[Elsevier] Rolling Contact Fatigue Analysis of Rails Inculding Numerical Simulations of the Rail...

International Journal of Fatigue 25 (2003) 547–558www.elsevier.com/locate/ijfatigue

Rolling contact fatigue analysis of rails inculding numericalsimulations of the rail manufacturing process and repeated wheel-

rail contact loads

Jonas W. Ringsberga,∗, Torbjorn Lindback b

a Chalmers University of Technology, Department of Applied Mechanics, SE-412 96 Go¨teborg, Swedenb Lulea University of Technology, Division of Computer Aided Design, SE-971 87 Lulea˚, Sweden

Received 21 June 2002; received in revised form 30 September 2002; accepted 3 October 2002

Abstract

The present work is an investigation on how an initially introduced residual stress-state affects the service life of a rail, i.e. thetime to fatigue crack initiation. The finite element (FE) method was used to make two-dimensional thermo-mechanical analyses ofthe rail cooling and roller straightening processes. The results became the initial conditions in a three-dimensional elastic-plasticrail model; the model is part of an FE tool developed for rolling contact fatigue (RCF) analysis of rails. The results from this toolwere analysed for fatigue, for eight wheel passages, according to a method which incorporates a critical plane approach that evaluatesfatigue damage on a cycle-by-cycle basis. A heavy-haul (30 tonne) train traffic situation on the Iron-ore Line in Sweden was studiedwith respect to subsurface fatigue crack initiation in straight track. Three examples using the rail model in the FE tool were assessed:(a) an initially stress-free rail, (b) a measured residual stress field in a newly manufactured rail, and (c) a calculated residual stressfield from the cooling and roller straightening analyses. The results from the thermo-mechanical FE analyses of the rail manufactur-ing process showed tensile residual stresses in the longitudinal direction of the rail; this was validated with experimental measure-ments on newly manufactured rails. The FE tool and fatigue calculations revealed only small differences in results for the threeexamples. It was concluded that, because of the very high axle load in the present traffic situation, the local wheel-rail contactloads governed the fatigue life to crack initiation. Additional FE tool calculations were made to show the axle load at which railmanufacturing stresses reduce the fatigue life to crack initiation. 2003 Elsevier Science Ltd. All rights reserved.

Keywords:Rolling contact fatigue; FE simulations; Critical plane approach; Residual stresses; Rail fatigue; Rail manufacturing

1. Introduction

Repeated railway wheel–rail rolling contacts cause anonproportional multiaxial stress–strain response in therail. This causes fatigue damage to the rail, which leadsto the initiation of cracks that originate on either the sur-face or subsurface of the rail. The site of the greatestfatigue damage is governed by factors including themagnitude of the traction forces, the axle load, and theresidual stress field in the rail. In addition, future

∗ Corresponding author. Chalmers University of Technology,Department of Applied Mechanics, SE-412 96 Go¨teborg, Sweden. Tel.:+46-31-772-1504; fax:+46-31-772-3827.

E-mail address: [email protected] (J.W.Ringsberg).

0142-1123/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0142-1123(02)00147-0

demands on train traffic will involve heavier axle loads,higher train speeds and increased traffic density. Eachof these demands increases rail rolling contact fatigue(RCF) damage, raises maintenance costs and affects traintraffic safety. To prevent the future demands fromresulting in catastrophic rail failures, it is urgent to con-duct investigations that identify and suggest improve-ments for the rails available today.

Ringsberg [1,2] developed a finite element (FE) toolfor RCF analysis of railway rails; see Section 3 fordescription of the FE tool. It comprises two coupled FEmodels, one for the track and one for the rail, whichtogether incorporate the global track response and thelocal wheel-rail contact loads in the analysis. The toolwas employed to analyse suburban train traffic andheavy-haul train traffic situations with respect to RCF

548 J.W. Ringsberg, T. Lindback / International Journal of Fatigue 25 (2003) 547–558

crack initiation. The fatigue calculations were madeusing a method proposed for life prediction of RCF crackinitiation [3,4]. Furthermore, the FE tool assumed thatthe rail was stress-free at the start of the analysis, i.e.residual stresses caused by the rail manufacturing pro-cess were not incorporated. Schleinzer [5], Lindback [6],Webster [7], and Urashima [8] have done experimentalwork and made numerical simulations of rail manufac-turing processes. They showed that the rail heads ofnewly manufactured rails often had a residual stress-stateof tensile stresses in the longitudinal direction near therunning surface. A residual stress-state of this type canreduce the time to fatigue crack initiation.

