Developing a hybrid, carbon/glass fiber-reinforced, epoxy composite automotive drive shaft

Article (PDF Available)inMaterials and Design 31(1):514-521 · January 2010with 887 Reads
DOI: 10.1016/j.matdes.2009.06.015
Abstract
In this study, a finite element analysis was used to design composite drive shafts incorporating carbon and glass fibers within an epoxy matrix. A configuration of one layer of carbon–epoxy and three layers of glass–epoxy with 0°, 45° and 90° was used. The developed layers of structure consists of four layers stacked as [+45glass°/-45glass°/0carbon°/90glass°]. The results show that, in changing carbon fibers winding angle from 0° to 90°, the loss in the natural frequency of the shaft is 44.5%, while, shifting from the best to the worst stacking sequence, the drive shaft causes a loss of 46.07% in its buckling strength, which represents the major concern over shear strength in drive shaft design.
Technical Report
Developing a hybrid, carbon/glass fiber-reinforced,
epoxy composite automotive drive shaft
A.R. Abu Talib
a
, Aidy Ali
b,*
, Mohamed A. Badie
a
, Nur Azida Che Lah
b
, A.F. Golestaneh
b
a
Department of Aerospace Engineering, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia
b
Department of Mechanical and Manufacturing Engineering, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia
article info
Article history:
Received 14 April 2009
Accepted 10 June 2009
Available online 14 June 2009
abstract
In this study, a finite element analysis was used to design composite drive shafts incorporating carbon
and glass fibers within an epoxy matrix. A configuration of one layer of carbon–epoxy and three layers
of glass–epoxy with 0°,45°and 90°was used. The developed layers of structure consists of four layers
stacked as ½þ45
glass
=45
glass
=0
carbon
=90
glass
. The results show that, in changing carbon fibers winding angle
from 0°to 90°, the loss in the natural frequency of the shaft is 44.5%, while, shifting from the best to the
worst stacking sequence, the drive shaft causes a loss of 46.07% in its buckling strength, which represents
the major concern over shear strength in drive shaft design.
Ó2009 Elsevier Ltd. All rights reserved.
1. Introduction
Drive shafts for power transmission are used in many applica-
tions, including cooling towers, pumping sets, aerospace, struc-
tures, and automobiles. In metallic shaft design, knowing the
torque and the allowable service shear stress for the material al-
lows the size of the shaft’s cross-section to be determined. In order
to satisfy the design parameter of torque divided by the allowable
shear stress [1], there is unique value for the shaft’s inner radius
because the outer radius is constrained by the space under the
car cabin. Metallic drive shafts have limitations of weight, low crit-
ical speed and vibration characteristics.
Composite drive shafts have proven that they can solve many
automotive and industrial problems that accompany the usage of
the conventional metal ones. Numerous solutions, such as fly-
wheels, harmonic dampers, vibration shock absorbers, multiple
shafts with bearings and couplings, and heavy associated hard-
ware, have shown limited success in overcoming the problems [2].
When the length of a steel drive shaft is beyond 1500 mm [3],it
is manufactured in two pieces to increase the fundamental natural
frequency, which is inversely proportional to the square of the
length and proportional to the square root of the specific modulus.
The nature of composites, with their higher specific elastic modu-
lus (modulus to density ratio), which in carbon/epoxy exceeds four
times that of aluminium, enables the replacement of the two-piece
metal shaft with a single component composite shaft which reso-
nates at a higher rotational speed, and ultimately maintains a high-
er margin of safety. A composite drive shaft offers excellent
vibration damping, cabin comfort, reduction of wear on drive train
components and increased tire traction. In addition, the use of sin-
gle torque tubes reduces assembly time, inventory cost, mainte-
nance, and part complexity. The first application of composite
drive shafts to automobiles was developed by Spicer U-joint divi-
sions of the Dana Corporation for the Ford Econoline van models
in 1985 [3].
Polymer matrix composites such as carbon/epoxy or glass/
epoxy offer better fatigue characteristics because micro cracks in
the resin do not freely propagate as in metals, but terminate at
the fibers. Generally, composites are less susceptible to the effects
of stress concentration, such as are caused by notches and holes,
compared with metals [4]. The filament winding process is used
in the fabrication of composite drive shafts. In this process, fiber
tows wetted with liquid resin are wound over a rotating male
cylindrical mandrel. In this technique, the winding angle, fiber ten-
sion, and resin content can be varied. Filament winding is relatively
inexpensive, repetitive and accurate in fiber placement [5].
