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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, Malaysiab Department of Mechanical and Manufacturing Engineering, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia

a r t i c l e i n f o

Article history:Received 14 April 2009

Accepted 10 June 2009

Available online 14 June 2009

a b s t r a c t

In this study, a finite element analysis was used to design composite drive shafts incorporating carbonand 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=0carbon=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.

2009 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 2009 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: [email protected] (A. Ali).

Materials and Design 31 (2010) 514–521

Contents lists available at ScienceDirect

Materials and Design

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m a t d e s

http://dx.doi.org/10.1016/j.matdes.2009.06.015

mailto:[email protected]

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http://dx.doi.org/10.1016/j.matdes.2009.06.015



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 1 j), in terms

of standard material constants, are as follows:

Q 11 ¼ E 1

1 t12t21

; Q 22 ¼ E 2

1 t12t21

; Q 66 ¼ G12;

Q 12 ¼ t12E 2

1 t12t21; and t21 ¼

E 2E 1 t

12;

ð1Þ

where E is modulus of elasticity, G is 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 ½Q of each layer and the respective thick-

nesses, represented as:

½ A ¼XN

K ¼1

½Q K

ð z K z K 1Þ: 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 ¼ 1

t A11

A2

12

A22" #; E h ¼ 1

t A22

A2

12

A11" #: 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 ¼ ð2pr 2t Þð0:272Þ½E xE 3h1=4 t

r

3=2

: ð4Þ

Here, r is the mean radius and t is 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 p2

ffiffiffiffiffiffiffiffiffiffi gE xI

WL4

s ; ð5Þ

where g is the acceleration due to gravity, W is the weight per unit

length, L is the shaft length and I is the second moment of inertiagiven, for a thin-walled tube, as:

Nomenclature

[ A] stiffness matrix A66 shear stiffness componentE modulus of elasticityE h average modulus in the hoop direction

E x average modulus in the axial directionG modulus of rigidity g acceleration due to gravityI second moment of inertiaI m mass moment of inertia J polar moment of inertiaK torsional spring rateL length[N ] force matrixQ 1 j reduced stiffness coefficientr radius

r 0 outer radiusr i inner radiusr m mean radiusT torque

T s failure torquet thickness.W weight per unit lengthe y strain in y direction/ angle of twistc xy shear stressg coefficient of mutual influenceh carbon fibers orientation anglesl in-plane shear strength of the laminatet Poisson’s rationT e torque coupling coefficient

A.R. Abu Talib et al. / Materials and Design 31 (2010) 514–521 515



I x ¼ p4

ðr 40

r 4i Þ pr 3t : 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 shaftwill fail, is directly related to the laminate shear strength through:

T s ¼ 2pr 2mt sl: ð7Þ

Here, T s is the failure torque, sl is the in-plane shear strength of the

laminate, r m is the mean radius and t is 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

YZ

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/m2

)

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 A1 matrix, multiplied by the laminate

thickness and the resultant forces matrix, N , gives the resultant

strain as follows:

feg ¼ A1

N x

N y

N xy

8><>: 9>=>; ¼ A1

0

0

N xy

8><>: 9>=>;; N xy ¼ T

2pr 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.

e1

e2

c12

8><>:

9>=>; ¼

m2 n2 mn

n2 m2 mn

2mn 2mn m2 n2

264

375

e x

eh

c xh

8><>:

9>=>;

r1

r2

s12

8><>:

9>=>;

¼

Q 11 Q 12 0

Q 21 Q 22 0

0 0 Q 33

264 375e1

e2

c12

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 u is the angle of twist

and T is the applied torque. The frequency of torsional vibration

can be presented as:

f t ¼ 1

2p

ffiffiffiffiffiK

I m

s ; ð10Þ

where K is 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]:

K ¼ T

/ ¼

G xy J

L ; ð11Þ

where J is the polar moment of inertia and L is 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, A66, by the total thickness of

the drive shaft as follows:

G xy ¼ A66

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 orientationangle in a hybrid drive shaft of stacking ½þ45

glass= 45 glass=0

carbon=90 glass.