The objective of the present investigation was toexamine whether the residual stresses caused by the railmanufacturing process influence the fatigue life to crackinitiation during multiaxial fatigue loading conditions.This was accomplished by further development andimprovement of the work done by Lindback [6] andRingsberg [1–4]. The work was divided into three parts:computational modelling of the rail manufacturing pro-cess; FE tool analyses of wheel–rail rolling–sliding con-tact loads subjected to rails during service conditions;presentation and utilisation of a method proposed forfatigue life prediction of RCF crack initiation.

Inspections of damaged pearlitic steel grade 1100rails, along the heavy-haul (30 tonne axle load) Iron-ore Line in Sweden, show early subsurface fatigue crackinitiation in straight track. The hypothesis was that theresidual stress-state caused by rail manufacturing affec-ted the time to fatigue crack initiation and the positionfor fatigue failure. The train traffic situation at the Iron-ore Line was therefore chosen for study in this investi-gation. Three FE tool analyses were made using differentinitial stress-state conditions of a rail: (a) stress-free rail,(b) measured residual stress field of newly manufacturedrails, and (c) calculated residual stress field from a railmanufacturing simulation. Each of the examples wasanalysed for fatigue. Hence, the influence of the initialstress-state of a rail, introduced during rail manufacture,on the fatigue life to crack initiation could be determ-ined.

2. Rail manufacturing simulation

The rail manufacturing process comprises three steps:hot-rolling; cooling; and roller straightening. In each ofthe three steps, residual stresses are introduced in the railbeam. The resultant stress field is complex to estimatein advance, as it is a result of successive steps each ofwhich affects the stress-state. The present investigationintegrates the simulation of the cooling and rollerstraightening steps.

Hot-rolling begins with an iron bloom that is heated toover 1000 °C. The bloom undergoes several hot-rolling

sequences through machinery that forms it to the correctrail profile. The finished hot-rolled rail profile has a tem-perature of approx. 900 °C just before it enters a coolingbed. The hot-rolled rail is pushed onto the cooling bedwhere it is laid on its side to cool almost to room tem-perature, see Fig. 1. When the rail is pushed onto thecooling bed, it is curved around an axis normal to thebed (the y-axis) to compensate for the change in curva-ture it will undergo during cooling. The curvature changeoccurs because different parts of the rail are cooled atvarying rates depending on rail shape and thickness.When the rail has cooled to room temperature, it entersa roller straightening process. The rail is straightenedvertically by running it through nine rollers. The positionof the rollers is designed to alternately bend the railbeam up and down, so that a straight rail passes theninth roller.

The rail manufacturing process was simulated with anin-house thermo-mechanical FE code. A two-dimen-sional rail FE model based on a generalised plane defor-mation formulation was used [9]. The out-of-plane strain(i.e. the strain in the longitudinal direction of the rail)was assumed to have the linear variation ex = b1 + b2y+ b3z, i.e. three unknowns were added. In contrast to athree-dimensional model, information on the out-of-plane shear strains is lost, i.e. exy = exz = 0. The railcross-section was divided into 576 four-node quadrilat-eral elements (see Fig. 2) with bilinear shape functions.

2.1. Simulation of cooling

The cooling simulation began when the hot-rolled rail,which has a temperature of 900 °C, was pushed onto thecooling bed. At this temperature, the yield limit of thematerial is low and its stress relaxation is fast. The mag-nitude of the residual stresses in the rail is, at this point,negligible when compared with the magnitude of theresultant residual stresses after cooling and rollerstraightening.

The rail was assumed fixed with respect to bendingaround its symmetry line in the z-direction (see Fig. 1).This assumption was based on the fact that the weightof the rail prevents it from moving out of the coolingbed plane. During cooling, heat was emitted from therail due to heat transfer to the cooling bed, convection,and radiation. The heat transfer to the cooling bed, whichacts as a heat sink that affects the cooling rate, was mod-elled by two extra elements at the contact points betweenthe rail and the cooling bed. The two elements weregiven a heat conduction coefficient of low value, corre-sponding to the heat transfer in the contact. The elementnodes in these two extra elements not connected to therail were given a temperature equivalent to that of thecooling bed. Convection and the magnitude of the con-vective heat transfer coefficient are highly dependent onthe properties and state of the cooling fluid (in this case

549J.W. Ringsberg, T. Lindback / International Journal of Fatigue 25 (2003) 547–558

Fig. 1. Hot-rolled rail profile pushed onto cooling bed; compensation for curvature changes is followed by roller straightening.

Fig. 2. FE mesh of rail cross-section in the cooling and roller straight-ening FE simulations.

“ free air” ). According to Kreith and Bohn [10], the con-vective heat transfer coefficient for convection betweensteel and free air varies between 6 and 30 W/m2 °C. Inthe present investigation, it was set to 12 W/m2 °C tofit cooling rates measured at the rail manufacturing plant.The heat loss due to radiation was calculated using theStefan-Boltzmann law with a unit emission factor. Therail was cooled for 8 hours, after which it had reachedroom temperature.

