Structural durability proof of aluminium safety components substituting steel safety components – criteria regarding testing time by the example of commercial vehicle wheels

The experimental structural durability proof bases on a damage equivalent shortening of the design spectrum into a test spectrum with reduced length taking statistics founded safety factors into account. Historically, test spectra were first derived for steel parts using Woehler‐lines for steel; the damage equivalence was calculated by applying the Palmgren–Miner rule with the modification according to Haibach. When aluminium parts were introduced later, the question arose, whether test spectra developed for steel parts could be applied without modifications for their durability proof. Caused by different slopes of aluminium before and after the knee‐point, the test spectra derived for steel wheels result for aluminium components a lower cumulated damage in comparison to the design spectra. This fact requires a longer test duration for aluminium components when test spectra or proving grounds developed for steel components are used.


| INTRODUCTION
The damage-equivalent densification of design spectra for safety components of vehicle engineering to time-reduced test spectra is a permanent central issue regarding the acceleration of development periods and of the experimental structural durability proof, Figure 1.Also, the energy consumption during testing plays here a significant role.The fixing of a reasonable testing time has to consider not only safety aspects but also material related influencing parameters.Historically, test spectra were first derived for steel parts.However, when aluminium for forged, cast or welded safety parts were introduced, e.g.wheels, suspension arms, wheel carriers, steering knuckles, the question arose, to what extent the substitution of steel should be considered regarding the test duration.In other words, can the test spectra and testing times established for steel safety components be kept or should they be shortened or prolonged?
There are recommendations already which are considered in the structural durability proof of safety components of vehicle engineering [1][2][3].In the present article, they will be presented by the example of commercial vehicle wheels of steel and aluminium having same highly stressed areas.The substitution of material and the consequences for the test duration will be discussed also in the context of load assumptions, cumulative damage and safety aspects.
Here, it should be pointed out that the structural durability proof of safety components does not solely consist of investigating the fatigue behaviour, Figure 2. Because of differences in the impact and elasto-plastic material behaviour of steel and aluminium, the component's structural integrity must also be assured by the design against misuse and special events before the final fatigue proof under derived test spectra.The present article treats only the fatigue aspect of the structural durability proof under spectrum loading and is a translation with slight modifications of [1][2][3][4].

| Transformation of a design spectrum into a test spectrum
The derivation of a damage-equivalent test spectrum from a design spectrum is carried out by application of the Palmgren-Miner law [5,6]: and the modification of the slope after the knee-point of the Woehler-line according to Haibach, Figure 3 [7]: with m = 1 for forged, wrought materials and m = 2 for welded joints, cast-and sintered materials.For obtaining an equivalence between the cumulative damage sums of the design and test spectra, D Test ¼D Design (3) amplitudes in the most damaging part of the test spectrum are amplified, Figure 1.A reduction of testing F I G U R E 1 Possibilities for reducing testing time (schematical) [2,3].

F I G U R E 2
Decisive loads in structural durability design and proof [2,3].
duration by an increase of the maximum loads of the spectrum above the maximum service loads should generally be avoided.Such an exceedance can change local residual stresses resulting a higher or lower fatigue life in contrast to the life which would result without the mentioned increase of the maximum loads [8].An omission of small amplitudes, which do not cause a damage, can be carried out, too.However, the damaging contribution of omitted amplitudes must be considered by a damage equivalent amplification of amplitudes below the maximum level of the spectrum, or the tolerable omission level under which fatigue life is not influenced, must be determined by time consuming experiments [8].
For the determination of the damage sum of the design spectrum for the intended service, e. g. 3 • 10 5 km for cars, 5 • 10 5 km for heavy-duty vehicles, 5 • 10 6 km for railway vehicles, following data are required: A design spectrum with the probability of load occurrence of P o, Load � 1 % and a Woehler-curve with a strength related probability of survival of P s, Strength to be selected, Figure 3 [2,3].For the cumulative damage calculation the slopes k and k' of the Woehler-curve before and after the kneepoint with N k cycles is of utmost importance.As the distance between the spectrum and the Woehler-curve is important for the comparison of the damage sums of the design and test spectra, the level of the fatigue strength amplitude σ a, k at the knee-point, i. e., the position of the Woehler-curve, should be adjusted as appropriate as possible.
When the size of the design spectrum L Design is reduced to the size of the test spectrum L Test , physical processes, which need a development time, such as environmental or fretting corrosion must be considered, too [9].For this, a test duration of L Test around 5 • 10 6 cycles is recommended [10,11].For the arrangement of the test spectrum, repeated service operations should be considered by means of sequences with the size L s ; the sum of these sequences result in the total spectrum size, Figure 3 [12].The damage equivalent test spectrum has the same probability of occurrence of P o, Load � 1 % as the design spectrum.The service related theoretical probability of failure P f, Service of the component or structure will be addressed in the next section.

