Recent

Published on June 2016 | Categories: Documents | Downloads: 108 | Comments: 0 | Views: 878
of 18
Download PDF   Embed   Report

Comments

Content


Review
Recent research advances on the dynamic analysis of composite shells: 2000–2009
Mohamad S. Qatu
a,
*
, Rani Warsi Sullivan
b
, Wenchao Wang
a
a
Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS 39762, United States
b
Department of Aerospace Engineering, Mississippi State University, Mississippi State, MS 39762, United States
a r t i c l e i n f o
Article history:
Available online 30 June 2010
Keywords:
Composite
Shells
Dynamic
Vibration
Review
a b s t r a c t
Laminated composite shells are frequently used in various engineering applications in the aerospace,
mechanical, marine, and automotive industries. This article follows a previous book and review articles
published by the leading author (Qatu, 2004, 2002, 1989, 1992, 1999 [1–5]). It reviews most of the
research done in recent years (2000–2009) on the dynamic behavior (including vibration) of composite
shells. This review is conducted with emphasis on the type of testing or analysis performed (free vibra-
tion, impact, transient, shock, etc.), complicating effects in material (damping, piezoelectric, etc.) and
structure (stiffened shells, etc.), and the various shell geometries that are subjected to dynamic research
(cylindrical, conical, spherical and others). A general discussion of the various theories (classical, shear
deformation, 3D, non-linear etc.) is also given. The main aim of this review article is to collate the
research performed in the area of dynamic analyses of composite shells during the last 10 years, thereby
giving a broad perspective of the state of art in this field. This review article contains close to 200
references.
Ó 2010 Published by Elsevier Ltd.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2. Shell theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1. Three-dimensional elasticity theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2. Thick shell theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1. Higher order shell theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2. Layer-wise shell theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3. Thin-shell theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4. Non-linear theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3. Shell geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1. Shells of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.1. Cylindrical and doubly curved shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.2. Conical shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.3. Spherical shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2. Shallow shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4. Dynamic analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1. Free vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2. Rotating shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3. Impact loading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4. Dynamic stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.5. Thermal and hygrothermal loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.6. General dynamic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5. Material complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.1. Piezoelectric shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2. Viscoelastic and damped shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
0263-8223/$ - see front matter Ó 2010 Published by Elsevier Ltd.
doi:10.1016/j.compstruct.2010.05.014
* Corresponding author. Fax: +1 662 325 7223.
E-mail address: [email protected] (M.S. Qatu).
Composite Structures 93 (2010) 14–31
Contents lists available at ScienceDirect
Composite Structures
j our nal homepage: www. el sevi er . com/ l ocat e/ compst r uct
6. Structural complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.1. Stiffened shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2. Shells with cracks and/or damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.3. Fluid filled and/or submerged shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.4. Imperfect shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.5. Other complexities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7. Other topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.1. Experimental analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.2. Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1. Introduction
The increased usage of laminated composite shells in many
engineering applications has generated much interest in composite
shell behavior. While very little attention was given to composite
shell vibration four decades ago [6], a recent article [2] on the sub-
ject listed about 400 papers in this area.
When compared to traditional metallic materials, laminated
composites offer advantages such as higher strength-to-weight
and stiffness-to-weight ratios, improved chemical and environ-
mental resistance and the ability to tailor properties. Additionally,
the advances in composite manufacturing methods have also con-
tributed to the increased usage of laminated composite materials
in many modern applications. Composite shells now constitute a
large percentage of aerospace, marine and automotive structures.
Literature on composite shell vibration research can be found in
many of the national and international conferences and journals.
Review articles like those by Qatu [2,4], Kapania [7], Noor and Bur-
ton [8,9], and Noor et al. [10] covered much of the research done on
the subject prior to the early 1990s. Liew et al. [11] reviewed the
literature on shallow shell vibrations. Soldatos [12] reviewed the
literature on non-circular cylindrical shells. Computational aspects
of the research were covered by Noor et al. [13] and Noor and Ven-
neri [14]. Recently, Carerra [15,16] presented a historical review of
zig-zag theories for multilayered plates and shells and reviewed
the theories and finite elements for multilayered, anisotropic, com-
posite plates and shells.
It should be mentioned that several books appeared on the sub-
ject of laminated shells during the past decade. Qatu [1] treats the
vibrations of laminated shells and plates and Soedel [17] deals
with vibrations of composite shells and plates. Reddy [18] focuses
on stress analysis and general mechanics of laminated composite
plates and shells. Ye [19] and Lee [20] focus on modeling of com-
posite shells and Shen [21] discusses non-linear analysis of func-
tionally graded materials.
This article aims to present a broad perspective of the recent re-
search (2000–2009) done on the dynamics of composite shells. Dif-
ferent types of dynamic analyses such as free vibrations, transient
analysis, impact, shock and other dynamic analyses are included.
Additionally, typically used shell theories (thin, or classical, and
thick shell theories, including shear deformation and three-dimen-
sional theories, shallow and deep theories, linear and non-linear
theories, and others) are reviewed. Most theories are classified
based on the thickness ratio of the shell being treated (defined as
the ratio of the thickness of the shell to the shortest of the span
lengths and/or radii of curvature), its shallowness ratio (defined
as the ratio of the shortest span length to one of the radii of curva-
ture) and the magnitude of deformation (compared mainly to its
thickness).
The literature is collated and categorized based on various as-
pects of research. First, a general overview regarding shell theories
is presented. Discussion will then be focused on shell geometries
that are typically used, such as the classical cylindrical, conical and
spherical shells and other shells of revolution as well as shallow
shells.
The second aspect will concentrate on types of analyses which
are typically performed: free vibration, which includes problems
in which various boundary conditions and/or shell geometries
are treated, rotating shells, transient, impact and shock loading,
dynamic stability and general dynamic behavior. The third aspect
of this review will focus on material-related complexities, which
include piezoelectric materials, viscoelastic or viscoplastic materi-
als with damping, braided materials and shape memory alloys,
thermoplastic or wood material. The fourth category of interest
in this paper will focus on the structural-related complexities
which include stiffened shells, shells with cut-outs, fluid filled or
submerged shells and shells with imperfections. Attention will
also be given to multi-scale analysis, sensitivity and robustness,
and optimization studies that may prove useful to design
engineers.
Many articles may be cited more than once in this survey. For
example, a research article that uses a shear deformation shell the-
ory to solve a conical shell dynamic problem can be cited under the
thick shell theory title of shell theories section, as well as under the
conical shells title of the shell geometries section.
2. Shell theories
Shells are three-dimensional bodies bounded by two, relatively
close, curved surfaces. The three-dimensional equations of elastic-
ity are complicated when written in curvilinear or shell coordi-
nates. Typically, researchers make simplifying assumptions for
particular applications. Almost all shell theories (thin and thick,
deep and shallow) reduce the three-dimensional (3D) elasticity
equations to the two-dimensional (2D) representation. This is done
usually by eliminating the coordinate normal to the shell surface in
the development of the shell equations. The accuracy of thin and
thick shell theories can be established if these theories are com-
pared to the 3D theory of elasticity.
A summary of equations for laminated composite shells is made
in this section. In particular, the strain–displacement equations,
the stress–strain equations and the equations of motion are de-
scribed. These equations and the associated boundary conditions
constitute a complete set of shell theory equations.
2.1. Three-dimensional elasticity theory
A shell is a three-dimensional body confined by two parallel
(unless the thickness is varying) surfaces. In general, the distance
between those surfaces is small compared with other shell param-
eters. In this section, the equations from the theory of 3D elasticity
M.S. Qatu et al. / Composite Structures 93 (2010) 14–31 15
in curvilinear coordinates are presented. The literature regarding
vibrations of laminated shells using 3D elasticity theory will then
be reviewed.
Consider a shell element of thickness h, radii of curvature R
a
and R
b
(a radius of twist R
ab
is not shown here) (Fig. 1). Assume that
the deformation of the shell is small compared to the shell dimen-
sions. This assumption allows us to neglect non-linear terms in the
subsequent derivation. It will also allow us to refer the analysis to
the original configuration of the shell. The strain displacement
relations can be written as [1]
e
a
¼
1
ð1 þ z=R
a
Þ
1
A
@u
@a
þ
m
AB
@A
@b
þ
w
R
a
_ _
e
b
¼
1
ð1 þ z=R
b
Þ
1
B
@m
@b
þ
u
AB
@B
@a
þ
w
R
b
_ _
e
z
¼
@w
@z
c
ab
¼
1
ð1 þ z=R
a
Þ
1
A
@m
@a
À
u
AB
@A
@b
þ
w
R
ab
_ _
þ
1
ð1 þ z=R
b
Þ
1
B
@u
@b
À
m
AB
@B
@a
þ
w
R
ab
_ _
c
az
¼
1
Að1 þ z=R
a
Þ
@w
@a
þ Að1 þ z=R
a
Þ
@
@z
u
Að1 þ z=R
a
Þ
_ _
À
m
R
ab
ð1 þ z=R
a
Þ
c
bz
¼
1
Bð1 þ z=R
b
Þ
@w
@b
þ Bð1 þ z=R
b
Þ
@
@z
m
Bð1 þ z=R
b
Þ
_ _
À
u
R
ab
ð1 þ z=R
b
Þ
ð1Þ
For the development of the constitutive relations, the laminated
composite thin shells are assumed to be composed of plies of uni-
directional long fibers embedded in a matrix material such as
epoxy resin. On a macroscopic level, each layer may be regarded
as being homogeneous and orthotropic. However, the fibers of a
typical layer may not be parallel to the coordinates in which the
shell equations are expressed. The stress–strain relationship for a
typical nth lamina (typically called monoclinic) in a laminated
composite shell made of N laminas, is shown in Fig. 2 and given
by Eq. (2) [1].
r
a
r
b
r
z
r
bz
r
az
r
ab
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
¼
Q
11
Q
12
Q
13
0 0 Q
16
Q
12
Q
22
Q
23
0 0 Q
26
Q
13
Q
23
Q
33
0 0 Q
36
0 0 0 Q
44
Q
45
0
0 0 0 Q
45
Q
55
0
Q
16
Q
26
Q
36
0 0 Q
66
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
e
a
e
b
e
z
c
bz
c
az
c
ab
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
ð2Þ
The positive notations of the stresses are shown in Fig. 1.
In order to develop a consistent set of equations, the boundary
conditions and the equations of motion will be derived using Ham-
ilton’s principle [1]. Substituting the equations for potential energy
(U), external work (W) and kinetic energy (T), performing the inte-
gration by parts, and setting the coefficients of the displacement
variations equal to zero, in a normal manner, yields the equations
of motion
@ðBr
a
Þ
@a
þ
@ðAr
ab
Þ
@b
þ
@ðABr
az
Þ
@z
þr
ab
@A
@b
þr
az
B
@A
@z
Àr
b
@B
@a
þ ABq
a
¼ q
@
2
u
@t
2
@ðBr
ab
Þ
@a
þ
@ðAr
b
Þ
@b
þ
@ðABr
bz
Þ
@z
þr
bz
A
@B
@z
þr
ab
@B
@a
Àr
a
@A
@b
þ ABq
b
¼ q
@
2
v
@t
2
@ðBr
az
Þ
@a
þ
@ðAr
bz
Þ
@b
þ
@ðABr
z
Þ
@z
Àr
b
A
@B
@z
Àr
a
B
@A
@z
þ ABq
z
¼ q
@
2
w
@t
2
ð3Þ
The above equations do not depend on the shell material. Ham-
ilton’s principle will also yield boundary terms that are consistent
with the other equations (strain–displacement and equilibrium
relations). The boundary terms for z = constant are:
r
0z
Àr
z
¼ 0 or w
0
¼ 0
r
0az
Àr
az
¼ 0 or u
0
¼ 0
r
0bz
Àr
az
¼ 0 or v
0
¼ 0
ð4Þ
Nomenclature
A, B Lame’ parameters
A
ij
;
´
A
ij
; A
ij
stretching and shearing stiffness parameters
A
ija
; A
ijb
; A
ijn
stiffness parameters
B
ij
;
´
B
ij
; B
ij
coupling stiffness parameters
B
ija
; B
ijb
; B
ijn
stiffness parameters
c
0
tracer
D
ij
;
´
D
ij
; D
ij
bending and twisting stiffness parameters
D
ija
; D
ijb
; D
ijn
stiffness parameters
E
ij
; E
ija
; E
ijb
; E
ijn
higher order stiffness parameters
h total thickness of the shell
h
k
distance from the middle surface to the kth layer
I
1
; I
2
; I
3
, I
i
inertia terms
k layer number
i
a
, i
b
, i
n
unit victors
K
i
, K
j
shear correction coefficients
M
a
, M
b
, M
ab
, M
ba
bending and twisting moment resultants
m
a
, m
b
body couples
N
a
, N
b
, N
ab
normal and in-plane shear force resultants
N total number of layers
P
a
, P
b
higher order stress resultants
q
a
, q
b
, q
n
body forces
Q
ðkÞ
ij
stiffness parameters for layer k
Q
a
, Q
b
transverse shear force resultants
R
a
, R
b
radii of curvature
R
ab
radius if twist
T kinetic energy
t time
U potential energy
u, v, w displacements at a point in the a, b and z directions,
respectively
u
0
, v
0
, w
0
displacements at the midsurface of the shell in the a, b
and z directions, respectively
W external work
z out of plane coordinate of the shell, also distance from
the middle surface
a, b in-plane coordinates of the shell
c
ab
, c
az
, c
bz
in-plane and transverse shear strain
c
0ab
, c
0az
, c
0bz
midsurface shear strain
e
a
, e
b
, e
z
normal strains
e
0a
, e
0b
midsurface normal strains
h, u coordinates used for shells of revolution
j
a
, j
b
curvature changes
j
ab
, j
ba
changes in twist
q density per unit volume
r
a
, r
b
, r
z
normal stress
r
0z
, r
0az
, r
0bz
surface tractions
r
ab
, r
az
, r
bz
shear stress
w
a
, w
b
midsurface rotations
16 M.S. Qatu et al. / Composite Structures 93 (2010) 14–31
where r
0z
, r
0az
and r
0bz
are surface tractions and u
0
, v
0
and w
0
are
displacement functions at z = constant. Similar results are obtained
for the boundaries a = constant and b = constant. A three-dimen-
sional shell element has six surfaces. With three equations at each
surface, a total of 18 equations can be obtained for a single-layered
shell.
The above equations are valid for single-layered shells. To use
3D elasticity theory for multilayered shells (the subject of this
study), each layer must be treated as an individual shell. Both dis-
placements and stresses must be continuous between each layer
(layer k to layer k + 1) in a n-ply laminate. These conditions must
be met to insure that there are no free internal surfaces (i.e.,
delamination) between the layers
uða; b; z ¼ h
k
=2Þj
k¼i
¼ uða; b; z ¼ Àh
k
=2Þj
k¼iþ1
vða; b; z ¼ h
k
=2Þj
k¼i
¼ vða; b; z ¼ Àh
k
=2Þj
k¼iþ1
wða; b; z ¼ h
k
=2Þj
k¼i
¼ wða; b; z ¼ Àh
k
=2Þj
k¼iþ1
; fork ¼ 1; . . . ; N À1
r
z
ða; b; z ¼ h
k
=2Þj
k¼i
¼ r
z
ða; b; z ¼ Àh
k
=2Þj
k¼iþ1
r
az
ða; b; z ¼ h
k
=2Þj
k¼i
¼ r
az
ða; b; z ¼ Àh
k
=2Þj
k¼iþ1
r
bz
ða; b; z ¼ h
k
=2Þ
¸
¸
k¼i
¼ r
bz
ða; b; z ¼ Àh
k
=2Þ
¸
¸
k¼iþ1
ð5Þ
The 3D theory of elasticity has been used to perform a dynamic
analysis of composite shells by Santos et al. [22,23] in which a fi-
nite element model for the free vibration analysis of 3D axisym-
metric laminated shells was developed. Also, Shakeri et al. [24]
ds
α
(z)
ds
β
(z)
R
α
R
β
α
β
z
i
z
i
α
i
β
h
2
h
(k)
σ
βz
σ
βα
σ
β
σ
αz
σ
α
σ
αβ
Fig. 1. Stresses in shell coordinates (free outer surfaces).
z
β
β
z
k
Middle Surface
h/2 Layer k
h/2 R
z
k-1
Fig. 2. Lamination parameters in shells.
M.S. Qatu et al. / Composite Structures 93 (2010) 14–31 17
performed dynamic analysis of thick laminated shell panel based
on the 3D elasticity solution. Malekzadeh et al. [25] conducted a
3D dynamic analysis on composite laminates under a moving load.
Saviz et al. [26] presented both 3D and 2D solutions of a layer-wise
theory in the investigation of thick laminated piezoelectric shells
subjected to dynamic loading.
2.2. Thick shell theory
Thick shells are defined as shells with a thickness smaller by at
least one order of magnitude when compared with other shell
parameters such as wavelength and/or radii of curvature (thick-
ness is at least 1/10th of the smallest of the wavelength and/or ra-
dii of curvature). The main differentiation between thick shell and
thin shell theories is the inclusion of shear deformation and rotary
inertia effects. Theories that include shear deformation are referred
to as thick shell theories or shear deformation theories.
Thick shell theories are typically based on either a displacement
or stress approach. In the former, the midplane shell displacements
are expanded in terms of shell thickness, which can be a first-order
expansion, referred to as first-order shear deformation theories.
Accurate shell equations based on a first-order shear deformation
theory are now presented.
The 3D elasticity theory is reduced to a 2D theory using the
assumption that the normal strains acting upon the plane parallel
to the middle surface are negligible compared with other strain
components. This assumption is generally valid except within the
vicinity of a highly concentrated force (St. Venant’s principle). In
other words, no stretching is assumed in the z-direction (i.e., e
z
= 0). Assuming that normals to the midsurface strains remain
straight during deformation but not normal, the displacements
can be written as [1]
uða; b; zÞ ¼ u
0
ða; bÞ þ zw
a
ða; BÞ
vða; b; zÞ ¼ v
0
ða; bÞ þ zw
b
ða; BÞ
wða; b; zÞ ¼ w
0
ða; bÞ
ð6Þ
where u
0
, v
0
and w
0
are midsurface displacements of the shell and
w
a
and w
b
are midsurface rotations. An alternative derivation can
be made with the assumption r
z
= 0. The subscript (
0
) will refer to
the middle surface in subsequent equations. The above equations
describe a typical first-order shear deformation shell theory, and
will constitute the only assumption made in this analysis when
compared with the 3D theory of elasticity. As a result, strains are
written as [1]
e
a
¼
1
ð1 þ z=R
a
Þ
ðe
0a
þ zj
a
Þ; e
b
¼
1
ð1 þ z=R
b
Þ
ðe
0b
þ zj
b
Þ
e
ab
¼
1
ð1 þ z=R
a
Þ
ðe
0ab
þ zj
ab
Þ; e
ba
¼
1
ð1 þ z=R
b
Þ
ðe
0ba
þ zj
ba
Þ
ð7Þ
c
az
¼
1
ð1 þ z=R
a
Þ
c
0az
À zðw
b
=R
ab
Þ
_ _
c
bz
¼
1
ð1 þ z=R
b
Þ
c
0bz
À zðw
a
=R
ab
Þ
_ _
where the midsurface strains are:
e
0a
¼
1
A
@u
0
@a
þ
v
0
AB
@A
@b
þ
w
0
R
a
e
0b
¼
1
B
@v
0
@b
þ
u
0
AB
@B
@a
þ
w
0
R
b
e
0ab
¼
1
A
@v
0
@a
À
u
0
AB
@A
@b
þ
w
0
R
ab
e
0ba
¼
1
B
@u
0
@b
À
v
0
AB
@B
@a
þ
w
0
R
ab
ð8aÞ
c
0az
¼
1
A
@w
0
@a
À
u
0
R
a
À
v
0
R
ab
þ w
a
; c
0bz
¼
1
B
@w
0
@b
À
v
0
R
b
À
u
0
R
ab
þ w
b
and the curvature and twist changes are:
j
a
¼
1
A
@w
a
@a
þ
w
b
AB
@A
@b
; j
b
¼
1
B
@w
b
@b
þ
w
a
AB
@B
@a
j
ab
¼
1
A
@w
b
@a
À
w
a
AB
@A
@b
; j
ba
¼
1
B
@w
a
@b
À
w
b
AB
@B
@a
ð8bÞ
The force and moment resultants (Figs. 3 and 4) are obtained by
integrating the stresses over the shell thickness considering the
(1 + z/R) term that appears in the denominator of the stress resul-
tant equations [5]. The stress resultant equations are:
i
z
i
α
i
β
Q
β
N
β
N
βα
Q
α
N
αβ
N
α
Q
β
+(∂ Q
β
/∂β)dβ
N
βα
+(∂ N
βα
/∂β)dβ
N
β
+(∂ N
β
/∂β)dβ
Q
α
+(∂ Q
α
/∂α)dα
N
αβ
+(∂ N
αβ
/∂α)dα
N
α
+(∂ N
α
/∂α)dα
Fig. 3. Force resultants in shell coordinates.
18 M.S. Qatu et al. / Composite Structures 93 (2010) 14–31
N
a
N
b
N
ab
N
ba
M
a
M
b
M
ab
M
ba
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
¼
A
11
A
12
A
16
A
16
B
11
B
12
B
16
B
16
A
12
´
A
22
A
26
´
A
26
B
12
´
B
22
B
26
´
B
26
A
16
A
26
A
66
A
66
B
16
B
26
B
66
B
66
A
16
´
A
26
A
66
´
A
66
B
16
´
B
26
B
66
´
B
66
B
11
B
12
B
16
B
16
D
11
D
12
D
16
D
16
B
12
´
B
22
B
26
´
B
26
D
12
´
D
22
D
26
´
D
26
B
16
B
26
B
66
B
66
D
16
D
26
D
66
D
66
B
16
´
B
26
B
66
´
B
66
D
16
´
D
26
D
66
´
D
66
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
e
0a
e
0b
e
0ab
e
0ba
j
a
j
b
j
ab
j
ba
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
ð9aÞ
Q
a
Q
b
P
a
P
b
_
¸
¸
¸
¸
_
_
¸
¸
¸
¸
_
¼
A
55
A
45
B
55
B
45
A
45
´
A
44
B
45
´
B
44
B
55
B
45
D
55
D
45
B
45
´
B
44
D
45
´
D
44
_
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
_
c
0az
c
0bz
Àw
b
=R
ab
Àw
a
=R
ab
_
¸
¸
¸
¸
_
_
¸
¸
¸
¸
_
ð9bÞ
where A
ij
, B
ij
, D
ij
, A
ij
; B
ij
; D
ij
;
´
A
ij
;
´
B
ij
and
´
D
ij
are
A
ij
¼

