Hindawi Publishing Corporation

Te Scientifc World Journal

Volume 2013, Article ID 261926, 11 pages
http://dx.doi.org/10.1155/2013/261926
Research Article
Explicit Nonlinear Finite Element Geometric Analysis of
Parabolic Leaf Springs under Various Loads
Y. S. Kong,
1,2
M. Z. Omar,
1,3
L. B. Chua,
2
and S. Abdullah
1,3
1
Department of Mechanical & Materials Engineering, Faculty of Engineering & Built Environment,
Universiti Kebangsaan Malaysia (UKM), 43600 Bangi, Selangor, Malaysia
2
APM Engineering & Research Sdn Bhd, Level 4, Bangunan B, Peremba Square, Saujana Resort, Seksyen U2, 40150 Shah Alam,
Selangor, Malaysia
3
Center for Automotive Research, Faculty of Engineering & Built Environment, Universiti Kebangsaan Malaysia (UKM),
43600 Bangi, Selangor, Malaysia
Correspondence should be addressed to M. Z. Omar; zaidi@eng.ukm.my
Received 30 July 2013; Accepted 15 September 2013
Academic Editors: J. Escolano, S. J. Rothberg, and B. F. Yousif
Copyright 2013 Y. S. Kong et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Tis study describes the efects of bounce, brake, and roll behavior of a bus toward its leaf spring suspension systems. Parabolic leaf
springs are designed based on vertical defection and stress; however, loads are practically derived from various modes especially
under harsh road drives or emergency braking. Parabolic leaf springs must sustain these loads without failing to ensure bus and
passenger safety. In this study, the explicit nonlinear dynamic fnite element (FE) method is implemented because of the complexity
of experimental testing Aseries of load cases; namely, vertical push, wind-up, and suspensionroll are introduced for the simulations.
Te vertical stifness of the parabolic leaf springs is related to the vehicle load-carrying capability, whereas the wind-up stifness
is associated with vehicle braking. Te roll stifness of the parabolic leaf springs is correlated with the vehicle roll stability. To
obtain a better bus performance, two new parabolic leaf spring designs are proposed and simulated. Te stress level during the
loadings is observed and compared with its design limit. Results indicate that the newly designed high vertical stifness parabolic
spring provides the bus a greater roll stability and a lower stress value compared with the original design. Bus safety and stability is
promoted, as well as the load carrying capability.
1. Introduction
Leaf springs are widely usedinthe automotive industry as pri-
mary components in suspension systems for heavy vehicles
because they possess advantages such as a simple structure,
excellent guiding efects, convenience in maintenance, low
cost, and prone to axle location. Leaf spring designs are
mainly based on simplifed equations as well as trial-and-
error methods. Simplifed equation models are limited to
three-link mechanism assumptions and linear beam theory.
According to beamdefection theory, the defection of a beam
is based on the dimensional, cross-sectional profle of the
current beam. Te thickness of the cross-sectional profle
of a parabolic leaf spring contributes to the stifness in the
vertical direction. Te higher vertical stifness of the leaf
spring provides vehicles additional load-carrying capabilities.
Leaf springs could be categorized into two types: multileaf
and parabolic leaf. From a geometric perspective, a parabolic
leaf spring has a constant width but decreasing thickness
from the center of its line of encasement in a parabolic
profle, whereas a multileaf spring maintains a constant
thickness along its length [1]. Parabolic springs are predicted
to perform more efciently compared with traditional multi-
leaf springs because the former is lightweight and has less
friction between steel leaves.
Leaf springs absorb and store energy and then release
it. Te characteristics of a spring suspension are chiefy
infuenced by the spring vertical stifness and the static
defection of the spring. Te ride frequency and the load-
carrying capabilities of the leaf spring vehicle are afected by
the vertical stifness of the installed leaf springs. Te vertical
stifness of a leaf spring is defned as the change in load per
2 Te Scientifc World Journal
unit defection in the vertical direction. Most leaf springs
are designed to operate with respect to the vertical loading
of the vehicle. However, leaf springs are practically loaded
not only by vertical forces but also by horizontal forces and
torques in the longitudinal directions. Te center of a spring
is elastically constrained against wind-up or rotation torque
along a longitudinal vertical plane because of its wind-up
stifness. Leaf spring wind-up usually occurs while the vehicle
brakes and accelerates. When a car suddenly starts or stops,
front-down or rear-down postures impose a rotational torque
on the spring, referred to as a wind-up torque [1]. In addition,
leaf springs also sustain torsional load where the moment
generated from the vertical lateral plane when the vehicle
rolls.
Several studies have been conducted on leaf spring
analysis such as defection and stress analysis by using the
fnite element method (FEM) [26]. Te vertical stifness and
stress analysis conducted is based on the vertical loading
of the leaf spring. Kong et al. performed a simulation of
leaf springs on the basis of vertical and longitudinal loading
[7]. Qin et al. published a research article on multi-leaf
spring and Hotchkiss suspension analysis [8]. Leaf spring
under varying load cases such as vertical push, wind-up,
roll, and cornering analysis was demonstrated in the analysis.
Te simulation results provided the vertical, wind-up, and
