International Journal of Fatigue 32 (2010) 434–442

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International Journal of Fatigue

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Simulation of dynamic cornering fatigue test of a steel passenger car wheel

Xiaofeng Wang a,*, Xiaoge Zhang b,1

a State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering, Tsinghua University, Beijing 100084, PR Chinab Dongfeng Automotive Wheel Co., Ltd., Shiyan 442042, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 March 2009Received in revised form 10 August 2009Accepted 2 September 2009Available online 7 September 2009

Keywords:Passenger car wheelFinite element analysisTransient strain historyRadial normal strainFatigue life prediction

0142-1123/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijfatigue.2009.09.006

* Corresponding author. Tel.: +86 10 62781338; faxE-mail addresses: wangxf60@mail.tsinghua.edu.cn

com (X. Zhang).1 Tel./fax: +86 719 8221399.

This paper aims at seeking a practical and comprehensive method for simulating the dynamic corneringfatigue test of the automotive wheels. The test of a steel passenger car wheel is simulated by combineduse of the linear transient dynamic finite element analysis and the local strain approach. A rotating forceof constant magnitude is applied to the moment arm tip to simulate the rotating bending effect on thewheel, with the wheel stationary. It is found that only a radial component of the rotating force is neededto obtain the sufficiently accurate radial normal strain histories of the elements located along the radialdirection. The strain history of the element whose local stress–strain characteristic keeps linear and clos-est to the critical element is applied to predict the fatigue life of the critical element with Neuber’s ruleand local strain approach, which is quite close to the test results.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The dynamic cornering fatigue test is a standard SAE test [1],which simulates cornering induced loads to the wheel. Fig. 1 showsthe test system applied by Dongfeng Automotive Wheel Co., Ltd., inwhich the test wheel is mounted to the rotating table, the momentarm is fixed to the wheel outer mounting pad with the bolts and aconstant force is applied at the tip of the moment arm by the load-ing actuator and bearing, thus imparting a constant rotating bend-ing moment to the wheel. If the wheel passes the dynamiccornering fatigue test, it has a good chance of passing all other re-quired durability tests [2]. So it is desirable to be able to estimatewhether the wheel under design passes the dynamic cornering fa-tigue test or not and make design improvements necessary accord-ingly even before the first prototype wheel is made in the wheeldesign stage. Therefore the method to simulate the dynamic cor-nering fatigue test of the wheel with finite element analysis meth-od and local stress–strain fatigue life prediction was studied.

Some papers dealing with the simulation of the dynamic cor-nering fatigue tests of automotive wheels have been published,which, such as those by Riesner and DeVries [2], Karandikar andFuchs [3], Shang, et al. [4], are valuable but generally do not pro-vide sufficient details for others to follow and make the simulation.This paper aims at seeking a practical and comprehensive methodfor simulating the test and predicting the wheel fatigue life in it

ll rights reserved.

: +86 10 62784655.(X. Wang), syxgchina@126.

and introducing the method in such a way that it is easy for othersto make the simulation following the method introduced.

2. Static finite element model of the wheel

The wheel analyzed is a steel passenger car wheel produced byDongfeng Automotive Wheel Co., Ltd. The static finite elementmodel of the wheel is first constructed, as shown in Fig. 2. The fi-nite element analysis software used in this paper is HyperWorks8.0. The rim and the disc of the wheel are modeled with the plateand shell elements (CQUAAD4 and CTRIA3) whose dimensions areabout 2 mm. The thicknesses of the elements for the rim and thedisc are determined according to their nominal thicknesses andare 2.8 mm and 3.5 mm, respectively. The thickness of the ele-ments in the connection zones of the disc and the rim is 6.3 mm,the sum of 2.8 mm and 3.5 mm. The moment arm is modeled withthe solid elements CTETRA whose dimensions are between 2 mmand about 25 mm. The forces and moments are transmittedthrough the common nodes in the interfaces of the disc and themoment arm. The bolt connections are not modeled for simplicityand the fact that comparatively small number of failures occurredat the bolt circle areas in the real dynamic cornering fatigue tests ofthe wheel studied. Although this simplified model is not realistic inthe interfaces, it may only introduce minor errors in the criticalareas because they are sufficiently far away from the interfacesof the moment arm and the wheel (as shown in Fig. 3). And it isbeneficial for the calculation speed.

