A Design Method for Rail Profiles in Switch Panel of

Research ArticleA Design Method for Rail Profiles in Switch Panel of TurnoutBased on the Contact Stress Analysis

Dilai Chen 1 Gang Shen2 Xin Mao2 and Buchen Chen2

1School of Railway Transportation Shanghai Institute of Technology Shanghai China2Railway and Urban Rail Transit Research Institute Tongji University Shanghai China

Correspondence should be addressed to Dilai Chen chendilai163com

Received 28 July 2019 Revised 2 October 2019 Accepted 8 October 2019 Published 9 October 2020

Academic Editor Dr Mahdi Mohammadpour

Copyright copy 2020 Dilai Chen et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Contact stress between wheel and rail is believed to cause damage to the rail (e relationship between the contact stress and theradius of the rail is initially based on the Hertz contact theory By adjusting its radius the rail profile is designed with an objectiveof reducing the maximal contact stress between wheel and rail (e rail profile of turnout is parameterized by defining severalcontrol cross sections along the switch(e experiment of dynamic vehicle-turnout interaction is also carried out to investigate theeffect of the improved rail profile on the dynamical behavior of the vehicle (e method is then verified through examples usingrail profile with a switch width of 20mm and LM worn-type tread at the CN60-350-112 turnout (e results show that thedesigned rail has a higher matching degree with the wheel profile It can reduce the contact stress improve the wheel-rail contactstate and prolong the service life of the rail without deteriorating the dynamic performance of the vehicle passing throughthe turnout

1 Introduction

Turnouts are mechanical installations enabling railwaytrains to be guided from one track to another which makesthem become integral components of the railway infra-structure When a vehicle passes through the switches at theturnouts the wheel-rail contact stress and contact geometryalso change along with the track width variation (ecomplex wheel-rail contact conditions at the switches ag-gravate the wheel-rail contact stresses wheel and rail wearand the rolling contact fatigue of rail [1] Rail grindingtechnology is an integral part of railway maintenance whichcan remove the rail surface defects with accumulated rollingcontact fatigue relieve the adverse wheel and rail wearreduce the wheel-rail contact stress and repair rail profile toprolong the service life of the rail [2 3]

(e optimization design of wheel and rail profiles iscrucial for the wheel-rail system performance and relevantissues were discussed in many previous literatures [4ndash14](e geometry characteristic of the wheel and rail profile isclosely related to the vehicle operational safety dynamic

performance and the service life of wheel and rail A rea-sonable improvement of the wheel-rail contact conditions isable to reduce wheel-rail contact stress and therefore pro-long the rail service life (us Shevtsov et al [4] and Shenand Zhong [5] proposed the optimization design of thewheel tread surface based on the rolling radius differencefunction (e obtained wheel tread surface proved to beeffective in reducing the wheel and rail wear without de-teriorating the dynamic performance Cui et al [6] putforward an optimization design method of the wheel profilewhich regarded the minimization of the normal dynamicgap between the wheel and rail as the optimization goal (isapproach took the modified sequential quadratic pro-gramming (SQP) as an iterative method for constrainednonlinear optimization however which turned out to bequite time-consuming Smallwood et al [7] adopted anoptimization method for minimizing the wheel-rail contactstress By optimizing the rail profile of the curved outertrack the maximumwheel-rail contact stress was reduced byabout 50 Besides it was reported that the contact stressbetween the wheel and rail had a large impact on the

HindawiShock and VibrationVolume 2020 Article ID 8575498 15 pageshttpsdoiorg10115520208575498

initiation of rolling contact fatigue cracks of the curved outerrail Mao and Shen [8] proposed a reverse design method forthe rail grinding profile which used the rolling radius dif-ference or contact angle difference as the objective functionof optimization (e optimized profile was able to reducewheel-rail contact stress and make the distribution ofcontact points more uniform but it was not suitable forturnout design Magel and Kalousek [9] proposed the cri-teria for rail profile design from five standpoints namelyantiwear antifatigue inhibition of wave corrugation de-velopment optimal stability and noise reduction Aftergrinding design the rail profile matched well with the wheelprofile the wheel-rail contact stress decreased obviously andthe service lives of the wheel and rail were increased Sincethe contact conditions between the wheel and rail at therailway switches are quite complicated few optimizationstudies have been conducted for the switches In order toimprove the dynamic performance of turnout crossingsWan [10] proposed a numerical method to optimize theshape of the rail to reduce rolling contact fatigue and wearbut its shortcoming is that this method needs lots of time forcomputation Wang et al [11] derived an optimized railprofile at the railway switches by SQP using the rollingradius difference function as the objective function of op-timization (e optimized profile could effectively improvethe geometry characteristic of wheel-rail contact and en-hance the operational stability of railway switch compo-nents but their work only included optimizing a sectionprofile in switch panel Palsson et al [12] introduced themultiobjective minimal optimization method for the energydissipation based on the contact pressure and wheel-railcontact estimations (e genetic optimization algorithm wasused to design the rail profile at the railway switches and toimprove the dynamic performance of the vehicle passing theturnout However this method increased the vertical load onthe rail at the switches Bugarın et al [13] suggested opti-mizing the impact load at the turnout by widening the trackgauge at the turnout switches As a result an appropriatewidening of the track gauge improved the dynamic action ofthe vehicle passing through the turnout and reduced therolling contact fatigue and wear at the turnout Oswald [14]proposed a design method for the geometric shape of theturnout and it is possible to reduce the forces created bywheel-rail contact up to 30 With the use of kinematicgauge optimization further reduction of the forces in aturnout by up to 50 was allowed

Unfortunately insulated rail joints and switch with gapscause stiffness discontinuity and stress singularity in railsections which significantly increases the contact force andcauses harmful vibrations of the vehicle and the track system[15] (ere is a problem that the rolling contact behaviorbetween wheels and rails in the switch panel is different fromthat in the tangent track rail system due to the discontinuityof rails Recently Zong and Dhanasekar [16] used the FEcode AbaqusStandard to analyze the wheel-rail contact andthen provided an idea of simplifying the design of the IRJsconsisting of only two pieces of insulated rails embeddedinto a concrete sleeper But it takes 35 h to calculate onetime Chen and Chen [17] assumed that the rail and the IRJ

are connected and discussed the Hertz and FEM model (edistributions of contact pressure significantly deviate fromHCT and Carterrsquos results as the wheelrail contact point isnear the IRJ Lu [18] conducted a comparative analysis ofnormal pressure of FEM and Hertzian theory indicatingthat the maximum pressure from FEM results is slightly lessthan that in Hertzrsquos theory and the contact area from FEMresults in the longitudinal direction is slightly larger thanthat with Hertzrsquos theory But the calculation time could bemore than ten times longer Chen and Kuang [19] andNirmal Kumar Mandal [20] carried out a 3D finite elementanalysis on an insulated rail joint to investigate the effect ofrail joint parameters on the contact pressure distributionand contact stress variation near wheel-rail contact zones(ey indicated that due to nonlinearity the traditional Hertzcontact theory (HCT) was no longer sufficient to predictstress contours near insulated rail joints Mandal [21] usedan FE model based on the modified HCT to determine thestress distribution on the railhead in the vicinity of end postHowever turnout is different from insulating rail joints inthe most control cross section the wheel-rail contact pointdoes not occur in the points of singularities (the point ofswitch rail working and nonworking side) Wiest and Kassa[22] assessed four models for wheel-rail rolling contact at agiven cross section in the crossing panel of a selected turnoutdesign and it is found that the contact pressure distributionscalculated using Hertz and CONTACT correlate well withthose results obtained from the finite element method aslong as no plasticization of the material occurs Li et al [23]carried out a comparative analysis of normal pressure aboutNo18 switch panel using Hertz simi-Hertz and Kalkerthree-dimensional non-Hertz rolling contact theory In theirstudy the difference between the maximum contact area andthe maximum contact stress was less than 10 In this paperthe wheel-rail contact stress distributions of No 12 turnoutunder Hertz and non-Hertz theories are compared andanalyzed

