 2009;76(17):2626-36.

29. A Kapoor, I Salehi, Asih AMS. Rolling contact Fatigue (RCF)‖, CRC

Encyclopaedia of Tribology.

30. Burstow MC, Watson AS, Beagles M, editors. Application of the whole life rail

model to control rolling contact fatigue. Proceedings of the Railway Engineering;

2002; London.

31. Kapoor A, editor. Wear fatigue interaction and maintenance strategies. In

Proceedings of the Advanced Railway Research Centre (ARRC) on Why

Failures Occur in Wheel-Rail Systems, The University of Sheffield, UK2001.

32. Bartle P, Alserda J, Mutton P J, Welsby D. Confidential report. Monash

University Institute of Railway Technology 2007.

33. Mutton PJ, Tan M. Confidential report. Monash University, Institute of Railway

Technology 2008.

http://dx.doi.org/10.1016/S0167-8922(03)80146-3

http://www.sciencedirect.com/science/article/pii/S0167892203801463;

References

205

34. Eisenmann J. Stress distribution in the permanent way due to heavy axle load and

high speeds. West Germany.: Techanical University of Munich. 1969 Contract

No.: 71-622-3.

35. Sugiyama T, Yamazaki T, Umeda S. Measurement of local stress on outer rail

head. Quarterly report of RTRI. 1971.p. 11-3;12(1).

36. Marich S. Major advances in rail techonologies achieved in the past 10-20 years.

8th International Heavy Haul Railway Conference; Brazil2005.

37. Marich S. Some facts and myths about rail grinding - The Australian experience.

8th International Heavy Haul Railway Conference; Brazil2005.

38. Marich S. Rail wear/fatigue limits: Thomas Telford Ltd, Track Technology,

London, UK 1985.

39. Jeong DY, Orringer O. Fatigue crack growth of surface cracks in the rail web.

Theoretical and Applied Fracture Mechanics. 1989;12(1):45-58.

40. Jeong DY, Tang YH, Orringer O. Damage tolerance analysis of detail fractures in

rail. Theoretical and Applied Fracture Mechanics. 1997;28(2):109-15.

41. Orringer O, Morris JM, Steele RK. Applied research on rail fatigue and fracture in

the United States. Theoretical and Applied Fracture Mechanics. 1984;1(1):23-49.

42. Salehi I, Kapoor A, Mutton PJ, Alserda J. Improving the Reliability of

Aluminothermic Rail Welds Under High Axle Load Conditions. In Proceedings

of the conference on Rail Engineering: rail rejuvenation and renaissance (CORE

2010); September 12-15, 2010, pp.16-24. ISBN:978-0-908960-55-2; Wellington,

New Zealand2010.

43. Salehi I, Kapoor A, Mutton P. Multi-axial fatigue analysis of aluminothermic rail

welds under high axle load conditions. International Journal of Fatigue.

2011;33(9):1324-36.

44. Salehi I, Mutton, P. and Kapoor, A. The effect of geometric features on multi-

axial fatigue behaviour of aluminothermic rail welds, Proceedings of the

Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit

226.4 (2012): 360-370.

References

206

.

45. Salehi I, Mutton, P.J., Kapoor, A., editor. Analysis of damaging factors in

thermite welds through multi-axial fatigue criterion, In proceedings of the 9th

International Heavy Haul Associations Conference (IHHA 2011); 19-22 June

2011; Calgary, Canada 2011.

46. Orringer O, Tang YH, Gordon JE, Jeong DY, Morris JM, Perlman AB. Crack

Propagation Life of Detail Fractures in Rails,Volpe National Transportation

System Center, Cambridge, Massachusetts, Report No. DOT/FRA/ORD-88/13.

1988.

47. Orringer O, Morris JM, Jeong DY. Detail fracture growth in rails: Test results.

Theoretical and Applied Fracture Mechanics. [doi: DOI: 10.1016/0167-

8442(86)90019-4]. 1986;5(2):63-95.

48. Lyons ML, Jeong DY, Gordon JE. Fracture Mechanics Approach to Estimate Rail

Wear Limits. ASME Conference Proceedings. 2009;2009(48944):137-46.

49. Salehi I. Fatigue and fracture behaviour of Aluminothermic rail weld under high

axle load conditions. PhD Thesis, Centre for Sustainable Infrastructure,

Swinburne Unoversity of Technology 2012.

50. Mutton PJ. Confidential report. Institute of Railway Technology, Monash

University2008.

51. Dang Van K, Cailletaud G, Flavenot JF, Le Douaron A, and Lieurade HP.

Criterion for high cycle fatigue failure under multiaxial loading, biaxial and

multiaxial fatigue,EGF 3. Mechanical Engineering Publications, London.

1989:459-78.

52. Dang Van K, Maitournam MH. Rolling contact in railways: modelling, simulation

and damage prediction. Fatigue & Fracture of Engineering Materials & Structures.

2003;26(10):939-48.

53. Dang Van K, Maitournam MH. On some recent trends in modelling of contact

fatigue and wear in rail. Wear. 2002;253(1–2):219-27.

References

207

54. Palmgren A. Die lebensdauer von kugella gern. Zeitschrift des Vereins Deutscher

Ingenieure, 1924, 68.14: 339-341.

55. Pombo J, Ambrósio JC. Application of a wheel–rail contact model to railway

dynamics in small radius curved tracks. Multibody Syst Dyn. 2008

2008/02/01;19(1-2):91-114.

56. Guidelines to Best Practices for Heavy Haul Railway Operations: Wheel and Rail

Interface Issues, Library of Congress Control No.: 2001-131901: International

Heavy Haul Association; 2001.

57. Hertz H. Uber die Bertihrung fester elastischer Korper. J. fur die reine u. angew.

Math., 1882, 92.

58. Johnson. KL. Contatct Mechanics: Cambridge University Press; 1985.

59. Dintwa E, Tijskens E, Ramon H. On the accuracy of the Hertz model to describe

the normal contact of soft elastic spheres. Granular Matter. 2008

2008/03/01;10(3):209-21.

60. Iwnicki S. Handbook of railway vehicle dynamics. Taylor and Francis. 2006.

61. Shabana AA, Zaazaa KE, Sugiyama H. Railroad vehicle dynamics a computaional

approach. CRC Press, Taylor and Francis Group, 13-978-1-4200-4581-9. 2007.

62. Ekberg A, Bjarnehed H, Lundbéan R. A fatigue model for general rolling contact

with application to wheel/rail damage. Fatigue & Fracture of Engineering

Materials & Structures. 1995;18(10):1189-99.

63. Zerbst U, Lundén R, Edel KO, Smith RA. Introduction to the damage tolerance

behaviour of railway rails - a review. Engineering Fracture Mechanics.

2009;76(17):2563-601.

64. Evans J, and Iwnicki S. Vehicle dynamics and the wheel/rail interface. In: Wheels

on Rails – An Update, Understanding and Managing the Wheel/Rail Interface.

Proceedings of the IMechE Seminar, London. . 2002.

65. Yan W, Fisher FD. Applicability of Hertz contact theory to rail-wheel contact

problems. Archives of Applied Mechanics. 2000;Vol. 70:255-68.

References

208

66. J. Piotrowski & H. Chollet (2005): Wheel–rail contact models for vehicle system

dynamics including multi-point contact, Vehicle System Dynamics: International

Journal of Vehicle Mechanics and Mobility, 43:6-7, 455-483.

67. Marshall MB, Lewis R, Dwyer-Joyce RS, Olofsson U, Bjorklund S. Experimental

Characterization of Wheel-Rail Contact Patch Evolution. Journal of Tribology.

2006;128(3):493-504.

68. Ringsberg JW, Loo-Morrey M, Josefson BL, Kapoor A, Beynon JH. Prediction of

fatigue crack initiation for rolling contact fatigue. International Journal of Fatigue.

[doi: 10.1016/S0142-1123(99)00125-5]. 2000;22(3):205-15.

69. Mandal NK, Dhanasekar M, Boyd P. Elasto-plastic stress analysis of an insulated

rail joint (IRJ) with a loading below shakedown limit, 8th International

Conference on Contact Mechanics and Wear of Rail/Wheel Systems Comitato

(CM2009), Firenze, Italy, September 15-18, 2009, ISBN. 978-88-904370-0-7

2009.

70. Ringsberg JW, Franklin FJ, Josefson BL, Kapoor A, Nielsen JCO. Fatigue

evaluation of surface coated railway rails using shakedown theory, finite element

calculations, and lab and field trials. International Journal of Fatigue.

2005;27(6):680-94.

71. Ringsberg JW, Skyttebol A, Josefson BL. Investigation of the rolling contact

fatigue resistance of laser cladded twin-disc specimens: FE simulation of laser

cladding, grinding and a twin-disc test. International Journal of Fatigue.

2005;27(6):702-14.

72. Kapoor A. A re-evaluation of the life to rupture of ductile metals by cyclic plastic

strain. Fatigue & Fracture of Engineering Materials & Structures. 1994;17(2):201-

19.

73. Tyfour WR, Beynon JH, Kapoor A. Deterioration of rolling contact fatigue life of

pearlitic rail steel due to dry-wet rolling-sliding line contact. Wear. 1996;197(1–

2):255-65.

References

209

74. Johnson KL, editor. A graphical approach to shakedown in rolling contact, Proc.

of the conference on Applied Stress Analysis, T. H. Hyde and E. Ollerton (eds),

Nottingham, 263-2741990.

75. Johnson KL. Contact mechanics and the wear of metals. Wear. 1995;190(2):162-

70.

76. Bower AF, Johnson KL. Plastic flow and shakedown of the rail surface in

repeated wheel-rail contact. Wear. 1991;144(1–2):1-18.

77. Bower AF, Johnson KL. The influence of strain hardening on cumulative plastic

deformation in rolling and sliding line contact. Journal of the Mechanics and

Physics of Solids. 1989;37(4):471-93.

78. Franklin F, Kapoor A. Modelling wear and crack initiation in rails. Proceedings of

the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit.

2007;221(1):23-33.

79. Welsby DR, Mutton PJ. Influence of rail material grade on wear and rolling

contact fatigue behaviour, Rail Wheel Interface, Track and Signal,Institute of

Railway Technology, Monash university, available online via

http://www.eng.monash.edu.au/railway/pdf/track-signal-volume.pdf2013.

80. AS1085.1. Railway track materials Part 1: Steel rails, Council of Standards

Australia, Standards Australia2002.

81. Manual for Railway Engineering, Rail, American Railway Engineering and

Maintenance-of-Way Association (AREMA); 2008, Chapter 4.

82. Alwahdi F, Franklin FJ, Kapoor A. The effect of partial slip on the wear rate of

rails. Wear. [doi: 10.1016/j.wear.2004.03.052]. 2005;258(7–8):1031-7.

83. Jun HK, You WH. Estimation of Crack Growth Life in Rail with a Squat Defect.

2010 Trans Tech Publications. 2010;417-418(2010):313-6.

84. Clayton P. Predicting the wear of rails on curves from laboratory data. Wear.

1995;181-183(Part 1):11-9.

http://www.eng.monash.edu.au/railway/pdf/track-signal-volume.pdf2013

References

210

85. Asih AMS, Ding K, Kapoor A. Modelling rail wear transition and mechanism due

to frictional heating. Wear. [doi: 10.1016/j.wear.2012.02.017]. (0).

86. Lim SC, Ashby MF. Overview no. 55 Wear-Mechanism maps. Acta Metallurgica.

1987;35(1):1-24.

87. Williams JA, Williams JA. Engineering tribology. New York: New York :

Cambridge University Press; 2005.

88. Fletcher DI, Franklin FJ, Kapoor A. Image analysis to reveal crack development

using a computer simulation of wear and rolling contact fatigue. Fatigue &

Fracture of Engineering Materials & Structures. 2003;26(10):957-67.

89. Cannon DF, Edel KO, Grassie SL, Sawley K. Rail defects: an overview. Fatigue

& Fracture of Engineering Materials & Structures. 2003;26(10):865-86.

90. Zerbst U, Beretta S. Failure and damage tolerance aspects of railway components.

Engineering Failure Analysis. [doi: 10.1016/j.engfailanal.2010.06.001].

2011;18(2):534-42.

91. Datsyshyn OP, Panasyuk VV. Pitting of the rolling bodies contact surface. Wear.

2001;251(1–12):1347-55.

92. Clayton P. Tribological aspects of wheel-rail contact: a review of recent

experimental research. Wear. 1996;191(1–2):170-83.

93. Sugino K, Kageyama H, Kuroki T, Urashima C, Kikuchi A. Metallurgical

investigation of transverse defects in worn rails in service. Wear. 1996;191(1–

2):141-8.

94. Farris TN. Effect of overlapping wheel passages on residual stress in rail corners.

Wear. 1996;191(1–2):226-36.

95. Ringsberg JW, Bergkvist A. On propagation of short rolling contact fatigue

cracks. Fatigue & Fracture of Engineering Materials & Structures.

2003;26(10):969-83.

References

211

96. Fletcher DI, Beynon JH. The effect of contact load reduction on the fatigue life of

pearlitic rail steel in lubricated rolling–sliding contact. Fatigue & Fracture of

Engineering Materials & Structures. 2000;23(8):639-50.

97. Sawley K, Kristan J. Development of bainitic rail steels with potential resistance

to rolling contact fatigue. Fatigue & Fracture of Engineering Materials &

Structures. 2003;26(10):1019-29.

98. Ishida M, Akama M, Kashiwaya K, Kapoor A. The current status of theory and

practice on rail integrity in Japanese railways—rolling contact fatigue and

corrugations. Fatigue & Fracture of Engineering Materials & Structures.

2003;26(10):909-19.

99. Dang Van K. Modelling of damage induced by contacts between solids. Comptes

Rendus Mécanique. 2008;336(1–2):91-101.