An efficient design of composite drive shaft could be achieved
by selecting the proper variables, which are specified to minimize
the chance of failure and to meet the performance requirements.
As the length and outer radius of drive shafts in automotive appli-
cations are limited due to spacing, the design variables include the
inside radius, layers thickness, number of layers, fiber orientation
angle and layer stacking sequence. In the optimal design of the
drive shaft, these variables are constrained by the lateral natural
frequency, torsional vibration, torsional strength and torsional
buckling of the shaft. In this study, another constraint is added that
relates to torsional fatigue and selection of the stacking sequence.
The ability to tailor the elastic constants in composites provides
numerous alternatives for the variables to meet the desired
stability and strength of the structure. At first, the fibers are
selected to provide the best stiffness and strength, together with
0261-3069/$ - see front matter Ó2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.matdes.2009.06.015
*Corresponding author. Tel.: +60 17 249 6293; fax: +60 38 656 7122.
E-mail address: aidy@eng.upm.edu.my (A. Ali).
Materials and Design 31 (2010) 514–521
Contents lists available at ScienceDirect
Materials and Design
journal homepage: www.elsevier.com/locate/matdes
their cost. Indeed, it is the best choice to use carbon fibers in all
layers to achieve desired stability. However, due to the cost con-
straint, a hybrid of layers of carbon–epoxy and E-glass–epoxy
was introduced. It is evident that the fiber orientation angle dic-
tates the maximum bending stiffness, in turn leading to the maxi-
mum natural frequency in bending. In this design, the fibers were
arranged longitudinally at the zero angle with respect to the shaft
axis. On the other hand, the angle of ±45°was used to obtain the
maximum shear strength, while 90°was the best for buckling
strength. The main goal was to achieve the minimum weight while
adjusting the parameters in order to meet a sufficient margin of
safety. The safety criteria specify that a critical speed (natural fre-
quency) must be higher than the operating speed, a critical torque
must be higher than the ultimate transmitted torque and a nomi-
nal stress (the maximum at fiber direction) must be less than the
allowable stress after applying the failure criteria.
In the shaft design, the shear strength could be increased by
increasing the diameter of the shaft. However, the crucial parame-
ter to consider was the buckling strength. The variable of the lam-
inate thickness has a big effect on the buckling strength and slight
effect on the natural frequency in bending. A discrete variable opti-
mization algorithm could be employed for optimization of layer
thickness and orientation. Vijayarangan et al. [6] used a Genetic
Algorithm, and Rastogi [3] used GENESIS/I-DEAS optimizers for
the optimization of variables in the design of a drive shaft for auto-
motive applications. In other work, Darlow and Creonte [7] em-
ployed the general-purpose package, OPT (version 3.2), in
optimizing the lay-up of a graphite–epoxy composite drive shaft
for a helicopter tail rotor.
2. Design procedure
The material properties of the drive shaft were analyzed with
classical lamination theory. The theory treats with the linear elastic
response of laminated composite under plane stress, and it incor-
porates the Kirchhoff-Love assumption for bending and stretching
of thin plates [8]. From the properties of the composite materials at
given fiber angles, the reduced stiffness matrix can be constructed.
The expressions for the reduced stiffness coefficients (Q
1j
), in terms
of standard material constants, are as follows:
Q
11
¼E
1
1
t
12
t
21
;Q
22
¼E
2
1
t
12
t
21
;Q
66
¼G
12
;
Q
12
¼
t
12
E
2
1
t
12
t
21
;and
t
21
¼E
2
E
1
t
12
;
ð1Þ
where Eis modulus of elasticity, Gis modulus of rigidity and
t
is
Poisson’s ratio.
The next step is to construct the extensional stiffness matrix [A].