roll stifness of the leaf spring suspension system. Savaidis
et al. evaluated the severe braking conditions of the axle
[9]. Te mechanical stress-strain behavior of the leaf springs
was calculated by FEM analysis. Another elastic leaf spring
model was also developed for multi-body vehicle systems of
a sport utility vehicle to simulate the axle wind-up under
severe braking [10]. A nonlinear FE formulation based on
the foating frame of reference approach was introduced
with a full FE model of leaf springs with contact and
friction. When contact and friction are considered, nonlinear
model analysis is considered instead of linear analysis. For
nonlinear model analysis, various models such as gun control
system were optimized through Pareto optimal solution [11]
and electrohydrostatic actuator through signal compression
method [12]. Te nonlinear model is preferred to be solved
in dynamic scheme where static analysis could not encounter
the friction, material, and geometric nonlinearities.
Te most implemented algorithms in dynamic FE analy-
sis (FEA) are the implicit and the explicit schemes. In implicit
dynamic simulation, an extension of the Newmark method
known as -HHT is used as a default time integrator [13].
Mousseau et al. implemented the implicit dynamic schemes
to predict the handling performance of a vehicle [14]. Tis
approach is time efcient and yields reasonable results. How-
ever, the explicit dynamic method derived fromthe Newmark
scheme was also widely adopted in dynamic analysis [15,
16]. An explicit dynamic simulation for the stamping part
of automotive components was performed [17]. Te explicit
method shows stability of convergence during simulation.
Both the implicit and the explicit methods have their pros
and cons. Te explicit technique entails a lower cost; however,
given a slow case, the solutions are unstable. Given the same
condition, the implicit methodprovides more accurate results
[18]. Te simulation of the crimping process, which uses
Solver
Postprocessing
Preprocessing
Geometry, meshing,
material, properties, and
boundary conditions
Select appropriate
solving method
Result review
and graph plot
Figure 1: Typical FEA procedures by commercial sofware.
both the implicit and the explicit techniques, was conducted
by Kugener [19]. Te simulation results indicated that the
explicit method is superior to the implicit method especially
when numerous contacts are considered. Other than the
mentioned two schemes, it is worth mentioning that the new
developed approach semi-implicit fnite diference scheme is
implemented to analyze the second law of thermodynamics
of fuid [20].
Te design of a parabolic leaf spring in a bus presents a
challenge to engineers given very complex and limited con-
siderations. Road conditions and the driving behavior of the
drivers subject the leaf springs to varying loading conditions,
at times severely damaging the leaf springs. Currently, leaf
spring designs focus solely on the load-carrying capabilities
or relative vertical stifness. As mentioned in other previous
studies, the design of the leaf spring with vertical stifness
only is insufcient when catastrophic failures have the pos-
sibility of occurrence. However, the experimental methods
verifying the stress under those varying loading modes are
too costly and complex to perform. Tis paper aims to present
the analysis of the stress level of the parabolic leaf springs
under diferent loading conditions by computer-aided engi-
neering. Te failure modes of the leaf springs normally occur
under harsh braking or suspension rolling while striking
a pothole. Te braking condition of the bus is associated
with the leaf spring wind-up, whereas pothole striking is
related to the suspension roll. To promote bus safety under
such conditions, newly designed parabolic leaf springs are
evaluated in simulations for their performance. Te new
leaf spring designs are expected to provide enhanced roll
resistance, improved load-carrying capability, and reduced
occurrence of potential failure.
2. FE Explicit Model
Te standard simulation setup for any commercial FEA sof-
ware is shown in Figure 1. As seen in Figure 1, the simulation
can be divided into three categories: preprocessing, solving,
and postprocessing. First, computer-aided design models are
generated for FE meshing. In this study, a manual hexahedra
element mesh is applied for the stress analysis of the parabolic
springs. To obtain good simulation results, the quality of
the mesh is optimized by the element quality index. Te
materials and properties of the leaf springs and silencers have
also been assigned, and these details are shown in Table 1.
Te Scientifc World Journal 3
Table 1: Materials and properties of leaf springs and silencers.
Leaf springs Silencers
Modulus of elasticity, , GPa 210 4
Density, kg/m
3
7850 900
Poisson ration 0.3 0.3
Boundary conditions to simulate the degree of freedomof the
leaf springs under varying loading conditions difer.
FE procedures need to be well developed to perform a
complex FE nonlinear analysis. Selection of the appropriate
solving method is signifcant. A conditionally stable explicit
integration scheme derived from the Newmark scheme from
the RADIOSS solver has been introduced (RADIOSS is a
copyright of Altair Hyperworks, Altair Engineering Inc.).
In dynamic analysis, the equation of motion for discrete
structural models is expressed as follows:
+ + = , (1)
where , , and represent the mass, viscous damping, and
stifness matrices. , , and denote the displacement, veloc-
ity, and acceleration vectors, respectively. is the external
force vector. In the general Newmark method, the state vector
is computed as follows:

