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Shape optimization analysis for corrugated wheels
M. Akama, T. Kigawa & E. Kimoto
Materials Technology Development Division,
Railway Technology Research Institute,
2838, Hikaricho, Kokubunjishi, Tokyo, Japan
Email: [email protected]
Abstract
A shape optimization analysis for corrugated wheels is performed using growthstrain method. The analysis consists of iteration of two steps. The first step is astress analysis under the condition of severe drag braking using generalpurposeFEM codes. The second step is a growthstrain analysis based on a growth law. Inthis analysis, bulk strain develops in proportion to the deviation of Mises equivalentstress from the average or permissible value in a plate part of the wheel. The resulting shapes indicate the effectiveness of this technique for improving strength and
reducing the weight of wheels.
1 Introduction
Corrugated wheels [1] are developed in order to raise the train speed by
reducing the unsprung mass. The plate part of the wheel is corrugated in the
circumferential direction to make the wheel more rigid. If the plate strength
is guaranteed, increases in the stiffness result in a reduction of plate thick
ness. This is the concept of reducing the wheel weight.
Recently, it has been recognized that stresses generated in the plate
region under the case of severe drag braking are much larger than those
under the case of mechanical loading which is a combination of vertical
load from rail and lateral load on the curved track. Therefore, a develop
ment of new shape that reduces such stresses without increasing the weight
is a current topic of research. In order to speed up more than ever, a further
Transactions on the Built Environment vol 34, © 1998 WIT Press, www.witpress.com, ISSN 17433509
384 Computers in Railways
reduction of weight is also desired.
In this paper, a shape optimization analysis of solid rolled corrugated
wheels is performed with uniform strength as a criterion for improving
strength and reducing the weight of wheels.
2 Procedure
The methodology is based on the iteration of two steps. The first step is an
analysis of the strength that is to be made uniform, and the second step is a
growthstrain analysis based on a growth law [2]. In this analysis, bulk
strain develops in proportion to the deviation of Mises equivalent stress from
the average or permissible value in the region that should be optimized.
Numerical analysis is accomplished by the finite element method.
2.1 Growth law
First, the bulk strain 8 * is formulated which develops in proportion to the
deviation of the strength o from the basic value a ̂ :
where d is the Kronecker delta, and h is a growth rate with which the
magnitude of the bulk strain at an iteration is given. As a strength measure,
the equivalent stress under Mises's criterion can be substituted. The basic
value a ̂ might be regarded as a design constant or an average in volume.
So the portion where the strength measure is larger than the basic value,
expanding bulk strain generates, and the portion where the strength measure
is smaller than the basic value, shrinking bulk strain generates.
The growth law which relates the growth strain 8 .® to the growth stress
a ..G generated in the step is given by Hook's law :
(2)
where 8 ̂ is the growth elastic strain generated in the process, and D..̂ is
the elastic constitutive tensor. The summation convention is used in this
equation.
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Computers in Railways 385
Conventional Shaj>e
Thermal Stress Analystsunder Drag Braking
Figure 1: Basic steps of shape optimization analysis for corrugated wheels
2.2 Growthstrain method
A Schematic flow chart for the shape optimization is shown in Fig. 1. In the
first step of strength analysis, the stress distribution is analyzed. The second
step of growthstrain analysis is intend to analyze the growth deformation
based on the growth law of equations (1) and (2), and to shift the nodal
positions in all parts of the body according to the result of this analysis. In
this step, the boundary condition comes from design restriction that is inde
pendent of the boundary condition in the first step. The computations are
terminated when the stress distribution has converged.
3 Shape optimization analysis
There are two types of corrugated wheels: Atype for trailer or diesel car and
Btype for motor car. In the plate part of Atype corrugated wheel, mechani
cal stresses due to vertical load offset those due to lateral load. So the thick
ness of plate can be thin. As a result, thermal stresses generated in the plate
region are large under the case of severe drag braking. In the plate part of B
Transactions on the Built Environment vol 34, © 1998 WIT Press, www.witpress.com, ISSN 17433509
386 Computers in Railways
type corrugated wheel, on the contrary, mechanical stresses due to vertical
load are superimposed on those due to lateral load. So the region has to be
thick and the weight is large. Considering these facts, shape optimization
analysis to increase strength without increasing weight is performed for A
type and that to reduce weight without decreasing strength below the per
missible value is performed for Btype corrugated wheel.