A staggered step-approach was used in the coolingsimulation. For each time step, the thermal field was cal-culated using an iterative procedure, followed by iter-ations to find a solution to the mechanical field. The coo-ling rate during the simulation was approximately0.5 °C/s, and the transformation from austenite to pearl-ite for the current steel takes place between 600 °C and650 °C. The thermal and mechanical fields were coupledin the FE simulation by thermal strains and temperature-

Fig. 3. The thermal dilatation (et) and the yield stress (sy) vs. thetemperature (T).

dependent mechanical properties for the pearlitic steelgrade 900A [11], see Figs. 3, 4 and 5 for the tempera-ture-dependent mechanical properties.

The material behaviour was modelled using a consti-tutive material model for linear kinematic hardening.

Fig. 4. The hardening modulus (H�), the elastic modulus (E), and thePoisson’s ratio (n) vs. the temperature (T).


Fig. 5. The heat capacity (c) and the thermal conductivity (l) vs. thetemperature (T).

The yield stress, sy, was defined by a von Mises yieldsurface, where the yield function, f, was defined as

f(s,a) � �32

|tdev|�sy with tdev � sdev�a, (1)

|tdev| � �tdev:tdev

In Eq. (1), sdev is the deviator stress tensor of the appliedstress tensor, s; a is the backstress tensor; and the oper-ator “ :“ defines the contraction a:b = aijbij. The materialis elastic when f(s,a) � 0 and plastic when f(s,a) =0. The backstress was determined by

da � H�·dep (2)

where H� is the hardening modulus and dep is the changein plastic strains. The normality for the plastic flow wasdefined as

dep � ldfds

(3)

where l is the plastic multiplier. The consistency con-dition was then defined as

df �dfds

:ds�H�dfds

:dfds

dl � 0 (4)

2.2. Simulation of roller straightening

The FE simulation of the roller straightening processrequired that the curvature of the rail be modelled asprescribed out-of-plane degrees of freedom (DOF)which were time-dependent [11]. These DOF werefinally released and the rail was allowed to take itsunloaded equilibrium position. In addition to these DOF,the rail was locally loaded on the rail head/rail foot bythe rollers in the roller straightening machine. The rollercontacts were represented by distributed contact loads,

which alternately operated on the rail head or rail footto simulate up and down rail bending; see Lindback [6]for details and the type of boundary conditions used.

An elastic beam model of the rail was used to calcu-late the displacements caused by the rollers, after whichthe curvature and contact loads could be estimated. Thecontact loads were applied to the rail head/rail foot as adistribution of time-dependent nodal forces, see Fig. 2for the FE mesh. These forces were defined as active,using ramp functions for increasing/decreasing magni-tude, while the rail was in contact with a roller; other-wise they were set to zero. The nodal forces had a Hertz-ian distribution that was calculated according to theHertz theory for elastic rolling of a cylinder on an infi-nite plane. Note that the Hertz theory presumes elasticdeformation in the contact zone. However, the plasticdeformation during rail straightening was known to besmall.

3. The FE tool developed for RCF analysis of rails

An FE tool was developed for the analysis of RCF ofrailway rails, see Ringsberg [1] for details. It mimicswheel–rail rolling–sliding contact on any track, since itincorporates both the dynamic global track response andthe three-dimensional local elastic–plastic contact con-ditions in the rail head. The FE tool was improved hereto include different types of initial stress-state conditionsin the rail. As a result, it was possible to apply it inanalysing the influence of residual stresses, caused bythe rail manufacturing process, on the fatigue life tocrack initiation. Section 3.1 gives a short description ofthe new FE tool; Section 3.2 specifies the contact loadsof the Iron-ore Line train traffic situation analysed; Sec-tion 3.3 presents three analyses with different initialstress-states of the rail. The FE tool analyses were madewith the FE code ABAQUS [12].

3.1. Descriptions of FE tool, and models of track andrail

Two FE models, for track and rail, which are coupledby time-dependent boundary conditions form the FEtool. An elastic FE analysis using the track model calcu-lates time-dependent displacements at two cross-sections12 cm apart; these are then used as boundary conditionsin an elastic-plastic FE analysis using the rail model. Asa result, the influences of both the dynamic global trackresponse and the three-dimensional local elastic-plasticmaterial response in the rail are incorporated in the railfatigue analysis. It was shown in Ringsberg [1] that theglobal track response predicted by the track modelaffects the longitudinal residual stress-state in the railhead by some 10 per cent, as compared with a situationwhere it is disregarded.


The track model is an elastic beam element one whichis used in dynamic analysis: 32 sleeper bays are designedto represent an arbitrary track. The ballast material, thepads, the sleepers, and the rail beam are all modelled byfinite elements to enable realistic simulations of traintraffic conditions. Distributed and moving force combi-nations of normal, longitudinal, and transverse/lateralrail force components represent a train vehicle travellingalong the track.

