reduction of weight is also desired

Shape optimization analysis for corrugated wheels

M. Akama, T. Kigawa & E. Kimoto

Materials Technology Development Division,

Railway Technology Research Institute,

2-8-38, Hikari-cho, Kokubunji-shi, Tokyo, Japan

Email: [email protected]

Abstract

A shape optimization analysis for corrugated wheels is performed using growth-strain method. The analysis consists of iteration of two steps. The first step is astress analysis under the condition of severe drag braking using general-purposeFEM codes. The second step is a growth-strain 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 result-ing 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 1743-3509

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

growth-strain 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.


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 Growth-strain 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 growth-strain 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: A-type for trailer or diesel car and

B-type for motor car. In the plate part of A-type 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-



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 B-type corrugated wheel.

3.1 Increasing strength of A-type 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 thermal-stress finite element analy-

sis is carried out using a general-purpose 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 A-type corrugated wheel



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 time-dependent 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 thermo-elastic-plastic 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 growth-strain 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.



3.2 Reducing weight of B-type corrugated wheel

The analysis procedure is almost the same as that described in the previous

sub-section 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 A-type 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 A-type, the

analysis is stopped.

4 Results

Figure 4 shows the Mises equivalent stress contours of the original A-type

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



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 growth-strain 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 growth-strain 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

»

" 4

1 |

, , . .

i — ' — ' — i — i —

»4

•

1 — i — i — i — i — | — i — i — i — i —

• a max/ crave 1 •• M / Mori §

»4

• •

, , . . i , . . .

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-

) 1 2 3 4 5Iteration number

Figure 6: Convergence rates



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 A-type, the ten-

dency of stress distribution is reversed. This is because the wave of plate is

reversed between A-type and B-type corrugated wheels.

Figure 8 shows convergence rates in this analysis, which consists of

the equivalent stress ratio of the maximum for B-type a ̂ ^ at the center of

element to the maximum for A-type a ̂ ^ in volume and the mass ratio of

the current mass of B-type M^ to the original A-type M^. It can be seen that

the weight, which is larger than that of the A-type 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 A-type. If the iteration proceeds to the 7th step, however,

the minimum safety factor becomes smaller.



<DCZ

"•

0.9

0.8

40.7

n A

_ ' ' ' '

! ,

i «

• ,

> <

i i

» 4

' i

» <

• jBmax/aAmax j• MB/MA ^

1 g

1 <

• i i i

| -* 1

» «

, , , .

.1

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• 1 1 1 "

0 1 2 3 4 5 6 7Iteration number

Figure 8: Convergence rates

5 Conclusions

A Shape optimization analysis for corrugated wheels is performed using

growth-strain method. The results are as follows:

(1) In the plate region of the A-type 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 B-type corrugated wheel, which is heavier than the A-type, 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. 18-26, (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.87-93, (1992)



( MPa)

Figure 4: Mises stress contours of conventional A-type corrugated wheel

(MPa)

Figure 5: Mises stress contours of new shape of A-type corrugated wheel


reduction of weight is also desired

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