1

Toward the wheels calculation and validation in multiaxial fatigue under service loads

F. COCHETEUX1, J. BENABES1,T. PALIN-LUC2, N. SAINTIER2, F. BUMBIELER1

1SNCF : Agence d’Essai Ferroviaire 21, avenue Salvador Allende - 94407 Vitry-sur-Seine Cedex France; 2ENSAM - Laboratoire Matériaux Endommagement Fiabilité et Ingeniérie des Procédés (EA2727),

Esplanade des Arts et Métiers - 33405 Talence Cedex France

Abstract

Dimensioning methods currently allow to design against fatigue the axisymmetric plate of a railway wheel. For drilled wheel plates, fatigue calculation methods enacted in UIC leaflets (International Union of the Railroads) [1] or European standards [2] are not adapted. Those are based on an assumption of often checked simplification: whatever the zone of the plate, the maximum principal stress is radial. A recent application, for which the plate of the wheel is drilled, showed that the principal stress field directions vary during the wheel rotation. Consequently, classical monoaxial fatigue criterion as GOODMAN or HAIGH [3] are not applicable. A reliable design of industrial parts against high-cycle multiaxial fatigue requires a fatigue criterion capable of predicting both the load-type and the stress gradient effects. Gradient effects are of prior importance for the applicability of fatigue models to real structures. By using the concept of volume influencing the fatigue crack initiation proposed by Palin-Luc and Banvillet [4], an extension of the criterion, based on an energy approach, is applied to design wheels against fatigue. To evaluate the fatigue strength in multiaxial stresses zones, it is initially necessary to estimate the stress fields according to the real service loadings. This is done by a numerical approach using a finite element model correlated by strain gauges measurment during on line tests. For most critical points of the modeled wheel, the application of the LAMEFIP volumetric criterion [4] to numerical results allowed to evaluate the theoritical safety level for the fatigue strength in theoretical way. In parallel, a validation test protocol is under development. It consists in applying vertical and transverse bidirectional forces (test bench with two actuators) to a full scale wheel with a drilled plate.

Introduction

New designs of railway wheels can present drillings at the core of their plate to allow the brake disc assembly, a sound-proofing system or any other element. Currently the failure risk evaluation, related to the use of such wheels, is only possible by the experience feedback. Within the framework of an optimized design and in order to estimate its safety margin, it is necessary to approach the service loading. Although calculations with finite element analysis (FEA) allow to evaluate the stress levels in a correct way, the assumptions of the current dimensioning methods in fatigue are not applicable because of the multiaxiality of the stress field near the geometrical singularities. The first part of the study consists in determining loading characteristic of a wheel at the time of various configurations of circulation. In a second stage, measurements of vertical and transverse forces allow to work out the sections of life which will be integrated in numerical simulations. The second part deals with the FE computation on a real wheel and presents the procedure used introduce the service loading records in the calculation. Finally, a postprocessing is applied to check if a fatigue life calculation must be implemented or if an equivalent stress calculation is enough [5]. An application of the method suggested for fatigue life calculation is finally presented.

2

In service measurements In order to determine the values of the vertical and transverse forces encountered on a wagon wheel, series of on line measurements were undertaken by the French Test Railway Agency (AEF), located at Vitry/Seine, on a truck carries containers equipped with Y25 type bogie and 9054 standard axles. The truck, of a 18.5t mass, circulated in empty configuration then with a 60t added load. Test circulations took place in France between Paris and Saint Jean de Maurienne in the Alps.

Figure 1: diagram of the tested train The tested wagon was equipped with a measurement axle allowing to record the vertical (FQ) and transversals (FY) forces. Moreover, strain gauges were stuck on the axle and on the internal and external wheel faces to measure the strains in some wheel zones. Data reductions of the measurements (forces and stresses) were led on portions of lines corresponding to the four most representative configurations of the European railway:

• Circulations with 120 km/h (alignments and large-radius curves). • Circulations with 80 km/h on sinuous way (short radius curve). • Passages of station (passage over trackside equipment at the maximum speed of the line). • Guard rail.

