A TOOL AND A METHOD FOR FE ANALYSIS OF WHEEL AND RAIL INTERACTION

Tanel Telliskivi, Ulf Olofsson, Ulf Sellgren and Patrik KruseMachine Elements, Department of Machine Design

Royal Institute of Technology (KTH), Stockholm, Sweden

ABSTRACT

Damage mechanisms such as surface cracks, plasticdeformation and wear can significantly reduce the service life ofrailway track and rolling stock. They also have a negative impacton the rolling noise as well as on the riding comfort. A properunderstanding of these mechanisms require a detailed knowledgeof physical interaction between wheel and rail. Furthermore,demands for higher train speeds and increased axle loads impliesthat the consequences of larger contact forces between wheel andrail must be thoroughly investigated.

Two methods have traditionally been used to investigate therail-wheel contact, that is the Hertz analytical method andsimplified numerical methods based on the boundary element(BE) method. These methods rely on a half-space assumption anda linear material model. To overcome these limitations, a tool forFE-based quasi-static wheel-rail contact simulations has beendeveloped. The tool is a library of Ansys macro routines forconfiguring, meshing and loading of a parametric wheel-railmodel. The meshing is based on measured wheel and rail profiles.In order to reduce the size of the computational model, thesuperelement technique is utilized. The wheel and rail materialsin the contact region are treated as elastic-plastic with kinematichardening. The kinematic constraints are enforced with the Ansyscontact element Contac49. By controlling the values of theconfiguration parameters, representations of various driving casescan be generated. The quasi-static loads are obtained from trainmotion simulations with special purpose software. Interactionphenomena such as rolling, spinning and sliding can be included.

The modeling tool and a methodology are described in thepresented paper. Simulation results are compared with Hertzianand BE solutions. Significant differences in the calculated statebetween the FE solution and the traditional approaches can beobserved. These differences are most significant in situations withflange contact.

1. INTRODUCTION

Most railway wheels are rigidly mounted on a steel shaft. Atypical wheelset on a straight track is shown in figure 1. The axleload may be as high as 220 kN and the contact area between awheel and the rail is roughly 1 cm2. The contact region is thusvery highly stressed. The interaction in the contact zone betweenwheel and rail is determined by the global dynamic behavior ofthe vehicle and by various physical phenomena that occur in thecontact zone. The profiles of wheel treads and railheads aretransformed by the wheel and rail interaction. Thistransformation, which may be severe in curves, has a significanteffect on the contact state.

FIGURE 1. A WHEELSET ON A STRAIGHT TRACK.

Since the early 1970s, numerical simulations of the dynamicbehavior of rail vehicles and the interaction between vehicle andtrack have been performed. Software, such as Vampire developedby British Rail, Medyna by Deutsche Luft und Raumfahrt, andNucars in the USA, were developed for that purpose. They wereall highly specialized and optimized for a reasonable a turn-around time for a simulation (Andersson et al., 2000). Anexample of recently developed commercially available software isGensys (1999). General-purpose software for dynamicsimulations of multi-body-systems (MBS), such as Adams,Simpack, and Dads, have recently included features that enableefficient dynamic simulations of railway vehicles and vehicle-track interaction. Adams, Gensys, Nucars, Simpack and Vampirehave recently been benchmarked (Iwnicki, 1998)(Iwnicki, 1999).One-dimensional beam models are usually sufficient for thefrequency range up to three kHz (Knothe et al., 1994). Softwarefor vehicle motion simulations, are normally concerned with theorientation of each wheel relative to the track, and thus the point-of-contact between wheel tread an rail head and the contact forcesthat are caused by the dynamic interaction (see figure 2).

FIGURE 2. CONTACT FORCES FROM AN MBS-SIMULATION.

Damage mechanisms such as surface cracks, plasticdeformation and wear, see for example Kalousek et al. (1999) fora survey, can significantly reduce the service life of railway trackand rolling stock. Furthermore, they can have a negative impacton the rolling noise as well as on the riding comfort. A properunderstanding of these mechanisms require a detailed knowledgeof the wheel and rail interaction. Five theories of rolling arepresently in use: the two-dimensional theory of Carter (1926), thelinear theory (Kalker, 1967), the complete contact theory (Kalker,1983), the theory of Shen et al. (1984), and the simplified theory(Kalker, 1982b). All of these theories have limitations and theycan be viewed as complementary.

