A beam ®nite element based on layerwise trigonometric shear
deformation theory
R.P. Shimpi *, A.V. Ainapure
Department of Aerospace Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
Abstract
A simple one-dimensional beam ®nite element, based on layerwise trigonometric shear deformation theory, is presented. The
element has two nodes and only three degrees of freedom per node. Yet, it incorporates through the thickness sinusoidal variation of
in-plane displacement such that shear-stress free boundary conditions on the top and bottom surfaces of the beam element are
satis®ed and the shear-stress distribution is realistic in nature. Constitutive relations between shear-stresses and shear-strains are
satis®ed in all the layers, and, therefore, shear correction factor is not required. Compatibility at the layer interface in respect of inplane displacement is also satis®ed. It is to be noted that the element developed is free from shear locking. The results obtained are
accurate and show good convergence. Unlike many other elements, transverse shear-stresses are evaluated directly using constitutive
relations. The ecacy of the present element is demonstrated through the examples of static ¯exure and free vibration.
Keywords: Beam ®nite element; Laminated beam; Layerwise theory; Shear deformation
1. Introduction
Laminated composite beams are ®nding increasing
use due to their high strength to weight and high sti€ness
to weight ratios.
It is to be noted that, the topic of ®nite elements
pertaining to laminated beams does not appear extensively in literature as has been noted by Abramovich and
Livshits [1], Abramovich et al. [2], Maiti and Sinha [3],
Eisenberger et al. [4].
Even in case of laminated plate, layerwise models have
shown the superiority of layerwise approach to predict
accurately static and dynamic response of thick structures [5]. In layerwise theories, appropriate separate
displacement ®eld expressions are assumed for each
material layer, providing a kinematically better representation of the strain ®eld in discrete layer laminates [6].
Yuan and Miller [7] derived a ®ve noded beam ®nite
element but, the number of degrees of freedom is dependent on number of layers (having 3N ‡ 7 total degrees of freedom per element for an N layered beam).
Davalos et al. [8] presented a one-dimensional three
noded laminated beam ®nite element having 2 ‡ N de-
grees of freedom at each node for an N-layered beam.
The layerwise constant shear-stresses obtained from
constitutive relations are transformed into parabolic
shear-stress distribution through tedious post-processing operation.
In the present paper, a simple one-dimensional beam
®nite element, based on variationally consistent layerwise trigonometric shear deformation theory [9] is presented. The element has two nodes and only three
degrees of freedom per node. Yet, it incorporates
through the thickness sinusoidal variation of in-plane
displacement such that shear-stress free boundary conditions on the top and bottom surfaces of the beam element are satis®ed and the shear-stress distribution is
realistic in nature. Compatibility at the layer interface in
respect of in-plane displacement is satis®ed. Present element is free from shear locking. Transverse shear-stresses
are evaluated directly using constitutive relations. The
e€ectiveness of the present element is demonstrated
through illustrative examples.
2. About layerwise trigonometric shear deformation
theory
Layerwise trigonometric shear deformation theory
(LTSDT) has been discussed in [9]. The theory has been
154
extended to deal with free vibration [10]. Here, essential
information about LTSDT will be outlined in brief.
2.1. About the beam
The beam under consideration consists of two layers,
layer-1 and layer-2.
Layer-1 occupies the region:
0 6 x 6 L;
b=2 6 y 6 b=2;
h=2 6 z 6 0:
b=2 6 y 6 b=2;
0 6 z 6 h=2;
…2†
2.2. Stress±strain relations
Based on the assumptions made in respect of LTSDT
[9], the stress±strain relations for the layers can be
written as:
rx…2† ˆ E…2† e…2†
x ;
…1†
…1†
szx
ˆ G…1† czx
;
…2†
…2†
szx
ˆ G…2† czx
;
where superscripts (1) and (2) refer to layer-1 and layer2, respectively.
Elastic moduli E…1† ; E…2† and shear moduli G…1† ; G…2†
can be expressed in the notation used by Jones [11] as
follows:
E…1† ˆ E1
and
G…1† ˆ G31
if fibers in the layer-1 are along x-direction:
E
…1†
ˆ E2
and
u…1† ˆ
u…2† ˆ
where, x; y; z are right-handed Cartesian co-ordinates, L
is the length, b is the width and h is the total depth of
beam.
The layers of beam are perfectly bonded to each
other.
Local principal material axes (Cartesian) for each
layer are denoted as 1, 2 and 3. Direction 3 is such that it
coincides with transverse z-direction, and direction 1 is
parallel to the ®bers.
The material densities of layer-1 and layer-2 are denoted by q…1† and q…2† , respectively.
