Advances in Engineering Software 92 (2016) 57–64

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Advances in Engineering Software

journal homepage: www.elsevier.com/locate/advengsoft

Simulation of biaxial wheel test and fatigue life estimation considering

the influence of tire and wheel camber
Xiaofei Wan a,b, Yingchun Shan a,b, Xiandong Liu a,b,∗, Haixia Wang a,b, Jiegong Wang c
a
Department of Transportation Science and Engineering, Beihang University, Beijing 100191, China
b
Beijing Key Laboratory for High-efficient Power Transmission and System Control of New Energy Resource Vehicle, Beihang University, Beijing 100191, China
c
Shandong Xingmin Wheel Company, Yantai 265700, China

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

Article history: The traditional fatigue test of wheel comprising the radial and cornering fatigue tests cannot simulate the
Received 13 July 2015 real stress state of wheel well. Biaxial wheel fatigue test combining these two traditional tests has become
Revised 7 October 2015
an internationally recognized method that can reproduce the real loading condition of the wheel in service.
Accepted 8 November 2015
Since the test is time- and cost-consuming, developing the simulation method on biaxial wheel fatigue test
Available online 9 December 2015
is urgently necessary. In this paper, a new method is proposed to evaluate the fatigue life of commercial
Keywords: vehicle wheel, in which the finite element model of biaxial wheel fatigue test rig is established based on the
Biaxial wheel fatigue test standards of EUWA ES 3.23 and SAE J2562, and the simulation of biaxial wheel test and fatigue life estimation
Finite element model considering the effects of tire and wheel camber is performed by applying the whole load spectrum specified
Numerical simulation in ES 3.23 to the wheel. The radial and cornering fatigue tests are also simulated, and the results are compared
Wheel camber with ones of the biaxial fatigue test. The research shows that the proposed method provides an efficient tool
Damage analysis
for predicting the fatigue life of the wheel in the biaxial fatigue test.
Stress fatigue theory
© 2015 Elsevier Ltd. All rights reserved.

1. Introduction tive Engineers [5] promote the advancement of the wheel industry
and show the great significance of biaxial wheel fatigue test in devel-
As one of the most important safety components in vehicle, wheel opment cycle of “Design–Evaluate–Improve”. Studies on cornering
plays a significant role on driving safety, handing stability and riding fatigue test [6,7] and radial fatigue test [8–10] have been conducted
comfort. Since the working conditions are random and complex, it is extensively, mainly focusing on the life evaluation of wheel under
essential to guarantee the required durability of wheel during service a certain uniaxial load, which cannot reflect the real working life of
life. Currently, the fatigue life of wheel is tested by radial and corner- wheel directly. Few researches were made to investigate the Virtual
ing fatigue tests in the lab. Because of the independence of these two Experimental Method (VEM) of wheel biaxial fatigue test. Wang [11]
tests, the real stress state of wheel cannot be well reproduced. Biaxial applied finite element analysis to calculate the damage of a steel
wheel fatigue test evaluates the durability of wheel under combined wheel in the biaxial fatigue test, but the effects of wheel camber and
variable radial and lateral loads, which is demanded by many Original the contact property between wheel and driving drum of test bench
Equipment Manufacturers (OEMs) especially in European market in were not considered. Firat et al. [12–15] had performed a series of
recent years. Moreover, the complexity of tire-wheel assembly cyclic research on multi-axial fatigue damage and proposed a prediction
loading distribution and wheel road conditions are considered syn- method of wheel fatigue life based on local strain-life applied to
thetically in the test. Therefore, the fatigue safety of the wheel can be estimate the fatigue life of an aluminum wheel. However, their work
tested and evaluated comprehensively by the biaxial wheel fatigue only focused on fatigue damage of wheel under one certain load
test [1–3]. Since the test is time- and cost-consuming, the develop- condition, without considering the load sequence prescribed in the
ment of simulation method on biaxial wheel fatigue test is necessary standard ES 3.23. To the best of authors’ knowledge, the research on
and meaningful. practical and exact VEM of wheel biaxial fatigue test is limited.
The two standards: ES 3.23 issued by Association of European To bridge these gaps, a simulation method is proposed for eval-
Wheel Manufacturers [4] and SAE J2562 issued by Society of Automo- uation of biaxial fatigue life of commercial vehicle wheel based
on EUWA standard ES 3.23 in this paper. The proposed technique
∗
Corresponding author at: Department of Transportation Science and Engineering,
involves the effects of tire model and wheel camber. The camber
Beihang University, Beijing 100191, China. Tel.: +86 10 82338123. angles of wheel under biaxial loads are firstly calculated through
E-mail address: liuxiandong@buaa.edu.cn (X. Liu). camber analysis, which is addressed in the next section. In Section 3,