The wheel is made of a material equivalent to St37, a steel,whose fatigue properties are shown in Table 1 [5]. The material

Fig. 2. Static finite element model of the wheel.

Fig. 1. Sketch of the dynamic cornering fatigue test system applied by Dongfeng Automotive Wheel Co., Ltd.

X. Wang, X. Zhang / International Journal of Fatigue 32 (2010) 434–442 435

properties for the elements above are as follows: modulus of elas-ticity, E = 214 GPa; Poisson’s ratio, l = 0.3; density, q = 7900 kg/m3.

The translational degrees of freedom along X, Y and Z directionsof all element nodes contacting with the rotating table of the testrig are restrained and a force of 1562.4 N is applied to the centernode in the end plane of the moment arm of the length of938 mm, as shown in Figs. 1 and 2.

The solver Optistruct is applied to process the wheel finite ele-ment model constructed and the results are shown in Fig. 3 (vonMises stress distribution in the wheel disc), with the critical areasbeing mainly located near the ventilation windows and the hat ofthe disc and the element No. 68225 having the maximum vonMises stress 239 MPa.

3. Static experimental stress analysis

The static experimental stress analysis was made to validate thestatic finite element model of the wheel constructed. Five 45�strain rosettes were pasted onto the surface of the wheel disc, asshown in Fig. 4, and the static bending test was done, in whichthe rotating table was held stationary and a constant force of1562.4 N was applied at the tip of the moment arm. And the strainswere measured. Fig. 4 also shows the local reference frame of eachstrain rosette, in which axis O1Y1 points outwards from the center

of the wheel along its radial direction. And the local referenceframe of each strain rosette is attached to the wheel.

The strains of each strain rosette were measured for four posi-tions by rotating the rotating table: (1) 0� position, in which theaxis O1Y1 of the local reference frame coincides with the axis OYof the fixed reference frame OXYZ (Fig. 4), which does not rotatewith the wheel; (2) 90� position, in which the axis O1Y1 coincideswith the axis OX of the fixed reference frame; (3) 180� position, inwhich the axis O1Y1 coincides with the negative direction of axisOY; (4) 270� position, in which the axis O1Y1 coincides with thenegative direction of the axis OX. Table 2 shows the strains mea-sured and Von Mises stresses calculated with the strains and thefinite element model of the wheel. Table 3 shows the correspond-ing normal stresses and shear stresses. The following formulas areused for processing the strain data measured:

rx1 ¼E

1� l2 ðe0 þ le90Þ ð1Þ

ry1 ¼E

1� l2 ðle0 þ e90Þ ð2Þ

sx1y1 ¼E

2ð1þ lÞ ðe0 þ e90 � 2e45Þ ð3Þ

r1 ¼rx1 þ ry1

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrx1 � ry1Þ2 þ 4s2

x1y1

q2

ð4Þ

Fig. 3. Von Mises stress distribution in the outside surface of the wheel disc.

Table 1St37 fatigue properties [4].

436 X. Wang, X. Zhang / International Journal of Fatigue 32 (2010) 434–442

r2 ¼rx1 þ ry1

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrx1 � ry1Þ2 þ 4s2

x1y1

q2

ð5Þ

rv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12½r2

1 þ r22 þ ðr1 � r2Þ2�

rð6Þ

Er ¼ rv ;c � rv

rv

�������� ð7Þ

Order no. Parameter Data

1 Cyclic modulus of elasticity, E, GPa 2142 Cyclic Strength coefficient, K0 MPa 9883 Cyclic strain hardening exponent, n0 0.2074 Fatigue strength coefficient, r0f , MPa 8735 Fatigue strength exponent, b �0.106 Fatigue ductility coefficient, e0f 0.5577 Fatigue ductility exponent, c �0.518

where rx1, ry1 and sx1y1 are normal stresses and shear stress in thereference frame O1X1Y1; e0, e45 and e90 are 0� strain, 45� strain and90� strain of a strain rosette, respectively; r1 and r2 are principalstresses; rv is the Von Mises stress calculated with the strains mea-sured; rv,c is the Von Mises stress calculated with the finite elementmodel of the wheel; Er is the percentage error of rv,c relative to rv.

