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BRITISH STEEL TECHNICAL Swinden Laboratories

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SUMMARY

AUTOMOTIVE WHEEL DESIGN CRITERIA FOR LOW CYCLE FATIGUE

Tim Wolverson

The automotive industries of Europe, the United States and Japan, have recently been developing

high strength alloy steels to produce lighter weight vehicle components by down gauging steel strip.

By designing towards a finite life, the expected loss in component stiffness due to down gauging has

been minimised by utilising the cyclic work hardening properties of higher strength steels.

A strategic investigation by British Steel Swinden Laboratories, in conjunction with Dunlop-Topy

Wheels Limited, has explored the relatively new approach of the low cycle fatigue technique, and

determined its accuracy in predicting the life of a wheel subject to the reverse bend test.

Finite element analysis was employed to calculate the nominal and local stress distributions in a

wheel due to the loading imposed by the reverse bend test rig. Three variations of Neuber analysis

were then performed to predict life at positions of highest stress.

When the variations in disc thickness, due to pressing, were included in the analysis models, it was

found that life estimations at the wheel vents were within 4% of the experimental mean of fatigue

performance data. Furthermore, it was shown that fatigue failure had occurred experimentally at

the vents.

By inputting the nominal thickness of the wheel material to the finite element models, the life

predictions at the wheel vents were found to be 20% less than the mean of experimental data. As

thicknesses due to pressing would not be known for a new wheel design, it was concluded that a

designer would be predicting wheel life conservatively.

A suggested design route was identified for designing towards low cycle fatigue. However, it was

stated that until the low cycle fatigue data for many materials are compiled and made readily

available to engineers, the implementation of the low cycle fatigue technique may not be successful.

KEYWORDS

35 Finite Element Method

Fatigue Life HSLA Steels

+ Hypress

Fatigue Properties Fatigue Tests Low Cycle Nippon Steel Stress Concentration Automobiles Wheels Lab Reports


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AUTOMOTIVE WHEEL DESIGN CRITERIA FOR LOW CYCLE FATIGUE

1. INTRODUCTION

There have been many developments in formable high strength low alloy steels for wheel

manufacture. The potential for these alloys is complicated by the complex shapes used in wheel disc

pressing. To optimise and develop these steels it is necessary to understand how the intrinsic

material properties are used within the complex shapes to achieve the performance specification in

fatigue tests. Therefore, a strategic research exercise was proposed to develop a suitable design

method so that material properties may be assessed.

Improvements in analytical techniques, over the last few years, have directed engineers towards

designing components for finite fatigue resistance rather than using the traditional approach of

infinite life calculations. Since the significant increase of fuel prices in the early 1970s, the

competitive automotive industry has attempted to reduce the weight of new vehicle models by

down gauging on steel strip and utilising higher strength steels to create lighter weight components.

The Chrysler Corporation(1) in the United States has employed such analytical methods to develop a

sophisticated design philosophy that is more advanced than in Europe, and many examples exist

that show how designing components for finite fatigue resistance has improved vehicle economy

through weight reduction.

The traditional method of fatigue design is based upon the high cycle S-N curve, where the number

of cycles to failure of a smooth test specimen is plotted against an applied nominal stress range.

Cycles to failure are greater than 100,000 when the stress range does not exceed the elastic limit of

the material. Materials such as iron or steel often exhibit a fatigue limit, a stress range below which

the fatigue life of the material appears to be infinite. To avoid fatigue failure, components are

usually designed to withstand a maximum stress range in service, considerably less than the fatigue

limit. Typically, the fatigue limit of a material is seldom greater than 50% of the tensile strength.

Designing to stress ranges lower than this value coupled with the introduction of safety factors

results in the full strength of the material being redundant. The disadvantage to component weight

is also an issue. Moreover, this design approach is considered unrealistic as most components have

stress raisers or discontinuities that allow the stress distribution to become in excess of the design

stress with the possibility of local plastic deformations occurring. As most engineered structures or

components are intended to have a finite life, this approach is inefficient and unnecessary.

A more reasonable procedure would be to design confidently, structures or components that will

not fail by fatigue within an expected service life. The aerospace industry, in particular, requires this

design approach as the aluminium based alloys used for airframe constructions do not possess

fatigue limits, and life predictions using S-N curves are only possible with reference to the endurance

limit of the material. Also, in automotive design, it has been found(2) that a large proportion of

fatigue damaged components are the result of cyclic plastic deformations arising from high stresses

caused by stress raisers or the geometry of the component itself. With low cycle fatigue, the number

of cycles to failure at a constant strain amplitude in such components is typically less than 100,000

cycles.


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Engineers designing for low cycle fatigue have a requirement to establish the cyclic characteristics of

different materials at high plastic strains. Extensive work(1) undertaken at the Chrysler Corporation

has shown that the cyclic work hardening properties of a material can differ from the work

hardening characteristics observed under static loading in a normal monotonic tensile test. It was

shown that cyclic work hardening was higher than monotonic work hardening in some materials,

lower in other materials and sometimes identical with the work hardening behaviour experienced

under normal tensile tests. It was found that from careful selection of high strength steels, the

expected loss in stiffness due to down gauging of a component could be minimised by utilising the

cyclic work hardening properties of the material grade by designing towards low cycle fatigue and a

finite life.

The Nippon Steel Corporation(2) has been developing high strength alloy steels for the Japanese

automotive industry. A recent fashionable trend in Japan has forced automotive engineers to design

wheels that are more decorative and aesthetically pleasing in appearance. Also, an increase in the

width to height ratio of tyres has led to a shallower rim form on the wheel that is compensated by a

larger wheel radius. Both trends have lead to a substantial increase in wheel weight. Furthermore,

an increase in the number and size of the suspension and tracking components has significantly

increased the overall weight of the automobile and attention has been focused on reducing the

weight of the unsprung mass. The high strength to weight ratio of the high strength alloys has

allowed designers to reduce the gauge of certain components that are subjected to dynamic and

fluctuating loads. One of the largest components of the unsprung mass is the car wheel sub-

assembly and there have been successful attempts to reduce the weight of the wheel pressing with

the use of high strength alloy steels and other materials. Weight savings in the unsprung mass of a

vehicle have been shown to be advantageous to the smoothness of the ride, and economy. Further

benefits are weight reductions of the suspension, steering support and sprung components.