(e optimization design of rail grinding profile caneffectively improve the geometry characteristic of the wheel-rail contact reduce the contact stress and alleviate dynamicinteraction and increase the rail service life (e wheel-railcontact stress has a significant impact on the generation anddevelopment of rolling contact fatigue cracks Excessivecontact stress will cause huge damage to the rail (ematching degree of wheel and rail profile directly influencesthe magnitude of contact stress Reducing the wheel-railcontact stress and minimizing wheel and rail damage are thecritical tasks in wheel and rail profile design In this study anoptimization design method of rail profile in a switch panelis proposed using the wheel-rail contact stress as the ob-jective function of optimization On this basis a reversedesign method of rail profile from the rail radius is estab-lished Since the rail profile of the switch panel variescontinuously along the longitudinal direction severalcontrol sections are selected for optimization design andother sections are obtained through interpolation [10] (ismethod can ensure the smoothness of the optimized railprofile An improved profile of the control section of No 12turnout is obtained by using the proposed method An

2 Shock and Vibration

example is given below to demonstrate the advantages of themethod

2 An Optimization Model of the Rail Profile

21 Wheel-Rail Normal Contact eory

211 e Hertz Contact eory (e Hertz contact theory isused as the theoretical basis for the contact stress compu-tation According to the classical Hertz contact theory thewheel-rail contact is simplified as an infinite elastic half-space(e wheel and rail deformation at the contact point isnegligible and the wheel and rail are two elastic cylinders thatare mutually perpendicular to each other (e contact surfacebetween the two is elliptical with the maximum stress p0appearing at the center of the elliptical contact spot [24]

p0 3P

2πab (1)

where P is the normal force applied to the elliptical contactspot while a and b are the major and minor semiaxes of theelliptical contact spot respectively

(e above values a and b are calculated as follows

a m32

timesGlowastP

2(A + B)1113890 1113891

13

b n32

timesGlowastP

2(A + B)1113890 1113891

13

(2)

wherem and n are the coefficients for computing the wheel-rail contact via the Hertz contact theory and A and B are thegeometrical factors

(e above value of Glowast is calculated as follows

Glowast

1 minus ]21

E1+1 minus ]22

E2 (3)

where E1 and E2 are the elastic moduli of the wheel and railmaterial respectively ]1 and ]2 are Poissonrsquos ratio of thewheel and rail material respectively

Geometrical factors A and B are obtained as follows

A + B 12

1R1

+1

R2+

1R3

1113888 1113889

A minus B 12

1R1

+1

R2minus

1R3

1113888 1113889

(4)

where R1 is the wheel rolling radius R2 is the profile radius ofthe wheel tread cross section and R3 is the profile radius ofthe rail cross section

According to the Hertz contact theory changes in radiusof wheel and rail profile directly influence the magnitude ofthe indirect wheel-rail contact stress

212 Kalker ree-Dimensional Non-Hertz Rolling Contacteory In order to analyze the contact between nonstan-dard wheel-rail profiles more accurately Kalker [24] pro-posed a three-dimensional non-Hertz rolling contact theory

Based on the assumption of elastic half-space combinedwith geometric equation constitutive equation boundaryconditions of force and displacement and Gauss integralthe principle of complementary energy of rolling contact istransformed into the expression of surface mechanics (etheory is the most perfect method to solve the rolling contactproblem between wheel and rail On the premise of satis-fying the basic assumptions the developed numerical pro-gram CONTACTcan be used to accurately calculate contactinformation such as contact area and contact stress distri-bution under arbitrary wheel-rail profile However the time-consuming and inefficient calculation of CONTACT is notsuitable for solving every time step of vehicle-track systemdynamics (erefore its calculation results are usually usedto verify the accuracy of other approximate calculationmethods which is known as the ldquogolden standardrdquo of wheel-rail contact (e contact problem is converted to a math-ematical minimization of the constrained function andimplemented in MATLAB

(e minimization function of the normal problem isgiven by

min F 12

1113944

N

i11113944

N

j1piAi3j3pj + 1113944

N

i1hipi

subject to p0 ΔxΔy 1113944N

i1pi

(5)

where N is the number of elements in the potential area pi

and pj are the pressure of the ith and jth elements hi is theundeformed distance and Δx and Δy are the dimensions ofthe element

213 Results and Discussion A dynamic load generates dueto the motion of the wheel over the roughness and thisvarying normal force is normally obtained from a wheel-track interaction model However the wheel-track inter-action is not addressed here and a constant normal force isinitially used [22]

In this section the results from Hertz and three-di-mensional non-Hertz methods are compared and discussed(e wheel-rail contact relationship between LM wheel treadand a switch point width of 20mm rail profile of No 12turnout is analyzed (e axle load is 17 tons (e calculatedwheel-rail contact stress distribution results based on Hertzand non-Hertz contact theories are shown in Figures 1ndash4(e corresponding wheelset transverse displacement is0mm minus3mm minus6mm and minus9mm respectively It can beseen that the contact area of wheel and rail calculated byHertz contact theory is ellipse under different transversedisplacements of wheelset and the contact stress is dis-tributed in the concentric ellipse form Using the non-Hertzcontact theory to calculate the contact area is more complexand it varies greatly with the transverse displacement of thewheelset (e difference between the maximum contactstresses calculated by the two theories is not significant andthe greater the transverse displacement of the wheelset has

Shock and Vibration 3

the greater the error between them is (e main reason isthat the contact area size required by Hertz contact theory ismuch smaller than the radius of the two contact objects andthe difference of the maximum contact stress is about 10However the calculation efficiency of Hertz contact theory ismore than 10 times faster than that of non-Hertz contacttheory At the same time Hertz contact theory can directlyestablish a relationship with rail radius which facilitates thecorrection of rail profile Considering comprehensivelyHertz contact theory is selected to calculate the normalcontact stress between wheel and rail

22 Objective Function We aim to reduce the wheel-railcontact stress without deteriorating other dynamic

performances (e optimization objective function is de-fined as the minimum total wheel-rail contact stress

minP α1 times P1 + α2 times P2 + middot middot middot αn times Pn + 1113944n

i1αi times Pi (6)

where Pi is the maximum wheel-rail contact stress and αi isthe weight coefficient To get the available weighting coef-ficient the lateral displacements of the wheelset are calcu-lated using the vehicle-turnout coupling dynamics model[25 26] (e studied turnout is straight-through and thelateral displacement is almost in the range from minus4 to 4mmso in this region the greatest weighting factors should beconsidered n is the number of points with maximumcontact stress within the optimization range

minus6 minus4 minus2 0 2 4 6minus5

0

5

times108

times108

Y Coo

rdina

tes(m

m)X Coordinates

(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

(a)

times108

times108

minus10 minus5 0 5 10 minus10minus5

05

10

Y Coordinates

(mm)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

(b)

Figure 1 Contact pressure distribution (wheelset transverse displacement is 0mm) based on (a) Hertz (b) non-Hertz

minus50

5minus5

0

5

Y Coordinates

(mm)

X Coordinates(mm)

0

2

4

6

8

10

12

14

16

18

times109

times108

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(a)

minus10 minus5 0 5 10 minus100

10

Y Coordinates

(mm)X Coordinates(mm)

0

2

4

6

8

10

12

14

16times109times108

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(b)

Figure 2 Contact pressure distribution (wheelset transverse displacement is minus3mm) based on (a) Hertz (b) non-Hertz


