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JOURNAL OF
C O M P O S I T E
M AT E R I A L S
Article
Enhanced thermal buckling of laminated
composite cylindrical shells with shape
memory alloy
H Asadi, Y Kiani, MM Aghdam and M Shakeri
Abstract
Thermal bifurcation behavior of cross-ply laminated composite cylindrical shells embedded with shape memory alloy
fibers is investigated. Properties of the constituents are assumed to be temperature-dependent. Donnell’s kinematic
assumptions accompanied with the von-Karman type of geometrical non-linearity are used to derive the governing
equations of the shell. Furthermore, the one-dimensional constitutive law of Brinson is used to predict the behavior
of shape memory alloy fibers through the heating process. Governing equilibrium equations are established by
employing the static version of virtual displacements principle. Linear membrane pre-buckling analysis is performed
to extract the pre-buckling deformations of the shell. Applying the well-known adjacent equilibrium criterion to the
pre-buckling state of the shell, stability equations are derived. The governing equations are solved via a semi-
analytical solution employing the exact trigonometric function in circumferential direction and the harmonic differ-
ential quadrature method in the longitudinal direction. Numerical results cover various cases of edge supports,
cross-ply lamination, shape memory alloy fibers volume fraction and shape memory alloy fiber pre-strain. It is shown
that, proper usage of shape memory alloy fibers results in considerable delay of the thermal bifurcation type of
buckling.
Keywords
Shape memory alloy, Brinson model, thermal buckling, temperature dependency, cylindrical shells, harmonic differential
quadrature method
Introduction
Due to their unique characteristics, such as high
strength-to-weight, high stiffness-to-weight, low coeffi-
cient of thermal expansion especially in fiber direction
and resistance to fatigue, composite materials have
gained considerable attention in the last decades.
Cylindrical shells are known as one of the major
structural components with vast applications in the
structural, mechanical, civil, aerospace and marine
engineering. When composite cylindrical shells are sub-
jected to thermal fields, there is a possibility of compres-
sive thermal forces through the shell due to the
constraints of the structure. Such case may result in
the buckling phenomenon. Consequently, investigation
of thermal stability of composite cylindrical shells and
presenting applicable procedures to enhance the buck-
ling resistance of the structure would be a vital step for
design purposes.
As it was stated by Thornton,1
investigations on the
subject of thermal stability of composite shells are rare
in comparison to isotropic shells. Thermal stability ana-
lysis of cylindrical shell-like structures in a composite
scheme is available through the open literature. To
study the thermal stability of composite cylindrical
shells with fiberglass-reinforced plastics, Radhamohan
and Venkataramana2
formulated the behavior of shells
within the framework of Sanders shell theory assump-
tions. Thangaratnam et al.3
performed an analysis on
the thermal stability behavior of circular cylindrical
shell in a composite laminated form. In this study,
Thermo-elasticity Center of Excellence, Mechanical Engineering
Department, Amirkabir University of Technology, Tehran, Iran
Corresponding author:
Asadi H, Thermo-elasticity Center of Excellence, Mechanical Engineering
Department, Amirkabir University of Technology, Tehran, Iran.
Email: hamed_asadi@aut.ac.ir
Journal of Composite Materials
0(0) 1–14
! The Author(s) 2015
Reprints and permissions:
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DOI: 10.1177/0021998315573287
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solution of the thermal stability equations is accom-
plished via the finite element technique. Shen4–7
exam-
ined the buckling and post-buckling behavior of
perfect/imperfect stiffened/unstiffened multi-layered
shells with arbitrary lamination scheme under the
action of thermal loads. A two-step perturbation tech-
nique is implemented to trace the equilibrium path of
the structure which may also be useful to detect the
bifurcation temperatures. Birman and Bert8
analyzed
the buckling and post-buckling behavior of a laminated
cylindrical shell at the elevated temperature. Ganesan
and Kadoli9
investigated the stability resistance of
cylindrical shells conveying hot fluid. In this research,
a semi-analytical finite elements solution is applied to
the governing stability equation.
Shape memory alloys (SMAs) are known as a branch
of smart materials with fascinating characteristics.
Capability of SMA materials to control the vibration,
snap-through, bifurcation and post-buckling resistance
of structures have been necessitated more investigations
on the analysis of SMA hybrid structures. A comprehen-
sive review of SMA phenomenological models was pre-
sented by Lagoudas et al.10
Various investigators
examined the influence of SMA fibers on the buckling
and post-buckling resistance of beams, circular and rect-
angular plates and cylindrical panels. An overview of
such investigations is presented in the next.