N
k¼1
Q
ðkÞ
ij
ðh
k
À h
kÀ1
Þ
B
ij
¼
1
2

N
k¼1
Q
ðkÞ
ij
ðh
2
k
À h
2
kÀ1
Þ
D
ij
¼
1
3

N
k¼1
Q
ðkÞ
ij
ðh
3
k
À h
3
kÀ1
Þ
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
i; j ¼ 1; 2; 6
A
ij
¼

N
k¼1
K
i
K
j
Q
ðkÞ
ij
ðh
k
À h
kÀ1
Þ
B
ij
¼
1
2

N
k¼1
K
i
K
j
Q
ðkÞ
ij
ðh
2
k
À h
2
kÀ1
Þ
D
ij
¼
1
3

N
k¼1
K
i
K
j
Q
ðkÞ
ij
ðh
3
k
À h
3
kÀ1
Þ
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
i; j ¼ 4; 5
A
ij
¼ A
ija
þ
B
ija
R
b
;
´
A
ij
¼ A
ijb
þ
B
ijb
Ra
;
B
ij
¼ B
ija
þ
D
ija
R
b
;
´
B
ij
¼ B
ijb
þ
D
ijb
Ra
;
D
ij
¼ D
ija
þ
E
ija
R
b
;
´
D
ij
¼ D
ijb
þ
E
ijb
Ra
_
¸
¸
¸
¸
_
¸
¸
¸
¸
_
i; j ¼ 1; 2; 4; 5; 6
ð9cÞ
where K
i
and K
j
are shear correction coefficients, typically taken at
5/6, and where
A
ijn
¼

N
k¼1
_
h
k
h
kÀ1
Q
ðkÞ
ij
dz
1þz=Rn
¼R
n

N
k¼1
Q
ðkÞ
ij
ln
Rnþh
k
Rnþh
kÀ1
_ _
B
ijn
¼

N
k¼1
_
h
k
h
kÀ1
Q
ðkÞ
ij
zdz
1þz=Rn
¼R
n

N
k¼1
Q
ðkÞ
ij
ðh
k
Àh
kÀ1
ÞÀR
n
ln
Rnþh
k
Rnþh
kÀ1
_ _ _ _
D
ijn
¼

N
k¼1
_
h
k
h
kÀ1
Q
ðkÞ
ij
z
2
dz
1þz=Rn
¼R
n

N
k¼1
Q
ðkÞ
ij
1
2
ðR
n
þh
k
Þ
2
ÀðR
n
þh
kÀ1
Þ
2
_ _
À2R
n
ðh
k
Àh
kÀ1
Þ
_
þR
2
n
ln
Rnþh
k
Rnþh
kÀ1
_ __
E
ijn
¼

N
k¼1
_
h
k
h
kÀ1
Q
ðkÞ
ij
z
3
dz
1þz=Rn
¼R
n

N
k¼1
Q
ðkÞ
ij
1
3
ðR
n
þh
k
Þ
3
ÀðR
n
þh
kÀ1
Þ
3
_ _
À
3
2
R
n
ðR
n
þh
k
Þ
2
ÀðR
n
þh
kÀ1
Þ
2
_ _
þ3R
2
n
ðh
k
Àh
kÀ1
Þ ÀR
3
n
ln
Rnþh
k
Rnþh
kÀ1
_ _
_
¸
_
_
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
n¼a; b
ð9dÞ
The above equations can be simplified by truncating the 1/
(1 + z/R) in a Taylor series [1]
A
ij
¼ A
ij
À c
0
B
ij
;
´
A
ij
¼ A
ij
þ c
0
B
ij
;
B
ij
¼ B
ij
À c
0
D
ij
;
´
B
ij
¼ B
ij
þ c
0
D
ij
D
ij
¼ D
ij
À c
0
E
ij
;
´
D
ij
¼ D
ij
þ c
0
E
ij
_
¸
¸
_
¸
¸
_
i; j ¼ 1; 2; 4; 5; 6 ð10aÞ
where all terms are as defined in (9c) and
E
ij
¼
1
4