100. Dang Van K. Introduction to fatigue analysis in mechanical design by the

multiscale approach. High-Cycle Metal Fatigue, from theory to applications.

1999:57-88.

101. Dang Van K. Introduction to fatigue analysis in mechanical design by the multi-

scale approach, in: K. Dang Van, I. Papadoupoulos (Eds.), High-Cycle Metal

Fatigue in the Context of Mechanical Design, CISM Courses and Lectures no.

392, Springer, Berlin, 1999, pp. 169–187.

102. Dang Van K. Macro–micro approach in high cycle multi-axial fatigue, A.S.T.M.

S.T.P., 1191 (1993), pp. 120–130.

103. Ekberg A. Rolling contact fatigue of railway wheels—a parametric study. Wear.

[doi: 10.1016/S0043-1648(97)00106-3]. 1997;211(2):280-8.

104. Jiang Y, Sehitoglu H. A model for rolling contact failure. Wear. 1999;224(1):38-

49.

105. Conrado E, Gorla C. Contact fatigue limits of gears, railway wheels and rails

determined by means of multiaxial fatigue criteria. Procedia Engineering.

2011;10(0):965-70.

References

212

106. Ciavarella M, Monno F. A comparison of multiaxial fatigue criteria as applied to

rolling contact fatigue. Tribology International. [doi: DOI:

10.1016/j.triboint.2010.06.003]. 2010;43(11):2139-44.

107. Ekberg A, Kabo E, Andersson H. Predicting rolling contact fatigue of railway

wheels: Proceedings of the 13th International Wheelset Congress, Rome.; 2001.

108. Ekberg A, Kabo E, Andersson H. An engineering model for prediction of rolling

contact fatigue of railway wheels. Fatigue & Fracture of Engineering Materials &

Structures. 2002;25(10):899-909.

109. Ekberg A, Bjarnehed H, Järnvägsmekanik KC. Rolling Contact Fatigue of

Wheel/rail Systems: A Literature Survey1995.

110. Meyers MA, Chawla KK. Mechanical behaviour of materials. Cambridge

University Press, Cambridge, ISBN:9780511455575 2009.

111. Sandström J, Ekberg A. Predicting crack growth and risks of rail breaks due to

wheel flat impacts in heavy haul operations Proceedings of the Institution of

Mechanical Engineers, Part F: Journal of Rail and Rapid Transit.

2009;223(2):153-61.

112. Sandström J, Ekberg A. Predicting crack growth and risks of rail breaks due to

wheel flat impacts in heavy haul operations. In proceedings of the conference of

International Heavy Haul associations (IHHA 2007); Specialist Techanical

Session (STS) Kiruna, Sweden. 2009(2).

113. Dowling NE. Mechanical behaviour of materials. 1st ed. Prentice Hall Englewood

Cliffs, New Jersey 07632, USA: 1993.

114. Ringsberg JW. Life prediction of rolling contact fatigue crack initiation.

International Journal of Fatigue. 2001;23(7):575-86.

115. Wong SL, Bold PE, Brown MW, Allen RJ. A branch criterion for shallow angled

rolling contact fatigue cracks in rails. Wear. 1996;191(1–2):45-53.

116. Fischer FD, Daves W, Pippan R, Pointner P. Some comments on surface cracks in

rails. Fatigue & Fracture of Engineering Materials & Structures. 2006;29(11):938-

48.

References

213

117. Fletcher D, Hyde P, Kapoor A. Investigating fluid penetration of rolling contact

fatigue cracks in rails using a newly developed full-scale test facility. Proceedings

of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid

Transit. 2007;221(1):35-44.

118. Fletcher DI, Hyde P, Kapoor A. Modelling and full-scale trials to investigate fluid

pressurisation of rolling contact fatigue cracks. Wear. 2008;265(9–10):1317-24.

119. Fletcher DI, Smith L, Kapoor A. Rail rolling contact fatigue dependence on

friction, predicted using fracture mechanics with a three-dimensional boundary

element model. Engineering Fracture Mechanics. [doi: DOI:

10.1016/j.engfracmech.2009.02.019]. 2009;76(17):2612-25.

120. Liu Y, Liu L, Mahadevan S. Analysis of subsurface crack propagation under

rolling contact loading in railroad wheels using FEM. Engineering Fracture

Mechanics. 2007;74(17):2659-74.

121. Bogdański S, Lewicki P. 3D model of liquid entrapment mechanism for rolling

contact fatigue cracks in rails. Wear. 2008;265(9–10):1356-62.

122. Farjoo M, Daniel W, Meehan PA. Modelling a squat form crack on a rail laid on

an elastic foundation. Engineering Fracture Mechanics. 2012;85(0):47-58.

123. Farjoo M, Pal S, Daniel W, Meehan PA. Stress intensity factors around a 3D squat

form crack and prediction of crack growth direction considering water entrapment

and elastic foundation. Engineering Fracture Mechanics. 2012;94(0):37-55.

124. Zerbst U, Schödel M, Heyder R. Damage tolerance investigations on rails.

Engineering Fracture Mechanics. [doi: DOI: 10.1016/j.engfracmech.2008.04.001].

2009;76(17):2637-53.

125. Ringsberg JW. Shear mode growth of short surface-breaking RCF cracks. Wear.

2005;258(7–8):955-63.

126. Jeong DY, Tang YH, Orringer O. Estimation of rail head wear limits based on rail

strength investigations, Office of research and development Washington, DC

20590, DOT/FRA/ORD-98/07, DOT-VNTSC-FRA-98-13, Final report,

December. 1998.

References

214

127. EN13674-1 Euronorm standard , - Track - Rail - Part 1: Vignole railway rails 46

kg/m and above, CEN. Railway applications. European Committee for

Standardization; 2011, .

128. Alwahdi, F., Wear and Rolling Contact Fatigue of Ductile Material, Doctoral

Thesis, Sheffield. University, Sheffield, UK 2004.

129. Alwahdi FAM, Kapoor A, Franklin FJ. Preliminary investigation of the effect of

roughness in Dynarat simulation. Wear. 2009;267(9–10):1381-5.

130. Ueda M, Iwano K, Yamamoto T. Rail Performance and Recent Developments of

Rail 10th International Heavy Haul Railway Conference; Calgary, CanadaJune

2011.

131. Deroche RY, Bourdon Y, Faessel A, Waeckerle R, Lieurade HP, Maillard-Salin

C, et al. Stress releasing and straightening of rail by stretching. Paper no 82-HH-

17, Proceedings of the secong International Heavy Haul Railway Conference pp

158-168; Colorado, Springs, CO, USA,1982.

132. Guidelines to Best Practices for Heavy Haul Railway Operations: Infrastructure

Construction and Maintenance Issues, ISBN 978-1-930566-74-3: International

Heavy Haul Association; 2009.

133. O'Rourke MD, Mair RI, Doyle NF. Towards the design of rail track for heavy

axle loads. Heavy Haul Railways Conference, Perth, Western Australia, 18th to

22nd September, 1978: conference papers; Perth: Institution of Engineers,

Australia, W.A. Division; 1978.

134. Xiao X, Jin X, Deng Y, Zhou Z. Effect of curved track support failure on vehicle

derailment. Vehicle System Dynamics: International Journal of Vehicle

Mechanics and Mobility. 2008;46(11):1029 - 59.

135. Abaqus user's manual - version 6.11-2 (2011).

136. Magiera J. Enhanced 3D analysis of residual stress in rails by physically based fit

to neutron diffraction data. Wear. 2002;253(1–2):228-40.

References

215

137. Greisen C. Estimation of rail bending stress from real-time vertical track

deflection measurement, Proceedings of the ASME/IEEE Joint Rail Conference

March 3-5, 2009, JRC2009, Pueblo, Colorado, USA2009.

138. Sheng Lu. "Real-time vertical track deflection measurement system" ETD

collection for University of Nebraska - Lincoln. Paper AAI3331436.

http://digitalcommons.unl.edu/dissertations/AAI3331436 2008.

139. Mutton PJ, Epp KJ, Alvarez E, Lynch M. A review of wheel-rail interaction and

component performance under high axle load conditions. In proceedings of the

6th International Conference on Contact Mechanics and Wear of Rail/Wheel

systems (CM 2003); Goteburg, Sweden2003.

140. Welsby D, Mutton PJ. Confidential report. Institute of Railway Technology,

Monash University. 2008.

141. Ekberg A. Rolling contact fatigue of railway wheels - Computer modelling and

in-field data. Chalmers University of Technology, Goteborg, Sweden. 1996.

142. Lee YL, Pan J, Hathaway R, Barkey M. Fatigue testing and analysis: theory and

practice. Massacheusetts: Butterworth-Heinemann; 2004.

143. Ekberg A. Rolling contact fatigue of railway wheels. Chalmers University of

Technology, Goteborg, Sweden. 2000.

144. Schleinzer G, Fischer FD. Residual stress formation during the roller straightening

of railway rails. International Journal of Mechanical Sciences. 2001;43(10):2281-

95.

145. Sasaki T, Takahashi S, Kanematsu Y, Satoh Y, Iwafuchi K, Ishida M, et al.

Measurement of residual stresses in rails by neutron diffraction. Wear.

2008;265(9–10):1402-7.

146. Liu Y, Stratman B, Mahadevan S. Fatigue crack initiation life prediction of

railroad wheels. International Journal of Fatigue. [doi: DOI:

10.1016/j.ijfatigue.2005.09.007]. 2006;28(7):747-56.

http://digitalcommons.unl.edu/dissertations/AAI3331436

References

216

147. Jacek M. Enhanced 3D analysis of residual stress in rails by physically based fit to

neutron diffraction data. Wear. [doi: 10.1016/S0043-1648(02)00106-0].

2002;253(1-2):228-40.

148. Mutton PJ. Confidential report. Institute of Railway Technology Monash

University2007.

149. Josefson BL, Ringsberg JW. Assessment of uncertainties in life prediction of

fatigue crack initiation and propagation in welded rails. International Journal of

Fatigue. [doi: 10.1016/j.ijfatigue.2009.03.024]. 2009;31(8–9):1413-21.

150. Ringsberg JW, Josefson BL. Finite element analyses of rolling contact fatigue

crack initiation in railheads. Proceedings of the Institution of Mechanical

Engineers -- Part F -- Journal of Rail & Rapid Transit. [Article]. 2001;215(4):243-

59.

151. Skyttebol A, Josefson BL, Ringsberg JW. Fatigue crack growth in a welded rail

under the influence of residual stresses. Engineering Fracture Mechanics. [doi:

10.1016/j.engfracmech.2004.04.009]. 2005;72(2):271-85.

152. Nielsen JCO, Ringsberg, J.W. Baeza, L. . Influence of railway wheel flat impact

on crack growth in rails. Proceedings of the 8th International Heavy Haul

Conference; Rio de Janeiro, Brazil2005. p. 789-97.

153. Bogdanski S. A rolling contact fatigue crack driven by squeeze fluid film. Fatigue

& Fracture of Engineering Materials & Structures. 2002;25(11):1061-71.

154. Bogdański S, Stupnicki J, Brown MW, Cannon DF. A two dimensional analysis

of mixed-mode rolling contact fatigue crack growth in rails. In: E. Macha WB,

Łagoda T, editors. European Structural Integrity Society: Elsevier; 1999. p. 235-

48.

155. Bogdański S, Brown MW. Modelling the three-dimensional behaviour of shallow

rolling contact fatigue cracks in rails. Wear. 2002;253(1–2):17-25.

156. Ringsberg JW, Lindbäck T. Rolling contact fatigue analysis of rails inculding

numerical simulations of the rail manufacturing process and repeated wheel-rail

contact loads. International Journal of Fatigue. 2003;25(6):547-58.

References

217

157. Goldberg M. Boundary Integral Methods - Numerical and Mathematical Aspects:

Computational Engineering, WIT Press; 1998.

158. Mellings S, Baynham J, Adey RA. Automatic crack growth prediction in rails

with BEM. Engineering Fracture Mechanics. 2005;72(2):309-18.

159. Akama M, Mori T. Boundary element analysis of surface initiated rolling contact

fatigue cracks in wheel/rail contact systems. Wear. 2002;253(1–2):35-41.

160. Fletcher D, Beynon J. A simple method of stress intensity factor calculation for

inclined fluid-filled surface-breaking cracks under contact loading. Proceedings of

the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology.

1999;213(4):299-304.

161. Rooke DP, Rayaprolu DB, Aliabadi MH. Crack-line and edge Green's functions

for stress intensity factors of inclines edge cracks. Fatigue & Fracture of

Engineering Materials & Structures. 1992;15(5):441-61.

162. Nowell D, Hills DA. Open cracks at or near free edges. The Journal of Strain

Analysis for Engineering Design. 1987 July 1, 1987;22(3):177-85.

163. Keer LM, Bryant MD. A pitting model for rolling contact fatigue. ASME J of

Lubriaction Technology. 1983 1983;105(2):198-205.

164. Bower AF. The influence of crack face friction and trapped fluid on surface

initiated rolling contact fatigue cracks. ASME Journal of Tribology. 1988 March

02, 1988; Online October 29, 2009;110(4):704-11.

165. Kaneta M, Yatsuzuka H, Murakami Y. Mechanism of Crack Growth in

Lubricated Rolling/Sliding Contact. A S L E Transactions. 1985

1985/01/01;28(3):407-14.

166. Kaneta M, Murakami Y. Propagation of semi-elliptical surface cracks in

lubricated rolling/sliding elliptical contacts. ASME Journal of Tribology. 1991

February 26, 1990; Revised June 19, 1990; Online June 05, 2008 113(2):270-5.