This matrix is the summation of the products of the transformed
reduced stiffness matrix ½Qof each layer and the respective thick-
nesses, represented as:
½A¼X
N
K¼1
½Q
K
ðz
K
z
K1
Þ:ð2Þ
The matrix, [A], is in (Pa m), and the thickness of each ply is calcu-
lated in reference to their coordinate location in the laminate. The A
matrix is used to calculate E
x
and E
h
, which are the average moduli
in the axial and hoop directions, respectively:
E
x
¼1
tA
11
A
2
12
A
22
"#
;E
h
¼1
tA
22
A
2
12
A
11
"#
:ð3Þ
2.1. Buckling torque
Since the drive shaft is considerably long, thin and hollow, there
is a possibility that it may buckle. The expression of the critical
buckling torque for thin-walled orthotropic tubes [9] is given as:
T
cr
¼ð2
p
r
2
tÞð0:272Þ½E
x
E
3
h
1=4
t
r

3=2
:ð4Þ
Here, ris the mean radius and tis the total thickness. It is obvious
that the stiffness modulus in hoop direction (E
h
) plays most sub-
stantial role in increasing the buckling resistance. The safety factor
is defined as the ratio of the buckling torque to the ultimate torque.
2.2. Literal bending natural frequency
The drive shaft is designed to have a critical speed of 60 times
larger than the natural frequency of the rotational speed. If these
become coincident, a large amplitude vibration (whirling) will oc-
cur. The drive shaft is idealized as either a simply-supported or a
pinned–pinned beam. The lowest natural frequency expression
[10,11] is given as:
f
n
¼
p
2ffiffiffiffiffiffiffiffiffi
gE
x
I
WL
4
s;ð5Þ
where gis the acceleration due to gravity, Wis the weight per unit
length, Lis the shaft length and Iis the second moment of inertia
given, for a thin-walled tube, as:
Nomenclature
[A] stiffness matrix
A
66
shear stiffness component
Emodulus of elasticity
E
h
average modulus in the hoop direction
E
x
average modulus in the axial direction
Gmodulus of rigidity
gacceleration due to gravity
Isecond moment of inertia
I
m
mass moment of inertia
Jpolar moment of inertia
Ktorsional spring rate
Llength
[N] force matrix
Q
1j
reduced stiffness coefficient
rradius
r
0
outer radius
r
i
inner radius
r
m
mean radius
Ttorque
T
s
failure torque
tthickness.
Wweight per unit length
e
y
strain in ydirection
/angle of twist
c
xy
shear stress
g
coefficient of mutual influence
hcarbon fibers orientation angle
s
l
in-plane shear strength of the laminate
t
Poisson’s ratio
n
T
e
torque coupling coefficient
A.R. Abu Talib et al. / Materials and Design 31 (2010) 514–521 515
I
x
¼
p
4ðr
4
0
r
4
i
Þ
p
r
3
t:ð6Þ
Here, r
0
is an outer radius, and r
i
is an inner radius.
2.3. Load carrying capacity
The composite drive shaft is designed to carry the torque with-
out failing. The torsional strength, the torque at which the shaft
will fail, is directly related to the laminate shear strength through:
T
s
¼2
p
r
2
m
t
s
l
:ð7Þ
Here, T
s
is the failure torque,
s
l
is the in-plane shear strength of the
laminate, r
m
is the mean radius and tis the thickness. The same for-
mula is used in the laboratory after a torsion tube test to determine
the shear modulus and shear strength of materials. Since the lami-
nate is assumed to have failed according to the first ply failure con-
vention, the maximum-stress failure criterion could be used after
finding the in-plane stresses at every ply to specify the safety factor
for torque transmission capacity. Again, the first steps are to con-
struct the transverse of the extensional stiffness matrix [A] and then
solve for the overall strains. Once complete, the stresses in each
layer can be examined by transforming these stresses into the
X
Y
Z
Fig. 1. The first bending mode as a function of natural frequency.
Table 1
Material properties [12].
Material E
11
(GPa) E
22
(GPa) E
12
(GPa) Ultimate strength (MPa) Weight density (kg/m
2
)
E-glass–epoxy 40.3 6.21 3.07 827 1910
Carbon–epoxy 126.9 11.0 6.6 1170 1610
Fig. 2. Set of the first six modes as a function of natural frequency along with their corresponding frequency values.