Material used for layers of beam is specially orthotropic and obeys Hooke's law [11].
The beam is subjected to lateral load q…x; t† on the top
surface (i.e. surface z ˆ h=2), where t denotes time.
rx…1† ˆ E…1† ex…1† ;
The displacement ®eld of the present layerwise beam
theory is given as follows:
…1†
Layer-2 occupies the region:
0 6 x 6 L;
2.3. The displacement ®eld
G…1† ˆ G23
ow
…z ah†
ox
p z=h a
‡ h C1 ‡ C2 sin
/…x; t†;
2 0:5 ‡ a
ow
…z ah†
ox
p z=h
‡ h C3 ‡ sin
2 0:5
a
a
…3†
/…x; t†;
w ˆ w…x; t†;
…4†
…5†
where u…1† and u…2† are the in-plane displacements in the
x-direction, superscripts (1) and (2) refer to layer-1 and
layer-2, w the transverse displacement in the z-direction,
t denotes time, and / is unknown function. Transverse
displacement w and unknown function / are same for
both the layers. C1 ; C2 ; C3 ; a are constants as given in
Appendix A.
2.4. Governing equations and boundary conditions
Using the principle of virtual work, variationally
consistent di€erential equations and associated boundary conditions for the beam under consideration at any
instant of time are obtained.
Governing di€erential equations are:
"
#
o2 M x
ou…1†
ou…2†
 ˆ q…x; t†;
I1
‡ I2
‡w
…6†
ox2
ox
ox
i
oMs h …1†
I3 u ‡ I4 u…2†
…7†
Vs ˆ 0;
ox
where overdot denotes di€erentiation w.r.t. time t, and
Mx ; Ms ; Vs ; I1 ; I2 ; I3 ; I4 are as given in Appendix A.
And, associated boundary conditions at x ˆ 0 and
x ˆ L are:
i
oMx h …1†
I1 u ‡ I2 u…2† ˆ 0 OR w is prescribed; …8†
ox
ow
is prescribed; …9†
OR
Mx ˆ 0
ox
Ms ˆ 0
OR / is prescribed: …10†
if fibers in the layer-1 are along y-direction:
E…2† ˆ E1
and
G…2† ˆ G31
if fibers in the layer-2 are along x-direction:
E
…2†
ˆ E2
and
G…2† ˆ G23
if fibers in the layer-2 are along y-direction:
3. Finite element formulation
The ®nite element formulation of two layered crossply laminated beam (based on LTSDT) using Galerkin
weighted residual method is outlined below.
155
3.1. Domain of the ®nite element under consideration
The domain of the beam is divided into number of
elements (say n ˆ 1 to N ). The typical nth beam ®nite
element under consideration consists of two layers, i.e.
layer-1 and layer-2.
Layer-1 occupies the region:
0 6 x n 6 Ln ;
bn =2 6 yn 6 bn =2;
hn =2 6 zn 6 0:
Layer-2 occupies the region:
0 6 x n 6 Ln ;
bn =2 6 yn 6 bn =2;
0 6 zn 6 hn =2;
where xn ; yn ; zn are element local co-ordinates (Cartesian), Ln is the length, bn is the width and hn is the total
height of the nth beam ®nite element. Element local coordinates are related to global co-ordinates by
xn ˆ x
ˆx
for n ˆ 1;
n 1
X
Lm
for n ˆ 2; 3; . . . N ;
for n ˆ 1; 2; 3; . . . N ;
zn ˆ z
for n ˆ 1; 2; 3; . . . N :
3.2. Weak forms of the di€erential equations
Using governing di€erential equations (6) and (7) and
Galerkin weighted residual method, we get the weak
form of di€erential equations for the nth element as
Z xn ˆLn o2 u 1
…1† ou1
…2† ou1
Mx 2 ‡ I1 
u
u
‡ I2 
oxn
oxn
oxn
xn ˆ0
oMx
 u1 q…xn ; t†u1 dxn ‡
‡w
u
oxn 1
xn ˆLn ou xn ˆLn
I1 
Mx 1
ˆ 0; …11†
u…1† ‡ I2 
u…2† u1
oxn xn ˆ0
xn ˆ0
xn ˆLn xn ˆ0
Ms
…1†
ou2
‡ Vs u2
oxn
…2†
‡ I3 
u u2 ‡ I4 
u u2
dxn
…13†
/ ˆ ‰N5 /1 ‡ N6 /2 Š sin xt;
…14†
where w1 ; h1 ; /1 are nodal degrees of freedom associated
with node 1 and w2 ; h2 ; /2 are nodal degrees of freedom
associated with node 2. In case of statics, the term sin xt
will be absent in expressions (13) and (14). Hermite
shape functions N1 ; N2 ; N3 ; N4 and Lagrange shape
functions N5 ; N6 are given as:
N 2 ˆ xn
The element is subjected to transverse normal load
q…x; t† on top surface (i.e. zn ˆ hn =2).