Dunlop-Topy is a customer of British Steel and is supplied with high strength steels for their current

wheel pressing for Rover. At present, new wheel designs are based upon the stresses that the wheel

is likely to expect in service, from intuition and experience. Prototypes of the wheel are pressed,

fabricated and eventually subjected to the cyclic loading of the reverse bend test, an acceptance test

for wheels. An assessment of the wheel design can then be formalised when fatigue failure occurs.

This design method is extremely expensive because of the costs incurred with tooling, manufacture

and testing if the wheel design does not meet specification, and it is difficult to yield an optimum

wheel design from such a slow iterative approach.

Dunlop differentiates wheel performance by the extensive testing of varying wheel designs on a

reverse bend test machine. Dunlop's current wheel design (LP1346) for Rover, is required to undergo

more than 50,000 cycles in the test machine that simulates harsh cornering of the wheel. Dunlop has

found, experimentally, that with a current wheel thickness of nominally 4.0 mm for this model, the

wheel life on the test machine was approximately 257,000 cycles.

However, Dunlop realises that a less expensive design route is necessary to remain competitive

within the automotive industry. As Dunlop-Topy is a valuable customer to British Steel, a strategic

investigation by British Steel Swinden Laboratories, in conjunction with Dunlop-Topy, has been


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undertaken to assess the viability and accuracy of the low cycle fatigue technique for wheels, by

predicting the low cycle fatigue life subject to the reverse bend test and comparing the predicted life

with that observed.

2. EXPERIMENTAL PROCEDURE

2.1 Wheel for Investigation

The wheel chosen for analysis was wheel LP1346. This is a design currently being developed for

Rover by Dunlop-Topy Wheels Limited. Fatigue life data, obtained from testing thirteen wheels in

the reverse bend test rig, was already available for this wheel and samples of wheel blanks and

actual pressings were obtained to assist the investigation. The wheel is shown in Figs. 1(a) to 1(c)

and the wheel nomenclature is presented in Fig. 2.

2.2 Wheel Material

Both the disc and rim of wheel design LP1346 are pressed from HYPRESS 23, a formable high

strength steel supplied by Brinsworth Strip Mill. Sample strips of HYPRESS 23 were obtained from

Dunlop-Topy and were representative of the product in its current form. The samples were

sectioned and machined into test specimens and then tested to ascertain the monotonic and cyclic

properties of the material. The test specimens produced for mechanical testing were British Steel

NFT2 test specimens. The specimen dimensions are shown in Appendix 2. For cyclic property

determination, the test specimens conformed to the specifications in BS7270 (Table 3). The cyclic

property determination of a material is explained in Appendix 1.

To obtain the elastic modulus of the material accurately, 4 samples of HYPRESS 23 were subjected to

flexural vibrations induced by acoustic means and the resonant frequency of each sample was

recorded in order that the elastic modulus could be calculated. Young's modulus determination by

this method has been shown(3) to be accurate to within 1%.

2.3 Reverse Bend Test Rig

Automobile wheel durability measurements are usually assessed with the reverse bend test. This

test simulates the fatigue strength of a wheel under a cyclic side loading which prevails during hard

cornering. Under a cyclic side loading in the elastic plastic region of a high strength steel, the reverse

bend test has been shown(2) to give the most reliable fatigue data for wheels. Therefore, it is this test

that Dunlop-Topy has adopted to evaluate wheel designs. Rover specifies that wheel design LP1346

must undergo more than 50,000 cycles in the test rig before failure occurs.

The reverse bend test rig at Dunlop-Topy is shown schematically in Fig. 3. Before a wheel is mounted

in the test rig, a moment arm, 0.276 m in length, is bolted to the back of the wheel centre. The

moment arm is substantial to provide rigidity at the wheel centre and to prevent bending occurring

in that region. This simulates the support that would be given from the wheel back plate on an

automobile. The wheel is then clamped to the chuck face plate of the machine by its rim, and a load

is adjusted to achieve the required moment load of 1579 Nm. The wheel is cycled at 510 rpm until

failure occurs.


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2.4 Finite Element Models

2.4.1 Geometric Models

An engineering drawing (No. WD 1478) of the disc pressing, from wheel design LP1346, was

obtained from Dunlop-Topy Wheels Limited. A detailed computer representation of the disc pressing

was constructed in Patran 2, a pre- and post-processor for the finite element solver, Abaqus. The

Patran software was capable of displaying wire frame and shaded plots of the constructed disc

geometry. A detailed wire frame plot of half the disc pressing is shown in Fig. 4. It can be seen that

the disc was modelled without the wheel rim.

The wheel rim was not modelled because the relevant boundary conditions were provided at the

disc to rim interface (see section 2.4.5). Also, preliminary analyses of the model indicated that stress

levels were low at this interface. It was therefore decided that the rim was insignificant to the

analysis and should not be modelled to save computational expense.

Three different computer models of the wheel disc were required so that three techniques of life

prediction could be employed.

A further computer model was analysed with the monotonic properties of the HYPRESS 23 material.

2.4.1.1 Finite Element Model 1

A model of the disc pressing was constructed with the vent details and stiffeners. The bolt holes and

the central hole shown in Fig. 4 however, were replaced with a flat surface to represent the steel

loading plate that is clamped to the back of the disc when the wheel is placed in the reverse bend

test rig. Preliminary computations, applying the load through the bolt holes of the model shown in

Fig. 5, had indicated that unrealistic stresses were prominent around the bolt holes because the

steel loading plate had not been considered in the analysis. Fig. 5 shows the high von Mises

equivalent stress distribution around the disc centre when the load was transmitted directly through

the bolt holes and no support to the centre of the disc was provided by the steel plate. Furthermore,

from experimental tests, it was found that fatigue failures did not occur in the centre of the disc

pressing where high stresses had been predicted. Therefore, it was considered that the centre

section detail of the pressing was insignificant to the analysis and the effect of the steel plate was

not minor and should be modelled. The analysis model 1 is shown in Fig. 6.