3.1 Increasing strength of Atype corrugated wheel
As mentioned above, stresses generated in the plate region under the case of
severe drag braking are much larger than those under the case of mechanical
loading. So the stress analysis for corrugated wheels under the condition of
severe drag braking is performed as a first step. The stress analysis is per
formed when the temperature at 10mm below the tread surface of wheel
reaches 230°C. Considering that the temperature under normal drag brak
ing is lower than about 200°C, this condition is considerably severe.
In this investigation, an uncoupled thermalstress finite element analy
sis is carried out using a generalpurpose nonlinear analysis code ABAQUS.
First, a nonlinear heat transfer analysis is performed using temperature de
pendant thermal properties. Only one twelfth of the wheel is enough for the
Figure 2: Finite element mesh of Atype corrugated wheel
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Computers in Railways 387
Figure 3: Finite element mesh for growth strain analysis
mesh, as shown in Fig.2. The heat input from a brake shoe is approximated
as timedependent surface heat generation by using the tread part of wheel
elements. Using the temperature distributions predicted from the heat trans
fer analysis as inputs, the transient thermal stresses in the wheel can be cal
culated by a thermoelasticplastic analysis. If this calculation is
accomplished, a and a ̂ in the equation (1) can be determined.
The strength a is assumed to be the equivalent stress in Mises's crite
rion. The average equivalent stress in volume is substituted for the basic
value a ̂ because the wheel weight should be constant. A value of 0.05 is
taken for the growth rate h. The portion where the shape changes should be
made as small as possible and the subject of current research is on the stress
in plate region. So the element mesh used for the growthstrain analysis
calculations is restricted to the plate region on which the previous stress
analysis is performed. This is shown in Fig.3. Nodes that are adjacent to theunchanged portion are restrained in all degrees of freedom.
First, bulk strains that are generated at each node are calculated by the
equation (1). These are used for the calculation which analyze the growth
deformation and the nodal positions in all parts of the body are shifted ac
cording to this result. Then the new shape of plate region is transformed
into the input together with the portions that do not change such as hub, rim
and flange region to perform the stress analysis again. The convergence isjudged after the analysis.
Transactions on the Built Environment vol 34, © 1998 WIT Press, www.witpress.com, ISSN 17433509
388 Computers in Railways
3.2 Reducing weight of Btype corrugated wheel
The analysis procedure is almost the same as that described in the previous
subsection except that the permissible value was substituted for o ^. For
the permissible value, the maximum value of the equivalent stress generated
in the plate region of the original Atype corrugated wheel under the condi
tion of severe drag braking is used. The weight decreases until the maxi
mum stress reaches the permissible values in this case.
In addition, restrictions that come from manufacture ability are also con
sidered. For example, a thickness of the plate region does not change in
circumferential direction for the case of solid rolled wheel. In analysis, this
restriction can be accomplished by limiting the bulk strains that are gener
ated in each node along a circle of the same radius to the minimum value
obtained by the calculation at a first step. Moreover, the thickness can not
decrease rapidly in radial direction toward the rim part. This can be accom
plished by setting the changing rate of bulk strain in radial direction at the
maximum permissible rate.
In this case, a safety factor that is used in a wheel design is considered
for judgement. The safety factor is defined as follows:
S=~, TT (3)
where o ̂ represents the mechanical stress amplitude due to a combination
of vertical and lateral load; o ̂ represents a stress of fatigue limit; a ̂ is a
mean stress including a thermal stress due to drug braking and a residual
stress and o ^ is an ultimate tensile strength. Mechanical stress analysis is
performed under the condition of simultaneous loads of 98kN in the vertical
direction and 59kN in the lateral direction. Once the maximum value in the
plate region of optimizing shape has reached that of original Atype, the
analysis is stopped.