The element has two nodes, i.e. node 1 and node 2.
Local co-ordinates of node 1 and node 2 are …0; 0; 0† and
…Ln ; 0; 0† respectively.
Z
w ˆ ‰N1 w1 ‡ N2 h1 ‡ N3 w2 ‡ N4 h2 Š sin xt;
N1 ˆ 1
mˆ1
yn ˆ y
shape functions for / should be di€erentiable only once.
Since, in reality, slope is continuous across the elements,
it is better to approximate not only w but also slope to
be continuous across the elements.
Therefore, Hermite cubic shape functions are used for
approximating transverse de¯ection w, and Lagrange
linear shape functions are used for approximating
function /.
Now w and / can be approximated as:
n
‰Ms u2 Šxxnn ˆL
ˆ0 ˆ 0;
…12†
where u1 ; u2 are weighing functions and are functions of
xn .
3.3. Shape functions
Examination of Eqs. (11) and (12) suggests that, the
shape functions for the transverse displacement w
should be twice di€erentiable w.r.t. xn , whereas the
3…xn =Ln †2 ‡ 2…xn =Ln †3 ;
2…x2n =Ln † ‡ …x3n =L2n †;
N3 ˆ 3…xn =Ln †2 2…xn =Ln †3 ;
N4 ˆ
x2n =Ln ‡ x3n =L2n ;
N5 ˆ 1 …xn =Ln †;
N6 ˆ …xn =Ln †:
…15†
…16†
…17†
…18†
…19†
…20†
3.4. Finite element equations
In Eq. (11), using expressions (13) and (14) and replacing weighing function u1 successively by shape
functions N1 ; N2 ; N3 ; N4 (given by expressions (15)±(18)),
one obtains four equations.
Similarly, in Eq. (12), using expressions (13) and (14)
and replacing weighing function u2 successively by
shape functions N5 ; N6 (given by expressions (19) and
(20)), two additional equations are obtained.
These six equations can be written in matrix form as
®nite element equation for the nth element:
 ˆ ff g ;
‰kŠn fdgn ‡ ‰mŠn fdg
n
n
…21†
where ‰kŠn is element sti€ness matrix, ‰mŠn the element

mass matrix, fdgn the element displacement vector, fdg
n
the element acceleration vector and ff gn is element force
vector. It should be noted that element sti€ness and
mass matrices are symmetric in nature.
The elements of the element sti€ness matrix ‰kŠn are
given by:
Z xn ˆLn 2
d Ni d2 Nj
kij ˆ Ic1
dxn
dx2n dx2n
xn ˆ0
for i ˆ 1; 2; 3; 4 and j ˆ 1; 2; 3; 4
Z xn ˆLn 2
d Ni dNj
ˆ Ic2
dxn
dx2n dxn
xn ˆ0
for i ˆ 1; 2; 3; 4
and
j ˆ 5; 6
156
Z
ˆ Ic3
xn ˆLn
xn ˆ0
dNi dNj
dxn ‡ Ic4
dxn dxn
for i ˆ 5; 6
and
Z
xn ˆLn
xn ˆ0
here, and these results hardly di€er from those corresponding results obtained using 30 and 40 elements.
Ni Nj dxn
j ˆ 5; 6;
…22†
where constants Ic1 ; Ic2 ; Ic3 ; Ic4 are given in Appendix A.
The elements of the element mass matrix ‰mŠn are
given by:
Z xn ˆLn
Z xn ˆLn
dNi dNj
mij ˆ Im1
dxn ‡ Im4
Ni Nj dxn
dxn dxn
xn ˆ0
xn ˆ0
for i ˆ 1; 2; 3; 4 and j ˆ 1; 2; 3; 4
Z xn ˆLn
dNi
ˆ Im2
Nj dxn
dxn
xn ˆ0
for i ˆ 1; 2; 3; 4 and j ˆ 5; 6
Z xn ˆLn
ˆ Im3
Ni Nj dxn
xn ˆ0
for i ˆ 5; 6 and j ˆ 5; 6;
…23†
where constants Im1 ; Im2 ; Im3 ; Im4 are given in Appendix A.