N
k¼1
Q
ðkÞ
ij
h
4
k
À h
4
kÀ1
_ _
i; j ¼ 1; 2; 6
c
0
¼
1
R
a
À
1
R
b
_ _
ð10bÞ
It has been shown [1,5] that the above Eqs. (9) and (10) yield
more accurate results when compared with those of plates and
those traditionally used for shells [18]. Hamilton’s principle can
be used to derive the consistent equations of motion and boundary
conditions. The equations of motion are [1]:
@
@a
ðBN
a
Þ þ
@
@b
ðAN
ba
Þ þ
@A
@b
N
ab
À
@B
@a
N
b
þ
AB
R
a
Q
a
þ
AB
R
ab
Q
b
þ ABq
a
¼ ABðI
1
€ u
0
þ I
2

w
a
Þ
@
@b
ðAN
b
Þ þ
@
@a
ðAN
ab
Þ þ
@B
@a
N
ba
À
@A
@b
N
a
þ
AB
R
b
Q
b
þ
AB
R
ab
Q
a
þ ABq
b
¼ ABðI
1
€ v
0
þ I
2

w
b
Þ
i
z
i
α
i
β
M
βα
M
β
M
α
M
αβ
M
β
+(∂ M
β
/∂β)dβ
M
βα
+(∂ M
βα
/∂β)dβ
M
α
+(∂ M
α
/∂α)dα
M
αβ
+(∂ M
αβ
/∂α)dα
α
β
Fig. 4. Moment resultants in shell coordinates.
M.S. Qatu et al. / Composite Structures 93 (2010) 14–31 19
À AB
N
a
R
a
þ
N
b
R
b
þ
N
ab
þ N
ba
R
ab
_ _
þ
@
@a
ðBQ
a
Þ þ
@
@b
ðAQ
b
Þ þ ABq
n
¼ ABðI
1
€ w
0
Þ
@
@a
ðBM
a
Þ þ
@
@b
ðAM
ba
Þ þ
@A
@b
M
ab
À
@B
@a
M
b
À ABQ
a
þ
AB
R
ab
P
b
þ ABm
a
¼ ABðI
2
€ u
0
þ I
3

w
a
Þ
@
@b
ðAM
b
Þ þ
@
@a
ðBM
ab
Þ þ
@B
@a
M
ba
À
@A
@b
M
a
À ABQ
b
þ
AB
R
ab
P
a
þ ABm
b
¼ ABðI
2
€ v
0
þ I
3

w
b
Þ ð11Þ
where the two dots over the terms represent the second derivative
of these terms with respect to time, and where:
I
i
¼ I
i
þ I
iþ1
1
R
a
þ
1
R
b
_ _
þ
I
iþ2
R
a
R
b
_ _
; i ¼ 1; 2; 3 ð12aÞ
and
½I
1
; I
2
; I
3
; I
4
; I
5
Š ¼

N
k¼1
_
h
k
h
kÀ1
q
ðkÞ
½1; z; z
2
; z
3
; z
4
Šdz ð12bÞ
The boundary terms for the boundaries with a = constant are
N
0a
À N
a
¼ 0 or u
0
¼ 0
N
0ab
À N
ab
¼ 0 or v
0
¼ 0
Q
0a
À Q
a
¼ 0 or w
0
¼ 0
M
0a
À M
a
¼ 0 or w
a
¼ 0
M
0ab
À M
ab
¼ 0 or w
b
¼ 0
ð13Þ
Similar equations can be obtained for b = constant.
Shear deformation theories have been used by many authors.
Qatu [5] used it to solve the free vibration problem of simply sup-
ported shells. Toorani and Lakis [27] discussed shear deformation
in dynamic analysis of laminated open cylindrical shells interact-
ing with a flowing fluid. Dong and Wang [28] analyzed the effect
of transverse shear and rotary inertia on wave propagation in
laminated piezoelectric cylindrical shells. Ribeiro [29] investi-
gated the influence of membrane inertia and shear deformation
on non-linear vibrations of open, cylindrical, laminated clamped
shells. Qatu [30] used the shear deformation shell theory de-
scribed here to study the free vibration of laminated cylindrical
and barrel thick shells. Wang et al. [31] were interested in the
wave propagation of stresses in orthotropic laminated thick-
walled spherical shells. Ding [32] employed a shear deformation
theory to study the thermoelastic dynamic response of thick
closed laminated shells. Ganapathi and Haboussi [33] studied
the free vibrations of thick laminated anisotropic non-circular
cylindrical shells. Other studies using thick shell theories will be
reviewed in other sections.
2.2.1. Higher order shell theories
The equations derived earlier for thick shells are called first-
order shear deformation theory because in Eq. (6), only the first-
order expansion is performed across the thickness for in-plane
displacements. If third-order terms are retained, the resulting the-
ory will be a third-order deformation theory. The group of theories
that are based on a cubic or higher expansion of the in-plane dis-
placements in terms of the thickness are referred to as higher order
theories.
Ganapathi et al. [34] used a higher order theory to perform dy-
namic analysis of laminated cross-ply composite non-circular
thick cylindrical shells. Khare and Rode [35] utilized a higher order
theory to develop closed-form solutions for vibrations of thick
shells. Balah and Al-Ghemady [36] employed a third-order theory
to develop an energy momentum conserving algorithm for non-
linear dynamics of laminated shells. Pinto Correia et al. [37] stud-
ied dynamics and statics of laminated conical shell structures
using higher order models. Qian et al. [38] studied active vibration
control of laminated shells using higher order layer-wise theory.
Lam et al. [39] combined displacements from transverse shear
forces and from thin-shell theory to modify a higher order theory
to study the vibration response of thick laminated cylindrical
shells.
2.2.2. Layer-wise shell theories
Other thick shell theories, such as layer-wise theories have also
been utilized. These theories typically reduce a 3D problem to a 2D
problem by expanding the 3D displacement field in terms of a 2D
displacement field and the through-the-shell thickness. Braga and
Rivas [40] used a layer-wise theory to study the high-frequency
response of cylindrical shells made of isotropic and laminated
materials. Basar and Omurtag [41] used a layer-wise model to
investigate free vibrations of shell structures. Other research
involving layer-wise shell theories include Lee et al. [42] who stud-
ied the dynamic behavior of cylindrical composite structures with
viscoelastic layers, Moreira et al. [43] who used a layer-wise theory
to formulate shell finite elements for dynamic modeling of com-
posite laminates and Oh, who used layer-wise mechanics to inves-
tigate dynamic response [44], damping characteristics [45], and
vibration characteristics [46] of cylindrical laminates. Qian used a
higher order theory for studying composite laminated shells under
active vibration control [38], Saravanan et al. [47] studied active
damping in a laminated shell and Saravanos and Christoforou
[48] investigated impacts of composite shells. Other works utiliz-
ing layer based techniques include Shin et al. [49] who investigated
aeroelastic analysis using zig-zag layer-wise theory, Varelis and
Saravanos [50] who studied the non-linear response of doubly
curved composite shells using a shear layer-wise shell theory,
and Wang et al. [51] who investigated the dynamic response of
laminated shells.
2.3. Thin-shell theory
If the shell thickness is less than 1/20th of the wavelength of the
deformation mode and/or radii of curvature, a thin-shell theory,
where shear deformation and rotary inertia are negligible, is gener-
ally acceptable. Depending on various assumptions made during
the derivation of the strain–displacement relations, stress–strain
relations, and the equilibriumequations, various thin shell theories
can be derived. Among the most common of these are Love’s, Reiss-
ner’s, Naghdy’s, Sander’s and Flugge’s shell theories. Descriptions
of these and other thin shell theories can be found [5]. All these
theories were initially derived for isotropic shells and expanded la-
ter for laminated composite shells by applying the appropriate
integration through laminas, and stress–strain relations. For very
thin shells, the following additional assumptions simplify the shell
equations and their order.
1. The shell is thin such that the ratio of the thickness compared to
any of the shell’s radii or any other shell parameter, i.e., width
or length, is negligible when compared to unity.
2. The normals to the middle surface remain straight and normal
when the shell undergoes deformation.
The first assumption assures that certain parameters in the shell
equations (including the z/R) term mentioned earlier in the thick
shell theory can be neglected. Due to the second assumption, the
shear deformation can be neglected in the kinematic equations
and this allows the in-plane displacement to vary linearly through
the shell’s thickness as given by
20 M.S. Qatu et al. / Composite Structures 93 (2010) 14–31
e
a
¼ e
0a
þ zj
a
e
b
¼ e
0b
þ zj
b
c
ab
¼ c
0ab
þ zs
ab
ð14Þ
where the midsurface strains, curvature and twist changes are
e
0a
¼
1
A
@u
0
@a
þ
v
0
AB
@A
@b
þ
w
0
R
a
e
0b
¼
1
B
@v
0
@b
þ
u
0
AB
@B
@a
þ
w
0
R
b
c
0ab
¼
1
A
@v
0
@a
À
u
0
AB
@A
@b
þ
1
B
@u
0
@b
À
v
0
AB
@B
@a
þ2
w
0
R
ab
ð15aÞ
j
a
¼
1
A
@w
a
@a
þ
w
b
AB
@A
@b
; j
b
¼
1
B
@w
b
@b
þ
w
a
AB
@B
@a
s ¼
1
A
@w
b
@a
À
w
a
AB
@A
@b
þ
1
B
@w
a
@b
À
w
b
AB
@B
@a
ð15bÞ
and where
w
a
¼
u
R
a
þ
v
0
R
ab
À
1
A
@w
@a
; w
b
¼
v
R
b
þ
u
0
R
ab
À
1
A
@w
@b
ð15cÞ
Applying Kirchhoff hypothesis of neglecting shear deformation
and the assumption that e
z
is negligible, the stress–strain equations
for an element of material in the kth lamina may be written as [1]
r
a
r
b
r
ab
_
¸
_
_
¸
_
k
¼
Q
11
Q
12
Q
16
Q
12
Q
22
Q
26
Q
16
Q
26
Q
66
_
¸
_
_
¸
_
k
e
a
e
b
c
ab
_
¸
_
_
¸
_
k
ð16Þ
where r
a
and r
b
are normal stress components, s
ab
is the in-plane
shear stress component [1], e
a
and e
b
are the normal strains, and c
ab
is the in-plane engineering shear strain. The terms Q
ij
are the elastic
stiffness coefficients for the material. If the shell coordinates (a, b)
are parallel or perpendicular to the fibers, then the terms Q
16
and
Q
26
are both zero. Stresses over the shell thickness (h) are integrated
to get the force and moment resultants as given by
N
a
N
b
N
ab
M
a
M
b
M
ab
_
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
_
¼
A
11
A
12
A
16
B
11
B
12
B
16
A
12
A
22
A
26
B
12
B
22
B
26
A
16
A
26
A
66
B
16
B
26
B
66
B
11
B
12
B
16
D
11
D
12
D
16
B
12
B
22
B
26
D
12
D
22
D
26
B
16
B
26
B
66
D
16
D
26
D
66
_
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
_
e
0a
e
0b
c
0ab
k
a
k
b
s
_
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
_
ð17Þ
where A
ij
, B
ij
, and D
ij
are the stiffness coefficients arising from the
piecewise integration over the shell thickness (Eq. (10b)). For shells
which are laminated symmetrically with respect to their midsurfac-
es, all the B
ij
terms become zero. Note that the above equations are
the same as those for laminated plates, which are also valid for thin
laminated shells. Using Hamilton’s principle yields the following
equations of motion
@
@a
ðBN
a
Þ þ
@
@b
ðAN
ba
Þ þ
@A
@b
N
ab
À
@B
@a
N
b
þ
AB
R
a
Q
a
þ
AB
R
ab
Q
b
þ ABq
a
¼ ABðI
1
€ u
0
Þ
@
@b
ðAN
b
Þ þ
@
@a
ðAN
ab
Þ þ
@B
@a
N
ba
À
@A
@b
N
a
þ
AB
R
b
Q
b
þ
AB
R
ab
Q
a
þ ABq
b
¼ ABðI
1
€ v
0
Þ
À AB
N
a
R
a
þ
N
b
R
b
þ
N
ab
þ N
ba
R
ab
_ _
þ
@
@a
ðBQ
a
Þ þ
@
@b
ðAQ
b
Þ þ ABq
n
¼ ABðI
1
€ w
0
Þ
where
ABQ
a
¼
@
@a
ðBM
a
Þ þ
@
@b
ðAM
ba
Þ þ
@A
@b
M
ab
À
@B
@a
M
b
ABQ
b
¼
@
@b
ðAM
b
Þ þ
@
@a
ðBM
ab
Þ þ
@B
@a
M
ba
À
@A
@b
M
a
ð18Þ
The following boundary conditions can be obtained for thin
shells for a = constant (similar equations can be obtained for
b = constant).
N
0a
À N
a
¼ 0 or u
0
¼ 0
N
0ab
À
N
0ab
R
b
_ _
À N
ab
À
N
ab
R
b
_ _
¼ 0 or v
0
¼ 0
Q
0a
À
1
B
@M
0ab
@b
_ _
À Q
a
À
1
B
@M
ab
@b
_ _
¼ 0 or w
0
¼ 0
M
0a
À M
a
¼ 0 or w
a
¼ 0
M
0ab