For each one of these typical circulations, the loading state (empty or full), as well as the direction of circulation (attack or in tail) are as many parameters taken into account in the measurement treatment corresponding to sixteen reference configurations. A train wheel travels between 700,000 km (for wagon) and 2,500,000 km (for high speed train) that corresponds for a Φ 920 mm wheel (2.89 m/turn) to 2.42 108 and 8.65 108 wheel rotation. This is equivalent to 230 or 830 repetitions of a 3,000 km section of life. From all the recordings carried out (Table 1), it is possible to reconstruct this representative life section by combining the elementary configurations in the following way:

iConfig16

1iia.n life) of(part .n life Wheel ∑

=== (1)

A standard life section for a wagon wheel would be, for example, to consider that the wagon circulates half of time in load and other half empty, that the axle is placed once on two at the head and that the train runs 83% of the course at maximum speed (V = 120 km/h), 14 % at speed limited (sinuous ways) with 1% of passage in station and 2% of guard rail. Table 1 summurises the extreme values (to 0,15 and 99,85% of probability of appearance according to UIC leaflet 518) and the average values of the forces measured for the sixteen reference configurations.

Tested wagon

Laboratory

G

testedwheelset

3

Table 1: Recapitulation of wagon measurments One can note that UIC leaflet 510-5, usually used for wheels dimensioning in Europe, indicates force values for a 78,5t wagon lower than the maxima measured in service (Table 2). Case 1 corresponds to circulation in alignment where only the vertical component is present (with Q=78500/8 kg and g=9.81 m/s²). Case 2 corresponds to curve circulation and case 3 in the passing in the trackside equipment.

Table 2: loadings according to UIC leaflet 510-5 [1]

Calculation of the stresses

The studied wheel was modeled and computed by FEA. The wheel meshing is carried out by 3D volumetric solid elements, making it possible to accurately describe the geometry of the drilled wheel. The nodes of the hub boring are blocked because it is not necessary to reproduce hooping conditions. Only the loading path induced by the dynamic stresses are required to compare with the dynamic on line measurements.

To determine the influence of the transverse (FY) and verticals (FQ) forces on the wheel in the three loading cases specified in UIC 510-5 card, five calculations in linear elasticity were carried out (Figure 2). The inertia forces are neglected. They allowed to determine the influence of each force on the wheel state of stress independently of the others. A unit force of 1kN is applied for both FQ and FY in order to determine the sensitivity (stresses tensors) in all points of the wheel. Assuming a linear elastic behavior, the real stress and strain tensors can be obtained for any loading case using the supperposition method. Stress tensors for all the angular positions of a wheel rotate were given for four localizations. Those four specific points, where strains calculations are carried out, are located at the sites where the strain gauges were stuck on the wheel. They correspond to a point on the internal face with plate/hub connection and to a point at the edge of a drilling and their opposite on the external face.

CASE 1 CASE 2 CASE 3

FQ (kN) 1,25 Q.g 120 120 120

FY (kN) 0,60 Q.g 58

-0,36 Q.g -35

LOAD ATTACK Case FY mean sy FY min FY max FQ mean sq FQ min FQ max Distance (km)

120 km/h 0.0 5.7 -13.7 17.4 100.6 12.6 69.8 140.0 5.40

80 km/h 1.5 6.3 -12.6 20.5 98.5 14.9 51.8 140.6 2.89

Station 11.6 17.4 -12.8 75.9 99.0 12.9 61.0 140.5 2.77

Service 1.2 7.8 -24.0 26.0 99.3 10.5 74.0 139.4 0.35

120 km/h 3.4 9.6 -8.5 45.4 97.6 14.5 60.3 139.9 6.4080 km/h 11.6 14.3 -9.0 57.5 100.3 17.6 50.1 146.3 5.97

Station 8.0 17.5 -10.9 77.1 98.4 14.6 64.5 141.1 1.99

Service 13.6 21.7 -41.3 83.0 97.9 26.8 34.5 185.8 0.42

120 km/h -0.3 1.9 -5.0 5.2 21.1 3.4 11.8 30.6 4.62

80 km/h 0.8 1.8 -4.4 7.1 21.2 3.5 10.6 30.5 9.07

Station -1.2 1.6 -4.2 3.9 20.8 2.9 10.7 35.1 1.06

Service 1.0 2.5 -6.2 9.4 20.7 3.8 9.3 34.0 2.32

120 km/h 2.0 3.6 -2.2 16.5 21.1 3.7 10.1 33.2 5.57

80 km/h 4.4 4.9 -2.7 19.6 20.5 4.5 8.3 33.9 8.52

Station 1.7 3.8 -7.7 16.6 23.2 4.9 9.2 38.2 3.15

Service 1.5 4.0 -13.0 17.3 24.5 4.6 10.3 39.3 1.36

kN

Yes

No

Yes

No

Yes

No

4

Case 1 Case 2 Case 3

Figure 2: diagram of the five unit loadings

To calculate stresses evolution for rail/wheel loads which turn, it is considered that the wheel is fixed and just forces application points on it vary (Figure 3). Then, by using symmetry conditions, the examination is made on eight equidistant sections making it possible to describe the loading path during the wheel rotation by fourty eight values.