+1
=

+ (
1
2
)
2

+
2

+1
,

+1
=

+ [(1 ) +
+1
] ,
(2)
where and are the specifed coefcients that govern the
stability, accuracy, and numerical dissipation of the integra-
tion method [16]. A conditionally stable explicit integration
scheme can be derived from the Newmark scheme given the
following:

+1
=

+
1
2

2
(

+
+1
) ,

+1
=

+
1
2

.
(3)
Te explicit central diference integration scheme can
be derived from the relationships. Te central diference
scheme is used when explicit analysis is selected. Te time
step must be smaller than the critical time step to ensure
the stability of the solution. Newmark nonlinear analysis
efciently captures energy decay and exhibits a satisfactory
long-term performance afer being tested [15].
To reduce the dynamic efects, dynamic relaxation is
used in the explicit scheme. A diagonal damping matrix
proportional to the mass matrix is added to the dynamic
equation
[] =
2

[] , (4)
where is the relaxation value and is the period to be
damped. Tus, a viscous stress tensor is added to the stress
translation X
translation X, Z
Vertical load
Rotation Y
Rotation Y,
Rotation Y,
(a)
Tire patch
distance
Rotation Y
translation X
Rotation Y,
translation X
Rotation Y,
(b)
Figure 2: Boundary conditions and loads applied: (a) vertical push,
(b) wind-up.
tensor. Inanexplicit code, the applicationof the dashpot force
modifes the velocity equation without relaxation

+/2
=
/2
+

(5)
to velocity equation with relaxation

+/2
= (1 2)
/2
+ (1 )

, (6)
where
=

. (7)
When this option is activated, the running time of the
whole simulation is increased. However, the damping period
for the system is controlled within acceptable limits.
3. Contacts and Load Cases
Tree diferent parabolic leaf spring designs were analyzed in
this study. Each design was simulated with diferent loading
cases. Terefore, diferent simulation boundary condition
setups for the vertical push, wind-up, and roll suitable to the
load case were conducted accordingly. First, the boundary
conditions for the vertical push were performed with free
rotation around the -axis for the front eye, whereas the
rear eye was constrained in the , translation and the
, rotation. Te boundary conditions are complied with
[21]. Te center of the spring was allowed only in the -
translation and the rotation. Te vertical push boundary
condition setup is shown in Figure 2(a). For the wind-up load
case setup, the applied boundary conditions for the eye were
similar to the vertical push with free rotation around the
axis for the front eye, whereas the rear eye was constrained
4 Te Scientifc World Journal
Tire patch
Parabolic leaf spring
Simplifed
beam axle
Shackle
Antiroll bar
Shock eye leaf
Figure 3: Roll load case simulation model.
in the -translation and the -rotation. Afer maximum
vertical loading is applied, a longitudinal force was created
and applied at the center of the parabolic leaf springs [22].
Te wind-up establishment of the parabolic leaf springs is
illustrated in Figure 2(b).
For the suspension roll study, loads are applied to push
the suspension to a curb position. A moment is subsequently
applied to the suspension by increasing the vertical load on
the lef side and decreasing the load on the right side [8]. Te
leaf spring is expected to hit the jounce stopper afer a 40 mm
displacement is imposed. In this case, the load is applied at
the tire patchthat represents the contacts of tire to the ground.
Te boundary condition of the parabolic leaf spring can freely
rotate around the axis for the front eye, whereas the rear eye
is attached to the shackle, and the shackle can rotate in the -
axis only. Te front module of conventional buses considered
in this study employed an antiroll bar to enhance the roll
stifness of the vehicle. Te antiroll bar can be idealized as
a torsional stifener connected between the sprung and the
unsprung masses. When the roll bar undergoes a relative
rotation between the two masses, a restoring moment,