The element displacement vector fdgn is such that:
T
fdgn ˆ ‰w1 h1 /1 w2 h2 /2 Š:
…24†
The elements of the element force vector ff gn are given
by:
Z xn ˆLn
fi ˆ
q…xn ; t†Ni dxn ‡ Q…i†
xn ˆ0
for i ˆ 1; 2; 3; 4
ˆ Q…i† for i ˆ 5; 6;
…25†
where expressions for Q…i† (i ˆ 1; 2; 3; 4; 5; 6) are given as:
oMx
u…1† ‡ I2 
u…2† u1
u1
;
Q…1† ˆ
I1 
oxn
xn ˆ0
ou
;
Q…2† ˆ Mx 1
oxn xn ˆ0
oMx
…1†
…2†
u
;
u ‡ I2 
u u1
I1 
Q…3† ˆ
oxn 1
xn ˆLn
ou
Q…4† ˆ Mx 1
;
oxn xn ˆLn
Q…5† ˆ ‰Ms u2 Šxn ˆ0 ;
Q…6† ˆ ‰Ms u2 Šxn ˆLn :
4.1. Example 1: Flexure of two layered simply supported
beam
A two layered cross-ply beam with layer-1 (90° layer)
and layer-2 (0° layer), occupies the regions given by
expressions (1) and (2), respectively. Both the layers are
of unidirectional graphite-epoxy. The beam is subjected
to transverse normal sinusoidal load q…x† ˆ q sin…px†=L
acting along z-direction, where q is magnitude of the
sinusoidal loading per unit length at midspan.
This is the same example, which was also considered
by Pagano [15] and many others (e.g. [3,12,14]). The
material properties of the beam material are such that:
E…2†
ˆ 25;
E…1†
G…1†
ˆ 0:20;
E…1†
G…2†
ˆ 0:02:
E…2†
For the discretised laminated beam, all the element
sti€ness matrices and force vectors are assembled using
routine procedure. This assembly gives global ®nite element equation for bending:
‰KŠfdg ˆ fF g;
…26†
where ‰KŠ; fdg; fF g are global sti€ness matrix, global
displacement vector and global force vector, respectively.
Solution of the global ®nite element equation (26) is
achieved after applying appropriate boundary conditions (Eqs. (8)±(10)). Results obtained for displacements
and stresses at salient points are presented using nondimensionalised parameters in Tables 1±3, and Figs. 1±3.
4.1.1. Non-dimensionalised parameters
The results in respect of transverse displacement, inplane normal bending stress, transverse shear-stress are
presented in the following non-dimensionalised form in
this paper.
wˆ
100 E…1† b h3 w
;
q L 4
rx ˆ
brx
;
q
szx ˆ
bszx
:
q
The percentage di€erence (% Di€) in results obtained by
models of various researchers with respect to the corresponding results obtained by the LTSDT±CFS are
calculated as follows:
% Diff ˆ f‰…value obtained by a model†
4. Illustrative examples
The e€ectiveness of the present element is demonstrated through following examples of static ¯exure and
free vibration. It may be noted that for obtaining numerical results of the examples, the beam was divided
into 2, 4, 10, 14, 20, 30 and 40 elements. For the sake of
brevity, results obtained using 20 elements are given
…value by LTSDT±CFS†Š
=…value by LTSDT±CFS†g 100:
4.2. Example 2: Free vibration of two layered simply
supported beam
A simply supported two layered cross-ply [90/0]
composite beam is considered. The two layered beam
157
Table 1
Comparison of non-dimensionalised maximum transverse displacement w of Example 1
At x ˆ 0:5L
Source and model
Sˆ4
S ˆ 10
w
Beam bending models
Present
LTSDT±FEM
Shimpi and Ghugal [9]
LTSDT±CFS
Manjunatha and Kant [12]
HOSTB5±FEM
Maiti and Sinha [3]
HST±FEM
Vinayak et al. [14]
HSDT±FEM
Euler-Bernoulli
ETB
Cylindrical plate bending models
Lu and Liu [13]
HSDT±CFS
Pagano [15]
Elasticity
a
a
% Di€
w
% Di€
4.7437
0.00
2.9744
0.00
4.7437
0.00
2.9744
0.00
4.2828
)9.71
2.8986
)2.55
3.5346
)25.49
)
)
4.5619
)3.83
)
)
2.6281
)44.60
2.6281
)11.64
4.7773
0.71
3.0000
0.86
4.6953
)1.02
2.9569
)0.59
a
% Di€ i.e. Percentage di€erence quoted is with respect to the corresponding value obtained using LTSDT±CFS.