| Component strength and theoretical probability of failure
The aim of the experimental structural durability proof is the verification of life for a required service related theoretical probability of failure P f, Service .For this, beside the knowledge of the load related probability of occurrence P o, Load of the spectrum, also the knowledge of the Gassner-line with the strength related probability of failure P f, Strength of the component or structure is necessary, Figure 4.This is determined from the strength related probability of survival P s, Strength : P f; Strength ¼ 1À P s; Strength (4) From the interaction between the load related probability of occurrence P o, Load of the spectrum and the strength related probability of failure P f, Strength the service related theoretical probability of failure P f, Service is calculated [13]: This empirical equation is valid for the prerequisite that the scattering of service loads is significantly higher than the scattering of strength [13].This is the case in vehicle engineering.
For obtaining a required, service related theoretical probability of failure P f, Service of the component, the Gassner-line with a defined failure criterion, e. g. for safety components of vehicles a first detectable technical crack of a surface length of l = 1 mm-2 mm, and a strength related probability of survival P s, Strength must be positioned on such a level: which results the design life L Design for the spectrum with the maximum value σ a, Service, max , Figure 4.This procedure for a service related theoretical probability of failure P f, Service � 10 À 4 and a strength related probability of survival P s, Strength = 99 % (P f, Strength = 10 À 2 ) is applied in many design cases, Figure 4.For safety components P f, Service -values of 10 À 3 to 10 À 6 and P s, Strength -values of 90 till 99.9999 % (P f, Strength of 10 À 1 till 10 À 6 ) are usual [3,[10][11][12].
The fixing of such probabilities also considers failure criteria and consequences of failure.
The positioning of a Gassner-line with the related probability of survival P s, Strength , Figure 4, is normally carried out by a cumulative damage calculation according to the Palmgren-Miner-rule with a suitable modification, e. g. after Haibach, an allowable damage sum, a component related Woehler-line for a defined failure criterion, e. g. a first technically detectable crack with a defined surface length or depth [7,13].If the fatigue life is based on a Woehler-line with P s, Strength = 50 %, the calculated Gassner-line owns the same probability of survival.Subsequently, the Gassner-line has to be reduced to a higher probability of survival as defined by Equation ( 6) corresponding to the probability of failure P f, Strength according to Equation (4).This reduction is carried out by the statistics-based safety factor: The normalized safety factor u o (P f, Strength ) in Equation ( 7) is nothing else than the Gauss'ian integral of the probability distribution, Table 1 [7].The standard deviation s N is determined by the scatter index T N : Relations between the probability of occurrence of the spectrum, the Gassner-line and the theoretical service-related probability of failure regarding the durability design of safety components (schematical) [3].
The scatter index T N depends on the material [7,14,15].With the knowledge of the scatter index and the normalized safety factor the fatigue life reads: If the calculated fatigue life is not yet equal with the spectrum length, i. e., the design life L Design , so the Woehler-line with P s, Strength = 50 % together with its appertaining scatter band must be shifted until N(P f, Service ) = L Design is rendered.
Finally, it has to be proved experimentally that the component or structure fulfills the design life L Design with the required, service related theoretical probability of failure P f, Service .As the performance of durability tests under the design spectrum and the statistical assurance by numerous tests require a big time effort, a reasonable damage equivalent testing time reduction applying condensed test spectra with L Test is well justified [16].Consequently, if with the test spectrum the life L Test is reached without a failure, the design life L Design is approved.

