Wear 258 (2005) 1255–1264

Structural integrity analysis of a tram-way: load spectra andmaterial damage

S. Beretta∗, F. Braghin, G. Bucca, H. Desimone

Dipartimento di Meccanica, Politecnico di Milano, Via La Masa 34, 20158 Milano, Italy

Received 13 June 2003; received in revised form 28 November 2003; accepted 1 March 2004

Abstract

Transportation authorities have to support high costs for track maintenance due to the wear of the rails and to crack generation at someparticular points along the line. To set up the planning of the maintenance and inspection operations, a predictive approach could be veryuseful. Moreover, such an approach could help tramcar designers to design tramcars less aggressive towards the line. This paper deals witha numerical approach for the prediction of damage phenomena on tramline rails. In order to be able to foresee maintenance and inspectioni multi-bodyt ach is veryg©

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ntervals, both wear and ratchetting analyses are carried out on the basis of load spectra that were determined with a validatedramcar model. Although the damage analysis is focused on a specific case (one single curve with a radius of 50 m), the approeneral thus allowing to be used for assessing tramcar design.2004 Elsevier B.V. All rights reserved.

eywords:Rolling contact fatigue; Ratchetting; Tramcar lines; Load spectra; Rail wear

. Introduction

Anybody who has ever taken a tramcar will have noticedhat at some points along the line the noise generation as wells the vibrations inside the vehicle are particularly annoying.his is due to the fact that, at these points, the track showsevere wear and fatigue of the rails. In order to avoid theseroblems, that are strictly related to ride comfort and runningafety, urban transportation authorities have to support highosts for track maintenance.

Tramcars cannot be considered as small railway vehicles.n fact, design criteria for tramcars are completely differentrom those of railway vehicles: the load per axle is less than0 kN, the vehicle speed is smaller than 80 km/h, usually in-ependent wheels with small radii and low-floor solutions aredopted and the track geometry is completely different (theurve radius values may reach 15 m and no super-elevations usually adopted). This means that problems that occur on

∗ Corresponding author. Tel.: +39 02 2399 8246; fax: +39 02 2399 8202.E-mail address:stefano.beretta@polimi.it (S. Beretta).

tramcar vehicles will usually not occur on railway vehicand vice versa.

The axle load, combined with a wheel radius lesshalf of that of common railway vehicles, leads to a dramincrease of normal surface stresses thus increasing theability of occurrence of plastic deformations in both wheand rails. The tramcar vehicle speed is small when compto railway vehicles due to the fact that the kinematic cstraints (and/or secondary suspensions) between carbobogie/semi-bogie are loose to allow the vehicles to runcurves characterised by very small radii. Thus, one woulpect smaller creepages and tractions inside the contacHowever, due to the high angle of attack of the wheelsetrespect to the rails (small-radius curves), complete sation of the contact region often occurs in curving conditioMoreover, the adoption of independent wheels, neceon low-floor vehicles, causes the disappearance of thetorque produced by the longitudinal contact forces on thewheels that allows to re-centre the wheelset. This meana wheel in flanging conditions will stick to these conditiountil a perturbation (e.g. due to track irregularity or cur

043-1648/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.wear.2004.03.034

1256 S. Beretta et al. / Wear 258 (2005) 1255–1264

Fig. 1. Typical failures in tramcar rails: (a, b) spalling in the tread; (c) spallingand wear in the counter-rail; (d) small cracks starting in the inner corner; (e)cracks and wear on tread and wear on the counter-rail; (f) crack on the outerside of the counter-rail propagated to whole section.

occurs. Being in flanging conditions usually means havingat least two contact regions, one on the tread and one onthe flange/flangeback.1 In this case, balanced longitudinal2

creepages on the two contacts may occur. Hence, even on astraight track and at low speed, full saturation of the contactarea may occur. The dynamic behaviour of tramcar vehiclesis therefore likely to produce enhanced surface damage of therails.