http://dx.doi.org/10.1016/j.advengsoft.2015.11.005
0965-9978/© 2015 Elsevier Ltd. All rights reserved.
58 X. Wan et al. / Advances in Engineering Software 92 (2016) 57–64

2.1. Camber analysis of the wheel under biaxial loads

The camber of the wheel assembly occurs when the radial and lat-
eral loads are simultaneously applied in the test. The direction of the
wheel camber depends on the direction of the lateral force as demon-
strated in Fig. 2. In the coordinate system of Fig. 2, negative lateral
load results in positive wheel camber (denoted as +θ ), while pos-
itive lateral force leads to negative wheel camber (denoted as −θ ).
When the wheel camber is positive, the bending moments acting on
spoke respectively generated by lateral force and radial force are in
the opposite direction. Conversely, the two bending moments are in
the same direction when the wheel camber is negative. This indicates
that wheel camber has a significant effect on stress magnitude and
distribution region on the wheel, which will be further discussed in
this paper. Therefore, the strength analysis and fatigue estimation of
the wheel under biaxial loads should involve the effect of the wheel
camber. However, the previous literatures [11–15] on the simulation
of wheel biaxial fatigue test did not consider this factor, which leads
to inevitable errors.
Fig. 1. Biaxial wheel fatigue test machine [16]. The simulation of biaxial wheel test using finite element method
can be performed nominally by directly applying the biaxial loads to
the wheel. However, because of the occurrence of wheel camber, the
simulation involves strong nonlinear problems including large dis-
the strength analysis of the wheel under biaxial loads is conducted. placement of the wheel, contact nonlinearity between the tire and
The fatigue life of the wheel is evaluated based on the stress life (S-N) drum, and tire material nonlinearity which may lead to divergence of
method in Section 4. The conclusions can be found in Section 5. The the simulation. On the other hand, the load program (Eurocycle) de-
proposed simulation method provides an efficient tool for predicting scribed in standard ES 3.23 only includes the lateral forces and radial
the fatigue life of the wheel in the biaxial fatigue test. forces, but not wheel camber angles generated automatically dur-
ing the test and determined by the lateral force, radial force, wheel
2. Camber status analysis of wheel under biaxial loads structure, loading structure and tire properties. To guarantee the ef-
ficiency and precision of the simulation, the analysis will be divided
The biaxial wheel test machine for commercial vehicle is illus- into two stages. The first stage is to calculate the camber angle for
trated in Fig. 1 [16]. It consists of the wheel assembly, driving drum, each load condition through camber analysis of the wheel and test
radial/lateral load actuators, kinematic links, main frame, servo mo- system based on the rigid model. Then the strength analysis is imple-
tor and so on. The drum is driven by the servo motor, and two inside mented based on the finite element model with wheel camber angles
curbs of the drum are designed to contact with the tire sidewall for obtained in first stage.
reaction of lateral load. The wheel assembly is installed on a dummy
vehicle axle inside the driving drum and is rotated by the drum 2.2. The calculation on camber angles of wheel under biaxial loads
while loaded by the actuators through kinematic links. The LBF load
program (Eurocycle) [3,4,16], including 98 loading events describing According to EUWA standard ES 3.23, a simplified but full scale
radial load, lateral load and wheel revolutions, running in a com- three-dimensional model of the biaxial wheel test machine for com-
puter is repeated until test termination as specified in the standard mercial vehicle is established using the software SolidWorks, which
ES 3.23. consists of a rotating drum, loading frame, flange, tire and wheel, as