| Test duration with condensed test spectra
The damage equivalent condensed spectrum, initially derived for steel components, can but doesn't have to be a priori the test spectrum.It can be applied as test spectrum if for example a statistically secured Gassner-line with many specimens should be determined.However, if further influencing factors, e. g. a low number of specimens, a compensation of corrosion, a material change, should be incorporated, then the size of the damage equivalent test spectrum must be enlarged with appropriate runtime (prolongation) factors for obtaining the final test spectrum.In the following the inclusion of afore mentioned factors will be presented.
As because of time and cost savings not so many tests with components and structures can be performed in contrast to small specimens, the statistical risk from carrying out few tests must be compensated, Figure 5 [2,3,7].The risk factor j C, n , which bases on the log-normal distribution, depends firstly on the amount n of the designated components or structures, and then on the assumed scatter T N or the standard deviation s N , respectively, of the basic population and of the probability of confidence C. As generally with big structures few tests, usually one to three, are performed, which do not allow the determination of the scatter of a bigger population, the scatter T N is taken from extensive experiences with specimens of same material being considered to be valid for the basic population [7,14].The scatter T N depends not only on the material, but also on manufacturing and quality control, which may restrict the scatter.The risk factor for a probability of confidence C = 90 %, which is usually sufficient and well-tried in the design and proof of safety components [2,3,7,8,11,12], is only one part of the total runtime factor: (12) with which the size of the damage equivalent test spectrum is enlarged up to the size of the final test spectrum.
The further parts are f t , by which possible manufacturing-related changes in geometry are considered, f Corrosion , with which the effect of a corrosion, if present, is captured, without performing the tests in a corrosive environment, and f St/Al , with which the influence of a material-substitution on fatigue life is taken into account [1,3,4,9,12].Consequently, the size of the damage equivalent test spectrum, which was derived for steel components, must be enlarged with the total runtime factor: T A B L E 1 Strength related theoretical probability of failure P f, Strength and normalized safety factor u o [7].
T A B E L L E 1 Festigkeitsbezogene rechnerische Ausfallwahrscheinlichkeit P A, Festigkeit und bezogener Sicherheitsfaktor u o [7].If after the run of the enlarged spectrum with the designated amount n of tests a failure, i. e., for safety components a technical detectable first crack with defined surface length or depth, didn't occur, the life requirement L Design is fulfilled.
The values of the individual factors are known.For example, for a life scatter of T N = 1 : 3 and with n = 3 components a risk factor of j C, n = 1.37 is obtained for a probability of confidence of C = 90 % [7].The damageequivalent condensed spectrum for steel wheels is then enlarged with the running factor f St = f t • j C, n = 2, Figure 6.The factor f t considers the influence of manufacturingrelated dimensional fluctuations on the load capacity and consequently on fatigue life.The substitution of steel by aluminium is respectively covered by f St/Al = 1.25 Figure 6, Table 2 [3,4,12].In case of a salt corrosion, for the durability proof of aluminium components in air without the corrosive environment a prolongation factor of f Corrosion = 2 to 3 is recommended for the compensation of the corrosive influence on the fatigue life, Figure 7 [9,11].
In following section, the derivation of the substitution factor f St/Al , which has hardly been discussed in literature, will be presented by the example of steel and aluminium commercial vehicle wheels with comparable highly stressed areas [2][3][4].