23 Design Constraints (e geometric shape of the railprofile is a strict convex curve (excluding the sag regionformed by the stock rail and nonworking side of switchpoints where there is no wheel-rail contact and no opti-mization needs to be performed) It satisfies the followingcondition

signd2zrl

dy2rl

1113888 1113889 signd2zrr

dy2rr

1113888 1113889 equiv 1 (7)

To avoid reciprocating points on the optimized profilethe horizontal coordinates of the optimized region should bekept as monotonic as possible (is yields the following

yi+1 gtyi (8)

As for grinding profile design the designed rail profilesshould not exceed the original one then

yref gtyopt

zref gt zopt(9)

where (yrlzrl) and (yrrzrr) are the coordinates of the left andright rails respectively

24 Optimization Algorithm As shown in Figure 5 the railprofile of turnout from B to C is chosen as an optimizationregion (e starting point B and the end point C are thepoints on the profile corresponding to the expected contactstress Pexp (e moving points (yizi) on the profile can be set


10

Y Coordinates


0

1

2

3

4

5

6

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)times108

times108

(a)


10

Y Coordinates


0

1

2

3

4

5

6

7times108

times108

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)

(b)


minus5 0 5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

2

4

6

8

10

12

14

16

18

0

05

1

15

2

Nor

mal

stre

ss (P

a)

times109

times108

(a)

minus10minus50510

minus10minus5

05

10Y Coordinates


0

2

4

6

8

10

12

14

16

18

times108

times109

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(b)



by dividing the tread from B to C into n+ 1 segments whereyi and zi are the lateral and vertical coordinates of themovingpoints respectively (e coordinate of the moving point isthe optimization variable (e latter moving point is re-cursively derived from the previous moving point

(e reverse design is performed to obtain the rail profilebased on the rail radius Given the coordinates of the startingpoint the coordinates of the next point are directly calcu-lated from the arc radius

yi+1 yi + Rri times minussin θi( 1113857 + sinθi + li

Rri

1113888 11138891113888 1113889 (10)

zi+1 zi + Rri times cos θi( 1113857 minus cosθi + liRri

1113888 11138891113888 1113889 (11)

where (yi zi) and (yi+1 zi+1) are the ith and (i+1)th points on

the rail respectively Rri is the radius of the ith point θi is the

angle of the ith point on the arc with a radius of Rri and li isthe arc length between the ith and (i+1)th points

25 Convergence Conditions To minimize the contact stressof the optimized profile the objective function needs to betested (is requires

ΔP Pk minus Pkminus11113868111386811138681113868

1113868111386811138681113868lt ε (12)

where Pk andPkminus1 are the total wheel-rail contact stresses forthe current iteration and the previous iteration respectivelywhile ε is the convergence tolerance (e tolerance ε shouldnot be too large otherwise it will lead to rapid convergenceof the program limiting the effectiveness of the optimizationfor rail profile It should not be too small either Too smallvalue will influence the convergence of the program (erecommendation value is about 3

In the optimization design if Pk gtPkminus1 the profile ob-tained from this iteration is considered invalid and then thenext iteration begins

26 Major Steps of the Optimization Design

(1) First the wheel tread and the referenced ungrindedrail profile (rail profile to be optimized) at theturnout and parameters of wheel-rail contact aregiven Geometric matching between wheel and rail is

computed Meanwhile the maximum wheel-railcontact stress under different lateral displacements iscalculated for the wheelset by using the Hertz contacttheory (e maximum wheel-rail contact stress is theobject of optimization

(2) (e position of the contact point between the railprofile and wheel profile at the turnout is determinedbased on the maximum contact stress Pmax (eradius of the contact point between the rail andwheel is also calculated (e expectation Pexp of thecontact stress is set based on the maximum contactstress Pmax in the following equation

Pexp σPmax (13)

where is the adjustment coefficient of the contactstress expectation (e optimization interval for therail profile is configured based on the contact stressexpectation as shown in Figure 6

(3) Assume that the position of the contact point on thewheel is fixed (e radius at the contact point on therail is adjusted and the maximum contact stressPHertz (Rr) under different radii can be calculatedby using the Hertz contact theory From stress ex-pectation Pexp and PHertz (Rr) in equation (11)the expectation of a radius at the contact point on therail is calculated as shown in Figure 7

(4) According to Step (1) it is assumed that the railradius varies linearly from the starting point B of theoptimization interval to point A with the maximumcontact stress (en the radius for the optimizationinterval from B to A is

Rri RrB +RrA minus RrB

nBA

times i (14)

(e radius for the optimization interval AC is

Rri RrA +RrA minus RrC

nAC

times i (15)

where nBA and nAC are the numbers of total points inthe optimization interval BA and AC respectivelywhile RrB RrA andRrC are the radii of rail at pointsB A and C respectivelyBased on the radius the optimized profile for theoptimization interval BC is reversely designed

(5) By splicing the optimum interval BC with the ref-erence profile the complete profile of the optimumdesign can be obtained Geometric matching be-tween the wheel and rail is computed for the railprofile design at the turnout (e current maximumcontact stress is calculated and compared to theinitial maximum value If the current maximumcontact stress is smaller than the initial one but largerthan the convergence tolerance the expectation ofthe maximum contact stress and optimization in-terval are adjusted and then (2) is executed If thecurrent maximum contact stress is larger than the

B(yi zi)D(yi+1 zi+1)

P

0 Y

B CA

li

Rri

Cont

act s

tres

s

PmaxPexp

θi

Figure 5 Schematic of rail profile design variables


initial one the optimization is considered ineffectiveand then the expectation of the maximum contactstress and optimization interval also need to beadjusted and then (2) is executed If the currentmaximum contact stress is smaller than the initialone and does not exceed the convergence tolerancethen the optimization design is performed for thenext peak point (e iteration is terminated when allpeak points have been optimized (e optimized railprofile at the turnout is obtained as output

When the adjustment coefficient of the expected contactstress and optimization interval are given the program will

be performed to obtain the optimized design When the firstoptimization design is completed the program will auto-matically recalculate the contact stress between wheels andrails Automatic judgment will be made according to Fig-ure 8 If it is necessary to adjust the rail profile the programwill first automatically increase the adjustment coefficientand then perform optimization calculation (e programwill not stop until the adjustment coefficient of the expectedcontact stress reaches 1 At that time it is necessary tomanually adjust the optimization interval and then theprogram proceeds to the optimization design calculationagain until the final optimized profile was generated asoutput

(e flowchart for the optimization design of the railprofile is presented in Figure 9 An iteration program inMATLAB has been developed and realized

3 Results and Discussion

(e proposed method is verified through a specific examplewhich is the optimization design of a key cross section of No12 ordinary single turnout (CN60-350-112 curve radius of350m frog angle of 1 12) Due to the limited space of thispaper only one key cross section with a switch point widthof 20mm and a wheel profile (type China LM worn type)was included in the case study(e inner-side distance of thewheelset is 1353mm track gauge is 1435mm and railbottom slope is 140 (e nominal rolling circle radius of thewheel is 4755mm and the axle load is 17 tons

As shown in Figure 10 during the dynamic lateraldisplacement of wheelset (lateral displacement of wheelsetfrom minus12 to +12mm) the contact stress between the wheeland rail gradually decreases from the working side to thecenterline of the rail (e wheel-rail maximum contact stressappears at the inner side of the railhead (closer to theworking side) Here the contact stress has far exceeded therail limit of stability(emain reason is that the radius at thiscontact point on the wheel is an anticircular arc withR 100mm while that on the rail is a circular arc withR 13mm (is implies that the inner side of the railhead isthe first region to suffer the rolling contact fatigue (iscoincides with the occurrence of an oblique crack at theinner side of the railhead as shown in Figure 11 and also withthe calculation of region with the maximum contact stress inFigure 10 (e calculation results hold high similarity withthe on-site inspection data To slow down the developmentof fatigue cracks it is necessary to optimize the rail profile ofthis region and reduce the respective contact stress