A second computer model was created without the vents and stiffeners in order that the finite

element analysis would provide a nominal stress distribution that could be compared to the local

stress distribution obtained from model 1. It was thought that stress concentration factors could be

calculated from models 1 and 2 and Neuber's method(4) of life prediction could be applied.


The third model was geometrically the same as model 1. However, the elastic properties given to the

first model were replaced with the cyclic properties of the material.


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The fourth model was identical to the first model but contained the monotonic properties of the

HYPRESS 23 material.

2.4.2 Mesh

All models were meshed with 2 dimensional shell elements. The elements were 8 noded and gave a

good fit to the curved surfaces of the models because the fit between nodes was a second order

interpolation rather than a linear fit. The aspect ratio of the elements was kept within 2:1 for

detailed areas of the models. Aspect ratios of 7:1 were allowed for regions not of interest and where

accuracy was unimportant.

2.4.2.1 Thickness Variations Due to Pressing

A non-painted wheel was sectioned and the thickness measured with a micrometer at various

regions on the disc. The recorded thickness measurements are tabulated in Table 1. Thickness

variations in a typical disc model were accounted by assigning the measured thicknesses to the

respective elements on the computer representations. Fig. 7 shows the 10 variations in thickness of

the detailed finite element model with vents and stiffeners (model 1). A thickness of 1000 mm was

given to the elements that described the steel loading plate so they would become stiff elements

and act in a rigid manner.

2.4.3 Material Models

As Neuber's method is valid only in the elastic regime, the material property assigned to the

elements on the wheel models 1 and 2 was a linear elastic definition.

On all models, the steel loading plate elements were given a high linear elastic modulus to ensure

that these elements would be infinitely stiff.

Fig. 8 indicates the two separate materials properties for model 1.

2.4.4 Loading

It was found that all the computer models could be described with 180° representations of the

actual disc pressing, since the nature of the loading was such that one plane of symmetry existed

through the centre of the wheel. However, it can be seen from fig. 9 that a full 360° disc was

produced for models 1, 3 and 4. A complete disc was modelled because further work is intended,

where a complete wheel model is necessary for results analysis.

2.4.4.1 Finite Element Models 1, 3 and 4

A pure moment of 1579 Nm was applied to the central node of the stiff loading plates common on

models 1, 3 and 4 since the discs had been modelled in their entirety.


Half the full moment load (789.5 Nm) was prescribed to the central node of finite element model 2

as only half the disc had been created for analysis.


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2.4.5 Boundary Conditions

During manufacture of the wheels, the disc pressing is spot welded to the rim in eight places

equidistant around the circumference. Since the rim was not constructed in any of the models, the

models of the disc were rigidly fixed at nodes that coincided with the spot welds. The nodes

restrained in all degrees of freedom are shown in Fig. 10 for models 1, 3 and 4.

Model 2 was supplied with the relevant boundary conditions at its half plane of symmetry.

2.4.6 Mathematical Solution

The Patran based models were transformed into Abaqus input decks and submitted for analysis. The

solutions were non linear and stringent convergence tolerances were set. The files generated by the

Abaqus finite element processing included text and binary files that were translated for post

processing in Patran.

2.4.7 Post Processing

The post-processing capabilities of Patran allowed coloured contour plots of stresses, strains and

displacements to be displayed to give a general view of the results. However, for accurate values to

be obtained from nodal positions, the text files were scanned for the appropriate nodal quantity

with a text editor.

2.5 Strain Measurements of Wheel

To verify the strains calculated from the finite element analysis of the wheel model that contained

the material's monotonic properties, ten strain gauge rosettes were applied to areas of the

predicted high strains. The rosettes contained three 5 mm strain gauges offset at 45°, and the centre

gauges were orientated radially to the wheel. Three rosettes were applied to the side wall and side

wall radii. Two rosettes were placed close to the edge of the vents. Five more strain gauge rosettes

were placed on the opposite side of the wheel, reflecting the positions of the first five rosettes. The

rosettes were fixed to the 'top' surface of the wheel since access to the underside of the wheel was

limited and made gauge installation difficult. Figs. 11(a) and 11(b) show the ten rosette positions on

the wheel. Fig. 12 indicates the rosette identification system adopted. Strain measurements from

each gauge were recorded with a Solartron Data Logger. The data logging system allowed excitation

and the internal bridge completion for each gauge. Strain readings were taken continuously as the

wheel made one complete revolution in the test rig. Angular displacement of the chuck was also

recorded. Figs. 13(a) and 13(b) show the gauged wheel in the reverse bend test rig.

2.6 Life Prediction Methods

2.6.1 Neuber's Method

Three methods of predicting the life of the wheel were considered. The first approach used Neuber's

method(4). This required two finite element models to calculate the stress concentration factors

around the wheel vents and other discontinuities. The nominal and local stress distributions

calculated with finite element analysis allowed the stress concentration factor and life to be


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predicted for regions of significantly high stress. Neuber's method of life prediction is explained in

detail in Appendix 1.

2.6.2. Obtaining Total Strain Range From Cyclic Properties and Finite Element Analysis

By including the cyclic properties of the material instead of the elastic properties with the finite

element model of the wheel, it was feasible(5,6) to use the total strain results from finite element

analysis output to predict the life of the wheel from the strain-life curve directly. The strain-life curve

is described in Appendix 1.

2.6.3 FATIMAS

A fatigue prediction software package known as FATIMAS was available and was capable of

analysing complex dynamic events. From a description of the load history that a component can

support in service, the cyclic properties of the material and the nominal and local conditions

obtained from finite element analysis, FATIMAS is able to calculate the local stress-strain response.