This research started from a survey on tramcar lines[1]that showed: (i) contact fatigue damages, as well as spalling,on tread corners (Fig. 1a and b) and on the counter-rails(Fig. 1c); (ii) contact fatigue damages characterised by thepresence of multiple cracks (Fig. 1d); and (iii) severe wear ofboth rail heads and counter-rails (Fig. 1c and e). Moreover,it has been observed that fatigue damage sometimes ends upin a crack affecting the whole rail cross-section (Fig. 1f).

The aim of this paper is to investigate the phenomena ofrail damage and to set up an integrated structural integrity

1 The tread contact on the wheel corresponds to the rail head contact onthe rail. The flange/flangeback contact on the wheel, instead, correspondsto the inner face of the rail/counter-rail contact on the rail. From now on,the contact position will be described with respect to the wheel (tread orflange/flangeback contact) even if we are talking about rails.

2 When talking about contact quantities, the longitudinal direction is as-sumed to be parallel to the track tangent while the transversal direction isn thatt

analysis of tramway lines. In particular, starting from dy-namic analyses of a modern tramcar vehicle under differentservice conditions, load spectra were achieved. Then, at eachtime instant of the vehicle’s numerical simulation and foreach contact point, a local contact analysis using Kalker’sCONTACT93 algorithm[9] was performed in order to deter-mine the surface creepage, pressure and traction distributionsas well as the subsurface stress distribution. Hence, ratch-etting and rolling contact fatigue phenomena were investi-gated while rail surface wear was estimated using a simplefrictional power–worn off mass relationship based on exper-imental data. Finally, the depth of wear was compared to thedepth of the material affected by rolling contact fatigue inorder to determine whether cracks have a chance to grow orthe material is worn off before cracks appear.

2. Dynamic behaviour of the tramcar vehicle

2.1. The vehicle model

In order to investigate rail damage phenomena due to thetransit of tramcar vehicles, a numerical model has been used.The model, called MO.N.S.TRAM (MOdular Non Station-ary TRAMcar dynamic simulator[2–4]), was developed atP hicletk mu-l s ofc peso

fig-u beend vingp

• gie;• een

ed asd ach:t ner-a exi-b ach,b inglec amp-i spen-s eanso andd s be-t

ttenw longt ntreo in-s and

ormal to the longitudinal direction and lies in the contact plane. Notehe direction normal to the contact plane is usually not vertical.

olitecnico di Milano and schematises the tramcar vehrough a multi-body, large displacement approach[5] whereinematic non-linearities are fully accounted for. The siation software has been thoroughly validated by meanomparison with on-line tests carried out on different tyf tramcars[2,4].

In order to be able to easily model a wide variety of conrations of modern tramcars, the numerical model hasesigned to allow combinations of two basic modules haarametric characteristics:

A module, which consists of one carbody and one boB module, which is a single carbody, suspended betwtwo A modules.

Carbodies, bogie frames and wheelsets are describeformable bodies through modal superposition appro

he equations of motion are written in terms of the gelised coordinates corresponding to the rigid (six) and flle (four) natural modes of each vehicle component (coogie and wheelsets), free from any constraint. The somponents are then coupled by means of elastic and dng elements, reproducing the primary and secondary suions. Different modules are linked one to the other by mf kinematic constraints and/or linear/non-linear elasticamping elements that reproduce the actual connection

ween carbodies.The equations of motion of each A module are wri

ith respect to a moving reference frame, travelling ahe ideal track centreline and following the coach’s cef gravity. The equations of motion of each B module,tead, are written considering a global coordinate system

S. Beretta et al. / Wear 258 (2005) 1255–1264 1257

a moving coordinate system, rigidly attached to the coach.Applying Lagrange’s formalism, it is possible to write theequations of motion of each module in matrix form:

[M]x- + [C]x- + [K]x- = Q(x-, x-, t) (1)

where:

- x- is a vector containing the independent variables of themechanical model;

- [M], [C] and [K] are, respectively, the mass, damping andstiffness matrices of the mechanical model;

- Q is a vector containing the generalised forces associatedwith:- the non-linear terms associated with the vehicle’s inertia;- the non-linear wheel–rail contact forces (see Section

2.2);- the connections between coaches;- the effects of the bumpstops;- the excitation due to track irregularity; and- the driving/braking torques applied to the wheels.