Continuum rolling contact theory started with a publicationby Carter (1926), where he approximated the wheel by a cylinderand the rail by an infinite half-space. The analysis was two-dimensional and the exact solution was found. Carter showed thatthe difference between the circumferential velocity VC of a drivenwheel and the translational velocity VT of the wheel has a non-zero value as soon as an accelerating or a braking couple isapplied to the wheel. This difference increases with increasingcouple until the maximum value according to Coulomb is

reached. Carter formulated a creepage-force law connecting thedrivingbraking couple and the velocity difference:

( )

+

=

sign

kk

F

F

N

T225.0

2

2

k

k(1)

where FT and FN is the total tangential and normal contactforce, respectively, per unit lateral length, is the coefficient offriction, =2(VT - VC)/(VT + VC) is the creepage, and k is thecreepage coefficient. is a function of , the radius of the wheel,and the semilength of the contact area measured in the rollingdirection.

Carters theory is sufficient when we consider the action ofdriven wheels, e.g. it is capable of predicting the frictional lossesin a locomotive driving wheel, but it is not sufficient for vehiclemotion simulations. For that we must treat forces in the lateraldirection together with the motion in the rolling direction(Kalker, 1991).

Johnson (1958) generalized Carters result to circular contactsand longitudinal and lateral creepage. Vermeulen and Johnson(1964) generalized this theory to elliptical contact areas. Shen etal. (1984) improved the results by replacing the approximatevalues for the creepage coefficients given by Vermeulen andJohnson with more accurate values. In these theories, which areall Hertzian-based, it turned out that the contact area is elliptic inform, with semiaxes a and b in the rolling and lateral directionsrespectively. The ratio of the axes, a/b depends only on thecurvatures of the wheel and the rail. Furthermore, the size of thecontact area depends on the normal force FN but it is independentof the tangential force FT.

FIGURE 3. CONTACT ON STRAIGHT RAIL (A) ANDFLANGE CONTACT ON HIGH RAIL IN A CURVE (B).

In vehicle motion simulations, usually only the global contactforces, as shown in figure 2, are required. A linearization of therelation between the tangential contact forces and the creepage, inrailway mechanics usually referred to as the linear theory (Kalker,

A

B

y

z

xFN

FYFX MZ

1967), is thus extensively used in this class of simulations. Thelinear theory is valid for small motions, without flange contact(see figure 3A). Flanging may occur on high rail, i.e. the outerrail in a curve, see the rightmost wheel in figure 3B. Due to theconicity of the wheel profile, flanging results in a large spin.

In the simplified theory, wheel and rail are modeled as rigidbodies with a set of three-orthogonal springs located at discretepoints on the interacting surfaces (Kalker, 1973). In thesimplified theory, the surface displacement at one unique pointdepends only on the surface traction at the same point, i.e. it is aso-called Winkler model. The simplified theory was earlyimplemented in special purpose computer codes (Kalker, 1982b).It has shown to be able to efficiently interpret a large number ofcontact phenomena as long as the contact is Hertzian (Kalker,1991).

Kalker (1979) generalized the three-dimensional rollingcontact problem of two elastic bodies for combinations oflongitudinal, lateral, and spin creepage. Spin creepage, or spin forshort, is a significant phenomenon in curves. It is caused by arotational velocity around the vertical and longitudinal axes of thewheelset due to different rolling radius of the two wheels in awheelset and the conicity of the wheel profile. Spin due toconicity is comparable to camber in the automotive industry.Kalker (1982) extended his theory of rolling contact betweenarbitrary bodies for the case where the shape of the contact area isnonelliptical, and thus non-Hertzian. This theory is often referredto as the complete theory, although it is limited to contactproblems between linear elastic bodies that can be described byhalf-spaces. In order to get an approximate solution, the contactarea is divided into rectangular elements. This theory wasimplemented in a computer program called Contact (Kalker,1983), which is based on the boundary element (BE) method.Contact which is roughly 400 times slower than routines based onthe simplified theory (Kalker, 1991) has been used to validate thelinear and simplified theories as well as to validate the theory byShen, Hendrick and Elkins (1984) which also is significantlyfaster than Contact.

In strength and fatigue calculations, a common practice is toassume an elliptic contact area and to use the Hertzian traction forthe normal pressure, while the tangential distribution is found bymultiplying the normal pressure distribution by the coefficient offriction (Kalker, 1991). This approach is also used in more recentstudies, e.g. by Ekberg (2000).