516 A.R. Abu Talib et al. / Materials and Design 31 (2010) 514–521
direction of fibers at each layer. The layers of fiber direction ±45°are
of special concern since they have a substantial contribution to the
load carrying capacity. The A
1
matrix, multiplied by the laminate
thickness and the resultant forces matrix, N, gives the resultant
strain as follows:
f
e
A
1
N
x
N
y
N
xy
8
>
<
>
:
9
>
=
>
;
¼A
1
0
0
N
xy
8
>
<
>
:
9
>
=
>
;
;N
xy
¼T
2
p
r
2
:ð8Þ
Here, the axial force is N
x
= 0, the centrifugal force, N
y
, is neglected
and N
xy
is the resultant shear force. The torque, T, is the peak torque
if the design involves fatigue considerations [12]. The resultant
strains are transformed to the fiber direction by multiplying these
strain matrices by the transformation matrix. From this, the plane
stress transformations can be obtained.
e
1
e
2
c
12
8
>
<
>
:
9
>
=
>
;
¼
m
2
n
2
mn
n
2
m
2
mn
2mn 2mn m
2
n
2
2
6
43
7
5
e
x
e
h
c
xh
8
>
<
>
:
9
>
=
>
;
r
1
r
2
s
12
8
>
<
>
:
9
>
=
>
;
¼
Q
11
Q
12
0
Q
21
Q
22
0
00Q
33
2
6
43
7
5
e
1
e
2
c
12
8
>
<
>
:
9
>
=
>
;
ð9Þ
2.4. Torsional frequency
The torsional frequency is another parameter and is directly re-
lated to the torsional stiffness (T/u), where uis the angle of twist
and Tis the applied torque. The frequency of torsional vibration
can be presented as:
f
t
¼1
2
p
ffiffiffiffi
K
I
m
s;ð10Þ
where Kis the torsional spring rate, and is equal to the torsional
stiffness, and I
m
is the mass moment of inertia at propeller. For a gi-
ven geometry in a specific drive shaft, the torsional stiffness is di-
rectly related to the modulus of rigidity (G
xy
) as follows [2]:
K¼T
/¼G
xy
J
L;ð11Þ
where Jis the polar moment of inertia and Lis the length. The shear
modulus can be tailored to its maximum value by orienting the fi-
bers at an angle equal to 45°. In some applications, like racing cars,
less torsional stiffness is required [2]. The shear modulus can be di-
rectly obtained from the extensional stiffness matrix [A], by divid-
ing the shear stiffness component, A
66
, by the total thickness of
the drive shaft as follows:
G
xy
¼A
66
t:ð12Þ
A practical application of torsional vibration systems is in engines.
These engines have damping (source of energy dissipation) in the
crankshafts (hysteresis damping), and damping in torsional vibra-
tion (in propellers). Since the damping present is normally small
in magnitude, it can be neglected when determining the natural fre-
quency [13].
3. Finite element analysis of drive shaft
Finite element models of the drive shaft were generated and
analyzed using LUSAS (version 13.5-7) commercial software. A
three-dimensional model was developed and meshed with three-
dimensional thick-shell elements (QTS8). This degenerate contin-
uum element is capable of modeling warped configurations,
accounting for varying thicknesses and supporting the definition
of anisotropic and composite material properties. Since it is quad-
rilateral, it uses an assumed strain field to define transverse shear,
which ensures that the element does not lock when it is thin. Such
elements can accommodate a broader range of curved geometries
than other element types can [14]. A cylindrical local coordinate
system was defined in order to align the material axis of the lay-
up, and to apply fixities and load cases.
Eigenvalue linear buckling analysis was performed to define the
critical buckling torque. The output from this analysis is a factor
that is multiplied by the applied load to determine the critical
buckling load. The linear analysis is considered satisfactory in com-
parison with nonlinear analysis due to the fact that cylindrical
shells under torsion are less sensitive to imperfections [15]. In this
study, the position of the buckling region along the axial length of
the shaft was detected as being shifted towards the end of the shaft
when a nonlinear analysis was performed. Modal analysis is a tech-
nique used to analyze structures dominated by global displace-
ment, such as in vibration problems. It was used to define the
natural frequency of the drive shaft. The Eigenvectors resulting
from the Eigenvalue analysis are the modes of the buckling defor-
mation and the natural frequency in bending, as presented in Figs.
1 and 6.
A composite drive shaft design example, presented by Swanson
[12], was taken as a reference model for all analyzes. In this exam-
ple, a shaft of length 1730 mm, mean radius 50.8 mm and consist-
Fig. 3. The effect of fiber orientation angle on the natural frequency as determined
by changing the first two ±45 layers angles.
Fig. 4. The effect on natural frequency of changing the carbon fiber orientation
angle in a hybrid drive shaft of stacking ½þ45
glass
=45
glass
=0
carbon
=90
glass
.
