Analysis and optimization of a composite leaf spring

This article from Desklib discusses the analysis and optimization of a composite leaf spring, including design constraints, stresses, and displacements. The article compares the results of a steel leaf spring to a composite one made from fiberglass with epoxy resin, and shows that the optimized composite spring has much lower stresses and a higher natural frequency. The article also discusses the advantages of using composite materials in leaf springs.

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
element results showing stresses and deflections verified the existing analytical and experimental solutions. Using the res
steel leaf spring, a composite one made from fiberglass with epoxy resin is designed and optimized using ANSYS. Main con
ration is given to the optimization of the spring geometry. The objective was to obtain a spring with minimum weight that
of carrying given static externalforces 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 linear
spring eyes towards the axle seat. Compared to the steel spring, the optimized composite spring has stresses that are mu
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
1. Introduction
Composite materials are now used extensively in the
automotive industry to take the place ofmetalparts.
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 elementson cars,
necessary to minimize the verticalvibrations,impacts
and bumps due to road irregularitiesand createa
comfortable ride.A leaf spring,especially the longitu-
dinaltype,is a reliable and persistent element in auto-
motive suspension systems.These springsare 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 resistbending and thin atthe ends
where it attachesto the body. A leaf spring should
support various kinds of external forces shown in Fig. 1,
but the mostimportanttask is to resistthe variable
vertical forces.
Vertical vibrations and impacts are buffered by vari-
ations in the spring deflection so thatthe potential
energy is stored in spring as strain energy and then re-
leased slowly.So, increasing theenergy storageca-
pability of a leaf spring ensuresa more compliant
suspension system.The amountof elastic energy that
can be stored by a leaf spring volume unit [6] is given by
Eq. (1).
S ¼1
2
r 2
E ð1Þ
where r is the maximum allowable stress induced into
the spring and E is the modulus of elasticity, both in the
longitudinaldirection.Considering thatthe dominant
loading on leaf spring is verticalforce [7],the Eq. (1)
shows that a material with maximum strength and
minimum modulus ofelasticity in the longitudinaldi-
rection is the mostsuitable materialfor 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
excellentfor 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 steelspring 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 steelspring 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 containingthe
stresses and deflections are listed in the Table 2.
Information in Table 2 is not sufficient to design a
composite leafspring.So, a stressanalysiswas per-
formed using finite element method. All the calculations
were done using the version 5.4 of ANSYS [9].In the
finite elementmodeling,every leafwas 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 coefficientof 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 fullbump loading.Another analyticalsolu-
tion was carried outusing the SAE standard design
formulas for leaf springs [7]. The results of experimental
analyticaland finite elementmethodsare shown and
compared in Table 3.
Maximum normal stressr xx from finite element
analysiswas compared to theexperimentalsolution
under static and full bump loading and has 23% and 3%
error, respectively.There is a good correlation for
maximum deflection from allthree 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 severaltypes ofvehicles thathave leaf
springs and different loading on them,various kinds of
composite leafspring 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 ofspring and towards the axle seatthe width
decreases hyperbolically and thickness increases linearly
In their design the curvature ofspring and fiber mis-
alignmentin the width and thicknessdirection 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 materialused directly affectsthe quantity of
storable energy in the leafspring.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 width50–60 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: longitudinalload, Tt: twisting torque,Tw: windup
torque.
318 M.M. Shokrieh, D. Rezaei / Composite Structures 60 (2003) 317–325

S ¼1
2
r 2
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 steelspring 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 ofloading 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 materialalso 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 costof this materialis another
drawback for practical use.
Compared to carbon fibers,glass fibers have lower
strength and stiffness,higher density,better corrosion
resistance,higherimpactstrength and lowercost. A
good combination between the materialproperties and
the cost is obtained with the glass fibers.
Glass fibers consistof two major types E and S2.
Although S2 fibers have better mechanicalproperties
than E fibers, but the cost of E fibers is much lower than
S2 fibers.So in the presentwork the E-glass/epoxy is
selected as the spring material. Mechanical properties of
this materialare listed in Table 4.This materialwas
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 longitudinaldirection.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 unidirectionalalong the
longitudinaldirection of the spring.The unidirectional
lay up may weaken the spring at the mechanicaljoint
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 optimaldesign 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 lessthan the width and the length).In the
middle of composite spring the thickness willincrease
(about a half of the spring width) to resist the maximum
bending moment applied in this area. Compared to the
brick elements,using the shellelementsneed greater
numberof elementsto representan exactmodel 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
Gxy (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

3.3.1. Design variables, constraints and objective function
Design variables (DVÕs)are independentquantities
that are varied in order to achieve the optimum design.