(e rail profiles before and after optimization are shownin Figure 12 (e respective differences are quite obviousand the optimized profile can be obtained by grinding withthe grindstone (e changes in the radius of the rail profilebefore and after optimization are shown in Figure 13 severalbroken straight lines are replaced by continuous curveswhich are good for the contact performance between thewheel and rail When the wheelset has a dynamic lateraldisplacement contact points will not jump greatly on thewheel tread or rail top surface Consequently the wheel-rail

P

Y0 Y Coordinates

A

B C

YB YA YC

Cont

act s

tres

s

Pmax

Pexp

Figure 6 Schematic for the maximum wheel-rail contact stressdistribution

Contact stress

Pk

Output the optimized rail profile

Automaticallyincreasing the

adjustment coefficientof the expected contact

stress

Manuallychanging theoptimization

interval

Adjusting the rail profilePk le Pkndash1

|Pk ndash Pkndash1| lt ε

Figure 8 Flowchart of optimal convergence condition of the railprofile

P

Rr0

Rail profile radiusRPmax RPexp

Cont

act s

tres

s

Pmax

Pexp

Figure 7 Schematic of relationship between contact stress andradius in rail profile design


contact stress will not fluctuate significantly when thecontact points move from one arc to another

A comparison of maximum wheel-rail contact stressbefore and after optimization is presented in Figure 14which implies that the maximum contact stress after theoptimization has been reduced significantly (us in the leftrail at the turnout it drops from 6140 to 5684MPa or by75When the lateral displacement is from minus8 to 3mm thecontact stress decreases by 52 in maximum(emaximumcontact stress on the right rail decreases from 5867 to5342MPa or by 89 when the lateral displacement is fromminus65mm to 25mm the contact stress decreases by 423 inmaximum(ese results prove that the optimized rail profilecan effectively reduce the wheel-rail contact stress

(e normal pressure distribution in the contact patcheson the left side (switch rail) of wheel-rail was calculated and

is plotted in Figure 15(emaximum normal pressure of theoptimized profile appears to be lower than that of thenominal profile when the transverse displacement of thewheelset is zero due to a larger area of the contact patch ontangent track (e optimized rail profile reduces the pos-sibility of fatigue damage of the rail contact surface andmight prolong the rail service life

Distributions of wheel-rail contact points for the LMworn-type tread before and after optimization with thewheel and rail match are shown in Figures 16 and 17 re-spectively Before optimization when the lateral displace-ment of the wheelset is from minus6 to minus7mm the contact pointsmove from the stock rail to the switch points after opti-mization that happens when the lateral displacement ofwheelset varies from minus8 to minus9mm In this case the wheelswill not come into contact with the switch points too early

Rail profileWheel profile Contact parameters

Contact stress

e peak contact stress is searched and the position of theoptimal point is determined

Expectation of contact stressand optimization interval

e expected radius of rail

Design the rail profile based onthe radius of the rail

Profile combination

Whether or not theconstraint conditions are

satisfied

Whether all thepeaks have been

optimized

Final designed profile

Next peak

Yes No

Yes

No

Yes

No


Figure 9 Flowchart of the rail profile optimization


which will reduce switch point wear When the lateraldisplacement of the right stock rail is close to zero thedistribution of contact points on the rail is more uniformafter optimization than before besides the contact zone onthe rail is wider which is conducive to reduce the contactfrequency at the same point relieving the rail wear andprolonging the service life of the rail

For the brevity sake we have only given an optimizedexample of the switch rail with a width of 20mm Accordingto the abovementioned optimization method other controlcross sections such as the width of 0mm 35mm 50mmand 70mm can be optimized in the same way Each rail

profiles were generated by longitudinal interpolation Twoadjacent control cross sections around the targeted locationare shown in Figure 18 [10 26]

Although turnouts can be designed separately by di-viding them into several control sections each section is notcompletely independent and the longitudinal relationshipof each section needs to be considered In order to reduce thedynamic impact of vehicles passing through the turnout thelongitudinal wheel-rail contact points should be as smoothas possible Along the longitudinal direction of the switchthe transverse width of the contact area may be very narrowand the influence of the jump of the contact point on thelateral operation of the wheel will be reduced which isbeneficial to enhancing the stability of the vehicle and re-ducing the lateral force of the wheel-rail(us another objectfunction can be defined as follows

obj min max ywl yw m( 1113857 minus ywl yw n( 11138571113868111386811138681113868

11138681113868111386811138681113872

+ max ywr yw m( 1113857 minus ywr yw n( 11138571113868111386811138681113868

11138681113868111386811138681113873(16)

where m is the minimum traverse of wheelset and n is themaximum traverse of wheelset

To ensure that the optimized profile will not appear as awavy shape in the vertical direction the maximum grindingdepth of the each optimized profile is set as a constant In theactual grinding operation the designed grinding profile iscompared with the measured profile and the noncriticalcross sections are interpolated to generate a three-dimen-sional grinding volume [27]

4 Dynamic Interaction of Vehicle and Turnout

In this verification study the standard MATLAB andSIMULINK software packages were used to construct apassenger rail vehicle (Chinese PW220-K) model andsimulate the dynamic responses of the vehicle passingthrough a straight turnout before and after the optimization

minus800 minus790 minus780 minus770 ndash760 minus750 minus740 minus730 minus720 minus710minus810Y Coordinates (mm)

0

5

10

Cont

act s

tres

s (Pa

)

minus50

0

50

Z Co

ordi

nate

s (m

m)

times109

(a)

times109

0

2

4

6

Cont

act s

tres

s (Pa

)

minus40

minus20

0

20

Z Co

ordi

nate

s (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)

(b)

Figure 10 (e maximum contact stress between the wheel and switch rail (a) and between the wheel and stock rail (b) (e green linescorrespond to the rail profile while blue ones depict the contact stress

Figure 11 Photo of the rail with an oblique crack at the inner sideof the railhead at the turnout


Using the LM-type wheel tread a rail vehicle passage of aCN60-350-112 turnout was simulated

(e matrix assembly method is used to construct thevehicle dynamic module It can be described as [28]

[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)

where [M] [C] and [K] are the system mass damping andstiffness matrix respectively F is the generalized forcevector x is the response vector

In the wheel-rail contact model the SHEN-Hedricktheory is used to solve the tangential contact problem Firstit is calculated according to Kalkerrsquos linear creep theory andthen the SHEN-Hedrick method is used to correct the creepsaturation When the suspension force and the wheel-rail

tangential force are already known the normal contact forceof the wheel-rail is obtained by solving the equation ofmotion of the wheelset (e force diagram of the wheelsetunder dynamic equilibrium is shown in Figure 19

(e variable names are explained as follows ϕw is the rollangle and ψw is the yaw angle δLδR are the leftright-sidecontact angle the left and right contact points (CL CR) aredefined as (xcL ycL zcL) and (xcR ycR zcR) respectivelyNLNR are the normal force of the leftright contact pointsTLTR are the tangential force of the leftright contact pointsMTxLMTxR are the rotation moment of the leftrightcontact points FszLFszR are the vertical component of leftright suspension forces MsxLMsxR are the longitudinalcomponent of leftright suspension moment FazMax are

Optimized profileNominal profile

40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)

minus810 minus790 minus780 minus770 minus760 minus750 minus740 minus730 minus720 minus710minus800Y Coordinates (mm)

(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)

Figure 12 Comparison of rail profiles before and after optimization (a) switch rail and (b) stock rail

0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)

Figure 13 Distribution of radius of the rail profile before and after optimization (a) switch rail and (b) stock rail


the inertial forcemoment of wheelset and FIGzMIGx arethe inertial forcemoment caused by the change of the railcoordinate system