The rail model is a three-dimensional (3D) solidelement model. The material volume near the wheel-railcontact undergoes elastic-plastic material response.Hence, high mesh density is used to resolve all the stressand strain gradients satisfactorily. The elastic–plasticmaterial behaviour was modelled with the nonlinearkinematic hardening model proposed by Jiang [13,14];Ekh [15] has coded this model as a user-supplied subrou-tine to the FE code ABAQUS. In addition, apart fromthe volume near the wheel-rail contact, the materialresponse is elastic and the mesh is modelled coarsely.Fig. 6 shows the rail model used in the present investi-gation.

Since train traffic on a straight track was investigated,the wheel-rail contact position was at the symmetry linein the rail longitudinal direction; this was verified byinspections at track. The wheel-rail contact loads actingon the rail model, in the normal, longitudinal, andtransverse/lateral directions, were represented by movingdistributed normal pressure and shear stress distri-butions. The distributions were calculated according tothe Hertz theory of rolling contact between two elasticnon-conforming solids with smooth and continuous con-

Fig. 6. The rail model of the FE tool.

tact surfaces [16]. The normal load was represented bya Hertzian contact pressure distribution, p(x,y), where(x,y) are the local coordinates that define the contactzone in the longitudinal and transverse/lateral directions,respectively. The traction forces, q(x,y), were modelledas proportional to p(x,y), that is q(x,y) = m(mx,my)·p(x,y)where (mx,my) are the friction coefficients in the (x,y)directions. Hence, regions of stick and slip were disre-garded, although they are known to occur.

3.2. Contact loads and rail steel material

The Iron-ore Line between Kiruna (Sweden) and Nar-vik (Norway) is two-way-trafficked on one track, i.e.loaded iron-ore cars run from Kiruna to Narvik wherethe ore is unloaded before the cars return (unloaded) toKiruna on the same track. The loaded iron-ore cars havean axle load of 30 tonnes and they run at 50 km/h on astraight track. Since these loaded iron-ore cars contributethe most to RCF damage of the rail, their contact loadswere used here. The 30 tonne axle load corresponded toa rail normal force of 147×103 N which, according tothe Hertz theory for the present wheel and rail geometry,equals a peak normal pressure of 1401 MPa. As straighttrack was modelled, only the in-plane rail longitudinalforce component due to traction was included. The fric-tion coefficient was taken as mx = 0.35, which is a rep-resentative value determined by material testing for drycontact for the current wheel and rail materials [4].Hence, there was a shear stress distribution for fully slip-ping contact only in the longitudinal direction of the rail.

The material for the rail FE model was pearlitic steelgrade 1100. Its constitutive material behaviour was rep-resented by the nonlinear kinematic hardening modelproposed by Jiang [13,14]. The Jiang model has thecapacity to simulate cyclic plasticity and ratchettingmaterial response with decaying rate. This type ofconstitutive model should be employed for the currentrepeated wheel–rail contact analyses, since a linear kine-matic hardening model, for example, cannot describe thematerial behaviour accurately for the current loadingsituation. The mechanical properties for the current steelgrade are presented in Table 1, while the parameters inthe Jiang model used to describe the cyclic behaviour ofthe material were taken from Ekh [15].

Table 1Mechanical properties for the pearlitic steel grade 1100

E sy b c e�f s�f ec J(GPa) (MPa) (%) (MPa)

209.82 400.1 �0.089 �0.559 10.3 936 11.5 0.2


3.3. Application of the FE tool to three examples

Three initial stress-state conditions of a rail were ana-lysed with the FE tool: (a) stress-free rail, (b) measuredresidual stress field of newly manufactured rails, and (c)calculated residual stress field from a rail manufacturingsimulation. The residual stress field (b) was measured tovalidate the computational model used to simulate therail manufacturing process (c), see Section 2.

A user-supplied subroutine was formulated in the FEcode ABAQUS, to transfer the results from the 2D FEanalysis (Section 2) to the 3D rail FE model. This wasdone by interpolation, between nodes, of node averagestresses in the longitudinal direction. These stressesbecame initial conditions to the first step of the 3D railFE model calculation. No contact loads were applied forthis step, and equilibrium was reached after some iter-ations.

Note that material properties for steel grade 900A wasused in the rail manufacturing simulation, while materialproperties for steel grade 1100 was used in the FE tool.The reason was that not all data for the 1100 materialwas known or found in the literature to make the railmanufacturing simulations for this material. However,these were available for the 900A material. Several railmanufacturing simulations were performed using knowndata for the steel grade 1100 together with additionalsteel grade 900 data, to check how the results were affec-ted. It was observed and concluded that the discrepancyin materials does not affect the conclusions drawn in thepresent investigation, since the residual stresses causedby the rail manufacturing process, for a steel of grade1100, are of the same magnitude as for 900A and havea similar distribution.