It can be seen, in Table 2, that the percentage errors, Er’s, of rv,c

calculated with the finite element model of the wheel relative to rv

based on the strains measured are less than 18% and most of them

Fig. 4. Five 45� strain rosettes on the outside surface of the disc and local referenceframe of each strain rosette.

Table 2Measured stains of each strain rosette and corresponding Von Mises stresses.

Strainrosette

Position(�) 0�strain(le)

45�strain(le)

90�strain(le)

rv

(MPa)rv,c

(MPa)Er(%)

A 0 358 569 737 176 197 1290 �9 �269 �62 68 63 7.4

180 �399 �568 �704 174 196 13270 8 266 56 68 68 0

B 0 322 581 756 176 187 6.390 �37 �309 �35 79 84 6.3

180 �282 �553 �763 174 179 2.9270 27 311 22 82 84 2.4

C 0 114 �240 �717 151 171 1390 �36 109 139 34 31 8.8

180 �149 256 847 180 173 3.9270 31 �111 �137 33 31 6.1

D 0 �305 �640 �644 160 172 7.590 �129 �37 �371 104 85 18

180 337 672 639 164 172 4.9270 121 13 346 100 85 15

E 0 �266 �456 �714 163 176 8.090 112 436 181 94 91 3.2

180 301 473 728 169 177 4.7270 �118 �446 �201 96 91 5.2

Table 3Stress states of five points for different positions based on the strains measured.

Point Position(�) r1

(MPa)r2

(MPa)rx1

(MPa)Drx1

(MPa)ry1

(MPa)Dry1

(MPa)sx1y1

(MPa)

A 0 199 136 136 199 �3.5180 �143 �194 �144 280 �194 393 2.7

B 0 201 128 129 201 �6.9180 �120 �200 �120 249 �199 400 5.0

C 0 �23 �161 �24 �161 �10.1180 190 23 25 49 189 350 15.3

D 0 �106 �184 �117 �173 27.2180 188 110 124 241 174 347 �30.3

E 0 �113 �187 �113 �187 �5.6180 193 121 122 235 192 379 6.8

X. Wang, X. Zhang / International Journal of Fatigue 32 (2010) 434–442 437

are less than 10%. Thus the finite element model of the wheel con-structed is quite accurate.

The strain rosette B, C, D and E were pasted almost in the samecircumferential circle (Fig. 4). And it can be seen, in Table 2, thatthe rv’s in the 0� and 180� positions are quite close to each other,which means that the bolt holes and ventilation windows onlyhave minor influence on the stresses in those two positions, inwhich the stresses reach their extreme values.

It can be seen, in Table 3, that the shear stresses, sx1y1’s in the 0�and 180� positions are generally much smaller than the corre-sponding normal stresses, rx1’s and ry1’s. And the principal stres-ses are quite close to the corresponding normal stresses. Thestress states on those points are basically tensile normal stress inboth O1X1 and O1Y1 direction in the 0� position and compressivenormal stress in both O1X1 and O1Y1 direction in the 180� position.And the normal stress ranges in O1Y1 direction, Dry1’s, are largerthan the corresponding normal stress ranges in O1X1 direction,Drx1’s, which indicates that the fatigue crack may occur first inthe circumferential direction of the wheel, which is parallel tothe O1X1 direction. And most fatigue cracks did occur in the cir-cumferential direction in the dynamic cornering fatigue tests madeby Dongfeng Automotive Wheel Co., Ltd. Fig. 5 shows an exampleof the wheel after the test, in which the circumferential fatiguecracks were formed in the test. Thus the fatigue crack of the wheeldisc may be mainly caused by the stress–strain cycles in the O1Y1

direction or radial direction.

4. Normal modes analysis of the wheel and moment armcombination

The modal transient dynamic analysis module of the Optistructwill be used to simulate the dynamic cornering fatigue test of thewheel, which is based on the normal modes analysis of the wheel.Thus the normal modes analysis of the wheel is done first with thenormal modes analysis module of the Optistruct, in which the fi-nite element model is the same as the static model introducedabove, except for the loading condition. Table 4 shows the modal

Fig. 5. Fatigue cracks in the wheel formed in the dynamic cornering fatigue test.

Table 4Modal frequencies lower than 1000 Hz.

