IJR International Journal of Railway

Vol. 7, No. 1 / March 2014, pp. 1-7

Vol. 7, No. 1 / March 2014 − 1 −

The Korean Society for Railway

Stress Analysis in the Elastic-Plastic Analysis of Railway Wheels

Roya Sadat Ashofteh† and Ali Mohammadnia*

Abstract

Fatigue and wear in wheels is often due to the forces and loading. These certainly have fundamental effects on reducing

the wheel life and increasing the costs related to repairing and maintenance. Modeling and stress analysis of a wheel

sample existing in the Iranian fleet have been performed in its contact with U33 and UIC60 rails. The results have been

reviewed and analyzed in elastic and elastic-plastic phase and under static (railcar weight) and quasi static loads. More-

over, effects of wheel diameter, axle load, wheel material, rail type are analyzed.

Keywords: Railway, Wheel, Rail, U33, UIC60, ABAQUS, Elastic-plastic analysis, Fatigue and wear

1. Introduction

Fatigue in wheels is concerned with Rolling Contact

Fatigue which is produced during rolling movement under

the effect of alternative contact stresses [1]. Contact forces

between the wheel and rail produce stresses which guide

material behavior to the elastic or elastic-plastic surface.

Stresses in the wheel and rail contact area at dynamic load

modes usually occur in the elastic-plastic areas and loading

this area usually leads to a breakage which is a result of

low cyclic fatigue. In wheel and rail contact, plastic defor-

mation gradually occurs. Wheel set is the most important

component of the car components and this component

plays a very important role from safety, ride comfort and

also economic issues point of view for the railway. Increas-

ing the axle load of cars and also increasing train speed in

tracks have increased the contact force between the wheel

and rail. Based on this study, the failure mechanism of the

wheels is divided into three categories [2]:

A) Superficial cracks:

This type of cracks usually occurs because of severe plas-

tic deformation which is a result of contact stresses. Defor-

mations can occur because of excessive loading (more than

the designed limit). Wheel failure including spalling, shell-

ing, severe deformation and etc. are all of this type of fail-

ure. This type of fatigue is of low cycle one.

B) Under surface cracks:

These cracks are produced because of under surface

stresses. Fatigue cracks usually start from some millime-

ters under the wheel surface where maximum shear stress

occurs. Wheel failures which are namely called deep shell-

ing and shattered rim are of this type of failures. This type

of fatigue is of long life cycle one.

C) Under surface cracks with a high depth:

Such cracks usually occur because of material impuri-

ties during production process.

Pressure in wheel/rail contact area is usually calculated

in two ways. The first method is calculating the contact

pressure applying the analytical method. The theory gov-

erning the wheel/rail contact is Hertz theory. This theory

describes this fact that when two solid materials are com-

pressed to each other by vertical loads, their contact area is

formed. Hertz theory is based on the assumption of the

elasticity of the materials in the contact area and at static

mode ignoring the friction coefficient [3]. The second

method is applying numerical method (F.E.M).

The advantage of numerical method compared with

Hertz analytical method is that the former is reliable when

the material behavior in the contact area is in elastic-plas-

tic range while the latter is acceptable if the stresses are

†

*

Corresponding author: Head Specialist at Studies & Research Group in RAJA

Rail Transportation Co., Iran

E-mail : roya_ashofteh@yahoo.co.uk

Head Specialist at Q.C Group in Raja Rail Transportation Co., Iran

ⓒThe Korean Society for Railway 2014

http://dx.doi.org/10.7782/IJR.2014.7.1.001

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Roya Sadat Ashofteh and Ali Mohammadnia / IJR, 7(1), 1-7, 2014

studied in the elastic area.

This study is to review elastically the wheel and rail

model under stable loading first based on static load of the

cars so that the stresses in the contact area are compared

with Hertz theory (to approve the accuracy of the soft-

ware responses, the study should be reduced based on

elasticity assumption of the hertz theory and static analy-

sis). The stresses are then analyzed under static load of the

car and elastic-plastic behavior. Last phase will be to con-

sider dynamic load (quasi-static analysis) elastic-plastic

phase and analysis of stress. To analyze ABAQUS 6.6

from standard and explicit package will be applied [4].

2. Modeling and Stress Analysis

For modeling the wheel and rail, the profile of passen-

ger wheel with the diameter of 920 mm (Fig. 1) which is

amongst the most common wheels in the Iranian fleet was

applied. UIC60 and U33 rail profiles [5] were modeled

with 1:20 inclination.