,
is generated, which is then related to its roll stifness

[23].
Te part of the antiroll bar that is connected to the vehicle
sprung mass is fxed in all degrees of freedom. Te total setup
of the suspension roll model is shown in Figure 3.
Some parabolic leaf springs are designed to endure
vertical load, whereas others are also designed to sustain
wind-up loads. Te vertical rate of the spring is calculated
based on the beam defection theory. Te formula for the
vertical rate for parabolic leaf springs is indicated [1] as
follows:
=

4
3

V
,
(8)
where is the spring material elastic modulus,

is the
thickness at center of the spring,

is the width at the center

of the spring, is the length of cantilever, and
V
is the
vertical rate factor. Besides that, lateral rate of the parabolic
leaf spring is also taken into design considerations. Te wind-
up stifness, is predicted through the vertical stifness of the
leaf spring as shown in equation [1] as follows:
=

2
4
. (9)
In geometric nonlinear analysis, components will
undergo large deformations. Te nonlinearities always come
from contact or materials. A general purpose contact is
introduced in Radioss which is FE commercial sofware. Te
interface stifness,

, is computed fromboth the masters,

,
and slaves segment,

. Te interface stifness relationship

between the master and slave is defned in equation

)
. (10)
Friction formulation is also being introduced in this
contact interface. Te most well-known friction law is the
Coulomb friction law. Tis formulation provides accurate
results with just one input parameter which is Coulomb
friction coefcient, [24].
4. Result and Discussions
Tree parabolic leaf spring designs were prepared and sim-
ulated for validation purpose. One of the front parabolic
leaf springs was obtained from the original bus model
as benchmark for the analysis. Te original parabolic leaf
spring was named as Baseline in the simulation case. Te
profle design of Baseline is shown in Figure 4(a). Te new
parabolic leaf spring designs are named as Iteration 1 and
Iteration 2, respectively where the designs are shown in
Figures 4(b) and 4(c), respectively. To obtain a proper spring
characteristic of the Baseline model parabolic leaf spring, an
experimental testing has been conducted. Te experimental
setup is shown in Figure 5 [25]. Avertical load is applied from
the centre of the leaf spring while the displacement at the
centre is measured. Te front and rear eye of the parabolic
spring are allowed to rotate in in lateral axis and translate in
longitudinal axis. Te gradient of the force versus defection
curve is the vertical stifness of the spring. Te simulation
result of Baseline model is compared to the experimental
result for correlation purpose as shown in Figure 6(a). From
Figure 6(a), the vertical stifness of the tested experimental
parabolic leaf spring is 311 N/mmwhile the simulation model
is 295 N/mm. It can be concluded that the simulation model
and experimental test have a 95% good correlation. Afer
that, vertical stifness of Iterations 1 and 2 parabolic leaf
springs is also plotted and compared to baseline model as
shown in Figure 6(b). As seen in Figure 6(b), parabolic leaf
spring of Iteration 1 has vertical stifness of 281 N/mm while
the parabolic leaf spring of Iteration 2 is 338 N/mm. Te
vertical stifness of the leaf springs plays important role
in determining the vehicle load-carrying capability. As the
vertical stifness of the leaf spring is higher, the load capacity
of the vehicle will also be greater. In order to examine the load
capabilities and stability of designed parabolic leaf springs
toward original design, the parabolic leaf springs in Iterations
1 and 2 should have diferent vertical stifnesses. Te parabolic
leaf spring in Iteration 1 has lower vertical stifness which
means lower load-carrying capability while the Iteration 2
has the greater vertical stifness compared to the original
parabolic leaf spring design (Baseline).
When a car suddenly starts or stops, front-down or rear-
down posture occurs, imposing a rotational torque or wind-
up torque on the leaf spring [22]. Leaf springs experience
Te Scientifc World Journal 5
S
p
r
i
n
g

c
e
n
t
r
e
Y
X
Z
C
e
n
t
r
e
C
e
n
t
r
e
C
e
n
t
r
e
1
8
.
9
1
8
.
1
1
7
.
3
1
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5
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.
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5
1
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6
1
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5
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4
1
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0
120 50 50 50 50 100 170 100 206
120 50 50 50 50 100 170 100 206
120 50 50 50 50 100 170 100 206
1
2
0
1
7
0
2
2
0
2
7
0
3
2
0
4
2
0
5
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0
(a)
(b)
(c)
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9
0
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6
1
2
0
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0
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0
2
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0
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0
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0
6
9
0
8
9
6