Fig. 2. Wheel camber under biaxial loads: (a) positive wheel camber; (b) negative wheel camber.
 X. Wan et al. / Advances in Engineering Software 92 (2016) 57–64 59

Fig. 3. 3D model of the biaxial wheel test system: (a) whole model; (b) cut view of the model.

Table 1
Camber angles and damage analysis results of wheel under different biaxial loading conditions.

Load Radial Lateral Camber The maximum Load Radial Lateral Camber The maximum
sequence force (kN) force (kN) angle(°) damage of wheel sequence force (kN) force (kN) angle(°) damage of wheel

1 27 −9 7.1 2.37E−12 28 51 0 0 1.27E−09

2 30 −9 7.3 8.24E−12 29 51 12 −2.8 1.99E−08
3 30 9 −4.2 4.70E−10 30 54 −7.2 0 5.71E−10
4 30 −12 15 7.22E−12 31 54 7.2 −0.1 6.27E−09
5 36 0 0 6.12E−10 32 54 25.2 −6.2 1.94E−07
6 36 −12 9.5 7.60E−12 33 55.2 25.2 −6.2 2.07E−07
7 37.2 0 0 6.44E−10 34 60 18 −4.2 1.27E−07
8 39 0 0 6.50E−10 35 60 27.6 −6.2 3.50E−07
9 39 −9 5.1 1.58E−11 36 60 32.4 −7.2 5.51E−07
10 40.8 −9 4.4 1.80E−11 37 60 33 −7.3 5.82E−07
11 42 0 0 6.65E−09 38 60 33.6 −7.5 6.13E−07
12 42 −4.44 0 2.12E−10 39 60 36 −7.9 7.56E−07
13 42 −7.44 2.1 5.27E−10 40 63 28.8 −6.2 6.09E−07
14 42 −12 6.1 1.09E−09 41 64.8 0 0 6.66E−09
15 42 12 −3.9 9.88E−09 42 66 30 −6.2 1.18E−06
16 45 0 0 7.39E−10 43 66 33 −6.7 1.17E−06
17 45 12 −3.5 1.29E−08 44 69.6 38.4 −7.4 2.58E−06
18 45 3 0 9.65E−10 45 72 33 −6.2 1.78E−06
19 45 6 −0.1 1.75E−09 46 75 34.8 −6.2 2.50E−06
20 46.5 18 −5.8 6.67E−09 47 78 0 0 6.80E−09
21 48 0 0 9.46E−10 48 78 −14 2.4 1.74E−09
22 48 12 −3.2 1.64E−08 49 78 14.4 −1.6 2.49E−07
23 48 18 −5.6 5.39E−08 50 78 36 −6.2 3.05E−06
24 48 21.6 −6.2 8.04E−08 51 84 24 −3.9 1.15E−06
25 48 24 −6.7 1.08E−07 52 84 38.4 −6.2 5.79E−06
26 48 27 −7.5 1.97E−07 53 102 46.8 6.2 1.98E−05
27 48 9 −1.7 7.35E−09