| RUNTIME FACTORS BY THE EXAMPLE OF COMMERCIAL VEHICLE WHEELS OF STEEL AND ALUMINIUM
The selected steel and aluminium wheels for the determination of the runtime factor f St/Al were designed for same service load conditions under application of the required fatigue strength (RFS) method for assessing the highest stressed areas [3,12,17].With this method the Woehler-curve with given slopes k and k' before and after the knee-point for a fixed number of cycles N k , is positioned against the design spectrum in that way that the allowable damage sum D al is reached by the cumulative damage calculation, Figure 8.The stress amplitude σ a, k at the knee-point is the required fatigue strength-value [3,12,17].The so determined required fatigue strength-Woehler-curve results the minimum fatigue strength, which has to be provided not only by the design, but also by the manufacturing and quality control, for fulfilling the required design life L Design .The failure criterion for the Woehler-curve and for the designed component is the first technically detectable, i. e., by the dye penetrant method, crack with a surface length of l = 1 mm to 2 mm.
The question posed is, whether the testing conditions for steel wheels can be maintained for aluminium wheels, or whether the substitution of the material has to be considered additionally.In the following, the results of a study treating this subject will be presented [4,18].

| Details about the wheels, materials, load spectra and loads
The investigated wheels of type 11.75×22.5 (IS 120) are designed for commercial vehicles with a maximum axle load of 10 t, i. e., for a static wheel load of 5 t, Figure 9.
The highest stressed areas are the ventilation holes.The welded steel wheel consists of two different structural steels for the disk and the rim and the forged aluminium wheel of an artificially aged aluminium alloy, Table 3.
The design spectrum contains as maximum value special events with a cumulative frequency of up to N SE = 100 cycles, Figure 8  Furthermore, the overall spectrum consists additionally of the partial spectra for straight driving and cornering, to which a further partial spectrum for parking and shunting maneuvers is superposed, Figures 10, 11, Table 4.The particular load factors and maximum load values of the partial spectra for the design cases static wheel load, cornering and straight driving, parking and shunting are compiled for an axle load of 9 to; they are based on extensive service measurements, Table 5 [1-3, 10, 13].These loads are used for flat-track based experimental stress analyses using strain gauge equipped components as well as for FE-calculations to determine the highly stressed areas of investigated wheels [19].
From the determined design spectrum with the probability of occurrence P o, Load � 1 % the damage equivalent test spectrum with same probability of occurrence is derived.

| Procedure of experimental stress analyses and highly stressed areas
For the local stress analysis, the steel wheel was equipped with 23 respective the aluminium wheel with 13 strain gauges having a measuring length of l o = 3 mm.For the available wheels the highly stressed areas are the ventilation holes; at the steel wheel the highest stresses were measured with gauge no. 4 and at the aluminium wheel with gauge no. 5, Figure 12.As at the edges of holes the stress state is uniaxial, the stresses were calculated from the measured strains linear-elastically according to Hooke's law: Firstly, the strains were measured in a flat track rig, which is specifically designed to apply vertical and  horizontal loads concurrently to the rotating wheel end, Figure 13, Table 5.The loads are introduced through the tire footprint which is approximately deformed such as on the road.Subsequently, the strains were measured with the same wheels in a biaxial durability test rig for the durability proof under a damage-equivalent test spectrum, Figure 14.As in the biaxial test rig the inner drum substitutes the flat track as the moving ground, it must be provided that the strains measured in the flat track rig are properly adjusted in the biaxial durability test rig, because the load introduction differs, Figures 13,  14.The load program for the durability test rig is adjusted iteratively in such a way that the same cumulative damage of the design spectrum and the same maximum stresses of the spectrum are realized.Load-time, camber angle-time and local strain-time histories are derived under the described considerations, Figure 15.The derived local stress spectra will be presented in next section.