Figure 4: Taper profle of (a) Baseline, (b) Iteration 1, and (c) Iteration 2.
Figure 5: Experimental vertical stifness test for leaf springs.
longitudinal loading, in addition to vertical stifness, espe-
cially when the vehicle brakes or accelerates. Meanwhile,
wind-up analysis is performed in two stages. In the frst stage,
the spring is pushed to a vertical curb position; in the second
stage, a longitudinal load is applied on the leaf spring center.
Te situation is considerably more difcult in case of braking.
Te acting brake force yields an S- shaped deformation of
the leaf spring. Tis S deformation changes the kinematics
of the front axle system, resulting in unwelcome swerving of
the vehicle [9]. Such deformation is particularly undesirable
because the moment of the inertia of the axle around the
axis can lead to periodic deformations, where the axle
accepts a torque higher than the friction limit for a short time
and then slips when the inertial force disappears. Vibration
and loss of braking efciency or traction then occur [26].
Terefore, the deformation of the S shape during braking is
undesirable. To predict the wind-up stifness of the parabolic
leaf spring, af load is applied to the tire patch to obtain
the wind-up moment versus the angle curve, as shown in
Figure 7. In Figure 7, the wind-up stifness of the parabolic
leaf spring in the Baseline is 1.82 kNm/degree, whereas that
in Iteration 1 is 2.04 kNm/degree. Te wind-up stifness of
the parabolic leaf spring in Iteration 2 is 2.42 kNm/degree,
indicating that Iteration 2 has a higher wind-up stifness
compared with Iteration 1 and the Baseline. Tis result
suggests that S deformation is reduced under the same
braking condition.
For the suspension roll study, a 1.5 g gravitational force
is applied to the lef side, and the load on the right side
is decreased to 0.5 g of the gravitational force. In this case,
the same antiroll bar, axle, and linkages are implemented to
ensure consistency in the simulation. To determine suspen-
sionroll stifness, the roll angle of the suspensionis measured,
as shown in Figure 8. Te roll angle is measured based on
the rotation of the solid axle in the -axis connecting the lef
and the right parabolic leaf springs. Te roll moment versus
the roll angle curve for Baseline, Iteration 1, and Iteration
2 is plotted in Figure 9. Te curve depicts an almost linear
relation. Te roll stifness indicated by the gradient of the
roll moment versus the roll angle curve and generated by the
parabolic leaf spring in the Baseline is 4.46 kNm/degree. Te
roll stifness in Iteration 1 is 4.60 kNm/degree, whereas that
in Iteration 2 is 4.73 kNm/degree. On the basis of the roll
6 Te Scientifc World Journal
Experimental
Baseline
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0 20 40 60 80 100 120 140
Displacement (mm)
V
e
r
t
i
c
a
l

l
o
a
d

(
N
)
(a)
Baseline Iteration 2
Iteration 1
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0 20 40 60 80 100 120 140
Displacement (mm)
V
e
r
t
i
c
a
l

l
o
a
d

(
N
)
(b)
Figure 6: Graph of vertical stifness comparison: (a) Baseline and experimental, (b) Baseline, Iteration 1, and Iteration 2.
12
10
8
6
4
2
0
0 1 2 3 4 5 6
Baseline Iteration 2
Iteration 1
Wind-up angle (deg)
W
i
n
d
-
u
p

m
o
m
e
n
t

(
k
N

m
/
d
e
g
)
Figure 7: Comparison of wind-up moment versus wind-up angle curve.
Antiroll bar
Axle
Leaf springs

524797
Global
Angle = 8.696
524809
Z
Y X
Figure 8: Roll angle measurement of suspension system.
Te Scientifc World Journal 7
30
35
40
25
20
15
10
5
0
0 1 2 3 4 5 6 7 8 9
Baseline Iteration 2
Iteration 1
Roll angle (deg)
R
o
l
l

m
o
m
e
n
t

(
k
N

m
)
Figure 9: Graph of roll moment versus roll angle.
stifness, the parabolic leaf spring in Iteration 2 contributes
most to the roll stifness of the suspension system, followed
by Iteration 1 and then the Baseline. Te suspension roll
stifness is closely associated with the vehicle body roll. Under
the infuence of the lateral inertia force, the vehicle body
produces a roll angle about the roll axis, approximately
determined by
=

roll

, (11)
where

is the vehicle sprung mass,

roll
is the height of the
center of gravity of the vehicle body above the roll axis, and

is the total roll stifness of suspension and tires [27]. Accord-

ing to (11), the vehicle body roll is inversely proportional to
the suspension roll stifness, with the suspension roll stifness
defned as follows:

