AC
1
finite element for flexural and torsional analysis of
rectangular piezoelectric laminated/sandwich composite beams
M. Ganapathi
1
, B. P. Patel
1, ∗,†
and M. Touratier
2
1
Institute of Armament Technology, Girinagar, Pune-411 025, India
2
LMSP, URA CNRS, ENSAM, 151, Bd de l’Hopital, 75013 Paris, France
This work deals with the development of a new C
1
finite element for analysing the bending and
torsional behaviour of rectangular piezoelectric laminated/sandwich composite beams. The formulation
includes transverse shear, warping due to torsion, and elastic–electric coupling effects. It also accounts
for the inter-layer continuity condition at the interfaces between layers, and the boundary conditions
at the upper and lower surfaces of the beam. The shear strain is represented by a cosine function of a
higher order in nature and thus avoiding shear correction factors. The warping function obtained from
a three-dimensional elasticity solution is incorporated in the present model. An exact integration is
employed in evaluating various energy terms due to the application of field consistency approach while
interpolating the transverse shear and torsional strains. The variation of the electric potential through
the thickness is taken care of in the formulation based on the observation of three-dimensional solution.
The performance of the laminated piezoelectric element is tested comparing with analytical results
as well as with the reference solutions evaluated using three-dimensional finite element procedure. A
detailed study is conducted to highlight the influence of length-to-thickness ratio on the displacements,
stresses and electric potential field of piezoelectric laminated beam structures subjected to flexural and
torsional loadings.
KEY WORDS: piezoelectric; flexural; torsional; sandwich; finite element; warping
1. INTRODUCTION
There has been an increasing interest in recent years in the development of lightweight smart
structures for many engineering applications. For the weight optimization, the structures made
of composite materials having high stiffness-to- and strength-to-weight ratios, and sandwiches
separating the stiff facings with a thick core of low density as load-bearing substrates are
preferred in aerospace engineering. Such structures are integrated with distributed piezoelectric
∗
Correspondence to: B. P. Patel, Institute of Armament Technology, Girinagar, Pune-411 025, India.
†
E-mail: badripatel@hotmail.com
Contract/grant sponsor: Indo French Centre for the Promotion of Advanced Research; contract/grant number:
2308-1
1
materials that act as sensors and actuators because
of the direct and converse piezoelectric
effects, respectively. These structural members while integrated with suitable control strategies
and circuits have the self-monitoring and self-controlling capabilities. For the effective utilization
of such structures, there is a growing appreciation among researchers in accurately modelling
and simulating the characteristics of smart composite structures.
Study on the behaviour of smart structures has received considerable attention in the literature
and reviewed by Tang et al. [1], Saravanos and Heyliger [2], Sunar and Rao [3], and Benjeddou
[4]. It is shown that the structures, in general, are characterized using classical/first-order shear
deformation theory coupled with either neglecting the electro-elastic interactions or introducing
constant electric field intensity over the piezoelectric layers with the substrate. It can be further
concluded that most of the studies are devoted to the flexural analysis of laminated smart
composite structures. Furthermore, a few studies have been devoted to the analysis of structures
with higher-order/layer-wise theory, and also assuming the through thickness variation for the
electric field of piezoelectric layers [5–10]. It is also observed that the available work of smart
structures is largely pertaining to laminated plates. Attempts are also made in deriving the
exact/analytical solutions for simple cases of geometry and boundary conditions [11–15].As
the exact solution is not possible/feasible for more general cases of loading and complicated
boundary conditions, the improved approximate techniques such as finite element method has
been explored for the simulation of the behaviour of smart structures [5, 8, 16–18]. However, in
analysing thick smart structures, 3D finite elements [19, 20] are used which are computationally
expensive. But it is highly desirable for the designers/analysts to have a model that can capture
the important 3D effects due to the through-thickness variations of displacements and stresses
in thick laminates but maintain the efficiency and convenience of a 1D model. To the best
of authors’ knowledge, however, the study on the torsion of laminated piezoelectric composite
beam structures has not received adequate attention in the literature [21–25].
