A new layerwise trigonometric shear deformation theory
for two-layered cross-ply beams
Rameshchandra P. Shimpi *, Yuwaraj M. Ghugal
Aerospace Engineering Department, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India
Abstract
A new layerwise trigonometric shear deformation theory for the analysis of two-layered cross-ply laminated beams is presented.
The number of primary variables in this theory is even less than that of first-order shear deformation theory, and moreover, it
obviates the need for a shear correction factor. The sinusoidal function in terms of thickness coordinate is used in the displacement
field to account for shear deformation. The novel feature of the theory is that the transverse shear stress can be obtained directly
from the use of constitutive relationships, satisfying the shear-stress-free boundary conditions at top and bottom of the beam and
satisfying continuity of shear stress at the interface. The principle of virtual work is used to obtain the governing equations and
boundary conditions of the theory. The effectiveness of the theory is demonstrated by applying it to a two-layered cross-ply laminated beam.
Keywords: Shear deformation; Laminated thick beam; Transverse shear stress; Interface shear continuity; Cross-ply beam
1. Introduction
The use of fiber-reinforced composite laminates has
greatly increased in weight sensitive applications such as
aerospace and automotive structures because of their high
specific strength and high specific stiffness. The increased
use of laminated beams in various structures has stimulated considerable interest in their accurate analysis. On
account of their low ratio of transverse shear modulus to
the in-plane modulus, shear deformation effects are more
pronounced in the composite beams subjected to transverse loads.
It is well-known that the classical Euler–Bernoulli
theory of beam bending, also known as elementary theory of bending (ETB), disregards the effects of the shear
deformation. The theory is suitable for slender beams
but not for thick or deep beams since it is based on the
assumption that the transverse normal to the neutral axis
remains so during bending and after bending, implying
that the transverse shear strain is zero. Since the theory
neglects the transverse shear deformation, it leads to less
accurate results in the case of isotropic thick beams and
more so in the case of laminated composite thick beams,
where shear effects are significant.
Bresse [1], Rayleigh [2], and Timoshenko [3] were the
pioneer investigators who included refined effects such as
the shear deformaton and rotatory inertia in the beam theory. Timoshenko showed that the effect of transverse shear
is much greater than that of rotatory inertia on the response
of transverse vibration of prismatic beams. This theory is
now widely referred to as the Timoshenko beam theory in
the literature. In this theory, transverse shear strain distribution is assumed to be constant through the beam
thickness and, thus, requires a shear correction factor to
appropriately represent the strain energy of deformation.
Kant and Manjunatha [4], Manjunatha and Kant [5],
Maiti and Sinha [6] and Vinayak et al. [7] used the
equivalent single layer, displacement based, higher-order
shear deformation theories (HSDT) in the analysis of
symmetric and unsymmetric laminated beams and
employed the finite-element method as a solution technique. These theories are the special cases of Lo et al. [8]
higher-order theory.
Levy [9] and Stein [10] developed refined plate theories expressing the displacement field in terms of
trigonometric functions to represent the thickness effect
and approximated the shear stress distribution through
the thickness.
Recently, Liu and Li [11] presented an overall comparison of laminate theories based on displacement
hypothesis emphasizing the importance of layerwise
1272
Nomenclature
Width of beam
Flexural rigidity
Modified flexural rigidity coefficient
as defined in Appendix
D1, D2, D3, D 3 Constants as defined in Appendix
E (1), E (2)
Young’s moduli of layer 1 and layer
2, respectively
G(1), G(2)
Shear moduli of layer 1 and layer 2,
respectively
h
Depth (i.e. thickness) of beam
L
Span of the beam
S
Aspect ratio (i.e. ratio of span to
depth of beam)
x, y, z
Rectangular coordinates
Neutral axis coefficient as defined in
Appendix
l1, l2
Coefficients as defined in Appendix
u
Non-dimensional inplane displacement
w
Non-dimensional transverse displacement
x
Non-dimensional inplane stress
CR
Non-dimensional transverse shear
zx
stress obtained from the constitutive
relationships
b
D
D
theories and also presented a series of quasi-layerwise
theories.
Lu and Liu [12] developed an interlaminar shear stress
continuity theory for composite laminates by using Hermite cubic shape functions. The theory is layer dependent
and the number of degrees of freedom involved is very
high and hence it is computationally complicated. Results
of single-layered, two-layered and three-layered crossply laminates for cylindrical bending were presented.
To improve the accuracy of the transverse stress prediction, layerwise higher-order theories based on
assumed displacements for individual layers, have been
proved to be very promising techniques in the flexural
analysis of thick laminates. Such theories were developed
and used by Lu and Liu [12], Li and Liu [13], Ambartsumyan [14], Reddy [15] and Reddy and Robbins [16].
A study of the literature indicates that, most of the
layerwise theories have been developed for symmetric
cross-ply laminates subjected to cylindrical bending.
Furthermore, higher order theories even with more than
three displacement variables (e.g. there are three in the
case of FSDT) appear to be insufficient and inefficient
when the laminated beam is unsymmetric, if the unsymmetry in the lay-up is not properly accounted for in the
theory. It is further noted that the research work dealing
with unsymmetric laminated beams using refined the-
EQL
zx
z
Superscripts
CR
EQL
Acronyms
ESCBP
CPT
ETB
FSDT
FEM
HSDT
HOST
HST
LTSDT-I
LTSDT-II
Non-dimensional transverse shear
stress obtained from the equilibrium
equations
Non-dimensional transverse normal
stress
Constitutive relationships
Equilibrium equations
Exact solution of cylindrical bending
of plate
Classical plate theory
Elementary theory of beam-bending
First-order shear deformation theory
Finite element method
Higher-order shear deformation
theory
Higher-order shear deformation
theory
Higher-order shear deformation
theory
Layerwise trigonometric shear deformation theory, Model I
Layerwise trigonometric shear deformation theory, Model II
ories is scarce. These facts have been commented upon
by Vinson and Chou [17] and Icardi [18]. Thus, there is
a need for a new layerwise refined shear deformation
theory with a minimum number of displacement variables, for shear-deformable laminated beams.
