Research Article
Advances in Mechanical Engineering
2017, Vol. 9(3) 1–11
Ó The Author(s) 2017
Study on the design method of equal DOI: 10.1177/1687814017692698
journals.sagepub.com/home/ade
strength rim based on stress and
fatigue analysis using finite element
method
Lei Chen1,2,3, Shunping Li2, Huiqin Chen3, David M Saylor2 and
Shuiguang Tong1
Abstract
Wheels are important safety components in the vehicle driving system. Automobile lightweight is the direction of the
modern automobile development. In this article, steel wheel lightweight was studied. The equal strength design of rim
was used to reduce the weight of the wheel. Stress analysis of the wheel was studied using UG/Nastran. A professional
software WheelStrength was used to predict the radial and cornering fatigue lives of the wheel. Sheet stamping process
was set up to analyze the interference fit between the disk and the rim. The assembly was simulated by axisymmetric
finite element method. After calculation and analysis, the stress distributions and fatigue lives for rim under different load
cases have been found. The thicknesses of wheel rim bead and the interface between rim and disk cannot be reduced.
Stress and fatigue simulation results were compared using different thicknesses of the optimized region. It was found that
the best thickness of the optimized region was 1.5 mm. Spinning was used to form the flared preform. The thickness of
the rim after spinning and rolling forming agreed well with the simulation. The results of fatigue tests indicated that the
lightweight wheel met the design requirement. The weight of the rim was reduced by about 14%.
Keywords
Equal strength, rim, fatigue analysis, finite element method
Date received: 23 October 2016; accepted: 16 January 2017
Academic Editor: Filippo Berto
Introduction finite element method was studied for many years.
Many papers have been published focusing on the
Wheel is a very important safety component of automo- simulation of the wheel fatigue test. Wang and Zhang5
bile. At present, there are two main types of wheels: studied the simulation of dynamic cornering fatigue test
steel wheels and aluminum alloy wheels. Steel wheels
still occupy a considerable market share due to its rela-
tively low price, high fatigue resistance, and high impact 1
School of Mechanical Engineering, Zhejiang University, Hangzhou, China
resistance. Fatigue life is the main performance of the 2
R&D Center, Zhejiang Jingu Co. Ltd., Hangzhou, China
wheel. There are two main kinds of fatigue testing 3
Institute of Mechanical Design, Hangzhou Dianzi University, Hangzhou,
methods: radial fatigue test and cornering fatigue test. China
It will waste a lot of cost and time using the test method
Corresponding author:
for wheel fatigue design. The numerical simulation Shuiguang Tong, School of Mechanical Engineering, Zhejiang University,
based on the finite element method is an effective tech- Hangzhou 310027, China.
nique in the design field.1–4 Fatigue prediction using Email: cetongsg@zju.edu.cn
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License
(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
2 Advances in Mechanical Engineering
Table 1. Materials parameters.
Item Value
E (GPa) 210
ss (MPa) 538
sb (MPa) 617
d (%) 22
v 0.3
Figure 1. The geometry of the 15 3 6 wheel. K (MPa) 940
n 0.16
of a steel passenger car wheel with Neuber’s rule and
local strain approach. Firat et al.6 investigated the simu-
lation of wheel radial fatigue tests based on the local
strain approach in conjunction with linear elastic finite
element stress analyses. Zheng et al.7 proposed a compu-
tational methodology to simulate wheel dynamic corner-
ing fatigue test based on the critical plane theory and
the finite element methods. Topac et al.8 predicted the
fatigue life of a heavy vehicle using the stress-life (S-N)
approach and finite element analysis. Meng et al.9 ana-
lyzed the radial fatigue of steel hub based on partitioned
seam weld model and a new pressure distribution regu-
lation. All of the studies have shown good agreement
with the experimental results of the fatigue life.
With the development of lightweight vehicle, light- Figure 2. Engineering stress–strain curves along 0° to the
weight steel wheel is also put on the agenda. Studies have rolling direction.
shown that the energy saving effect of the lightweight of
the wheel is 1.2–1.3 times as much as that of the other shapes of the disk and the rim have great effects on the
parts. Two methods have been used in the lightweight of strength and fatigue life. In this article, these factors
wheel: material and geometry optimization. The studies were not considered. Only the effects of the rim thick-
of material improvement have focused on the application ness on the fatigue life and strength were studied. The
of the high strength steel (HSS) sheet10 while the research thickness of the disk is fixed to 3.8 mm.
on geometry optimization of the wheel is little.