The matrices of the whole vehicle are obtained by assem-bling the ones of the A and B modules. The whole set ofequations of motion, in matrix form, is integrated in the timedomain.

refert ndb le ismTD thed s arem to ani f ther

F n; (b)t

2.2. The contact model

In order to reproduce the real contacting conditions occur-ring between wheel and rail even for worn wheel/rail profiles,the wheel–rail contact model has to consider the real wheeland rail profiles, i.e. measured profiles defined through dis-crete points. Moreover, the contact model has to take intoaccount the forward/backward movement of the (eventually)multiple contact points occurring at the flange/flangebackwith respect to the vertical plane containing the axle.

To satisfy these two requirements, the estimation of thecontact forces is divided into two steps: at first, a geometricanalysis of the contacting bodies is carried out before the dy-namic simulation. The results of this analysis are stored inthe form of a contact table that may be used during railwayvehicle dynamic simulations in which the same wheel–railprofiles and the same set of geometric parameters (like trackgage and wheel flange spacing) are considered. During thesimulation, at each integration step, the normal contact prob-lem, as well as the tangential contact problem, is solved usingthe contact parameters stored in the contact table.

The geometric analysis is carried out under the assump-tion of rigid motion of the wheel and rail pairs. Wheel andrail profiles are defined through Cartesian points thus allow-ing to consider measured (worn) profiles. In order to pre-c higha ge-o rela-t yawa et,t backm lanec ef-f d asf filesa rela-t rollr nde-f n thel thew s ofa d bya files,l thec rma-t theg evenw ulti-p , ther an-g tactp rmald avedi tivew le ofa

The analyses described in the following paragraphso a typical ‘modern’ tramcar with low-floor coaches aogies equipped with independent wheels. This vehicade up of four A modules and three B modules (Fig. 2).he bogie of this tramcar is schematically depicted inFig. 2b.ue to the described configuration, the vertical loads onifferent wheelsets are not equal: the two central bogieore heavily loaded than the external ones. This leads

ncrease in the severity of wear and fatigue damage oails as will be described in the following chapters.

ig. 2. Scheme of a modern tramcar: (a) typical tramcar configuratioypical bogie with independent wheels.

isely evaluate contact parameters also in the case ofngle of attack that may occur in low-radius curves, themetrical analysis is carried out considering both the

ive wheel–rail lateral displacement and the wheelsetngle[6]. In fact, due to the yaw motion of the wheels

he generic contact point occurring at the flange/flangeoves forward/backward with respect to the vertical p

ontaining the axle, thus determining an uplift/downliftect. The table of the contact parameters is determineollows: for a given angle of attack, the yawed wheel prore determined. Then, for several different values of

ive wheel–rail lateral displacement, the correspondingotation of the wheelset is determined by equating the uormed distance, measured in vertical direction, betweeeft and right wheel–rail couples. Once the position ofheelset relative to the rails is determined, the positionll those ‘potential’ contact points that are characterisedistance, measured normal to the wheel and rail pro

ower than the maximum allowed elastic deformation ofontacting bodies are searched for. In fact, due to the defoion of the adjacent bodies, contact may occur not only ineometric contact point (single contact point that existsith zero contact force) but also at different locations (mle contact points). For all these potential contact pointsolling radius, the local curvature, the inclination of the tential contact plane, the forward movement of the conoint and the undeformed distance taken along the noirection to the contacting bodies are determined and s

nto a bi-dimensional contact table as function of the relaheel–rail lateral displacement and of the wheelset angttack.