More recently, Shimpi and Ghugal [19] developed a simple layerwise trigonometric shear deformation theory
(designated here as LTSDT-I) for flexural analysis of crossply laminated beams. However, the displacement model of
the theorysuffered fromthe defectinthattherewas anunbalancedsmallresultantforcealongthelengthwisedirectionof
the beam. Removing this deficiency, an improved layerwise
displacementmodelis presented in this paper and will be
referred to as LTSDT-II.
The constitutive relationships in respect of transverse
shear stress and shear strain in each layer are satisfied
and also interface shear stress continuity is satisfied in
the proposed theory. In order to verify the accuracy of
the theory, it has been applied to two-layered cross-ply
[90/0] laminated beam.
2. Theoretical formulation
The theoretical formulation of a cross-ply laminated
beam based on certain kinematical and physical
1273
assumptions is presented. The variationally correct
forms of differential equations and boundary conditions,
based on the assumed displacement field are obtained
using the principle of virtual work.
The beam under consideration consists of two layers:
layer 1 and layer 2.
Layer 1 (90 layer) occupies the region:
04x4L;
b=24y4b=2;
h=24z40
ð1Þ
dw
uð1Þ ðx; zÞ ¼ ðz hÞ
dx
z=h þ h C1 þ C2 sin
ðxÞ
2 0:5 þ dw
uð2Þ ðx; zÞ ¼ ðz hÞ
dx
z=h þ h C3 þ sin
ðxÞ
2 0:5 ð3Þ
ð4Þ
Layer 2 (0 layer) occupies the region:
wðxÞ ¼ wðxÞ
04x4L;
b=24y4b=2;
04z4h=2
ð2Þ
where x, y, z are Cartesian coordinates, L is the length,
b is the width and h is the total depth of beam. The
beam is subjected to transverse load of intensity q(x) per
unit length of the beam. The beam can have any meaningful boundary conditions.
ð5Þ
Here u(1) and u(2) are the inplane displacement components in the x direction, superscripts 1 and 2 refer to
layer 1 and layer 2, w(x) is the transverse displacement
in the z direction and C1, C2, C3 and are the constants
(given in the Appendix). The function (x) is a rotation
function or the warping function of the cross-section of
the beam.
2.1. Assumptions made in the theoretical formulation
2.3. Superiority of the present theory
1. The axial displacement consists of two parts:
2.2. The displacement field
The present theory is a displacement-based layerwise
theory, and layerwise higher-order or refined sheardeformation theories are known to be successful techniques for improving the accuracy of displacement and
stresses [11].
The kinematics of the present theory is much richer
than those of the higher order shear deformation theories available in the literature, because if the trigonometric term (involving thickness coordinate z) is expanded
in power series, the kinematics of higher order theories
(which are usually obtained by power series in thickness
coordinate z) are implicitly taken into account to good
deal of extent.
Also, it needs to be noted that every additional power
of thickness coordinate in the displacement field of
other higher-order theories of Lo et al. [8] type not only
introduces additional unknown variables in those theories but these variables are also difficult to interpret
physically [16]. Thus use of the sine term in the thickness coordinate (in the kinematics) enhances the richness of the theory, and also results in the reduction of
the number of unknown variables as compared to other
theories [5,7] without loss of physics of the problem in
modelling.
The number of primary variables in this theory is even
less than that of first-order shear-deformation theory,
and moreover, it obviates the need of a shear correction
factor. Thus, the displacement field chosen is superior to
those of others.
Based on the before mentioned assumptions, the displacement field of the present layerwise beam theory is
given as below:
2.3.1. Strains
Normal and transverse shear strains for layers 1
and 2
(a) displacement which is analogous to that given
by elementary theory of bending;
(b) displacement due to shear deformation, which
is assumed to be sinusoidal in nature with respect
to thickness coordinate, such that maximum shear
stress occurs at neutral axis as predicted by the
elementary theory of bending of beam.
2. Displacement u is such that the resultant of
inplane stress, x, acting over the cross-section
should result in only bending moment and should
not result in force in the x direction.
3. The transverse displacement is assumed to be a
function of longitudinal length coordinate of beam.
4. The displacements are small compared to beam
thickness.
5. The layers are perfectly bonded to each other.
6. The stacking sequence of layers is such that there
is no bending-twisting coupling.
7. The body forces are ignored in the analysis. (The
body forces can be effectively taken into account
by adding them to the external forces.)
8. One-dimensional constitutive laws are assumed for
each layer.
9. The beam is subjected to lateral load only.
1274
@uð1Þ
d2 w
¼ ðz hÞ 2
@x
dx z=h d
þ h C1 þ C2 sin
2 0:5 þ dx
ðx1Þ ¼
ð2Þ
zx
2
ð2 Þ
@u
dw
¼ ðz hÞ 2
@x
dx
z=h d
þ h C3 þ sin
2 0:5 dx
@uð1Þ dw
C2
z=h þ
cos
¼
¼
@z
dx 1 þ 2
2 0:5 þ @uð2Þ dw
z=h þ
¼
cos
¼
@z
dx 1 2
2 0:5 ðx2Þ ¼
ð1Þ
zx
ð6Þ
ð7Þ
ð8Þ
ð9Þ
2.3.2. Stresses
According to assumption (8), one-dimensional constitutive laws are used to obtain the normal bending and
transverse shear stresses for layers 1 and 2.