167. Kaneta M, Suetsugu M, Murakami Y. Mechanism of surface crack growth in

lubricated rolling/sliding spherical contact. ASME Journal of Applied Mechanics,

References

218

1986 February 07, 1985; Revised August 02, 1985; Online July 21, 2009

53(2):354-60.

168. Cruse T. Boundary element analysis in computational fracture mechanics.

Kluwer-Dordrecht1988.

169. Guinea GV, Planas J, Elices M. KI evaluation by the displacement extrapolation

technique. Engineering Fracture Mechanics. 2000;66(3):243-55.

170. Rice JR. Mathematical analysis in the mechanics of fracture. Fracture: an

advanced treatise (Vol 2, Mathematical Fundamentals) (ed H Liebowitz),

Academic Press, NY. 1968;2:191-311.

171. Krueger R. Virtual crack closure technique: History, approach, and applications.

Appl Mech Rev. 2004;57(2).

172. Smith SA, Raju IS. Evaluation of stress-intensity factors using general finite-

element models. Fatigue and Fracture Mechanics (Twenty-Ninth Volume) ASTM

STP. 1998;1332:176-200.

173. Moës N, Dolbow J, Belytschko T. A finite element method for crack growth

without remeshing. International Journal for Numerical Methods in Engineering.

1999;46(1):131-50.

174. Carter BJ, Wawrzynek PA, Ingraffea AR. Automated 3-D crack growth

simulation. International Journal for Numerical Methods in Engineering.

2000;47(1-3):229-53.

175. Maligno AR, Rajaratnam S, Leen SB, Williams EJ. A three-dimensional (3D)

numerical study of fatigue crack growth using remeshing techniques. Engineering

Fracture Mechanics. 2010;77(1):94-111.

176. Henshell RD, Shaw KG. Crack tip finite elements are unnecessary. International

Journal for Numerical Methods in Engineering. 1975;9(3):495-507.

177. Shi J, Chopp D, Lua J, Sukumar N, Belytschko T. Abaqus implementation of

extended finite element method using a level set representation for three-

dimensional fatigue crack growth and life predictions. Engineering Fracture

Mechanics. 2010;77(14):2840-63.

References

219

178. Ekberg A, Kabo E. Fatigue of railway wheels and rails under rolling contact and

thermal loading--an overview. Wear. 2005;258(7-8):1288-300.

179. Marich, S, Development of improved rail and wheel materials [database on the

Internet]2011. Available from: http://vanitec.org/wp-

content/uploads/2011/09/Development-of-Improved-Rail-and-Wheel-

Materials.pdf.

180. Tillberg J, Larsson F, Runesson K. A study of multiple crack interaction at rolling

contact fatigue loading of rails. Proceedings of the Institution of Mechanical

Engineers, Part F: Journal of Rail and Rapid Transit. 2009 July 1,

2009;223(4):319-30.

181. Guo J, Li W, Wen ZF, Jin XS. Mutual Interaction of Two Rail Surface Cracks

under Thermo-Mechanical Contact Loading. Advanced Materials Research.

2010;97:543-6.

182. Lam KY, Phua SP. Multiple crack interaction and its effect on stress intensity

factor. Engineering Fracture Mechanics. 1991;40(3):585-92.

183. Kuo CH, Keer LM, Bujold MP. Effects of Multiple Cracking on Crack Growth

and Coalescence in Contact Fatigue. Journal of Tribology. 1997;119(3):385-90.

184. Wang YZ, Atkinson JD, Akid R, Parkins RN. Crack interaction, coalescene and

mixed mode fracture mechanics. Fatigue & Fracture of Engineering Materials &

Structures. 1996;19(4):427-39.

185. Rooke DP, Cartwright DJ. (Eds) Compendium of Stress Intensity Factors,

(Procurement Executive, Ministry of Defence, Her Majesty‘s Stationery Office,

London).1976.

186. Erdogan F, Sih G. On the Crack Extension in Plates Under Plane Loading and

Transverse Shear. Journal of Basic Engineering. 1963;85:519.

187. Schöllmann M, Kullmer G, Fulland M, Richard H, editors. A new criterion for 3D

crack growth under mixed-mode (I+ II+ III) loading. Proceedings of the 6th

International Conference on Biaxial/Multiaxial Fatigue and Fracture; 2001.

http://vanitec.org/wp-content/uploads/2011/09/Development-of-Improved-Rail-and-Wheel-Materials.pdf



References

220

188. Hussain M, Pu S, underwood J. Strain energy release rate for a crack under

combined mode I and mode II. ASTM, STP. 1974;560:2-28.

189. Richard HA, Fulland M, Sander M. Theoretical crack path prediction. Fatigue &

Fracture of Engineering Materials & Structures. 2005;28(1-2):3-12.

190. Richard HA, Sander M, Fulland M, Kullmer G. Development of fatigue crack

growth in real structures. Engineering Fracture Mechanics. 2008;75(3–4):331-40.

191. Richard HA, Buchholz FG, Kullmer G, Schöllmann M. 2D- and 3D-Mixed Mode

Fracture Criteria. Key Engineering Materials. 2003;251-252(1):251-60.

192. Schöllmann M, Fulland M, Richard HA. Development of a new software for

adaptive crack growth simulations in 3D structures. Engineering Fracture

Mechanics. 2003;70(2):249-68.

193. Kerr A, Wilson A, Marich S. The Epidemiology of Squats and Related Defects

[online] In: Conference on Railway Engineering (2008 : Perth, WA) CORE 2008:

Rail; The Core of Integrated Transport Perth: Railway Technical Society of

Australasia: Engineers Australia, 2008: 83-96 Availability:

<http://searchinformitcomau/documentSummary;dn=564088292893778;res=IEL

ENG> ISBN: 0858257831 [cited 08 Apr 13].

194. Cook RD, Young WC. Advanced mechanics of materials. N.J.: 2nd ed. Upper

Saddle River, Prentice Hall; 1999.

195. Hay WW, William WH. Railroad engineering. New York: New York : Wiley;

1982.

http://searchinformitcomau/documentSummary;dn=564088292893778;res=IELENG

http://searchinformitcomau/documentSummary;dn=564088292893778;res=IELENG

Appendix A. Equations for rail bending stresses

221

Appendices


The most simple track model to estimate the bending moment and the deflection of a

loaded beam laid on an elastic foundation was presented by Winkler in 1867, who

derived the governing equations as has been reported in [60, 194, 195]. Another

approach based on Timoshenko and Langer beam theory analysis is considered as

acceptable railroad engineering practice for the calculation of bending stresses [15, 46,

83]. In order to calculate longitudinal bending stress components, dimensions of a

generic rail section with a reference point (stress point) set near the lower gauge corner

(underhead radius) has been reported by Jeong et al [7] as shown in Figure A.1.

A.1 Vertical and lateral bending

Jeong et al [7] and Jeong [15] reported the analytical solution for the longitudinal

bending stresses due to vertical and lateral bending as given in Equations A.1 and A.2

respectively.

( ) ( )

(A.1)

( ) ( )

(A.2)

Where

(A.3)

( ) ∑

| | , ( ) | |- (A.4)


222

( ) ∑

| | , ( ) | |- (A.5)

(

)

(A.6)

.

/

(A.7)

Where is distance from the neutral axis of entire rail to the stress point as shown in

Figure A.1. and are the vertical and lateral bending inertia of the entire rail,

and are the vertical and lateral foundation modulus respectively as shown in Figure

1.7 and E is the modulus of elasticity for rail steel. is the wheel point and ( ) is

the distance of stress measuring point from wheel point.

Figure A.9.1 Dimension of generic rail section with a reference point set near the lower gauge corner (underhead radius position) [7]


223

A.2 Head-on-web bending

Jeong et al [7] and Jeong [15] reported that the rail head can be assumed as a separate

beam bending on elastic foundation formed by web as shown in Figure A.1. The

corresponding analytical solutions for the longitudinal bending stresses due to vertical

and lateral head-on-web-bending are given below.

( )

( )

(A.8)

( )

( )

(A.9)

Where

(A.10)

( ) ∑

| | , ( ) | |- (A.11)

( ) ∑

| | , ( ) | |- (A.12)

(

)

(A.13)

.

/

(A.14)

(A.15)

.

/

(A.16)

Where is the thickness of web and is height of web. is distance from the

neutral axis of rail head to the stress point as shown in Figure A.1. and are the

vertical and lateral bending inertia of the rail head only.


224

A.3 Constrained warping

Jeong et al [7] and Jeong [15] reported that the eccentric vertical loading and lateral

wheel loading cause the rail to twist and the corresponding warping stresses are given as

(A.17)

Where

(A.18)

(A.19)

∑

, ( ) ( )- (A.20)

Figure A.9.2 Eccentric vertical loading and lateral loading of rail [7, 15]


225

Where ‗e‘ and ‗f‘ are the locations of vertical and lateral load with reference to shear

centre of rail as was shown in Figure A.2. The detailed description regarding these

parameters is reported by Orringer et al [47], Jeong et al [7] and Jeong [15].


Appendix B. Rail underhead radius stresses data

227

Appendix B. Rail underhead radius (UHR) stresses data

Figure B.1 Longitudinal stresses at UHR, 26 km, 41 km, 74 km-parent rail (gauge and field side) [32, 148]


228


-200

-150

-100

-50

0

50

100

150

0 100 200 300 400 500 600 700 800 900 1000

Stre

ss (M

Pa)

Wheel no

Longitudinal stresses at UHR, parent rail

26km gauge compression

26km gauge tension


229


-200

-150

-100

-50

0

50

100

150

0 100 200 300 400 500 600 700 800 900 1000

Stre

ss (M

Pa)

Wheel no


41km high gauge compression

41km high gauge tension


230


-200

-150

-100

-50

0

50

100

150

0 100 200 300 400 500 600 700 800 900 1000

Stre

ss (M

Pa)

Wheel no





231

Table B.1 Rail stress data - Longitudinal stresses at UHR, 26 km, 41 km, 74 km - parent rail (high gauge tension / compression) [32, 148]

Wheel pass


26km gauge tension





1 -60.89 10.37 -143.59 24.36 -69.26 21.73 2 -47.29 11.54 -96.52 24.95 -43.69 35.6 3 -43.41 9.21 -89.88 24.4 -63.14 9.33 4 -58.53 7.29 -116.17 21.26 -44.81 22.02 5 -46.49 8.84 -65.99 23.21 -53.34 46.42 6 -43.38 9.04 -105.62 25.19 -68.3 11.37 7 -39.3 13.12 -94.86 66.27 -78.15 23.95 8 -42.21 13.71 -57.9 45.31 -45.96 20.29 9 -32.89 41.67 -86.25 26.47 -65.01 7.84 10 -45.1 11.21 -66.67 48.01 -59.35 32.42 11 -34.03 11.99 -45.65 32.1 -52.68 13.76 12 -27.81 15.87 -101.6 28.43 -71.54 9.11 13 -35.38 12.77 -86.78 27.53 -61.71 40 14 -29.36 19.37 -85.58 29.48 -50.95 18.22 15 -38.87 14.52 -92.14 26.05 -73.51 11.42 16 -31.29 20.74 -92.12 29.55 -55.18 37.56 17 -36.34 13.37 -93.44 30.56 -45.59 9.94 18 -37.5 10.85 -74.17 27.08 -58.41 10.74 19 -34.78 20.17 -102.5 25.8 -48.4 61.49 20 -30.31 25.03 -83.04 27.55 -66.29 9.29 21 -45.45 14.93 -79.89 35.76 -44.79 57.11 22 -41.17 15.71 -89.78 28.39 -69.5 10.99 23 -38.26 9.7 -85.26 37.19 -38.08 73.18 24 -27.57 18.44 -94.18 26.52 -61.81 8.32 25 -42.52 13.2 -84.23 30.84 -45.18 60.43 26 -38.63 9.13 -79.73 25.61 -73.98 11.19 27 -43.29 9.52 -53.83 34.22 -61.07 38.5


232

Wheel pass


26km gauge tension





28 -31.83 10.11 -88.21 27.05 -59.47 12.02 29 -31.83 12.44 -73.79 72.77 -53.75 32.95 30 -27.94 11.08 -99.81 25.55 -58.58 10.76 31 -47.93 9.73 -61.27 44.47 -61.65 27.78 32 -36.67 10.7 -86.52 25.84 -62.2 12.6 33 -35.89 13.43 -91.54 86.71 -48.88 41.92 34 -40.35 9.35 -96.18 25.49 -59.75 11.74 35 -45.78 11.1 -75.33 87.56 -60.48 30.9 36 -37.24 10.72 -98.04 26.95 -58.69 12.61 37 -38.4 12.66 -81.68 46.99 -49.27 35.88 38 -30.24 16.55 -90.98 26.99 -65.4 11.54 39 -40.91 9.37 -90.16 35.6 -56.39 36.35 40 -29.84 11.32 -94.21 27.45 -59.47 12.02 41 -37.02 12.68 -91.85 76.48 -50.63 30.62 42 -41.87 10.16 -85.99 25.18 -58.97 11.93 43 -36.43 8.61 -119.57 28.16 -59.7 26.61 44 -28.66 16.77 -82.22 28.76 -61.03 12.61 45 -39.14 13.86 -84.52 51.54 -47.32 39.39 46 -47.1 11.73 -92.85 27.07 -58 11.58 47 -42.24 9.4 -62.28 49.29 -60.87 25.25 48 -29.62 18.14 -94.52 27.34 -60.25 12.22 49 -39.13 22.03 -82.83 90.94 -52.58 32.57 50 -46.7 10.58 -94.27 26.23 -60.14 11.58 51 -36.6 8.25 -80.22 88.69 -59.7 24.47 52 -34.65 11.56 -100.8 26.5 -59.27 13.78 53 -45.13 12.53 -90.66 51.62 -58.81 31.2 54 -37.36 19.13 -88.69 24.62 -58 11.39 55 -36.58 10.6 -82.42 86.69 -72.56 22.52 56 -29.01 19.73 -112.13 28.78 -51.09 26.46 57 -41.04 12.54 -76.14 65.56 -53.94 26.53 58 -41.04 9.83 -86.03 26.31 -57.99 12.91 59 -37.15 13.91 -91.81 50.67 -40.02 70.46 60 -33.46 15.47 -91.78 26.78 -61.22 10.86