Here, a new finite element for the analysis of laminated smart beam with rectangular
cross-section, having only the independent generalized displacements and electric potential,
is proposed by extending the recent work of Ganapathi et al. [24]. The element developed is
one-dimensional model from the point of view of structural behaviour and utilizes C
1
con-
tinuous function for the transverse displacement associated with bending in accordance with
the refined shear deformation theory, and the torsional warping of the beam is accounted for
based on 3D elasticity solution. The electric field variation is taken as three-dimensional one
in which quadratic form of through-thickness variation in the piezoelectric layer is considered.
The element has good features for all the standard requirements such as free from locking,
spurious rigid modes, etc. The formulation includes electro-elastic coupling effects, and has
no requirement of introducing arbitrary shear correction factor as the shear strain is defined
through the cosine function of a higher order nature. The efficacy of the present formulation
is tested comparing the solutions with those of three-dimensional analysis. A detailed study is
carried out to bring out the effect of length-to-thickness ratio on the variation of displacements,
stresses and electric potential fields due to both bending and torsional loads.
2. FORMULATION
A laminated composite beam is considered with the co-ordinates x along the length, y along
the width and z along the thickness directions as shown in Figure 1. The displacements in
2
Figure 1. Laminated beam co-ordinate system.
kth layer u
k
, v
k
and w
k
at point (x,y,z) from the median surface are expressed as functions
of mid-plane displacements u
0
, v
0
, w
0
, independent shear bending rotations
x
and
y
of the
normal in xz and yz planes. They are also the functions of torsional rotation and independent
parameter for torsional rotation gradient in the length direction as
u
k
(x,y,z)= u
0
(x) − yv
0
,
x
(x) + f
2
(y)[v
0
,
x
(x) +
y
(x)]−zw
0
,
x
(x)
+[f
3
(z) + g
k
(z)][w
0
,
x
(x) +
x
(x)]+
k
(y, z)(x)
v
k
(x,y,z)= v
0
(x) − z (x)
w
k
(x,y,z)= w
0
(x) + y(x) (1)
where the subscript comma denotes the partial derivative with respect to spatial co-ordinate
succeeding it. The functions f
2
(y), f
3
(z) and g
k
(z) are defined as
f
2
(y) = b/ sin(y/b) (2a)
f
3
(z) = h/ sin(z/ h) − h/b
55
cos(z/h) (2b)
g
k
(z) = a
k
z + b
k
(2c)
b and h are width and total thickness of the beam.
In Equation (2), coefficients b
k
are determined such that the contribution to the displacement
component u
k
, due to bending in xz plane, is continuous at the interface of adjacent layers and
is zero at the mid-point of the cross-section. Finally, coefficients b
55
and a
k
in Equation (2)
are computed from the requirement that the transverse shear stress due to bending in xz plane
is continuous at the interface of the adjacent layers and vanishes at the top and bottom surfaces
of the beam. The detailed derivation of these constants b
55
, a
k
and b
k
can be obtained from
the work of Beakou and Touratier [26]. The kinematics shown in Equation (1), in particular
for torsion, allows one to represent the constrained torsion where axial stress is not zero, for
instance near the clamped support, and free torsion i.e. Saint-Venant torsion when approaches
,
x
, which may be realized far away from the support of a thin beam.
The torsional warping function
k
used in defining the kinematics in Equation (1) is the
solution derived from three-dimensional elasticity equations in conjunction with Saint-Venant
assumption of torsion, for composite beam of rectangular cross-section made of different layers.
3
The general expression for
k
is taken in the form of a harmonic function and is expressed as
k
=
∞
N=1,3,...
(C
k
N
sinh(z) + D
k
N
cosh(z)) sin(y) + yz (3)
where is defined as N/b.
The coefficients C
k
N
and D
k
N
, in Equation (3), while defining the warping function for
the rectangular cross-section, are determined such that the contribution to the displacement
component u
k
due to torsion is continuous at the interface of adjacent layers, and the transverse
shear stress associated with torsion, is continuous at the interface of the adjacent layers and
vanishes at the top and bottom surfaces of the beam [27].