, (12)
where the

is the front suspension roll stifness,

is
the device roll stifness such as antiroll bar, and

is the
rear suspension roll stifness. Te suspension front, rear roll
stifness, and the contribution of the antiroll bar constitute
the amount of the vehicle body roll; thus, an increase in
any of them reduces the vehicle body roll. Te vehicle body
roll reduces the stabilizing moment because of insufcient
roll stifness, leading to vehicle instability. Terefore, the
parabolic leaf springs in Iteration 2 exhibit the highest
suspension roll stifness compared with those in Iteration 1
and the Baseline, thereby providing the vehicle the highest
roll stability.
External loads applied to a component, particularly
springs that undergo repeated cyclic loading, produce stress.
In real-life settings, stresses would not be uniaxial, biaxial,
and/or even multiaxial for most cases. Alternatively, an
equivalent stress can be calculated from multiaxial stresses.
Te von Mises stress is a widely known equivalent stress,
which is implemented for stress analysis of the leaf spring in
this study. Te stress levels of machine components are ofen
monitored and controlled within the limit of the material
that can sustain stress to prevent component failure. Te von
Mises stress contours of the Baseline, Iteration 1, and Iteration
2 of parabolic leaf springs under vertical and wind-up load
cases are illustrated in Figure 10. To improve the visualization
of stress analysis, a comparison of von Mises stress across the
length of the leaf spring for vertical push is plotted and shown
in Figure 11. Te von Mises stress of parabolic leaf springs
under wind-up loading is plotted in Figure 12. Te stress level
of each leaf in the Baseline, Iteration 1, and Iteration 2 can
be clearly visualized and compared. As shown in Figures 10
and 11, the overall von Mises stress level of the parabolic
leaf springs ranges from 500 MPa to 800 MPa at the region
200 mm to 400 mm away from the center of the spring. Te
highest von Mises stress level of the frst leaf until the fourth
leaf of the parabolic leaf spring in Iteration 1 ranges from
700 MPa to 800 MPa. Te stress level of the Baseline ranged
from 650 MPa to 750 MPa in the high-stress region. Iteration
2 exhibits the lowest von Mises stress from about 600 MPa
to 700 MPa, under the same load, followed by the Baseline;
however, the highest stress is shown by Iteration 1. For wind-
up analysis, the von Mises stress for the Baseline ranged
from 1000 MPa to 1200 MPa for all leaves of the parabolic
leaf spring. Te stress is evenly distributed during the wind-
up load case for Baseline. Under the same load, the von
Mises stress for Iteration 2 is also distributed from 1000 MPa
to 1200 MPa. Te stress level for Iteration 1 ranged from
1040 MPa to 1080 MPa. Te variationinstress level is typically
small when the Baseline is compared with Iteration 1. In the
wind-up cases, the parabolic leaf spring of Iteration 1 has a
narrower stress range and amplitude compared with those
of the Baseline and Iteration 2. Te entire stress distribution
can be afected by the design taper profle of the cantilever
of the parabolic spring itself. However, the entire simulation
model for Baseline, Iteration 1, and Iteration 2 remains within
acceptable limits with an even stress distribution. Iteration 2
contributes the highest value of wind-up stifness.
Figure 13 shows the von Mises stress contours of the
parabolic leaf springs when the roll load case is applied.
Te highest stress level is observed at the outer edge of the
parabolic leaf spring during suspension under roll loading.
Te stress levels of all parabolic leaf spring variants are then
plotted into a graph in Figure 14, which reveals that the main
leaf and leaf 4 of Iteration 1 obtain the maximum range
of the von Mises stress amplitude ranging from 1200 MPa
to 1450 MPa. Te remaining leaves ranged from 1000 MPa
to 1200 MPa. By comparing the von Mises stress of the
simulation, the level of stress of this roll loading approaches
the yield strength of the material, which is 1502 MPa [28].
Iteration 1 is found to possess a very low safety factor under
this condition. For the Baseline, the stress values of leaves
1 and 4 ranged from 1200 MPa to 1400 MPa, whereas those
of leaves 2 and 3 ranged from 1000 MPa to 1200 MPa. Te
8 Te Scientifc World Journal
Contour plot
Stress (vonMises)
Analysis system
7.930E + 02
7.049E + 02
6.168E + 02
5.287E + 02
4.406E + 02
3.525E + 02
2.644E + 02
1.763E + 02
8.818E + 01
7.380E 02
Y X
Z
(a)
Y X
Z
Contour plot
No result
Stress (vonMises)
Analysis system
7.987E + 02
7.100E + 02
6.212E + 02
5.325E + 02
4.437E + 02
3.550E + 02
2.662E + 02
1.775E + 02
8.875E + 01
0.000E + 00
(b)
Y X
Z
Contour plot
Stress (vonMises)
Analysis system
7.027E + 02
6.246E + 02
5.465E + 02
4.685E + 02
3.904E + 02
3.123E + 02
2.342E + 02
1.562E + 02
7.808E + 01
0.000E + 00
(c)
Y X
Z
Contour plot
Stress (vonMises)
Analysis system
1.206E + 03
1.072E + 03
9.379E + 02
8.039E + 02
6.700E + 02
5.360E + 02
4.020E + 02
2.680E + 02
1.340E + 02
3.285E 02
(d)
Y X
Z
Contour plot
Stress (vonMises)
Analysis system
1.087E + 03
9.667E + 02
8.458E + 02
7.250E + 02
6.042E + 02
4.833E + 02
3.625E + 02
2.417E + 02
1.209E + 02
2.570E 02
(e)
Y X
Z
Contour plot
Stress (vonMises)
Analysis system
9.836E + 02
8.743E + 02
7.650E + 02
6.557E + 02
5.465E + 02
4.372E + 02
3.279E + 02
2.186E + 02
1.094E + 02
8.877E 02
(f)
Figure 10: von Mises stress contour of parabolic leaf springs: (a) Baseline model vertical push, (b) Iteration 1 vertical push, (c) Iteration 2
vertical push, (d) Baseline model wind-up, (e) Iteration 1 wind-up, and (f) Iteration 2 wind-up.
900
800
700
600
500
400
300
200
100
0
1000 800 600 400 200 0 200 400 600 800 1000
Distance across length (mm)
v
o
n