233

Wheel pass


26km gauge tension





61 -42.97 14.89 -100.1 66.86 -47.12 58.49 62 -40.44 15.28 -98.52 27.81 -62.28 12.56 63 -37.91 12.37 -71.65 75.49 -42.74 59.74 64 -29.95 23.06 -103.5 27.13 -60.24 10.67 65 -41.98 12.38 -65.95 60.97 -37.37 60.44 66 -41.79 12.19 -92.56 25.23 -69.68 12.36 67 -34.4 15.69 -67.63 34.81 -43.33 60.91 68 -32.46 19.19 -101.62 28.22 -62.19 11.06 69 -45.07 12.98 -66.01 94.54 -44.39 56.54 70 -35.56 10.65 -93 26.72 -64.81 12.95 71 -35.55 16.48 -72.93 66.63 -48.59 67.15 72 -38.27 11.83 -104.78 27.01 -56.93 10.67 73 -38.26 13.77 -84.73 90.99 -47.5 52.06 74 -38.65 9.11 -95.39 24.92 -64.22 12.56 75 -41.36 9.51 -59.58 69.1 -40.21 63.25 76 -33.4 11.26 -93.57 26.96 -64.52 11.64 77 -50.29 13.4 -102.09 47 -48.67 57.91 78 -42.32 11.07 -90.79 26.22 -65.39 13.15 79 -47.37 9.52 -70.52 70.98 -48.19 61.7 80 -39.21 10.69 -102.37 26.9 -56.14 10.48 81 -40.18 11.86 -78.43 82.12 -45.55 71.74 82 -37.65 10.69 -94.73 24.8 -69.87 12.96 83 -45.41 9.54 -69.41 69.57 -40.4 75.92 84 -38.03 15.95 -105.93 28.59 -57.51 9.9 85 -36.67 12.65 -84.51 67.88 -51.2 62.2 86 -35.5 15.56 -93.62 24.55 -66.94 12.57 87 -43.84 14.02 -67.33 67.17 -45.07 62.68 88 -32.19 23.53 -96.66 27.15 -56.92 10.68 89 -46.55 13.05 -85.35 76.95 -46.52 61.81 90 -36.26 22.76 -93.1 26.04 -64.6 12.77 91 -36.84 12.48 -65.64 68.09 -46.24 73.01 92 -36.45 16.75 -95.74 27.5 -58.86 10.68 93 -34.69 13.26 -79.77 79.62 -51.78 62.79


234

Wheel pass


26km gauge tension





94 -46.34 11.13 -101.51 25.6 -67.33 13.35 95 -41.29 9 -68.23 82.41 -52.66 50.21 96 -36.24 12.5 -103.58 28.03 -54.77 11.07 97 -49.24 14.25 -84.88 71.78 -48.07 59.09 98 -35.84 14.64 -96.32 25.54 -66.55 12.77 99 -35.25 9.98 -72.17 48.15 -46.81 68.14

100 -26.32 19.7 -102.08 28.16 -60.22 11.85 101 -45.34 13.29 -91.16 96.41 -59.37 51.49 102 -39.9 15.62 -101.63 25.48 -64.21 12.77 103 -41.06 10.58 -64.65 77.24 -49.73 66 104 -30.77 19.52 -96.5 26.94 -57.69 10.88 105 -47.27 13.69 -104.05 77.69 -51.18 52.86 106 -41.05 8.45 -93.91 25.42 -63.62 13.17 107 -41.44 12.93 -61.01 73.3 -44.28 61.72 108 -39.88 11.38 -98.11 26.69 -63.14 12.44 109 -37.16 14.1 -122.77 45.76 -52.15 62.02 110 -41.23 8.47 -96.3 27.11 -61.47 12.98 111 -39.87 11.78 -70.6 51.47 -44.27 67.18 112 -36.95 15.86 -104 27.02 -57.88 10.5 113 -41.61 12.56 -106.5 70.39 -55.46 62.22 114 -37.34 16.06 -93.25 26.47 -64.97 13.17 115 -35.39 19.36 -74.15 93.2 -54.79 61.33 116 -28.4 23.64 -107.37 27.54 -62.16 12.06 117 -46.45 13.54 -79.34 93.85 -49.42 59.89 118 -52.46 13.74 -101.08 25.83 -63.22 13.37 119 -41 9.67 -65.66 84.79 -48.16 59 120 -38.48 11.03 -102.37 27.87 -57.09 11.09 121 -42.94 12.98 -106.81 80.77 -53.12 59.69 122 -38.66 9.29 -97.84 25.38 -63.8 12.79 123 -41.18 10.85 -63.38 56.74 -43.68 66.99 124 -37.49 11.44 -93.29 26.45 -64.1 11.68 125 -41.37 13.19 -123.19 31.92 -50.19 57.75 126 -34.57 10.86 -96.34 26.69 -63.21 13.19


235

Wheel pass


26km gauge tension





127 -44.47 11.45 -65.58 44.24 -48.93 54.53 128 -33.98 12.04 -91.8 26.18 -58.25 10.71 129 -34.17 13.98 -102.65 63.15 -46.87 60.87 130 -45.24 11.85 -102.04 25.07 -63.79 12.8 131 -37.46 21.56 -68.94 62.45 -55.36 59.21 132 -47.95 12.83 -105.27 27.31 -59.02 11.3 133 -36.29 13.6 -100.76 91.86 -37.13 60.29 134 -41.34 11.08 -97.82 26.57 -66.9 13.39 135 -24.83 43.51 -74.06 66.87 -49.31 61.74 136 -43.46 10.51 -98.13 26.08 -62.14 11.69 137 -21.52 51.09 -129 38.94 -52.13 62.05 138 -50.64 10.13 -88.35 25.54 -67.48 13.78 139 -44.81 8.58 -86.75 76.72 -42.1 79.67 140 -31.41 14.02 -104.8 27.97 -62.52 11.31 141 -44.8 12.86 -80.66 100.3 -56.02 45.88 142 -38.2 9.56 -98.91 24.51 -63.19 12.62 143 -46.16 10.15 -74.37 75.69 -41.51 60.58 144 -47.32 12.1 -101.94 27.72 -61.74 11.31 145 -43.43 14.43 -99.38 86.25 -53.29 63.42 146 -35.86 9.58 -96.05 25.81 -62.99 12.62 147 -42.06 9.97 -69.18 53.47 -48.91 52.02 148 -45.17 11.91 -102.78 26.49 -61.73 10.73 149 -38.37 13.08 -78.45 61.5 -47.83 55.44 150 -38.56 10.76 -100.39 24.2 -65.13 12.43 151 -46.51 10.57 -64.76 56.33 -47.54 65.27 152 -31.76 23.77 -97.98 26.43 -58.81 10.35 153 -36.41 14.65 -85.5 101.87 -51.91 65.19 154 -39.71 9.41 -99.28 25.11 -67.46 13.02 155 -40.29 10.19 -68.32 36.25 -48.51 65.67 156 -41.06 10.98 -102.7 26.57 -57.05 10.94 157 -34.07 11.95 -85.56 65.08 -50.35 59.15 158 -30.57 17.78 -102.25 24.66 -65.31 12.25 159 -51.34 10.98 -71.88 64.77 -40.13 65.09


236

Wheel pass


26km gauge tension





160 -46.68 11.57 -105.48 27.29 -61.13 10.36 161 -35.6 13.52 -95.53 80.38 -44.11 62.86 162 -32.69 15.66 -96.48 24.99 -65.5 12.26 163 -44.33 9.83 -69.99 52.66 -42.65 77.17 164 -35.79 12.75 -103.6 26.26 -62.88 10.95 165 -35.01 13.53 -101.62 77.99 -53.45 59.16 166 -48.21 12.37 -93.82 24.16 -64.33 12.46 167 -35.19 15.68 -79.77 49.29 -44.99 52.43 168 -36.16 11.8 -113.96 27.37 -58.2 9.79 169 -42.95 12.57 -98.18 60.43 -54.81 50.4 170 -33.44 13.36 -93.88 25.27 -64.91 12.66 171 -45.86 10.45 -69.92 71.2 -46.34 62.57 172 -34.4 16.86 -109.35 26.53 -63.84 10.38 173 -37.31 15.31 -81.32 86.81 -53.25 59.56 174 -39.05 9.49 -97.63 23.07 -60.61 12.66 175 -48.37 10.85 -70.56 71.72 -52.77 58.87 176 -42.15 11.05 -98.91 25.5 -57.02 11.16 177 -36.32 12.6 -85.27 73.34 -52.85 65.61 178 -37.29 11.25 -98.46 23.01 -67.62 12.67 179 -47.77 10.29 -69.84 70.69 -50.03 73.69 180 -41.94 11.65 -110.83 26.22 -63.25 10.39 181 -34.75 13.78 -92.13 81.25 -53.43 45.55 182 -42.71 11.85 -99.69 22.95 -64.69 12.29 183 -45.62 10.3 -64.46 56.83 -41.84 68.24 184 -33.58 11.27 -100.97 25.38 -58.37 9.62 185 -42.7 12.83 -78.2 90.91 -36.28 71.08 186 -40.95 9.92 -97.8 22.12 -69.56 12.1 187 -43.27 9.54 -69.37 52.89 -66.97 40.58 188 -34.73 13.04 -114.05 26.69 -54.66 11.98 189 -39.19 12.26 -68.73 57.42 -61.6 58.62 190 -34.14 13.24 -101.75 24.2 -62.93 12.3 191 -48.5 11.11 -71.76 62.35 -49.62 60.66 192 -39.57 14.22 -104.2 26.04 -56.21 10.21


237

Wheel pass


26km gauge tension





193 -32.57 22.96 -84.34 94.1 -49.32 69.73 194 -40.92 8.2 -111.33 24.53 -66.82 11.72 195 -50.04 10.35 -69.49 57.63 -47.28 56.18 196 -36.06 17.14 -97.26 25.57 -61.07 10.03 197 -34.89 16.18 -95.09 88.01 -53.4 59.41 198 -37.8 9.58 -99.53 24.47 -63.88 11.54 199 -46.72 9.58 -76.55 79.15 -56.82 47.81 200 -38.18 11.53 -100.24 25.34 -54.64 10.04 201 -38.17 13.28 -70.85 83.87 -44.04 60.39 202 -34.87 12.89 -101.54 23.44 -68.36 11.93 203 -46.9 9.99 -66.89 56.35 -44.73 68.08 204 -36.8 12.13 -108.07 25.48 -56.38 9.46 205 -37.38 14.07 -74.99 102.86 -59.23 50.27 206 -48.83 11.74 -105.09 23.77 -69.32 11.94 207 -38.15 32.72 -71.03 83.11 -45.89 65.75 208 -46.49 11.95 -115.13 25.42 -56.57 8.88 209 -40.67 12.92 -90.21 78.7 -47.92 63.53 210 -38.92 10.01 -95.43 23.13 -71.26 11.76 211 -42.6 15.65 -71.67 61.87 -49.58 67.51 212 -40.85 11.96 -101.77 25.36 -59.29 10.06 213 -39.87 13.13 -93.77 88.36 -53.37 61.39 214 -38.71 8.67 -98.41 22.68 -69.31 12.15 215 -43.94 11 -73.09 61.42 -43.15 64.41 216 -31.13 16.25 -108.83 25.69 -57.33 8.7 217 -38.89 13.73 -98.69 74.5 -45.18 72.7 218 -38.69 9.26 -99.83 24.18 -66.77 11.77 219 -42.57 10.63 -76.84 57.28 -46.25 65.19 220 -41.01 11.6 -105.19 26.61 -64.34 9.69 221 -40.04 12.38 -95.63 94.85 -51.01 65.89 222 -30.13 17.24 -101.05 23.15 -66.95 11.59 223 -45.85 10.06 -72.04 51.97 -39.42 62.48 224 -39.06 12.01 -101.56 24.6 -61.99 8.92 225 -36.33 13.37 -92.19 63.89 -52.37 70.38


238

Wheel pass


26km gauge tension





226 -37.3 8.52 -98.19 24.06 -71.42 10.43 227 -46.03 10.08 -75.4 41.61 -60.46 27.61 228 -30.89 16.1 -108.22 25.32 -57.89 10.1 229 -40.98 13.77 -93.8 51 -75.74 24.41 230 -38.26 9.12 -100.19 24 -52.9 48.81 231 -39.03 17.47 -70.6 45.05 -45.06 67.37 232 -36.11 12.62 -101.67 24.29 -66.45 9.72 233 -36.89 13.79 -92.89 91.77 -41.43 59.49 234 -36.3 11.85 -100.06 23.56 -67.12 10.45 235 -41.54 9.91 -67.74 56.85 -48.75 69.33 236 -37.65 12.25 -102.51 24.63 -63.52 9.53 237 -36.29 25.65 -134.16 40.78 -53.31 60.87 238 -42.89 8.37 -86.12 26.22 -63.99 10.46 239 -46.18 10.32 -66.44 78.57 -38.02 69.92 240 -32.78 11.87 -105.48 25.15 -59.61 9.15 241 -38.02 13.23 -93.98 83.29 -65.96 46.46 242 -36.27 8.77 -93.76 22.47 -57.35 33.05 243 -34.91 22.37 -72.91 86.48 -47.36 69.15 244 -33.35 12.28 -110.59 26.65 -61.16 9.36 245 -43.05 13.25 -91.7 100.34 -53.09 58.74 246 -37.03 9.95 -104.9 23.97 -66.11 11.26 247 -46.93 10.93 -67.91 57.85 -35.27 63.32 248 -36.83 12.49 -103.85 24.84 -51.21 22.05 249 -26.73 42.78 -106.53 51.3 -41.39 72.59 250 -47.5 9.38 -107.29 24.3 -71.36 11.08 251 -42.84 8.61 -74.97 73.15 -58.44 59.82 252 -34.68 12.11 -104.68 25.95 -58.41 8.8 253 -42.44 11.92 -113.2 64.46 -52.49 55.07 254 -33.9 15.42 -98.98 25.21 -67.26 10.31 255 -45.93 9.79 -76.19 79.11 -55.51 51.46 256 -35.25 11.35 -104.16 25.7 -55.08 9.01 257 -40.1 12.52 -117.14 51.96 -54.42 58 258 -31.55 16.41 -101.76 24.77 -61.4 10.91