demonstrated in Fig. 3. The solid model is imported into the finite Table 1. Due to some groups of loads are reduplicative, the camber
element analysis software ABAQUS for further analysis. For the calcu- angles of 53 different groups of load sequences are listed in Table 1.
lation of the camber angle through camber analysis, all parts of the
model are set to be rigid bodies to improve the convergence speed
of the simulation. The drum is completely fixed. The friction contact 3. Strength analysis of the wheel
interaction is defined between the tire and the inside surface of the
drum. The tire, wheel, loading frame and flange are all tied together 3.1. Finite element model
to constrain their relative motion.
The calculation process of the camber angle of wheel under bi- The finite element model (FEM) of a rotating drum and kinematic
axial loads contains two steps. At the first step, the lateral loader is link for loading in the biaxial wheel test machine is established (as
connected to the loading frame by the two-force bar and the radial shown in Fig. 4) for wheel strength analysis. The drum and load-
load is applied to the wheel. As shown in Fig. 3, the lateral loader is ing frame are assumed to be rigid bodies but other parts are set to
fixed on all six degrees of freedom (DOFs) at reference point 1 (RP-1, be elastic. The spoke and rim are modeled using improved tetrahe-
built at the endpoint of lateral loader), and the loading frame is free dron element with intermediate node (C3D10I). Because of the in-
on DOF of Z axis but constrained on other DOFs at RP-2 (built at mid- compressible property of the rubber material, the tire is discretized
point of radial loading frame). Radial load is applied on the loading using hexahedral-hybrid element with reduced integration (C3D8R).
frame at RP-2. The second step is to maintain the radial load and ap- The tire–wheel assembly tilts with a camber angle corresponding to
ply the lateral load to the wheel. In this step, the constraints on X each loading condition. In order to avoid the interference between
axis and rotational DOFs of the loading frame and the X axis DOF of the tire and the inside surface of the drum, an initial clearance needs
lateral loader are released. At this status, the wheel simultaneously to be set between the tire and drum. The constraint and interaction
undertakes the given radial and lateral loads. The simulation results conditions of the assembly are the same to that set for camber calcu-
of camber angles of wheel under various biaxial loads are shown in lation in Section 2.2.
60 X. Wan et al. / Advances in Engineering Software 92 (2016) 57–64

80
test data
70 curve fitting of test data
simulation data 1
60

Deflection of tire (mm)

simulation data 2
50 simulation data 3

40

30

20

10

0
0 10 20 30 40 50 60
Radial force (KN)

Fig. 6. Radial load-deflection curves of tire.

depict the tire. The form of the Mooney–Rivlin model [17,18] is:
    1 el
U = C10 I¯1 − 3 + C01 I¯2 − 3 + ( J − 1 )2 (1)
D1
Fig. 4. FEM of the biaxial wheel test system for wheel strength analysis.
where U is the strain energy per unit of reference volume; I¯1 and I¯2
are the first and second deviatoric strain invariants, respectively; C10 ,
Table 2
The material properties of wheel.
C01 and D1 are temperature-dependent material parameters; Jel is the
elastic volume ratio. Fig. 6 shows the relationship between deflection
Parts Material Thickness Young’s Poisson’s Ultimate tensile of tire and radial loads, which were obtained by using both the test
(mm) modulus (GPa) ratio strength (MPa)
and simulation methods, and the tire parameters used in the simu-
Spoke BG380L 12.5 198 0.3 398.7 lation are selected from the related literature [19]. Comparing with
Rim BG380L 6.25 189 0.3 413.5 the simulation data and the fitting curve of the test data, the simula-
tion data 1 meets the fitting curve well, and the parameters listed in
formula (1) are valued as C10 = 30, C01 = 0, D1 = 0.
420
400 3.3. Loading process
For strength analysis of the wheel, the 53 load sequences of Eu-
380 rocycle and the corresponding wheel camber angles obtained in
360 Section 2.2 are used to achieve the real working condition as much
Stress (MPa)