4. Life prediction of RCF crack initiation

All materials contain microcracks or defects, etc., thatgrow during fatigue loading. The fatigue growth ofcracks can be studied using fracture mechanics basedapproaches which are divided into three groups accord-ing to length scales [17]: microstructural fracture mech-anics (MFM), elastic-plastic fracture mechanics (EPFM),and linear-elastic fracture mechanics (LEFM). Theobjective of the present investigation, however, empha-sises calculating the time to fatigue crack initiation. Thiscould be done using a strain-based total-life approach,for which the growth of fatigue cracks does not need tobe modelled. In this approach, it is important to definewhat is meant by crack initiation. From an engineeringpoint of view, fatigue crack initiation is often used asthe definition of a crack that has completed its Stage Igrowth, i.e. crack sizes of approx. 5 to 10 grains [17].The material used here was a pearlitic steel grade 1100.The size of a fatigue-initiated crack in this material, fol-

lowing the above definition, was 0.1–0.5 mm. Therefore,fatigue crack initiation was defined here as a crack thathas a grown to a size of approx. 0.5 mm.

Ringsberg [2–4] proposed a method for fatigue analy-sis and life prediction of fatigue crack initiation in rollingcontacts; Fig. 7 shows the flow chart of the steps in themethod. The method was designed as an iterative pro-cedure by which each step should be critically reviseduntil good agreement with test results was achieved. Ithas been successfully used in previous work to deter-mine the time to fatigue crack initiation in rails and twindisc specimens used for accelerated RCF testing [2]. Theresults from the numerical analyses, presented in Sec-tions 2 and 3, were analysed for RCF crack initiationusing this method; they are presented in Section 5.

In the method proposed, it was presumed that the railmaterial was homogeneous and free from microcracks ordefects, etc. The wheel–rail rolling–sliding contact loadswere assumed to cause fatigue damage to the rail due tothe elastic-plastic material response in and near thewheel-rail contact zone. Hence, the RCF damage wasinduced by low-cycle fatigue and ratchetting damagemechanisms. A strain-based total-life approach wasadopted in the fatigue life predictions. The strain-lifeapproach used together with FE analyses forms a power-ful combination. For example, arbitrary geometries canbe analysed for fatigue, for any material and loading,without presuming at the outset the positions and sizes ofprimary fatigue cracks. Macroscopic material parametersthat are determined from (standard) tests can be utilisedin the fatigue analysis; they can also describe thematerial by a constitutive material model in the FEanalysis. Hence, a variety of materials can easily becompared in parametric studies by a substitution ofmaterial parameters [4].

A fatigue analysis was carried out as a post-processusing the results from an FE analysis. Hence, the FE andfatigue analyses were uncoupled. A damage summationrule similar to Palmgren-Miner’s, presented in Ringsberg[2], was used to calculate fatigue damage in the rail forevery wheel passage, n, from low-cycle fatigue (LCF)and ratchetting failure (RF). The number of wheel pass-ages to crack initiation for the current load and wheelpassage is denoted as NLCF

f for LCF and NRFf for ratchet-

ting damage, respectively.A fatigue damage parameter proposed by Jiang [18],

see Eq. (5), was used to calculate the fatigue damagecaused by low-cycle fatigue for every wheel passage n,i.e. DLCF = 1 /NLCF

f for wheel passage n. The materialplane with the largest value of the fatigue parameter inEq. (5a), FPmax, identified the crack plane and Eq. (5b)was used to calculate NLCF

f for the current wheel passagen, crack plane, and FPmax as

FPmax � ��smax��e2

� J�t�g�max

(5a)


Fig. 7. A method for fatigue life prediction of crack initiation [4].

FPmax �(s�f)2

E(2NLCF

f )2b � s�fe�f(2NLCFf )b+c (5b)

In Eq. (5), �� denotes the MacCauley bracket, �x� =0.5( |x| + x); smax and �e are the maximum stress andthe strain range normal to the crack plane; �t and �gare the shear stress range and the (engineering) shearstrain range on the crack plane; J is a material and loaddependent constant; E is the elastic modulus; s�f ande�f are the axial fatigue strength and the axial fatigue duc-tility coefficients; b and c are the fatigue strength andthe fatigue ductility exponents. The energy-density basedmodel in Eq. (5) was used together with the critical planeconcept to identify the position and orientation of thecrack plane.