| Presentation of the design and test spectra with appertaining required fatigue strength-Woehler-curves
The normalized design spectra of the steel and aluminium wheels for a design range of L Design = 5 • 10 5 km, respectively the corresponding design life of N Design = 1.54 • 10 8 cycles, result from the superposition of the mentioned driving conditions, Figures 16, 17.Further, these figures contain also the related RFS-Woehlercurves, with the modified slopes k' according to Haibach after the knee-points for the cumulative damage calculation.The required fatigue strength-Woehler-curves were derived using the allowable damage sum of D al = 0.5, which is proven and established in the fatigue design of vehicle wheels [2,3,[10][11][12].For these wheels a theoretical service related probability of failure of P f, Service � 10 À 3 is postulated, which is calculated by the Equation ( 5) from the interaction between the probability of occurrence of the design spectrum P o, Load � 1 % and the strength related theoretical probability of failure of P f, Strength � 10 %, corresponding to P s, Strength � 90 %.
The normalized design spectra and the related required fatigue strength-Woehler-curves for both wheel variants reveal a difference, Figure 18.The divergency of both spectra above 10 4 cycles is caused by different local geometries and cross-sections, Figure 9, 12.However, a congruent coverage of both normalized design spectra would be expected for the same geometrical design.
The damage equivalent reduction of the design life of steel wheels was arranged from L Design = 5 For the investigated steel and aluminium wheels, the normalized test spectra with the related required fatigue strength-Woehler-curves are, in contrast to the design  spectra, very similar, Figures 19-21.The test spectra were determined by a rainflow-evaluation of the measured local strain-time histories during testing.As in case of rotating components mean-stress changes are neglectable, based on experiences with wheels they were not considered [8].
The test spectra contain already the runtime prolongation by the factor f n, t = j C, n • f t = 2, considering the risk factor and possible manufacturing-related changes in geometry, as explained in section 2.3 and resulting the required cumulative damage sum D Test, req = 2 D al = 1.0.
In contrast to the above mentioned, geometry-dependent divergencies of the design spectra of the steel and aluminium wheels, the test spectra differ very little from each other despite the differences in wheel designs, local geometries and the places where the strain gauges F I G U R E 1 5 Cut-outs of forces-time, camber angle-time and strain-time histories at strain gauge no. 5 at ventilation hole of the aluminium wheel in the biaxial test machine (ZWARP).were applied, Figures 18,21,22.The differences between the normalized amplitudes lie below 10 %.This is a consequence of the damage equivalent compression of the longer design spectra to shorter and fuller test spectra using Woehler-curves with different slopes and kneepoints.Further, Figure 22

| Determination of the partial runtime factor for proofs with aluminium wheels substituting steel wheels
For deriving the partial runtime factors f St/Al , the knowledge of the normalized design and test spectra of aluminium and of steel together with the related required fatigue strength-Woehler curves is necessary, Figures 23,  24.The test spectrum for aluminium is not compensated with f St/Al .For investigating the influence of the kneepoint on the damage compensating substitution factor f St/Al , a further Woehler-curve with N k = 2 • 10 6 cycles was involved.For this, the slopes k = 5 and k' = 9 before and after the knee-point of the Woehler-curve with N k = 5 • 10 6 cycles were maintained.The decisive material data for determining the interaction between the spectra and the Woehler-curves, i. e., the damage sums according to Equation ( 1), are the mentioned slopes and the position of the knee-points.These data were provided by fatigue tests with coupons, which were removed from the wheels, and also by loading selected areas of the wheels [20].
First, the damage sums D Al and D St are determined according to Equation ( 1) by contrasting the test spectra for aluminium, without the factor f St/Al , and for steel with the related required fatigue strength-Woehler-curves.Because of the different slopes before and after the different knee-points and different levels of the curves, smaller damage sums result for aluminium compared to steel: D Al < D St .These damage sums render ratios: (15) which are larger than 1.0, Figures 23, 24 significantly the outcome reported above for forged aluminium wheels.Therefore, the suggested substitution factor f St/Al is also applied for the durability proof of cast wheels [12].

