Analysis and optimization of a composite leaf spring

Analysis and optimization of a composite leaf spring
Mahmood M. Shokrieh * , Davood Rezaei
Composites Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology,
Narmak, Tehran 16844, Iran
Abstract
A four-leaf steel spring used in the rear suspension system of light vehicles is analyzed using ANSYS V5.4 software. The finite
element results showing stresses and deflections verified the existing analytical and experimental solutions. Using the results of the
steel leaf spring, a composite one made from fiberglass with epoxy resin is designed and optimized using ANSYS. Main conside-
ration is given to the optimization of the spring geometry. The objective was to obtain a spring with minimum weight that is capable
of carrying given static external forces without failure. The design constraints were stresses (Tsai–Wu failure criterion) and dis-
placements. The results showed that an optimum spring width decreases hyperbolically and the thickness increases linearly from the
spring eyes towards the axle seat. Compared to the steel spring, the optimized composite spring has stresses that are much lower, the
natural frequency is higher and the spring weight without eye units is nearly 80% lower.
Ó 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Leaf spring; Composite; Shape optimization; Finite element; Composite joints; Natural frequency; Suspension system; Composite beam
1. Introduction
Composite materials are now used extensively in the
automotive industry to take the place of metal parts.
Several papers were devoted to the application of com-
posite materials for automobiles. Some of these papers
are reviewed here, with emphasis on those papers that
involve composite leaf springs. Breadmore [1,2] studied
the application of composite structures for automobiles.
Moris [3] concentrated on using composites in the rear
suspension system. Daugherty [4] studied the applica-
tion of composite leaf spring in heavy trucks. Yu and
Kim [5] designed and optimized a double tapered beam
for automotive suspension leaf spring. Corvi [6] inves-
tigated a preliminary approach to composite beam de-
sign and used it for a composite leaf spring.
Springs are crucial suspension elements on cars,
necessary to minimize the vertical vibrations, impacts
and bumps due to road irregularities and create a
comfortable ride. A leaf spring, especially the longitu-
dinal type, is a reliable and persistent element in auto-
motive suspension systems. These springs are usually
formed by stacking leafs of steel, in progressively longer
lengths on top of each other, so that the spring is thick
in the middle to resist bending and thin at the ends
where it attaches to the body. A leaf spring should
support various kinds of external forces shown in Fig. 1,
but the most important task is to resist the variable
vertical forces.
Vertical vibrations and impacts are buffered by vari-
ations in the spring deflection so that the potential
energy is stored in spring as strain energy and then re-
leased slowly. So, increasing the energy storage ca-
pability of a leaf spring ensures a more compliant
suspension system. The amount of elastic energy that
can be stored by a leaf spring volume unit [6] is given by
Eq. (1).
S ¼ 1
2
r2
E ð1Þ
where r is the maximum allowable stress induced into
the spring and E is the modulus of elasticity, both in the
longitudinal direction. Considering that the dominant
loading on leaf spring is vertical force [7], the Eq. (1)
shows that a material with maximum strength and
minimum modulus of elasticity in the longitudinal di-
rection is the most suitable material for a leaf spring.
Fortunately, composites have these characteristics [8].
One of the most advantageous reasons for considering
composites instead of steel is their weight. Another im-
portant characteristics of composites which make them
excellent for leaf spring are: higher strength-to-weight
* Corresponding author. Fax: [phone].
E-mail address: [e-mail] (M.M. Shokrieh).
0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0263-8223(02)00349-5
Composite Structures 60 (2003) 317–325
www.elsevier.com/locate/compstruct

ratio (up to five times that of steel), no interleaf friction,
superior fatigue strength, ‘‘fail-safe’’ capabilities, excel-
lent corrosion resistance, smoother ride, higher natural
frequency, etc.
In the present work, a four-leaf steel spring used in
passenger cars is replaced with a composite spring made
of glass/epoxy composites. The main objective was the
shape optimization of the spring to give the minimum
weight.
2. Steel leaf spring
Parameters of the four-leaf steel spring used in this
work are shown in Table 1. This spring is unsymmetrical
so that the length of the front half is 559 mm and the
rear half is 686 mm. Every leaf is 50 mm wide and 7 mm
thick.
Experimental results from testing the steel leaf spring
under static and full bump loading containing the
stresses and deflections are listed in the Table 2.
Information in Table 2 is not sufficient to design a
composite leaf spring. So, a stress analysis was per-
formed using finite element method. All the calculations
were done using the version 5.4 of ANSYS [9]. In the
finite element modeling, every leaf was modeled with
eight-node 3D brick elements (SOLID 45) and then five-
node 3D contact elements (CONTACT 49) were used to
represent contact and sliding between adjacent surfaces
of leaves. An average coefficient of friction 0.03 was
taken between surfaces [7]. The axle seat of spring was
assumed to be fixed and loading was applied at the eyes
corresponding to the length of each half of spring.
A finite element stress analysis was performed under
static and full bump loading. Another analytical solu-
tion was carried out using the SAE standard design
formulas for leaf springs [7]. The results of experimental,
analytical and finite element methods are shown and
compared in Table 3.
Maximum normal stress rxx from finite element
analysis was compared to the experimental solution
under static and full bump loading and has 23% and 3%
error, respectively. There is a good correlation for
maximum deflection from all three methods. The max-
imum deflection given in Table 3 is related to the rear
half deflection. The front half of spring has a deflection
about 78 mm.
3. Composite leaf spring
Considering several types of vehicles that have leaf
springs and different loading on them, various kinds of
composite leaf spring have been developed. In some
designs the thickness and width of the spring are fixed
along the longitudinal axis [10]. In some types, the width
is kept fixed and thickness is variable along the spring
[11]. In other types width is fixed and in each section the
thickness is varying hyperbolically so that at two edges
the thickness is minimum and in the middle is maximum
[12]. Another design is presented by Yu and Kim [5] so
that the width and thickness are fixed from eyes to the
middle of spring and towards the axle seat the width
decreases hyperbolically and thickness increases linearly.
In their design the curvature of spring and fiber mis-
alignment in the width and thickness direction are
neglected. Therefore, in this study the simplified assump-
tions are removed and the spring is designed using a
more realistic situation.
3.1. Material selection
The material used directly affects the quantity of
storable energy in the leaf spring. The specific strain
energy can be written as Eq. (2).
Table 1
Parameters of steel leaf spring
Parameter Value
Total length 1245 mm
Front half (the arc length between
the axle seat and the front eye)
559 mm
Arc height at axle seat 120.4 mm
Spring rate 20.76 N/mm
Normal static loading 2500 N
Full bump loading 4660 N
Available space for spring width 50–60 mm
Spring weight 9.2 kg
Table 2
Results of experiments on the steel leaf spring
Static loading Load (N) 2500  115
Deflection (mm) 120.4
Stress (MPa) 483.3  22.2
Full bump Load (N) 4660  460
Deflection (mm) 209.3
Stress (MPa) 844.4  83.3
Fig. 1. Forces acting at the axle seat of a leaf spring. FV : vertical load,
Fs : side load, Ft : longitudinal load, Tt : twisting torque, Tw : windup
torque.
318 M.M. Shokrieh, D. Rezaei / Composite Structures 60 (2003) 317–325