The steel wheel consists of two parts: the rim and the
disk. Due to different forces in different parts of the Materials parameters
wheel, the design of variable cross section is an impor- The rim is made of SPFH590 from POSCO. Table 1
tant method for the lightweight of the wheel. Roll form- displays the mechanical properties of these materials
ing is widely used in the fabrication of the rim. The from the factory. In Table 1, E is Young’s modulus, ss
thickness of the rim changes little after roll forming. In is the yield strength, sb is tensile strength, K is the
order to reduce the thickness, spinning (flow forming) strength coefficient, n is strain-hardening exponent,
is often used. The equal strength disk has been studied and v is Possion’s ratio. In order to verify these para-
in the heavy vehicle steel wheel using spinning11 while meters, the tensile specimens were made to do tensile
the study on equal strength rim is little. In this article, a test. The specimens were made along 0°, 45°, and 90°
typical wheel was used to study the optimization of the to the rolling direction. Figure 2 shows the stress–strain
equal strength rim. The strength and fatigue tests were curves along 0° to the rolling direction. Table 2 shows
simulated using finite element method. The simulation the measured materials parameters. From Table 2, we
results were compared with the experiment. can see that the yield strengths and tensile strengths are
smaller than those of Table 1. These should be consid-
Finite element model ered in the fatigue prediction.
Geometry
Figure 1 shows the geometry of the 15 3 6 J wheel. The
Finite element model for static analysis
steel wheel includes two parts: the disk and the rim. The The static finite element analysis of the wheel includes
disk is made using sheet metal stamping, and the rim is radial fatigue test and cornering fatigue test. The finite
made using sheet metal rolling forming. The fillets and element analysis software used here is UG 9.0. The
Chen et al. 3
Table 2. Measured materials parameters.
No. ss (MPa) sb (MPa) d (%)
00-1 503 589 24.5
00-2 514 597 25.5
00-3 508 592 24.5
45-1 511 578 29
45-2 513 581 30
45-3 507 575 29
90-1 540 611 24
90-2 534 612 24
90-3 523 611 23
Figure 4. Finite element model for static radial fatigue test.
Figure 3. Finite element model for static cornering fatigue test.
finite element solver is NX/Nastran. The rim and the
disk of the wheel are modeled with the 10 nodes tetra-
hedron elements. The length of the tetrahedron is set as
6 mm. The finite element model for static cornering Figure 5. Radial loading schematic.
fatigue test is shown in Figure 3. The moment arm is
modeled with the rigid body element 2 (RBE2). The
length of the arm is 1 m. The rim and the disk are fixed where Wr denotes the distributed pressure, W is a total
together. The force is transmitted through the arm. The radial load, b is total width, rb is the radius of the bead
force is 2400 N. The cornering fatigue design cycle is seat, u is the loading angle, and u0 is the angle at maxi-
set as 400,000. The real arm and bolt connections are mum load.
not modeled. Bolt holes were removed in order to sim-
plify the simulation. This can save a lot of computation
time and hard disk space. The inner edge of the rim is Fatigue model
fixed. The finite element model for static radial fatigue In the design of steel wheel, the fatigue prediction plays
test is shown in Figure 4. The tire dimension is an important role. Many researchers have studied the
195/65/R15. The tire pressure is 4.5 Bar. The applying fatigue prediction.5–9,13–17 Many softwares, such as
force on the wheel is 15,100 N. The radial fatigue Msc.Fatigue, Ncode, Ansys, and so on, have been
design cycle is 500,000. Contact pressure distribution developed. In this article, a professional software
between tire and wheel is shown in Figure 5. Contact WheelStrength developed by LBF of Germany was
pressure is calculated as follows12 used to predict the radial and cornering fatigue tests.
The finite element mesh is the same as the model set up
pu by UG9.0. First, the mesh model (*.dat) was imported
Wr = W0 cos ð1Þ
2 u0 into WheelStrength. Then, the boundary conditions
such as loading, fatigue cycle, and materials were
Wp
W0 = ð2Þ imposed by WheelStrength. The model *.bdf was
brb 4u0 exported using WheelStrength. Finally, the model
4 Advances in Mechanical Engineering
Figure 6. The finite element model for interference analysis: (a) model and (b) mesh.