Damage caused by ratchetting was computed forevery wheel passage, n, as DRF = 1 /NRF

f . A criterion forratchetting failure as proposed by Kapoor [19] was usedfor this purpose to calculate NRF

f as

NRFf � ec /�er where �er � �(�e)2 � ��g /�3�2

(6)

In Eq. (6), ec is a material constant that represents thematerial ductility when it is subjected to rolling contactloading [2]; �er is an equivalent ratchetting strain perwheel passage estimated from the average normal, �e,and shear, �g, ratchetting strains per wheel passage. Theratchetting deformation in rolling contacts is, in mostinstances, governed by the ratchetting shear strain.

Damage summation was calculated for every wheelpassage, n, as

D � �Nf

n � 1

max((dDLCF /dN)n, (dDRF /dN)n) (7)

The proposed damage summation rule accounts for avarying damage rate in the damage summation, whichmay be caused by, for example, a decaying ratchettingrate due to constant or variable amplitude loads. Failureis assumed to occur when D = 1.

Table 1 presents the mechanical properties used in

Eqs. (5) and (6), for the pearlitic steel grade 1100 whichhas properties to the pearlitic steel grade BS11 used inprevious work. Note, however, that the yield stress forthe pearlitic steel grade 1100 is normally 690 MPa. Thelower value of the yield stress in Table 1 was the yieldstress used in the current constitutive material model andit was obtained by parameter optimisation to fit experi-mental data for ratchetting material behaviour from cycle1 to 600, see Ekh [15] for details.

5. Results

5.1. Rail manufacturing simulation

The contour plots of the calculated residual stress fieldin the longitudinal direction, sx, after cooling and rollerstraightening can be seen in Figs. 8 and 9, respectively.

Fig. 8. Contour plot of the rail longitudinal residual stress distribution(sx) after cooling.


Fig. 9. Contour plot of the rail longitudinal residual stress distribution(sx) after cooling and roller straightening.

Note that the residual stress-state calculated in the railafter cooling (Fig. 8) seems to have no effect on theresidual stress-state (Fig. 9) after roller straightening.The result in Fig. 9 is also shown in Fig. 10 as a graphthat shows the calculated stress distribution together withresults from residual stress measurements made in newlymanufactured rails. The discontinuities in the graph,especially in the web region, are associated with numeri-cal errors from the calculation. The measurements weremade on the surface of a rail using a hole-drilling straingauge method [11]; the measurement uncertainty accord-ing to this method was 10 per cent. The agreement wasgood in the rail head but poor in the rail foot. The dis-

Fig. 10. Calculated (solid line) and measured residual stresses innewly manufactured rails.

agreement in the rail foot was related to the rather simplematerial model (linear kinematic hardening model) usedin the rail manufacturing simulation. This material modelcould not describe the material behaviour properly, dur-ing cyclic loading, in the roller straightening simulation,see Schleinzer [20]. A nonlinear kinematic hardeningmodel was recommended for use in future work to achi-eve better agreement between measurements and simul-ation in the entire rail cross-section. However, as RCFanalysis of the rail head was the main interest of thepresent investigation, the poor agreement in results forthe rail foot had no influence on the results or con-clusions from the subsequent FE tool and RCF analyses.

5.2. FE tool and RCF analyses—comparison of initialconditions to rail FE model

Eight wheel passages were simulated using the FEtool. The material response in the rail head was ratchet-ting, which decayed for every wheel passage. The RCFassessments were made following the proposed evalu-ation method given in Section 4. The three FE toolanalyses were compared with respect to the position ofthe greatest fatigue damage, the orientation of thematerial plane most damaged (crack plane), and thefatigue life to crack initiation on this plane.

Table 2 presents results from the three FE tool andRCF analyses. The results are presented for the eighthwheel passage alone: the maximum residual von Miseseffective stress (smax

eff ); the maximum accumulated effec-tive plastic strain according to von Mises (emax

eff, pl); themaximum value of the fatigue parameter (FPmax),recorded during the eighth wheel passage on a crackplane defined by an orientation j and by the distance dfrom the rail head surface (see Fig. 11). This type ofcomparison revealed whether the initial stress field ofthe rail model was redistributed, or if fatigue damageaccumulation was still influenced by the initial con-ditions of the model.

In all of the three FE tool analyses d was calculatedas 2.8 mm, i.e. the greatest damage was subsurface. Thiswas expected due to the very high axle load and the lowfriction coefficient. The largest values of smax

eff andemax

eff, pl were also found at this position. The orientationof j was in accordance with that observed in damagedrails that had been taken out from service for inspection.

Fig. 12 shows the calculated residual von Mises effec-tive stress in the rail head after the rail manufacturingsimulation, i.e. it shows the initial residual stress distri-bution to example (c). The residual von Mises effectivestress after the eighth wheel passage is presented in Fig.13. A contour plot of the accumulated effective plasticstrain according to von Mises after the eighth wheelpassage is shown in Fig. 14, and Fig. 15 shows theaccumulation of this strain for every wheel passage forthe examples (a) and (c).