239

Wheel pass


26km gauge tension





259 -44.75 10.78 -72.17 53.01 -46.14 66.28 260 -33.48 12.53 -99.55 23.7 -61.89 9.21 261 -38.72 12.92 -98.93 94.08 -46.42 50.8 262 -39.11 8.85 -100.26 23.35 -69.18 11.11 263 -50.36 11.19 -68.14 59.95 -51.39 57.72 264 -34.63 15.85 -98.63 24.59 -58.56 9.63 265 -32.3 22.84 -90.24 72.06 -63.36 56.08 266 -39.48 8.47 -96.44 21.93 -60.2 10.54 267 -49.18 10.42 -73.64 82.05 -43.2 70.39 268 -42.97 12.17 -111.13 24.75 -65.96 8.85 269 -33.84 12.76 -73.58 87.55 -49.9 67.78 270 -33.84 15.87 -104.07 23.04 -66.03 9.58 271 -51.5 10.83 -72.73 82.58 -57.21 46.83 272 -39.65 11.8 -107.11 24.89 -61.27 24.86 273 -39.26 12.77 -93.85 63.59 -47.74 63.9 274 -32.27 15.3 -93.83 23.37 -63.88 9.4 275 -42.75 10.26 -66.57 28.87 -58.37 42.56 276 -37.69 12.59 -88.7 24.42 -57.94 10.65 277 -33.03 25.8 -75.06 88.41 -62.93 51.25 278 -38.85 12.21 -95.63 22.34 -61.92 10.19 279 -47.97 11.05 -70.12 76.83 -55.04 59.33 280 -38.07 12.22 -110.72 24.96 -56.95 9.28 281 -31.46 21.35 -83.86 99.05 -45.58 60.81 282 -36.9 8.34 -110.85 23.25 -70.28 9.82 283 -40.97 9.9 -72.32 73.85 -44.51 58.17 284 -34.36 12.62 -106.31 24.52 -63.56 8.12 285 -36.88 22.14 -94.22 45.15 -53.74 52.44 286 -43.68 9.33 -96.53 22.03 -63.64 10.03 287 -39.79 14.38 -76.65 80.21 -38.84 71.82 288 -33.57 12.06 -105 23.68 -62.19 7.16 289 -41.14 13.42 -81.06 89.41 -61.91 42.91 290 -36.67 10.7 -99.31 22.17 -62.27 10.23 291 -40.94 16.53 -70.49 62.46 -47.79 58.39


240

Wheel pass


26km gauge tension





292 -37.83 11.68 -101.95 23.43 -65.68 8.15 293 -34.72 22.36 -117.85 63.69 -46.51 72.34 294 -35.88 10.14 -81.87 23.28 -69.66 9.86 295 -44.42 11.31 -74.24 76.6 -47.78 60.75 296 -36.84 12.67 -101.04 22.99 -61.38 7.58 297 -31.21 25.87 -126.66 28.26 -50.98 54.63 298 -43.82 9.18 -76.87 22.64 -71.01 9.29 299 -38.38 17.92 -67.69 59.82 -49.32 64.66 300 -31.97 12.88 -101.68 22.54 -64.48 7.2 301 -43.42 12.69 -126.72 29.17 -55.25 54.64 302 -36.43 11.33 -82.37 23.16 -64.17 9.89 303 -38.17 13.86 -71.44 74.35 -59.05 40.51 304 -35.26 11.92 -102.51 23.85 -55.51 8.99 305 -28.46 31.34 -118.03 46.22 -50.17 58.16 306 -40.88 7.66 -82.23 23.69 -69.03 10.68 307 -37.19 11.35 -68.38 50.96 -46.76 61.37 308 -34.08 11.36 -100.04 24.37 -60.17 7.82 309 -42.42 11.75 -96.51 102.34 -47.42 63.83 310 -38.15 15.05 -100.37 22.66 -67.65 9.53 311 -29.79 22.43 -82.44 88.81 -41.1 70.55 312 -34.65 13.12 -93.1 22.95 -63.85 6.66 313 -29.21 22.63 -94.42 59.71 -56.96 62.48 314 -41.63 8.27 -104.7 22.02 -66.08 9.54 315 -37.16 9.63 -65.97 59.79 -41.28 68.81 316 -35.21 11.19 -96.07 23.09 -62.48 7.07 317 -34.63 11.97 -97.79 89.2 -72.14 40.48 318 -33.07 8.87 -95.43 22.35 -54.18 32.92 319 -31.32 15.67 -69.53 43.79 -40.09 67.85 320 -42.19 12.95 -89.33 22.45 -63.82 7.28 321 -46.65 14.31 -105.03 80.79 -53.03 57.64 322 -40.24 11.21 -88.68 22.49 -66.63 9.77 323 -32.08 9.67 -74.44 57.54 -53.52 56.56 324 -26.25 14.91 -100.27 23.17 -65.56 7.29


241

Wheel pass


26km gauge tension





325 -38.48 11.81 -97.51 94.14 -48.53 69.15 326 -40.81 13.75 -91.85 21.85 -67.59 9.59 327 -31.68 12.4 -62.45 44.46 -66.17 50.93 328 -28.37 12.98 -86.13 22.15 -54.05 17.46 329 -34.58 10.85 -92.12 86.31 -46.76 67.41 330 -34.77 17.07 -102.98 21.6 -67.96 10.19 331 -32.63 18.24 -65.03 67.15 -47.45 66.34 332 -35.54 11.64 -92.02 23.45 -63 6.94 333 -34.95 14.17 -90.04 92.09 -50.26 53.79 334 -33.01 17.28 -94.29 21.16 -67.75 9.03 335 -32.23 16.31 -66.25 82.25 -43.34 67.72 336 -33.39 15.73 -94.41 23 -65.91 6.95 337 -38.43 11.85 -96.71 82.9 -52.97 57.89 338 -33.19 18.07 -95.71 20.52 -68.91 9.24 339 -32.99 14.97 -66.89 62.95 -47.61 61.69 340 -36.09 12.25 -100.88 23.73 -66.47 6.38 341 -37.25 11.67 -78.1 95.28 -59.97 35.89 342 -40.55 8.18 -92.46 21.04 -64.02 9.26 343 -32.97 15.18 -80.94 44.62 -42.34 66.77 344 -30.25 19.84 -87.72 23.26 -68.41 7.37 345 -38.98 11.11 -80.88 62.96 -57.42 48.96 346 -38.98 10.53 -104.37 22.93 -68.1 8.89 347 -36.65 10.53 -66.23 57.01 -44.46 69.52 348 -30.04 20.63 -102.55 22.62 -62.55 7.39 349 -35.09 13.84 -104.84 68.93 -59.74 49.76 350 -36.25 19.47 -85.77 21.32 -62.62 9.29 351 -48.28 10.36 -79.11 84.16 -45.42 66.61 352 -38.38 10.74 -106.88 23.36 -66.23 6.23 353 -25.95 23.37 -84.88 80.53 -58.75 54.65 354 -33.52 16.38 -98.66 21.46 -69.23 8.92 355 -38.17 26.48 -73.92 63.5 -40.73 60.39 356 -44.38 11.34 -109.85 23.09 -65.05 6.64 357 -44.18 13.09 -81.63 88.45 -47.04 71.03


242

Wheel pass


26km gauge tension





358 -35.44 9.41 -85.3 18.1 -71.75 9.13 359 -57.37 12.52 -81.36 35.85 -50.84 49.3 360 -34.07 38.35 -86.78 21.31 -62.11 4.71 361 -33.1 12.33 -98.01 54.76 -60.47 54.88 362 -35.23 10.59 -83.21 23.68 -64.13 8.33 363 -42.41 12.15 -52.45 50.56 -50.63 59.65 364 -41.05 13.71 -90.33 21.45 -62.09 7.06 365 -38.13 14.87 -99.43 50.43 -54.61 60.54 366 -36.96 9.63 -87.74 19.54 -71.52 9.16 367 -37.15 17.6 -97.41 62.75 -51.2 62.98 368 -40.84 11 -85.72 21.56 -61.29 5.13 369 -32.1 21.88 -90.35 87.5 -53.03 54.72 370 -32.87 18.39 -101.02 20.07 -66.63 8.21 371 -37.53 15.09 -59.56 57.45 -45.73 73.13 372 -40.24 10.82 -84.03 22.11 -63.61 9.26 373 -36.55 11.8 -82.05 85.89 -58.86 54.15 374 -37.13 11.02 -86.88 19.24 -61.55 8.42 375 -50.71 12.2 -77.7 72.36 -45.51 55.8 376 -41.97 13.17 -87.2 21.47 -67.49 6.73 377 -34.98 14.34 -85.61 92.64 -53.77 60.99 378 -40.8 10.65 -78.97 18.98 -67.96 8.24 379 -36.52 10.85 -78.53 65.5 -58.36 59.52 380 -38.07 11.43 -99.5 22.58 -64.75 11.63 381 -35.35 12.8 -76.14 70.81 -64.28 24.57 382 -52.82 12.99 -89.52 20.09 -60.54 8.23 383 -44.08 11.06 -74.12 52.03 -65.94 25.05 384 -29.13 32.42 -93.72 23.5 -57.52 17.11 385 -39.8 11.65 -97.19 80.67 -49.26 66.67 386 -38.83 10.1 -83.36 19.45 -69.09 8.67 387 -38.05 15.15 -87.2 60.72 -66.31 20 388 -43.48 11.08 -101.36 22.86 -55.16 12.64 389 -38.23 12.05 -71.39 100.05 -67.75 24.21 390 -35.71 8.95 -97.61 32.04 -56.02 7.71


243

Wheel pass


26km gauge tension





391 -39.97 19.63 -82.2 67.27 -58.69 19.44 392 -39.19 11.87 -97.34 20.44 -54.95 8.16 393 -32.98 12.07 -117.71 64.61 -64.03 44.3 394 -33.36 9.55 -79.78 19.53 -57.56 8.32 395 -46.75 11.88 -84.59 47 -47.18 71.48 396 -35.1 27.61 -77.57 20 -66.82 7.6 397 -27.33 23.15 -102.02 82.44 -41.21 61.27 398 -33.35 16.94 -82.56 20.06 -61.05 7.56 399 -42.66 10.54 -89.31 75.13 -39.75 62.34 400 -32.17 17.34 -93.37 20.93 -72.64 6.84 401 -37.8 14.82 -98.39 74.22 -53.66 60.9 402 -39.74 6.86 -88.25 21.56 -57.13 8.35 403 -39.93 9.58 -75.18 58.94 -48.89 72.88 404 -27.5 21.04 -90.71 22.43 -77.69 8.41 405 -45.74 12.31 -94.56 97.1 -44.68 56.83 406 -36.42 8.43 -85.59 18.97 -55.75 6.81 407 -34.28 15.23 -71.94 54.8 -48.87 59.26 408 -27.29 16.79 -90.38 21.01 -62.67 8.06 409 -38.93 11.55 -96.56 86.93 -47.97 61.13 410 -36.41 16.4 -90.9 19.11 -63.52 8.2 411 -38.15 19.32 -69.66 28.69 -51.19 74.87 412 -37.76 16.22 -95.1 21.71 -62.84 5.14 413 -30.18 14.86 -79.32 95.04 -44.64 58.43 414 -33.48 8.07 -85.71 19.05 -63.11 8.61 415 -42.4 18.37 -70.11 41.46 -48.05 68.46 416 -42.98 10.6 -79.6 20.68 -66.72 5.94 417 -31.33 11.58 -82.87 102.75 -51.44 49.68 418 -31.91 15.46 -88.29 20.35 -65.24 8.04 419 -41.42 11 -82.02 103.8 -60.89 56.79 420 -43.55 14.5 -105.32 23.37 -59.88 7.34 421 -35.97 12.76 -84.49 95.5 -40.51 64.51 422 -30.34 21.11 -94.37 20.1 -73.01 6.89 423 -32.28 36.07 -92.77 43.88 -48.79 49.99