b
2
b
1
¼ 0
ð19Þ
The equations presented thus far are complete in the sense that
the number of equations is equal to the number of unknowns for
each of the theories presented. The number of equations using
3D elasticity theory is 15 written in three coupled variables (a, b
and z). Other shell theories use only two variables (a and b). For
3D elasticity theory, three boundary conditions must be satisfied
at each of the six external boundaries, leading to a total of 18 con-
ditions whereas for thin-shell theory, five conditions (four along
the edges and one for the corners) at each of the four edges of
the shell are required.
Much research using thin-shell theory has been performed. Ray
and Reddy [52] studied control of thin circular cylindrical lami-
nated composite shells using active constrained layer damping
treatment. Evseev and Morozov [53] discussed the aeroelastic
interaction of shock waves with thin-walled composite shells.
The transient vibration of a composite cylindrical shell due to an
underwater shock wave was studied by Li and Hua [54]. Ruotolo
[55] determined the natural frequencies of stiffened shells and
compared the results obtained from Donnell’s, Love’s, Sanders’
and Flugge’s thin shell theories.
Numerous studies dealing with modal analysis on laminated
composite shells using thin-shell theory appear in the literature.
Free vibration analyses based on Love’s thin-shell theory have been
performed by Civalek [56,57] for laminated conical and cylindrical
shells. Free vibration analyses have also been performed by Sakiy-
ama et al. [58] and Hu et al. [59] who considered composite conical
shells with twist and by Lee et al. [60] who determined the vibra-
tion characteristics of a laminated cylindrical shell with an interior
rectangular plate. Other free vibration studies have been con-
ducted by Kimand Lee [61] on the modal response of ring-stiffened
laminated cylindrical shells, and by Shang [62] on composite cylin-
drical shells with hemispherical end caps.
2.4. Non-linear theories
The magnitude of transverse displacement compared to shell
thickness is the third criterion used in classifying shell equations.
It can be shown that if the transverse displacement approaches
the thickness of the shell (often earlier than that), the results can
be in gross error. The non-linear terms are neglected in thin, thick,
and 3D shell theories described earlier. In non-linear shell theories,
such terms are retained. In many cases, they are expanded using
perturbation methods, and smaller orders of the rotations are re-
tained. Most frequently the first-order is retained and occasionally
third-order terms have been included in non-linear shell theories.
The material used can also be non-linear (e.g., rubber, plastics and
others). Theories that include non-linear material constants are
also called non-linear shell theories. However, the vast majority
of shell theories deal only with geometric nonlinearity.
Krejaa and Schmidt [63] used a first-order shear deformation
theory in a finite element analysis to describe large rotations of
laminated shells. Abea et al. [64] studied the non-linear dynamic
behavior of clamped laminated shallow shells with internal
M.S. Qatu et al. / Composite Structures 93 (2010) 14–31 21
resonance. Wang et al. [65] were interested in the non-linear dy-
namic response of laminated cylindrical shells with an axial shal-
low groove. Zhou and Wang [66] developed a non-linear theory
of dynamic stability for laminated composite cylindrical shells.
Kundu and Sinha [67] and Swamy and Sinha [68] studied the
non-linear transient phenomenon of laminated composites. Lom-
boy et al. [69] developed a co-rotational shell element for progres-
sive non-linear dynamic failure analysis. Fatigue delamination
growth for a piezoelectric laminated cylindrical shell considering
non-linear contact effect was studied by Fuhui et al. [70]. Balah
[71] analyzed laminated shells undergoing finite rotations and
large motion. Amabili [72] evaluated the elastic strain energy ob-
tained fromnon-linear shell theories to investigate large amplitude
vibrations of circular cylindrical shells.
3. Shell geometries
Shells may have different geometries based mainly on their cur-
vature characteristics. In most shell geometries, the fundamental
equations have to be treated at a very basic level and are depen-
dent upon the
1. choice of the coordinate system (Cartesian, Polar, curvilinear or
spherical),
2. characteristics of the Lame parameters (constant or a function
of the coordinates),
3. curvature (constant or varying curvature).
Some shell geometries can be obtained from a more general
derivation of the fundamental equations. For example, equations
for cylindrical, spherical, conical and barrel shells can be derived
from the equations of the more general case of shells of revolution.
Equations for cylindrical, barrel, twisted and shallow shells can
also be derived from the general equation of doubly curved shells
with constant Lame parameters.
The shell geometry being addressed by most researchers is the
closed cylindrical shells. The widespread usage and ease of manu-
facturing of cylindrical shells are undoubtedly the reason for such
attention. Other shell geometries have also been investigated.
Among those receiving considerable attention are shallow shells
and conical shells. Shallow shells are open shells with rectangular,
triangular, trapezoidal, circular, elliptical, rhombic or other plan-
forms. They are used frequently as panels in aerospace and sub-
marine industries. General equations for specific shell geometries
will be described in the following sections.
3.1. Shells of revolution
Consider a shell of revolution as shown in Fig. 5. The fundamen-
tal form can be written as (e.g., Soedel [17]):
ðdsÞ
2
¼ ðdaÞ
2
þsin
2
ð/ÞðdbÞ
2
where da ¼ R
a
d/; db ¼ R
b
dh ð20Þ
The Lame’ parameters are:
A ¼ 1; B ¼ sinð/Þ ð21Þ
Examples of the dynamic analysis of laminated shells of revolu-
tion is the work of Pinto Correia et al. [73] who modeled and opti-
mized laminated adaptive shells of revolution, Saravanan et al. [47]
who studied active damping and Shang [62] who presented an ex-
act analytical solution for the free vibration of composite capsule
structures.
3.1.1. Cylindrical and doubly curved shells
Consider further a shell with the following characteristics:
1. Constant radii of curvature R
a
, R
b
, and R
ab
.
2. Constant Lame’ parameters.
This is shown to be the case for twisted plates (R
a
= R
b
= 0),
cylindrical shells (R
a
= R
ab
= 0), and barrel shells (R
ab
= 0). Constant
Lame’ parameters cannot be applied to general shells. For cylindri-
cal shells, the angle / (Fig. 5) is 90°. This leads to unit Lame’ param-
eters (sin(/) = 1) if the in-plane coordinates are used. For barrel
shells of revolution, the angle / varies between 75° and 105°. Thus,
the approximation of sin(/) % 1 is reasonable. The fundamental
form can then be approximated as
ðdsÞ
2
% ðdaÞ
2
þ ðdbÞ
2
ð22Þ
Lame’ parameters can be written as A = 1 and B % 1. The same
equations can be used for open doubly curved shells (Fig. 6). For
the complete list of equations, the reader is referred to Ref. [1].
Saravanos and Christoforou [48] researched the impact of adap-
tive cylindrical piezoelectric composite shells. Dong and Wang
[28,74,75] studied the wave propagation in piezoelectric laminated
cylindrical shells. Thamburaj and Sun [76] used conformal map-
ping to perform modal analysis of a non-circular cylindrical lami-
nated shell. Toorani and Lakis [27] studied shear deformation in
dynamic analysis of anisotropic laminated open cylindrical shells.
Saravanan et al. [47] performed analysis of active damping in lam-
inated cylindrical shells of revolution. Optimal stacking sequence
for the vibrations of laminated cylindrical shells using genetic algo-
a
r θ r
φ R
R
β
α
Fig. 5. Cross-section of a shell of revolution shells.
22 M.S. Qatu et al. / Composite Structures 93 (2010) 14–31
rithms was performed by Shakeri et al. [77]. Fan et al. [78] dis-
cussed the critical speed in creep buckling of viscoelastic laminated
plates and circular cylindrical shells. Han et al. [79] determined the
material constants of laminated cylindrical shells based on an in-
verse optimal approach. Dynamic stability of laminated cylindrical
shells was the subject of many studies [66,80,81]. Krishnamurthy
et al. [82,83] studied the impact response and damage in laminated
composite cylindrical shells. Dong and Wang [84] found influences
of large deformation and rotary inertia on wave propagation in pie-
zoelectric cylindrically laminated shells. Toorani [85] conducted a
geometric non-linear analysis of anisotropic composite shells.
Other dynamic analyses of cylindrical shells include the aforemen-
tioned studies by Ganapathi et al. [34] and Wang et al. [65].
3.1.2. Conical shells
Conical shells constitute another special type of shells of revo-
lution. They are formed by revolving a straight line around an axis
that is not parallel to the line. If the axis of revolution is parallel to
the line, the special case of cylindrical shells is obtained. Fig. 7
shows a typical closed conical shell of revolution. Conical shells
can have circular and elliptic cross-sections (where they will not
be shells of revolutions). Open conical shells can be produced by
cutting a segment of the closed conical shell. An example of an
open conical shell can be obtained by considering the shell be-
tween h = h
1
and h
2
(Fig. 7). If the side lengths of the open conical
shell (for the fundamental mode) are less than half the minimum
radius of curvature, a shallow conical shell theory can be
developed.
The equations of conical shells can be obtained by deriving and
substituting the proper Lame’ parameters of conical shells in the
general shell equations. Consider Fig. 8, which is a side view of
the conical shell described in Fig. 7. The fundamental form can
be written as:
ðdsÞ
2
¼ ðdaÞ
2
þa
2
sin
2
ðuÞðdhÞ
2
ð23Þ
Z
α
β
R
α