Eð1Þ ðx1Þ
ð1Þ
ð1 Þ
zx
¼ Gð1Þ zx
¼
ð2Þ
ð2 Þ
zx
¼ Gð2Þ zx
¼
ð1Þ
G ð1 Þ C 2
z=h cos
2 0:5 þ 1 þ 2
ð11Þ
ð16Þ
The associated boundary conditions obtained are of
the following form:
d3 w
d2 D1 2 ¼ 0 or w
3
dx
dx
d2 w
d
dw
¼ 0 or
D1
2
dx
dx
dx
is prescribed
is prescribed
ð17Þ
ð18Þ
ð19Þ
where D, D1, D2, D3 are the constants as defined in
Appendix.
Thus the variationally consistent governing differential equations and boundary conditions are obtained.
The flexural behaviour of beam is given by the solution
of these equations and simultaneously satisfaction of
the associated boundary conditions.
3. Illustrative example
ð13Þ
Using the expressions (6)–(13) for strains and stresses
and principle of virtual work, variationally consistent
differential equations and boundary conditions for the
beam under consideration are obtained. The principle of
virtual work when applied to the beam leads to:
ð x¼L ð z¼0
ð1 Þ ð1 Þ
ð1Þ ð1Þ
b
x x þ zx
zx dx dz
ð2 Þ ð2 Þ
xð2Þ ðx2Þ þ zx
zx dx dz
þb
x¼0 z¼0
ð x¼L
q w dx ¼ 0
d3 w
d2 D
þ D3 ¼ 0
2
dx3
dx2
ð12Þ
2.4. Governing equations and boundary conditions
x¼0 z¼h=2
ð x¼L ð z¼þh=2
ð15Þ
ð10Þ
Gð2Þ z=h cos
1 2
2 0:5 d4 w
d3 q
D
¼0
1
dx4
dx3 D
d2 w
d
¼ 0 or is prescribed
D2
2
dx
dx
d2 w
ðz hÞ 2
¼
¼E
dx
z=h d
þ h C1 þ C2 sin
2 0:5 þ dx
d2 w
xð2Þ ¼ Eð2Þ ðx2Þ ¼ Eð2Þ ðz hÞ 2
dx
z=h d
þ h C3 þ sin
2 0:5 dx
xð1Þ
(i.e. w and ), we obtain the governing equations and
the associated boundary conditions. The governing
equations are as follows:
ð14Þ
x¼0
where the symbol denotes the variational operator.
Employing Green’s theorem in Eq. (14) successively
and collecting the coefficients of the primary variables
A simply supported two-layered [90/0] composite
beam, wherein layers 1 and 2 occupy the regions given
by expressions (1) and (2), respectively, is considered for
detailed numerical study. The beam is subjected to single sinusoidal load q(x)=qo sin (x=L), acting in the z
direction, where qo is magnitude of the sinusoidal loading at midspan. This is the same problem, which was
also considered by Pagano [20]. The material of the
beam layers is a carbon-fibre/epoxy uni-directional
composite. The following has been assumed:
Eð2Þ
¼ 25;
Eð1Þ
G ð1 Þ
¼ 0:20;
Eð1Þ
G ð2 Þ
¼ 0:02
Eð2Þ
Superscripts (1) and (2) refer to layers 1 and 2,
respectively.
3.1. The solution scheme
The following is the form of the solution for w(x) and
(x) that satisfies boundary conditions perfectly:
1275
wðxÞ ¼ w1 sin
x
L
ðxÞ ¼ 1 cos
x
L
where w1 and 1 are constants.
Using the assumed solution scheme and the governing
equations, the unknown functions w(x) and (x) can be
found. Expressions for transverse displacement w and
function are given as follows:
w¼
¼
q0 hS4 l1
x
sin
L
Eð2Þ b4 D
q0 S 3 l2
Eð2Þ b3 D
cos
x
L
ð20Þ
ð21Þ
where b, D , E(2), h, S, l1 and l2 are defined in the
Nomenclature and/or Appendix. Substituting expressions for w and given by Eqs. (20) and (21) in Eqs.
(3)–(5), final expressions for in-plane displacements can
be obtained.
q0 S2 Eð1Þ z
l1 l2
xð1Þ ¼
h
b 2 D Eð2Þ
z=h x
C1 þ C2 sin
sin
2 0:5 þ L
ð24Þ
q0 S2 z
l1 l2
¼
b 2 D h
z=h x
C3 þ sin
sin
2 0:5 L
ð25Þ
xð2Þ
3.4. Expressions for transverse shear stresses derived
from the constitutive relationships, tCR
zx
It may be noted that it is possible to obtain transverse
shear stress, zx, by using either the constitutive relaCR
tionships or the equilibrium equations. Notation zx
denotes zx obtained by using constitutive relationships.