244

Wheel pass


26km gauge tension





424 -44.7 11.02 -90.61 21.76 -63.95 5.2 425 -32.08 17.43 -102.82 28.39 -75.37 43.68 426 -43.33 9.47 -91.32 24.13 -58.38 7.3 427 -31.87 12.01 -60.37 17.97 -47.8 67.35 428 -29.54 11.62 -84.05 19.17 -65.88 6.97 429 -39.63 12.59 -100.93 40.77 -45.34 68.06 430 -41.96 10.27 -84.77 21.54 -63.81 8.1 431 -33.61 11.83 -59.06 22.19 -80.71 10.66 432 -33.99 12.61 -78.28 19.31 -54.36 13.83 433 -31.07 18.82 -95.93 47.52 -56.82 32.81 434 -38.06 7.95 -81.33 19.35 -67.3 7.89 435 -34.37 18.06 -68.45 16.67 -67.63 17.51 436 -34.36 15.92 -90.58 19.83 -52.39 7.03 437 -37.85 10.3 -73.25 88.67 -55.04 23.67 438 -37.07 9.72 -92.27 19.29 -63.19 7.72 439 -45.81 11.08 -70.26 61.14 -57.48 24.16 440 -35.12 11.28 -104.44 22.5 -58.41 7.64 441 -34.54 11.86 -98.96 81.42 -56.38 20.19 442 -37.84 8.95 -81.64 18.26 -56.54 7 443 -44.43 9.74 -93.45 54.28 -41.87 62.56 444 -39.57 11.1 -83.51 21.64 -70.08 5.5 445 -35.69 11.88 -89.3 99.82 -60.45 28.59 446 -43.26 9.16 -94.13 19.17 -57.88 7.41 447 -41.7 9.37 -70.57 79.69 -48.08 35.11 448 -32.96 16.94 -85.12 20.44 -58.56 7.86 449 -38.2 12.87 -108.99 25.31 -57.7 49.65 450 -33.53 8.99 -72.03 32.15 -65.46 8.21 451 -40.52 9.19 -72.76 54.55 -40.66 65.34 452 -35.27 13.85 -103.06 21.74 -60.49 11.6 453 -41.09 12.69 -90.39 67.44 -41.12 67.21 454 -35.85 10.36 -90.17 18.48 -59.79 7.83 455 -40.7 10.95 -73.6 27.48 -43.56 62.43 456 -34.67 9.79 -94.18 22.07 -71.57 5.37


245

Wheel pass


26km gauge tension





457 -41.27 11.34 -99.19 24.61 -41.88 42.88 458 -39.13 10.38 -80.31 23.86 -54.5 7.26 459 -46.7 9.41 -59.27 41.22 -37.1 61.29 460 -34.85 19.9 -86.85 19.1 -71.94 4.62 461 -40.29 10.58 -87.59 96.48 -52.18 59.85 462 -38.54 9.81 -113.03 19.72 -64.22 6.52 463 -44.74 9.43 -63.8 65.07 -44.88 58.19 464 -32.9 15.64 -98.37 20.99 -63.74 3.08 465 -35.03 10.21 -103.97 59.3 -37.55 61.63 466 -35.8 9.05 -87.81 20.64 -62.45 6.16 467 -46.86 9.83 -63.47 36.25 -58.69 46.14 468 -30.55 17.6 -88.91 19.76 -64.49 5.64 469 -43.75 13.14 -72.54 87.43 -66.16 48.79 470 -38.31 9.84 -88.65 19.41 -50.54 34.03 471 -31.7 41.11 -75.19 51.74 -47.75 62.33 472 -42.57 12.57 -92.66 21.07 -71.68 6.05 473 -40.82 12.96 -89.12 99.81 -50.55 62.84 474 -41.79 11.41 -90.26 20.13 -59.67 7.37 475 -35.96 8.11 -78.55 37.69 -56.5 52.22 476 -28.19 13.55 -95.24 21.2 -65.22 14.08 477 -34.59 10.64 -64.49 66.52 -51.89 62.28 478 -33.81 8.9 -90.51 21.05 -57.7 6.03 479 -43.9 9.29 -80.36 42.68 -43.81 65.69 480 -33.22 23.28 -103.66 21.34 -67.34 18.78 481 -36.52 11.63 -100.12 52.85 -55.57 49.64 482 -34.57 9.11 -89.4 18.27 -54.36 17.34 483 -32.05 25.04 -83.14 66.92 -43.2 65.52 484 -38.64 12.23 -100.41 20.89 -69.27 7.11 485 -28.93 27.18 -78.02 54.15 -45.42 61.16 486 -39.02 8.54 -90.63 19.18 -58.43 5.89 487 -35.33 21.36 -77.95 56.95 -52.33 61.26 488 -38.63 11.47 -96.19 21.42 -72.95 5.36 489 -30.86 11.86 -104.51 78.98 -30.39 64.69


246

Wheel pass


26km gauge tension





490 -38.43 9.92 -87.58 19.51 -58.99 17.19 491 -45.22 10.7 -67.7 38.82 -38.67 67.13 492 -35.12 11.48 -88.28 20.19 -70 4.61 493 -37.06 12.06 -75.8 67.06 -40.11 61.6 494 -37.83 10.71 -96.19 18.68 -68.32 6.72 495 -44.62 10.52 -74.76 62.86 -35.72 68.91 496 -37.63 10.91 -99.22 20.14 -69.97 4.83 497 -28.5 11.5 -109.68 23.26 -47.88 62.79 498 -34.32 14.61 -79.33 21.73 -67.32 7.13 499 -38.78 15.97 -68.01 78.74 -56.74 40.29 500 -34.89 11.9 -94.23 20.27 -64.88 39.75 501 -44.79 12.87 -105.47 25.54 -45.9 69.83 502 -33.53 22.97 -89.31 20.89 -61.06 5.6 503 -34.1 33.46 -58.74 39.42 -35.67 60.58 504 -37.98 11.72 -96.62 20.02 -72.07 5.46 505 -41.47 12.89 -132.54 31.7 -31.07 63.23 506 -31.96 10.76 -94.03 20.25 -60.64 5.82 507 -40.88 11.35 -66.96 26.34 -62.34 41.51 508 -39.71 12.51 -90.46 20.74 -64.64 6.67 509 -43.98 14.07 -91 95.01 -51.31 59.75 510 -41.65 10.77 -90.01 20.2 -59.84 7.21 511 -35.63 10.59 -122.22 24.14 -61.34 25.95 512 -38.73 11.75 -99.26 21.07 -57.4 12.94 513 -36.39 12.53 -89.7 79.99 -53.03 45.55 514 -34.45 9.23 -92.79 18.19 -68.19 5.09 515 -49.01 12.93 -91.38 45.27 -42.22 55.59 516 -38.91 44.39 -105.35 20.62 -67.9 4.18 517 -31.72 18.37 -114.25 30.17 -38.79 76.56 518 -36.76 9.06 -77.29 18.72 -60.57 6.67 519 -40.25 8.28 -76.47 33.55 -47.84 64.58 520 -34.62 19.36 -87.13 20.17 -69.63 9.48 521 -42.97 11.01 -112.37 82.59 -35.84 70.93 522 -34.23 15.67 -89.99 18.27 -61.71 16.81


247

Wheel pass


26km gauge tension





523 -38.1 16.84 -104.32 57.2 -47.62 62.07 524 -35.19 20.54 -103.32 22.43 -69.21 23.15 525 -38.48 11.61 -107.95 68.34 -60.94 55.18 526 -38.67 11.42 -83.05 18.99 -55.45 28.33 527 -39.06 9.48 -81.25 76.19 -52.86 45.54 528 -40.03 11.04 -99.5 22.97 -57.69 3.47 529 -40.02 12.01 -107.24 76.06 -56.05 60.47 530 -37.11 10.46 -94.96 20.68 -64.97 5.58 531 -40.79 10.08 -93.55 47.95 -56.92 59.98 532 -43.7 9.89 -98.58 21.75 -63.31 2.92 533 -36.9 11.83 -64.53 70.75 -65.57 49.19 534 -39.81 8.53 -99.69 19.07 -60.66 5.03 535 -41.94 9.71 -67.76 17.58 -39.16 41.89 536 -33.59 17.67 -98.64 20.7 -64.65 4.89 537 -37.08 14.76 -107.16 24.62 -73.92 42.98 538 -39.02 9.13 -80.7 21.34 -50.5 7.2 539 -37.46 13.02 -67.04 18.11 -37.58 67.24 540 -31.24 24.68 -80.62 20.83 -68.52 4.92 541 -36.48 13.03 -115.77 24.17 -56.55 51 542 -31.43 10.51 -85.61 21.48 -58.85 5.86 543 -39 12.84 -74.87 68.77 -57.43 57.92 544 -40.35 10.13 -99.34 20.02 -65.57 4.36 545 -35.69 13.43 -120.3 22.95 -44.25 49.86 546 -39.18 8.39 -81.59 21.61 -58.82 5.69 547 -37.81 9.95 -94.37 89.32 -50.77 51.32 548 -29.46 20.63 -112.61 24.63 -64.96 5.75 549 -34.31 12.67 -88.48 75.38 -43.44 59.82 550 -41.49 9.37 -90.01 19.03 -63.66 5.91 551 -35.47 9.77 -80.82 52.33 -50.16 55.25 552 -39.15 11.72 -103.54 20.85 -56.74 4.42 553 -37.99 12.69 -66.76 43.06 -48.48 57.12 554 -37.21 10.17 -96.48 19.36 -65.19 5.94 555 -49.43 11.15 -92.93 42.36 -44.87 66.97


248

Wheel pass


26km gauge tension





556 -40.11 11.93 -103.98 22.96 -73.28 5.03 557 -33.7 12.9 -94.04 76.24 -42.6 56.76 558 -38.55 11.93 -86.82 18.91 -60.29 5.97 559 -45.15 12.13 -87.54 80.4 -52.05 54.91 560 -41.26 12.71 -99.57 21.34 -67.4 3.7 561 -42.61 12.33 -95.45 53.83 -49.39 60.3 562 -39.89 10.59 -88.82 19.05 -65.72 5.61 563 -44.74 10.79 -78.86 57.21 -58.26 49.49 564 -38.91 11.76 -100.6 21.48 -62.5 4.31 565 -44.15 12.93 -77.63 69.51 -56.38 62.28 566 -32.89 10.41 -99.76 18.41 -64.13 6.42 567 -34.44 40.51 -86.49 50.54 -53.36 61.21 568 -45.31 10.42 -103.77 21.42 -66.37 4.53 569 -32.88 11 -92.46 82.28 -76.42 14.57 570 -30.93 8.09 -90.49 18.74 -51.83 20.85 571 -41.6 27.32 -92.19 63.51 -50.79 56.17 572 -40.44 10.82 -105.77 22.92 -68.87 4.76 573 -33.44 11.99 -93.68 64.73 -69.96 39.15 574 -34.99 8.89 -92.49 18.88 -49.46 15.81 575 -40.62 24.43 -87.78 61.7 -51.35 60.88 576 -39.84 11.42 -103.69 21.31 -62.61 4.98 577 -39.25 12 -87.91 77.89 -50.64 56.13 578 -37.89 9.48 -98.38 19.21 -61.9 7.48 579 -39.44 9.69 -83.75 42.98 -49.37 60.32 580 -29.53 30.85 -102.78 21.64 -64.53 11.85 581 -25.06 26.97 -100.6 84.25 -57.04 46.03 582 -51.27 9.11 -90.86 19.54 -59.92 6.15 583 -41.17 9.51 -108.11 71.88 -49.93 66 584 -39.61 10.68 -110.61 21.77 -67.23 4.07 585 -43.88 13.59 -87.05 92.35 -49.8 58.14 586 -34.17 9.52 -100.05 18.7 -59.7 5.98 587 -43.29 11.07 -75.12 37.81 -65.29 39.34 588 -35.71 12.24 -114.94 23.08 -57.06 24.58


249

Wheel pass


26km gauge tension





589 -35.71 12.44 -90.81 92.3 -54.64 51.74 590 -38.81 10.7 -103.22 19.03 -63.95 6.6 591 -38.42 8.76 -84.32 70.21 -39.34 63.73 592 -33.76 10.9 -107.23 20.3 -67.17 4.52 593 -36.47 11.29 -142.18 25.17 -41.17 63.66 594 -33.55 7.8 -78.59 22.47 -66.46 5.46 595 -49.08 10.13 -67.66 83.18 -53.93 47.19 596 -41.31 10.53 -107.28 21.99 -65.58 3.58 597 -31.6 16.74 -86.26 92.37 -67.64 40.7 598 -37.42 8.59 -105.28 18.92 -55.12 6.08 599 -43.44 9.57 -78.99 74.57 -41.62 61.25 600 -34.11 26.46 -103.84 24.26 -64.77 5.56 601 -33.91 12.88 -112.94 48.78 -57.28 47.16 602 -38.57 8.61 -99.7 18.66 -60.94 7.08 603 -37.01 25.31 -81.96 64.4 -40.42 54.86 604 -37.79 14.05 -110.51 22.45 -71.94 7.34 605 -38.36 12.5 -82.29 82.73 -42.63 61.8 606 -45.54 9.98 -97.62 18.02 -59.15 6.92 607 -34.28 10.96 -101.07 81.84 -47.99 46.31 608 -33.89 12.13 -114.07 22.4 -60.81 7.18 609 -35.82 12.71 -115.59 28.64 -51.56 61.05 610 -42.23 11.36 -85.82 20.49 -61.46 6.37 611 -38.34 11.56 -75.86 48.54 -50.88 59.6 612 -29.99 11.37 -134.34 23.7 -66.23 4.68 613 -35.42 11.57 -102.82 64.54 -56.4 48.61 614 -34.45 10.4 -95.6 19.07 -58.89 6.79 615 -34.83 30.8 -81.75 41.87 -36.23 66.25 616 -43.37 10.03 -111.66 21.89 -61.33 24.79 617 -35.99 12.36 -86.94 74.2 -48.58 53.32 618 -38.31 11.58 -101.1 20.18 -63.15 6.43 619 -44.52 8.68 -87.64 51.53 -45.55 64.92 620 -35.78 25.96 -120.27 22.42 -69.48 4.93 621 -39.08 11.6 -86.22 89.5 -49.91 62.32