Techniques for cyclic counting are then performed and damage summation from each event predicts

the life of the component. It was considered that FATIMAS could be used to verify the results

obtained from the two other methods described, and that FATIMAS would give the most realistic

predictions of life.

3. RESULTS OF INVESTIGATION

3.1 Wheel Life

Thirteen wheels were subjected to a cyclic moment loading of 1579 Nm in the reverse bend test rig .

It was found that the number of cycles to failure for wheel design LP1346 were normally distributed

about a mean of 257,000 cycles to failure. The fatigue performance data for the wheel are shown in

Fig. 14. Areas of failure were common around the wheel vents. Cracks causing failure were

discovered and made prominent by spraying the wheel with a coloured dye penetrant. Figs. 15(a) to

15(d) indicate excessive crack growth, and failure around the vents of a tested wheel.

3.2 Material Testing

3.2.1 Monotonic Properties

The monotonic properties of the HYPRESS 23, obtained mechanically and acoustically, are shown in

Table 2. The mean elastic modulus for the material was calculated to be 210 KN/mm2 from acoustic

testing and 200 KN/mm2 from mechanical testing. The Young's modulus determined by acoustic

means was considered the most accurate(3).

The material was found to have a 0.2% proof strength and tensile strength of 358 N/mm2 and 461

N/mm2 respectively.


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3.2.2 Cyclic Properties

3.2.2.1 Cyclic Stress-Strain Curve

The cyclic stress-strain curve was plotted from the stable hysteresis loop tips produced from the

testing of several test specimens at different strain amplitudes (see Appendix 1). The cyclic stress-

strain relationship for the HYPRESS 23 material is shown in Fig. 16, where;

K‘ = 701 N/mm2

E‘ = 210,000 N/mm2

n' = 0.09

and relate to equation (A 1.2) shown in Appendix 1.

3.2.2.2 Strain-Life Curve

The resistance of a material to strain cycling may be described with the contribution of both the

elastic and plastic strains (see Appendix 1). The method of least squares regression was performed

on the cyclic plastic data to obtain the Coffin-Manson relationship(7). This relates the strain range and

cycles to failure for the material in the plastic regime. The experimental data and line of best fit can

be seen in Fig. 17. Similarly, the regression for the elastic portion of the strain amplitude versus life

plot is presented in Fig. 18. Table 3 presents the relevant coefficients and exponents derived from

the two regressions. The summation of the elastic and plastic cyclic data was used to give the total

strain amplitude. Fig. 19 shows the total strain amplitude versus life plot with the elastic and plastic

experimental data points. The total strain amplitude was expressed against life as;

∆𝜀𝑡𝑜𝑡𝑎𝑙

2= 0.4169(2𝑁𝑓)−0.667 +

794.85(2𝑁𝑓)−0.085

𝐸

3.3 Finite Element Analysis

3.3.1 Finite Element Model 1

Patran was capable of interpolating the Abaqus results for the top and bottom surfaces of the 2

dimensional shell elements. The ‘top' surface was defined as the surface visible in all disc figures.

Figs. 20 and 21 show the differences in the von Mises equivalent stress distributions for the top and

bottom surfaces of the disc model. It was found that the highest stresses were predicted on the

bottom surface of the disc model.

Five nodal positions, shown in Fig. 22, were shown to be the highest points of maximum stress.

Three positions were located at the vent edges where wheel failures had been shown to originate.

As stress values could not be obtained accurately from the contour plots, the exact values were

extracted from the Abaqus text files with a text editor. The von Mises and Tresca equivalent stresses

were noted at the five nodal positions.

--- (1)


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An exaggerated displaced plot of the disc mesh is shown in Fig. 23. The geometric distortions

indicate that the steel plate was modelled with ‘stiff’ elements and that the applied moment and

boundary conditions were satisfactory.


Fig. 24 shows the von Mises equivalent stress contour plot of the bottom surface for the model

without any discontinuities. It was found that the stresses on the vent platform radius (Fig. 2) were

more uniform in the absence of the wheel vents. However, a high stress was prominent at the radial

crest of the disc which was identical in value to that of model 1. The von Mises and Tresca equivalent

stresses were recorded at nodal positions similar to those on model 1. Division of the stresses at the

respective nodal positions from models 1 and 2 gave the theoretical stress concentration factor, Kt

for the five areas of interest. The von Mises and Tresca criterion were used for Kt calculation. Table 4

shows the five calculated stress concentration factors. A significant Kt value was apparent at the

radial crest and had a value of 1.0. This implied that the stress concentration occurring at that

position was due to the geometry and loading arrangement of the wheel and not a discontinuity.

This is known as a no notch effect.


Contour plots of the von Mises and Tresca equivalent stresses are presented in Figs. 25 and 26 for

the bottom surface of the disc pressing. Both distributions predict maximum stresses around the

vents and at the radial crest in similar locations to model 1. However, as this model contained the

cyclic stress-strain material data instead of the linear elastic definition given to models 1 and 2, the

contour plots indicated areas of cyclic plasticity. Indeed, the five nodal positions chosen for fatigue

prediction were found to be just within the plastic portion of the cyclic stress-strain curve. However,

one nodal position was noted within the elastic regime of the cyclic stress-strain curve. The results

suggested that the predicted strains spanned the transition point of the plastic to elastic

relationships which govern the strain-life plot (see Appendix 1).

Figs. 27, 28 and 29 show the strain contours of the bottom surface in the x, y and xy planes. Strains

in the x plane were caused directly by the moment load being applied. The strain in the y plane was

a consequence of the Poisson's ratio effect in the material, and the strain in the xy plane was shear

strain. The three values of strain were found in the Abaqus text file for the locations at the five nodal

points, and were used to calculate the equivalent strain(7) by using the formula;

𝜀𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 = (2

3 𝜀𝑥2 + 𝜀𝑦2 + 2𝜀𝑥𝑦 2 )1/2

It was found that the equivalent strain at the radial crest was 1340 micro strain. It was also

calculated that an equivalent strain of the same value had been predicted on the opposite radial

crest. The strain amplitude at the radial crest was therefore considered as 1340 micro strain, since

the strain history on experimental wheels had been a constant sinusoidal cyclic load.