as possible.
340
The strength analysis of wheel under biaxial loads can be carried
320 out in three steps. In the first step, the loading frame is fixed at RP-
300 3 (the same location as RP-2) and the air pressure is applied to the
inside of wheel and tire. The air pressure is set to 0.9 MPa accord-
280 ing to the standard ES 3.23. The second step is to apply the radial
spoke force to the wheel. In this step, the Z axis DOF of the loading frame
260
rim is released and the radial force is applied at RP-3. Under the action of
240 the radial force, the wheel assembly and the loading frame move in
0 2 4 6 8
Strain -3 the radial direction until the tire contacts the drum and the contact
x 10
force between the tire and drum is equal to the radial force. In the last
Fig. 5. Hardening characteristics of spoke and rim material. step, the radial force is maintained at RP-3 and the lateral load is ap-
plied on the loading frame at RP-4. It is noted that the lateral loading
position (RP-4) and the contact location between the tire and drum
3.2. Material properties should be always on the same horizontal line in order to avoid extra
bending moment acting on wheel at the contact position. This indi-
The spoke and rim of wheel are both made of steel mate- cates that the location of RP-4 varies with the radial load, and can be
rial BG380L (Chinese Standard). The mechanical properties of steel determined based on the results in step 2.
BG380L with different thickness are obtained by static tensile test ac-
cording to the standard GB/T 228.1-2010 as shown in Table 2. Because 3.4. Stress analysis
plastic deformation of the spoke or rim may take place during the
test, the hardening characteristic of the material is taken into account Stress analysis of the wheel under biaxial loads is conducted for
in the simulation. The effective stress-strain curves of the material af- 53 groups of loading conditions [5]. For convenience, the simulation
ter tensile yielding are shown in Fig. 5. results of two loading conditions are presented and compared as ex-
Tire rubber is assumed to be incompressible elastic material, and amples. In order to demonstrate the effect of wheel camber on stress
compressive stress cannot be calculated with displacement fields. distribution, the two loading conditions which lead to positive and
The isotropic hyperelastic Mooney–Rivlin material model is used to negative wheel camber respectively are selected. The two loading
 X. Wan et al. / Advances in Engineering Software 92 (2016) 57–64 61

Fig. 7. Stress results of the spoke: (a) Negative wheel camber; (b) Positive wheel camber.

Fig. 8. Stress results of the rim: (a) Negative wheel camber; (b) Positive wheel camber.

Table 3
The stress value of dangerous point on three groups of biaxial loads.

Radial force Lateral force Camber angle Maximum stress in spoke Maximum stress in rim well Maximum stress in hump