(*.bdf) was computed with NX/Nastran. The finite ele- Finite element model for interference analysis
ment analysis result of WheelStrength is the required In the assembly of the disk and the rim, there is an
fatigue strength (RFS). If the computed RFS is smaller interference fit between the disk and the rim. The inter-
than the RFS of the material, it is qualified. Otherwise, ference value has large effect on the wheel. During the
the geometry or material of the wheel should be assembly of the disk and the rim, the disk is pressed
modified. into the rim by a punch. A stamping process was intro-
The S-N was used to predict the fatigue of the wheel. duced to analyze the interference fit. The assembly pro-
The cumulative damage calculation method is the cess was simulated by axisymmetric finite element. The
Miner hypothesis,18,19 which can be written as follows symmetry axis is X-axis. The thickness of the rim is
X ni 1 mm. The thickness of the disk is 3.8 mm. The element
=D ð3Þ size of the rim is set as 0.3 mm 3 0.3 mm. The element
Ni
size of the disk is set as 1 mm 3 1 mm. Full Newton–
s k Raphson iteration strategy was employed. Real stress–
D
Ni = ND if s.sD ð4Þ
s strain curve was used based on the experiment data.
s 2k1 Coulomb’s frictional law with a constant die friction
D
Ni = ND if s sD ð5Þ coefficient 0.1 was used in the simulation. Figure 6
s shows the finite element model and mesh for interfer-
where D is the damage sum, here D = 0.5, Ni is the ence analysis.
endurance, ND is the design number of cycles, ni is
the step frequency, k is the slope of the S-N line, s is Results and discussion
the stress amplitude, and sD is the knee point of the
S-N line (see Figure 1 of Schoenborn et al.18). Stress analysis with uniform thickness rim
The RFS of the wheel can be calculated by iteration In the design of steel wheel, the thickness of the rim
as follows and disk is the most important factor. With the increase
1=k in the thickness, the weight and cost of the wheel
Di increase and the fuel economy decreases. There are two
RFS = sD, i + 1 = sD, i ð6Þ
D most important factors including material type and the
sheet thickness in the lightweight design of vehicle. In
where i is the step number, sD,i + 1 is step (i + 1) knee this study, the material used is fixed. The thickness of
point of the S-N line. sD,i is step i knee point of the the rim was studied. First, the cornering and radial fati-
S-N line. Di is the damage sum of i step. First, the load gue tests were simulated using a 2-mm-thick rim.
cycle is divided into n1, n2, ni, ., a start RFS is given Figure 7 shows the simulation results of the cornering
such as 200 MPa. Then, Ni is computed by equation (4) fatigue with 2-mm-thick rim. It can be seen that the
or (5). D is computed by equation (3). If D is not equal largest Von Mises stress region is in the disk. The stress
to 0.5, a new RFS is computed using equation (6). The in the rim is not even. The Von Mises stress is larger in
iteration starts. When the D is equal to 0.5, the RFS is places 1, 3, and 4 than other places (see Figure 7(b)).
arrived. The thickness of the rim with the small stress can be
Chen et al. 5
Figure 7. Simulation Von Mises stress distributions of cornering fatigue with 2-mm-thick rim: (a) lateral view and (b) frontal view.
Figure 8. Simulation Von Mises distributions of radial fatigue: (a) lateral view and (b) frontal view.
reduced. Figure 8 shows the simulation results of the cornering fatigue test. It can be seen from Figure 9 that
radial fatigue with 2-mm-thick rim. It can be seen that the largest RFS (234.2 MPa) is in the disk, and this area
the largest stress region is in the rim, and as well the is the region that was suffered the largest stress. Owing
stress in the rim is not even. The stress is larger in places to the wheel rotation and homogeneous material, the
1, 2, 3, and 4 than other places. The largest stress of the fatigue results are circumferential symmetrical distribu-
radial fatigue test (see Figure 8) is smaller than that of tion. And in stress analysis (see Figures 7 and 8), the
the cornering fatigue test (see Figure 7). The Von Mises applying force is in one direction, so the stress distribu-
stress of the rim in radial and cornering fatigue test is tion is not symmetrical. Figure 10 shows the simulation
generally small. The largest stress is place 1. And place results of the radial fatigue test. It can be seen that the
5 is the symmetric region of place 1. The stress in place RFS values are not even in the rim and disk. The larg-
5 is small because no force is imposed on the cornering est value is in the wind hole. The RFS value in the
fatigue test. In reality, the tire air pressure is applied on radial test at the rim flange is larger because the force is
places 1 and 5. In the stress analysis, the tire air pres- applied directly in this region, while it is nearly zero in
sure is not considered. So the thickness of the rim can cornering fatigue test (see Figure 9) owing to not apply-
be reduced except places 1 and 5 (rim flange) through ing force and being fixed. The areas with the larger
the stress analysis. RFS agree well with the areas with larger stress.