The strains in terms of mid-plane deformation for kth layer can be written as
{
k
}=
p
0
0
+
k
xx
2
k
xz
2
k
xy
b
+
k
xx
2
k
xz
2
k
xy
t
(4)
where superscripts b and t denote the strain contributions due to bending and torsion, respec-
tively.
The mid-plane strains,
p
, strain terms associated with bending and torsion in Equation (4)
are written as
p
= u
0,x
(5a)
k
xx
2
k
xz
2
k
xy
b
=
−zw
0,xx
+[f
3
(z) + g
k
(z)][w
0,xx
+
x,x
]−yv
0,xx
+ f
2
(y)(v
0,xx
+
y,x
)
(f
3,z
+ g
k
,z
)(w
0,x
+
x
)
f
2,y
(v
0,x
+
y
)
(5b)
k
xx
2
k
xz
2
k
xy
t
=
k
,x
k
,z
+ y
,x
k
,y
− z
,x
(5c)
The total strain can be rewritten as
{
k
}=
xx
2
xz
2
xy
=[Z]{¯} (6a)
4
where
[Z]=
1 yf
2
zf
3
+ g
k
0000
k
00 0 0 0 0 f
3,z
+ g
k
,z
y
k
,z
0
00 0 0 0 f
2,y
0 −z
k
,y
0
(6b)
{¯}={u
0,x
−v
0,xx
v
0,xx
+
y,x
−w
0,xx
w
0,xx
+
x,x
v
0,x
+
y
w
0,x
+
x
,x
,x
}
T
(6c)
For a composite laminated beam of layer thickness h
k
(k = 1, 2, 3 ...), and the ply-angle
k
(k = 1, 2, 3,...), the necessary expressions for computing the stiffness coefficients, available
in the literature [28], are used.
The stress–strain relation, incorporating the piezoelectric effect, for kth layer is written as
{
k
}=
Q
k
11
0 Q
k
16
0 Q
k
44
0
Q
k
16
0 Q
k
66
{
k
}−
00¯e
k
31
¯e
k
14
¯e
k
24
0
00¯e
k
36
E
k
x
E
k
y
E
k
z
(7a)
D
k
x
D
k
y
D
k
z
=
0 ¯e
k
14
0
0 ¯e
k
24
0
¯e
k
31
0 ¯e
k
36
{
k
}+
¯
k
11
¯
k
12
0
¯
k
21
¯
k
22
0
00¯
k
33
E
k
x
E
k
y
E
k
z
(7b)
where Q
k
ij
(i, j = 1, 4, 6) are the reduced stiffness coefficients of kth layer, ¯e
k
ij
are reduced
piezoelectric coefficients, ¯
k
ij
are reduced dielectric coefficients, D
k
x
, D
z
y
and D
k
z
are electrical
displacements.
The electrical field intensities E
k
x
, E
k
y
and E
k
z
can be related to the electric potential
k
as
{E
k
x
E
k
y
E
k
z
}
T
=−
*
k
*x
*
k
*y
*
k
*z
T
(8)
The total potential energy functional U of the system is given as
U() = (1/2)
L
0
b/2
−b/2
k
h
k+1
h
k
[{
k
}
T
{
k
}−{D
k
}
T
{E
k
}] dx dy dz
−
L
0
{d}
T
{f
x
f
y
f
z
m
y
m
z
m
x
}
T
dx (9)
5
![Figure 12. Distribution of displacements, electrical potential and stresses along the width [at (x, z) = (0.5 m, 0.0375 m)] due to 1 Nm torque applied at free end in Problem No. 3.](/figures/figure-12-distribution-of-displacements-electrical-potential-1ogso0v4.webp)
![Figure 13. Distribution of displacements, electrical potential and stresses along the thickness [at (x, y) = (0.5 m, 0.0375 m)] due to electrical load in Problem No. 3.](/figures/figure-13-distribution-of-displacements-electrical-potential-3q04tuqx.webp)