3.4 Strain Measurements Of Wheel

Figs. 30 to 39 show the strain cycles, during one revolution of the wheel, for the ten strain gauge

rosettes. Reference to the rosette identification system is given in Fig. 12. The results indicated that

---(2)


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the maximum strain amplitude at the vent areas (positions l and 2) was 750 to 800 micro strain. The

strain amplitude at position 7 (radial crest) was calculated at 790 micro strain. The principal strains

were calculated for each rosette position using the formula(7);

𝜀1,2 =𝜀1+𝜀3

2 ±

1

2[ 𝜀1 − 𝜀3

2 + 2𝜀2 − 𝜀1 − 𝜀3 2]1/2

where 𝜀1,2 are the principal strains and 𝜀1, 𝜀2 and 𝜀3 are the strains from the three gauges on a 45°

strain gauge rosette. From the principal strains, the equivalent strain at each rosette position was

calculated with modification to equation (2);

𝜀𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 = [2 𝜀1

2+𝜀22+𝜀3

2

3]1/2

where 𝜀3 is the third principal strain normal to the principal plane. Since the strains were measured

in 2 dimensions only, then 𝜀3 = 0;

𝜀𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 = [2 𝜀1

2 + 𝜀22

3]1/2

The equivalent strain amplitudes for each rosette are displayed in Table 5 and are compared with

the results from the finite element analysis model that contained the monotonic properties of the

material. It can be seen that the measured strains agreed closely with those predicted, the maximum

difference being 7.6% at rosette number 6.

3.5 Life Prediction

Since a notch, or discontinuity has less effect during fatigue conditions than the theoretical stress

concentration factor would imply(5,6,7), the fatigue notch factor, Kf should be used for life prediction.

The fatigue notch factor is explained in Appendix 1 and its value is significantly less than the

theoretical stress concentration factor at high plastic strains. As the fatigue notch factor could not be

calculated, the theoretical stress concentration factor, Kt was used for each prediction method. This

was thought to be satisfactory, because at low plastic strains the fatigue notch factor Kf has been

shown to be almost equal to Kt(5,6,7).

3.5.1 Neuber's Method

Table 6 contains the variables required to create Neuber's hyperbola, the predicted strain

amplitudes and life for the five positions that were of interest. Figs 40 to 44 show the intersections

of the cyclic stress-strain curve with the hyperbolas to give local strain amplitudes. It can be seen

that the intersections of the curves are within the elastic portion of the cyclic stress-strain curve. The

low strains suggested that using Kt in place of Kf would give valid life predictions(5, 6, 7).

Table 6 shows that the life predictions generally fell within the distribution of experimental data for

the areas of interest.

3.5.2 Using The Total Strain Obtained From Finite Element Analysis To Predict Life Directly

It is generally assumed(5, 6) that the intersection of the cyclic stress-strain curve with Neuber's

hyperbola gives the strain amplitude for a particular stress range (see Appendix 1). Since model 3

---(3)

---(4)

---(5)


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contained the cyclic stress-strain characteristics of the HYPRESS 23 material, it was considered

valid(5, 6) to obtain the equivalent strain at the point of high stress. This value was treated as the

predicted strain amplitude and the life was estimated accordingly.

The analysis showed that the equivalent strain at the radial crest was 1340 micro strain. On the

strain amplitude log scale, 1340 micro strain was equivalent to -2.87. This gave a life estimation of

261,000 cycles for the wheel, which was 13% greater than the experimental mean of 257,000 cycles

to failure. The estimation of life from the strain-life plot is shown in Fig. 45.

At the vent areas, where experimental tests had shown failures to occur, predictions of life ranged

from 157,000 to 283,500 cycles to failure. The results are tabulated in Table 6.

3.5.3 FATIMAS

From inputs of strain history both elastic modulus and cyclic properties of the material and stress

concentration factors, FATIMAS was able to predict life. Fig. 46 shows the software's interpretation

of the strain history at the radial crest where the strain amplitude was found to be 1340 micro strain.

FATIMAS verified data input by plotting the cyclic stress-strain curve for the material and the strain

life relationship. These plots are shown in Figs. 47 and 48. With the fatigue notch factor equal to 1.0,

F ATIMAS calculated that there would be 25,062 repeats of strain history or blocks. FATIMAS

indicated that 9 strain reversals had been used in 1 block. Therefore, FATIMAS predicted, where Kt

had been found to equal 1.0, that failure would occur after 9 x 25,062 = 225,558 cycles. Life

predictions for the five nodal positions are shown in Table 6.

The FATIMAS software plotted the local stress-strain response at the radial crest and the hysteresis

loops may be seen in Fig. 49.

The variation in life against Kt was calculated readily by FATIMAS with the strain amplitude of 1340

micro strain. Fig. 50 shows that above a life of approximately 104 cycles to failure, a small change in

the stress concentration factor is detrimental to life.

4. DISCUSSION.

Three variations on the low cycle fatigue technique have been performed to predict the life of wheel

design LP13-46 due to the loading imposed by the dynamic cornering fatigue test. Three variations

on the low cycle fatigue technique were followed using the method defined by Neuber(4). The

methods of life prediction were;-

1. Neuber analysis.

2. Determining equivalent strain amplitude from FEA using the cyclic stress-strain properties of the

material.