Fatigue analysis Interference analysis
The fatigue analysis includes cornering and radial fati- Figure 11 shows the simulation process of the interfer-
gue test. Figure 9 shows the simulation RFS of the ence analysis. The base and punch contact the rim and
6 Advances in Mechanical Engineering
Figure 9. Simulation RFS results of the cornering fatigue test: (a) lateral view and (b) frontal view.
RFS: required fatigue strength.
Figure 10. Simulation RFS results of the radial fatigue test. (a) lateral view and (b) frontal view.
RFS: required fatigue strength.
disk first (see Figure 11(a)). Then, the base is fixed. The than 1.2 mm, the deformation of the rim is very large
punch moves and the disk is pressed into the rim until it and the shape of the rim is destroyed (see the part in
reaches the position (see Figure 11(b) and (c)). Finally, the rectangle of Figure 13(d)). So the interference fit
the punch moves reversely and the residual stress is should be smaller than 1 mm.
released.
Figure 12 shows the Von Mises stress distribution
with an interference fit of 0.7 mm. It can be seen from Optimal regions of rim
Figure 12 that the stress is concentrated at the interface According to the stress and fatigue analysis above, the
of the rim and disk, and the stress at the other areas is thickness of the rim can be reduced except the rim
nearly zero. So the interference fit has great effects on flange. In the interference fit analysis, the stress is
the interface. The interface of the rim and disk will be focused in the interface of the disk and rim. And these
welded after interference connection. So the thickness regions cannot be thinned as well. The optimal regions
of the part near the interface should not be reduced. of rim are shown in Figure 14. The regions of A, C,
Figure 13 shows the stress distribution with different and E of the rim as shown in Figure 14 are not thinned.
interference fits. The interference fits are 0.8, 1, 1.2, and The other two regions of B and D are thinned. In order
1.5 mm. The interference fit is changed through moving to study the effect of the thinning amount on the fati-
the disk along the Y-axis when fixing the rim. It can be gue and strength of the rim, different thicknesses of the
seen from Figure 13 that with the increase in the inter- regions B and D (1.75, 1.5, and 1.25 mm) were ana-
ference fit the deformation of the rim increases. When lyzed. The thickness of A, C, and E is 2 mm.
the interference is smaller than 0.8 mm, the stress of the The stress and RFS distributions are similar to the
rim is nearly zero. When the interference fit is larger distributions of the rim with 2 mm thickness, as shown
Chen et al. 7
Figure 11. Simulation process of the interference analysis: (a) 0 step, (b) 30 step, (c) 50 step, and (d) 55 step.
taken from the rim. The sampling distribution is shown
in Figure 15. Figure 16 shows the Von Mises distribu-
tions of the rim with different thinning amounts. It was
found that the stress was not even in the rim. There are
five peaks in the curves of both the cornering and radial
fatigue tests. The five peaks of the cornering fatigue test
are in the sample points of 2, 4, 10, 12, and 14. It is
nearly zero at points 17–20 because of not applying
force in simulation. Initially, the largest stress is at the
point of sample 2. With the decrease in the thickness of
the rim, the Von Mises stress increases and the largest
stress is at the point of sample 4. The five peaks of the
radial fatigue test are in the sample points of 2, 4–6, 10,
12, and 16–18.
Fatigue is the main property of the wheel. Figure 17
Figure 12. Von Mises stress distribution with an interference
shows the RFS distribution of the rim with different
fit of 0.7 mm.
thinning amounts. There are four peaks in the corner-
ing fatigue test. The four peaks are in the sample points
in Figures 7–10. In order to comparing the results of of 7, 10, 12, and 14. The maximum RFS is about
the stress and RFS distributions, the sample points were 100 MPa at point 12. There are five peaks in the radial
8 Advances in Mechanical Engineering
Figure 13. Von Mises stress distribution with different interference fits: (a) 0.8 mm, (b) 1 mm, (c) 1.2 mm, and (d) 1.5 mm.
Figure 15. Sampling distribution of the rim.