Order 1 2 3 4 5 6 7 8

Frequency,Hz

22.2 22.2 104.3 269.4 271.2 506.8 509.7 804.0

Fig. 7. The wheel is held fixed and the force of a constant magnitude Fo rotates.

438 X. Wang, X. Zhang / International Journal of Fatigue 32 (2010) 434–442

frequencies lower than 1000 Hz. And Fig. 6 shows the 1st and the5th order of the modes (eigenvectors).

5. Transient dynamic analysis of the dynamic cornering fatiguetest of the wheel

The wheel rotates and the force is constant in both magnitudeand direction in the dynamic cornering fatigue test (Fig. 1). Butin order to simplify the construction of the finite element model,the wheel is held fixed and the force of constant magnitude Fo ro-tates, which is applied at the tip of the moment arm, as shown inFig. 7. The rotating force can be resolved into the componentsalong OX and OY axis of the fixed reference frame OXYZ,

Fx ¼ Fo cos xt ð8ÞFy ¼ Fo sinxt ð9Þ

where Fx is the force component along OX axis; Fy is the force com-ponent along OY axis; x is the angular speed of the wheel in thetest; t is time.

The modal transient dynamic analysis module of the Optistructis used to simulate the dynamic cornering fatigue test of the wheel,in which the finite element model is the same as the static modelintroduced above, except for the loading condition. Two finite ele-ment models are constructed. Only Fy is applied at the tip of themoment arm in the Model 1, as shown in Fig. 8, and only Fx is ap-plied at the tip in the Model 2, as shown in Fig. 9. The stress andstrain at any point in the wheel structure are obtained by adding

Fig. 6. The first five orders of

up the stresses and strains at the same point, which are obtainedwith those two models, respectively. Thus a linear hypothesis ismade. And this hypothesis is reasonable because the finite elementmodel constructed is a linear one and the wheel structure basicallyremains linear in the test except for some very small areas of highstress concentration.

the modes (eigenvectors).

Fig. 8. Model 1 (only Fy is applied at the tip of the moment arm).

Table 5Results of the accelerated fatigue tests of the wheel.

Test moment(Nm)

Fatigue life or test time 104

cyclesFailure mode

2199 4.8 Circumferential crack in thedisc

2199 5.6 Circumferential crack in thedisc

2199 4.2 Crack at the bolt circle inthe disc

2199 4.3 Circumferential crack in thedisc

2199 4.0 No failure (suspended)2199 4.0 No failure (suspended)

X. Wang, X. Zhang / International Journal of Fatigue 32 (2010) 434–442 439

Dongfeng Automotive wheel Co., Ltd. found that the test timefor initiating a fatigue crack in the wheel was too long if a constantforce of 1562.4 N (refer to 2) was applied. Thus they did acceler-ated fatigue tests by applying a larger force of 2344.4 N to thewheels. The overall number of the wheels tested was six and Table5 shows the results. It can be seen, in the table, that there were fourwheels failed out of the six and the three wheels failed had the cir-cumferential fatigue cracks in their discs, as shown in Fig. 5. Thus itis decided to focus on studying the method for predicting the fati-gue life of the wheel for this failure mode. The simulation of theaccelerated test mentioned above is studied as an example. Thusthe constant force Fo is 2344.4 N in the formulas (8) and (9). Andthe angular speed x in the formulas should be the same as thatof the rotating table of the test rig (Fig. 1) in the dynamic corneringfatigue tests, which is about 429 rpm. So.

f ¼ 42960¼ 7:143 Hz

T ¼ 1f¼ 1

7:143¼ 0:14 s

x ¼ 2pf

where f is the frequency of the rotating table; T is the periodic timeof it.

The time interval selected is 0.007 s or one twentieth of T, in or-der to obtain sufficiently good calculation accuracy of the peak and

Fig. 9. Model 2 (only Fx is applied a

valley values of the transient responses. And 800 transient re-sponse points are calculated, which corresponds to the time lengthof 5.6 s. The modes whose modal frequencies are below 1000 Hzare used to calculate the transient responses. A constant dampingvalue of 0.02 is set for each mode in the above frequency range.Fig. 10 shows the transient displacement history of the center nodein the end surface of the moment arm in the OY direction (refer toFig. 7), which is obtained with the Model 1 shown in Fig. 8. And itcan be seen that the transient displacement basically reaches itssteady state after about 3 s. The similar situation occurs for thetransient displacement history of the same center node in the OXdirection, which is obtained with the Model 2 shown in Fig. 9.