3. The software package, FATIMAS.

Empirical fatigue data for the wheel were obtained from Dunlop-Topy Wheels Limited. Thirteen

wheels had been subjected to the reverse bend test and fatigue failures ranged from 180,000 to

390,000 cycles. The fatigue failures were found to be distributed normally about a mean value of


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257,000 cycles. The distribution is shown in Fig. 14. Five lives, for the five main areas of highest

stress were predicted using Neuber's method. Finite element analysis revealed that the highest

stresses were around the wheel vents and at the radial crest. The life predictions ranged from

214,500 up to 460,000 cycles. A life of 247,000 cycles was calculated at the vent position which was

situated in the direct radial direction of the applied moment (nodal position 110). This prediction of

fatigue failure differed from the mean of experimental data by 3.9%. Figs. 15(a) to 15(d) indicate

that fatigue failures occurred at this discontinuity experimentally. The lower life predictions were

found to be at the vents offset to the applied moment (nodal position 61) and at the radial crest. The

prediction at the radial crest indicated that the cycles to failure was approximately 214,500 cycles

and was 16.5% lower than the experimental mean value of 257,000 cycles. It is significant that the

radial crest of a disc pressing is often the cause of highest stress and fatigue failure in a wheel

design. However, crack growth at the radial crest was not apparent in this wheel design at

inspection.

Fatigue life predictions, obtained from the finite element model containing the cyclic properties of

the material, were comparable to predictions of life gained from Neuber analysis. However, this

method of life prediction indicated that fatigue failure at the radial crest was 261,000 cycles and that

the lowest life of 157,000 cycles was at the vents. This value of life fell outside the distribution of

empirical data.

The life predictions from the FATIMAS software package agreed closely with the life values obtained

with the two other methods. Both methods relied on manual life estimation from the strain-life plot.

FATIMAS however, used an iterative algorithm to determine the life from the strain-life relationship,

and so was considered the most precise. FATIMAS indicated that the number of cycles to failure for

this wheel design ranged from 216,055 to 302,391 cycles for the vent areas. The life at the radial

crest was calculated to be approximately 13% lower than the mean of the experimental data at

225,558 cycles to failure.

The overall predictions from the three methods considered, suggested that values of life generally

fell within the normal distribution curve of experimental data. The estimated life at the radial crest

had a value close to the mean of the empirical test data, as did the predicted life at the vents.

Moreover, it was found that the corresponding individual life results, from the three prediction

methods were agreeable. They differed slightly because of manual errors in estimating life from the

log scale of the strain-life plot. A graphical comparison of the predicted lives is given in Fig. 51.

Neuber analysis indicated that the estimated strain amplitudes, at the five positions of high stress,

were just within the plastic regime of the cyclic stress-strain curve (Figs. 40 to 44). Indeed, the

corresponding fatigue lives were estimated within the transitional region of the strain-life curve. This

suggested that the traditional approach of high cycle fatigue prediction may have been applicable

for this wheel design.

However, the overall results, from the three methods, illustrated that the low cycle fatigue

technique gave reliable estimations of life, either way of the transition from elastic to plastic strain

amplitude. Furthermore, the accuracy of each life prediction could be confirmed by analysing a

series of stress concentration factor versus life plots for nodal positions of different strain

amplitudes. Shown in Fig. 50 is one such plot for the radial crest position, where it was found that


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the local strain amplitude was 1340 micro strain. The graph shows clearly, that above approximately

104 cycles, the gradient of the curve is less steep and life becomes more sensitive to a change in Kt.

Therefore, any errors incurred in calculating the stress concentration factor from finite element

analysis would have affected the accuracy of life significantly above 104 cycles. To ensure the

accuracy of the calculated stress concentration factor, both the von Mises and Tresca equivalent

stresses were used to calculate the Kt values at areas of high stress. It was found that both criteria

gave similar values of Kt, with any differences being minimal so as not to affect life seriously.

Ten strain gauge rosettes were applied to the surface of wheel design LP1346 at areas of predicted

high stress. The wheel was placed in the reverse bend test rig and the moment of 1579 Nm applied.

The wheel was rotated incrementally and the strains at each rosette recorded. The equivalent

strains at each of the strain gauge rosettes were calculated and compared to the results of the

analysis model containing the monotonic properties of the material. Table 5 compares the measured

equivalent strains with those predicted by the finite element analysis model. The measured strains

compared well with those predicted. Indeed, at the radial crest, measured strains differed by 0.1%.

The largest error of 7.6% was found at rosette position 6 which had been placed on the side wall of

the disc pressing. It was noted however, that finite element analysis had shown this area to be

relatively high in stress, hence the positioning of this rosette. The stress gradients at this region were

thought to be high and were concentrated within a small area. It is possible that the positioning of

this particular rosette may have been inaccurate, or the 5 mm gauges may have been too large to be

affected by the small area of high stress and actually recorded the nominal stress at this position.

Generally though, the measured strains were comparable with those predicted by finite element

analysis. The degree of accuracy indicated the validity of the geometric computer models and the

monotonic property determination of the HYPRESS 23 material.

The low cycle fatigue technique was shown to give reasonable estimations of life when the actual

thickness variations in the pressed disc were modelled. If the life prediction for a new wheel design

was to be calculated, it would be inevitable that the thickness variations due to the forming of the

disc would not be known. It was considered that the designer would have to base the life prediction

on the nominal thickness of the original blank and disregard thinning of the material due to the

forming process. It was noted that thickness in areas of the wheel, that gave the most accurate life

predictions, differed from the nominal thickness of 4.0 mm by 2.7% at the radial crest and by less

than 0.1% at the vents. However, to determine whether the accuracy of the low cycle fatigue

technique would be retained by supplying the nominal thickness of the original disc blank, finite

element models 1 and 2 were modified and given a uniform thickness of 4.0 mm. As before, the

elastic stress concentration factors, at points of maximum stress, were calculated and analysed with

FATIMAS. The results from the analysis are compared to the thickness variation results in Table 7.

It can be seen that the fatigue predictions were similar to those predicted when the thickness

variations due to pressing were included in the analysis models. An increase in life at the radial crest

was calculated at 4.7%. However, reductions in the lives predicted at the four other nodes were

shown to be as much as 10.9% at node 110. The lowest life predicted was at the vent area of the

model (node 61) and was estimated at 207,225 cycles to failure. This life is approximately 20% less

than the mean of the experimental test data. This procedure illustrated that the accuracy of life

estimation had been substantially reduced by neglecting the thickness variations in the disc pressing.