ð1Þ CR
CR
zx
¼ zx
ifh=24z40
ð2Þ CR
if 04z4h=2
¼ zx
ð1Þ CR
ð2Þ CR
where zx
and zx
are as follows:
3.2. Expressions for inplane displacement, u
u ¼ uð1Þ
¼u
ifh=24z40
ð2Þ
ð1Þ CR
zx
¼
if 04z4h=2
q0 S3 Gð1Þ
b 3 D Eð2Þ
C2 l2 z=h x
cos
cos
1 þ 2
2 0:5 þ L
ð26Þ
where u(1) and u(2) are as follows:
u
ð1 Þ
q0 h S3
¼
Eð2Þ b 3 D z
z=h x
l1 þ C1 þ C2 sin
l2 cos
h
2 0:5 þ L
ð22Þ
q0 h S3
u ð2 Þ ¼
Eð2Þ b 3 D z
z=h x
l1 þ C3 þ sin
l2 cos
h
2 0:5 L
ð23Þ
Substituting expressions for w and given by Eqs.
(20) and (21) in Eqs. (10)–(13), the final expressions for
in-plane and transverse shear stresses can be obtained.
3.3. Expressions for inplane stress, sx
x ¼
¼
xð1Þ
xð2Þ
ifh=24z40
if 04z4h=2
where xð1Þ and xð2Þ are as follows:
ð2Þ CR
zx
¼
q0 S3 Gð2Þ
b 3 D Eð2Þ
l2 z=h x
cos
cos
1 þ 2
2 0:5 L
ð27Þ
3.5. Expressions for transverse shear stress, tEQL
zx , and
transverse normal stress, z, obtained from the
equilibrium equations
The following equilibrium equations of two-dimensional elasticity, ignoring body forces, are used to
obtain transverse shear and transverse normal stresses.
@x @zx
þ
¼0
@x
@z
ð28Þ
@zx @z
þ
¼0
@x
@z
ð29Þ
To obtain zx, we integrate Eq. (28) layerwise, w.r.t
the thickness coordinate z and impose the following
boundary condition at bottom of the beam
1276
ð2 Þ
zx
z¼h=2
¼0
ð30Þ
and the stress-continuity condition
ð1 Þ
zx
z¼0
ð2Þ ¼ zx
z¼0
ð31Þ
q0 S
ð2Þ EQL
¼
zx
9
8 b D 2
z
1
z
1 >
>
>
>
>
>
=
< h 2 h þ 8 2 l1
x
cos
>
>
L
z
1
1
2
z=h
>
>
> þ C3 cos
l2 >
;
:
h 2
2 0:5 ð35Þ
at interface to get constants of integrations and to
maintain the continuity of transverse shear stress at
layer interface.
To obtain z, we substitute the expressions obtained
for shear stresses Eq. (29) and integrate layerwise, w.r.t
the thickness coordinate z and impose the following
boundary condition at bottom of the beam
ð2 Þ
zz
z¼h=2
¼0
ð32Þ
and the stress-continuity condition
ð1 Þ
zz
z¼0
ð2 Þ ¼ zz
z¼0
ð33Þ
at interface to obtain constants of integrations and to
maintain the continuity of transverse normal stress at
layer interface.
Using the above procedure, expressions are obtained
EQL
for transverse shear zx
and transverse normal z
stresses in their following regions of through-thickness
variation:
EQL
(Notation zx
denotes shear stress zx as obtained by
using equilibrium equations).
ð1Þ EQL
EQL
¼ zx
zx
ð2Þ EQL
¼ zx
z ¼ zð1Þ
¼ zð2Þ
ð36Þ
q0
b D
8 9
1 z 3 z 2 1 z 1
>
>
>
>
>
þ
l1 >
>
>
>
>
6
h
2
h
8
h
8
24
>
>
>
>
= x
< C z2 z 1
3
þ
l2
sin
h
h
4
2
>
>
L
>
>
>
>
2
>
>
>
>
1
2
z=h
>
>
>
1 sin
l2 >
;
:
2 0:5 zð2Þ ¼ if h=24z40
ð37Þ
if 04z4h=2
if h=24z40
if 04z4h=2
ð1Þ EQL ð2Þ EQL ð1Þ
where zx
, zx
, z and zð2Þ are as follows:
q0 Eð1Þ
zð1Þ ¼ Eð2Þ
9
8 b D 1 z 3 z 2 z 1 z
>
>
>
>
l1
>
>
>
>
>
>
6
h
2
h
2
h
8
h
>
>
>
>
>
>
>
>
ð
2
Þ
2
>
>
E
1
C
z
z
1
>
>
>
>
þ
þ
l
l
>
>
1
2
>
>
ð
1
Þ
>
>
24
8
h
h
E
2
>
>
>
>
>
>
>
>
2
= x
<
1 þ 2
z=h þC2
sin
l2
sin
2 0:5 >
>
L
>
>
>
>
>
>
2
>
>
1
þ
2
>
>
>
>
>
>
C2
sin
l2
>
>
>
>
2 0:5 þ >
>
>
>
"
#
>
>
2 n
>
> ð2Þ >
>
o
>
>
E
C
1
2
3
>
>
>
>
þ
1
sin
l
>
>
2
;
:
ð
1
Þ
2 0:5 E
8
q0 S Eð1Þ
ð1Þ EQL
¼
zx
b D Eð2Þ
9
8 z
1 z2 Eð2Þ 1 >
>
>
>
þ
l
>
>
1
ð1Þ
>
>
h
2
h
8
2
E
>
>
>
>
>
>
>
>
>
>
z
1
þ
2
z=h
>
>
>
>
>
þ
C
C
cos
l
1
2
2>
=
<
h
2 0:5 þ x
cos
>
>
L
1 þ 2
>
>
>
>
þC2
cos
l2
>
>
>
>
2
0:5
þ
>
>
>
>
> >
>
>
>
>
>
>
Eð2Þ C3
1 2
>
>
;
:
þ
l
cos
2
2 0:5 Eð1Þ
2
ð34Þ
Thus expressions for displacements and stresses have
been obtained in their explicit forms.