250

Wheel pass


26km gauge tension





622 -32.67 9.85 -106.99 18.57 -69.35 6.07 623 -42.18 11.02 -83.23 40.4 -60.91 57.74 624 -40.62 11.61 -109.83 21.39 -61.65 3.41 625 -36.34 12.39 -118.15 58.92 -47.54 65.86 626 -34.59 7.93 -87.8 21.42 -58.6 5.32 627 -39.05 10.46 -83.87 69.3 -65.36 45.89 628 -38.08 10.46 -102.7 21.13 -57.92 31.71 629 -40.6 11.24 -102.27 57.5 -37.37 61.6 630 -36.71 7.55 -108.86 20.39 -51.56 36.9 631 -41.56 8.72 -76.15 75.46 -54.81 52.55 632 -34.37 18.24 -99.84 21.27 -66.26 4.45 633 -47.18 12.61 -103.3 89.52 -49.23 63 634 -37.67 7.96 -101.53 20.53 -63.8 6.95 635 -40.96 9.71 -76.6 59.66 -78.74 19.07 636 -37.85 13.02 -100.68 21.01 -50.64 21.26 637 -45.62 13.02 -83.93 71.96 -62.64 25.82 638 -37.65 11.08 -105.67 17.75 -63.96 6.79 639 -40.95 8.37 -77.82 51.82 -71.5 44.62 640 -32.21 10.7 -105.79 21.54 -52.75 22.07 641 -40.94 10.32 -99.15 65.88 -37.08 68.72 642 -31.23 19.06 -99.12 19.83 -59.64 22.39 643 -45.01 11.1 -81.58 51.38 -46.91 63.17 644 -32.77 21.4 -105.46 21.29 -70.25 3.57 645 -41.7 13.05 -85.4 79.43 -44.26 61.93 646 -36.26 8.01 -94.9 18.41 -64.09 6.47 647 -39.75 9.76 -98.55 61.23 -60.33 39.82 648 -37.42 11.12 -126.32 23.17 -60.87 5.56 649 -33.34 17.92 -107.82 79.37 -58.83 48.91 650 -37.02 20.64 -96.71 20.49 -57.62 6.31 651 -36.05 14.82 -82.28 62.73 -51.91 53.69 652 -34.29 11.91 -116.27 22.53 -61.61 4.03 653 -30.8 14.44 -87.27 85.33 -53.15 67.85 654 -36.04 7.85 -100.27 19.26 -61.1 5.75


251

Wheel pass


26km gauge tension





655 -44.57 9.79 -85.45 57.81 -48.18 61.91 656 -37.19 11.74 -105.05 21.11 -67.03 4.46 657 -31.17 16.4 -95.3 67 -67.73 46.25 658 -45.14 8.64 -96.05 18.43 -55.41 6.37 659 -38.34 9.61 -86.87 75.83 -34.11 69.35 660 -34.46 15.83 -110.55 22.8 -65.83 17.17 661 -38.92 11.56 -93.61 62.47 -48.99 65.19 662 -34.26 16.03 -110.88 19.14 -55.77 6.41 663 -40.85 15.45 -77.79 24.84 -40.31 56.13 664 -36.58 14.1 -96.62 22.35 -72.03 5.7 665 -33.85 21.09 -110.19 72.52 -38.63 62.5 666 -35.79 10.61 -106.47 22.19 -62.74 6.64 667 -43.94 13.53 -74.74 34.5 -53.92 67.08 668 -37.53 12.56 -102.31 21.1 -64.01 5.55 669 -39.86 13.34 -85.95 67.99 -53.79 58.83 670 -39.66 9.46 -106.73 19.22 -59.2 6.48 671 -44.51 11.02 -90.35 69.23 -40.05 70.04 672 -43.15 12.58 -117.34 23.79 -73.12 6.55 673 -33.82 12.97 -86.4 93.01 -50.83 62.76 674 -46.05 10.45 -131.28 19.94 -59.36 6.13 675 -45.66 9.29 -107.71 78.12 -46.25 58.77 676 -37.31 12.4 -119.34 22.95 -69.19 5.41 677 -38.85 11.82 -81.02 84.2 -49.82 70.2 678 -36.13 17.26 -98.1 17.55 -65.36 6.75 679 -42.34 13.19 -81.14 62.12 -63.75 50.24 680 -37.48 15.33 -102.3 21.53 -62.14 4.86 681 -37.29 12.42 -83.6 90.94 -42.18 54.65 682 -35.15 14.75 -100.88 18.26 -61.63 6 683 -39.61 16.7 -74 39.9 -55.33 50.27 684 -37.27 14.76 -98.08 21.86 -62.11 4.72 685 -38.43 13.02 -81.72 68.73 -49.75 63.06 686 -36.68 8.16 -94.91 18.59 -67.43 5.85 687 -39.4 14.38 -100.11 84.55 -44.38 63.36


252

Wheel pass


26km gauge tension





688 -37.65 12.84 -106.11 22.97 -65.58 3.38 689 -33.56 12.64 -100.44 77.22 -59.06 58.42 690 -40.74 10.9 -96.53 20.28 -54.15 8.98 691 -39.96 10.13 -77.04 42.89 -58.57 48.4 692 -36.85 12.66 -97.03 22.13 -66.13 11.42 693 -25.39 37.12 -98.94 87.85 -53.96 54.56 694 -45.19 8.97 -100.67 19.06 -57.62 6.89 695 -44.61 8.79 -76.52 49.83 -51.33 60.32 696 -37.42 13.64 -101.37 21.68 -58.3 11.46 697 -38 14.23 -98.03 101.59 -55.09 49.73 698 -33.14 14.04 -99.37 19.58 -63.04 6.15 699 -37.6 28.41 -82.02 51.91 -43.1 57.63 700 -41.09 12.3 -110.76 22.4 -61.77 17.54 701 -26.33 39.1 -83.32 91.42 -41.61 69.25 702 -43.22 8.62 -99.62 18.55 -57.55 12.79 703 -38.36 22.02 -80.13 45.04 -56.32 50.46 704 -42.63 12.51 -115.29 23.7 -58.03 19.91 705 -37.58 13.48 -95.23 101.86 -36.7 66.17 706 -43.2 9.8 -98.51 19.66 -59.85 12.25 707 -39.12 8.06 -76.31 62.67 -76.15 15.03 708 -34.65 11.56 -104.66 23.84 -55.26 11.57 709 -45.91 12.72 -123.29 87.42 -69.98 9.12 710 -37.95 10.98 -102.46 21.73 -57.08 5.87 711 -48.81 12.74 -75.39 52.31 -67.93 8.83 712 -49.97 12.54 -98.11 21.44 -50.74 35.57 713 -34.05 11.96 -99.05 81.14 -47.54 55.14 714 -37.35 13.33 -106.02 19.34 -63.86 6.5 715 -49.57 9.06 -72.93 50.11 -48.6 63.04 716 -31.13 12.17 -94.87 22.35 -66.68 4.42 717 -34.81 14.7 -84.53 70.78 -51.01 59.86 718 -38.3 10.24 -103.36 20.84 -59.34 6.73 719 -30.92 39.37 -77.65 61.52 -53.05 24.3 720 -47.22 12.38 -105.81 22.1 -59.82 7.38


253

Wheel pass


26km gauge tension





721 -30.52 12.38 -139.99 25.8 -62.27 41.77 722 -40.62 10.06 -89.42 21.75 -56.97 7.16 723 -48.38 12.01 -77.52 63.02 -40.73 65.84 724 -40.61 11.81 -105.09 22.62 -66.99 5.08 725 -24.49 35.31 -75.13 77.46 -51.51 64.61 726 -44.48 7.35 -100.37 19.74 -68.81 7.39 727 -52.24 12.41 -68.83 57.12 -44.2 69.19 728 -39.82 12.99 -104.57 22.95 -66.76 4.53 729 -39.03 12.42 -104.54 82.45 -47.77 60.56 730 -39.23 8.73 -100.04 20.66 -67.61 7.82 731 -35.73 20.97 -74.72 71.06 -52.15 57.93 732 -39.41 11.85 -105.99 22.7 -67.11 4.01 733 -39.99 13.01 -102.26 56.73 -53.97 65.27 734 -48.34 11.27 -99.71 21.18 -69.71 7.86 735 -45.81 9.53 -76.14 37.95 -52.7 66.93 736 -42.7 12.25 -106.25 22.83 -66.88 4.22 737 -39.39 12.45 -86.58 98.27 -67.38 53.23 738 -33.37 11.67 -95.5 19.76 -61.1 7.31 739 -49.67 9.54 -77.95 65.69 -44.86 57.43 740 -40.54 12.27 -110.19 22.97 -68.2 4.46 741 -35.3 12.85 -113.46 54.86 -55.26 63.99 742 -43.84 11.3 -99.83 21.06 -63.98 7.74 743 -45.97 11.31 -76.65 68.74 -57.29 62.73 744 -44.41 11.5 -102.87 22.71 -64.27 5.66 745 -38.19 13.06 -92.53 84.15 -49.37 60.91 746 -43.05 10.93 -104.17 19.25 -66.86 7.78 747 -49.64 10.94 -78.27 53.91 -53.75 65.69 748 -42.45 11.72 -101.96 22.26 -66.17 4.34 749 -35.85 15.41 -111.64 65.24 -55.37 48.28 750 -39.53 8.42 -99.76 21.13 -65.07 7.82 751 -47.68 9.98 -78.33 58.51 -48.84 67.48 752 -40.11 11.73 -105.52 23.17 -67.89 5.55 753 -40.69 12.7 -107.81 67.71 -47.73 60.01


254

Wheel pass


26km gauge tension





754 -36.61 12.71 -98.65 22.24 -63.47 7.86 755 -34.85 19.32 -74.31 47.76 -44.7 74.73 756 -35.43 12.14 -101.69 22.72 -71.74 5.98 757 -40.48 13.89 -104.18 47.82 -47.5 60.25 758 -46.1 10.2 -98.91 21.4 -65.38 7.12 759 -45.71 10.79 -79.62 76.08 -45.44 65.81 760 -35.22 15.26 -113.41 23.05 -64.49 4.27 761 -40.46 11.38 -88.69 66.03 -44.34 56.78 762 -38.52 10.02 -106.55 20.17 -68.07 7.36 763 -44.14 9.44 -79.29 62.02 -46.18 66.05 764 -38.31 11.39 -115.22 23.96 -64.84 3.92 765 -43.36 12.17 -126.27 48.48 -50.53 66.57 766 -36.56 13.34 -86.2 22.06 -67.25 8.18 767 -41.6 8.68 -80.13 64.49 -48.67 67.84 768 -37.13 11.21 -110.81 24.1 -64.41 4.93 769 -40.62 11.99 -114.28 67.85 -42.89 69.34 770 -37.51 10.64 -102.39 20.44 -70.52 8.03 771 -43.72 10.26 -72.22 28.86 -45.9 67.3 772 -36.92 10.84 -102.9 22.87 -69.83 5.17 773 -41 11.43 -121.53 79.26 -51.23 59.25 774 -35.36 11.43 -100.51 21.35 -64.24 7.68 775 -45.65 8.91 -74.03 63.98 -46.45 68.31 776 -40.4 11.63 -98.3 21.84 -69.98 5.21 777 -39.04 12.41 -100.99 85.03 -48.46 60.26 778 -34.18 8.53 -99.99 20.51 -68.29 7.72 779 -45.82 11.65 -71.37 64.5 -48.94 66.02 780 -41.36 11.46 -96.03 22.16 -66.43 5.45 781 -38.63 12.04 -101.63 76.61 -59.72 39.65 782 -38.83 9.91 -101.6 21.03 -62.21 7.18 783 -44.06 9.33 -74.54 67.16 -48.7 53.59 784 -36.29 11.47 -97.84 23.27 -66.19 5.3 785 -34.15 19.24 -130.27 62.94 -59.87 54.7 786 -39.78 7.4 -99.72 21.17 -64.31 7.42


255

Wheel pass


26km gauge tension





787 -47.35 9.93 -73.43 67.88 -47.68 56.36 788 -37.44 11.68 -107.23 24.57 -67.9 5.73 789 -41.13 12.07 -93.01 101.76 -48.53 67.79 790 -34.72 10.33 -101.34 20.52 -64.46 7.26 791 -42.86 17.32 -79.52 47.21 -46.28 59.33 792 -43.06 11.5 -110.6 22.76 -66.3 5.97 793 -39.36 11.5 -87.82 98 -46.14 70.17 794 -35.87 10.73 -106.45 19.49 -66.36 7.31 795 -47.9 8.99 -79 46.18 -47.99 65.02 796 -36.44 11.71 -103.27 22.5 -69.96 4.84 797 -40.32 12.1 -105.76 90.36 -43.57 73.72 798 -33.71 13.27 -96.6 20.99 -67.49 7.74 799 -44.58 9.39 -73.62 26.49 -45.99 64.67 800 -37.78 11.34 -98.67 21.84 -68.16 5.67 801 -40.69 11.73 -100.58 85.64 -53.85 39.09 802 -35.64 9.99 -96.28 19.56 -65.3 7.2 803 -50 9.99 -74.07 33.23 -52.77 62.77 804 -38.74 11.55 -99.7 22.96 -65.2 19.56 805 -39.51 12.72 -113.47 92.18 -59.65 58.81 806 -37.37 9.23 -105.28 21.83 -62.34 7.24 807 -43.77 9.43 -78.99 69.9 -47.47 66.9 808 -38.91 11.57 -107.15 23.29 -69.05 5.17 809 -38.91 11.76 -92.15 83.76 -68.96 51.84 810 -36 11.57 -109.42 21 -57.81 18.18 811 -47.45 10.41 -85.85 56.04 -50.54 65.58 812 -39.68 12.17 -114.4 24.59 -63.16 24.91 813 -39.28 12.55 -112.42 63.88 -68.72 51.3 814 -35.59 10.62 -103.85 19.96 -54.45 27.77 815 -41.8 9.26 -78.92 63.95 -67.25 50.04 816 -39.08 12.18 -106.88 23.36 -56.49 22.23 817 -35 14.71 -115.98 73.14 -54.06 66.93 818 -37.71 7.53 -90.49 21.46 -64.35 8.16 819 -44.89 8.89 -84.23 40.95 -55.12 56.71