After ignoring some high-order small quantities thewheelset equilibrium equation becomes

cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL

1113896 1113897 Faz minus FIG minus FszL minus FszR minus TL minus TR

Maz minus MIG minus MsxL minus MsxR minus MTxL minus MTxR

1113890 1113891 (18)

where S cos(δR + ϕw)(ψwxcR + ycR minus ϕwzcR) +

sin(δR + ϕw)(ϕwycR + zcR) and W cos(δL + ϕw)(ψwxcL +

ycLminus ϕwzcL) + sin(δL + ϕw)(ϕwycL+ zcL)

(e normal force of wheel and rail can be obtained bysolving equation (18) (e overall flowchart of wheel-railcreep calculation is shown in Figure 20 (e simulation is


0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)

Figure 15 Distribution of the normal pressure (a) before optimization and (b) after optimization

times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)

minus10 minus5 0 5 10 15minus15Y Coordinates (mm)


(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)

Figure 14 Maximum wheel-rail contact stress before and after optimization (a) switch rail and (b) stock rail


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

minus780 minus760 minus740 minus720 minus700 minus680minus800Y Coordinates (mm)

minus12minus11minus10minus9minus8minus7minus6minus5minus4minus3minus2minus10123456789101112Yw (mm)

(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)

Figure 17 Distribution of wheel-rail contact points after optimization (a) switch rail and (b) stock rail


0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)

Figure 16 Distribution of wheel-rail contact points before optimization (a) switch rail and (b) stock rail

0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II

Cross section i(interpolated)

(b)

Figure 18 Rail profile in the switch (a) control cross sections of the rail along the switch (b) interpolation of the rail profile


NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw

Figure 19 Force decomposition of wheelset

Wheel-rail geometriccontact calculation

Contact stress

Calculation ofwheelset dynamic

equation


equilibrium equation

Calculation ofcreepages

e radius ofcurvature at the

contact point

Kalkercoefficient

Creepcoefficient

Normal force

Figure 20 Flowchart of wheel-rail creep

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3


Figure 21 Lateral displacement of the first wheelset

0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)


Figure 22 Wear indexes


carried out at the speed of 120 kmh for the vehicle and notrack irregularity is imposed

(e lateral displacements of the first wheelset passingthrough this turnout with and without optimization areshown in Figure 21 When passing through the optimizedturnout the wheelset shows a much better performance ofrunning back to the track center than before and the lateraldisplacement of the wheelset is a little lower (e wear in-dexes [29] of the left wheel of the first wheelset before andafter optimization are shown in Figure 22 After optimi-zation the maximumwear index is 31 vs 376 which impliesa reduction of 175 (is indicates that the optimizedprofile better matches the LM worn-type tread withoutdeteriorating the dynamic performance of the vehiclepassing the turnout

5 Conclusions

A new direct numerical method is proposed to optimize therail profile in the switch panel minimizing the wheel-railcontact normal stress as the design objective (e rela-tionship between the radius of rail profile and contact stresswas initially established based on Hertz theory (en the railprofile is designed by directly adjusting the radius of thecurve Although the rails in the turnout switch panel varycontinuously along the longitudinal direction severalcontrol sections can be selected for optimization design andother rail profiles can be obtained through interpolation

(e design method not only ensures that the designedrail profile is smooth but also makes it easy to control thewidth of the design interval and the designed grinding depth(e optimization example shows that when the optimizedrail profile at the turnout matches the LM worn-type wheelwell both the contact stress and geometry characteristic ofwheel-rail contact are significantly improved (is is of higheconomic importance in terms of prolonging rail service life(e above findings may guide the selection of the targetprofile in the rail grinding operation (is optimizationmethod is also applicable to the design of wheel and railprofile of the railway segment

(e profile optimization of the railway turnout is a long-term and complicated process the rail profile in switch panelin diverging line and the profile of crossing nose are notconsidered in this work and they should be studied in thefuture As to high-speed railway turnout the stability of thevehicle should be guaranteed first

Data Availability

(e profile data and software code used to support thefindings of this study are currently under embargo while theresearch findings are commercialized Requests for data 6months after publication of this article will be considered bythe corresponding author

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e present work was supported by the Project of Scienceand Technology Research and Development Plan of ChinaRailway Corporation (No 2017G003-A)

References

[1] N Burgelman Z Li and R Dollevoet ldquoA new rolling contactmethod applied to conformal contact and the train-turnoutinteractionrdquo Wear vol 321 pp 94ndash105 2014

[2] P A Cuervo J F Santa and A Toro ldquoCorrelations betweenwear mechanisms and rail grinding operations in a com-mercial railroadrdquo Tribology International vol 82 pp 265ndash273 2015

[3] Y Satoh and K Iwafuchi ldquoEffect of rail grinding on rollingcontact fatigue in railway rail used in conventional line inJapanrdquo Wear vol 265 no 9-10 pp 1342ndash1348 2008

[4] I Y Shevtsov V L Markine and C Esveld ldquoOptimal designof wheel profile for railway vehiclesrdquo Wear vol 258 no 7-8pp 1022ndash1030 2005

[5] G Shen and X Zhong ldquoA design method for wheel profilesaccording to the rolling radius difference functionrdquo Pro-ceedings of the Institution of Mechanical Engineers Part FJournal of Rail amp Rapid Transit vol 225 no 5 pp 457ndash4622011

[6] D B Cui L Li and X S Jin ldquoOptimal design of wheel profilesbased on weighed wheelrail gaprdquo Wear vol 271 pp 218ndash226 2001

[7] R Smallwood J C Sinclair and K J Sawley ldquoAn optimi-zation technique to minimize rail contact stressesrdquo Wearvol 144 no 1-2 pp 373ndash384 1991

[8] X Mao and G Shen ldquoA design method for rail profiles basedon the geometric characteristics of wheel-rail contactrdquo Pro-ceedings of Institution of Mechanical Engineers Part F Journalof Rail amp Rapid Transit vol 232 no 5 pp 1ndash11 2017

[9] E E Magel and J Kalousek ldquo(e application of contactmechanics to rail profile design and rail grindingrdquo Wearvol 253 no 1-2 pp 308ndash316 2002

[10] C Wan V L Markine and I Y Shevtsov ldquoImprovement ofvehicle-turnout interaction by optimising the shape ofcrossing noserdquo Vehicle System Dynamics vol 52 no 11pp 1517ndash1540 2014

[11] PWang XMa JWang J Xu and R Chen ldquoOptimization ofrail profiles to improve vehicle running stability in switchpanel of high-speed railway turnoutsrdquo Mathematical Prob-lems in Engineering vol 2017 no 2 pp 1ndash13 2017

[12] B A Palsson ldquoDesign optimisation of switch rails in railwayturnoutsrdquo Vehicle System Dynamics vol 51 no 10pp 1619ndash1639 2013

[13] M R Bugarın J M Garcıa and D D Villegas ldquoImprove-ments in railway switchesrdquo Institution of Mechanical Engi-neers Part F Journal of Rail amp Rapid Transit vol 216 no 4pp 275ndash286 2002

[14] J R Oswald ldquoTurnout geometry optimization with dynamicsimulation of track and vehiclerdquo in Proceedings of the AnnualConference American Railway Engineering and Maintenanceof Way Association pp 297ndash302 Dallas TX USA September2000

[15] H M El-sayed M Lotfy H N El-Din Zohny and H S RiadldquoA three dimensional finite element analysis of insulated railjoints deteriorationrdquo Engineering Failure Analysis vol 91pp 201ndash215 2018


[16] N Zong and M Dhanasekar ldquoSleeper embedded insulatedrail joints for minimising the number of modes of failurerdquoEngineering Failure Analysis vol 76 pp 27ndash43 2017