4. Numerical results
The results obtained for displacements and stresses at
salient points are presented in Tables 1–6 and in Figs. 1–
3 in non-dimensional parameters.
4.1. Non-dimensional displacements and stresses
The results for inplane displacement, transverse displacement, inplane and transverse stresses are presented
in the following non-dimensional form in this paper.
u ¼
Eð1Þ bu
;
qo h
w ¼
100Eð1Þ bh3 w
;
qo L 4
x ¼
bx
;
qo
1277
zx ¼
bzx
;
qo
z ¼
bz
qo
The percentage difference (% Diff.) in results obtained
by models of other researchers with respect to the corresponding results obtained by the present theory
(LTSDT-II) is calculated as follows:
%Diff: ¼
value obtained by other modelcorresponding value by LTSDT-II
100
value by LTSDT-II
5. Discussion
For the problem under consideration, it may be noted
that the exact results are not available in the literature
to the best of the authors’ knowledge. In the absence of
exact results, researchers have compared their results
with the exact solution for the cylindrical bending of
plate (ESCBP) obtained by Pagano [20] for composite
laminates.
Kant and Manjunatha [4], Manjunatha and Kant [5],
Maiti and Sinha [6] and Vinayak et al. [7] used the
equivalent single layer (ESL) theories in the analysis of
laminated beams and provided the finite element solutions.
It is well known that the cylindrical-plate-bending
problem, though near to a beam-bending problem, is a
different problem altogether, in which the Poisson’s
effect figures. In this paper also, results obtained by Lu
and Liu [12] and Pagano [20] are quoted for the sake of
the record. Also, results using elementary theory of
beam bending are quoted.
In LTSDT-II, stress/strain relationships are satisfied
in both the layers and the interface continuity for
transverse shear stress is maintained, also the closed
form solution has been obtained for the problem under
consideration. Because of these reasons, it is believed
that the LTSDT-II results are superior.
effect in beam theory. Results of present
LTSDT-II model and HOSTB5 beam model of
Kant and Manjunath [4] are comparable within
3.46%. The maximum value of this displacement
using LTSDT-II has been found to decrease by
0.43% as compared to that of LTSDT-I for the
aspect ratio 4.
(b) For the aspect ratio 10, the value of this displacement obtained by LTSDT-I is found to
increase by 0.11% when compared to LTSDT-II.
The values by Lu and Liu [12] and Pagano [20]
are found to decrease by 2.58% for aspect ratio
10.
(c) The ETB underestimates the value by 23.28% as
compared to that of LTSDT for the aspect ratio
4. For aspect ratio 10, ETB underestimates the
result by 4.66% compared to LTSDT-II. In general, ETB underpredicts the inplane displacements as compared to those given by higher
order models.
5.2. Transverse displacement
The variation of maximum nondimensionalised
transverse displacement with various aspect ratios, is
shown in Fig. 1. The comparison of maximum transverse displacements for the aspect ratios of 4 and 10 is
presented in Tables 2 and 3. It should be noted that the
deflection of cylindrical bending of a plate strip is
somewhat smaller than the corresponding beam deflection due to the plate action in plate strip.
(a) It is seen from the comparison that the results for
transverse displacement of present model are
comparable with the results of higher order
5.1. In-plane displacement
The comparison of results of maximum nondimensionalised inplane displacement and the percentage difference in results of other models with respect to
results of present model is shown in Table 1 for the
aspect ratios 4 and 10. These displacements are calculated at left end of the beam (i.e. at x=0).
(a) Results of in-plane displacement of other higherorder models and the present model are on the
high side compared to the exact results of
cylindrical bending of plate for the aspect ratio 4
as a consequence of the neglect of the Poisson’s
Fig. 1. Variation of maximum transverse deflection w at x=0.5 L with
aspect ratio S.
1278
placement by 11.64% as compared to that of the
present model. It is seen from the Fig. 2 that the
LTSDT-II converges to the ETB results asymptotically as the aspect ratio increases.
models of Lu and Liu [12], Vinayak et al. [7] and
the finite element solution of first order shear
deformation theory by Maiti and Sinha [6].
However, higher order model of Maiti and Sinha
underestimates the deflection by 25.49% as
compared to that given by the present model for
the aspect ratio 4 and this may be due to the lack
of C1 continuity in the finite element formulation.
(b) For aspect ratio 10 results of LTSDT-II model,
HOSTB5 model of Manjunatha and Kant [5], Lu
and Liu [12], model and Pagano [20] are in close
agreement with each other. No significant difference is observed between the value of transverse
displacement obtained by LTSDT-II and
LTSDT-I for the aspect ratio 4 and 10.
(c) The ETB underestimates the results for transverse displacement by 44.60% as compared to
result of LTSDT-II for the aspect ratio 4; and for
the aspect ratio 10, ETB underestimates the dis-
5.3. In-plane stress
This component of stress is directly evaluated using
the constitutive law and strain displacement relations.
The comparison of maximum nondimensionalised
inplane stresses obtained by the present theory and
other refined theories is presented in the Tables 2 and 3
for the aspect ratios of 4 and 10 respectively. Fig. 2
shows through the thickness variation of inplane stress
for the aspect ratio 4, at midspan of beam.