Figure 3: detail of the six angular positions

Thus thee obtained loading path are completely different from one position to another as well as loading case to the other.

Figures 4 and 5 illustrate these differences in stresses evolution versus time for one wheel rotation at a point of the plate/hub connection on the external face and at a point at the edge of a drilling when a transverse force of 1KN is applied in the roll case.

-1.E+06

-5.E+05

0.E+00

5.E+05

1.E+06

2.E+06

2.E+06

3.E+06

-60 0 60 120 180 240 300

Pa

sRR stt sZZ sRt stZ sZR

Figure 4: plate/hub connection Figure 5: edge of drilling

-1.E+06

-1.E+06

-1.E+06

-8.E+05

-6.E+05

-4.E+05

-2.E+05

0.E+00

2.E+05

4.E+05

6.E+05

-60 0 60 120 180 240 300 360°

Pa

sRR stt sZZ sRt stZ sZR

FFQQ == 11 kkNN FFQQ == 11 kkNN FFYY == 11 kkNN FFQQ == 11 kkNN FFYY == 11 kkNN

5

To illustrate the loading path of the selected positions and to visualize their multiaxial character, it is preferable to be placed in the octahedral coordinate system (τoct,PH) where τoct is the octahedral shear stress and PH the hydrostatic pressure (Figures 6 and 7).

In alignment, one can note that stress levels are relatively low. This confirms that alignment is not the most severe loading case for the wheel dimensioning. On the other hand in curve, the stress levels are higher. It should be noted that the plate/hub connection position is where the stresses are highest but also quasi uniaxial. The position at the edge of a drilling presents a more complex loading path but with lower stresses. For this position and to be able to take into account this type of loading path, a multiaxial approach is essential. Tests carried out on various portions of line, described previously, showed that forces between the wheel and the rail strongly varied (Figure 8). Indeed wheel boundary conditions change throughout a circulation in service (case 1, 2 or 3), therefore the calculation model to consider exchange during the time. If FY is null or quasi null, one is located in case 1, if it is positive it is case 2, finally if FY is negative it is case 3.

l Figure 8: example of temporal evolution of the Y and Q forces

A data-processing development has been elaborated in order to establish the link between measurements of forces recorded during the tests and the sensitivities calculated by finite elements. The objective is to constitute a matrix FYFQ counting the couples transverse-vertical forces. Such matrix (Figure 9) is then combined linearly with the unit stresses loading path calculated in order to determine the wheel stress respons. Section 1 gathers the values of FY close to zero, that corresponds to FEA where the wheel is in alignment. Section 2 gathers the positive FY , when the wheel is located in a curve. Lastly, the values identified by section 3 are associated to guard rail.

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

7.E+05

8.E+05

9.E+05

1.E+06

-6.E+05 -4.E+05 -2.E+05 0.E+00 2.E+05 4.E+05 6.E+05 8.E+05 1.E+06 1.E+06

PH (Pa)

Toct

(Pa)

Figure 6 : connection plate/hub Figure 7 : edge of drilling

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

7.E+05

-6.E+05 -5.E+05 -4.E+05 -3.E+05 -2.E+05 -1.E+05 0.E+00 1.E+05 2.E+05 3.E+05

PH (Pa)

Toct

(Pa)

FQ FY

6

Figure 9: FYFQ matrix of one configuration

The good agreement between the stress measurements carried out in service and those calculated starting from force measurements can be easily illustrated by comparing, for example, the radial stress temporal evolution of a wheel point (Figures 10 and 11).

Design against fatigue

Except some recent publications [6], few works concern the fatigue wheel dimensioning under complex loadings. In this part, a new wheel dimensioning method respecting the multiaxial fatigue strength is presented in two main steps.