R
β

a
b
w
o


u
o

v
o


Fig. 6. Parameters of the middle surface for a doubly curved shell.
1

2
1

2
Fig. 7. A closed conical shell.
M.S. Qatu et al. / Composite Structures 93 (2010) 14–31 23
which yields the following Lame’ parameters and radii of curvature.
A ¼ 1; B ¼ asinðuÞ;
R
a
¼ 1; R
b
¼ atanðuÞ
ð24Þ
Tzou et al. [86] studied the dynamics of conical shells laminated
with full and diagonal actuators. Pinto Correia et al. [37] studied
dynamics and statics of laminated conical shell structures using a
higher order theory. Aksogan and Sofiyev [87] researched the dy-
namic stability of a laminated truncated conical shell with variable
elastic moduli and densities. Wu and Chiu [88] studied thermally
induced dynamic instability of laminated conical shells. Moreira
et al. [43] used finite elements to perform dynamic analysis on a
conical sandwich shell. Mamalis et al. [89] studied crushing loads
of conical shells. Effects of conical shell characteristics on the fun-
damental cyclic frequencies was investigated by Sofiyev [90] and
Senthil et al. [91] performed a dynamic analysis on cones of vary-
ing cone angles and boundary conditions. Civalek [56,57] used
thin-shell theory to determine vibration characteristics of conical
shells. Modal characteristics of twisted conical shells were deter-
mined by Hu et al. [59] and Sakiyama et al. [58].
3.1.3. Spherical shells
Spherical shells are another special case of shells of revolution.
For these shells, a circular arc, rather than a straight line, revolves
about an axis to generate the surface. If the circular arc is half a cir-
cle and the axis of rotation is the circle’s own diameter, a closed
sphere will result. If a segment of this shell is taken, an open spher-
ical shell will be produced. If the dimensions of the segment are
small when compared with the radius, a shallow spherical shell
will result. Shallow spherical shells can have rectangular, circular
(spherical caps) or other planforms. These shallow shells can have
both rectangular orthotropy as well as spherical orthotropy.
The equation for spherical shells can be obtained by deriving
and substituting the proper Lame’ parameters in the general shell
R
2
R
2
*
α
R
β
=α tan( )
ϕ
ϕ
ϕ
Fig. 8. Side view of a closed conical shell.
R
θ
Fig. 9. A closed spherical shell.
24 M.S. Qatu et al. / Composite Structures 93 (2010) 14–31
equations. Considering Fig. 9, the fundamental form can be written
as
ðdsÞ
2
¼ R
2
ðduÞ
2
þ R
2
sin
2
ðuÞðdhÞ
2
: ð25Þ
which yields the following Lame’ parameters and radii of curvature
A ¼ R
B ¼ Rsin u ð Þ
R
a
¼ R
R
b
¼ R
ð26Þ
Wang et al. [31] studied wave propagation of stresses in ortho-
tropic thick-walled spherical shells. Dai and Wang [92] analyzed
stress wave propagation in laminated piezoelectric spherical shells
under thermal shock and electric excitation. Lellep and Hein [93]
did an optimization study on shallow spherical shells under impact
loading. Dynamic stability of spherical shells was studied by Gana-
pathi [94] and Park and Lee [95].
3.2. Shallow shells
The theory of shallow shells is based on the following
assumptions:
1. Both in-plane displacements and transverse shear forces are
very small compared to the radius of curvature, i.e.,
u
i
R
i
<< 1 and
Q
i
R
i
<< 1 ð27Þ
where u
i
can be any of the in-plane displacement components u
and v, Q
i
can be any of the shear forces Q
a
, and Q
b
, and R
i
can be
any of the radii R
a
, R
b
, and R
ab
2. The term (1 + z/R
i
) can be neglected. R
i
can be any of the radii R
a
,
R
b
, and R
ab
.
3. The curvilinear coordinates can be replaced by the Cartesian or
Polar coordinates (for Cartesian coordinates a = x, b = y).
Frequently, many analysts employ only assumptions 2 and 3
above. The resulting equations will still be valid for shallow shells
and will not generally apply to deep shells. It should be noted that
only shells on a circular planform produce a shell of revolution.
Abea et al. [64] studied the non-linear dynamic behaviors of
clamped laminated shallow shells. Zang et al. [96] analyzed dy-
namic buckling of laminated shallow spherical shells. Guo et al.
[97] used finite elements to study the non-linear random response
of laminated shallow shells. Optimization studies were performed
by Lellep and Hein [93] on clamped shallow spherical shells under
impulsive loading.
4. Dynamic analyses
Both the theory and the analysis of shell structures can become
more complex if one or more of the several complicating effects,
such as transverse shear deformation and rotary inertia, are in-
cluded. Including the effects of shear deformation, which is neces-
sary for most composites, increases the order of the differential
equations used for shell analysis from 8 to 10 or higher as have
been discussed earlier. In this section, complexities arising from
the type of dynamic analysis, such as free vibration, impact load-
ing, dynamic stability, rotating shells, etc. are covered.
4.1. Free vibration
Santos et al. [22] performed vibration analysis of 3D axisym-
metric laminated shells with piezoelectric actuators. Ganapathi
and Haboussi [33] studied the free vibrations of thick laminated
anisotropic non-circular cylindrical shells. Civalek [57] numerically
solved the free vibrations of laminated composite conical and
cylindrical shells using discrete singular convolution. Lee et al.
[60] analyzed the free vibrations of laminated composite cylindri-
cal shells with an interior rectangular plate. Korhevskaya and
Mikhasev [98] studied the free vibrations of a laminated cylindrical
shell subjected to nonuniformly distributed axial forces. Toorani
and Lakis [99] analyzed the free vibrations of nonuniform compos-
ite cylindrical shells. Umur and Varlik [100] studied the free vibra-
tions of bonded single lap joints in composite shell panels whereas
Singh [101] studied the non-linear vibrations of fiber reinforced
laminated shallow shells. Hu and Ou [102] maximized the funda-
mental frequency of laminated truncated conical shells with re-
spect to fiber orientation. Timarchi and Soldatos [103] analyzed
the vibrations of angle-ply laminated circular cylindrical shells
with different edge boundary conditions. Topal [104] performed
a mode-frequency analysis of laminated spherical shells. Tizzi
[105] presented a Ritz procedure for optimization of cylindrical
shells with a nearly symmetric angle-ply composite laminate.
Ferreira et al. [106] presented natural frequencies of cross-ply
composite shells by multiquadrics using a shear deformation shell
theory. Iqbal and Qatu [107] studied the transverse bending vibra-
tion of circular two-segment composite shafts. Tetsuya [108] sim-
plified the analytical method for calculation of natural frequencies
of laminated composite cylindrical shells using equivalent
curvature.
4.2. Rotating shells
Zhang [109] used a wave propagation approach to study the f
frequency of rotating composite cylindrical shells. Zhao et al.
[110] studied the vibrations of rotating cross-ply circular cylindri-
cal shells with ring stiffeners. Hua and Lam [111] researched the
orthotropic influence on frequency characteristics of a rotating
laminated conical shell by the differential quadrature method.
Gong and Lam [112] studied the transient response of rotating
multilayer cylindrical shells to impact loading. Shi et al. [113] stud-
ied the effects of dynamic characteristics (centrifugal force, trans-
verse shear force, etc.) on the natural frequencies of a rotating
composite shell.
4.3. Impact loading
Krishnamurthy et al. [82,83] performed a parametric study of
the impact response of laminated cylindrical shells. Huang and
Lee [114] studied composite shells subjected to low-velocity im-
pact loads. Kim et al. [115] investigated the behavior of laminated
composite shells under transverse impact loading. Saravanos and
Christoforou [48] studied the low-energy impact of adaptive cylin-
drical piezoelectric–composite shells. Johnson and Holzapfel [116]
investigated the response of laminated composite shells to soft
body impacts such as bird strikes on aircraft surfaces and contin-
ued their study by investigating high velocity impacts [117]. John-
son et al. [118] reviewed recent developments regarding the
impact response of composite structures. Other research on com-
posite shells under impact loading includes studies by Kim et al.
[119,120] who characterized the response and resulting damage
from impact on curved composite shells, Krishnamurthy et al.
[82,83] who used finite elements to investigate impact response
and damage, and Lee et al. [121], who used neural networks to
investigate impact loading parameters on composite shells. Low-
velocity impact of composite shells was also investigated by
Rastorguev and Snisarenko [122], Smojver et al. [123] (impact
damage modeling), Tiberkak et al. [124] (impact damage predic-
tion), and Zhao and Cho [125–127] (impact damage analysis and
M.S. Qatu et al. / Composite Structures 93 (2010) 14–31 25
prediction). Wan et al. [128], studied the failure behavior of fila-
ment wound cylindrical shells subjected to axial impact loading.
Yang et al. [129], performed penetration experiments on laminated
composite shells to determine penetration characteristics of vari-
ous stacking sequences.
4.4. Dynamic stability
Sahu and Datta [130] reviewed recent research activity on the
dynamic stability behavior of plates and shells. Zhou and Wang
[66] presented a theory of non-linear dynamic stability for lami-
nated composite cylindrical shells. Liew et al. [80] applied the
mesh-free kp-Ritz method to perform a dynamic stability analysis
on composite cylindrical shells subjected to periodic axial forces.
Yang and Fu [81] analyzed the dynamic stability for composite
laminated cylindrical shells with delaminations. Birman and Sim-
itses [131] studied cylindrical sandwich shells subjected to peri-
odic lateral pressure; Darabi et al. [132] and Ng et al. [133]
analyzed functionally graded cylindrical shells subjected to peri-
odic axial loading; Darvizeh et al. [134] investigated the general
dynamic behavior of circular cylindrical shells including both
shear deformation and rotary inertia terms. Ganapathi [94] studied
the dynamic behavior of functionally graded shallow spherical
shells; Kamat et al. [135] investigated the dynamic instability of
a joined conical and cylindrical composite shell undergoing a peri-
odic in-plane load; Kasuya and Yamagishi [136] studied the behav-
ior of cross-ply laminated cylindrical shells subjected to impact
hydrostatic pressure; the dynamic stability under an external
pressure loading on composite cylindrical shells was also investi-
gated by Nemoto et al. [137]. Dynamic stability analysis was per-
formed by Park and Lee [95] on composite spherical shells
subjected to in-plane pulsating forces and by Peng et al. [138] on
cylindrical shells under axially harmonic loads. Sofiyev [139] stud-
ied the dynamic stability of cross-ply laminated cylindrical shells
with a focus on torsional buckling and also investigated the dy-
namic stability of functionally graded conical shells under external
pressure [90].
4.5. Thermal and hygrothermal loading
Dong and Wang [84] studied wave propagation in piezoelectric
cylindrical shells in a thermal environment. Swamy and Sinha [68]
analyzed the non-linear transient phenomenon of laminated shells
in hygrothermal environments. Khdeir [140] studied thermally in-
duced vibrations of cross-ply laminated shallow shells. Dynamic
thermal analyses have also been studied by Dong and Wang [28],
Wu and Chiu [88], and Kim et al. [115]. Tylikowski [141] studied
the dynamic stability of shape memory alloys subjected to thermal
loads. Oh [45] used a finite element method to determine the
vibration and damping characteristics as functions of temperature
and frequency.
4.6. General dynamic behavior
Several other studies have addressed the general dynamic
behavior of laminated composite shells. Pinto Correiaa et al.
[142] presented a finite element model for the dynamic analysis
of laminated axisymmetric shells. Wu and Lo [143] discussed an
asymptotic theory for dynamic response of laminated piezoelectric
shells. Prusty and Satsangi [144] performed finite element tran-
sient dynamic analysis of laminated stiffened shells. Park et al.
[145] conducted a linear static and dynamic analysis of laminated
composite plates and shells using finite elements. Fares et al. [146]
studied the dynamic response of laminated doubly curved compos-
ite shells for optimization. Yang and Shen [147] performed dy-
namic instability analysis of laminated piezoelectric shells. Sahu
and Datta [148] discussed parametric resonance characteristics of
laminated composite shells subjected to nonuniform loading. Sof-
iyev [139] investigated the dynamic loading and its impact on
the torsional buckling of cross-ply laminated orthotropic cylindri-
cal shells. Tetsuya et al. [149] studied the vibration characteristics
of laminated composite cylindrical shells. Lee [150] presented a
dynamic variational asymptotic procedure for laminated compos-
ite shells. Tzou et al. [86] investigated the dynamics of conical
shells laminated with full and diagonal actuators. Vu-Quoc and
Tan [151] used optimal solid shells for non-linear dynamic analy-
ses of laminated composites. Birman et al. [152] studied the dy-
namic behavior of composite spherical shells reinforced with
active piezoelectric composite stiffeners. Lee and Hodges [153]
performed a low frequency vibration analysis of laminated com-
posite shells and Tetsuya [154] determined the resonant frequen-
cies and deformations of composite cylindrical shells. Ip and Tse
[155] located damage in circular cylindrical composite shells based
on frequency sensitivities and mode shapes. Ribeiro and Jansen
[156] studied the non-linear vibrations of laminated cylindrical
shallow shells under thermomechanical loading. Jansen [157]
investigated the effects of static loading and imperfections on the
non-linear vibrations of laminated cylindrical shells. Li and Hua
[54] studied the transient vibrations of laminated composite cylin-
drical shells exposed to underwater shock waves.
5. Material complexity
The analysis of composite shells is significantly more complex
than that of isotropic or metallic shells. Most metallic structures
exhibit material properties that are related only to the modulus
of elasticity and Poisson’s ratio of the material. Composite materi-
als exhibit the properties of orthotropy where several material
constants exist. A typical orthotropic layer can have nine material
constants in the analysis of laminated composites. In addition,
unsymmetric laminates exhibit coupling, i.e., stretching-bending,
stretching-twisting, shearing-bending and shearing-twisting,
