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 : [email protected]
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
− 2 −
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
− 7 −
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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