The reason for choosing a constant damping value of 0.02 is thatthe ratio of the excitation frequency 7.143 Hz to the first ordermodal frequency 22.2 Hz is smaller than 0.75, with the damping ra-tios having little effect on the steady response amplitudes. In viewof the test system (Fig. 1), the damping ratios are small and a com-paratively small damping ratio 0.02 is chosen. The simulations withthe model show that the steady displacement amplitudes of themoment arm tip are 10.025 mm, 10.020 mm, 9.996 mm and9.955 mm, respectively, for the damping ratios of 0.02, 0.04, 0.08and 0.12, with the maximum relative error being 0.7%.

6. Fatigue life prediction of the wheel in the dynamic corneringfatigue test

It is decided to make use of the local strain approach [6–8] topredict the fatigue life of the wheel, whose basic equations arethe cyclic stress–strain curve,

t the tip of the moment arm).

440 X. Wang, X. Zhang / International Journal of Fatigue 32 (2010) 434–442

De2¼ Dr

2Eþ Dr

2K 0

� � 1n0

ð10Þ

and the strain–life equation,

De2¼

r0f � r0

Eð2Nf Þb þ e0f ð2Nf Þc ð11Þ

where E is cyclic modulus of elasticity; K0 is cyclic strength coeffi-cient; n0 is cyclic strain hardening exponent; r0f is fatigue strengthcoefficient; b is fatigue strength exponent; e0f is fatigue ductilitycoefficient; c is fatigue ductility exponent. Table 1 shows the valuesof these parameters of the wheel material (st37). De is the strainrange of the stress–strain hysteresis loop caused by the load historyin the material; Dr is the stress range of the loop; r0 is the meanstress of the loop. De, Dr and r0 for each stress–strain hysteresisloop in the material can be obtained by the local stress–strain re-sponse analysis [7].

Fig. 10. Transient displacement history of the center node in

Fig. 11. Critical eleme

The strains of the critical zones in the wheel structure should bedetermined to predict the wheel fatigue life. Fig. 11 shows thecritical element of the wheel, Element A, and other two elements,Element B and Element C, which are located along the OY axis.Fig. 12 shows the normal Y strain history of the Element A withthe amplitude larger than those of other elements. It should benoted that the Element A is the element No. 68225 having the max-imum von Mises stress 239 MPa, as shown in Fig. 3. So, the staticfinite element model is effective in searching the critical areas ofthe wheel tested.

The amplitudes of the normal Y strain histories of the ElementA, which are caused only by Fy and by both Fx and Fy, are1925.0 le and 1924.8 le, respectively. So the contribution of theforce Fx ¼ Fo cos xt along the OX axis to the amplitude of the resul-tant normal Y strain history only account for �0.010%. The similarsituations occur for the Element B and the Element C. Thus theamplitudes of the resultant normal Y strain histories of the ele-ments which are located along the OY axis can be accurately

the end surface of the moment arm in the OY direction.

nts of the wheel.

X. Wang, X. Zhang / International Journal of Fatigue 32 (2010) 434–442 441

obtained only with the Model 1 (Fig. 8), in which only the forceFy ¼ Fo sin xt is applied. So only the Model 1 is needed to calculatethe transient normal Y strain history of an element by rotating thewheel to make the element be located along the OY axis.

A special software based on the local strain approach [7] ismade and used to process the strain histories of the Element A, Ele-ment B and Element C, which are obtained with the Model 1(Fig. 8) and have the amplitudes of 1925 le, 564.40 le and332.00 le, respectively. And Fig. 13 shows the local stress–strainhysteresis loops formed. It indicates that the local stress versusstrain characteristic of the Element A is highly non-linear. Butthe modal transient dynamic analysis applied is based on linearhypothesis. Thus the finite element model used can not give anaccurate and realistic transient strain history of the Element A. Itis necessary to find an element along the OY axis, which is closeto the critical element (the Element A) and whose local stress–strain characteristic remains basically linear, in order for the Model1 (Fig. 8) be able to obtain accurate transient strain history of it.