| CONCLUSIONS AND OUTLOOK
The ratios D St /D Al = 1.18 and 1.39, D St /D Al = 1.60 and 2.02, respectively, suggest that the testing duration   6, 7.However, if the damage sums D Al are related to the required, i. e., sufficient, damage sum D Test, req.= 1.0, smaller values than the afore mentioned ones result in: D Test,req./D Al = 0.88 and 1.04 for the investigated aluminium wheel, and for the investigated steel wheel 1.08 and 1.37.From this it can be concluded that the application of the substitution factor f St/Al, req.= D Test, req./D Al suffices for fulfilling the targeted damage sum D Test, req.= 2 D al = 1.0.These values are well comparable with f St/Al = 1.25, Tables 6, 7 [3].Here, it should be noted that a damage equivalent test spectrum, derived from the design spectrum and the required fatigue strength-Woehler-curve for the aluminium part, does not require the consideration of the discussed prolongation with the substitution factor f St/Al .This is necessary only if the test spectrum derived for steel components is applied in the biaxial test facility or if a vehicle test track developed for steel components is used.
It can be summarized that the presently applied prolongation factor of f St/Al = 1.25 for the durability proof of an aluminium wheel does not always generate the complete damage intensity, as obtained for the steel wheel.Nevertheless, this factor provides an acceptable cumulative damage at an aluminium wheel.The necessity for a longer testing time for aluminium wheels is primarily founded by the lower cumulative damage due to the differences in the slopes of the Woehler-curves of aluminium and steel, Figure 25.In principle, the interaction between the material and local geometry, i. e., stress concentration and stress distribution, determines the slopes of the Woehler-curve before and after the knee-point and the shape of the spectrum decides in which part of the Woehler-curve the cumulative damage will be more intense.
A single durability proof with the aluminium wheel, used for the presented stress analysis, was carried out by applying the prolongation factor of f St/Al = 1.25.The load program was matched to the static wheel load of F v,stat = 50.0kN.Crack detections by dye penetrant method were carried out at 100 % (= ^2 • 10 4 km) and 150 % (= ^3 • 10 4 km) runtime.The minimum durability performance requirement of 2 • 10 4 km was achieved while having no fatigue cracks.At the extended test duration of 3 • 10 4 km cracks were detected on the inner side of the wheel along the hub contour.By cutting out the cracked areas and separating the crack edges, a maximum crack depth of about 1 mm was found, corresponding to the definition of the first technical crack occurring far above the required fatigue life, Figure 26.These cracks were caused by fretting fatigue between the attached hub and wheel  surfaces.Even after the 150 %-extension of the required test duration, cracks were not detected at the highly stressed ventilation holes.It can be concluded that for the structural durability proof of forged as well as cast aluminium wheels the inclusion of the substitution factor f St/Al = 1.25 extending the test spectrum for steel wheels to the test spectrum for aluminium components is well justified.

F I G U R E 5
Testing requirement for covering the risk resulting from few specimens for determining fatigue life[3].B I L D 5Versuchsforderung zum Abdecken des Risikos der Lebensdauerbestimmung bei wenigen Versuchen[3].
. The maximum stresses should not exceed the structural yield point for achieving the intended design life L Design = 5 • 10 5 km [3, 8, 12].

F I G U R E 8 10 F I G U R E 9
Determination of the required fatigue strength-value (RFS) (schematical).B I L D 8 Bestimmung des geforderten Schwingfestigkeits-Wertes (RFS) (schematisch).T A B L E 3 Mechanical properties of the materials used.T A B E L L E 3 Mechanische Kennwerte der verwendeten Werkstoffe.Details about commercial vehicle wheels 11.75×22.5 (IS 120) of steel and aluminium.B I L D 9 Angaben zu den Nutzfahrzeugrädern 11.75×22.5 (IS 120) aus Stahl und Aluminium.T A B L E 4 Data for deriving stress spectra for different conditions on wheels and hubs of semi-trailer axles.T A B E L L E 4 Beziehungen zur Ableitung der Kollektive für relevante Fahrbetriebszustände von Aufliegerachsen.