when loaded dynamically. Also, these composites can have active
piezoelectric layers, be braided or made of wood or natural fibers
or other more complex materials. This paper will focus on the stud-
ies concerning piezoelectric or damping layers used for vibration
control.
5.1. Piezoelectric shells
Pinto Correia et al. [158] studied active control of axisymmetric
shells with piezoelectric layers using a mixed lamination theory.
Wu and Lo [143] presented an asymptotic theory for the dynamic
response of laminated piezoelectric shells. Varelis and Saravanos
[50] used coupled mechanics and finite elements to investigate
non-linear laminated piezoelectric shallow shells undergoing large
displacements and rotations. Dong and Wang [74] studied the
influences of large deformation and rotary inertia on wave propa-
gation in piezoelectric cylindrically laminated shells in a thermal
environment. Ray and Pradhan [159] analyzed the performance
of vertically and obliquely reinforced piezoelectric composites for
active damping of laminated composite shells. Fuhui et al. [70]
analyzed fatigue delamination growth for piezoelectric laminated
cylindrical shells. Zemcık et al. [160] presented a shell element
with piezoelectric coupling for the analysis of smart laminated
structures. Yao and Lu [161] developed a solid-shell model of lam-
inated composite piezoelectric structures under a non-linear distri-
bution of electric potential through the thickness. Kozlov and
Karnaukhova [162] presented basic equations for viscoelastic lam-
inated shells with distributed piezoelectric inclusions intended to
control nonstationary vibrations. Rasskazov et al. [163] studied
26 M.S. Qatu et al. / Composite Structures 93 (2010) 14–31
the forced vibrations of shallow viscoelastic laminated shells with
a piezoelectric effect. Santos et al. [22] developed a finite element
model to analyze the bending and free vibration behavior of lami-
nated shells with piezoelectric sensor and actuators. Shakeri et al.
[24] performed a dynamic analysis, based on the 3D elasticity the-
ory, on a thick composite shell with a piezoelectric layer. Dong and
Wang [28,74,75] studied the effects of transverse shear, rotary
inertia and large deformations on the wave propagation behavior
of laminated piezoelectric cylindrical shells. Saravanos and Chris-
toforou [48] investigated the low impact behavior of adaptive pie-
zoelectric composite shells and Yang and Shen [147] studied the
dynamic instability of composite piezoelectric shells.
5.2. Viscoelastic and damped shells
Babeshko and Prokhorenko [164], Galishin [165–167] and Gali-
shin and Shevchenko [168] studied the thermoviscoelasticplastic
stress state of laminated shells. Fan et al. [78] studied creep buck-
ling of viscoelastic laminated plates and circular cylindrical shells.
Sprenger et al. [169] studied the delamination of laminated struc-
tures with continuum-based 3D shell elements and a viscoplastic
softening model. Yan et al. [170] described the behavior of angle-
ply laminated cylindrical shells with viscoelastic interfaces. Teters
[171] presented and analyzed a method for estimating the reliabil-
ity of optimal viscoelastic composite shells. Studies on viscoelastic
damping include Moreira et al. [43], Saravanan [47], Shin et al. [49],
Ray and Reddy [52], Shina et al. [172], Ray and Pradhan [159], Koz-
lov and Karnaukhova [162], Rasskazov et al. [163]. Shin et al. [173]
included viscoelastic damping in their analysis of thermal post-
buckled behaviors of cylindrical composite shells. Vamsi and Gane-
sa [174] investigated fluid-filled and submerged composite shells
having a viscoelastic layer. Several studies that address dynamic
characteristics of composite shell structures with viscoelastic lay-
ers include Lee et al. [42], Oh [44,45], Oh and Cheng [46], Shin et
al. [49] and Peng et al. [138].
6. Structural complexity
Structural complexity arises when the shell geometry or bound-
ary conditions deviates from those of the classical shells described
earlier. These include stiffened shells, shells with internal bound-
aries from cracks, shells submerged in or filled with fluids, imper-
fect shells as well as other types of complexities.
6.1. Stiffened shells
Prusty and Satsangi [144] performed finite element transient
dynamic analysis of laminated stiffened shells. Zhao et al. [110]
studied the vibrations of rotating cross-ply circular cylindrical
shells with stringer and ring stiffeners. Birman et al. [152] investi-
gated the axissymmetric dynamics of composite spherical shells
with active peizoelectric-composite stiffeners.
6.2. Shells with cracks and/or damage
Krishnamurthy et al. [82,83] performed parametric studies of
the impact response and damage of laminated shells. Galishin
[165] studied the axisymmetric thermoviscoelastoplastic state of
thin laminated shells made of a damageable material. Ip and Tse
[155] were able to locate damage in circular cylindrical composite
shells based on frequency sensitivities and mode shape analysis.
Smojver et al. [123] and Tiberkak et al. [124] used finite elements
to perform damage analysis of laminated shells subjected to im-
pact. Zhao and Cho [125–127] studied the initiation and evolution
of damage due to low-velocity impact of composite shells.
6.3. Fluid filled and/or submerged shells
Toorani and Lakis [27] showed the effects of shear deformation
in dynamic analysis of anisotropic laminated open cylindrical shells
filled with or subjected to a flowing fluid. Amabili [72,175] per-
formed a non-linear vibration analysis of fluid filled cylindrical
shells; Kuo et al. [176] investigated acoustic-structure interaction
of sound by considering various fluid filled composite shells of dif-
ferent stacking sequences. Okazaki et al. [177,178] experimentally
determined the modal characteristics of a cross-ply composite
cylindrical shell partially filled with fluid and followed the study
by analyzing the free vibrations using finite elements. Senthil and
Ganesan [91] performed a dynamic analysis on composite conical
shells filled with fluid. Toorani and Lakis [27,179] explored the free
vibration of composite cylindrical shells filled and submerged with
fluid as did Vamsi and Ganesan [174] with viscoelastic cylindrical
shells. Xi et al. investigated wave scattering due to a crack in a
fluid-filled laminated cylindrical shell [180] and studied the disper-
sion of waves in immersed composite hollow cylinders [181]. Yu et
al. [182,183] performed damage detection of partially fluid-filled
laminated composite shells. Zhang and Yang [184] studied the
behavior of a fluid-filled polymer-metal composite cylindrical shell.
6.4. Imperfect shells
Hasheminejad and Maleki [185] studied the acoustic wave
interaction with a laminated transversely isotropic spherical shell
with imperfect bonding. Interlaminar bonding imperfection in
composite shells was also investigated by Librescu and Schmidt
[186]. Jansen [157] investigated the effect of static loading and
imperfections on the dynamic behavior of laminated cylindrical
shells.
6.5. Other complexities
Rasskazov et al. [163] discussed forced vibrations and vibro-
heating of shallowviscoelastic laminated shells with the piezoelec-
tric effect. Lee et al. [60] studied a complex structure made of
laminated composite cylindrical shells with an interior rectangular
plate. Umur and Varlik [100] studied the free vibrations of bonded
single lap joints in composite shallow cylindrical shell panels. Zhu
et al. [187] discussed multi-scale analysis including strain rate
dependency for transient response of composite laminated shells.
Turkmen [188] studied the effects of blast loading on composite
shells. Other studies with complexities in shell theory include, Vel-
murugan and Gupta [189], Sofiyev et al. [190], Icardi and Ruotolo
[191], Piskunov and Rasskazov [192], and Cugnoni et al. [193].
7. Other topics
7.1. Experimental analyses
To validate calculations and models, experimental data is used
by Amabili [72,175] to investigate large amplitude vibrations of
circular cylindrical shells. Guedes [194] performed an experimen-
tal study to evaluate the three-point bend testing of curved panels.
To assess a dynamic model, Kwak et al. [195] conducted modal
experiments for a cylindrical shell withpiezoelectric sensors and
actuators. Lee et al. [60] compared measured data from vibration
testing of composite cylindrical shells to a finite element model.
7.2. Optimization
Optimization studies on dynamic responses have been conducted
by Cho [196] for an orthotropic composite shell in a hygrothemal
M.S. Qatu et al. / Composite Structures 93 (2010) 14–31 27
environment. Sahoo and Chakravorty [197] investigated various
features (laminations, boundary conditions, stiffener depth) of
composite hypar shells to select optimized shell configurations.
Teters [198] conducted an optimization based on natural frequen-
cies and thermal stresses of a composite shell subjected to thermal
and dynamic stresses. Roy and Chakraborty [199] implemented a
genetic algorithm to obtain optimal control of smart fiber compos-
ite shell structures subjected to combined mechanical and thermal
loading.
8. Concluding remarks
Researchers in the field of composite shell vibration agreed that
for moderately thick shells, shear deformation and rotary inertia
should be included in the analysis. As a result of these inclusions,
the differential equations necessary for shell theory are increased
to at least a degree of 10. This requirement facilitated the develop-
ment of thick shell theories.
Cylindrical shells are still the subject of research of most recent
articles. Doubly curved shallow shells as well as twisted plates
have also received considerable interest. These shells can be spher-
ical, cylindrical, or hyperbolic-paraboloidal in shape.
The dynamic response of composite shells with complicating ef-
fects of various kinds has received considerable attention. The use
of piezoelectric shells, necessitated by various applications, where
vibrations need to be controlled, resulted in considerable literature
in the field. Damping characteristics of composite shells has also
received much attention; this could be due to the increasing sensi-
tivity of the public to noise and vibrations in engineering products.
Research is also being conducted on the response of shells with
elastic or fluid media.
Due to its practical application, the finite element method is
increasingly used to analyze the behavior of laminated shells of
various geometries, which are subjected to a variety of boundary
and loading conditions. For simple geometries, the Ritz method
for the linear analysis of composite shells and the Galerkin method
for the non-linear analysis of such shells are frequently used.
Recent research is directed towards the interpretation of results
and finding new applications for composite shell structures. The
tailorability of composites (material, fiber orientation, ply se-
quence, curvature) for specific applications has given engineers
the ability to design superior and more economical structures.
Attention is needed for experimental verification and correlation
with current analytical methods and theories. As is shown here,
the articles that are dedicated for obtaining experimental results
are limited.
Acknowledgments
The authors thank Mr. Imran Aslam and Ms. Jutima Simsiri-
wong for their help in locating key articles.
References
[1] Qatu MS. Vibration of laminated shells and plates. San Diego, CA: Elsevier;
2004.
[2] Qatu MS. Recent research advances in the dynamic behavior of shells, part 1:
laminated composite shells. Appl Mech Rev 2002;55:325–50.
[3] Qatu, MS. Free vibration and static analysis of laminated composite shallow
shells. PhD Dissertation, Ohio State University; 1989.
[4] Qatu MS. Review of shallow shell vibration research. Shock Vib Dig
1992;24:3–15.
[5] Qatu MS. Accurate theory for laminated composite deep thick shells. Int J
Solids Struct 1999;36:2917–41.
[6] Leissa AW. Vibration of shells. NASA SP388. The Government Printing Office,
Washington, DC; 1973. [Republished 1993, the Acoustical Society of America].
[7] Kapania PK. Review on the analysis of laminated shells. J Pressure Vessel
Technol 1989;111:88–96.
[8] Noor AK, Burton WS. Assessment of computational models for multilayered
composite shells. Appl Mech Rev 1990;43:67–97.
[9] Noor AK, Burton WS. Computational models for high-temperature
multilayered composite plates and shells. Appl Mech Rev 1992;45:419–46.
[10] Noor AK, Burton WS, Peters JM. Assessment of computational models for
multilayered composite cylinders. Int J Solids Struct 1991;27:1269–86.
[11] Liew KM, Lim CW, Kitipornchai S. Vibration of shallow shells: a review with
bibliography. Appl Mech Rev 1997;50:431–44.
[12] Soldatos KP. Mechanics of cylindrical shells with non-circular cross-section.
Appl Mech Rev 1999;52:237–74.
[13] Noor AK, Burton WS, Bert CW. Computational models for sandwich panels
and shells. Appl Mech Rev 1996;49:155–200.
[14] Noor AK, Venneri SL. High-performance computing for flight vehicles.
Comput Syst Eng 1992;3:1–4.
[15] Carrera E. Historical review of zig-zag theories for multilayered plates and
shells. Appl Mech Rev 2003;56:287–309.
[16] Carrera E. Theories and finite elements for multilayered, anisotropic,
composite plates and shells. J Arch Comput Methods Eng 2002;9:87–140.
[17] Soedel W. Vibrations of shells and plates. 3rd ed. New York, NY: Marcel
Dekker; 2004.
[18] Reddy JN. Mechanics of laminated composite plates and shells: theory and
analysis. second ed. Boca Raton, FL: CRC Press; 2003.
[19] Ye J. Laminated composite plates and shells: 3D modeling. London: Springer-
Verlag; 2003.
[20] Lee CY. Geometrically correct laminated composite shell modeling. Ikoyi,
Lagos, Nigeria: VDM Verlag; 2008.
[21] Shen HS. Functionally graded materials: nonlinear analysis of plates and
shells. Boca Raton, FL: CRC Press; 2009.
[22] Santos H, Mota Soares CM, Mota Soares CA, Reddy JN. A finite element model
for the analysis of 3D axisymmetric laminated shells with piezoelectric
sensors and actuators: bending and free vibrations. Comput Struct
2008;86:940–7.
[23] Santos H, Mota Soares CM, Mota Soares CA, Reddy JN. A finite element model
for the analysis of 3D axisymmetric laminated shells with piezoelectric
sensors and actuators. Compos Struct 2006;75:170–8.
[24] Shakeri M, Eslami MR, Daneshmehr A. Dynamic analysis of thick laminated
shell panel with piezoelectric layer based on three dimensional elasticity
solution. Comput Struct 2006;84:1519–26.
[25] Malekzadeh P, Fiouz AR, Razi H. Three-dimensional dynamic analysis of
laminated composite plates subjected to moving load. Compos Struct