Table 2
Comparison of non-dimensionalised transverse shear stress szx of Example 1
For S ˆ 4 at x ˆ 0:0 and z ˆ ah
Source and model
Beam bending models
Present
LTSDT±FEM
Shimpi and Ghugal [9]
LTSDT±CFS
Manjunatha and Kant [12]
HOSTB5±FEM
Maiti and Sinha [3]
HST±FEM
Vinayak et al. [14]
HSDT±FEM
Euler±Bernoulli
ETB
Cylindrical plate bending models
Lu and Liu [13]
HSDT±CFS
Pagano [15]
Elasticity
a
szx CR
% Di€
2.9951
a
szx EQM
% Di€
0.19
)
)
2.9895
0.00
2.6784
0.00
1.9270
)35.54
2.8240
5.44
)
)
2.4252
)9.45
)
)
2.7500
2.67
)
)
2.9453
9.96
2.7925
)
)6.59
)
)
2.7000
a
)
0.81
% Di€ i.e. Percentage di€erence quoted is with respect to the corresponding value obtained using LTSDT±CFS.
with layer-1 and layer-2, occupies the regions given by
expressions (1) and (2), respectively. The layer properties
of the beam material under consideration are:
E…1† ˆ 9:65 GPa;
E…2† ˆ 144:80 GPa;
G…1† ˆ 3:45 GPa;
G…2† ˆ 4:14 GPa;
…1†
q
…2†
ˆq
3
ˆ 1389:23 kg=m :
For the discretised laminated beam, all the element
sti€ness matrices and force vectors are assembled using
routine procedure. The assembly gives global ®nite element equation for free vibration
 ˆ f0g;
‰KŠfdg ‡ ‰MŠfdg
…27†
 are global sti€ness matrix, glowhere ‰KŠ; ‰MŠ; fdg; fdg
bal mass matrix, global displacement vector and global
acceleration vector, respectively. The global ®nite element equation (27), after applying appropriate boundary conditions (Eqs. (8)±(10)), is solved for eigenvalues
158
Table 3
Comparison of non-dimensionalised in-plane stress rx of Example 1
Source and model
For S ˆ 4, at x ˆ 0:5L and
zˆ
h=2
rx
Beam bending models
Present
LTSDT±FEM
Shimpi and Ghugal [9]
LTSDT±CFS
Manjunatha and Kant [12]
HOSTB5±FEM
Maiti and Sinha [3]
HST±FEM
Vinayak et al. [14]
HSDT±FEM
Euler-Bernoulli
ETB
Cylindrical plate bending models
Lu and Liu [13]
HSDT±CFS
Pagano [15]
Elasticity
a
z ˆ h=2
% Di€
a
rx
% Di€
)3.9589
0.11
30.6167
0.17
)3.9547
0.00
30.5650
0.00
)3.7490
)5.20
27.0500
)11.50
)2.3599
)40.33
25.7834
)15.64
)4.0000
1.15
27.0000
)11.66
)3.0386
)23.16
27.9540
)8.54
)3.5714
)9.69
30.0000
)1.85
)3.8359
)3.00
30.0290
)1.75
a
% Di€ i.e. Percentage di€erence quoted is with respect to the corresponding value obtained using LTSDT±CFS.
Fig. 1. Plot of maximum non-dimensional transverse de¯ection w at
x ˆ 0:5L versus aspect ratio S.
Fig. 2. Variation of non-dimensionalised transverse shear stress szx
through the thickness at x ˆ 0 (for aspect ratio S ˆ 4).
using general solution procedure. Results obtained for
natural frequencies are presented in Table 4.
4.3. Example 3: Free vibration of single layered simply
supported beam
A simply supported single layered orthotropic beam,
which is a special case of two layered orthotropic [0/0]
composite beam, is considered here. This is the same
example, which was also considered in [16,17].
This example can be posed as a two layered beam
example. Layer-1 and layer-2 of the beam occupy the
regions given by expressions (1) and (2), respectively.
The layer properties are:
Fig. 3. Variation of non-dimensionalised in-plane stress rx through the
thickness at x ˆ 0:5L (for aspect ratio S ˆ 4).
159
Table 4
Comparison of natural frequencies x of (90/0) beam Example 2
S
10
15
120
Mode no.
Natural frequency x in kHz
Present
LTSDT
FSDT
ETB
1
2
3
1
2
3
1
2
3
0.2096
0.7410
1.4382
0.1436
0.5393
1.1115
0.0184
0.0734
0.1649
0.2093
0.7395
1.4340
0.1435
0.5386
1.1093
0.0184
0.0734
0.1648
0.2106
0.7509
1.4647
0.1440
0.5434
1.1264
0.0184
0.0734
0.1650
0.2192
0.8604
1.8782
0.1467
0.5817
1.2906
0.0184
0.0735
0.1654
Table 5
Comparison of natural frequencies x of orthotropic beam, Example 3
S
Natural frequency x in kHz
Mode no.