The first one consists in determining, from the stresses calculated by the data-processing development previously presented, an equivalent fatigue stress. This variable is computed thanks to a multiaxial fatigue criterion and is then compared with the endurance limit of the wheel material under simple loading. This threshold is determined for a given number of cycles. In the illustrated example figure 12, the selected multiaxial fatigue criterion is the Crossland one, which regards as damage parameter the linear combination of the greatest amplitude of octaedral shear stress Τ

oct,a,Δ,J2 and the maximum

hydrostatic stress Σ H, max

. The given equivalent stress, in this case, is compared with the endurance limit under fully reversed torsion. If the damage parameter is below the asymtotic line of the Wöhler’s curve, the fatigue life of the wheel is infinite; otherwise, a fatigue life calculation method is necessary.

FQ max

36 44 52 61 69 77 85 93 101 109 117 125 134 142 150 158FY min -29

-21-14 2 5 2 2

-6 3 25 93 176 106 13 2 12 2 6 23 34 31 22 2 19 8 13 16 20 18 5 2

17 1 2 12 28 24 8 16 10 2 124 5 16 25 3 8 13 532 1 2 3 6 3 440 6 7 15 11 347 5 12 15 5 255 1 5 4 10 262 1 3 4 6 2 170 2 3 177 1

FY max 85

FQ min FQ

FY

1

3

2

173 MPa

0 MPa

-67 MPa

167 MPa

0 MPa

-67 MPa

Figure 10: Stresses measured on line Figure 11: Calculated stresses

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Multiaxial stress state

Multiaxial fatigue criterion (ex : Crossland) [7]

maxHΔJaoct aστ ,2,, +

Uniaxial equivalent stress

Comparison: torsion S-N curve

Case n°1 : σmax1 infinite fatigue life Case n°2: σmax2 fatigue life calculation method

Figure 12: example of the Crossland fatigue criterion application in a multiaxial stress state The second stage is: if the previous test is not checked, a fatigue life calculation method is used to estimate the number of cycles to fatigue crack initiation in the wheel. The proposal for fatigue life calculation is based on the energy and volumetric approach of LAMEFIP [4], making it possible on the one hand to distinguish the types of loading; and in addition to consider the effect of stress-strain gradients. In this paragraph, the multiaxial fatigue criterion of the LAMEFIP is presented first, then the fatigue life calculation method itself is detailed. Banvillet et al. [4] proposed to consider the strain work density given to each elementary volume of material per loading cycle, Wg, as the driving force for damage initiation and growth.

∑ ∑ ∫∫Δ−

•><=><=

ji ji

eijijij

Tijij

ij

ijij

dtMdttMtMMWg, ,

max,

max,

)),((),( ),()(ε

εεεεσεσ (2)

This parameter is equal to the integral over a loading cycle of the positive part of the strain power density, this is also equal to the positive variation of the potential strain energy density over a loading period. Wg is calculated from elastic strains after elastic shakedown of the material. Indeed since usually the endurance limit (infinit life) is low enough to consider that the material remains elastic at the macroscopic scale. It has to be noticed that for a uniaxial stress state Wg is not shape dependent (sine, triangle, square...). As proposed by Palin-Luc and Lasserre [8], to predict the effect, on the fatigue limit, of the stress-strain distribution inside the component, a volume influencing fatigue crack initiation, V* is considered. For a fully reversed uniaxial stress state, this volume is defined by the set of points loaded by a stress amplitude higher than a material dependent threshold stress σ*. From fully reversed fatigue experiments on a spheroidal graphite cast iron, these authors show that a threshold stress, σ*, can be defined below the conventional endurance limit σD (Palin-Luc and Lasserre [8]). At a considered point, a stress amplitude lower than this threshold does not initiate observable damage at the microscale (no micro-crack). A stress amplitude between σ* and σD contributes to micro-damage initiation only, which could develop if, either near this point or in the course of time there is a stress amplitude higher than σ*. The stress limit σ* is considered as a threshold stress of no damage initiation at the microscale, whereas the usual endurance limit σD corresponds to a macrocrack initiation. By reference to an homogenous fully reversed uniaxial stress state (tension on smooth specimen) the threshold value corresponding to σ* is Wg* = (σ*)²/E, where E is the Young modulus of the material. The volume influencing fatigue crack initiation is then defined by (3) around each critical point Ci (where Wg has a local maximum). Over this volume, the damaging parameter is the strain work density, exceeding the threshold Wg*, given to the material (4). This damaging strain work density is supposed to be constant for a given material loaded at its endurance limit in a uniaxial stress state.