A.R. Abu Talib et al. / Materials and Design 31 (2010) 514–521 517
ing of three layers of (±45°,90°) glass–epoxy and 0°carbon–epoxy
layer was used. The ultimate torque was 2030 Nm and the mini-
mum natural frequency in bending is 90 Hz. The material proper-
ties are listed in Table 1.
4. Specimens fabrication
Four layers of carbon/epoxy, glass/epoxy and a hybrid of both
were wrapped around aluminium tubes of length equal to
216 mm and outside diameter equal to 12.7 mm. For easily remov-
ing the aluminium tubes, thin film of oil was formed then a thin
plastic sheet wrapped around. The epoxy impregnated carbon
and glass fabrics had been wrapped with plastic sheet at outside
surface for the purpose of producing smooth surface. These tubes
are removed after curing under room temperature. The ends of
the specimens were reinforced by the winding of carbon or glass
fibers tows used in filament winding. The stacking of these speci-
mens is as follows:
1. [45°]
4
All layers are of glass/epoxy.
2. [45°]
4
All layers are of carbon/epoxy.
3. [90°]
4
All layers are of glass/epoxy.
4. [90°]
4
All layers are of carbon/epoxy.
5. [(45°)
2
glass/(90°)
2
carbon].
6. [(45°)
2
carbon/(90°)
2
glass].
Woven roving Fabric fibers used in both [0/90] and [±45] lay-up.
The thicknesses of the composites were measured to be: Carbon/
epoxy layer thickness = 0.35 mm and glass/epoxy layer thickness =
0.37 mm.
5. Results and discussion
5.1. Effect of fiber orientation angle on natural frequency
The vibration problem is described by a set of equations, and
there is a natural vibration mode for every equation that can be ex-
tracted by using an eigenvalue extraction analysis. The displace-
ment behavior dominating any structure subject to vibration is
global; therefore, modal analysis is utilized in these types of prob-
lems. In modal analysis, the model of the drive shaft does not need
a fine mesh because the stress output is not required. Additionally,
there is no requirement to input an applied load, because the nat-
ural frequency is only a function of mass and stiffness.
The ends of the drive shaft model were modeled as simply-sup-
ported, and the boundary condition was varied until the value of
the natural frequency became nearly coincident with that pre-
sented by a reliable example. It was recognized that the end sup-
port conditions must also be applied to the edges of the drive
shaft. For simplicity, the contact area between the shaft tube and
the yoke joint, as well as the join itself, were not considered in
the calculations of the natural frequency.
In any structural design with vibration concerns, only the first
mode is of concern for engineering applications. Fig. 1 presents
the shape of the first bending mode based on natural frequency.
Fig. 2 shows a set of the first six natural frequencies in bending.
The drive shaft specified in the previous section was used to
investigate the effect of fiber orientation angle on the natural fre-
quency. This structure consists of four layers stacked as
½þ45
glass
=45
glass
=0
carbon
=90
glass
. From Figs. 3 and 4, it is clear that
the fibers must be oriented at zero degrees to increase the natural
frequency by increasing the modulus of elasticity in the longitudi-
nal direction of the shaft. This explains why the carbon fibers, with
their high modulus were oriented at the zero angle. In Fig. 3, de-
spite the configuration [0, 0, 90, 0] resulting in the highest natural
frequency, it is not a good selection when an optimization with
other parameters, such as buckling resistance and fatigue strength,
is made.
From Fig. 4, the drive shaft loses 44.5% of its natural frequency
when the carbon fibers are oriented in the hoop direction at 90°in-
stead of 0°. The cost factor plays a role in selecting only one layer of
carbon/epoxy.
The analysis was conducted on comparatively thin composite
tubes, and shows that the behavior of the thinner tube is different.
Specifically, the critical speed and the natural frequency did not in-
crease as the orientation angle approached the value of zero. As
seen in Fig. 5, three models of the same material (carbon/epoxy)
and different thicknesses were constructed. It was found that the
critical speeds for all models were the same when the fibers of
all layers were oriented at 38–90°. The fiber angle of 38°,or37°,
as mentioned in the literature (Herakovich, 1998) imparts special
properties, since, at this angle, unidirectional off-axis tubes under
pure torque loading exhibit the maximum coupling between shear
strain and axial strain. The axial strain reaches as much as 50% of
the shear strain. However, from this figure, it is clear that, for the
tubes of smaller thickness, the membrane stress plays an effective
role in the lateral stiffness of the tube. At 38°, the torque coupling
coefficient (n
T
e
) is at the maximum, and hence the axial strain is at
the maximum. This directly leads to the highest bending stiffness,
implying a higher natural frequency. The stacking sequence has no
effect on the natural frequency because the matrix form of the
equation of dynamic equilibrium for an elastic body only contain
stiffness and mass matrices when no damping and external forces
are applied. The mass matrix is a function of the total density and
the absence of loads make the stacking sequence irrelevant.