15
120
1
2
3
4
5
1
2
3
4
5
Present
LTSDT [10]
FSDT [17]
HSDT [16]
CLT [16]
0.753
2.548
4.740
7.048
9.404
0.051
0.202
0.451
0.795
1.230
0.753
2.548
4.740
7.048
9.403
0.051
0.202
0.451
0.795
1.230
0.755
2.548
4.716
6.960
9.194
0.051
0.203
0.454
0.804
1.262
0.756
2.554
4.742
7.032
9.355
0.051
0.202
0.453
0.799
1.238
0.813
3.250
7.314
13.002
20.316
0.051
0.203
0.457
0.812
1.269
E…1† ˆ E…2† ˆ 144:80 GPa;
G…1† ˆ G…2† ˆ 4:14 GPa;
3
q…1† ˆ q…2† ˆ 1389:23 kg=m :
Following similar procedure as that of Example 2,
natural frequencies of the simply supported beam can be
obtained. Results obtained for natural frequencies are
presented in Table 5.
5. Discussion
The exact results in respect of static ¯exure and free
vibration examples studied here, are not available in the
literature to the best of the knowledge of authors.
In the absence of exact results, in respect of static
¯exure, many researchers (e.g. [3,12,14]) have compared
their results with the exact solution of cylindrical
bending of composite plate by Pagano [15]. However, it
is to be noted that the di€erence between cylindrical
bending and beam bending is analogous to the di€erence
between plane strain and plane stress in classical theory
of elasticity [18]. Therefore, cylindrical plate bending
results are quoted here only for the sake of information.
In case of free vibration, researchers (e.g. [16,19])
have compared their results with the ®rst order shear
deformation theory (FSDT) results, probably in the
absence of exact results.
In this paper, results obtained in various references
are compared with respect to the close form solution
(CFS) results obtained by layerwise trigonometric shear
deformation theory in [9,10] because of the following:
1. In both the layers of the beam, constitutive relations
are satis®ed between:
1.1. in-plane stress rx and in-plane strain ex ,
1.2. transverse shear-stress szx and transverse shearstrain czx .
2. Continuity conditions between the layers are satis®ed
a priori for:
2.1. in-plane displacement u,
2.2. transverse shear-stress szx .
3. In-plane displacement u is such that the resultant of
in-plane stress rx acting over the cross-section is zero.
Whereas, higher order shear deformation theories
(HSDT) by Maiti and Sinha [3], Manjunatha and Kant
[12], Vinayak et al. [14], Chandrashekhara and Bangera
[16] do not satisfy all the characteristics.
From the illustrative examples studied (i.e. Examples
1±3), for bending and free vibration, following observations are made for displacement, stresses and natural
frequencies:
1. Transverse Displacement w:
(a) For aspect ratio S equal to 4, it is observed from
Table 1, that the maximum values of non-dimensionalised transverse displacement obtained by the present
approach (LTSDT±FEM) are matching (upto three
160
Table 6
Comparison of di€erent beam ®nite elements in respect of degrees of freedom (dof) and method of obtaining shear stresses
Source
Type of element
Present
Shi and Lam [19]
Manjunatha and Kant [12]
Vinayak et al. [14]
Maiti and Sinha [3]
1D
1D
2D
1D
2D
±
±
±
±
±
2
2
4
3
9
noded,
noded,
noded,
noded,
noded,
C1
C1
C0
C0
C0
type
type
type
type
type
dof per
element
Shear stress obtained by using
6
8
20
21
108
Constitutive relations
)
Constitutive relations or equilibrium equations
Equilibrium equations
Equilibrium equations
decimal places) with LTSDT±CFS. Whereas, results
obtained by other researchers using higher order theories are di€ering by 3% to 9% w.r.t. LTSDT±CFS.
(b) From Fig. 1, it is seen that LTSDT±FEM results
converge to elementary theory of bending (ETB) results as the aspect ratio increases, which demonstrates, the present element is free from shear locking.
2. Transverse shear-stress szx : To the best of the
knowledge of authors, very few ®nite elements are
available for beams which evaluate shear-stresses using
constitutive relations. Evaluation of transverse stresses
from integration of equilibrium equations is a vexing
problem as per [12]. But Vinayak et al. [14] expressed
di€erence of opinion regarding this evaluation method.
In this paper transverse shear-stresses are evaluated directly using constitutive relations only.
From Table 2 and Fig. 2, the following needs to be
noted about transverse shear-stress szx :
(a) Results of maximum non-dimensionalised transverse shear-stress, obtained directly using constitutive
relation, for the present element are typically di€ering
by about 0.19% w.r.t. LTSDT±CFS for aspect ratio
4, whereas, results of Manjunatha and Kant [12] are
di€ering by 35.54%.