For the leaf spring, the DVÕs must identify the complete
geometry.Spring length and curvature are as the same
as the steelspring,so the width and thickness at each
cross section along the spring are obvious candidates as
the DVÕs (Fig.3). On the other hand,the cross-section
area must be equalalong the spring to ensure that the
fiber volume fraction and consequently the mechanical
properties of the composites are constant over the length
of spring. So to reduce the number of design variables,
the cross-section area and the width at each point are
selected asDVs. Dividing the constantcross-section
area by the width at each point derives the thickness. If
Wi is the width of ith cross section and A is the cross-
section area, the Eq. (3) represents the vector of design
variables.
X ¼ ½W1; W2; . . . ; Wi ; . . . ; Wn; A ð3Þ
Constraints are dependent variables and functions of the
DVÕs that constrain the design.Stresses and deflections
are limited in the spring and mustbe considered as
constraints.The stress constraint is an inequality con-
straint to guarantee that the maximum applied stress in
the spring is not greater than the materialstrength.In
this work the Tsai–Wu [8] theory was selected as failure
criterion to evaluate the stress constraint because it is a
simple, versatile, analytical criterion and includes inter-
action among the stress components (Eq. (4)).
FxxR2r 2
x þ 2FxyR2r xr y þ FyyR2r 2
y þ FssR2r 2
s þ FxRrx
þ FyRry 6 1 ð4Þ
where the rÕs are stress components, the F Õs are strength
parameters and R is the safety factor.Failure occurs if
the magnitude of the left-hand side of Eq. (4) is greater
than one.
The constrainton the deflection isan inequality
constraint that ensures the spring under a given load has
a specified maximum deflection (Eq. (5)).
d ¼ Const: ð5Þ
Regarding the very closed feasible region of deflection
and numericalerrors in the finite elementmethods,
using Eq.(5) in the optimization process causes some
problems in converging and may notbe satisfied.To
solve this problem, two constraints dl, du can be defined
as illustrated below:
dl P d e ð6Þ
du 6 d þ e ð7Þ
where e is a very small value. The effective feasible regio
is now 2e, but each constraint has a wide enough range
for smooth converging.
Another constraintrefers to the variation ofwidth
along the spring so that the difference between any two
sequent widths (from spring end to the axle seat) must
be a positive value (Eq. (8)). This constraint is necessary
to prevent converging to an impractical and undesirable
local minimum value of the width.
Wiþ1 W i P 0 i ¼ 1; 2; . . . ; n 1 ð8Þ
Objective function is the dependent variable that must
be minimized.One of the mostadvantageous reasons
for considering the composite leafspring over steelis
their very low weight. At the present work the weight of
spring is selected as an objective function.
Our optimization problem can be expressed as fol-
lows:
Minimize
W ¼ W ðW1; W2; . . . ; Wn; AÞ
Subject to:
FxxR2r 2
x þ 2FxyR2r xr y þ FyyR2r 2
y þ FssR2r 2
s
þ FxRrx þ FyRry 6 1
dl P d e
du 6 d þ e
Wiþ1 W i P 0 i ¼ 1; 2; . . . ; n 1
W 6 Wi 6 W i ¼ 1; 2; . . . ; n
3.3.2. Modeling and optimization
According to the SAE standard [7] a leaf spring can
be considered as two cantilevers. Therefore,in the pre-
sent work the rear half of spring was optimized first as a
cantilever beam and then the front half was optimized
based on the results obtained from the rear half.The
rear half was divided into 18 segments.Width and
thickness at each segment were defined parametrically.
Verticalstatic loading on the spring is 2500 N,so the
corresponding reaction on the rearhalf eye is about
1123 N. Design variables,constraintand objective
function were introduced in the software.The ANSYS
5.4 has several techniques for optimization. In this work
the first-order method [17]was selected.This method
Fig. 3. Choosing DVs for the leaf spring.
320 M.M. Shokrieh, D. Rezaei / Composite Structures 60 (2003) 317–325

usesderivative information,that is, gradientsof the
dependent variables with respect to the design variables.
It is highly accurate and works well for problems having
dependent variables that vary widely over a large range
of design space.