(a) It can be seen that the maximum value of inplane
stress obtained by LTSDT-II is comparable with
the values of Manjunatha and Kant [5], Vinayak
et al. [7], Lu and Liu [12] and Pagano [20] for the
Table 1
Comparison of non-dimensional in-plane displacement (u)
For S=4 when x=0 and z= h=2
For S=10 when x=0 and z= h=2
a
Source
Model
u
Percentage difference
u
Present
Shimpi and Ghugal [19]
Kant and Manjunatha [4]
Lu and Liu [12]
Pagano [20]
Euler–Bernoulli
LTSDT-II
LTSDT-I
HOSTB5
HSDT
ESCBP
ETB
5.0353
5.0571
4.8611
4.7143
4.5000
3.8627
0.00
0.43
3.46
6.37
10.63
23.28
63.3050
63.3790
–
61.6667
61.6667
60.3540
a
Percentage differencea
0.00
0.11
–
2.58
2.58
4.66
Percentage difference quoted is with respect to the corresponding value obtained using LTSDT-II.
Table 2
Comparison of non-dimensional transverse displacement (w ) and in-plane stress ( x )
For S=4
Source
Model
when x=0.5 L
w
when x=0.5 L and z= h=2
a
% Diff.
x
a
% Diff.
when x=0.5 L and z=h=2
x
% Diff.a
Present
Shimpi and Ghugal [19]
Manjunatha and Kant [5]
LTSDT-II
LTSDT-I
HOSTB5
FEM
4.7437
4.7431
4.2828
0.00
0.01
9.71
3.9547
3.9650
3.7490
0.00
0.26
5.20
30.5650
30.2980
27.0500
0.00
0.87
11.50
Maiti and Sinha [6]
HST
FEM
3.5346
25.49
2.3599
40.33
25.7834
15.64
Maiti and Sinha [6]
FSDT
FEM
4.7898
0.97
3.1514
20.31
29.1144
4.75
Vinayak et al. [7]
HSDT
FEM
4.5619
3.83
4.0000
1.15
27.0000
11.66
Lu and Liu [12]
Pagano [20]
Euler and Bernoulli
HSDT
ESCBP
ETB
4.7773
4.3276
2.6281
0.71
8.77
44.60
3.5714
3.8359
3.0386
9.69
3.00
23.16
30.0000
30.0290
27.9540
1.85
1.75
8.54
a
Percentage difference quoted is with respect to the corresponding value obtained using LTSDT-II.
1279
Table 3
Comparison of non-dimensional transverse displacement (w ) and in-plane stress ( x }
For S=4
Source
Model
when x=0.5 L
w
when x=0.5 L and z= h=2
% Diff.a
when x=0.5 L and z=h=2
x
% Diff.a
x
% Diff.a
Present
Shimpi and Ghugal [19]
Manjunatha and Kant [5]
LTSDT-II
LTSDT-I
HOSTB5
FEM
2.9744
2.9743
2.8986
0.00
0.00
2.55
19.8880
19.8999
19.7100
0.00
0.06
0.90
177.140
176.870
173.000
0.00
0.15
2.34
Lu and Liu [12]
Pagano [20]
Euler and Bernoulli
HSDT
ESCBP
ETB
3.0000
2.9569
2.6281
0.86
0.59
11.64
20.0000
20.0000
18.9910
0.56
0.56
4.51
175.000
175.000
174.710
1.21
1.21
1.37
a
Percentage difference quoted is with respect to the corresponding value obtained using LTSDT-II.
Fig. 2. Variation of inplane stress x through the thickness for aspect
ratio 4 at x=0.5 L.
aspect ratio of 4. Results of higher order model
of Maiti and Sinha [6] in both the layers deviate
significantly (by 40.33 and 15.64%) from those
of the present model, however, their FSDT solution gives comparable value of this stress in the
0o layer. LTSDT-I has exhibited an increase in
value of this stress in the 90 layer by 0.26% and
decrease in the value by 0.87% in 0 layer as
compared to LTSDT-II for the aspect ratio 4.
(b) For the aspect ratio of 10, the stresses given by
LTSDT-II, higher order models and cylindrical
bending of plate (Pagano [20]) are in good
agreement with each other with negligible variation in the results. However, ETB underestimates
the value of inplane stress by 4.51% in the 90
layer and by 1.37% in the 0 layer as compared
to the maximum value of inplane stress given by
present model. No significant difference is found
between the values of inplane stress for the
aspect ratio 10 in both the layers.
(c) It is observed from the Tables 2 and 3 that the
ETB underestimates the maximum value of
Fig. 3. Variation of transverse shear stress zx through the thickness
aspect ratio 4 at x=0.0.
inplane stress by 23.16% as compared to the
value obtained by the present model for the
aspect ratio of 4 in the 90 layer. In the 0 layer,
ETB underpredicts the value of inplane stress by
8.54% as compared to value of LTSDT-II. The
values of inplane stress in 90 and 0 layers given
by the present model are in close agreement with
those of Pagano [20].
5.4. Transverse shear stress
The transverse shear stresses are obtained directly
from the constitutive relationships ( CR
zx ) or, alternatively, from the equilibrium equations of two dimensional elasticity ( EQL
zx ). The transverse shear stress
satisfies the stress free boundary conditions on the top
and bottom surfaces of the beam and continuity at layer
interface when these stresses are obtained by both the
above mentioned approaches.
1280
0.81%, however, the model of Maiti and Sinha
[6] underpredicts this value by 4.77% for the
aspect ratio 4. LTSDT-I has exhibited a decrease
in this value by 2.1% as compared to that of
LTSDT-II for the aspect ratio 4. The use of constitutive relations showed the underestimation of
shear stresses in 90 layer and overestimation of
same in the 0 layer as compared to those obtained
by direct integration of equilibrium equations for
the aspect ratio 4.