1258 S. Beretta et al. / Wear 258 (2005) 1255–1264

During the tramcar vehicle dynamic simulation, the con-tact forces on each ‘potential’ contact point have to be de-termined at each integration step. These contact forces aredetermined in two separate steps: at first, the normal con-tact problem is solved (i.e. the contact area shape and ex-tension and the pressure distribution are determined); then,the tangential contact problem is computed (i.e. the tractiondistribution inside the contact area are calculated). Let us as-sume that the position of the wheelset, in terms of yaw angleand relative wheel–rail lateral displacement, is known at agiven time instantt. Hence, it is possible to interpolate thebi-dimensional geometric tables thus determining for eachith ‘potential’ contact the values of the contact parameters.The normal contact problem is solved using a multi-Hertzianapproach[7]: the real contact area is approximated throughsuperposition of elliptic contact areas, and for each ellipse, thenormal contact force is determined using the Hertzian theory.Whether the ‘potential’ contact point is active or not dependson the relative distance between wheel and rail profiles atthe considered point. Once the normal contact problem hasbeen solved, the contact forces acting in the tangential con-tact plane can be evaluated. For each active contact, the localtangent plane is considered. Along this plane, the local longi-tudinal (parallel to the vertical plane that contains the rollingdirection) and transversal creepages are determined from theg andl pagec risticS ,t deter-m el o thev mbero

then distri-b tant,K taa po-n oint.K lingc heelsa

2

renti agep arer and5 evel(

thea sid-e gentt On

Fig. 3. Wheel–rail normal contact forces: (a) left wheel of first axle; (b) rightwheel of first axle; (c) left wheel of fourth axle; (d) right wheel of fourthaxle.

the tangent track, only one contact is active (tread contact).On the curved track, instead, a second contact arises (flangeor flangeback contact). Due to the dynamic behaviour of thetramcar vehicle, the second contact occurs at different posi-tions for the first and fourth axles: the outer wheel of the firstaxle shows up a second contact point on the flange while forthe same wheel of the fourth axle the second contact occurson the flangeback. Vice versa, for the inner wheel. Moreover,it can be observed that the tread contact of the fourth axle ismore loaded (higher normal force component) than the oneof the first axle. Thus, the passage of the fourth axle will leadto a higher damage of the rails.

As previously explained, using Kalker’s CONTACT93 al-gorithm, pressure as well as shear stress distributions are de-termined for each active contact and at each time instant.As an example,Fig. 4 shows the stress distributions on thecontact area for a typical tread and flange contact. Althoughthe normal contact force on the two contacts is almost equal(Fig. 3), the normal pressure values are much higher on theflange than on the tread (nearly 10 times as much). This is dueto the fact that the flange contact area is much smaller. Thesame conclusions can be drawn for both the shear stresses per-pendicular to the rails’ tangent direction (τyz) and the shearstresses parallel to the rails’ tangent direction (τxz): the flangecontact is characterised by higher shear stresses than the treadc ssesh ue tob ettingc

lobal kinematics of the considered wheel. The lateralongitudinal contact forces are determined from the creeomponents and the normal contact force using the heuhen–Hedrick–Elkins contact model[8]. The longitudinal

ransversal and normal contact force components areined in the local reference system of theith contact. Thes

ocal contact forces have, therefore, to be projected intehicle’s reference system and then summed over the nuf active contacts.

After the tramcar vehicle simulation, in order to obtainormal pressure and the tangential traction/creepageutions for each active contact point and at each time insalker’s CONTACT93[9] algorithm is used. The input dare the time histories of the normal contact force coment and of the creepages of each ‘potential’ contact palker’s CONTACT93 outputs are used to analyse rolontact fatigue phenomena and to estimate wear on wnd rails.

.3. Results: contact loads and contact stresses

Simulations have been carried out considering diffen-service conditions in order to fully investigate damhenomena of the rail. In particular, the shown resultseferred to a fully loaded vehicle running on a right-h0-m-radius curve at 15 and 25 km/h considering a low laccording to ORE B176,[26]) of rail irregularity.

Fig. 3 shows the normal contact force component onctive contacts for the first and fourth axles of the conred tramcar vehicle. The track consists of a short tan

rack followed by a transition curve and the full curve.

ontact. Moreover, for both contacts, the two shear streave almost equal modulus. This is a very important isse taken into account when selecting the damage ratchriteria, as it will be see in the following section.

S. Beretta et al. / Wear 258 (2005) 1255–1264 1259

Fig. 4. Contact stresses (expressed in MPa) for two simulated contacts: (a)typical tread contact; (b) typical flange contact.

2.4. Results: occurrence of contact loads

The time histories of the contact force components on thevarious wheels can be easily converted into load values alongthe track since the tramcar vehicle’s speed is constant. Thus,for any given rail section, it is possible to determine the timehistories of the contact forces due to the passage of the wholevehicle.