However, since the life predictions were slightly lower than the fatigue life of the wheel obtained


SL/SF/R/S2330/1/93/C

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experimentally, it was concluded that a designer would be predicting life conservatively if the

nominal thickness of the blank was used in the analysis.

The flow chart in Fig. 52 shows the suggested(8) design route that should be followed when designing

for low cycle fatigue. The three input variables for the design process are the service loading, the

plastic modulus and cyclic properties of the material, and the predicted nominal and local elastic

stress conditions derived from finite element analysis. Software packages are available that will

perform rainflow counting on complex service histories in order to calculate local stress-strain

responses. FATIMAS is one such package that proceeds to predict life with damage summation(7, 8, 9)

of the hysteresis loops. For very simple strain histories such as constant amplitude sinusoidal

histories, the main strain range is constant and rainflow counting is not needed. However, FATIMAS

removes any manual errors that may occur from estimating life from the strain-life plot.

The iterative design approach shown in Fig. 52 indicates that the geometry of the design should be

modified or the material type changed if the initial life predictions are not satisfactory.

Implementation of the finite element method can provide a wealth of data for particular component

designs under cyclic loading. Correct interpretation of the data would indicate where modifications

are required in the design to prolong life. However, material selection is currently based upon press

formability, hole expansivity and fatigue resistance(2) . One important criterion that is often over

looked is the material's sensitivity to a concentration of stress at a constant strain amplitude. Fig. 50

shows the variation in life with the stress concentration factor Kt, at a constant strain amplitude of

1340 micro strain. The graph shows that the HYPRESS 23 steel becomes more sensitive to an

increase in Kt above approximately 104 cycles, where the gradient changes significantly and the curve

becomes less steep. The curve is also influenced by the degree of strain amplitude. Shown in Fig. 53

are six graphs of Kt against fatigue life at different strain amplitudes. The graph explains how

optimum material selection could be accomplished by comparing the theoretical stress

concentration factor, fatigue life and strain amplitude data for a range of high strength steels in

order to choose the material least sensitive to discontinuities at particular strain amplitude. The

justification for this selection method is relevant when the blanked edge condition of the vents is

considered. Tool wear of the punch and die during batch production will result in a progressively

degrading edge condition of the vents that will raise the stress concentration factor and affect life.

Judicious selection of a material less sensitive to changes in Kt, at a constant strain amplitude, would

result in less statistical scatter of actual fatigue life if tool wear was expected.

Although this method of material selection can provide optimum wheel designs, the interaction

between Kt, nominal thickness, disc shape, strain amplitude and geometry sensitivity should be

recognised. Modifications to disc shape and nominal thickness, during the design process, will affect

Kt and strain amplitude. Since the strain amplitude and stress concentration factor will change with

each modification, it may be difficult to identify the ideal material that would be less sensitive to

stress concentrations at known strain amplitudes. However, if the actual material thickness due to

pressing was known, then only the shape of the design model could be modified during the iterative

design process. To estimate pressing thickness, finite element analysis could be used to model multi-

stage press working. It is recommended that further work should be undertaken to assess the

accuracy of thickness pressing calculations using finite element analysis. It would also be prudent to

investigate sensitivity to geometry changes in a disc pressing to determine the effect on life.


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Low cycle fatigue data for most materials are not readily available to designers. There is an urgent

requirement(2) for the compilation of low cycle fatigue data in order that the low cycle fatigue

technique may be implemented efficiently. Vehicle manufacturers are becoming aware of the

advantages of fatigue life prediction in the automobile design process, but they are cautious that the

availability of fatigue properties for materials are limited. The reluctance of automotive engineers to

design for low cycle fatigue is reflected by the absence of such data. Until such a data base is made

available, automotive engineers may remain hesitant in adopting the low cycle fatigue design

process.

5. CONCLUSIONS AND RECOMMENDATIONS.

1. Finite element analysis has been used to implement successfully three variations of the low cycle

fatigue technique for wheels.

2. It was found that Neuber's method of life prediction could be performed by calculating the elastic

stress concentration factor Kt, from two finite element analysis models. One model contained the

disc pressing detail, such as the vents and stiffeners, which produced areas of stress concentration.

The second model was constructed to reveal the nominal stress distribution in the absence of the

detail.

3. By supplying the finite element model with the cyclic stress-strain characteristics of the material,

it was viable to use the equivalent strain values, at positions of stress concentrations, to estimate life

directly from the material's strain-life relationship.

4 The software package FATIMAS was used to predict life from inputs received from finite element

analysis. The FATIMAS life predictions were considered the most precise due to the elimination of

manual errors in determining life from the log scale of the strain-life plot.

5. The accuracy of the FATIMAS life predictions were confirmed with the generated Kt versus life

plots for the different nodal positions. It was generally found that for predicted lives above 104

cycles, a small change in the stress concentration factor would affect life significantly. The von Mises

and Tresca equivalent stresses were used to determine Kt values from the two finite element

models. It was found that corresponding pairs of Kt values differed slightly but the errors were not

detrimental to the accuracy of predicted life.

6. The three techniques of life prediction gave comparable and accurate estimations of life when

variations in thickness, due to pressing of the disc, were included in the analyses.

7. Life estimations at the vents of the disc pressing were calculated to be within 3.9% of the mean of

experimental test data by Neuber analysis. Moreover, it was found that excessive crack growth at

the vents was the cause of cyclic fatigue failure in this wheel design.

8. A high stress concentration, due to a no notch effect, was visible at the radial crest. Life

predictions at this region suggested that fatigue failure would occur at 214,500 to 261,000 cycles.

However, the main mode of failure was at the vents and crack growth at the radial crest was not

noted at inspection.