Higher train speeds and increased axle loads have led tolarger contact forces between wheel and rail. The half-spaceassumption in the traditional approaches puts geometricallimitations on the contact i.e. the significant dimensions of thecontact area must be small compared to the relative radii ofcurvature of each body. Especially in the gauge corner of the railprofile the half-space assumption is not valid since the contactradius here is of the same order of size to the contact zone i.e.approximately around 1cm. The form change of rail can be largeover time, see Olofsson and Nilsson (1998) and Olofsson (1999).Figure 4 shows the transformation of the profile of a UIC 60 highrail over a period of two years in a narrow curve with a radius of303m and trafficked by commuter trains. These experimentalresults show that plastic deformation must be treated thoroughlyin wheel-rail contact analysis. Previously presented results byCassidy (1996) and Knothe et al. (1999) reveal contact stresses

up to and above 3 GPa when the wheel is in contact with thegauge corner of the rail. The former result was based on Hertziantheory and the latter was calculated with Contact. Such resultshighlight the need for a nonlinear elastic-plastic material modeland a thorough three-dimensional approach.

FIGURE 4. FORM CHANGE OF AN UIC HIGH RAIL IN ANARROW CURVE. THREE YEAR OLD RAIL AT START

(A). NEW RAIL AT START (B).

To overcome the limitations inherent in the traditionalapproaches, a tool for FE-based quasi-static wheel-rail contactmodeling and simulations has been developed. The tool is alibrary of macro routines for configuring, meshing and loading aparametric wheel-rail model. The routines are written in theAnsys programming language. The meshing can be based onmeasured wheel and rail profiles, i.e. worn profiles. Thekinematic constraints are enforced with the Ansys contact elementContac49 (Ansys, 1997). The material models are treated aselastic-plastic with kinematic hardening. By controlling thevalues of the configuration parameters, representations of variousdriving cases can be generated. The quasi-static loads areobtained from train dynamic calculations with special purposeMBS software. Interaction phenomena such as rolling, spinningand sliding can be included. In order to reduce the size of thecomputational model, the superelement technique is utilized forthe linear model features.

The modeling and simulation tool and a methodology aredescribed below, and simulation results are compared withsolutions obtained with traditional methods.

0 20 40 60

0

10

20

30

rail new at test start

rail three years old at test start

wear

plasticdeformation

hei

ght (

mm

)

length (mm)

0 20 40 60

0

10

20

30

plasticdeformation

wear

hei

ght (

mm

)

length (mm)

initial profile (3 years)one year of traffictwo years of traffic

initial profile (new)one year of traffictwo years of traffic

A

B

Center node

SuperelementWheel

SuperelementRail

Contact region

600 mm

2. FE MODELLING OF WHEEL-RAIL INTERACTION

A complete FE model of one wheel, a piece of rail, a set ofcontact elements and the quasi static loads on the wheel areinteractively created with a library of Ansys macro routines thatare accessed with the Ansys Graphical User Interface (GUI), seefigure 5. Sets of keypoints that define the rail head and wheeltread profiles are assumed to exist as separate files. Keypointsthat describe the shape of a profile are preferably generated with astandard Miniprof instrument. The two sets of keypoints areconverted to 2-D spline curves by an Ansys macro routine.

FIGURE 5. THE MODELING TOOL IS ACCESSEDTHROUGH THE ANSYS GUI.

The basic steps in the creation of a wheel-rail interactionmodel for an X1 motor coach in a curve with a radius of303m and a standard UIC60 rail is presented below. Thethree-dimensional geometric model of the wheel is generatedby revolving the two-dimensional spline curves that describethe profile of the wheel tread. The rail model is created byextruding railhead profile curves a distance of 600 mm, whichis the distance between the sleepers. Two sets of curves for anew wheel and a new rail are shown in figure 6. The twosolid bodies are shown in figure 7. To get a reasonableconfigure model, the wheel is spatially oriented relative to therail according to the quasi-static state calculated by an MBSsoftware, e.g. Gensys or Medyna. To aid an efficientdiscretization of the contact region, a measure of the expectedcontact length and the number of expected contact patcheshave to be supplied by the user.

FIGURE 6. SPLINE CURVES GENERATED FROMMEASURED POINT SETS.

FIGURE 7. SOLID MODEL GENERATED FROM THEPROFILE CURVES.

The contact region, i.e. the small portion of the two bodiesthat are close to the anticipated contact patch, is meshed with theAnsys linear isoparametric element Solid45 (Ansys, 1997). Forthese elements, which almost exclusively are hexahedrons (seefigure 8), an elastic-plastic material model with kinematichardening is defined (see figure 9).

50 mm

Rolling direction

10

Contact regionWheel

Contact regionRail

Constraint equations connectcenter node to hub

Contact region

Angel 1/30

The size of the contact zone is approximately 30mm in thelateral direction of the rail, 50mm in the longitudinal direction,and 10mm in the normal direction. The size in the longitudinaldirection is based on the size of the contact zone and the distanceof rolling required for the simulation.