256

Wheel pass


26km gauge tension





820 -39.26 12.39 -106.94 23.11 -62.87 6.88 821 -34.4 14.53 -90 91.55 -57.14 53.73 822 -39.05 8.51 -99.89 20.23 -61.77 7.62 823 -44.1 9.68 -72.24 42.64 -50.21 64.94 824 -39.43 12.02 -106.62 22.66 -68.09 5.93 825 -35.94 11.63 -103.28 84.1 -54.75 61.96 826 -39.62 7.95 -98.2 21.53 -62.11 8.25 827 -48.35 10.28 -75.41 73.29 -53.87 59.53 828 -37.67 11.45 -103.37 23.76 -66.29 6.56 829 -39.8 12.04 -94.79 108.53 -41.85 68.63 830 -36.89 11.07 -110.51 21.66 -68.69 8.49 831 -45.23 8.55 -82.27 61.56 -53.04 55.29 832 -39.4 11.66 -104.02 24.09 -65.08 6.02 833 -34.35 18.85 -99.71 93.31 -47.45 68.28 834 -39.78 7.78 -99.68 21.21 -67.48 7.56 835 -43.85 10.51 -77.86 81.52 -60.2 64.3 836 -34.14 13.04 -104.27 23.25 -65.81 6.07 837 -38.8 12.65 -78.97 89.75 -54.42 44.36 838 -39.38 11.88 -103.05 20.95 -55.93 8.19 839 -41.12 11.5 -106.5 48.03 -50.22 64.74 840 -39.56 11.5 -106.67 24.74 -62.84 5.34 841 -38.98 11.7 -108.77 55.86 -48.33 69.94 842 -32.76 13.45 -97.86 22.25 -68.55 8.43 843 -44.21 15.01 -79.93 51.08 -40.04 66.34 844 -42.66 13.07 -106.54 24.68 -64.16 5.97 845 -35.86 26.66 -109.03 97.4 -46.14 67.06 846 -37.6 7.64 -105.12 21.02 -64.8 8.09 847 -42.06 15.6 -77.47 72.78 -37.85 59.37 848 -42.64 12.11 -105.43 23.06 -67.42 6.6 849 -40.31 12.7 -110.06 43.69 -54.67 65.74 850 -40.11 8.62 -96.24 21.93 -60.86 7.94 851 -46.32 10.57 -75.39 68.06 -53.58 60.01 852 -40.1 11.74 -108.02 24.17 -64.45 6.26


257

Wheel pass


26km gauge tension





853 -40.87 11.74 -102.74 104.08 -52.48 65.4 854 -36.6 10.19 -106.8 21.48 -62.37 8.38 855 -44.94 10.2 -84.39 69.16 -48.86 66.09 856 -35.04 13.7 -112.55 24.49 -64.21 5.92 857 -37.17 12.15 -84.72 107.51 -51.26 66.23 858 -37.94 10.4 -99.47 20.64 -65.05 8.82 859 -42.59 14.68 -75.9 44.22 -39.27 60.88 860 -42.59 12.16 -96.87 23.26 -62.6 5.96 861 -34.82 17.79 -104.81 67.8 -58.03 59.65 862 -36.95 16.83 -104.59 22.14 -62.47 8.67 863 -46.85 9.84 -72.27 68.07 -51.49 65.8 864 -33.84 23.24 -105.29 23.79 -63.92 6.21 865 -34.81 11.6 -79.99 89.7 -56.81 61.65 866 -34.61 18.4 -107.37 21.88 -59.5 16.88 867 -43.92 9.08 -77.19 68 -36.05 65.07 868 -33.63 24.23 -111.19 25.08 -73.61 8.4 869 -36.93 13.55 -128.65 71.95 -55.79 66.76 870 -38.09 11.62 -103.16 23.18 -63.54 8.57 871 -39.64 21.52 -72.79 60.56 -51.01 63.95 872 -38.08 16.28 -98.42 23.08 -64.41 17.23 873 -33.03 17.07 -118.41 55.75 -56.91 57.07 874 -37.88 12.21 -98.36 22.14 -63.3 8.23 875 -43.5 18.05 -75.96 69.82 -73.56 20.94 876 -37.48 14.17 -106.25 23.99 -55 22.93 877 -33.79 17.28 -90.67 71.05 -48.87 57.9 878 -34.76 16.11 -99.39 20.91 -63.05 8.67 879 -41.16 20 -90.21 83.76 -51.49 41.44 880 -39.02 11.27 -114.29 24.51 -61.77 6.79 881 -35.71 19.23 -91.12 93.34 -62.66 22.68 882 -35.52 17.49 -96.35 19.69 -59.5 8.53 883 -45.22 12.45 -78.22 61.15 -42.87 73.45 884 -37.25 15.56 -112.41 25.03 -64.45 6.65 885 -33.56 26.04 -78.94 68.98 -49.16 62.48


258

Wheel pass


26km gauge tension





886 -38.8 14.98 -107.29 22.15 -65.68 9.16 887 -44.43 10.71 -78.67 53.5 -47.11 64.54 888 -34.13 14.99 -109.55 24.19 -60.12 8.27 889 -28.11 22.57 -87.55 90.88 -47.36 64.87 890 -39.76 8.98 -97.05 19.95 -65.24 8.82 891 -43.05 15.97 -79.9 58.3 -60.11 52.9 892 -39.94 15.78 -109.03 24.32 -60.46 6.75 893 -35.08 16.37 -75.95 92.57 -62.7 58.49 894 -34.11 18.51 -100.61 22.03 -58.37 9.26 895 -43.23 15.02 -89.88 76.32 -58.7 58.21 896 -37.4 17.74 -106.18 23.68 -58.85 6.8 897 -37.4 11.92 -100.9 82.79 -51.35 66.14 898 -37.39 8.43 -101.07 21.96 -67.48 9.51 899 -48.65 9.79 -78.27 79.75 -58.45 30.98 900 -39.13 11.93 -117.51 24.78 -57.63 8.99 901 -36.8 11.93 -108.73 30.63 -51.69 68.91 902 -32.72 19.32 -101.13 23.85 -60.02 8.97 903 -38.15 22.23 -80.67 65.69 -64.44 26.55 904 -45.33 11.17 -104.36 23.16 -55.63 9.84 905 -37.95 11.95 -105.3 66.92 -47.35 67.99 906 -37.94 8.07 -100.02 22.42 -63.09 8.63 907 -45.32 12.93 -77.82 63.3 -74.91 26.4 908 -40.07 10.99 -105.59 24.07 -56.36 8.51 909 -36.77 13.13 -101.28 28.17 -63.28 54.21 910 -36.18 9.83 -99.7 24.69 -56.22 9.44 911 -42.97 9.26 -83.32 79.37 -55.96 28.4 912 -35.2 15.47 -100.4 23.04 -55.72 9.53 913 -34.42 17.42 -82.1 87.4 -49.39 59.13 914 -36.55 17.03 -106.56 21.33 -64.16 9.13 915 -41.01 15.68 -82.61 40.82 -57.47 27.48 916 -36.35 18.01 -99.1 23.95 -57.03 8.61 917 -37.32 12.58 -115.98 28.82 -65.71 25.09 918 -34.21 7.73 -93.6 25.35 -60.01 9.57


259

Wheel pass


26km gauge tension





919 -44.1 8.51 -97.83 20.75 -57.81 23.25 920 -35.95 24.82 -94.11 22.72 -56.79 9.44 921 -40.41 13.37 -95.63 26.04 -58.06 24.94 922 -35.94 9.3 -90.16 24.12 -57.82 8.99 923 -46.42 9.31 -103.53 22.44 -56.97 37.13 924 -35.93 12.03 -96.12 23.8 -58.1 8.91 925 -38.45 11.06 -95.31 37.65 -54.69 45.45 926 -35.54 17.47 -99.56 24.06 -63.42 8.89 927 -41.16 11.46 -92.9 21.6 -53.8 38.35 928 -37.27 18.45 -97.34 24.32 -58.63 8.57 929 -36.49 11.46 -107.61 69.07 -56.59 30.5 930 -33.58 20.2 -101.56 23.99 -62 9.14 931 -37.26 15.35 -91.41 20.18 -59.99 35.29 932 -40.56 11.09 -95.46 22.53 -57.21 8.43 933 -35.51 18.86 -115.64 27.6 -62.77 35.82 934 -32.59 20.61 -91.13 24.32 -59.61 9.19 935 -44.43 18.86 -90.69 64.81 -50.19 44.11 936 -40.93 16.54 -105.83 22.08 -59.3 8.29 937 -34.33 19.07 -76.64 50.87 -54.54 46.59 938 -36.85 16.16 -94.1 19.4 -62.28 8.47 939 -43.44 12.47 -110.97 30.73 -46.63 66.38 940 -38.2 12.28 -101.42 19.67 -63.73 7.17


Appendix C. FORTRAN-code

261

Appendix C. FORTRAN-code based on the Dang Van fatigue criterion

SUBROUTINE UVARM(UVAR,DIRECT,T,TIME,DTIME,CMNAME,ORNAME, 1 NUVARM,NOEL,NPT,LAYER,KSPT,KSTEP,KINC,NDI,NSHR,COORD, 2 JMAC,JMATYP,MATLAYO,LACCFLA) C INCLUDE 'ABA_PARAM.INC' C CHARACTER*80 CMNAME,ORNAME CHARACTER*3 FLGRAY(15) DIMENSION UVAR(NUVARM),DIRECT(3,3),T(3,3),TIME(2) DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),COORD(*) C CC Error counter: JERROR = 0 CC C CALL STRESS STATE FROM EACH POINT IN THE DEFINED SETS C============================================================================ C Stress tensor: FOR EACH TIME INCREMENT AT THE MATERIAL POINT CALL GETVRM('S',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP, 1 MATLAYO,LACCFLA) C JERROR = JERROR + JRCD SIG11=ARRAY(1) SIG22=ARRAY(2) SIG33=ARRAY(3) SIG12=ARRAY(4) SIG23=ARRAY(5) SIG31=ARRAY(6) C Press stress:FOR EACH TIME INCREMENT AT THE MATERIAL POINT CALL GETVRM('SINV',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP, 1 MATLAYO,LACCFLA) Mise=ARRAY(1) Shh=ARRAY(3) INV3=ARRAY(4) C Principal stress:FOR EACH TIME INCREMENT AT THE MATERIAL POINT CALL GETVRM('SP',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP, 1 MATLAYO,LACCFLA) JERROR = JERROR + JRCD SP1=ARRAY(1) SP2=ARRAY(2) SP3=ARRAY(3) C Note (SP1<SP2<SP3) Tmax=0.5*(SP3-SP1) Tmin=0.5*(SP2-SP1) Tmid=0.5*(Tmax+Tmin) C=============================================================================


262

C DEFINING THE DANG VAN PARAMETERS: C Enter the fatigue limit from the tension-compression test (in MPa): Sfl=353.0 C Enter the fatigue limit from the torsion test (in MPa): Tfl=205.0 aDV=3*((Tfl/Sfl)-0.5) bDV=Tfl C START TO CHECK SHEAR PLANE Do I=1,360 C LL START TO CHECK EVERY 1 DEGREE IN THE Nx DIRECTION Do J=1,360 DO K=1,360 C AT EACH 1 DEGREE OF Nx, TO CHECK ALL DEGREES OF Ny DIRECTION at=I-1.0 bt=J-1.0 ct=k-1.0 aL=COSD(at) aM=COSD(bt) aN=COSD(ct) Tal=aL**2+aM**2+aN**2 If (Tal.eq.1.0) then C TOTAL STRESS IN ‘N’ DIRECTION Snn=SIG11*aL**2+SIG22*aM**2+SIG33*aN**2+2*(SIG12*aL*aM+ +SIG23*aM*aN+SIG31*aL*aN) C Sxx=SIG11*aL+SIG12*aM+SIG31*aN Syy=SIG12*aL+SIG22*aM+SIG23*aN Szz=SIG31*aL+SIG23*aM+SIG33*aN Tsp=(abs(Sxx**2+Syy**2+Szz**2-Snn**2))**0.5 C DANG VAN SHEAR STRESS Tt=ABS(Tsp-Tmid) BB=Tt+abs(aDV*Shh) If (BB.GT.Tfl) THEN Du=5*(Tfl-BB)/(Tfl-Sfl)-6 Dl=10**Du aNf=1/Dl UVAR(1)=BB UVAR(2)=ACOSD(aL) UVAR(3)=ACOSD(aM) UVAR(4)=ACOSD(aN) UVAR(5)=Tsp UVAR(6)=Tt UVAR(7)=Dl UVAR(8)=aNf


263

Else UVAR(9)=BB ENDIF EndIF Enddo ENDDO ENDDO write(*,*)'====================================' write(*,*)'Time INC=',DTIME,'Total time=',time(2) write(*,*)'ELE No:',NOEL write(*,*)'Tsp=',Tsp write(*,*)'Tmax=', Tmax write(*,*)'Dang Van Shear_BB=',BB write(*,*)'Plane n to X axis=',UVAR(2),'Deg' write(*,*)'Plane n to Y axis=',UVAR(3),'Deg' write(*,*)'Plane n to Z axis=',UVAR(4),'Deg' C If error, write comment to .DAT file: C IF(JERROR.NE.0)THEN C WRITE(*,*) 'REQUEST ERROR IN UVARM FOR ELEMENT NUMBER ', C 1 NOEL,'INTEGRATION POINT NUMBER ',NPT C ENDIF RETURN END

Documents

The effect of head wear on rail underhead radius stresses and ......Sagheer Abbas Ranjha B.Eng. (Hons), M. Eng. A thesis submitted in fulfilment of the requirements for the degree