Wheel is considered as a mass of deformable-solid type.

Rail length is considered as 600 mm. Next step is to con-

sider mechanical properties for the wheel and rail. First,

wheel and rail are considered to have a completely elastic

behavior. Solid wheel with R7T material [6,7] is with elastic-

ity module of 206 GPa, yield stress 545 MPa, Poisson coeffi-

cient 0.27 [1], and rail UIC60 and U33 are with elasticity

module 210 GPa, yield stress 550 MPa and Poisson coeffi-

cient of 0.3 [8]. Next step is to load the wheel on rail. The

assumption is that the wheel contact area on rail has a dis-

tance of 70 mm from the flange. Based on the weight of pas-

senger cars when loaded (with passengers) which is 51 tons,

the weight applied on each wheel is 63.75 kN. Boundary

conditions are the applied as a further step. Rail bed is con-

sidered completely solid and restrained. Wheel is com-

pletely restrained in all directions except the vertical direction

which is completely free for applying load on wheel. Next

step is meshing. Wheel and rail element shape was both cho-

sen of hex. Solid linear type C3D8R: An 8-node linear brick,

reduced integration, hourglass control, 8 nodes square one.

Element type is explicit and 3D stress. Smaller meshing was

applied in the contact area of the wheel and rail (Fig. 2).

Contact pressure value achieved 485 MPa (Fig. 3).

Since the contact pressure in contact area is less than

yield limit of wheel steel (545 MPa). Wheel does not enter

the plastic limit.Fig. 1 Passenger wheel drawing with the diameter of 920 mm

Fig. 2 Wheel and UIC60 rail meshing

Fig. 3 Wheel contact pressure

Stress Analysis in the Elastic-Plastic Analysis of Railway Wheels

− 3 −

2.1 comparing the result achieved from F.E

method and Hertz theory

Based on Hertz theory (analytical method) contact pres-

sure value for passenger wheel and UIC60 rail is 497

MPa. The F.EM method results are similar to results of

Hertz theory method having a 2% diversion. The Labora-

tory at TU Berlin University in Germany had provided test

results for S1002 wheel profile[9] and UIC60 rail speci-

men that indicate the contact pressure is 502 MPa for

60 kN load on each wheel.

Accordingly, it can be concluded that the normal stress

value obtained during laboratory works in Germany com-

pares well with that of Hertz Theoretical Analysis with

even the limit element results of both being very similar to

each other.

2.2 passenger wheel and U33 rail

The second step was modeling and analysis of the same

wheel with U33 rail. Since the curve radius of the surface

of such a rail is less than UIC60 rail, contact ellipse should

be naturally smaller and contact pressure should be more.

The model was made in computer (Fig. 4). In static analy-

sis and applying standard package, plastic deformation of

the rail was ignored and rail was considered as discrete

rigid. Wheel element shape was of hex solid linear type

C3D8R: An 8-node linear brick, reduced integration, hour-

glass control. Number of wheel elements is 28968 ele-

ments. Rail element shape was of linear quadratic type

R3D4: A 4-node 3-D bilinear rigid quadrilateral. Number

of rail elements is 1718 elements.

After the analysis of the program, maximum pressure (in

node number 187) achieved 870 MPa and Von-Misses

stress was 489 MPa (Fig. 5).

After reviewing the response convergence, elastic-plas-

tic static analysis is performed. For the passenger wheel

which is of R7T steel type, the coordinates of engineering

stress-strain of the points are as the Table 2.

Based on this, the real plastic stress-strain is achieved as

Table 3.

Table 1 Lab Test Work at TU Berlin University of Germany,

Contact Ellipse and Stress Calculation [10]

Fig. 4 Wheel as a deformable mass and rail as a solid mass

Fig. 5 Maximum Pressure and Von-Misses stress values in

wheel and U33 rail contact (Static load, elastic analysis)

Table 2 Engineering (elastic) stress-strain specification of

wheel

Engineering stress (MPa) Engineering strain

745 0.025

845 0.05

875 0.095

Table 3 Specifications of real (plastic) stress- strain of the

wheel

Plastic stress (MPa) Plastic strain

545 0

763.625 0.02099

887.25 0.0863

958.125 0.0863

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Roya Sadat Ashofteh and Ali Mohammadnia / IJR, 7(1), 1-7, 2014

Through the analysis of the program, maximum pres-

sure (in node number 187) will be the same 870 MPa

value. The reason is that the Von-Misses stress in the elas-

tic-plastic step was 489 MPa. This value is less than yield

stress of wheel steel. Thus, the stresses do not enter the

plastic limit (Fig. 6).