In presence of track irregularity, the vehicle’s behaviouris non-stationary. However, in the special case of low trackirregularity, as defined by ORE B176 standards[26], the ve-hicle’s behaviour can be assumed as nearly stationary: thevehicle’s dynamic contact force components, excited by thetrack irregularity, are small if compared to the stationary val-ues. The vehicle’s dynamic behaviour on a given rail sectionwill be function of the initial conditions, and therefore con-tact forces acting on this section will be slightly differentfrom one vehicle’s passage to the other. To take this randomphenomenon into account, several simulations with differ-ent initial conditions and fixed track could be performed or,

Fig. 5. Scheme of the computation of the load spectra.

Fig. 6. Exceedance probability: (a) first axle; (b) fourth axle.

1260 S. Beretta et al. / Wear 258 (2005) 1255–1264

equivalently, a single simulation could be used assuming thatthe contact forces in quasi-stationary conditions acting onany rail section will also act on the considered section witha probability of occurrence equal to the probability of oc-currence along the track. Under this assumption, the wholetrack (along which the vehicle’s behaviour is almost station-ary) can be reduced to a single section and the time historiesof the contact forces can be applied to this single section.Fig. 5 shows the scheme of the procedure used to calculatethe load spectra starting from the distribution of the contactforces along the line.

Fig. 6shows the exceedance probability of the normal (Fn)and tangential (Ft =

√FL

2 + FT2,FL being the longitudinal

contact force andFT being the transversal contact force) con-tact forces acting on the left wheel of the first and fourth axleof a fully loaded modern tramcar driving along a 50-m-radiuscurve with low track irregularity. The vehicle speed is equalto 25 km/h and the simulation has been carried out consid-ering new profiles for both wheels and rails. Load spectrathat take into account the passage of all eight wheels maybe obtained by multiplying the exceedance probability timesthe number of axle passages. Supposing that the number oftramcar passages per day is equal to 150, the estimated loadspectra after 10 years are shown inFig. 7 for both normal

and tangential contact forces on the left rail. The correlationbetween tangential and normal loads is determined by thetramcar vehicle’s dynamic simulation and is accounted forin the load spectra. It can be observed that the ratio betweentangential and normal loads is equal to the friction coefficient(0.36). This is due to the fact that, for these working condi-tions, macro-sliding occurs on all contacts for the wheels onthe left side of the tramcar.

3. Surface damage analysis

3.1. Shakedown maps

In order to determine the mechanisms of rolling contactfatigue, the shakedown map[10] proposed by Johnson andco-workers can be used. As already observed, for the con-sidered working conditions theFt/Fn ratio (Ft andFn beingthe tangential and normal contact force components, respec-tively) is always equal to the friction coefficient (0.36) due tothe fact that macro-sliding occurs on each contact of the eightleft wheels. Comparing the simulated results (Fig. 8b–d) withrailway vehicle results reported by Ekberg et al. (Fig. 8a,[11])on the shakedown map, it can be observed that, for tramcarvehicles, ratchetting occurs on all contacts (tread, flange andflangeback), at least for the considered working conditions.E indexp et ofr ge

Fig. 7. Load spectra for the left rail: (a) normal force; (b) tangential force.

F .f latedpNp

ven if loads exceed the Hertzian elastic approach, the0/k can be considered as a ‘severity index’ for the onsatchetting[19]. In particular, it can be noticed that flan

ig. 8. Shakedown maps: (a) working points reported by Ekberg et al[11]or railway systems; (b) simulated points for a tread contact; (c) simu

oints for a flange contact; (d) simulated points for a flangeback contact.ote: in (b–d) only the simulated points over the shakedown limit are dis-layed.

S. Beretta et al. / Wear 258 (2005) 1255–1264 1261

and flangeback contacts are characterised by very highp0/kratios: the maximum values are equal to 18.5 and 26.3 forflange and flangeback contact, respectively.