F I G U R E 1 0B I L D 1 0
Stress spectra for different driving conditions and resulting design spectrum for wheels and hubs of a semi-trailer axle of commercial vehicles.Spannungskollektive für unterschiedliche Fahrbetriebszustände und resultierendes Bemessungskollektiv für Nutzfahrzeugräder und -naben einer Aufliegerachse.
• 10 5 km to F I G U R E 1 1 Forces to the wheel.B I L D 1 1 Kräfte auf das Rad.F I G U R E 1 2 Steel and aluminium wheels with strain gauges at the ventilation holes.B I L D 1 2 Stahl-und Aluminiumräder mit Dehnungsmessstreifen an den Belüftungslöchern.L Test = 1.6 • 10 4 km, corresponding to about N Test = 1 • 10 7 cycles.The edited damage equivalent load-time sequence for the durability tests performs in the biaxial test facility, a round of about 70 km, Figure 14.This means, for reaching the testing distance of L Test = 1.6 • 10 4 km for the steel wheels, 229 rounds must be driven in the test rig.However, for the aluminium wheels the test distance for steel wheels is extended by the substitution factor f St/Al = 1.25 to L Test = 2 • 10 4 km with 286 rounds in the test rig.

F I G U R E 1 3B I L D 1 3
Rolling of a wheel on a flat track facility (Type LBF) under biaxial loading for experimental stress analysis with strain gauges.Abrollen eines Rades unter biaxialer Belastung im Abrollprüfstand (Bauart LBF) und experimentelle Spannungsanalyse mit Dehnungsmessstreifen (DMS).
contains additionally the F I G U R E 1 6 Normalized design spectrum for steel wheels -measuring point 4 at the ventilation hole and required fatigue strength-Woehler-curve.B I L D 1 6 Bezogenes Bemessungskollektiv für Stahlräder -Messstelle 4 am Belüftungsloch und RFS-Wöhlerkurve.F I G U R E 1 7 Normalized design spectrum for aluminium wheels -measuring point 5 at the ventilation hole and required fatigue strength-Woehler-curve.B I L D 1 7 Bezogenes Bemessungskollektiv für Aluminiumräder -Messstelle 5 am Belüftungsloch und RFS-Wöhlerkurve.extended test spectrum with the presently in the industry used substitution factor of f St/Al = 1.25 for aluminium wheels [3].Hence, the cumulative damage at the aluminium component is more approached to the cumulative damage of the steel component, justifying the practice of the factor f St/Al = 1.25.