S ¼ 1
2
r2
t
qE ð2Þ
where rt is the allowable stress, E is the modulus of
elasticity and q is the density. The specific strain energy
of steel spring and some composites are compared in
Fig. 2, when the ultimate static strength is used for rt .
The S2-Glass/epoxy value is set to 1 and other values are
expressed as their relative percentages to it. Regarding
the dynamic nature of loading on spring, the hatched
regions identify the quantity of specific strain energy in
dynamic loading when the fatigue strength is used for rt .
Considering the Fig. 2, in dynamic loading the HT-
carbon/epoxy is capable of storing the greatest amount
of energy. This material also has high strength and
stiffness and low weight. But on the other hand, it has
low impact strength and in the case of contact with a
metal, galvanic the corrosion would cause some prob-
lems. The very high cost of this material is another
drawback for practical use.
Compared to carbon fibers, glass fibers have lower
strength and stiffness, higher density, better corrosion
resistance, higher impact strength and lower cost. A
good combination between the material properties and
the cost is obtained with the glass fibers.
Glass fibers consist of two major types E and S2.
Although S2 fibers have better mechanical properties
than E fibers, but the cost of E fibers is much lower than
S2 fibers. So in the present work the E-glass/epoxy is
selected as the spring material. Mechanical properties of
this material are listed in Table 4. This material was
assumed to be linearly elastic and orthotropic.
3.2. Lay up selection
According to the Eq. (1), the stored energy in a leaf
spring varies directly with the square of maximum al-
lowable stress and inversely with the modulus of elas-
ticity both in the longitudinal direction. Composite
materials like the E-glass/epoxy in the direction of fibers
have good characteristics for storing strain energy. So,
the lay up is selected to be unidirectional along the
longitudinal direction of the spring. The unidirectional
lay up may weaken the spring at the mechanical joint
area and require strengthening the spring in this region.
3.3. Design and optimization
With the extensive use of laminated composite ma-
terials in all engineering fields, the optimal design of
laminated composites has been an extensive subject of
research in recent years. Some of the papers published in
this area are given in Refs. [13–16].
Since the composite leaf spring is a mono leaf, it is
necessary to optimize the shape of the spring. The de-
signer must make decisions relating to the selection of
optimum geometry. This requires a verification of a wide
variety of different complicated solutions. Therefore the
optimization is carried out with the finite element
method using ANSYS software.
3D brick element can be used for modeling the thick,
curved and orthotropic structures. 3D shell elements are
used for thin structures (the thickness should be one-
tenth or less than the width and the length). In the
middle of composite spring the thickness will increase
(about a half of the spring width) to resist the maximum
bending moment applied in this area. Compared to the
brick elements, using the shell elements need greater
number of elements to represent an exact model of
spring and consequently the computing time increases.
Considering the unidirectional lay up of composite leaf
spring, 3D eight-node brick element (SOLID 45) was
selected to develop a finite element model of the spring.
Table 3
Stress analysis of leaf spring using experimental, analytical and finite element methods
Loading (kg) Analytical solution Experimental method Finite element method
d (mm) r (MPa) d (mm) r (MPa) d (mm) r (MPa)
250  11.5 120.4 471.22 120.4 483.3  22.2 127.8 578.6
466  46 209.3 878.90 209.3 844.4  83.3 209.7 905
Fig. 2. The specific strain energies of the spring materials [5].
Table 4
Mechanical properties of the E-glass/epoxy
(E-glass/epoxy)
Exx (GPa) 27.7 XT (MPa) 589.8
Eyy (GPa) 8.4 XC (MPa) 450
G xy (GPa) 2.3 YT (MPa) 48.68
mxy 0.237 YC (MPa) 120.7
q (kg/m3 ) 1608 Sxy (MPa) 43.54
M.M. Shokrieh, D. Rezaei / Composite Structures 60 (2003) 317–325 319