[17] Y C Chen and L W Chen ldquoEffects of insulated rail joint onthe wheelrail contact stresses under the condition of partialsliprdquo Wear vol 260 no 11-12 pp 1267ndash1273 2006

[18] C Lu I J Nieto and J M J Martınez-EsnaolaMelendezldquoFatigue prediction of rail welded jointsrdquo InternationalJournal of Fatigue vol 113 pp 78ndash87 2018

[19] Y C Chen and J H Kuang ldquoContact stress variations nearthe insulated rail jointsrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 216 no 4 pp 265ndash273 2002

[20] N K Mandal ldquoFEA to assess plastic deformation of railheadmaterial damage of insulated rail joints with fibreglass andnylon end postsrdquo Wear vol 366-367 pp 3ndash12 2016

[21] N K Mandal ldquoRatchetting of railhead material of insulatedrail joints (IRJs) with reference to endpost thicknessrdquo Engi-neering Failure Analysis vol 45 pp 347ndash362 2014

[22] M Wiest E Kassa W Daves J C O Nielsen andH Ossberger ldquoAssessment of methods for calculating contactpressure in wheel-railswitch contactrdquo Wear vol 265 no 9-10 pp 1439ndash1445 2008

[23] J Li J Ding Y Niu et al ldquoAnalysis of wheel and rail rollingcontact theory of switchrdquo Journal of Southwest JiaotongUniversity vol 54 no 1 pp 1ndash9 2019 in Chinese

[24] J J Kalker and K L Johnson ree-Dimensional ElasticBodies in Rolling Contact Kluwer Academic Publishers Delft(e Netherlands 1990

[25] W M Zhai Vehicle-Track Coupling Dynamics China RailwayPublishing House Beijing China 2nd edition 2002

[26] Z S Ren WheelRail Multi-Point Contacts and Vehicle-Turnout System Dynamic Interactions Science Press BeijingChina (in Chinese) 2014

[27] (e Committee of Routine Turnout Main ParametersHandbook Routine Turnout Main Parameters HandbookRoutine Turnout Main Parameters Handbook China RailwayPublishing House Beijing China 2007 in Chinese

[28] F-S Liu Z-P Zeng and W-D Shuaibu ldquoStochastic analysisof nonlinear vehicle-track coupled dynamic system and itsapplication in vehicle operation safety evaluationrdquo Shock andVibration vol 2019 pp 1ndash23 2019

[29] J A Elkins and R A Allen ldquoVerification of a transit vehiclersquoscurving behavior and projected wheelrail wear performancerdquoJournal of Dynamic Systems Measurement and Controlvol 104 no 3 pp 247ndash255 1982


initiation of rolling contact fatigue cracks of the curved outerrail Mao and Shen [8] proposed a reverse design method forthe rail grinding profile which used the rolling radius dif-ference or contact angle difference as the objective functionof optimization (e optimized profile was able to reducewheel-rail contact stress and make the distribution ofcontact points more uniform but it was not suitable forturnout design Magel and Kalousek [9] proposed the cri-teria for rail profile design from five standpoints namelyantiwear antifatigue inhibition of wave corrugation de-velopment optimal stability and noise reduction Aftergrinding design the rail profile matched well with the wheelprofile the wheel-rail contact stress decreased obviously andthe service lives of the wheel and rail were increased Sincethe contact conditions between the wheel and rail at therailway switches are quite complicated few optimizationstudies have been conducted for the switches In order toimprove the dynamic performance of turnout crossingsWan [10] proposed a numerical method to optimize theshape of the rail to reduce rolling contact fatigue and wearbut its shortcoming is that this method needs lots of time forcomputation Wang et al [11] derived an optimized railprofile at the railway switches by SQP using the rollingradius difference function as the objective function of op-timization (e optimized profile could effectively improvethe geometry characteristic of wheel-rail contact and en-hance the operational stability of railway switch compo-nents but their work only included optimizing a sectionprofile in switch panel Palsson et al [12] introduced themultiobjective minimal optimization method for the energydissipation based on the contact pressure and wheel-railcontact estimations (e genetic optimization algorithm wasused to design the rail profile at the railway switches and toimprove the dynamic performance of the vehicle passing theturnout However this method increased the vertical load onthe rail at the switches Bugarın et al [13] suggested opti-mizing the impact load at the turnout by widening the trackgauge at the turnout switches As a result an appropriatewidening of the track gauge improved the dynamic action ofthe vehicle passing through the turnout and reduced therolling contact fatigue and wear at the turnout Oswald [14]proposed a design method for the geometric shape of theturnout and it is possible to reduce the forces created bywheel-rail contact up to 30 With the use of kinematicgauge optimization further reduction of the forces in aturnout by up to 50 was allowed

Unfortunately insulated rail joints and switch with gapscause stiffness discontinuity and stress singularity in railsections which significantly increases the contact force andcauses harmful vibrations of the vehicle and the track system[15] (ere is a problem that the rolling contact behaviorbetween wheels and rails in the switch panel is different fromthat in the tangent track rail system due to the discontinuityof rails Recently Zong and Dhanasekar [16] used the FEcode AbaqusStandard to analyze the wheel-rail contact andthen provided an idea of simplifying the design of the IRJsconsisting of only two pieces of insulated rails embeddedinto a concrete sleeper But it takes 35 h to calculate onetime Chen and Chen [17] assumed that the rail and the IRJ

are connected and discussed the Hertz and FEM model (edistributions of contact pressure significantly deviate fromHCT and Carterrsquos results as the wheelrail contact point isnear the IRJ Lu [18] conducted a comparative analysis ofnormal pressure of FEM and Hertzian theory indicatingthat the maximum pressure from FEM results is slightly lessthan that in Hertzrsquos theory and the contact area from FEMresults in the longitudinal direction is slightly larger thanthat with Hertzrsquos theory But the calculation time could bemore than ten times longer Chen and Kuang [19] andNirmal Kumar Mandal [20] carried out a 3D finite elementanalysis on an insulated rail joint to investigate the effect ofrail joint parameters on the contact pressure distributionand contact stress variation near wheel-rail contact zones(ey indicated that due to nonlinearity the traditional Hertzcontact theory (HCT) was no longer sufficient to predictstress contours near insulated rail joints Mandal [21] usedan FE model based on the modified HCT to determine thestress distribution on the railhead in the vicinity of end postHowever turnout is different from insulating rail joints inthe most control cross section the wheel-rail contact pointdoes not occur in the points of singularities (the point ofswitch rail working and nonworking side) Wiest and Kassa[22] assessed four models for wheel-rail rolling contact at agiven cross section in the crossing panel of a selected turnoutdesign and it is found that the contact pressure distributionscalculated using Hertz and CONTACT correlate well withthose results obtained from the finite element method aslong as no plasticization of the material occurs Li et al [23]carried out a comparative analysis of normal pressure aboutNo18 switch panel using Hertz simi-Hertz and Kalkerthree-dimensional non-Hertz rolling contact theory In theirstudy the difference between the maximum contact area andthe maximum contact stress was less than 10 In this paperthe wheel-rail contact stress distributions of No 12 turnoutunder Hertz and non-Hertz theories are compared andanalyzed