By comparing Figures 5 and 6, it can be seen that under

static load, the elastic and elastic-plastic analysis results

are identical.

The other adopted measure was that the maximum

weight of the car in dynamic mode (quasi-static load) was

1.25 times bigger than the railcar weight in the analysis of

the static mode [11]. The behavior of wheel and rail in

elastic-plastic mode is considered here. The friction coeffi-

cient has been considered as 0.3 [12,13]. Since the

involvement length of the wheel-rail is very little com-

pared with the total surface of the wheel, and also consid-

ering the 160 km/h linear speed of the wheel, the length

for a wheel to have a complete rotation shall be 0.07 sec-

onds. Considering this fact that the diameter of Hertz

ellipse is at most 15 mm in this mode, the involvement

time of wheel and rail is about 0.000364 seconds. Cyclic

quasi-static load with the maximum range of 100 kN shall

be according to Fig. 7.

Through dedicating variable forms of force to the soft-

ware (according to Fig. 7) in the elastic mode, the maxi-

mum pressure and Von-Misses stresses are achieved. In

elastic mode with cyclic quasi-static load, maximum pres-

sure and Von-Misses stress shall be 1173 and 635 accord-

ingly (Fig. 8).

In elastic- plastic mode with cyclic quasi-static load,

maximum pressure and Von-Misses stress shall be 1103

and 553 accordingly (Fig. 9).

Fig. 6 Maximum Pressure and Von-Misses stress values in

wheel and U33 rail contact (Static load, elastic-plastic analysis)

Fig. 7 Cyclic load based on time (based on quasi-static load)

Fig. 8 Contact pressure and Von-misses stress values in wheel

and U33 rail contact (quasi-static load, elastic analysis)

Fig. 9 Contact pressure and Von-misses stress values in wheel

and U33 rail contact (quasi-static load, elastic-plastic analysis)

Stress Analysis in the Elastic-Plastic Analysis of Railway Wheels

− 5 −

Based on this mode, Hertz ellipse shall also change with

the application of cyclic quasi-static load. When the load

increases at first, Hertz ellipse shall also grow (Fig. 10).

By comparing Figs. 8 and 9, it can be seen that under

dynamic load (quasi-static), the elastic and elastic-plastic

analysis results discord with one another.

In the Fig. 11, the Von-Misses stress at various points on

a cross-sectional surface of the wheel is illustrated (results

taken from F.E.M).

As shown in Fig. 11, the inside of the wheel, stresses

attenuate from near the surface to the center of the wheel

and a maximum stress is manifested at a point (about 3

mm) below the wheel surface. Table 4 shows a compari-

son of the maximum pressure and Von-Misses stress for

wheel and U33 rail.

3. Discussion

The F.E.M. modeling results with the assumption of

engineering behavior of wheel material (Eng. stress-strain

curve) signify that the R7T wheel in contact with U33 rail

under static loading (weight of railcar) is within the elastic

range. In other words, the stresses at contact area are lower

than the steel wheel yield stress. By taking into consider-

ation the real material behavior of R7T wheel (real stress-

strain curve); the R7T wheel in contact with U33 rail

under static loading is also within the elastic limit.

Thus, under static load, the elastic and elastic-plastic

analysis results are identical.

However, under dynamic load (quasi-static) and by tak-

ing into consideration the real material behavior of R7T

wheel (real stress-strain curve) due to high forces applied

and stresses exceeding the yield stress, the wheel material

behavior is within the elastic-plastic limit.

Thus, under dynamic load (quasi-static), the elastic anal-

ysis (with the assumption of engineering behavior of

wheel material-Eng. stress-strain curve) and elastic-plastic

analysis (by taking into consideration the real material

behavior- real stress-strain curve) results discord with one

another and the elastic-plastic behavior analysis of the

wheel is an apt method and the results can be further uti-

lized.