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V*(Ci) = {points M(x,y,z) around Ci so that Wg(M) ≥ Wg*(Ci)} (3)

( ) ( ) ( )[ ]( )∫∫∫ −=

iCVM

iCiC dvWfWgV

g*

**1ϖ (4)

Thus, the condition to avoid fatigue crack initiation is ( ) D

giCg ϖϖ ≤ , where Dgϖ is the value of gϖ at the

endurance limit for an uniaxial stress state. By applying this hypothesis with the fatigue limits under fully reversed tension D

tens,1−σ and rotating bending Drotbend,1−σ on smooth specimens, the threshold stress σ*

can be identified.

DD−∗=σσ ( ) ( )2

,12

,12 Drotbend

Dtens −−

∗ −= σσσ (5)

It is assumed that for any loading and stress state at the endurance limit the value of Dgϖ , noted

( )uniaxDgϖ , is constant where E is the Young modulus(6, 7).

( ) ( ) ( ) ( )... bendrotbendplanetensuniax Dg

Dg

Dg

Dg ϖϖϖϖ === (6)

( ) ( ) ( )E

tensbendrotuniaxDD

Dg

21

21 )(.).( −− −

=σσϖ (7)

For a multiaxial stress state at the fatigue critical point a “multiaxial correction” is used. The triaxiality degree of stresses dT(M) is defined by the ratio between the strain work density WgSph due to the spherical part of the stress tensor and Wg. This ratio varies between 0 (pure distorsion) and 1 (pure equitriaxial stress state).

( ) ( )( )MW

MWMdTg

Sphg= (8)

In order to take into account the influence of the stress triaxiality, the strain work density given a point M which is equivalent to a uniaxial stress state, Wgeq(M), is introduced:

( )( )β

β,

),().()(M

eq dTFdTuniaxFMWgMWg = (9)

This quantity depends on the stress state at M and on the volumetric distribution of stresses inside V*(M). The proposed energy based volumetric criterion distinguishes simple and combined loadings. We show that it can also be applied in limited endurance to predict the S-N curves corresponding to any load type for fatigue life higher than approximatively 5.104 cycles. These predictions are possible, from a reference experimental median S-N curve (crack probability 50%). From a median experimental S-N curve on smooth specimens, σa = f(N), it is possible to link ϖgeq and the median fatigue life N (Figure 13). Indeed, as the proposed criterion distinguishes loadings, it is possible, by using the relations developed before and with the assumption that ϖgeq is constant for a fixed number of cycles whatever the load type and the stress state, to identify a master curve ϖgeq = G(N).

9

σa

Nτa

N

ϖfeq

N

Tension

Tension + Torsion

Damage parameter : ϖfeq Master curve

+

+…

σa

Nτa

N

ϖfeq

N

Tension

Tension + Torsion

Damage parameter : ϖfeq Master curve

+

+… Figure 13 : Relation between any experimental S-N curve and ϖgeq(N) as damage parameter

To apply this proposal for a fatigue life calculation method, an experimental curve SN on smooth specimens: σa = f(N) is necessary. By transforming this S-N curve in a curve in damaging work provided in V* versus N, the identification of the 'master' curve ϖgeq = G(N) is possible. The equivalent parameter, rising from the application of criterion of the energy and voluminal multiaxial fatigue ϖfeq(Ci), is then

injected into the equation of this main curve. By reversing the relation ( ) ( )uniaxCf Dfieq ϖϖ = , a prediction

of the number of cycles to crack initiation is obtained. The application of this fatigue life calculation method within the framework of a usage in an engineering and design department is illustrated in figure 14.

Figure 14 : Synoptic of the proposal

Application of the method The taking into account of the sixteen reference configurations defining the life section to be applied to a wheel with a complex geometry, to compare the fatigue life predictions of the proposed method compared to the experiments determined fatigue lifes to full scale wheel is in hand. This work is not completed to date, therefore elements used to show the good predictions of the fatigue life calculation method applied to railway real loadings are extracted from committee ERRI B169 / RP19 report [5]. The steel ER7 caracteristics constituting the test specimen are included in tables 3 and 4.