(b) Results of non-dimensionalised transverse shearstress, obtained indirectly by using equilibrium equations, by Maiti and Sinha [3], Manjunatha and Kant
[12], Vinayak et al. [14] are di€ering by about 6.0%
w.r.t. LTSDT±CFS for aspect ratio 4.
Thus, it is seen that the present element gives accurate
results for transverse shear-stress directly using constitutive relations (and thus avoiding the tedious way of
obtaining the shear-stress indirectly by the use of equilibrium equation for evaluating transverse shear-stress).
Thus, the element has the added advantage of simplicity.
3. In-plane stress rx : It is observed, from Table 3 and
Fig. 3, that the maximum values of in-plane stress rx
obtained by LTSDT±FEM are typically di€ering by
0.10% with LTSDT±CFS. Whereas, results obtained
by other researchers [12,14,15] are di€ering typically by
about 10% with LTSDT±CFS.
Thus, it is seen that the present element gives accurate
results for in-plane stress.
4. Natural frequency x:
(a) It can be seen from Table 4, results of natural frequencies using the present element (LTSDT±FEM)
are in close agreement (matching upto three decimal
places) with LTSDT±CFS results for aspect ratios
10, 15, 120. Whereas, results of ®rst order shear deformation theory are matching upto second decimal
point of LTSDT±CFS and as expected this di€erence
is increasing in case of higher modes, which indicates
the e€ect of shear deformation.
(b) LTSDT±FEM results converge to ETB results as
the aspect ratio increases, which demonstrates, the
present element is free from shear locking.
(c) It can be seen from Table 5, among the available
natural frequency results in respect of single layer orthotropic simply supported beam, present element
gives most accurate results (matching upto three decimal places) w.r.t. LTSDT±CFS. It is to be noted that
in Table 5 results of natural frequencies for CLT are
taken from [16].
(d) From Tables 4 and 5, it can be seen that, as expected, the elementary theory of bending overestimates the natural frequencies.
In addition to above discussion, it is interesting to note
the comparison of present beam ®nite element and
various beam ®nite elements in respect of degrees of
freedom and method of obtaining shear-stresses. It can
be clearly seen from Table 6 (wherein present element
and various elements are compared) that:
1. The present element has only six degrees of freedom.
All other elements have substantially more degrees of
freedom per element. Therefore, the present element
is computationally more ecient.
2. Only two references (the present one and the reference [12]) have reported obtaining shear-stresses directly by using constitutive relations. Therefore,
cumbersome post-processing for obtaining shearstresses (in an indirect way by using equilibrium
equations) is avoided.
6. Concluding remarks
In this paper, a new displacement type beam ®nite
element, which takes into account shear deformation
e€ects, is developed for two layered cross-ply composite
beams. The present ®nite element has following features:
1. It is one-dimensional, two noded ®nite element with
only three degrees of freedom at each node. Yet,
161
1.1. the constitutive relations between shear-stress
and shear-strain are satis®ed in both the layers,
1.2. shear-stress free boundary conditions on the top
and bottom surfaces of the beam element are satis®ed,
1.3. compatibility at the layer interface in respect of
in-plane displacement, as well as transverse
shear-stress is satis®ed, and
1.4. the shear-stress distribution is realistic in nature.
2. The governing di€erential equations and boundary
conditions, on which the ®nite element formulation
is based, are variationally consistent.
3. From the illustrative examples (dealing with static
¯exure and free vibration of two layered simply supported beams), it can be seen that:
3.1. Present element gives accurate results.
3.2. Unlike other elements, transverse shear-stresses
are obtained directly using constitutive relations
(instead of obtaining them in a tedious and indirect manner using equilibrium equations).
3.3. The use of element results in fast convergence.
3.4. It does not su€er from shear locking.
In general, present element is simple, accurate and easy
to use.