(e optimization design of rail grinding profile caneffectively improve the geometry characteristic of the wheel-rail contact reduce the contact stress and alleviate dynamicinteraction and increase the rail service life (e wheel-railcontact stress has a significant impact on the generation anddevelopment of rolling contact fatigue cracks Excessivecontact stress will cause huge damage to the rail (ematching degree of wheel and rail profile directly influencesthe magnitude of contact stress Reducing the wheel-railcontact stress and minimizing wheel and rail damage are thecritical tasks in wheel and rail profile design In this study anoptimization design method of rail profile in a switch panelis proposed using the wheel-rail contact stress as the ob-jective function of optimization On this basis a reversedesign method of rail profile from the rail radius is estab-lished Since the rail profile of the switch panel variescontinuously along the longitudinal direction severalcontrol sections are selected for optimization design andother sections are obtained through interpolation [10] (ismethod can ensure the smoothness of the optimized railprofile An improved profile of the control section of No 12turnout is obtained by using the proposed method An






p0 3P

2πab (1)



a m32

timesGlowastP

2(A + B)1113890 1113891

13

b n32

timesGlowastP

2(A + B)1113890 1113891

13

(2)



Glowast

1 minus ]21

E1+1 minus ]22

E2 (3)



A + B 12

1R1

+1

R2+

1R3

1113888 1113889

A minus B 12

1R1

+1

R2minus

1R3

1113888 1113889

(4)






min F 12

1113944

N

i11113944

N

j1piAi3j3pj + 1113944

N

i1hipi


i1pi

(5)










i1αi times Pi (6)



0

5

times108

times108

Y Coo

rdina

tes(m

m)X Coordinates

(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

(a)

times108

times108


05

10

Y Coordinates

(mm)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

(b)


minus50

5minus5

0

5

Y Coordinates

(mm)

X Coordinates(mm)

0

2

4

6

8

10

12

14

16

18

times109

times108

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(a)


10

Y Coordinates


0

2

4

6

8

10

12

14

16times109times108

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(b)




signd2zrl

dy2rl

1113888 1113889 signd2zrr

dy2rr

1113888 1113889 equiv 1 (7)


yi+1 gtyi (8)


yref gtyopt

zref gt zopt(9)




10

Y Coordinates


0

1

2

3

4

5

6

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)times108

times108

(a)


10

Y Coordinates


0

1

2

3

4

5

6

7times108

times108

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)

(b)


minus5 0 5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

2

4

6

8

10

12

14

16

18

0

05

1

15

2

Nor

mal

stre

ss (P

a)

times109

times108

(a)

minus10minus50510

minus10minus5

05

10Y Coordinates


0

2

4

6

8

10

12

14

16

18

times108

times109

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(b)






Rri

1113888 11138891113888 1113889 (10)


1113888 11138891113888 1113889 (11)






1113868111386811138681113868lt ε (12)







Pexp σPmax (13)





nBA

times i (14)



nAC

times i (15)




P

0 Y

B CA

li

Rri

Cont

act s

tres

s

PmaxPexp

θi











P

Y0 Y Coordinates

A

B C

YB YA YC

Cont

act s

tres

s

Pmax

Pexp


Contact stress

Pk




stress


interval




P

Rr0


Cont

act s

tres

s

Pmax

Pexp









Contact stress





Profile combination


satisfied


optimized


Next peak

Yes No

Yes

No

Yes

No









11138681113868111386811138681113872


11138681113868111386811138681113873(16)






0

5

10

Cont

act s

tres

s (Pa

)

minus50

0

50

Z Co

ordi

nate

s (m

m)

times109

(a)

times109

0

2

4

6

Cont

act s

tres

s (Pa

)

minus40

minus20

0

20

Z Co

ordi

nate

s (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)

(b)






[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)






40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)


(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)


0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)





cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References




































p0 3P

2πab (1)



a m32

timesGlowastP

2(A + B)1113890 1113891

13

b n32

timesGlowastP

2(A + B)1113890 1113891

13

(2)



Glowast

1 minus ]21

E1+1 minus ]22

E2 (3)



A + B 12

1R1

+1

R2+

1R3

1113888 1113889

A minus B 12

1R1

+1

R2minus

1R3

1113888 1113889

(4)






min F 12

1113944

N

i11113944

N

j1piAi3j3pj + 1113944

N

i1hipi


i1pi

(5)










i1αi times Pi (6)



0

5

times108

times108

Y Coo

rdina

tes(m

m)X Coordinates

(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

(a)

times108

times108


05

10

Y Coordinates

(mm)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

(b)


minus50

5minus5

0

5

Y Coordinates

(mm)

X Coordinates(mm)

0

2

4

6

8

10

12

14

16

18

times109

times108

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(a)


10

Y Coordinates


0

2

4

6

8

10

12

14

16times109times108

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(b)




signd2zrl

dy2rl

1113888 1113889 signd2zrr

dy2rr

1113888 1113889 equiv 1 (7)


yi+1 gtyi (8)


yref gtyopt

zref gt zopt(9)




10

Y Coordinates


0

1

2

3

4

5

6

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)times108

times108

(a)


10

Y Coordinates


0

1

2

3

4

5

6

7times108

times108

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)

(b)


minus5 0 5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

2

4

6

8

10

12

14

16

18

0

05

1

15

2

Nor

mal

stre

ss (P

a)

times109

times108

(a)

minus10minus50510

minus10minus5

05

10Y Coordinates


0

2

4

6

8

10

12

14

16

18

times108

times109

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(b)






Rri

1113888 11138891113888 1113889 (10)


1113888 11138891113888 1113889 (11)






1113868111386811138681113868lt ε (12)







Pexp σPmax (13)





nBA

times i (14)



nAC

times i (15)




P

0 Y

B CA

li

Rri

Cont

act s

tres

s

PmaxPexp

θi











P

Y0 Y Coordinates

A

B C

YB YA YC

Cont

act s

tres

s

Pmax

Pexp


Contact stress

Pk




stress


interval




P

Rr0


Cont

act s

tres

s

Pmax

Pexp









Contact stress





Profile combination


satisfied


optimized


Next peak

Yes No

Yes

No

Yes

No









11138681113868111386811138681113872


11138681113868111386811138681113873(16)






0

5

10

Cont

act s

tres

s (Pa

)

minus50

0

50

Z Co

ordi

nate

s (m

m)

times109

(a)

times109

0

2

4

6

Cont

act s

tres

s (Pa

)

minus40

minus20

0

20

Z Co

ordi

nate

s (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)

(b)






[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)






40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)


(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)


0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)





cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References




































i1αi times Pi (6)



0

5

times108

times108

Y Coo

rdina

tes(m

m)X Coordinates

(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

(a)

times108

times108


05

10

Y Coordinates

(mm)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

(b)


minus50

5minus5

0

5

Y Coordinates

(mm)

X Coordinates(mm)

0

2

4

6

8

10

12

14

16

18

times109

times108

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(a)


10

Y Coordinates


0

2

4

6

8

10

12

14

16times109times108

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(b)




signd2zrl

dy2rl

1113888 1113889 signd2zrr

dy2rr

1113888 1113889 equiv 1 (7)


yi+1 gtyi (8)


yref gtyopt

zref gt zopt(9)




10

Y Coordinates


0

1

2

3

4

5

6

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)times108

times108

(a)


10

Y Coordinates


0

1

2

3

4

5

6

7times108

times108

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)

(b)


minus5 0 5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

2

4

6

8

10

12

14

16

18

0

05

1

15

2

Nor

mal

stre

ss (P

a)

times109

times108

(a)

minus10minus50510

minus10minus5

05

10Y Coordinates


0

2

4

6

8

10

12

14

16

18

times108

times109

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(b)






Rri

1113888 11138891113888 1113889 (10)


1113888 11138891113888 1113889 (11)






1113868111386811138681113868lt ε (12)







Pexp σPmax (13)





nBA

times i (14)



nAC

times i (15)




P

0 Y

B CA

li

Rri

Cont

act s

tres

s

PmaxPexp

θi











P

Y0 Y Coordinates

A

B C

YB YA YC

Cont

act s

tres

s

Pmax

Pexp


Contact stress

Pk




stress


interval




P

Rr0


Cont

act s

tres

s

Pmax

Pexp









Contact stress





Profile combination


satisfied


optimized


Next peak

Yes No

Yes

No

Yes

No









11138681113868111386811138681113872


11138681113868111386811138681113873(16)