3.2. Damage criterion

The ratchetting process involves strain accumulation un-der cyclic directional loads. A material subjected to ratchet-ting fails either by low cycle fatigue or after accumulating acritical strain comparable to the strain of failure in a mono-tonic test (so-called ‘ratchetting’ failure). These two failuremodes are in competition. Thus, the actual failure mode cor-responds to the one that occurs first. Kapoor and co-workers[12,13] have shown that materials of tribological surfacesusually fail due to the second mode, i.e. due to the accumula-tion of strain. This also agrees with shakedown maps obtainedfor high friction coefficient values[10,14].

In order to determine the accumulation of strain for thesimulated manoeuvres, a (local) 3D grid below each con-tact area is considered (Fig. 9). At each node of this grid thestress tensor is calculated using Boussinesq–Cerruti equa-tions (elastic half-spaces). The ratchetting criterion[15–17]adopted is:

τ > keff (2)

w al,f ion,k , duet ,c itudeo g ise planep

τ

U yv eredt s acti witha cur-

Fc

Fig. 10. Scheme of the ratchetting analysis: (a) all simulated contacts arereported to one section; (b) map of the damaged section.

rence along the track. It is therefore possible to reduce the(local) 3D grids below each contact area to a (global) 2D gridover the considered section. The reduction is carried out intwo separate and sequential steps: at first, each (local) 3Dgrid is reduced to a (local) 2D grid by considering only one(y, z) section and assigning to each node of this (local) 2Dgrid the maximum shear stressτ according to Eq.(3) alonga line in thex-direction parallel to the surface, i.e.:

τ(y, z) = max{τ(x, y, z)} (4)

Then, the (local) 2D grids determined for each contact and ateach time instant have to be combined on the (global) 2D grid.This combination process is carried out by re-gridding themaximum shear stress values, defined on the (local) 2D grids,on the nodes of the (global) 2D grid through bi-linear inter-polation.Fig. 10schematically depicts the above-describedapproach.

3.3. Damage results

The rail damage due to the passage of a tramcar vehiclecan be represented through two different graphs: a first graphshowing the probabilityp(y, z) of occurrence of ratchettingover the rail cross-section and a second graph showing howmuch the limit ratchetting condition has been exceeded (av-e

ert

p

w

R

R

a thata the( edw e orfl atr

herekeff is the ‘current’ yield limit in shear of the materiunction of the strain history. In fact, as a first approximateff is assumed to be constant. As previously observedo the peculiar dynamic behaviour of modern tramcarsτyzannot be neglected, being of the same order of magnf τxz. For this reason, the accumulation of ratchettinstimated considering the maximum shear stress on aarallel to the surface[18]:

=√

τxz2 + τyz

2 > keff (3)

nder the assumption made in Section2.4(quasi-stationarehicle behaviour), one single rail section can be consido analyse track damage phenomena: the contact forceng on any rail section will act on the considered section

probability of occurrence equal to the probability of oc

ig. 9. The three-dimensional grid under the contact. All values ofτ areomputed and reduced to one (y, z) section.

- rage ratchetting intensity).The probabilityp(y, z) of occurrence of ratchetting ov

he rail cross-section is defined as:

(y, z) =∑Nstep

i=1 Ri(y, z)

Nstep(5)

hereRi(y, z) is an indicator function:

i(y, z) = 1 if τi(y, z) > keff

i(y, z) = 0 if τi(y, z) ≤ keff

ndNstepis the number of simulated contact cases. Notevalue ofp(y, z) equal to 1 means that, at this node of

global) 2D grid, the limit ratchetting condition is reachhenever contact occurs in this position (tread, flangangeback). Instead, a value ofp(y, z) equal to 0 means thatchetting never occurs at this node.

1262 S. Beretta et al. / Wear 258 (2005) 1255–1264

Fig. 11. Damaged left rail cross-section under the first axle: (a) probabilityof occurrence of ratchetting; (b) average ratchetting intensity〈τ/keff〉.