78 KN 14.4 KN Negative (−) 264.6 MPa 310.1 MPa 212.1 MPa

78 KN −14.4 KN Positive (+) 181.8 MPa 271.7 MPa 287.9 MPa

conditions and the stress results of the wheel are shown in Figs. 7 by the function
and 8 and Table 2.
Sm N = C (2)
It can be observed from Fig. 7 that the wheel camber does not af-
fect the distribution of stress in the spoke, but the value of stress in where, S is the amplitude of the cyclic stress; N is the fatigue life; m
the spoke, and we may see that the dangerous areas of spoke mostly and C are the unknown parameters, and their value are determined
locate at the air ventilation hole for both loading conditions. The max- empirically by the ultimate tensile strength of material and the pre-
imum stress of the spoke with negative wheel camber angle is much vious fatigue test of the wheel. As the ultimate tensile strength (Su ) of
larger than that of the spoke with positive wheel camber angle, the spoke material is 398.7 MPa, the fatigue strength (Sf ) defined by the
values of which are 264.6 MPa and 181.8 MPa, respectively. However, stress amplitude when the fatigue life is infinite can be formulated
the stress value and distribution of the rim are both affected greatly empirically using the following formula [17]:
by the wheel camber as illustrated in Fig. 8. The maximum stress in S f = kSu (3)
the rim with negative camber angle is larger than that in the rim with
positive camber even if the equal lateral forces are applied. The high where k is a coefficient within the range of 0.3∼0.6. According to the
stress areas of the rim mostly locate on both the rim well and the results of radial fatigue test and cornering fatigue test on this type
hump. The stress on these two areas are approximately equal for the of wheel presented in the literatures [22,23], the coefficient k can be
case of negative wheel camber, while the stress in the hump are much determined as 0.52. Then the fatigue strength Sf can be obtained as
larger than that in the rim well for the case of positive wheel camber 207.3 MPa, and the fatigue life of spoke (N) under the fatigue strength
as shown in Table 3. Sf is approximately considered to be 107 . Based on the empirical for-
mula, the fatigue life (N) of spoke under 90% of the ultimate tensile
strength is about 103 cycles. Therefore, the S-N curve of the spoke can
4. Life estimate of wheel in biaxial fatigue test
be approximately calculated according to Eq. (2). The S-N curve of the
rim can also be obtained in the same way. The S-N curves are linear
In this section, the fatigue life of the wheel is estimated using
in logarithm relationship as shown in Fig. 9.
stress life (S-N) method [20,21] in the software Fe-safe. The damage of
the wheel under one biaxial load condition is firstly evaluated based
4.2. Cycle loads in wheel damage analysis
on the stress results obtained in Section 3. Then the total damage of
the wheel after going through one Eurocycle is calculated and accu-
As a rotating part of vehicle, the wheel is subjected to the cyclic
mulated using Miner method.
loads. To evaluate the damage of the wheel using the commercial
software Fe-safe, the cyclic loads acting on the whole circumference
4.1. S-N curve of the wheel of the wheel are required. However, in modeling wheel’s rolling pro-
cess, dynamic explicit approach generates ratcheting effect which
According to the fatigue theory, the fatigue life of the wheel can be greatly influences analysis precision [24]. As the wheel speed in fa-
evaluated using the stress fatigue theory when the maximum stress is tigue test is not high enough to excite the first order dynamic mode
less than the yield stress. The S-N curve can be commonly described of the wheel, it is reasonable to replace the dynamic process by static
62 X. Wan et al. / Advances in Engineering Software 92 (2016) 57–64

2.65
S-N curve of spoke
2.6 S-N curve of rim

2.55
Stress lg(σ)

2.5

2.45

2.4

2.35

2.3
3 4 5 6 7
Cycles lg(N)

Fig. 9. S-N curves of the spoke and rim.

Fig. 11. The damage of wheel under one loading condition (with negative wheel cam-
ber) for one cycle of load.
one in finite element analysis [22,23]. In the static simulation, the
wheel and drum are both fixed, the biaxial loads are applied to the
wheel separately on 10 locations over a span of 36° with uniform well and air ventilation hole. The results of damage analysis of all bi-
space in a circle to simulate the movement of the wheel, which means axial loads specified in Eurocycle are listed in Table 1. It can be found
that ten groups of simulations need to be performed in sequence for that the lateral force has a great influence on the durability of the
each biaxial loading condition. After each simulation, the wheel is ro- wheel. As the maximum stress of wheel with positive camber is much
tated by an angle of 36° and the biaxial loads maintain for the next less than that of wheel with negative camber, the fatigue life of the
simulation. Therefore, 10 groups of stress distributions are obtained wheel with positive camber is much longer than that of the wheel
for one loading condition to simulate the cyclic loads acting on the with negative camber.
wheel in one rotation period, which can be used for the damage anal-
ysis of the wheel. With the stress distributions for all biaxial loading 4.4. Damage analysis of the wheel under multi-groups of loads
conditions, the fatigue life of the wheel under biaxial loads can be
predicted. The fatigue life of the wheel going through all the biaxial loads
specified in Eurocycle [4] can be estimated based on the integrated
stress distribution obtained through the simulations under all these
4.3. Damage analysis of wheel under one biaxial loading condition
load sequences. Since the wheel revolutions for each loading condi-
tion in Eurocycle are different, Miner method [25] is used to calculate
The simulation results for one loading condition, where the ra-
the total damage of the wheel by accumulating the damage generated
dial and lateral forces are 78 kN and 14.4 kN respectively, are taken
under each biaxial load and specified revolution. Miner method can
as an example of damage analysis of the wheel. Under this loading
be expressed by the following formula:
condition, the wheel appears in negative camber. According to the
method presented above, ten groups of stress distribution data of the 
n
ni
wheel are imported into the software Fe-safe. The obtained logarith- D= (4)
Ni
i=1
mic fatigue life distribution of the wheel is shown in Fig. 10. It can
be seen that the low life areas of rim and spoke mostly locate at the where, D is the total damage of the wheel; i is the index of biaxial
rim well and the air ventilation hole, respectively. The wheel may fail load sequence; ni is the specified revolutions of the wheel under the
after 527,229 (105.722 ) revolutions under this loading condition, and ith biaxial loads; Ni is the fatigue life of wheel under the ith biaxial
the initial crack may emerge at the area of air ventilation hole. load. When D is accumulated to 1, the wheel fails due to fatigue.
Fig. 11 illustrates the damage (defined as the reciprocal of fatigue The total damage of the wheel after one Eurocycle is shown in
life) generated after acting one cycle of load on wheel. It also demon- Fig. 12. The dangerous areas of rim and spoke are concentrated on
strates that the fatigue failure most likely occurs at the area of rim the location of rim hump and air ventilation hole, respectively. The