0

5

10

Cont

act s

tres

s (Pa

)

minus50

0

50

Z Co

ordi

nate

s (m

m)

times109

(a)

times109

0

2

4

6

Cont

act s

tres

s (Pa

)

minus40

minus20

0

20

Z Co

ordi

nate

s (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)

(b)






[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)






40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)


(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)


0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)





cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References

































signd2zrl

dy2rl

1113888 1113889 signd2zrr

dy2rr

1113888 1113889 equiv 1 (7)


yi+1 gtyi (8)


yref gtyopt

zref gt zopt(9)




10

Y Coordinates


0

1

2

3

4

5

6

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)times108

times108

(a)


10

Y Coordinates


0

1

2

3

4

5

6

7times108

times108

0

1

2

3

4

5

6

7

Nor

mal

stre

ss (P

a)

(b)


minus5 0 5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

2

4

6

8

10

12

14

16

18

0

05

1

15

2

Nor

mal

stre

ss (P

a)

times109

times108

(a)

minus10minus50510

minus10minus5

05

10Y Coordinates


0

2

4

6

8

10

12

14

16

18

times108

times109

0

05

1

15

2

Nor

mal

stre

ss (P

a)

(b)






Rri

1113888 11138891113888 1113889 (10)


1113888 11138891113888 1113889 (11)






1113868111386811138681113868lt ε (12)







Pexp σPmax (13)





nBA

times i (14)



nAC

times i (15)




P

0 Y

B CA

li

Rri

Cont

act s

tres

s

PmaxPexp

θi











P

Y0 Y Coordinates

A

B C

YB YA YC

Cont

act s

tres

s

Pmax

Pexp


Contact stress

Pk




stress


interval




P

Rr0


Cont

act s

tres

s

Pmax

Pexp









Contact stress





Profile combination


satisfied


optimized


Next peak

Yes No

Yes

No

Yes

No









11138681113868111386811138681113872


11138681113868111386811138681113873(16)






0

5

10

Cont

act s

tres

s (Pa

)

minus50

0

50

Z Co

ordi

nate

s (m

m)

times109

(a)

times109

0

2

4

6

Cont

act s

tres

s (Pa

)

minus40

minus20

0

20

Z Co

ordi

nate

s (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)

(b)






[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)






40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)


(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)


0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)





cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References



































Rri

1113888 11138891113888 1113889 (10)


1113888 11138891113888 1113889 (11)






1113868111386811138681113868lt ε (12)







Pexp σPmax (13)





nBA

times i (14)



nAC

times i (15)




P

0 Y

B CA

li

Rri

Cont

act s

tres

s

PmaxPexp

θi











P

Y0 Y Coordinates

A

B C

YB YA YC

Cont

act s

tres

s

Pmax

Pexp


Contact stress

Pk




stress


interval




P

Rr0


Cont

act s

tres

s

Pmax

Pexp









Contact stress





Profile combination


satisfied


optimized


Next peak

Yes No

Yes

No

Yes

No









11138681113868111386811138681113872


11138681113868111386811138681113873(16)






0

5

10

Cont

act s

tres

s (Pa

)

minus50

0

50

Z Co

ordi

nate

s (m

m)

times109

(a)

times109

0

2

4

6

Cont

act s

tres

s (Pa

)

minus40

minus20

0

20

Z Co

ordi

nate

s (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)

(b)






[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)






40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)


(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)


0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)





cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References








































P

Y0 Y Coordinates

A

B C

YB YA YC

Cont

act s

tres

s

Pmax

Pexp


Contact stress

Pk




stress


interval




P

Rr0


Cont

act s

tres

s

Pmax

Pexp









Contact stress





Profile combination


satisfied


optimized


Next peak

Yes No

Yes

No

Yes

No









11138681113868111386811138681113872


11138681113868111386811138681113873(16)






0

5

10

Cont

act s

tres

s (Pa

)

minus50

0

50

Z Co

ordi

nate

s (m

m)

times109

(a)

times109

0

2

4

6

Cont

act s

tres

s (Pa

)

minus40

minus20

0

20

Z Co

ordi

nate

s (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)

(b)






[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)






40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)


(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)


0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)





cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References






































Contact stress





Profile combination


satisfied


optimized


Next peak

Yes No

Yes

No

Yes

No









11138681113868111386811138681113872


11138681113868111386811138681113873(16)






0

5

10

Cont

act s

tres

s (Pa

)

minus50

0

50

Z Co

ordi

nate

s (m

m)

times109

(a)

times109

0

2

4

6

Cont

act s

tres

s (Pa

)

minus40

minus20

0

20

Z Co

ordi

nate

s (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)

(b)






[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)






40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)


(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)


0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)





cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References





































11138681113868111386811138681113872


11138681113868111386811138681113873(16)






0

5

10

Cont

act s

tres

s (Pa

)

minus50

0

50

Z Co

ordi

nate

s (m

m)

times109

(a)

times109

0

2

4

6

Cont

act s

tres

s (Pa

)

minus40

minus20

0

20

Z Co

ordi

nate

s (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)

(b)






[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)






40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)


(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)


0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)





cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References


































[M]xmiddotmiddot

+[C]xmiddot

+[K]x F (17)






40

35

30

25

20

15

10

5

0

minus5Z

Coor

dina

tes (

mm

)


(a)


40

35

30

25

20

15

10

5

0

minus5

Z Co

ordi

nate

s (m

m)

710 730 740 750 760 770 780 790720Y Coordinates (mm)

(b)


0

200

400

600

800

1000

1200

Radi

us (m

m)



(a)

0

100

200

300

400

500

600

700

800

900

1000

Radi

us (m

m)

720 730 740 750 760 770 780 790710Y Coordinates (mm)


(b)





cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References


































cos δR + ϕw( 1113857 cos δL + ϕw( 1113857

S W1113890 1113891

NR

NL



1113890 1113891 (18)






0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

2

4

6

8

10

12

0

2

4

6

8

10

12

14

Nor

mal

stre

ss (P

a)

times108

times108

(a)

minus50

5minus5

0

5

Y Coo

rdina

tes(m

m)

X Coordinates(mm)

0

1

2

3

4

5

6

7

8

9

10

times108

times108

02468

101214

Nor

mal

stre

ss (P

a)

(b)


times109

0

1

2

3

4

5

6

7Co

ntac

t str

ess (

Pa)



(a)

times109

0

1

2

3

4

5

6

Cont

act s

tres

s (Pa

)



(b)



30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References
































30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)



(a)

30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)


(b)



0

30

20

10

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)


(a)


30

20

10

0

minus10

minus20

minus30

Z Co

ordi

nate

s (m

m)

700 720 740 760 780 800680Y Coordinates (mm)

(b)


0mm

1185mm 1232mm 1232mm 1790mm

20mm 35mm 50mm 70mm

(a)

Cross section I

Cross section II


(b)



NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References
































NL

NR

CRTR

TL

Ψw

δL

δR

CL

MTxL

MsxL

FszR

MsxR

MTxR

MIGx ndash Max

FIGz ndash Faz

FszL

ϕw



Contact stress


equation





contact point

Kalkercoefficient

Creepcoefficient

Normal force


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

15

Late

ral d

ispla

cem

ent (

mm

)

05 06 07 08 09 1 11 1204Time (s)

times10ndash3



0

5

10

15

20

25

30

35

40

Wea

r ind

ex

05 06 07 08 09 1 11 1204Time (s)






5 Conclusions




Data Availability




Acknowledgments


References


































5 Conclusions




Data Availability




Acknowledgments


References















































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