Table 2Results from three FE tool and RCF analyses for the eighth wheel passage

Examples smaxeff (MPa) emax

eff, pl (%) FPmax (MPa) j (deg.) d (mm)

(a) Stress-free; reference example 270 1.62 1.32 40 2.8(b) Measured stress distribution 265 1.63 1.25 41 2.8(c) Calculated stress distribution 273 1.69 1.33 40 2.8

Fig. 11. Definition of the crack plane angle j and the distance d tothe position of the greatest fatigue damage.

Fig. 12. The residual von Mises effective stress in the rail head afterthe rail manufacturing simulation.

Fig. 13. The residual von Mises effective stress in the rail head afterthe eighth wheel passage.

Fig. 14. The accumulated effective plastic strain according to vonMises after the eighth wheel passage.

Fig. 15. The maximum value of the accumulated effective plasticstrain according to von Mises vs. the number of wheel passages.

The results in Table 2 show that the differencebetween the three examples was small, which indicatesthat the residual stresses introduced initially werealready redistributed during the first wheel passages. Inrelation to the reference example, the largest percentagedifference in the maximum von Mises effective stresswas 2%; the maximum effective plastic strain accordingto von Mises, 4%; the maximum fatigue parameterrecorded during the eighth wheel passage, 5%; there wasa 2% difference in j.

A calculation of the train traffic time (in days) to sub-surface fatigue crack initiation was made using theresults from the complete RCF analyses and statistics for


the annual train traffic density at the Iron-ore Line. Notehere that the residual stress in the rail head for theexamples (b) and (c) in Fig. 10 is of almost the samemagnitude. The small discrepancy gives rise to a differ-ence in the fatigue parameter FPmax during wheel–railcontact loading, and hence, the time to fatigue crackinitiation is also influenced. In addition, the damage ratefor subsequent wheel passages was extrapolated usingthe values calculated for the sixth to eighth wheel pass-ages. As a result, subsurface fatigue crack initiation wasdetermined to be 2 days of train traffic for the referenceexample (a), example (b) showed 19% more time tofatigue crack initiation, and example (c) 2% less. Theseresults have not been validated because the fatiguecracks were subsurface-initiated, which meant it was notpossible to acquire results for validation from visualinspections.

5.3. When do initial residual stresses affect the railfatigue life?

The results presented for the 30 tonne axle load inSection 5.2 showed that the initial residual stress-statehad negligible influence on the fatigue life to crackinitiation. The question is then if, or when, an initialresidual stress-state, here caused by the rail manufactur-ing process, affects the fatigue initiation life. Accordingto the results presented in Fig. 10, the longitudinalresidual stress in the rail head after rail manufacturingis tensile. The initiation of surface-initiated cracks calledhead checks is governed by a ratchetting shear strain onthe rail head surface, see Ringsberg [4]. However, thefatigue initiation life is shorter when a tensile stress actsnormal to the crack plane, see Eq. (5), than it is whenthere is compressive stress. Consequently, the fatiguelife to crack initiation of head checks can be expectedto be lower, for example (c), as compared with (a). TheFE tool was utilised for six axle loads to determine theaxle load for which the longitudinal residual stresschanges from tension to compression. Examples (a) and(c) were compared in separate FE tool calculations forthe axle loads 10, 12, 15, 20, 25, and 30 tonnes, withmx = 0.35. The calculations were done for eight wheelpassages, and the results are presented in Fig. 16 as thelongitudinal residual stress after the eighth wheel pass-age. The results show that the longitudinal residual stresschanges from tension to compression at a 13 tonne axleload, for example (a), and at 16 tonnes, for example (c).For both of these axle loads, the greatest fatigue damagewas found at the rail head surface for the prevailing con-tact load conditions. The conclusion was that, for an axleload of the same magnitude, example (c) has a shorterfatigue initiation life than example (a).

Fig. 16. Longitudinal residual stress (sx) after eight wheel passagesvs. axle load.

6. Discussion

The influence of residual stresses caused by the railmanufacturing process on the rail fatigue life to crackinitiation was investigated. A comparison of threeexamples showed that the initial residual stress-statedoes not have any great influence on the fatigue crackinitiation life for the train traffic situation analysed. Itwas concluded that the local wheel-rail contact load gov-erns fatigue damage, in particular for very high axleloads. Note, however, that the RCF life predictionmethod identified the greatest fatigue damage at thedepth where the start of fatigue cracking was found incracked rails taken out of service.

The discrepancy in magnitude of the longitudinalstresses in the rail head, between examples (b) and (c)shown in Fig. 10, was within the error margin of themeasurements. Note that this error was transferredthroughout the numerical analyses, see the results inTable 2.