The optimization process for the rear half converged
after about 100 iterations. Fig. 4 shows the variation of
objective function (weight) with respect to the iteration
number. The minimum obtained weight for the rear half
was 888.1 g.
Variation of width and thickness along the spring are
shown in Fig. 5. It can be seen that the width increases
hyperbolically and the thicknessdecreaseshyperboli-
cally near the axle seat (about 120 mm along the spring)
and then linearly towards the spring eye.
As shown in Fig. 5, at spring end the width and
thickness are kept constant. This has been done to avoid
the problems that may occur in joining the eyes.
In order to ensure that the optimization process
converged to the global optimum point, the spring was
optimized using severalstart points.Table 5 shows the
first selected start point and three different start points
with their corresponding weight and cross-section area
after optimization.It can be seen thatthe maximum
difference between the obtained weights is about 8 g or
0.09% and between the cross sections is about 5.67 mm2
or 0.65%.Therefore,there is confidence thatthe con-
vergence to the global optimum point is achieved.
Having the cross-section area and the width at axle
seatand spring end from the optimized rear half,the
front half was optimized in the same way.The results
showed that the variation of width and thickness is the
same as the rear half. The minimum weight of front half
was obtained about 719 g.Therefore the weight of full
composite spring without considering eye units is nearly
1607 g, which is 80% lower that the steel spring (9.2 kg).
3.3.3. Stress analysis results
The leaf spring is analyzed under bending loading
condition and the normalstresses are important,but
because of the anisotropic properties of composites, the
other components of the stress tensor must be studied.
The longitudinalcompression strength ofcomposite
used in thiswork is lessthan its longitudinaltensile
strength,so failure always occurs atthe lower (com-
pression)surfaceof spring. Therefore,in the stress
analysis this surface is taken into more consideration.
Fig. 6 shows the longitudinalcompression stress at the
middleof lower surface.Compression stresshas its
maximum value about 217 MPa near the axle seat and
decreases towards the spring eye. At a same loading the
maximum stress applied in the steel leaf spring is about
0.850
0.900
0.950
1.000
0 10 20 30 40 50 60 70 80 90 100
Iteration Number
Weight (kg)
Fig. 4. Variation of spring weight during the optimization loops.
40
45
50
55
60
65
0 100 200 300 400 500 600 700
Spring Length (mm) Spring Length (mm)
Width (mm)
14
16
18
20
22
24
0 100 200 300 400 500 600 700
Thickness (mm)
Fig. 5. Variations of width and thickness along the optimized spring.
Table 5
Weight and cross-section area obtained from different start points
Start point I II III IV
Weight (g) 888.1 890.1 895.98 890.86
A (mm2) 873.29 875.28 878.96 873.94
0
50
100
150
200
250
0 100 200 300 400 500 600 700
Spring Length (mm)
Stress (M Pa)
Fig. 6. Longitudinalstress rx at lower surface ofcomposite spring
under static vertical loading.
M.M. Shokrieh, D. Rezaei / Composite Structures 60 (2003) 317–325 321

505 MPa from Table 2.The compression strength of
fiber glass/epoxy is 450 MPa and the yielding stress of
the steel spring is 1175 MPa. So the safety factor in steel
spring is 2.3 while in the composite spring it is 2.1. Due
to conservative nature of Tsai–Wu failure criterion the
actualfailure load may be greater than the value cal-
culated by this method.The deflection of spring under
static loading (Fig. 7) is 120.7 mm, which is very close to
the desired value 120.4 mm.
In order to verify the stress analysis, an approximate
analyticalsolution is presented asfollows. In this
method the curvature of the spring and misalignment of
fibers are ignored and the spring is assumed to be a
direct cantilever beam as shown in Fig. 8. The deflection
d at loading point P is:
d ¼ou
op¼ o
op
Z l
0
M2
ðxÞ
2EðxÞIðxÞ
dx ¼o
op
Z l
0
p2ðl xÞ 2
2ExxIðxÞ
dx
¼ o
op
p2
2Exx
Z l
0
ðl xÞ 2
WðxÞt3
ðxÞ
12
dx
2
4
3
5
¼12p
Exx
Z l
0
ðl xÞ 2
WðxÞA3
t3
ðxÞ
dx ¼ 12p
ExxA3
Z l
0
ðl xÞ2W2
ðxÞdx ð9Þ
where U is the stored strain energy in the beam,l the
length of beam, WðxÞand tðxÞthe width and thickness of
cross section,P the applied load,EðxÞ the modulus of
elasticity in the longitudinal direction which in this case
is equal to the modulus of elasticity along the fibers Exx,
M ðxÞ the bending moment at a cross section, A the cross-
section area and IðxÞis the second moment of area. It is
assumed that the spring width WðxÞis hyperbolically and
is obtained from Fig.5. Substituting the related value,
the deflection of the spring is
d ¼ 147:6 mm ð10Þ
The maximum induced normal stress at axle seat is
r ¼ plt=2
wt3=12
¼6pl
wt2 ¼ 242:3 MPa ð11Þ
where w and tare the width and thickness of spring at
axle seat.
