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BRITISH STEEL TECHNICAL Swinden Laboratories
SL/SF/R/S2330/1/93/C
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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.
2.4.1.2 Finite Element Model 2
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.
2.4.1.3 Finite Element Model 3
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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2.4.1.4 Finite Element Model 4
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.
2.4.4.2 Finite Element Model 2
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.
3.3.2 Finite Element Model 2
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.
3.3.3 Finite Element Model 3
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
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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.
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SL/SF/R/S2330/1/93/C
93