2009;90(2):105–14.
[26] Saviz MR, Shakeri M, Yas MH. Electrostatic fields in a layered piezoelectric
cylindrical shell under dynamic load. Smart Mater Struct 2007;16:1683–95.
[27] Toorani MH, Lakis AA. Shear deformation in dynamic analysis of anisotropic
laminated open cylindrical shells filled with or subjected to a flowing fluid.
Comput Methods Appl Mech Eng 2001;190:4929–66.
[28] Dong K, Wang X. The effect of transverse shear, rotary inertia on wave
propagation in laminated piezoelectric cylindrical shells in thermal
environment. J Reinforc Plast Compos 2007;26:1523–38.
[29] Ribeiro P. On the influence of membrane inertia and shear deformation on the
geometrically nonlinear vibrations of open, cylindrical, laminated clamped
shells. Compos Sci Technol 2009;69:176–85.
[30] Qatu MS. Theory and vibration analysis of laminated barrel thick shells. J Vib
Control 2004;10:319–41.
[31] Wang X, Lu G, Guillow SR. Stress wave propagation in orthotropic laminated
thick-walled spherical shells. Int J Solids Struct 2002;39:4027–37.
[32] Ding KW. The thermoelastic dynamic response of thick closed laminated
shell. Shock Vib 2005;12:283–91.
[33] Ganapathi M, Haboussi M. Free vibrations of thick laminated anisotropic non-
circular cylindrical shells. Compos Struct 2003;60:125–33.
[34] Ganapathi M, Patel BP, Pawargi DS. Dynamic analysis of laminated cross-ply
composite non-circular thick cylindrical shells using higher-order theory. Int
J Solids Struct 2002;39:5945–62.
[35] Khare RK, Rode V. Higher-order closed-form solutions for thick laminated
sandwich shells. J Sandwich Struct Mater 2005;7:335–58.
[36] Balah M, Al-Ghemady HN. Energy–momentum conserving algorithm for
nonlinear dynamics of laminated shells based on a third-order shear
deformation theory. J Eng Mech 2005;131:12–22.
[37] Pinto Correia IF, Mota Soares CM, Mota Soares CA, Herskovits J. Analysis of
laminated conical shell structures using higher order models. Compos Struct
2003;62:383–90.
[38] Qian W, Liu GR, Chun L, Lam KY. Active vibration control of composite
laminated cylindrical shells via surface-bonded magnetostrictive layers.
Smart Mater Struct 2003;12:889.
[39] Lam KY, Ng TY, Qian W. Vibration analysis of thick laminated composite
cylindrical shells. AIAA J 2000;38:1102.
[40] Braga AMB, Rivas ACE. High-frequency response of isotropic-laminated
cylindrical shells modeled by a layer-wise theory. Int J Solids Struct
2005;42:4278–94.
[41] Basar Y, Omurtag MH. Free-vibration analysis of thin/thick laminated
structures by layer-wise shell models. Comput Struct 2000;74:409–27.
[42] Lee I, Oh IK, Shin WH, Cho KD, Koo KN. Dynamic characteristics of cylindrical
composite panels with Co-cured and constrained viscoelastic layers. JSME Int
J, Ser C: Mech Syst, Mach Elem Manuf 2002;45:16.
[43] Moreira RAS, Rodrigues JD, Ferreira AJM. A generalized layerwise finite
element for multi-layer damping treatments. Comput Mech 2006;37:426.
28 M.S. Qatu et al. / Composite Structures 93 (2010) 14–31
[44] Oh IK. Dynamic characteristics of cylindrical hybrid panels containing
viscoelastic layer based on layerwise mechanics. Composites Part B: Eng
2007;38:159.
[45] Oh IK. Damping characteristics of cylindrical laminates with viscoelastic layer
considering temperature- and frequency-dependence. J Therm Stresses
2009;32:1.
[46] Oh IK, Cheng TH. Vibration analyses of cylindrical hybrid panel with
viscoelastic layer based on layerwise finite elements. Key Eng Mater
2006;324–325(II):699.
[47] Saravanan C, Ganesan N, Ramamurti V. Analysis of active damping in
composite laminate cylindrical shells of revolution with skewed PVDF
sensors/actuators. Compos Struct 2000;48:305–18.
[48] Saravanos DA, Christoforou AP. Low-energy impact of adaptive cylindrical
piezoelectric-composite shells. Int J Solids Struct 2002;39:2257–79.
[49] Shin WH, Oh IK, Han JH, Lee I. Aeroelastic characteristics of cylindrical hybrid
composite panels with viscoelastic damping treatments. J Sound Vib
2006;296:99.
[50] Varelis D, Saravanos DA. Coupled mechanics and finite element for nonlinear
laminated piezoelectric shallow shells undergoing large displacements and
rotations. Int J Numer Methods Eng 2006;66:1211–33.
[51] Wang A, Yang G, Zhang X. Vibration and transverse stresses of
laminated cylindrical shells. Chin J Appl Mech (Ying Yong Li Xue Xue Bao)
2001;18:34.
[52] Ray MC, Reddy JN. Optimal control of thin circular cylindrical laminated
composite shells using active constrained layer damping treatment. Smart
Mater Struct 2004;13:64–72.
[53] Evseev EG, Morozov EV. Aeroelastic interaction of the shock waves with the
thin-walled composite shells. Compos Shells 2001;54:153–9.
[54] Li J, Hua H. Transient vibrations of laminated composite cylindrical shells
exposed to underwater shock waves. Eng Struct 2009;31:738–48.
[55] Ruotolo R. A comparison of some thin shell theories used for the dynamic
analysis of stiffened cylinders. J Sound Vib 2001;243:847.
[56] Civalek Ö. Free vibration analysis of composite conical shells using the
discrete singular convolution algorithm. Steel Compos Struct 2006;6:353.
[57] Civalek Ö. Numerical analysis of free vibrations of laminated composite
conical and cylindrical shells: discrete singular convolution (DSC) approach. J
Comp Appl Math 2007;205:251.
[58] Sakiyama T, Hu XX, Matsuda H, Morita C. Vibration of cantilevered laminated
composite conical shells with twist, vol. 4. Seville, Spain: WIT Press; 2002.
[59] Hu XX, Sakiyama T, Matsuda H, Morita C. Vibration of twisted laminated
composite conical shells. Int J Mech Sci 2002;44:1521–41.
[60] Lee YS, Choi MH, Kim JH. Free vibrations of laminated composite cylindrical
shells with an interior rectangular plate. J Sound Vib 2003;265:795–817.
[61] Kim YW, Lee YS. Transient analysis of ring-stiffened composite cylindrical
shells with both edges clamped. J Sound Vib 2002;252:1–17.
[62] Shang XC. An exact analysis for free vibration of a composite shell structure-
hermetic capsule. Appl Math Mech 2001;22:1035–45 [English Edition].
[63] Krejaa I, Schmidt R. Large rotations in first-order shear deformation FE
analysis of laminated shells. Int J Non-lin Mech 2006;1:101–23.
[64] Abea A, Kobayashib Y, Yamada G. Nonlinear dynamic behaviors of clamped
laminated shallow shells with one-to-one internal resonance. J Sound Vib
2007;304:957–68.
[65] Wang TL, Tang WN, Zhang SK. Nonlinear dynamic response and buckling of
laminated cylindrical shells with axial shallow groove based on a semi-
analytical method. J Shanghai University 2007;11:223–8 [English Edition].
[66] Zhou CT, Wang LD. Nonlinear theory of dynamic stability for laminated
composite cylindrical shells. Appl Math Mech 2001;22:53–62.
[67] Kundu CK, Sinha PK. Nonlinear transient analysis of laminated composite
shells. J. Reinforc Plast Compos 2006;25:1129–47.
[68] Swamy NV, Sinha PK. Nonlinear transient analysis of laminated composite
shells in hygrothermal environments. Compos Struct 2006;72:280–8.
[69] Lomboy GR, Kim KD, Onate E. A co-rotational 8-node resultant shell element
for progressive nonlinear dynamic failure analysis of laminated composite
structures. Mech Adv Mater Struct 2007;14:89–105.
[70] Fuhui Z, Yiming F, Deliang C. Analysis of fatigue delamination growth for
piezoelectric laminated cylindrical shell considering nonlinear contact effect.
Int J Solids Struct 2008;45:5381–96.
[71] Balah M. Analysis of laminated general shells undergoing finite rotations and
large motion. PhD Thesis, KFUPM, Saudi Arabia; 2000.
[72] Amabili M. A comparison of shell theories for large-amplitude vibrations of
circular cylindrical shells: Lagrangian approach. J Sound Vib 2003;264:
1091–125.
[73] Pinto Correia IF, Martins PG, Mota Soares CM, Mota Soares CA, Herskovits J.
Modelling and optimization of laminated adaptive shells of revolution.
Compos Struct 2006;75:49–59.
[74] Dong K, Wang X. Wave propagation in piezoelectric laminated cylindrical
shells under large deformations and rotary inertias. J Mech Eng Sci
2006;220:1537–48.
[75] Dong K, Wang X. Wave propagation characteristics in piezoelectric cylindrical
laminated shells under large deformation. Compos Struct 2007;77:171–81.
[76] Thamburaj P, Sun JQ. Modal analysis of a non-circular cylindrical laminated
shell using conformal mapping. J Sand Struct Mater 2001;3:50–74.
[77] Shakeri M, Yas MH, Ghasemi Gol M. Optimal stacking sequence of laminated
cylindrical shells using genetic algorithm. Mech Adv Mater Struct
2005;12:305–12.
[78] Fan P, YiMing FU, YiFan L. On the durable critic load in creep buckling of
viscoelastic laminated plates and circular cylindrical shells. Sci China Ser G –
Phys Mech Astron 2008;51:873–82.
[79] Han X, Xu D, Yap FF, Liu GR. On determination of the material constants of
laminated cylindrical shells based on an inverse optimal approach. Inverse
Probl Eng 2002;10:309–22.
[80] Liew KM, Hu YG, Zhao X, Ng TY. Dynamic stability analysis of composite
laminated cylindrical shells via the mesh-free kp-Ritz method. Comput
Methods Appl Mech Eng 2006;196:147–60.
[81] Yang J, Fu Y. Analysis of dynamic stability for composite laminated cylindrical
shells with delaminations. Compos Struct 2007;78:309–15.
[82] Krishnamurthy KS, Mahajan P, Mittal PK. Impact response and damage in
laminated composite cylindrical shells. Compos Struct 2003;59:15–36.
[83] Krishnamurthy KS, Mahajan P, Mittal RK. A parametric study of the impact
response and damage of laminated cylindrical composite shells. Compos Sci
Technol 2001;61:1655–69.
[84] Dong K, Wang X. Influences of large deformation and rotary inertia on wave
propagation in piezoelectric cylindrically laminated shells in thermal
environment. Int J Solids Struct 2006;43:1710–26.
[85] Toorani MH. Dynamics of the geometrically nonlinear analysis of anisotropic
laminated cylindrical shells. Int J Non-lin Mech 2003;38:1315–35.
[86] Tzou HS, Wang DW, Chai WK. Dynamics and distribution control of conical
shells laminated with full and diagonal actuators. J Sound Vib
2002;256:65–79.
[87] Aksogan O, Sofiyev AH. The dynamic stability of a laminated truncated
conical shell with variable elasticity moduli and densities subject to a time-
dependent external pressure. J Strain Anal 2002;37:201–10.
[88] Wu CP, Chiu SJ. Thermally induced dynamic instability of laminated
composite conical shells. Int J Solids Struct 2002;39:3001–21.
[89] Mamalis AG, Manolakos DE, Demosthenous GA, Ioannidis MB. Analytical
modelling of the static and dynamic axial collapse of thin-walled fiberglass
composite conical shells. Int J Impact Eng 1997;19:477–92.
[90] Sofiyev AH. The vibration and stability behavior of freely supported FGM
conical shells subjected to external pressure. Compos Struct 2009;89:356–66.
[91] Senthil KD, Ganesan N. Dynamic analysis of conical shells conveying fluid. J
Sound Vib 2008;310:38–57.
[92] Dai HL, Wang X. Stress wave propagation in laminated piezoelectric spherical
shells under thermal shock and electric excitation. Eur J Mech A/Solids
2005;24:263–76.
[93] Lellep J, Hein H. Optimization of clamped plastic shallow shells subjected to
initial impulsive loading. Eng Optim 2002;34:545–56.
[94] Ganapathi M. Dynamic stability characteristics of functionally graded
materials shallow spherical shells. Compos Struct 2007;79:338–43.
[95] Park T, Lee SY. Parametric instability of delaminated composite spherical
shells subjected to in-plane pulsating forces. Compos Struct 2009;91:
196–204.
[96] Zang YQ, Zhang D, Zhou HY, Mab HZ, Wang TK. Nonlinear dynamic buckling
of laminated composite shallow spherical shells. Compos Sci Technol
2000;60:2361–3.
[97] Guo X, Lee YY, Mei C. Nonlinear random response of laminated composite
shallow shells using finite element modal method. Int J Numer Methods Eng
2006;67:1467–89.
[98] Korhevskaya EA, Mikhasev GI. Free vibrations of a laminated cylindrical shell
subjected to nonuniformly distributed axial forces. Mech Solids
2006;41:130–8.
[99] Toorani MH, Lakis AA. Free vibrations of non-uniform composite cylindrical
shells. Nucl Eng Des 2006;236:1748–58.
[100] Umur Y, Varlik O. Free vibrations of bonded single lap joints in composite,
shallow cylindrical shell panels. AIAA J 2005;43:2537–48.
[101] Singh AV. Linear and geometrically nonlinear vibrations of fiber reinforced
laminated plates and shallow shells. Comput Struct 2000;76:277–85.
[102] Hu HT, Ou SC. Maximizations of fundamental frequency of laminated
truncated conical shells with respect to fiber orientation. Compos Struct
2001;52:265–75.
[103] Timarchi T, Soldatos KP. Vibrations of angle-ply laminated circular cylindrical
shells subjected to different sets of edge boundary conditions. J Eng Math
2000;37:211–30.
[104] Topal U. Mode-frequency analysis of laminated spherical shell. In:
Proceedings of the 2006 IJME. INTERTECH International Conference-
Session ENG P501-001, Kean University, NJ; 19–21 October 2006.
[105] Tizzi S. A Ritz procedure for optimisation of cylindrical shells, formed by a
nearly symmetric and balanced angle-ply composite laminate, with fixed
minimum frequency. Comput Struct 2006;84:2159–73.
[106] Ferreira AJM, Roque CMC, Jorge RMN. Natural frequencies of FSDT cross-ply
composite shell by multiquadrics. Compos Struct 2007;77:296–305.
[107] Iqbal J, Qatu MS. Transverse vibration of circular two-segment composite
shafts. Compos Struct 2010;92:1126–31.
[108] Tetsuya N. Simplified analytical method for calculation of natural frequencies
of laminated composite cylindrical shells using equivalent curvature. Reports
of the Tokyo Metropolitan Technical College 2000;35:25–30.
[109] Zhang XM. Parametric analysis of frequency of rotating laminated composite
cylindrical shells with the wave propagation approach. Comput Methods
Appl Mech Eng 2002;191:2057–71.
[110] Zhao X, Liew KM, Ng TY. Vibrations of rotating cross-ply circular cylindrical
shells with stringer and ring stiffener. Int J Solids Struct 2002;39:529–45.
M.S. Qatu et al. / Composite Structures 93 (2010) 14–31 29
[111] Hua L, Lam KY. Orthotropic Influence on frequency characteristics of a
rotating composite laminated conical shell by the generalized differential
quadrature method. Int J Solids Struct 2001;38:3996–4015.
[112] Gong SW, Lam KY. Rotating multilayered cylindrical shells to impact loading.
AIAA J 2003;41:139–42.
[113] Shi Y, Hong J, Wu W. Dynamic characteristic analysis of rotating composite
shell. Beijing Hangkong Hangtian Daxue Xuebao/J Beijing Univ Aeronaut
Astronaut 2004;30:31.
[114] Huang CH, Lee YJ. Quasi-static simulation of composite-laminated shells
subjected to low-velocity impact. J Reinforc Plast Compos 2005;24:763–74.
[115] Kim YN, Im KH, Lee KS, Cho YJ, Kim SH, Yang IY. Experimental approach on
the behavior of composites laminated shell under transverse impact loading.
Rev Quant Nondestruct Anal 2005;24:1100–6.