Fig. 10 Maximum stress – Hertz ellipse in the maximum mode

of quasi-static range

Fig. 11 Von-Misses stress graph of the under study nodes

Table 4 Comparison of the maximum pressure and Von-Misses stress in the wheel U33 rail contact

Level of loading (N) Type of loading Behavior of contact area Pressure (MPa) Von-Misses (MPa)

63750 Static(weight of railcar) Elastic 870 489

63750 Static(weight of railcar) Elastic-plastic 870 489

100000 Quasi-static (cyclic) Elastic 1173 635

100000 Quasi-static (cyclic) Elastic-plastic 1103 553

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Roya Sadat Ashofteh and Ali Mohammadnia / IJR, 7(1), 1-7, 2014

4. Effect of Wheel and Rail Parameters

The most important factor which affects wheel life is the

increase of stresses. Thus, contact pressure is very important.

Parameters mentioned below affect on contact pressures.

4.1 effects of wheel diameter

Given a load of 100 kN exerted on the S1002 wheel on

U33 rail, the contact stresses would vary given the wheel

has varying diameters. Fig. 12 illustrates the compared

effect of wheel diameter on contact pressure at area of

wheel and rail contact for wheel diameters of 890, 900,

910 and 920 mm.

Increase in wheel diameter causes contact pressure to

decrease, thereby, the stresses will decrease.

4.2 effect of axle load

For a quasi-static load of 100 kN in proportion to a

scope of 63.75 kN, Von-Misses stress shall reduce from

553 MPa to 488 MPa. By changing the scope of quasi-

static load to 90 kN, Von-Misses stress shall be 530 MPa

(Fig. 13).

4.3 effect of wheel material

By changing the material from steel R7T to R9T, Von-

Misses stress shall change from 553 MPa to 581 MPa.

4.4 effect of rail type

Fig. 14 is based on the lateral displacement of the wheel

on rail. However, by comparing the stresses in every sin-

gle moment, it can be understood that the stresses of con-

tact area for the passenger wheels in contact with UIC60

rail is less than the stresses of the wheel and U33 rail con-

tact. At the wheel tread and rail contact area (70 mm from

the wheel flange), the contact pressure for UIC60 rail

equals 497 MPa while this value for U33 rail is about 876

MPa.

5. Conclusion

The F.E.M. modeling results with the assumption of

engineering behavior of wheel material (Eng. stress-strain

curve) signify that the R7T wheel in contact with U33 rail

under static loading is within the elastic range. By taking

into consideration the real material behavior of R7T wheel

(real stress-strain curve); the R7T wheel in contact with

U33 rail under static loading is also within the elastic

limit.

Thus, under static load, the elastic and elastic-plastic

analysis results are identical.

However, under dynamic load (quasi-static) and by tak-

ing into consideration the real material behavior of R7T

wheel (real stress-strain curve) due to high forces applied

and stresses exceeding the yield stress, the wheel material

behavior is within the elastic-plastic limit.

Thus, under dynamic load (quasi-static), the elastic analy-

sis (with the assumption of engineering behavior of wheel

material-Eng. stress-strain curve) and elastic-plastic analy-

sis (by taking into consideration the real material behavior-

real stress-strain curve) results discord with one another

and the elastic-plastic behavior analysis of the wheel is an

apt method and the results can be further utilized.

Fig. 12 Effect of wheel diameter in the pressure

of the contact area

Fig. 13 Effect of changing weight (load) applied on each

wheel on the stress of contact area

Fig. 14 Effect of rail type

Stress Analysis in the Elastic-Plastic Analysis of Railway Wheels

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Calculation of stresses at wheel/rail contact (tribology)

using finite element method (F.E.M) is a better approach

than Hertz Theory method. The latter approach makes an

assumption about wheel material elastic behavior that is

without any friction coefficient between the two materials

at the contact surface. Thus, the Hertz Theory method can-

not be considered an accurate way of analyzing the con-

tact stresses.

With due respect to the rail track condition including the

rail joints, rail corrugations, dynamic condition of the rail-

car, etc., it can be concluded that the forces acting at

wheel/rail contact zone are high and due to acting high

stresses (exceeding wheel steel yield stress), the tribology

behavior are within the elastic-plastic behavior limit. Also,

the Hertz Theory method can never take into account vari-

ous parameters involved.

Diameter of wheel, axial load, wheel material (chemical

and mechanical characteristics), type of rail (radius of rail

curvature in contact with the wheel), type of wheel profile

(curve radius at the wheel tread) are types of factors that

due to their effect on wheel/rail contact forces play a sig-

nificant role for determining the life cycle and reliability of

wheel tread.

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