ϖgeq

10

E (MPa) ν Re0.2 (MPa) Rm (MPa) A (%) 195000 0,29 440 795 19

Table 3 : Static mechanical characteristics

. σD

tens,-1 (MPa) σDbend,-1 (Mpa) σD

rot.bend,-1 (MPa) τD-1 (MPa)

272 296 283 198

Table 4 : Experimental endurance limit at 106 cycles on cylindrical smooth specimens

Fatigue tests presented in this part were carried out on smooth specimens made of ER7 steel with a 8 mm diameter [5]. The loading applied to the specimens corresponds to a multiaxial loading exerted by the AEF calibration bench for measurement axles called bench Y (Figure 15). Strain Gauges, stuck on the wheel critical zones (fillets and edge of drilling), allow to know the stresses evolution in a point during a wheel rotation. The loading path is then restored on smooth specimens with an equivalent loading of combined tension-torsion. The variable amplitude loading is illustrated in figure 16. In order to obtain different fatigue life, this loading is amplified by a magnification factor k. Several factors of amplification are applied to the specimens. The experimental numbers of cycles to failure and the life predicted by the proposed method are shown in table 5.

k Nexp (cycles) Npredicted (cycles) Comments

2.5 > 1 000 000 2 384 258 No crack during test 4 894 766 715 276 5 57 843 52 935 5 67 731 52 935

Table 5 : Predictions and experimental data of the variable amplitude combined tension-torsion tests.

Figure 17 presents a comparison between the fatigue life from experiments on specimens under combined tension-torsion loadings with variable amplitude restoring the loading path coming from the bench Y, and those laid down by the proposed fatigue life calculation method. One can note that predictions of the method remain in an interval of more or less twice the experimental results, which is completely correct.

Figure 16 : Variable amplitude combined tension-torsion load history and illustration of the magnification factor k

1 rotation of the wheel

Stre

ss

TensionTorsionTension * kTorsion * k

kReference

11

k

Reference

Figure 15 : bench Y

11

104 105 106104

105

106

N p

redi

cted

(cyc

les)

N experiment (cycles)

x2÷2

Load magnification factor : 2.5Load magnification factor : 4Load magnification factor : 5

Figure 17. Comparison of the predicted and experimental fatigue life for the variable amplitude combined

tension-torsion tests

Conclusions and prospects

Literature offers many formulations for multiaxial fatigue criteria with constant amplitude under various approaches. One can also find some damage calculation methods in variable amplitude but their industrial exploitation is to date the object of only few publication. Although the study is not finished yet, the confrontation of first experimental results obtained within the framework of this work to the predictions of the proposed method, confirms the relevance to resort to such fatigue method for the dimensioning of the wheels presenting strongly multiaxial stress zones. Unfortunately there is not universal criterion and methods unanimously recognized in the industrial world, even if some of them are already implemented in computer fatigue codes. Designing wheels against multiaxial fatigue is only at its beginnings and other studies are still to be carried out like seeking modeling criteria, having better knowledge of the stresses in service for differents types of trains (High speed trains, coaches, locomotives...), to ensure the thermal and mechanical coupling or to take into account the residual stresses. The results presented and to come justify these ambitious prospects which will allow, may be, to better know what is surely more the hurdy-gurdy invention, appeared in Mesopotamie towards 3500 before J.C, which one names the wheel.

References

[1] UIC leaflet 510-5 : monobloc wheel technical approval procedure. [2] EN 13979-1 standard : wheelsets and bogies – monobloc wheels - technical approval procedure [3] GOODMAN, J Mechanics applied to engineering (1st ed.). Longmans, Green and Co, 1899. [4] A. BANVILLET,T. PALIN-LUC, S. LASSERRE : A volumetric energy based high cycle multiaxial fatigue criterion. IJF 26,8 (2003), 755-769. [5] RP19 from ERRI B169 committee : Wheels designing, multiaxial fatigue criteria.

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[6] A. BERNASCONI, M. FILIPPINI, D. VAUDO : Multiaxial fatigue of railway wheel under non-proportional loading, IJF (accepted 6 july 2005). [7] CROSSLAND B., Effect of a large hydrostatic pressure on the torsional fatigue strength on a alloy steel, Int. Conf. on Fat. Of Metals, London, 1959. [8] T. PALIN-LUC, S. LASSERRE, An energy based criterion for high cycle multiaxial., A/Solids, 17, 1998, pp. 237-251. fatigue, Eur. J. Mech