Figure 14. Optimal sections of rim.
fatigue test. The five peaks are in the sampling points Figure 18 shows the spinning process of the rim. The
2, 5, 12, 14, and 17. With the decrease in the thickness spinning die contains two parts. One part has a groove.
of the rim, the RFS value increases. In order to assess In the spinning process, the blank is first spun into the
the strength and fatigue, date mining and experimental groove and fixed. Other parts of the blank are spun
data are used. For the rim, the RFS can be written as according to the optimal thinning region of the rim.
follows19 Here, it is 1.5 mm, as shown in Figure 14. Figure 19(a)
0:436ss + 77 shows the photo of the spinning part. Figure 19(b)
RFSRim \ ð7Þ
1:65 shows the photo of rolling part after line cutting for
thickness measurement. Figure 20 shows the thickness
where RFSRim is the RFS of the rim and ss is the yield
distribution of the rim. The horizontal coordinate-axis
strength.
(section length) is the length of the cross-sectional
According to the data of Tables 1 and 2, the maxi-
length of the rim of Figure 19 (along the points of
mum allowable RFS for the rim is 179.6 MPa. The RFS
Figure 19(a) and the cross section of Figure 19(b)).
of the 1.25-mm rim is larger than 179.6 MPa, so the
After spinning, the thickness distribution of the rim was
1.5-mm-thick rim is better.
changed. The minimum thickness after spinning is
1.6 mm. The thickness of rolling part (finial shape) is
Experiments smaller than that of the spinning part. The minimum
The HSS rim is very hard to be formed. In this article, a thickness of the rolling part is 1.55 mm. The thickness
new flow forming die is designed to form the HSS rim. of the rolling part changed little.
Chen et al. 9
Figure 16. Von Mises stress distributions of the rim with different thinning amounts: (a) cornering fatigue test and (b) radial
fatigue test.
Figure 17. RFS distribution of the rim with different thinning amounts: (a) cornering fatigue test and (b) radial fatigue test.
Figure 19. The photo of the rim: (a) spinning part and
(b) rolling part.
Figure 18. The spinning process.
10 Advances in Mechanical Engineering
Table 3. Fatigue test results.
Wheel Radial test Cornering test
1 1,058,588 1,275,247
2 865,553 1,744,101
3 1,010,895 1,187,799
4 768,326 1,502,085
areas with the larger RFS agreed well with the areas
with larger stress.
A stamping process was applied to analyze the inter-
ference fit. The assembly process was simulated by axi-
symmetric finite element. With the increase in the
interference fit, the deformation of the rim increases.
Figure 20. The thickness distribution of the blank.
When the interference is smaller than 0.8 mm, the
deformation of the rim is nearly zero. When the inter-
ference is larger than 1.2 mm, the deformation of the
rim is very large and the shape of the rim is destroyed.
So the interference fit should be smaller than 1 mm.
The thicknesses at the rim bead and interface
between rim and disk cannot be reduced because the
stress and fatigue are larger. The optimal section of rim
was obtained. The comparisons of stress and fatigue
simulation results were carried out by different thinning
amounts. The best thickness of the optimized zone
1.5 mm was gotten.
Spinning was used to form the blank of the rim. The
thickness of the rim after spinning and rolling forming
agrees well with the design. The fatigue experimental
results showed that the wheels met the design require-
ment. The weight of the rim was reduced by about
14%.
Figure 21. The photo of the wheel after assembly.
Acknowledgements
Figure 21 shows the wheel after assembly. Before
application, the wheel was verified using fatigue test. The authors would like to thank Professor M. Ding for gui-
Table 3 shows the fatigue test results. It can be seen dance and the help of Dong Ping and Hao Bin during the
work.
from Table 3 that the fatigue cycles of the radial and
cornering tests were larger than the design. The parts
meet the design requirements. The rim after thinning is Declaration of conflicting interests
3.46 kg. The weight of the rim was reduced by about The author(s) declared no potential conflicts of interest with
14% (0.55 kg). respect to the research, authorship, and/or publication of this
article.
Conclusion Funding
In this work, we summarized the design method of The author(s) disclosed receipt of the following financial
equal strength rim based on stress and fatigue analysis support for the research, authorship, and/or publication of
using finite element method. On the basis of above- this article: The research is supported by the Zhejiang
reported results, the following statements can be Provincial Top Key Discipline of Mechanical Engineering
derived: (GK160203201003/010).
Stress analysis with equal thickness shows that the
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