F I G U R E 1 8F I G U R E 2 1
Comparison of normalized design spectra on highest stressed points of the ventilation holes and required fatigue strength-Woehler-curves for steel and aluminium wheels.B I L D 1 8 Vergleich der bezogenen Bemessungskollektive an den höchstbeanspruchten Stellen der Belüftungslöcher und RFS-Wöhlerlinien für die Stahl-und Aluminiumräder.F I G U R E 1 9 Normalized test spectrum for steel wheels -measuring point 4 at the ventilation hole and required fatigue strength-Woehler-curve.B I L D 1 9 Bezogenes Versuchskollektiv für Stahlräder -Messstelle 4 am Belüftungsloch und RFS-Wöhlerkurve.F I G U R E 2 0 Normalized test spectrum for aluminium wheels -measuring point 5 at the ventilation hole and required fatigue strength-Woehler-curve.B I L D 2 0 Bezogenes Versuchskollektiv für Aluminiumräder -Messstelle 5 am Belüftungsloch und geforderte Schwingfestigkeits-Wöhlerkurve.Comparison of normalized test spectra (aluminium not damage compensated) in highest stressed points of the ventilation holes and required fatigue strength-Woehler-curves for the steel and aluminium wheels.B I L D 2 1 Vergleich der bezogenen Versuchskollektive (Aluminium unkompensiert) an den höchstbeanspruchten Stellen der Belüftungslöcher und geforderte Schwingfestigkeits-Wöhlerkurven für die Stahl-und Aluminiumräder.
. The test spectrum for aluminium wheels with the related required fatigue strength-Woehler-curves delivers factors f St/Al = 1.18 and 1.39 and the test spectrum for steel with the related required fatigue strength-Woehler-curve results in f St/Al = 1.60 und 2.02, Tables6, 7.However, basing on the required target-damage sum D Test, req .= 1.0 (= 2 D al ) to be at least reached and further from the course of the Woehler-curve of aluminium (Al1) with N k = 5 • 10 6 cycles, the factors f St/Al decrease toward the sufficient (required) values:f St=Al; req: ¼ D Test; req: D Al (16)for the aluminium wheel from 1.39 to 1.04 and for the steel wheel from 2.02 to 1.37, Tables6, 7.The here presented substitution factors f St/Al for forged aluminium wheels were derived with the slopes k = 5 and k' = 9 before and after the knee-point at N k = F I G U R E 2 2 Comparison of normalized test spectra (aluminium not damage compensated and damage compensated) in highest stressed points of the ventilation holes and required fatigue strength-Woehler-curves for the steel and aluminium wheels.B I L D 2 2 Vergleich der bezogenen Versuchskollektive (Aluminium unkompensiert und kompensiert) an den höchstbeanspruchten Stellen der Belüftungslöcher und geforderte Schwingfestigkeits-Wöhlerkurven für die Stahl-und Aluminiumräder.5 • 10 6 cycles.In case of cast wheels, the slope k = 5 and the knee-point at N k = 5 • 10 6 cycles remain, but the slope after the knee-point with k' = 2 kÀ 2 = 8 changes slightly [7].However, this does not influence F I G U R E 2 3 Damage of the normalized and not damage compensated test spectrum for aluminium confronted with the required fatigue strength-Woehler-curves for steel and aluminium with different knee-points.B I L D 2 3 Schädigung des bezogenen, unkompensierten Versuchskollektives für Aluminium bei Zuordnung zu geforderte Schwingfestigkeits-Wöhlerkurven für Stahl und Aluminium mit verschiedenen Abknickpunkten.F I G U R E 2 4 Damage of the normalized test spectrum for steel confronted with the required fatigue strength-Woehler-curves for steel and aluminium with different knee-points.B I L D 2 4 Schädigungen des bezogenen Versuchskollektives für Stahl bei Zuordnung zu geforderte Schwingfestigkeits-Wöhlerkurven für Stahl und Aluminium mit verschiedenen Abknickpunkten.

F I G U R E 2 5
Compensation of damage regarding test duration for the substitution of steel wheels by aluminium wheels (schematical).B I L D 2 5 Schädigungskompensation bei der Versuchslaufzeit für die Substitution von Stahlrädern durch Aluminiumräder (schematisch).

F I G U R E 2 6
Cracks along the hub contour of a commercial vehicle wheel 11.75×22.5 (IS 120) of aluminium after 3 • 10 4 km (150 %) proof test caused by fretting fatigue.B I L D 2 6 Risse an der Nabenkontur eines Nutzfahrzeugrades 11.75×22.5 (IS 120) aus Aluminium nach 3 • 10 4 km (150 %) Prüfstanderprobung infolge Reibermüdung.should be prolonged by the factor f St/Al = D St /D Al for obtaining the same cumulative damage content as for steel wheels, Figures 23, 24, Tables

T A B L E 6
Design and test spectra for aluminium wheels and required fatigue strength-Woehler-curves for determining damage sums.T A B E L L E 6Bemessungs-und Versuchskollektive für Aluminiumräder und RFS-Wöhlerkurven zur Bestimmung der Schadenssummen.
Design load cases with load factors and maximum load values of partial spectra.Bemessungslastfälle mit Lastfaktoren und Höchstlasten der Teilkollektive.
e Distribution L Design = Design life in km; r dyn = Dynamic rolling radius in m; z = Number of cycles per wheel revolution; N b = Cumulative frequency of the stress s a,stat under the static wheel load F z,stat ; N e = Cumulative frequency of maximum values N t = Total cumulative frequency N t ¼ T A B E L L E 5 Design and test spectra for steel wheels and required fatigue strength-Woehler curves for determining damage sums.Bemessungs-und Versuchskollektive für Stahlräder und geforderte Schwingfestigkeits-Wöhlerkurven zur Bestimmung der Schadenssummen.