Fig. 10. Fatigue life of wheel under one loading condition: (a) Fatigue life of rim; (b) Fatigue life of spoke.
 X. Wan et al. / Advances in Engineering Software 92 (2016) 57–64 63

The standard ES 3.23 requires that the wheel should be in full

functionality or only has allowable short technical cracks after run-
ning 16000 km in the biaxial fatigue test. According to the results of
the simulation, the above wheel cannot meet the durability perfor-
mance requirements.
To demonstrate the difference in fatigue life predicted by different
fatigue tests, the simulations of radial wheel fatigue test and corner-
ing fatigue tests are respectively performed, and the fatigue life dis-
tribution diagrams of the wheel obtained through these two uniaxial
fatigue test simulation are shown in Fig. 14. It can be obtained that
the durability performance of wheel in radial and cornering fatigue
tests can satisfy each requirement well according to EUWA standard
ES 3.11, and the fatigue life of wheel is much higher than that in bi-
axial fatigue test, meanwhile the wheel can pass the two fatigue test
freely on bench tests [22,23]. Obviously, the biaxial fatigue test has
provided a more rigorous but more practical standard for the devel-
opment of wheel.
As there is not yet the biaxial fatigue test machine for commercial
Fig. 12. The total damage of wheel after one Eurocycle.
vehicle wheels in China at present, to validate the FEM of biaxial fa-
tigue test is difficult. However, a great deal of calculation on uniaxial
maximum damage of the wheel locates at the air ventilation hole in- fatigue test of the wheel by using the similar method presented in
side of the spoke, with the value of 7.393 × 10−3 as shown in Fig. 13. the paper had been carried out, and the results validated by bench
According to the total damage after one Eurocycle of biaxial loads, test [22,23] had shown the feasibility of the approach used in the pa-
the maximum driving distance of wheel in biaxial fatigue test can be per.
calculated using the following formula: The improved design of the wheel is now in progress based on
the simulation results, and the sample of the wheel will be produced
π Dr
S≈ (5) and tested at LBF (in Germany) after the improved wheel passes the
Dmax virtual biaxial wheel fatigue test.
where, S is the maximum distance of wheel; D is outside diameter of
the tire, which is 1.05 m in the simulation; r is the total revolutions 5. Conclusions
for one Eurocycle, which is 15960; Dmax is the calculated maximum
damage for one Eurocycle. Therefore the test life of the above wheel In this paper, a method of the biaxial fatigue test of a wheel
under biaxial loads can be approximately calculated as S = 6980.6 km. according to EUWA ES 3.23 is proposed based on the FE-integrated

Fig. 13. The maximum damage of wheel after one Eurocycle.

Fig. 14. Fatigue life of wheel in fatigue test: (a) Radial fatigue test; (b) Cornering fatigue test.
64 X. Wan et al. / Advances in Engineering Software 92 (2016) 57–64

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