5.2. Effect of fiber orientation angle on buckling torque
A linear eigenvalue buckling analysis was conducted to esti-
mate the maximum torque that can be supported prior to losing
stability. In this analysis, the specified load must be closer to the
collapse load in order to obtain accurate results. The output from
the analysis is a factor that can be multiplied by the actual magni-
tude of the applied load in order to obtain an estimate of the crit-
ical torque. Fig. 6 presents the contour of maximum shear stress,
and the deformed shape after linear eigenvalue analysis.
From the contour of maximum shear stress, it can be observed
that higher shear stress accumulated at the two bands that heli-
0.00
1000.00
2000.00
3000.00
4000.00
5000.00
6000.00
7000.00
0 102130385060708090
Fiber Orientation Angle (deg)
Critical Spead (r.p.m)
glass ( 0.762)
carbon (0.3022)
carbon (0.532)
carbon (0.762)
Fig. 5. Comparisons between the critical speeds of composite tubes having four
layers and stacking of [h]
4
(glass and carbon are abbreviations of glass/epoxy and
carbon/epoxy, respectively, and parentheses indicate thickness in millimeters).
518 A.R. Abu Talib et al. / Materials and Design 31 (2010) 514–521
cally wrapped the cylindrical tube at 45°. The tubes buckled when
they lost their stability, and the circular cross-sections became
ovoid.
The best fiber orientation angle for maximum buckling strength
is 90°. At this angle, the fibers are oriented in the hoop direction,
thereby increasing the hoop modulus (E
h
). Fig. 7 presents the effect
on buckling torque of changing the fiber orientation angles with
only two glass/epoxy layers.
As shown in Fig. 8, there is little dependence of the buckling tor-
que on the fiber orientation. It can be observed that, by changing
the angles of the 3rd or the 4th layer, the critical buckling torque
of the drive shaft is not substantially affected by the fiber orienta-
tion angles. This is attributed to the fact that the modulus, E
x,
has
its maximum value at the zero degree fiber orientation angle,
Fig. 6. First buckling mode shape, and the corresponding contour of maximum shear stress.
y = 0.00x - 0.01x + 0 .79x - 7.24x + 1784.11
R = 1.00
700
1200
1700
2200
2700
3200
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Fiber Orientation Angle (Deg.)
Critical Buckling Torque (N.m)
Fig. 7. Effect of fiber orientation angle on buckling torque by changing the first two
layers of a ½h
glass
=0
carbon
=90
glass
stack.
0
500
1000
1500
2000
2500
3000
0 102030405060 708090
Fiber Orientation An
g
le (De
g
.)
Buckling Torque (N.m)
2
[
]
45, 45,
θ
−±
1
[
]
,0,90
θ
±
4
[]
45, 45, ,90
θ
3
[
]
45, 45, 0,
θ
2
1
Fig. 8. Effect of fiber orientation angle on the buckling torque of a drive shaft having
stacking of ½45
glass
=0
carbon
=90
glass
.
Table 2
Five selected laminates with different stacking sequences and their corresponding
stiffness component D
22
(the bulking strength descending from top to bottom).
Layers stacking sequence D
22
(Pa m
3
)
[45,45,0,90] 58.8
[0,45,45,90] 55.4
[0,45,90,45] 42.9
[45,45,90,0] 36.31
[0,90,45,45] 36.23
A.R. Abu Talib et al. / Materials and Design 31 (2010) 514–521 519
and the modulus, E
h,
has its maximum value at a 90°angle. Since
the expression of buckling torque is related to both moduli, then
the peak value for this torque is realized when the fibers are ori-
ented at 0°and 90°.
5.3. Effect of layers stacking sequence on buckling torque
The stacking sequence of the layers has an effect on the buck-
ling strength. Although the [A] matrix is independent of the stack-
ing sequence, both the [B] and [D] matrices are dependent upon it.