Regarding therough approximationsused in this
solution, it can be seen that the results of stress analysis
with the finite element method (d ¼ 120:7 mm and r ¼
217 MPa) are acceptable.
3.3.4. Case studies
3.3.4.1.Maximum loading analysis.Maximum loading
on a leaf spring is shown in Fig.1. Because the forces
applied in the accelerating and braking situations are
more critical in the front half of spring, therefore
the front half is analyzed under maximum loading. The
verticalforce in the fullbump is about 4660 N,so the
front half should carry 2568 N. The longitudinal forces
are produced by the change in the linear momentum of
vehicle during the brake or accelerating processes.As-
suming that the velocity of car from 120 km/h reaches to
0 km/h in 5 s,the maximum longitudinalforce willbe
3000 N.The side loads are produced by the change in
the angular momentum of vehicle in the bends of road.
This force is usually assumed to be 75% of the vertical
loads [5]. Therefore in this work it is 1927.5 N. The wind
up torque is produced when one ofthe rear wheels
moved up and the other moved down,so the axle will
twist.The twistangle of9° is the maximum possible
twist angle between the axle seat and each eye [5]. Fig.
shows the fullloading on the front half of spring with
the magnitude of every load.Figs. 10–12 show the re-
sults of front half stress analysis.As shown in Fig.10,
the maximum normal stress in the longitudinal direction
r xx is about 314 MPa. This stress is induced at axle seat
near the inside edge of the spring, because at this point
the stresses due to the verticaland side loads are com-
Fig. 9. Full loading applied to the front half of spring.
0
20
40
60
80
100
120
140
0 100 200 300 400 500 600 700
Spring Length (mm)
Deflection (mm)
Fig. 7. Deflection of composite spring under static vertical loading.
Fig. 8. Approximated modelof rear half of spring for analyticalso-
lution.
322 M.M. Shokrieh, D. Rezaei / Composite Structures 60 (2003) 317–325

bined. At a same loading the maximum normal stress in
the steel spring is about 928 MPa. So the safety factor of
the steel spring is 1.3 while in the composite spring that
is 1.4.The transverse stress (normalto the direction of
fibers) ryy is shown in the Fig. 11. The stress ryy varies a
little along the spring and its maximum value is about
32.6 MPa. The shear stress rxy is shown in Fig. 12. Also
in this case because of the combination of the loads, the
stress rxy has its maximum value about 43 MPa in the
middle of the spring at the axle seat.
3.3.4.2.Naturalfrequency.The road irregularities usu-
ally have the maximum frequency of 12 Hz [5],so the
leaf spring should have a higher naturalfrequency to
avoid the resonance.The stiffnessof compositeleaf
spring is the same as the steel leaf spring but its weight is
lower than the steelspring.Therefore,we can expect
that the first natural frequency of composite leaf spring
will be higher than that of the steel one. Using ANSYS
the first five naturalfrequencies of steeland composite
leaf springs are listed in Table 6.
Considering the Table 6,it is obvious that first nat-
ural frequency ofleaf spring is nearly three times the
road frequency and resonance will not occur.
3.3.4.3.Large deformations. Due to the large deflection
of spring it may be seem that the leaf spring is not under
smalldeflection conditions and has large deformation
and smallstrains behavior.Applying this behavior in
the spring design causes a high degree of nonlinearly to
the problem,so the optimization process willbe com-
putationally intense and may not converge.To ensure
that the results oflinearly elastic assumption are reli-
able, the optimized leaf spring is analyzed assuming the
large deformations and smallstrains behavior.Table 7
shows the results of the two methods.With respect to
the small differencebetween theresultsof the two
methods it is more reasonable to use the linearly elastic
assumption and avoid large deformation theory.