(b) It may be seen from the Fig. 3 that the through
thickness distribution of this stress obtained by
use of constitutive relations is more closer to that
of ETB in the 0 layer. The results of LTSDT-II
using constitutive relations are comparable with
those of FSDT and ETB for the aspect 4. However, ETB overestimates this value by 9.96%
when compared with that of LTSDT-II obtained
The through thickness variation of shear stress for the
aspect ratio 4 is given in the Fig. 3, in which the variation obtained by using constitutive relations of the present theory is denoted by LTSDT-II- CR curve and the
same obtained by using equilibrium equations is denoted by LTSDT-II- EQL curve. The comparison of
maximum nondimensionalised shear stresses for aspect
ratios 4 and 10 is presented in the Tables 4 and 5
respectively.
(a) The maximum transverse shear stress obtained
by HOSTB5 model of Manjunatha and Kant [5]
via constitutive relations underpredicts the value
by 35.54% while that obtained by direct integration overestimates the value by 5.44% for the
aspect ratio 4. The model of Vinayak et al. [7]
overpredicts the value by 2.67%, and that is
given by Pagano [20] is comparable within
Table 4
EQL
Comparison of maximum transverse shear stress CR
and transverse normal stress ( z )
zx , zx
For S=4
Source
Model
when x=0.0 and z= h
when x=0.0 and z= h
when x=0.5 L and z=h=2
CR
zx
% Diff.a
EQL
zx
% Diff.a
z
% Diff.a
Present
Shimpi and Ghugal [19]
Manjunatha and Kant [5]
LTSDT-II
LTSDT-I
HOSTB5
FEM
2.9895
2.9886
1.9270
0.00
0.03
35.54
2.6784
2.6222
2.8240
0.00
2.10
5.44
1.00
0.98
1.00
0.00
2.00
0.00
Maiti and Sinha [6]
HST
FEM
–
–
2.4252
9.45
–
–
Maiti and Sinha [6]
FSDT
FEM
2.8467
–
–
–
–
Vinayak et al. [7]
HSDT
FEM
HSDT
ESCBP
ETB
–
–
2.7500
2.7925
–
–
6.59
–
–
–
2.7000
2.9453
Lu and Liu [12]
Pagano [20]
Euler and Bernoulli
a
4.77
2.67
1.00
0.81
9.96
–
1.00
1.00
–
0.00
–
0.00
0.00
Percentage difference quoted is with respect to the corresponding value obtained using LTSDT-II.
Table 5
EQL
and transverse normal stress ( z )
Comparison of non-dimensional transverse shear stress CR
zx , zx
For S=4
Source
Present
Shimpi and Ghugal [19]
Manjunatha and Kant [5]
Lu and Liu [12]
Pagano [20]
Euler and Bernoulli
a
Model
LTSDT-II
LTSDT-I
HOSTB5
FEM
HSDT
ESCBP
ETB
when x=0.0 and z= h
when x=0.0 and z= h
when x=0.5 L and z=h=2
CR
zx
% Diff.
EQL
zx
% Diff.
z
% Diff.a
7.6948
7.6945
4.9130
0.00
0.00
36.15
7.2638
7.2406
7.2820
0.00
0.32
0.25
1.00
0.99
1.00
0.00
1.00
0.00
7.3000
–
–
5.13
–
–
–
7.3000
7.3643
–
–
1.00
1.00
–
a
0.49
1.38
Percentage difference quoted is with respect to the corresponding value obtained using LTSDT-II.
a
0.00
0.00
1281
Table 6
Comparison of variation of non-dimensional transverse normal stress ( z ) through the thickness of beam ( z at x=0.5 L and S=4)
LTSDT-II
Present
LTSDT-I
Shimpi and Ghugal [19]
HOSTB5
Manjunatha and Kant [5]
ETB
z/h
ESCBP
Pagano [20]
0.5
0.4
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
0.5
1.0000
0.9886
0.9571
0.9097
0.8500
0.7815
0.6525
0.4574
0.2499
0.0768
0.0000
0.9803
0.9688
0.9373
0.8897
0.8298
0.7609
0.6393
0.4500
0.2467
0.0759
0.0000
1.0000
0.9869
0.9552
0.9087
0.8508
0.7848
0.6581
0.4578
0.2432
0.0716
0.0000
1.0000
0.9911
0.9660
0.9274
0.8777
0.8197
0.6951
0.4869
0.2590
0.0754
0.0000
1.0000
0.9737
0.9605
0.9211
0.8684
0.7894
0.6711
0.4605
0.2500
0.0789
0.0000
by equilibrium equations, for the aspect ratio 4.
(c) For aspect ratio 10, the model of Manjunatha
and Kant [5], using constitutive relations, underestimates the maximum value of this stress by
36.15% and that of Lu and Liu [12] underpredicts it by 5.13% as compared to that
obtained by LTSDT-II. The value of this stress
in LTSDT-II, obtained by equilibrium equations, is in excellent agreement with other solutions as can be seen from Table 5 for aspect ratio
10.
This difference in transverse shear stresses obtained
by use of constitutive relations (CR) and equilibrium
equations (EQL) is well known in the bending analysis
of laminated beams/plates [5,21]. Since the displacement
solution can not give these stresses accurately when
constitutive relations are used, many authors [22] suggested use of stress equilibrium equations of elasticity to
obtain the transverse shear and transverse normal
stresses through the thickness of the laminated beams
more accurately, but their approach has been adversely
commented upon by Liu and Li [11], Bisegna and Sacco
[23]. Furthermore, this approach is more tedious compared to the approach using constitutive relations.