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 bothwere 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, andthere 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 andthe 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=0carbon=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 of0. 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, or 37,

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 (nT 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 fromthe 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 10 21 30 38 50 60 70 80 90

Fiber Orientation Angle (deg)

C r i t i c a l S p e a d (

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 andcarbon/epoxy, respectively, and parentheses indicate thickness in millimeters).




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.)

C r i t i c a l B u c k l i n g T o r q u e ( N . m

)

Fig. 7. Effect of fiber orientation angle on buckling torque by changing the first twolayers of a ½h glass=0

carbon=90 glass stack.

0

500

1000

1500

2000

2500

3000

0 10 20 30 40 50 60 70 80 90

Fiber Orientation Angle (Deg.)

B u c k l i n g T o r q u e ( N . m

)

2[ ]45, 45, θ − ±

1[ ], 0 , 9 0θ ±

4[ ]45, 45, ,90θ −

3[ ]45, 45,0,θ −

2

1

34

Fig. 8. Effect of fiber orientation angle on thebucklingtorque of a drive shaft having

stacking of ½45 glass=0

carbon=90 glass.

Table 2

Five selected laminates with different stacking sequences and their corresponding

stiffness component D22 (the bulking strength descending from top to bottom).

Layers stacking sequence D22 (Pa m3)

[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




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, D22, of the

bending stiffness matrix [D].Therefore, the value of D22 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 D22 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 D16 and

D26 in the bending stiffness matrix [D]. The D16 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

¼ S 26

S 66

: ð13Þ

Here, e y is strain in y direction 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 thesign is positive, the diameter of the cross-section tends to decrease,

or the curvature in the hoop direction (D26) 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 [±h2n]

has a zero coupling or zero for D16 and D26, because for every +h

there is h at the same distance from the mid-plane. However, a

laminate of the configuration [hn]S has a coupling. Here, the compo-nent D26 contributes to the buckling strength of the drive shaft.

Fig. 10 exhibits the effect of coupling on the buckling torque for

twoconfigurations having the same value of the component of nor-

mal bending stiffness in the hoop direction, D22.

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 stackingsequence, the shaft loses 46.07% of its buckling strength compared

0

500

1000

1500

2000

2500

B u c k l i n g T o r q u e ( N m )

[ 4 5 ,

- 4 5 ,

0 , 9 0

]

[ 9 0 ,

0 , 4 5

, - 4 5

]

[ 9 0 ,

0 , - 4 5 ,

4 5 ]

[ 9 0 ,

4 5 , 0

, - 4 5 ]

[ 0 , 4 5 ,

- 4 5 ,

9 0 ]

[ 0 , - 4 5 ,

4 5 , 9 0 ]

[ 9 0 ,

- 4 5 ,

0 , 4 5

]

[ 0 , 4 5 ,

9 0 , - 4 5

]

[ 9 0 ,

4 5 , - 4 5

, 0 ]

[ 0 , - 4 5 ,

9 0 , 4 5 ]

[ 4 5 ,

- 4 5 ,

9 0 , 0 ]

[ - 4 5 ,

4 5 , 9 0 ,

0 ]

[ 0 , 9 0 ,

4 5 , - 4 5

]

[ 0 , 9 0 ,

- 4 5 ,

4 5 ]

Fig. 9. Effect of stacking sequence on buckling torque.

0

1000

2000

3000

4000

5000

6000

0 10 20 30 40 50 60 70 80 90

Fiber Orientation Angle

B u c k l i n g T o r q u e ( N . m

)

Carbon-epoxy

Glass-epoxy

Carbon-epoxy

Glass-epoxy

- - - [±θº]2 [θº]4

Fig. 10. The effect on the buckling torque of coupling between twisting moment

and normal curvature in hoop direction.




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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[12] Swanson SR. Introduction to design and analysis with advanced composite

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[13] Ramamurti V. Computer-aided mechanical design and analysis. 3rd ed. New

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[14] Version 13.7-5. User’s Manual Lusas, In.

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