4. Joints design
Heretofore,the composite leafspring was designed
and optimized. In order to provide a reliable suspension
system, the leaf spring should have appropriate joints to
be fixed to the axle and the vehicle body. Four types of
joints are shown in Fig. 13 that can be used to attach the
spring to the vehicle body.
Considering Fig.13,the joint type (a) consists of a
steeleye thatcan be bolted or pinned to the spring.
Becauseof drilling, a region of stressconcentration
-8
-6
-4
-2
0
2
4
6
8
0 100 200 300 400 500 600 700
Spring Length (mm)
Stress (MPa)
Fig. 10.Longitudinalstress rx at lower surface of composite spring
under full loading.
0
100
200
300
400
0 100 200 300 400 500 600 700
Spring Length (mm)
Stress (MPa)
Fig. 11. Transverse stressr y at lower surface ofcomposite spring
under full loading.
0
5
10
15
20
25
0 100 200 300 400 500 600 700
Spring Length (mm)
Stress (MPa)
Fig. 12.Longitudinalstress rxy at lower surface of composite spring
under full loading.
Table 6
The first five natural frequencies of composite leaf spring
f (HZ) 1 2 3 5 6
Steel leaf
spring
29.6 51.8 94.9 102.5 134.3
Composite
leaf spring
33.3 57.6 173.1 248.9 464.2
Table 7
Results of stress and deflection analysis of leaf spring from the two
methods: linearly elastic behavior, large deformations behavior
Material behavior Max. longitudinal
stress (MPa)
Max. deflection
(mm)
Linearly elastic 217 120.7
Large deformations
and small strains
224 131
M.M. Shokrieh, D. Rezaei / Composite Structures 60 (2003) 317–325 323

will exist around the holes in this type of attachment.
Simplicity and low costare the advantageous ofthis
attachment.In joint type (b) the eye and spring are
manufactured simultaneously from the same material.
In this type there is no stress concentration.But rein-
forcement of composite at the junction of eye and spring
is necessary to avoid the splitting ofunidirectionalfi-
bers.Difficultiesof manufacturing and high costare
disadvantages of this method. In joint types (c) and (d)
the spring end has a conical or concave width profile so
that the steeleye fittings with the same conicalor con-
cave profile can be mounted easily and reliably, together
with rubber pads. In these joints there is no stress con-
centration due to the drilling,but the cost of manufac-
turing of the conicalor concave parts of spring should
be considered.On the other hand the angle of conical
parts should be selected great enough to avoid squash-
ing the composites that may cause some problems due to
available space. Joint type (d) is used by Yu and Kim [5].
The joint type (a)is the simplestand cheapestone
from the manufacturing point of view. Although in this
case the compositesshould be drilled,in the present
work this joint was analyzed if it satisfies the strength
requirements,it should be used.Due to the critical
loading on the fronthalf, the eye in this area is con-
sidered.
The leafspring under loading is nearly flat,so the
longitudinal and side loads are the forces that act at the
spring eyes. The spring end at front half with the max-
imum loading condition is shown in Fig.14.For sim-
plicity, it is assumed that the dimensions of spring are 60
mm wide,60 mm long and 12.8 mm thick.Two holes
with the diameter of 8 mm are considered in order to
attach the steel eye.
Among the main failure modes in mechanicaljoints
in composites[18–20],tension and shearout failures
may occur.But it should be considered that the fibers
are unidirectionalalong the spring.In this casethe
amount of actual failure stresses are considerably lower
that the calculated stresses.In fact, in these cases the
failure does not occur due to tension or shear out but
rather the failure occur because ofsplitting the com-
posite in the direction of fibers. To avoid this failure and
reinforcement of composite,additionallayers with dif-
ferentdirectionswere laid around the holes.At the
presentwork the lay up of [45°/0°/90°/45°/0°/90°/
45°] is used at the upper and lower surface of spring
ends around the holes to strengthen the composite ma-
terial in these areas.
The spring is analyzed with the additionallay up at
spring end. The resultsshowed thatthe maximum
stresses(rxx ¼ 6:82 MPa, r yy ¼ 11:7 MPa, r xy ¼ 13:7
MPa) are acting around the holes at an angle of 45° to
the direction of fibers.By substituting these stresses in
the Tsai–Wu failure criterion (Eq. (4)), the left-hand side
of the inequality becomes0.2. Therefore,the critical
loading on the eye is supportable with the composite.