The drive shaft buckled when its bending stiffness along the hoop
direction could not support the applied torsion load. This normal
bending stiffness is correspondent to the component, D
22
, of the
bending stiffness matrix [D].
Therefore, the value of D
22
specifies the buckling strength. Fig. 9
presents the effect of stacking sequence on the buckling strength
and it is concluded that the best case scenario stacking sequence
is [45/45/0/90], and the worst case scenario is [0/90/45/45].
Table 2 shows the correspondent D
22
components for five lam-
inates with different stacking sequences. The best stacking offers a
buckling torque of 2303.1 Nm and the worst stacking offers a tor-
que of 1242 Nm, with a loss in buckling resistance capability equal
to 46.07%.
5.4. Effect of coupling between the twisting moment and normal
curvature
The twisting moment resulted from a pure torque loading cou-
pled with a normal curvature in terms of the components D
16
and
D
26
in the bending stiffness matrix [D]. The D
16
component repre-
sents the curvature in the longitudinal direction, and as its value
increases the drive shaft tends to bend, and its natural frequency
in bending decreases. The coupling between the twisting moment
and the normal curvature in the hoop direction can be directly re-
lated to the coefficient of mutual influence (
g
), which is a normal-
shear coupling. One form of this coupling is:
g
x;xy
¼
e
y
c
xy
¼S
26
S
66
:ð13Þ
Here, e
y
is strain in ydirection and
c
xy
is the shear stress.
This coefficient represents the radial strain resulting from a tor-
que loading, and it may have a negative or positive value. If the
sign is positive, the diameter of the cross-section tends to decrease,
or the curvature in the hoop direction (D
26
) tends to increase. An-
gle-ply laminates that consist of 0°and 90°angles do not experi-
ence such coupling. On the other hand, a configuration of [±h
2n
]
has a zero coupling or zero for D
16
and D
26
, because for every +h
there is hat the same distance from the mid-plane. However, a
laminate of the configuration [h
n
]
S
has a coupling. Here, the compo-
nent D
26
contributes to the buckling strength of the drive shaft.
Fig. 10 exhibits the effect of coupling on the buckling torque for
two configurations having the same value of the component of nor-
mal bending stiffness in the hoop direction, D
22
.
6. Conclusions
The present finite element analysis of the design variables of fi-
ber orientation and stacking sequence provide an insight into their
effects on the drive shaft’s critical mechanical characteristics and
fatigue resistance. A model of hybridized layers was generated
incorporating both carbon–epoxy and glass–epoxy. Buckling,
which dominates the failure modes, has a value does not increases
regularly with increasing the winding angle. For the worst stacking
sequence, the shaft loses 46.07% of its buckling strength compared
0
500
1000
1500
2000
2500
Buckling Torque (Nm)
[45,-45,0,90]
[90,0,45,-45]
[90,0,-45,45]
[90,45,0,-45]
[0,45,-45,90]
[0,-45,45,90]
[90,-45,0,45]
[0,45,90,-45]
[90,45,-45,0]
[0,-45,90,45]
[45,-45,90,0]
[-45,45,90,0]
[0,90,45,-45]
[0,90,-45,45]
Fig. 9. Effect of stacking sequence on buckling torque.
0
1000
2000
3000
4000
5000
6000
0 102030405060708090
Fiber Orientation Angle
Buckling Torque (N.m)
Carbon-epoxy
Glass-epoxy
Carbon-epoxy
Glass-epoxy
- - - [±
θ
º]
2
[θº]
4
Fig. 10. The effect on the buckling torque of coupling between twisting moment
and normal curvature in hoop direction.
520 A.R. Abu Talib et al. / Materials and Design 31 (2010) 514–521
to what it achieves with the best stacking sequence. On the other
hand, the stacking sequence has an obvious effect on the fatigue
resistance of the drive shaft.
Acknowledgement
The authors would like to thank the University Putra Malaysia
for supporting these research activities.
References
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[8] Herakovich TC. Mechanics of fibrous composites. New York: John Wiley &
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[9] Column Research Committee of Japan. Handbook of structural stability. Tokyo:
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[10] Young DH, Timoshenko S, Weaver W. Vibration problems in engineering. 4th
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River (New Jersey): Prentice Hall; 1998.
[12] Swanson SR. Introduction to design and analysis with advanced composite
material. Upper Saddle River (New Jersey): Prentice Hall; 1997.
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York: McGraw-Hill; 1998.
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