[116] Johnson AF, Holzapfel M. Modelling soft body impact on composite
structures. Compos Struct 2003;61:103–13.
[117] Johnson AF, Holzapfel M. Influence of delamination on impact damage in
composite structures. Compos Sci Technol 2006;66:807–15.
[118] Johnson AF, Pickett AK, Rozycki P. Computational methods for predicting
impact damage in composite structures. Compos Sci Technol 2001;61:
2183–92.
[119] Kim YN, Im KH, Yang IY. Characterization of impact damages and responses
in CFRP composite shells, vol. 465–466. Kumamoto, Japan: Trans Tech
Publications Ltd.; 2004. p. 247–52.
[120] Kim YN, Yang IY. Impact response and damage of composite shell with
various curvatures, vol. 270–273. Jeju Island, Korea: Republic of Trans Tech
Publications Ltd.; 2004. p. 1911.
[121] Lee YS, Ryu CH, Myung CM. Identification of impact loading characteristics of
composite laminated cylindrical shells using neural networks. Prague, Czech
Republic: Civil-Comp Limited; 2002. p. 195–6.
[122] Rastorguev GI, Snisarenko SI. Physical relations for problems of impact
loading and unsteady deformation of composite structures. J Appl Mech Tech
Phys 2009;50:155–62.
[123] Smojver I, Soric J, Bathe KJ. On damage modelling of laminated composite
shells subjected to low velocity impact. Computational fluid and solid
mechanics. Oxford: Elsevier Science Ltd.; 2003.
[124] Tiberkak R, Bachene M, Rechak S, Necib B. Damage prediction in composite
plates subjected to low velocity impact. Compos Struct 2008;83:73–82.
[125] Zhao G, Cho CD. On impact damage of composite shells by a low-velocity
projectile. J Compos Mater 2004;38:1231–54.
[126] Zhao GP, Cho CD. Progressive damage analysis of composite shell under
impact loading. Key Eng Mater 2004;274–276:111–6.
[127] Zhao GP, Cho CD. Damage initiation and propagation in composite shells
subjected to impact. Compos Struct 2007;78:91–100.
[128] Wan ZM, Wang L, Du XW. Failure behavior of laminated composite
cylindrical shells under axial impact loading. Harbin Gongye Daxue
Xuebao, J Harbin Inst Technol (China) 2001;33:304–8.
[129] Yang IY, Cho YJ, Kim YN, Heo U, Park SG, Im KH, et al. The penetration
characteristics of CF/epoxy curved sheller according to stacking sequence.
Key Eng Mater 2006;321–323(Pt 2):885–8.
[130] Sahu SK, Datta PK. Research advances in the dynamic stability behavior of
plates and shells: 1987–2005 – part I: conservative systems. Appl Mech Rev
2007;60:65.
[131] Birman V, Simitses GJ. Dynamic stability of long cylindrical sandwich shells
and panels subject to periodic-in-time lateral pressure. J Compos Mater
2004;38:591–607.
[132] Darabi M, Darvizeh M, Darvizeh A. Non-linear analysis of dynamic stability
for functionally graded cylindrical shells under periodic axial loading.
Compos Struct 2008;83:201–11.
[133] Ng TY, Lam KY, Liew KM, Reddy JN. Dynamic stability analysis of functionally
graded cylindrical shells under periodic axial loading. Int J Solids Struct
2001;38:1295–309.
[134] Darvizeh M, Haftchenari H, Darvizeh A, Ansari R, Sharma CB. The effect of
boundary conditions on the dynamic stability of orthotropic cylinders using a
modified exact analysis. Compos Struct 2006;74:495–502.
[135] Kamat S, Ganapathi M, Patel BP. Analysis of parametrically excited laminated
composite joined conical-cylindrical shells. Comput Struct 2001;79:65–76.
[136] Kasuya H, Yamagishi Y. An analysis of dynamic stability of cross-ply
laminated cylindrical shells under impact hydrostatic pressure. Zairyo/J Soc
Mater Sci, Jpn 2003;52:1357–62.
[137] Nemoto K, Kasuya H, Yamagishi Y. An analysis of dynamic stability of
composite laminated cylindrical shells subjected to periodic external
pressure. Nippon Kikai Gakkai Ronbunshu, A Hen/Trans Jpn Soc Mech Eng,
Part A 2003;69:545–51.
[138] Peng F, Xiang H, Fu YM. Dynamic instability of viscoelastic cross-ply
laminated plates and circular cylindrical shells. Zhendong Gongcheng
Xuebao/J Vib Eng 2006;19:459.
[139] Sofiyev AH. Torsional buckling of cross-ply laminated orthotropic composite
cylindrical shells subject to dynamic loading. Eur J Mech A/Solids
2003;22:943–51.
[140] Khdeir AA. Thermally induced vibrations of cross-ply laminated shallow
shells. Acta Mech 2001;151:135–47.
[141] Tylikowski A. Dynamic stability of rotating composite shells with
thermoactive shape memory alloy fibers. J Therm Str 1998;21:327–39.
[142] Pinto Correiaa IF, Barbosa JI, Mota Soares CA, Mota Soares CA. A finite element
semi-analytical model for laminated axisymmetric shells: statics, dynamics
and buckling. Comput Struct 2000;76:299–317.
[143] Wu CP, Lo JY. An asymptotic theory for dynamic response of laminated
piezoelectric shells. Acta Mech 2006;183:177–208.
[144] Prusty BG, Satsangi SK. Finite element transient dynamic analysis of
laminated stiffened shells. J Sound Vib 2001;248:215–33.
[145] Park T, Kim K, Han S. Linear static and dynamic analysis of laminated
composite plates and shells using a 4-node quasi-conforming shell element.
Composites: Part B 2006;37:237–48.
[146] Fares ME, Youssif YG, Alamir AE. Minimization of the dynamic response of
composite laminated doubly curved shells using design and control
optimization. Compos Struct 2003;59:369–83.
[147] Yang XM, Shen YP. Dynamic instability of laminated piezoelectric shells. Int J
Solids Struct 2001;38:2291–303.
[148] Sahu SK, Datta PK. Parametric resonance characteristics of laminated
composite doubly curved shells subjected to non-uniform loading. J
Reinforce Plast Compos 2001;20:1556–76.
[149] Tetsuya N, Masanori K, Kohei S. A study on vibration characteristics of
laminated composite cylindrical shells. Natural frequencies of FW
antisymmetrically laminated composite cylindrical shells. Trans Jpn Soc
Mech Eng 2000;66:1747–55.
[150] Lee CY. Dynamic variational asymptotic procedure for laminated composite
shells. PhD Thesis, Georgia Inst Tech; 2007.
[151] Vu-Quoc L, Tan XG. Optimal solid shells for nonlinear analyses of multilayer
composites. II. Dynamics. Comput Methods Appl Mech Eng 2003;192:
1017–59.
[152] Birman V, Griffin S, Knowles G. Axissymmetric dynamics of composite
spherical shells with active peizoelectric-composite stiffeners. Acta Mech
2000;141:71–83.
[153] Lee CY, Hodges DH. Dynamic variational-asymptotic procedure for laminated
composite shells – part I: low-frequency vibration analysis. J Appl Mech
2009;76:77–84.
[154] Tetsuya N. Natural frequencies and transverse deformations of laminated
cylindrical shells. Dyn Des Conf 2000;907–10.
[155] Ip KH, Tse PC. Locating damage in circular cylindrical composite shells based
on frequency sensitivities and mode shapes. Eur J Mech A/Solids 2002;21:
615–28.
[156] Ribeiro P, Jansen E. Nonlinear vibrations of laminated cylindrical shallow
shells under thermomechanical loading. J Sound Vib 2008;315:626–40.
[157] Jansen EL. The effect of static loading and imperfections on the
nonlinear vibrations of laminated cylindrical shells. J Sound Vib 2008;315:
1035–46.
[158] Pinto Correia IF, Mota Soares CM, Mota Soares CA, Herskovits J. Active control
of axisymmetric shells with piezoelectric layers: a mixed laminated theory
with a high order displacement field. Comput Struct 2002;80:2265–7.
[159] Ray MC, Pradhan AK. Performance of vertically and obliquely reinforced 1–3
piezoelectric composites for active damping of laminated composite shells. J
Sound Vib 2008;315:816–35.
[160] Zemcık R, Rolfes R, Rose M, Teßmer J. High-performance four-node shell
element with piezoelectric coupling for the analysis of smart laminated
structures. Int J Numer Methods Eng 2007;70:934–61.
[161] Yao LQ, Lu L. Hybrid-stabilized solid-shell model of laminated composite
piezoelectric structures under nonlinear distribution of electric potential
through thickness. Int J Numer Methods Eng 2003;58:1499–522.
[162] Kozlov VI, Karnaukhova TV. Basic equations for viscoelastic laminated shells
with disturbed piezoelectric inclusions intended to control nonstationary
vibrations. Int Appl Mech 2002;38:1253–60.
[163] Rasskazov AO, Kozlov VI, Karnaukhova TV. Forced vibrations and
vibroheating of shallow viscoelastic laminated shells with piezoelectric
effect. Int Appl Mech 2000;36:769–78.
[164] Babeshko ME, Prokhorenko IV. The thermoviscoelastoplastic axisymmetric
stress–strain state of laminated flexible orthotropic shells. Int Appl Mech
2000;36:1088–96.
[165] Galishin AZ. Axisymmetric thermoviscoelastoplastic state of thin laminated
shells made of a damageable material. Int Appl Mech 2008;44:431–41.
[166] Galishin AZ. Determination of axisymmetrical geometrical nonlinear
thermoviscoelaticplastic state of laminated medium-thickness shells. Int
Appl Mech 2001;37:271–8.
[167] Galishin AZ. Determining the axisymmetrical geometrically nonlinear
thermoviscoelastoplastic state of laminated shells by the theory of
deformation along paths of small curvature. Int Appl Mech 2003;39:848–55.
[168] Galishin AZ, Shevchenko YN. Determining the axisymmetric, geometrically
nonlinear, thermoelastoplastic state of laminated orthotropic shells. Int Appl
Mech 2003;39:56–63.
[169] Sprenger W, Gruttmann F, Wagner W. Delamination growth analysis in
laminated structures with continuum-based 3D-shell elements and a
viscoplastic softening model. Comput Methods Appl Mech Eng 2000;185:
123–39.
[170] Yan W, Ying J, Chen WQ. The behavior of angle-ply laminated cylindrical
shells with viscoelastic interfaces in cylindrical bending. Compos Struct
2007;78:551–9.
[171] Teters G. Reliability estimation of optimal viscoelastic composite shells in
critical-time calculations. Mech Compos Mater 2003;39:553–8.
[172] Shina WH, Oh IK, Lee I. Nonlinear flutter of aerothermally buckled composite
shells with damping treatments. J Sound Vib 2009;324:556–69.
[173] Shin WH, Lee SJ, Oh IK, Lee I. Thermal post-buckled behaviors of cylindrical
composite shells with viscoelastic damping treatments. J Sound Vib
2009;323:93–111.
30 M.S. Qatu et al. / Composite Structures 93 (2010) 14–31
[174] Vamsi KB, Ganesan N. Studies on fluid-filled and submerged cylindrical shells
with constrained viscoelastic layer. J Sound Vib 2007;303:575–95.
[175] Amabili M. Nonlinear vibrations of circular cylindrical shells with different
boundary conditions. AIAA J 2003;41:1119–30.
[176] Kuo YM, Lin HJ, Wang CN, Wang WC, Chang CL. Study of the interaction of
two acoustical fields and one FRP structure with FEM-BEM coupling analysis.
J Reinforce Plast Compos 2009;28:2083–96.
[177] Okazaki K, Tani J, Qiu J, Kosugo K. Vibration test of a cross-ply laminated
composite circular cylindrical shell partially filled with liquid. Nihon Kikai
Gakkai Ronbunshu, C Hen/Trans Jpn Soc Mech Eng, Part C 2007;73:724–31.
[178] Okazaki K, Tani J, Sugano M. Free vibrations of a laminated composite coaxial
circular cylindrical shell partially filled with liquid. Nippon Kikai Gakkai
Ronbunshu, C Hen/Trans Jpn Soc Mech Eng, Part C 2002;68:1942–9.
[179] Toorani MH, Lakis AA. Swelling effect on the dynamic behaviour of composite
cylindrical shells conveying fluid. Int J Numer Methods Fluids 2006;50:
397–420.
[180] Xi ZC, Liu GR, Lam KY, Shang HM. A strip-element method for analyzing wave
scattering by a crack in a fluid-filled composite cylindrical shell. Compos Sci
Technol 2000;60:1985–96.
[181] Xi ZC, Liu GR, Lam KY, Shang HM. Dispersion of waves in immersed laminated
composite hollow cylinders. J Sound Vib 2002;250:215–27.
[182] Yu L, Cheng L, Yam LH, Yan YJ, Jiang JS. Experimental validation of vibration-
based damage detection for static laminated composite shells partially filled
with fluid. Compos Struct 2007;79:288–9.
[183] Yu L, Cheng L, Yam LH, Yan YJ, Jiang JS. Online damage detection for
laminated composite shells partially filled with fluid. Compos Struct
2007;80:334–42.
[184] Zhang L, Yang Y. Modeling of a fluid-filled ionic polymer-metal composite
cylindrical shell. In: Proceedings SPIE, Adelaide, Australia; 6214: 11
December 2006.
[185] Hasheminejad SM, Maleki M. Acoustic wave interaction with a laminated
transversely isotropic spherical shell with imperfect bonding. Arch Appl
Mech 2009;79:97–112.
[186] Librescu L, Schmidt R. A general theory of laminated of laminated composite
shells featuring interlaminar bonding imperfections. Int J Solids Struct
2001;38:3355–75.
[187] Zhu L, Chattopadhyay A, Goldberg RK. Multiscale analysis including strain
rate dependency for transient response of composite laminated shells. J
Reinforce Plast Compos 2006;25:1795–831.
[188] Turkmen HS. Structural response of laminated composite shells subjected to
blast loading: comparison of experimental and theoretical methods. J Sound
Vib 2002;249:663–78.
[189] Velmurugan R, Gupta NK. Energy absorption characteristics of metallic and
composite shells. Def Sci J 2003;53:127–38.
[190] Sofiyev H, Sofiyev AH, Yusufoglu E, Karaca Z. Semi-analytical solution of
stability of composite orthotropic cylindrical shells undertime dependant a-
periodic axialcompressive load. Iran J Sci Technol 2006;30:343–7.
[191] Icardi U, Ruotolo R. Laminated shell model with second-order expansion of
the reciprocals of Lame coefficients Ha;Hb and interlayer continuities
fulfillment. Compos Struct 2002;56:293–313.
[192] Piskunov VG, Rasskazov AO. Evolution of the theory of laminated plates and
shells. Int Appl Mech 2002;38:135–66.
[193] Cugnoni J, Gmür T, Schorderet A. Identification by modal analysis of
composite structures modelled with FSDT and HSDT laminated shell finite
elements. Composites: Part A 2004;35:977–87.
[194] Guedes RM, Sá A. Numerical analysis of singly curved shallow composite
panels under three-point bend load. Compos Struct 2008;83:212–20.
[195] Kwak MK, Heo S, Jeong M. Dynamic modelling and active vibration controller
design for a cylindrical shell equipped with piezoelectric sensors and
actuators. J Sound Vib 2009;321:510–24.
[196] Cho HK. Optimization of dynamic behaviors of an orthotropic composite shell
subjected to hygrothermal environment. Finite Elem Anal Des
2009;45(11):852–60.
[197] Sahoo S, Chakravorty D. Stiffened composite hypar shell roofs under
free vibration: behaviour and optimization aids. J Sound Vib 2006;295:
362–77.
[198] Teters G. Multicriteria optimization of a composite cylindrical shell
subjected to thermal and dynamic actions. Mech Compos Mater 2004;40:
489–94.
[199] Roy T, Chakraborty D. Genetic algorithm based optimal control of smart
composite shell structures under mechanical loading and thermal gradient.
Smart Mater Struct 2009;18(11):5006.
M.S. Qatu et al. / Composite Structures 93 (2010) 14–31 31

Sponsor Documents

Or use your account on DocShare.tips

Hide

Forgot your password?

Or register your new account on DocShare.tips

Hide

Lost your password? Please enter your email address. You will receive a link to create a new password.

Back to log-in

Close