FIGURE 9. THE WHEEL-RAIL CONTACT REGION.

SuperelementsForm Element Number of Number MasterFeature type elements of nodes DOFsRail SOLID45 35131 14796 1701Wheel SOLID45 284 212741 2709Hub SHELL63 52785 171 0

Contact regionRail SOLID45 4560 5439 16317Wheel SOLID45 5920 7056 21168Center MASS21 1 1 6Contact CONTAC49 2978 1391 0

TABLE 1. TYPICAL SIZE OF A WHEEL-RAIL MODEL.

The main parts of the wheel and rail bodies are meshed withdegenerated linear isoparametric elements. The hub surface iscovered with shell elements. These two submodels are condensedto superelements. The size of a typical wheel-rail model is givenin table 1. The nodes on the hub surface are connected to a centernode with constraint equations (see figure 10).

0.0 0.2 0.4 0.6 0.8 1.0 (%)

800

640

480

320

160

0

A

B

FIGURE 9. NON-LINEAR MATERIAL MODELS FOR THERAIL (A) AND WHEEL (B) PORTIONS OF THE CONTACT

REGION.

3. A METHODOLOGY

Here, the term methodology is used for a collection ofmethods and tools, the use of which is governed by a processsuperimposed on the whole (Coleman, 1994). Generally, amethod, which is an organized, single purpose discipline orpractice, evolve as a distillation of the best-practices experiencein a particular domain of cognitive or physical activity (IDEF4,1995). A tool refers to a software system, such as a FE system,designed to support the method.

FIGURE 10. THE WHEEL HUB IS CONNECTED TO THE CENTER NODE WITH CONSTRAINT EQUATIONS.

Rail Wheel

FIGURE 11. A FE MODELING PROCESS AND THE SUPPORTING TOOLS.

FE modeling of the wheel-rail interaction requires the shapeof the geometric, domains, a material model, a value for thecoefficient of friction, and knowledge about the contact forces.

The material model is based on stress-strain curves suppliedby the manufacturers of wheel and rail. The coefficient of frictionis achieved from field instruments such as the Salient systemtribometer. The profiles of railheads and wheel treads aremeasured with the Miniprof instrument. MBS-simulationsprovide contact point locations and quasi-static contact forces.These contact forces are transformed to global forces at the centernode. With these data as input, the macro routines describedabove is capable of defining a complete FE model and the properboundary conditions for a quasi-static simulation that capture thephysical behavior that is caused by the combined rolling andsliding interaction. In a final step, the contact state history isextracted from the Ansys result database an exported in ASCII-format for further analysis and manipulation.

4. A COMPARISON WITH TRADITIONAL METHODS

A sharp curve in a track trafficked by commuter trains servingthe Stockholm area is chosen for a comparison between the FEsimulation results and results obtained with traditional methods.The chosen track carries almost exclusively unidirectionalcommuter trains with an average speed of 75 km/h. Two types ofvehicles are used: the X1 and the X10 both operating in pairswith one powered unit and one trailing unit. This track and the

rolling stock have been studied in a national Swedish program.Both rail and wheel profiles have been measured over a couple ofyears, see Nilsson (2000). Furthermore, has the X1 and the X10vehicles been modeled in the train dynamic simulation softwareGensys, see Jendel (2000), giving access to the necessary inputdata in form of wheel attitude with optional contact locations andforces resulted from simulations.

Two cases were used to study the model. Both cases werefrom simulation of the X1 powered unit and represent the firstand second wheel set in the leading boogie. The coordinatesystem is chosen similar to the Deutche Industrial Norm (DIN)with positive vertical (z) co-ordinate upwards, y to left and x ispositive to the train motion direction. Figure 12 presents thecontact points location on the wheel and the rail and table 1shows the forces in the center point of the wheel. From ageometrical perspective the two load cases represents the contactpoints with a large difference in the curvature of contactingbodies. In case 1, the minimum contact radius was about 300 mmand in case 2 it was about 20 mm. New rail profiles and a wheelprofile from a X1 train that has been in traffic for two years wereused in the two cases. In both cases, the normal force was80377N.

The tool for the FE analysis is made and the preliminaryresults along with the comparison to the main concurrentlyavailable methods is outlined. In the first stage the differencesbetween the methods are presented (see table 2 and figure 13).

Ansys,Macros

Macros

FE modelingof wheel-railinteraction

FE simulation

Ansys,Macros

Rail profile

Wheel profile

Wheel location

Contact forces

Rail measurement

Miniprof

Miniprof

Wheel measurement

measurement

Salienttribometer

Stress-strain curves

FE modelContact state history

Simulationconditiondefinition

Simulation control

Boundaryconditions

Motion simulation

Gensys/Medyna

Case 1 Case 2

FIGURE 12. CONTACT POINT LOCATION FOR THE TWO TEST CASES.