The average ratchetting intensityτ/keff (indicated as〈τ/keff〉), instead, is determined by dividing the sum of theτ/keff values by the number of contacts that occur in the con-sidered position (tread, flange or flangeback):

〈τ/keff〉 (y,z) =∑Nstep

i=1 Ri(y, z)(τ(y, z)/keff)

p(y, z)Nstep(6)

Figs. 11 and 12show the damaged area over the rail cross-section due to the passage of the first and fourth left (outer)wheels. These numerical results are related to a fully loadedtramcar vehicle running at 25 km/h over a 50-m-radius curve.In particular,Figs. 11a and 12ashow the map of the proba-bility of occurrence of ratchetting over the rail cross-sectionwhile Figs. 11b and 12bshow the map of〈τ/keff〉 over the railcross-section.

It can be observed that the ratchetting intensity (damage)on the tread is higher and more spread for the fourth wheelthan for the first one. This, again, is due to the fact that thenormal contact force on the tread contact is higher for thefourth wheel than for the first one (Fig. 3). Moreover, asalready observed, damage occurs on the inner face of the railwhen the first wheel transits while it occurs on the counter-rail

Fig. 12. Damaged left rail cross-section under the fourth axle: (a) probabilityof occurrence of ratchetting; (b)average ratchetting intensity〈τ/keff〉.

when the fourth wheel passes. Obviously, the global damageof the considered rail section is obtained by summing up thedamages produced by all eight wheels.

3.4. Rail life prediction

Knowing the probability of occurrence of ratchettingp(y,z) and the average〈τ/keff〉 ratio, the number of cycles to failurefor each discrete point (y, z) of the (global) 2D grid is givenby:

N(y, z) = γr

C(〈τ/keff〉(y,z) − 1

)p(y, z)

(7)

γ r being the shear strain to failure andC being an empiri-cal constant independent of the stress. These two constantswere taken from literature[19] and are related to BS11 steel:γ r = 11.5 andC= 0.00237. As a first approximation, in thepresent studykeff is assumed to be a constant. In accordancewith an elastic–perfectly plastic material behaviour and acyclic yield stress,keff is equal toσcyc/

√3, (σcyc being the

cyclic yield stress[20]).For the considered manoeuvre, the number of cycles (vehi-

cle passages) to failure goes from 6× 102 cycles in the worst

S. Beretta et al. / Wear 258 (2005) 1255–1264 1263

condition (flange/flangeback contact) to 6× 103 cycles fortread contact (here one cycle corresponds to one vehicle pas-sage). The RCF-affected zone has a depth of approximately1.5 mm and 1 mm for tread and flange/flangeback contact,respectively.

It appears from these values that rail surface is severelydamaged in terms of plastic strain accumulation even duringvery first cycles. Moreover, RCF damage is likely to pre-vail over surface wear, since the depth of worn-off material(see Section4) is lower than RCF-affected zone. It followsthat the estimated depth of the damaged zone (material sub-jected to ratchetting) can be assumed as the ‘initial flaw size’for rail life prediction analyses based on fracture mechanics([21–23]).

4. Rail wear analysis

Rail wear is determined assuming a proportionality rela-tion between frictional work and amount of worn-off material([24,25]):

ma = ka · Lf (8)

wherema is the amount of worn-off material,ka = 1.7×10−9 kg/J the proportionality coefficient (determined exper-i her

L

τ ndl ,η geso dl ss

achc e be-h eag ayb s de-s con-t thef ongt dedo ter-m railt lueso andt acts.N fied,n r oft s ofc

hec y of

Fig. 13. Wear distribution on left rail after one day: (a) tramcar speed,25 km/h; (b) tramcar speed, 15 km/h.

service (150 tramcar passages per day). These results wereobtained with a fully loaded tramcar vehicle running on aright-hand 50-m-radius curve with low track irregularityat 15 km/h (Fig. 13a) and 25 km/h (Fig. 13b). The abradedmaterial depth is normal to the rail profile. The assumedtime span for wear calculations (one day) comes from acompromise between ‘visible’ wear results and the necessityof keeping constant contacting conditions. The left verticalscale refers to the profile’s dimension while the right onegives the depth of worn-off material (represented through avertical bar plot). The railhead shows three zones of wear,two almost overlapping regions on the central part of the rail-head and one very small region near the inner face of the rail.Moreover, one wear zone on the inner face of the rail and oneon the counter-rail are visible. These two wear zones are char-acterised by a very small extension and by a high wear depth.ComparingFigs. 13a and b, it can be seen that increasingthe tramcar speed produces a much higher wear rate on thecounter-rail. The wear depth of the inner face of the rail andof the railhead contacts, instead, remains almost the same.