The simulation of the rail manufacturing process canbe improved in future work. In the present investigation,the roller straightening process was simulated using a2D FE model. Schleinzer [5] simulated the rollerstraightening process in three-dimensional FE calcu-lations. An advantage of using Schleinzer’s 3D FEmodel is that all stress and strain components are incor-porated in the FE calculation. On the other hand, thecomputational time is excessive when compared with the2D FE model used in the present investigation. Using theSchleinzer approach would not change the conclusionsdrawn in the present investigation, since the local wheel-rail contact load still governs the rail fatigue damage.

The fatigue damage accumulated in the rail from theroller straightening process was not incorporated in thepresent investigation. Here, only the stress-state causedby the cooling and roller straightening steps was trans-ferred to the rail model of the FE tool where it was usedas an initial condition. The reason it was disregarded


was that, in the authors’ opinion, its contribution to totalaccumulated fatigue damage is negligible. However, forcomplete RCF analysis of rail fatigue life, fatigue dam-age accumulation calculations could start by recordingthe accumulation of plastic strain from the rail manufac-turing process. To keep track of changes in residualstress, plastic strain, material parameters, and the evol-ution of fatigue damage, without loss of informationbetween the different steps, would require using thesame 3D rail FE model and the same constitutivematerial model through the entire simulation.

The present work emphasised on RCF evaluation ofrails at straight track, where the contact load motion wasalong the symmetry line in the longitudinal direction. Atcurved track, the contact position/positions are closer tothe rail head gauge corner. The tangential loads and nor-mal contact load are often higher, as compared withthose at straight track, for the same train traffic con-ditions. As a result, the initial residual stresses causedby the rail manufacturing will almost immediately redis-tribute, and thus, have less influence on the RCF life tocrack initiation. The results presented in Fig. 16 illustratehow the axle load/normal contact load affects the longi-tudinal residual stress. In addition, the magnitude of thefriction coefficient used to model the tangential loads inthe longitudinal direction is important for the outcomeof the results. A higher value would have increased theplastic shear deformation near the surface and reducedthe time to fatigue initiation, while a lower value reducessurface plastic flow and the risk/time for surface fatigueof rails.

7. Conclusions

The cooling and roller straightening sequences of therail manufacturing process were simulated in this inves-tigation. The objective was to calculate the residualstress field caused by the rail manufacturing process andto determine whether this influences the RCF life ofrails. Three examples of initial conditions in a rail werecompared using an FE tool developed for RCF analysisof rails: (a) stress-free rail, (b) introduction of a meas-ured residual stress-state in a newly manufactured rail,and (c) introduction of a residual stress-state as calcu-lated from a rail manufacturing simulation. With the FEtool and fatigue calculations, the three examples werecompared. The three sets of initial conditions weredesigned in accordance with the heavy-haul train trafficsituation at the Iron-ore Line (in Sweden) where subsur-face crack initiation has been observed. The conclusionscan be summarised as follows.

The results from the numerical simulation of a railmanufacturing process showed acceptable agreement forthe longitudinal residual stress field in the rail head,which was the main interest in this investigation, when

compared with residual stresses measured in newlymanufactured rails.

The FE tool and fatigue analyses of the three examplesshowed that there was little difference in results withrespect to position of the greatest fatigue damage, orien-tation of the crack plane, and time to fatigue crackinitiation.

The initially introduced stress-states of the rail modelin the FE tool were already redistributed after the firstwheel passages. Hence, it was concluded that the localwheel load governs the fatigue damage of rails.

The position of the greatest fatigue damage was calcu-lated to be 2.8 mm below the surface. This depth corre-sponded to the depth where cracks have been found toinitiate.

The FE tool calculations made for several axle loads,for example (a) and (c), showed that the longitudinalresidual stress caused by the rail manufacturing processshortens the onset of fatigue initiation of surface-initiated cracks, i.e. head checks.

In future work, the fatigue damage calculations couldalso incorporate the damage introduced during the railmanufacturing process. Although this contribution maybe negligible in total, it offers a more complete RCF lifeprediction model which can be important in assessmentsof different types of rail materials and train traffic situ-ations. Calculating this damage would require using thesame 3D rail FE model and the same constitutivematerial model through the entire simulation, to incor-porate the multiaxial effects of the evolution of fatiguedamage.

Acknowledgements

The work presented was funded by the SwedishNational Board for Industrial and Technical Develop-ment (NUTEK, now VINNOVA), Lulea RailwayResearch Centre (JVTC), the Swedish National Centreof Excellence CHARMEC (CHAlmers RailwayMECHanics), and the Polhem Laboratory at Lulea Uni-versity of Technology. Professor Lennart Josefson at theDepartment of Applied Mechanics, Chalmers Universityof Technology, and Mats Nasstrom at the Division ofComputer Aided Design, Lulea University of Tech-nology, are acknowledged.

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