Method Case 1 Case 2FEM with plasticity 606 MPa 577 MPaContact 3057 MPa 715 MPaHertzian max stress - 1080 MPa

TABLE 2. MAXIMUM CONTACT PRESSUREWITH THE FE AND TRADITIONAL METHODS.

The main scope for this work was to enhance the knowledgeof the contact pressure and the maximum stresses in bulkmaterial. This should give an appropriate basis to study thedegradation mechanism along with the wear simulation. Theresults in case 1 could not be compared with the Hertz methodassuming one-point contact. The Hertzian solution showedapproximately twice the contact length in y-direction comparedwith the other methods. This remained the same for the regions ofthe surface radii of curvature in the range of 0.01 .. 0.015m forthe rail and -0.02.. -0.1m for the wheel. So, the normal force wassplit similar way as in program Contact to three parts. After thatthe results were very similar for these two classical methods (seefigure 13). Compared to Ansys, the difference was approximately300% for the contact area and more than 200% for the maximumpressure. For case 2 the result show that the difference in themaximum contact pressure and the size of the contact area wassmall when the minimum contact radius is large compared withthe significant dimensions of the contact area, i.e. the half spaceassumption is valid.

Using the linear-elastic model the differences in results aresignificant in regards especially to the relatively new wheel andrail shapes. Significant flattening of the contact pressure profilewas found with increasing plastic deformation. The maximumpressure and plastic work moved outward in the direction of thecontact edge along with the increase of friction coefficient.

The distribution of the equivalent vonMises stress shows that,even with the three contact patches, significant yielding willoccur (see figure 14).

FIGURE 13. RESULTS FROM THECOMPARISON BETWEEN THREE DIFFERENTCONTACT MECHANICS METHODS, MAXIMUMCONTACT PRESSURE AND CONTACT AREA.

Experiments have shown that the losses in the rail cross-sectional area remains approximately constant in time, seeNilsson (2000). Assuming a constant rate of degradation therelationship between the plastic flow and wear changes. In theinitial phase, i.e. case 1, the plastic work is very large. Themaximum equivalent von Mises stress exceeds even the ultimatestress limit which for the actual plasticity model was 606MPa(see table 2). Thus, the material in this phase of the degradationprocess behaves perfectly plastic. The plastic flow hardens thematerial and makes the contact more conform. In the continuingprocess, other wear mechanisms will thus be significant.

FIGURE 14. DISTRIBUTION OF VON MISESEQUIVALENT STRESS IN THREE YEARS OLD RAIL AND

WHEEL.

5. CONCLUSIONS

A tool for contact mechanics modeling and simulation of thewheel rail contact has been developed. The geometry of thecontact can easily be changed. The model can be generated frommeasured wheel and rail profiles. Traditional methods andcomputational tools are limited by an half space assumption and alinear material model. The results from two test cases show thatthe difference in maximum equivalent stress between traditionalmethods and the FE model is small when the minimum contactradius is large compared to the significant dimensions of thecontact area, i.e. when the half space assumptions is valid.However, in the test case where the minimum contact radius wasof the same order as the significant dimensions of the contact areathe difference between the FE results and results obtained withtraditional Hertzian and BE methods was as large as 3GPa. Thislarge difference was probably due to limitations in both the halfspace assumption and the linear elastic material model in thetraditional methods.

6. ADDITIONAL WORK TO BE PERFORMED

The regime for a rail and wheel contact is frequentlycharacterized as either stress related or wear related (Tournay,1996). Crack initiation and propagation are examples of damagemechanisms that are driven by stresses.

In many situations, wear is a significant damage mechanism.The wear can be integrated if the the contact state history isknown. A wear model includes a wear coefficient, which has ascientific base, see for example Archard (1953) and Lim andAshby (1987). If the contact pressure distribution and theaccumulated sliding distances are known, the right wear regimecan be predicted for each point that has passed the contact zone.For a given wear regime, the wear coefficient can be obtainedfrom lab-tests.

Already the pure static tests along with the degraded profiles

give a lot of information necessary for the wear-plasticityanalysis.

7. ACKNOWLEDGEMENT

This work was performed within the Swedish researchprogramme SAMBA. The work was financially supported by theSwedish National Board for Industrial an d TechnicalDevelopment (NUTEK), Adtranz Sweden AB, Stockholm LocalTraffic, the Swedish National Rail Administration and theSwedish State Railways.

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