mentally) andLf the frictional work determined through telation:

f(i, j) = [τx(i, j)ηx(i, j) + τy(i, j)ηy(i, j)] v dx dy dt (9)

x (i, j) andτy (i, j) being, respectively, the longitudinal aateral tractions on the (i, j)th element of CONTACT93 gridx(i, j) andηy(i, j) being the longitudinal and lateral creepan the same grid element, dxand dybeing the longitudinal an

ateral dimensions of the grid element,v being the vehiclepeed and dt being equal tov/dx.

Thus, the material loss distribution is determined for eontact. Under the assumption of quasi-stationary vehiclaviour (see Section2.4), these (local) 2D grids (contact arrids), on which the material loss distribution is defined, me reduced to a (global) 1D grid with a similar approach acribed for the stress distributions: at first, the (local) 2Dact grids are reduced to (local) 1D grids by summing uprictional work/material loss calculated for each node alhe x-direction; then, these (local) 1D grids are re-gridver the (global) 1D grid by linear interpolation. To deine the mean rail abrasion (rail regular wear) over the

ransversal direction, the frictional work/material loss vaf each node on the (global) 1D grids are summed up

he resulting values are divided by the number of contote that, if the quasi-stationary assumption is not satiso reduction is possible. In this case, the irregular wea

he rail surface, i.e. the formation of non-uniform patternorrugation, could be easily estimated[24].

Fig. 13 shows the left rail profiles together with talculated mean depth of worn-off material after one da

1264 S. Beretta et al. / Wear 258 (2005) 1255–1264

5. Concluding remarks

This paper deals with a numerical method to analyse thedifferent factors that influence wear and fatigue phenom-ena of rails. The proposed numerical model combines dif-ferent algorithms: a fully validated tramcar dynamic modelthat account for the deformability of carbodies, bogies andwheelsets, an accurate contact model that considers the elas-tic deformation of the wheels and rails, a surface damageaccumulation model based on ‘ratchetting’ criterion and awear model that allows to determine both regular and irreg-ular wear patterns along the rails.

Applying the proposed methodology in quasi-stationaryconditions, load spectra on a generic rail section, surfacedamage due to rolling contact fatigue and rail wear wereestimated.

The model is presently being applied to different paths andspeeds in order to derive comprehensive load spectra andto assess their influence on rail integrity. Moreover, designparameters of tramcar vehicles are analysed to determine theireffect on track damage and wear in order to set up designcriteria for less aggressive tramcar vehicles.

Acknowledgements

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[10] A. Ponter, A. Hearle, K. Johnson, Application of the kinematicalshakedown theorem to rolling and sliding point contacts, J. Mech.Phys. Sol. 41 (1985) 487–505.

[11] A. Ekberg, E. Kabo, H. Andersson, Predicting rolling contact fatigueor railway wheels, in: 13th International Wheelset Congress, Rome,2001.

[12] A. Kapoor, F.J. Franklin, Tribological layers and the wear of ductilematerials, Wear 245 (2000) 204–215.

[13] A. Kapoor, A re-evaluation of the life to rupture of ductile metalsby cyclic plastic strain, Fatigue Fract. Eng. Mater. Struct. 17 (1994)201–219.

[14] K.L. Johnson, Contact Mechanics, Cambridge Press University,1997.

[15] J.W. Ringsberg, Life prediction of rolling contact fatigue crack ini-tiation, Int. J. Fatigue 23 (2001) 575–586.

[16] X. Su, P. Clayton, Ratchetting strain experiments with a pearliticsteel under rolling/sliding contact, Wear 205 (1997) 137–143.

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The authors wish to thank all the people who coopero this research, especially Prof. G. Diana and Prof. F. Coreover, A.T.M. (Azienda Trasporti Milanesi) is gratefucknowledged for its cooperation. Financial support foresearch project was provided by ‘Cofinanziamento MIsercizio Finanziario 2000’.

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