4 Results
Figure 4 shows the Mises equivalent stress contours of the original Atype
shape under drag braking. In the convex portion of the plate region near the
hub, stresses are very high and plasticity occurs in several elements due to a
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Computers in Railways 389
high temperature gradient of the wheel. In the plate region near the back
rim fillet, high stresses are also generated. As compared with these por
tions, the stresses in the rim fillet regions of both front and back are low.
Figure 5 also shows the Mises equivalent stress contours under the
same condition for the shape after the growthstrain analysis has been per
formed to increase strength. The iteration calculation was repeated four
times. In the plate region, the thickness increases near the boss region and
decreases near the rim region. Large reduction of stresses, about 20%, can
be seen in high stress regions whereas stresses increase in the rim fillet re
gions where they are low in the original shape. There exists no yield region
any more, and it is seen that the stress distribution gradually becomes uni
form. If the number of iterations increases, the stress distribution can be
expected to be more uniform in the plate region.
Figure 6 shows convergence rates in this analysis, which consists of
the equivalent stress ratio of the maximum a ̂ at the center of element to
the average a ̂ in volume and the mass ratio of the current mass M to the
original mass M̂ . As is expected, the results in this figure show that stresses
become steadily more uniform after iterations and the weight is about thesame after each iteration.
Figure 7 shows principal stress tensors under drag braking of the B
type shape after the growthstrain analysis has been performed to reduce the
weight. The iteration calculation was repeated six times. In the convex
3 
2 
) , , , , •
4

1 
i i i i
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1 
, , . .
i — ' — ' — i — i —
»4
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1 — i — i — i — i —  — i — i — i — i —
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) 1 2 3 4 5Iteration number
Figure 6: Convergence rates
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390 Computers in Railways
Max : 562MPa
Min : 257MPa
Figure 7: Principal stress tensors under drag braking of new shape of B
type corrugated wheel
portion of plate region near the back hub and in the plate region near the rim
fillet, the circumferential stress is high. As compared with Atype, the ten
dency of stress distribution is reversed. This is because the wave of plate is
reversed between Atype and Btype corrugated wheels.
Figure 8 shows convergence rates in this analysis, which consists of
the equivalent stress ratio of the maximum for Btype a ̂ ^ at the center of
element to the maximum for Atype a ̂ ^ in volume and the mass ratio of
the current mass of Btype M^ to the original Atype M^. It can be seen that
the weight, which is larger than that of the Atype by about 10%, becomes
smaller after the 6th iteration. Even at this moment, the stress generated in
the plate region is smaller, and the minimum safety factor is still larger than
that of the original Atype. If the iteration proceeds to the 7th step, however,
the minimum safety factor becomes smaller.
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Computers in Railways 391
<DCZ
"•
0.9
0.8
40.7
n A
_ ' ' ' '
! ,
i «
• ,
> <
i i
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' i
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Figure 8: Convergence rates
5 Conclusions
A Shape optimization analysis for corrugated wheels is performed using
growthstrain method. The results are as follows:
(1) In the plate region of the Atype corrugated wheel where large thermal
stresses are generated under the drag braking, an about 20% reduction of
stresses is achieved without increasing weight.
(2) With the Btype corrugated wheel, which is heavier than the Atype, an
about 10% reduction of weight is achieved without decreasing strength be
low the permissible value.
It is expected that this technique can easily be applied not only to other
railway wheels but also to general structural components even if their shapes
are much more complicated.
References
[1] Yamamura, Y, Nakata M. & Anjiki, M., Development of brakeheat
resistant corrugated wheel, Sumitomo Search, No.57, pp. 1826, (1995)
[2] Azegami, H., Okitsu, A. & Takami, A., An adaptive growth method for
shape refinement: Methodology and applications to pressure vessels
and piping, Transactions oftheASME, Journal of Pressure Vessel Tech
nology, Vol.114, No.l, pp.8793, (1992)
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392 Computers in Railways
( MPa)
Figure 4: Mises stress contours of conventional Atype corrugated wheel
(MPa)
Figure 5: Mises stress contours of new shape of Atype corrugated wheel
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