It can be seen in Fig. 13 that the Element C is the element closestto the critical element (the Element A), in which the local stressversus strain characteristic keeps nearly linear. And Neuber’s ruleis applied to predict the local stress–strain response in the critical

Fig. 12. Normal Y strain histories of the Element A.

Fig. 13. Local stress–strain hysteresis loops formed in the Element A, Element B andElement C.

element (the Element A) from the transient normal Y strain historyof the Element C, which is obtained with the Model 1 (Fig. 8). TheNeuber’s rule applied is

K2f DrCDeC ¼ DrBDeB ð12Þ

where DrC and DeC are the local stress range and strain range of thehysteresis loop formed in the Element C; DrA and DeA are the localstress range and strain range of the corresponding hysteresis loop inthe Element A; Kf is the fatigue notch factor. The fatigue notch factorof the Element A relative to the Element C, Kf, is estimated with thefollowing formula:

Kf ¼AA

ACð13Þ

where AA and AC are the amplitudes of the normal Y strain historiesof the Element A and the Element C, respectively, which are ob-tained with the finite element model. Thus Kf = 1925/332 = 5.80.

The special software based on the local strain approach [7] isapplied to process the transient normal Y strain history of the Ele-ment C to predict the local stress–strain response in the Element Aand predict the fatigue life of it. The predicted fatigue life of theElement A is 38,271 cycles. As shown in Table 5, there are threewheels having the failure mode of circumferential crack in the discin the accelerated fatigue tests. The fatigue lives of these wheelsare 43,000, 48,000 and 56,000 cycles, respectively. The average lifeis 49,000 cycles. Thus the percent error of the predicted life relativeto this average test life is (38,271 � 49,000)/49,000 = �22%. So thislife prediction of the wheel in the dynamic cornering fatigue testshas the engineering significance, which indicates that the methodintroduced and the assumption that the circumferential crack inthe wheel disc is mainly caused by the radial normal strain historyare reasonable.

7. Conclusions

The linear transient dynamic finite element analysis is appliedhere to obtain the transient strain histories of the wheel structurein the dynamic cornering fatigue test. Thus accurate responsescan only be obtained for the elements whose local stress–straincharacteristics keep linear in the test. It is important to find the lin-ear element which is closest to the critical element. And Neuber’s

442 X. Wang, X. Zhang / International Journal of Fatigue 32 (2010) 434–442

rule can be applied to predict the local stress–strain response inthe critical element from the strain history of the linear element,based on which the local strain approach can be used to predictthe fatigue life of the critical element. And the assumption thatthe circumferential crack of the wheel is mainly caused by the ra-dial normal strain history is proved reasonable.

The amplitudes of the resultant normal Y strain (a form of radialnormal strain) histories of the elements which are located alongthe OY axis can be accurately obtained by only applying the dy-namic force component along the axis. So only the Model 1(Fig. 8) is needed to calculate the transient normal Y strain historyof an element by rotating the wheel to make the element be lo-cated along the OY axis.

It should be pointed out that this paper does not take the man-ufacturing effects into account, which may have significant effectson the wheel life.

References

[1] SAE handbook. Warrendale, PA: Society of Automotive Engineers, Inc.; 1982. p.30.05.

[2] Riesner M, I.’DeVries R. Finite element analysis and structural optimization ofvehicle wheels. SAE technical paper, No. 830133; 1983.

[3] Karandikar H, Fuchs W. Fatigue life prediction for wheels by simulation of therotating bending test. SAE technical paper, No. 900147; 1990.

[4] Shang R, Altenhof W, Hu H, Li N. Rotary fatigue analysis of forged magnesiumroad wheels. SAE technical paper, No. 2008-01-0211; 2008.

[5] Radaj D. Design and analysis of fatigue resistant welded structures. Abington,Cambridge: Woodhead Publishing Ltd.; 1990.

[6] Dowling N. A discussion of methods for estimating fatigue life. SAE technicalpaper, No. 820691; 1982.

[7] Reemsnyder H. Constant amplitude fatigue life assessment models. SAEtechnical paper, No. 820688; 1982.

[8] Socie D. Variable amplitude fatigue life estimation models. SAE technical paper,No. 820689; 1982.