The discrepancy in the shear stresses obtained by
constitutive relations and by integration of equilibrium
equations is much less in the present case as compared
to other attempts [5]. Furthermore, overall distribution
of the shear stress obtained via constitutive relations is
reasonably accurate.
The difference between results is due to the two different approaches used to evaluate the stresses.
5.5. Transverse normal stress
The transverse normal stress is obtained by direct
integration of equilibrium equations of two dimensional
elasticity. The comparison of through thickness variation of this stress component is presented in the Table 6
for the aspect ratio of 4. An improved through the
thickness variation of this stress is obtained by the present model as compared to that given by the LTSDT-I
model. The results for this stress component obtained
by the present model LTSDT-II are in good comparison
with those of HOSTB5 model of Manjunatha and Kant
[5], ETB, and exact solution of cylindrical bending of
plate obtained by Pagano [20].
The above discussion in respect of the numerical
example, validates the efficacy of the LTSDT-II model.
6. Conclusions
In this paper a new model has been presented for
cross-ply [90/0] laminated composite beam. This model
is an improvement over the earlier model developed by
the authors. The earlier model suffered from the defect
in that there was an unbalanced small resultant force
along the lengthwise direction of the beam. This deficiency has been removed in the present model. The displacement based model LTSDT-II presented in this
paper has following advantages:
1. It is a displacement based layerwise model.
2. The number of displacement variables is less than
that in first order shear deformation theory. It
contains only two variables.
3. Transverse shear stress satisfies continuity conditions at layer interface and zero shear stress conditions at top and bottom surfaces of the beam.
4. The model does not require the use of shear correction factor.
5. Constitutive relations are satisfied in both the layers in respect of inplane stress and transverse shear
stress.
6. It is possible to get reasonable accuracy of transverse shear stress even when the shear stress is
obtained by the use of constitutive relations.
1282
7. The governing differential equations and the
boundary conditions are variationally consistent.
8. The use of the present model gives accurate results,
as has been seen from the numerical example studied.
Appendix
(a) Layer 1 integration constants A1, A2, A3, A4 are as
follows:
3
A1 ¼ b h E
ð1Þ
1 2
þ þ
24 4 2
9
8 1 >
>
>
>
>
>
C1 þ
>
>
>
>
8
2
>
>
>
2 >
=
<
1 þ 2
3
ð1Þ
A2 ¼ b h E
þC2
1 þ sin
>
>
>
1 þ2 >
>
>
>
>
>
>
1
þ
2
>
>
>
>
þ
C
cos
;
:
2
1 þ 2
9
8 2
C1
1 þ 2
>
>
>
>
2C1 C2
cos
=
<
1 þ 2
2 3
ð1Þ
A3 ¼ b h E
2
>
>
C
1 þ 2
>
>
;
:þ
1þ
sin
0:5 þ 4
b h Gð1Þ C2 2
1 þ 2
A4 ¼
1þ
sin
1 þ 2
0:5 þ 4
(b) Layer 2 integration constants B1, B2, B3, B4 are as
follows:
B1 ¼ b h3 Eð2Þ
1 2
þ
24 4 2
9
8 1 1
>
>
>
>
>
>
C3 >
>
>
>
8
2
>
>
>
>
2 h
=
< i
1 2
3
ð2Þ
B2 ¼ b h E
1 þ sin
þ
>
1 2 >
>
>
>
>
>
>
>
>
1
2
>
>
>
>
cos
;
: 1 2
9
8 2
C3
1 2
>
>
>
>
þ 2 C3
cos
=
<
1 2 2 3
ð2Þ
B3 ¼ b h E
1
1 2
>
>
>
>
;
:þ 1 sin
4
0:5 b h Gð2Þ 2
1 2
B4 ¼
1
sin
1 2
0:5 4
(c) Constants C1, C2, C3 are as follows:
Eð2Þ
C1 ¼
Eð1Þ þ Eð2Þ
9
8 >
>
>
>
sin
sin
C
2
>
>
>
>
1
2
1
þ
2
>
>
>
ð1Þ >
=
<
E
1 þ 2
þ2 C2 ð2Þ
cos
>
1 þ 2 >
E >
>
>
>
>
>
>
>
1
2
>
> 2
;
:
cos
1 2
cos
G 0:5 þ 1 2 C 2 ¼ ð1 Þ
G 0:5 cos
1þ2 ð2 Þ
9
8
>
>
>
>
sin
C
sin
2
>
>
>
>
1 2 1 þ 2
>
>
>
>
=
<
ð1 Þ
1 þ 2
E
þ2
C
cos
C3 ¼
2
Eð1Þ þ Eð2Þ >
>
>
1 þ 2 >
>
> ð2Þ >
>
>
E
1
2
>
>
>
;
: 2
cos
ð
1
Þ
1 2
E
(d) Constants D, D , D1, D2, D3, D 3 are as follows:
D ¼ ðA1 þ A2 Þ ¼ D Eð2Þ b h3
ð1Þ E
1 2
1 2
þ þ
þ
D¼
þ
24 4 2
Eð2Þ 24 4 2
A2 þ B2
A3 þ B3
D1 ¼
; D2 ¼
;
A1 þ B1
A2 þ B2
A4 þ B4
D3 ¼
A2 þ B2
D 3 ¼ D3 h2
(e) Constants S, , l1, l2, are as follows:
S ¼ L=h
1 Eð2Þ Eð1Þ
¼
4 Eð2Þ þ Eð1Þ
h
i
l1 ¼ 1 þ D1 = D2 D1 þ D 3 2
h
i
l2 ¼ 1= D2 D1 þ D 3 2
¼ S=
1283
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