M
i
s
e
s

s
t
r
e
s
s

(
M
P
a
)
Iteration 1: leaf 1
Iteration 1: leaf 2
Iteration 1: leaf 3
Iteration 1: leaf 4
Baseline: leaf 1
Baseline: leaf 2
Baseline: leaf 3
Baseline: leaf 4
Iteration 2: leaf 1
Iteration 2: leaf 2
Iteration 2: leaf 3
Iteration 2: leaf 4
Figure 11: von Mises stress across length plot of vertical push.
Te Scientifc World Journal 9
1040
1170
1300
910
780
650
520
390
260
130
0
1000 750 500 250 0 250 500 750 1000
Distance across length (mm)
v
o
n

M
i
s
e
s

s
t
r
e
s
s

(
M
P
a
)
Iteration 1: leaf 1
Iteration 1: leaf 2
Iteration 1: leaf 3
Iteration 1: leaf 4
Baseline: leaf 1
Baseline: leaf 2
Baseline: leaf 3
Baseline: leaf 4
Iteration 2: leaf 1
Iteration 2: leaf 2
Iteration 2: leaf 3
Iteration 2: leaf 4
Figure 12: von Mises stress across length of wind-up loading.
Contour plot
Stress (vonMises)
Analysis system
Y X
Z
s)
1.354E + 03
1.204E + 03
1.053E + 03
9.029E + 02
7.524E + 02
6.019E + 02
4.514E + 02
3.010E + 02
1.505E + 02
0.000E + 02
(a)
Contour plot
Stress (vonMises)
Analysis system
Y X
Z
1.420E + 03
1.262E + 03
1.104E + 03
9.466E + 02
7.888E + 02
6.311E + 02
4.733E + 02
3.155E + 02
1.578E + 02
0.000E + 00
(b)
Contour plot
Stress (vonMises)
Analysis system
Y X
Z
1.274E + 03
1.133E + 03
9.910E + 02
8.495E + 02
7.079E + 02
5.663E + 02
4.247E + 02
2.832E + 02
1.416E + 02
0.000E + 00
(c)
Figure 13: von Mises stress contour of parabolic leaf springs under roll load case: (a) Baseline model, (b) Iteration 1, and (c) Iteration 2.
stress ranges of the Baseline and Iteration 1 are almost the
same. Te design of the Baseline model has a lowsafety factor.
Finally, the stress contour of leaves 1 and 4 of Iteration 2 is also
plotted, with the stress amplitude ranging from 1100 MPa to
1300 MPa. Meanwhile, the stress levels of leaves 2 and 3 range
from about 900 MPa to 1100 MPa in the high-stress region.
A 100 MPa stress reduction is observed when Iteration 1 and
the Baseline are compared. Te safety factor of the parabolic
leaf spring of Iteration 2 is higher compared with those of
Iteration 1 and the Baseline in this case. Te parabolic leaf
spring in Iteration 2 has a lower probability of failure under
this load case compared with those of Iteration 1 and the
Baseline.
In a vertical load case, Iteration 2 exhibits higher vertical
stifness compared with both Iteration 1 and the Baseline. In
addition, Iteration 2 possesses a higher resistance to longi-
tudinal loading compared with Iteration 1 and the Baseline
during wind-up loading. Te roll stifness of Iteration 2 is also
slightly greater than those of Iteration 1 and the Baseline. Te
stress level of Iteration 2 is lower than those of Iteration 1 and
the Baseline even in the case of vertical and roll loads, as listed
in Table 2. Te parabolic leaf spring in Iteration 2 must be able
to successfully sustain the load for a longer period. However,
a low-stifness spring is favorable for the ride dynamics of
any ground vehicle, which is ofen a compromise with vehicle
handling. Te latter usually prefers a high-stifness spring.