Consequently,the joint of type (a) was selected to join
the spring to the vehicle.
5. Summaryand results
A steel leaf spring used in the rear suspension of light
passenger cars was analyzed by two analytical and finite
element methods.The experimentalresults verified the
analytical and the finite element solutions.
The steel leaf spring was replaced with an optimized
composite one.Main consideration wasgiven to the
optimization of the leaf spring geometry.The objective
was to obtain a spring with minimum weightthat is
capable of carrying given static externalforces by con-
straintslimiting stresses(Tsai–Wu criterion)and dis-
placements.
The results showed thatthe optimum spring width
decreaseshyperbolicallyand the thicknessincreases
linearly from spring eye towards the axle seat.
The stresses in the composite leafspring are much
lower than that of the steel spring.
Compared to the steelleaf spring (9.2 kg) the opti-
mized composite leaf spring without eye units weights
nearly 80% less than the steel spring.
The naturalfrequency ofcomposite leafspring is
higher than that of the steel leaf spring and is far enough
from the road frequency to avoid the resonance.
To join the spring to the vehicle body, an additional
lay up was used on the spring end and the steel eyes we
mounted through bolts.Fig. 14. The spring end at front half with maximum loading.
Fig. 13. Differentjoints to attach the composite leafspring to the
vehicle body.
324 M.M. Shokrieh, D. Rezaei / Composite Structures 60 (2003) 317–325

References
[1] BeardmoreP, Johnson CF. The potentialfor compositesin
structuralautomotive applications.Comp Sci Technol1986;26:
251–81.
[2] Beardmore P. Composite structures for automobiles. Comp Struct
1986;5:163–76.
[3] Morris CJ. Composite integrated rear suspension.Comp Struct
1986;5:233–42.
[4] Daugherty RL.Composite leafsprings in heavy truck applica-
tions.In: Composite Materials,Proceedings of Japan–US Con-
ference, Tokyo, 1981, p. 529–38.
[5] Yu WJ, Kim HC. Double taperedFRP beam for auto-
motive-suspensionleaf spring. Comp Struct 1988;9:279–
300.
[6] Corvi A. A preliminary approach to composite beam design using
fem analysis. Comp Struct 1990;16:259–75.
[7] Manual on design and applicationof leaf springs, Spring
Design Manual, AE-11, Society of AutomotiveEngineerHs
788,1990.
[8] Tsai SW, Hahn HT. Introduction to compositematerials.
Technomic Publishing; 1980.
[9] ANSYS 5.4, ANSYS Inc., 1997.
[10]Ryan WE. Method of making a molded fiber reinforced plastic
leaf spring, US patent 4,560,525 (24/12/1985).
[11]Richard DS, Mutzner JE, Eilerman JF, et al. Method of forming
a composite leaf spring with fabric wear pad,US patent 4,894,
108 (16/1/1990).
[12]Nickel HW. Bushing construction for a fiber reinforced plastic
leaf spring, US patent 4,565,356 (21/1/1986).
[13]Abrate S.Optimaldesign of laminated plates and shells.Comp
Struct 1994;29:269–86.
[14]Kam TY, Snyman JA. Optimaldesign oflaminated composite
plates using a global optimization technique. Comp Struct 1991;19:
350–71.
[15]Conecicao CA,Torres Marques A,Soeiro AV. Optimization of
laminated composite structures using a bilevelstrategy.Comp
Struct 1995;33:193–200.
[16]Liu IW, Lin CC. Optimum design of composite wing structures by
a refined optimality criterion. Comp Struct 1991;17:51–65.
[17]Nanderpleats GN.Numericaloptimization techniques for engi-
neeringdesign:with applications.New York: McGraw-Hill
Publishers; 1984.
[18]Mallick DK, editor.Composite engineering handbook.Marcel
Dekker; 1997. p. 720–36.
[19]Dreger DR. Design guideline for joining advanced composites,
Machine Design, 8 May 1980, pp. 89–93.
[20]Hart-Smith LJ. Mechanically fastened joints for advanced com-
posites,phenomenologicalconsideration and simpleanalysis,
Fibrous Composites in Structural Design, Premium Press, 1993.
M.M. Shokrieh, D. Rezaei / Composite Structures 60 (2003) 317–325 325