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9. If the life prediction for a new wheel design was to be calculated using this technique, then the

variations in thickness due to pressing would not be known. It was shown that by supplying the

original blank thickness to the analysis models, the lives predicted were approximately 11% lower

than the lives predicted with the models including details of thickness variation in the disc pressing.

10. It was shown that by including the nominal thickness of the blank instead of the thickness

variations due to pressing, a designer would be predicting life conservatively.

11. The calculated equivalent strains at the ten rosette positions agreed accurately with the

equivalent strains that were predicted by finite element analysis. The computer models and

monotonic material data were validated with the agreement of the strain measurements.

12. A suggested design route was identified for designing towards low cycle fatigue. It has been

shown that the FATIMAS software package can be used in conjunction with this approach to replace

the need for manual calculations. However, for the sinusoidal strain histories encountered, rainflow

counting was not required.

13. The iterative design approach indicated that the geometry of the disc pressing must be modified

or the material type changed if initial estimations of life were found to be unsatisfactory.

14. It was illustrated how tool wear could progressively deteriorate the edge condition of the vents

and bolt holes and consequently affect wheel life. It was shown that prudent selection of a material,

least sensitive to changes in the stress concentration factor Kt at a calculated strain amplitude, could

result in less statistical scatter of actual fatigue data.

15. Further work is required to assess the accuracy of determining disc thickness by modelling multi-

stage press working using finite element analysis.

16. It was recommended that a sensitivity study of geometry changes in a disc pressing should be

undertaken.

17. It was suggested that the compilation of low cycle fatigue data was required so that material

selection could be performed efficiently.

18. It was concluded that automotive engineers would be reluctant to design for low cycle fatigue

without a library of material fatigue data.

D.J . Naylor Research Manager Special Steel Products

Tim Wolverson Specialist Technician A.F. Turner Manager Steel Fabrication Department


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ACKNOWLEDGEMENTS

Dr R. Baker, Director Research and Development, British Steel Technical and Dr M.J. May, Manager

Swinden Laboratories are acknowledged for allowing this work to be submitted as a final project to

Sheffield Hallam University.

Many thanks to Mr J. Brennan and Mr K. Thompson at Dunlop-Topy Wheels Limited for their help in

supplying wheels and wheel material to assist the course of the investigation, and allowing the use

of the reverse bend test rig for the testing of the gauged wheel.

Dr A. Yazdanpanah at Sheffield Hallam University gave valuable assistance by demonstrating the

FATIMAS software and suggesting variations of the low cycle fatigue technique for investigation. Dr

T. D. Campbell is also acknowledged for his help in the planning of the project and the preparation of

this paper.

British Steel Swinden Laboratories is thanked for allowing the use of the Silicon Graphics computers

and the finite element analysis software. Thanks to Mr C.S. Betteridge who offered guidance

throughout the project. Many thanks are also due to Mr N. Bennett for the strain gauge installation

of the wheel.

REFERENCES

1. M.J. Godwin: "Influence Of The Degree And Mode Of Forming Strain On The Cyclic Stress-Strain

And Stress-Life Behaviour Of High Strength Sheet Gauge Formable Steels", Welsh Laboratories, Port

Talbot, BS Ref:FR W181 7 882, June 1989.

2. M. Mizui: and M. Takahashi: "High Strength Steels For Automotive Wheels", Mechanical Working

and Steel Processing (MWSP) Conference, October 1991.

3. E. Schreiber, O.L. Anderson and N. Soga: "Elastic Constants and Their Measurement", McGraw-

Hill, Pages 82-103, 1973.

4. H. Neuber: "Theory Of Stress Concentration for Shear-Strained Prismatical Bodies With Arbitrary

Nonlinear Stress-Strain Law", Journal Of Applied Mechanics, Vol. 28, Trans. ASME, Vol.83, Series E,

December 1961.

5. I. Bela: "Fundamentals Of Cyclic Stress And Strain", Sandor, 1972.

6. H.O. Fuchs and R.I. Stephens: "Metal Fatigue In Engineering", Wiley-Interscience, 1980.

7. E.J. Hearn: "Mechanics Of Materials, 2nd Edition", 1985.

8. D. Cowburn, B.J. Dabell and S.P. Rawlings: "Design Optimization Of Front Axle Beams", GKN

Technology Limited, Birmingham New Road, Wolverhampton, Published By IMechE, 1985.

9. C.M. Davies: "Low Cycle Fatigue Testing Of Strip And Sheet Steels For Automotive Applications - A

Review.", Swinden Laboratories Report SH/EM/3577/-/85/A, 30th October 1985.


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10. C.M. Davies, I.M. Austen, and E.F. Walker: "Low Cycle Fatigue Testing Of 8mm Gauge Hypress 23,

26 and 29 and 4 mm Gauge Hypress 29", Swinden Laboratories Contract Report RSC/7216/2/85,

13th September 1985.

11. K. Hatanaka, T. Fujimitsu and S. Shiraishi: "Low Cycle Fatigue Life Prediction Of Circumferentially

Notched Component Based On Elastic-Plastic Stress-Strain Analysis", Fracture And Strength '90 Key

Engineering Materials v 51-52, Published By Trans Tech Publishing, Zuerich, Switzerland, Pages 7-12,

1991.

12. "Method For Constant Strain Amplitude Strain Controlled Fatigue Testing", BS 7270: 1990.

13. J. Samuelsson, D.K. Holm and A.F. Blom: "Applications Of Damage Tolerance Concepts and

Conventional Fatigue Life Estimation Methods to an Axle Housing in a Wheel Loader", Measurement

and Fatigue, EIS-86.

14. T.G. Ivanina and A.D. Trukhnii: "Correlating The Characteristics Of Low Cycle Service Life Of

B2MoA Rotor Steel", Thermal Engineering (English Translation Of Teploenergetika), Pages 370-372,

July 1990.

15. PDA Engineering, "PATRAN Plus User Manual", Volumes I and II, P/N - 2191001, 1989.

16. Hibbitt, Karlsson and Sorensen, "ABAQUS User Manual", HKS Inc.


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