To identify the most suitable parabolic leaf spring design,
many other factors should be considered, depending on the
application and user perception of the vehicle. Nevertheless,
the parabolic leaf spring inIteration2 withthe highest vertical
10 Te Scientifc World Journal
Table 2: Summary of vertical, wind-up, and roll stifness and stress for Baseline, Iteration 1, and 2.
Vertical stifness
(N/mm)
Wind-up stifness
(kNm/degree)
Roll stifness
(kNm/degree)
Maximum vertical
stress
(MPa)
Maximum
wind-up stress
(MPa)
Maximum roll
stress
(MPa)
Baseline 295 1.82 4.46 800 1198 1450
Iteration 1 281 2.04 4.60 750 1083 1400
Iteration 2 338 2.42 4.73 700 1198 1300
1000
1250
1500
750
500
250
0
1000 750 500 250 0 250 500 750 1000
Distance across length (mm)
v
o
n

M
i
s
e
s

s
t
r
e
s
s

(
M
P
a
)
Iteration 2: leaf 1
Iteration 2: leaf 2
Iteration 2: leaf 3
Iteration 2: leaf 4
Iteration 1: leaf 1
Iteration 1: leaf 2
Iteration 1: leaf 3
Iteration 1: leaf 4
Baseline: leaf 1
Baseline: leaf 2
Baseline: leaf 3
Baseline: leaf 4
Figure 14: von Mises stress across length plot of roll load case.
stifness is shown to be the most suitable based on the load
case simulation results.
5. Conclusions
An explicit dynamic nonlinear geometric scheme was
adopted to simulate the vertical push, wind-up, and roll load
cases of the parabolic leaf spring of a bus. An FE-based
procedure dealing with the evaluation and assessment of the
parabolic leaf spring of the bus was presented. Modeling
details for an accurate calculation of the spring are discussed.
Newparabolic leaf spring designs are included in the analysis
to obtain an improved bus load-carrying capability, brak-
ing resistance, and roll resistance, which were determined
through the analysis of vertical stifness, wind-up stifness,
and roll stifness. In addition to the vertical, wind-up, and
roll stifness provided by the parabolic leaf springs, the stress
level of the spring component itself is plotted and monitored
to ensure falling within the controlled limit. Hence, no
failures are expected when the new parabolic leaf spring
designs are implemented in the vehicle. In this analysis, the
designed parabolic leaf spring with higher vertical stifness
leads to higher wind-up and roll stifness. Te new parabolic
leaf spring design with the highest vertical stifness should
possess higher load-carrying capability, braking instability
resistance, and roll stability compared with the others. Te
stress level observed for the new leaf spring designs under
these circumstances is lower compared with the original
design. Te chances of failure are reduced, and vehicle safety
is enhanced under a braking or pothole strike condition.
Vehicle safety is increased because of the increase in suspen-
sion reliability.
Acknowledgments
Tis work is fnancially supported by Universiti Kebangsaan
Malaysia a.k.a Te National University of Malaysia under
research Grant code Industri-2012-037 and APMEngineer-
ing and Research Sdn Bhd.
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