Z
Mx ˆ
pa E…2†
pa
C1 ˆ
C2 sin
sin
1 2a
1 ‡ 2a
E…1† ‡ E…2†
…1† E
1 ‡ 2a
pa
‡ 2C2
cos
…2†
p
1 ‡ 2a
E
1 2a
pa
2
cos
;
p
1 2a
G2 0:5 ‡ a cos 1 pa
2a ;
C2 ˆ
pa
G1 0:5 a cos
1‡2a
C3 ˆ
pa E…1†
sin
1 2a
E…1† ‡ E…2†
pa
‡ 2C2 sin
1 ‡ 2a
…2† pa E
1 2a
2
cos
p
1 2a
E…1†
1 ‡ 2a
pa
‡ 2C2
cos
:
p
1 ‡ 2a
Mx ; Ms ; Vs ; I1 ; I2 ; I3 ; I4 as referred in Eqs. (6) and (7) are
given as:
zˆ h=2
Z
‡
Z
Ms ˆ
yˆb=2
yˆ b=2
zˆh=2
Z
zˆ0
yˆb=2
r…2†
x …z
yˆ b=2
Z
zˆ0
zˆ h=2
yˆb=2
yˆ b=2
ah† dy dz
rx…1† …z
ah† dy dz;
rx…1† h C1
Z zˆh=2 Z yˆb=2
p z=h a
dy dz ‡
2 0:5 ‡ a
zˆ0
yˆ b=2
p z=h a
r…2†
dy dz;
x h C3 ‡ sin
2 0:5 a
‡ C2 sin
Z
Vs ˆ
zˆ0
Z
zˆ h=2
yˆb=2
yˆ b=2
pC2
1 ‡ 2a
Z
I1 ˆ
zˆ0
zˆ h=2
Z
I2 ˆ
Z
zˆh=2
zˆ0
Z
zˆ0
yˆb=2
yˆb=2
p z=h a
2 0:5 ‡ a
Z zˆh=2 Z yˆb=2
zˆ0
yˆ b=2
a p z=h
2 0:5
yˆ b=2
Z
…1†
szx
cos
dy dz ‡
yˆ b=2
Z
s…2†
zx cos
Appendix A
Constants associated with displacement ®eld (as referred in Eqs. (3)±(5)):
1 E…2† E…1†
aˆ
;
4 E…2† ‡ E…1†
Z
zˆ0
a
p
1
2a
q…1† …z
ah† dy dz;
q…2† …z
ah† dy dz;
dy dz;
yˆb=2
q…1†
p z=h a
h C1 ‡ C2 sin
dy dz;
2 0:5 ‡ a
Z zˆh=2 Z yˆb=2
p z=h a
…2†
q h C3 ‡ sin
I4 ˆ
dy dz:
2 0:5 a
zˆ0
yˆ b=2
I3 ˆ
zˆ h=2
yˆ b=2
Integration constants associated with element sti€ness
matrix (refer Eq. (22)) are
8 9
Ic1 >
>
>
=
<I >
c2
ˆ
>
I >
>
;
: c3 >
Ic4
9
8 …1†
E …zn ahn †2
>
>
>
>
h
i
>
>
>
>
…1†
p zn =hn a
>
>
>
E
h
…z
ah
†
C
‡
C
sin
Z zn ˆ0 >
n
n
n
1
2
=
<
2 0:5‡a
h
i2
dzn
z
=h
a
n
n
…1†
2
p
>
zn ˆ hn =2 >
>
> E hn C1 ‡ C2 sin 2 0:5‡a
>
>
>
>
>
>
>
>
;
: G…1† p2 C22 cos2 p zn =hn a
…1‡2a†2
2 0:5‡a
9
8 …2†
E …zn ahn †2
>
>
>
>
h
i
>
>
>
>
…2†
p zn =hn a
>
>
>
>
Z zn ˆhn =2 < E hn …zn ahn † C3 ‡ sin 2 0:5 a
=
h
i
2
dzn :
‡
=hn a
>
>
E…2† h2n C3 ‡ sin p2 zn0:5
zn ˆ0
>
>
a
>
>
>
>
>
>
>
>
;
: G…2† p2 cos2 p zn =hn a
2
2 0:5 a
…1 2a†
162
Integration constants associated with element mass
matrix (refer Eq. (23)) are
8 9
Im1 >
>
>
=
< >
Im2
ˆ
I >
>
>
;
: m3 >
Im4
9
8
>
>
q…1† …zn ahn †2 h
>
>
i
>
>
>
z=hn a
>
…1†
p
Z zn ˆ0 >
=
< q hn …zn ahn † C1 ‡ C2 sin 2 0:5‡a >
h
i2
dzn
>
…1† 2
p zn =hn a
zn ˆ hn =2 >
>
>
>
> q hn C1 ‡ C2 sin 2 0:5‡a
>
>
>
>
;
: …1† hn
q bn 2
9
8
2
>
q…2† …zn ahn †
>
>
h
i >
>
>
>
> …2†
=hn a
>
Z zn ˆhn =2 >
=
< q hn …zn ahn † C3 ‡ sin p2 zn0:5
a
h
i2
dzn :
‡
>
>
…2† 2
p zn =hn a
zn ˆ0
>
>
q
h
C
3 ‡ sin 2 0:5 a
>
>
n
>
>
>
>
;
: …2† hn
q bn 2
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