Birman11
studied the stability loss phenomenon in a
rectangular plate embedded with SMA fibers. Only the
case of a plate with all edges simply supported and
subjected to uni-axial loading is investigated. As con-
cluded, non-uniform distribution of fibers through the
width is more influential than the conventional uniform
fiber dispersion. To examine the influence of SMA
fibers on the stability characteristics of SMA hybrid
plates, a series of experimental studies was carried out
by Thompson and Loughlan12
and Loughlan et al.13
Results of these studies accept the high influence of
SMA fibers on the buckling delay and alleviation of
post-buckling deflections of SMA hybrid composite
plates. To analyze the behavior of panels with
embedded SMA fiber, a finite element formulation
was developed by Tawfik et al.,14
which takes into
account the geometrical non-linearity of von-Karman
type. The unique feature of SMA wire to recover the
large pre-strain in temperatures above the austenite
finish temperature is found as an influential parameter
on the stability of SMA fiber reinforced panels. Park
et al.15,16
examined the thermal buckling, post-
buckling and small-amplitude vibration of a thermally
post-buckled SMA hybrid plate using the finite elem-
ents formulations. This study concluded in the consid-
erable interference of the SMA fiber volume fraction
and SMA fiber pre-strain on the equilibrium path and
vibration characteristics. Ibrahim17,18
implemented the
finite elements method to study the thermal buckling
and post-buckling resistance of initially imperfect com-
posite laminated rectangular plates reinforced with
SMA fibers. For more investigations on the subject of
SMA hybrid composites, one may refer to Choi
et al.,19
Lee et al.,20
Lee and Lee,21
Kumar and
Singh22,23
and Panda and Singh.24
It is worth noting that, in most of the previous
works, the behavior of SMA fibers is read from some
experimentally validated results or extracted from the
simulated-based analysis. Only a few investigations deal
with the mathematical modelling of SMA fibers.25–35
In the following investigations, a mathematical model
is used to obtain the kinematics of the SMA fibers.
Thermal post-buckling analysis of circular plates was
developed by Li et al.25
based on the shooting method.
The thermal snapping phenomenon in cylindrical
panels was studied by Roh et al.26
based on the finite
elements method. Asadi et al.27–29
developed analytical
closed-form solutions to determine efficiency and reli-
ability of SMA fibers for improving the free and forced
vibration characteristics of smart composite beams.
Asadi et al.30,31
investigated the thermal post-buckling
behaviors of geometrically imperfect and perfect lami-
nated composite beam consisting of SMA fibers based
on exact analytical solutions. Results of these studies
indicate that the recovery stresses of SMA fibers may
enhance the structural responses of laminated beam.
Furthermore, the SMA fibers reinforced composite
beam reveals superior behavior of the thermal post-
buckling compared to the conventional composite
beam. A comprehensive study on the thermally induced
pre/post buckling deformation of geometrically imper-
fect hybrid composite plates with embedded SMA
fibers was conducted by Asadi et al.32
The orientation
of SMA fibers was reported as a significant param-
eter in critical buckling load and buckling configur-
ations. In all of these studies, the well-known
one-dimensional constitutive law of Brinson is used to
estimate the behavior of SMA fibers through the heat-
ing process.36
As the above literature survey reveals, there is no
work supplied on the thermal stability analysis of
SMA hybrid composite cylindrical shells, especially
when the behavior of SMA fibers is dictated by
means of a mathematical constitutive law. Present art-
icle examines the influence of SMA fibers on the ther-
mal bifurcation phenomenon of cylindrical shells. The
behavior of SMA fibers is modelled by means of the 1D
constitutive law of Brinson. It is assumed that cylin-
drical shell obeys the Donnell type of kinematic
assumptions incorporated with the von-Karman type
of geometrical non-linearity. The complete set of non-
linear governing equations for such conditions is
obtained. Solution of the pre-buckling state is
2 Journal of Composite Materials 0(0)
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accomplished by means of proper simplifications and
the stability loss equations are established by means of
the adjacent equilibrium criterion. A semi-analytical
solution using Fourier expansion and the harmonic dif-
ferential quadrature (HDQ) discretization is adopted to
detect the fundamental buckling temperatures and
buckled shape of the structure. The main contribution
of the present work is to reveal the efficient application
of SMA fibers for improving the thermal stability
behavior of composite cylindrical shell. Considerable
delay of the bifurcation is observed through the numer-
ical results, which reveal the SMA fibers as the influen-
tial parameter to control the thermal stability of the
structures.
Equilibrium equations
Consider an SMA fiber reinforced hybrid composite
cylindrical shell with total thickness h, length L and
mean radius R, referred to the coordinates system (x,
, z), as shown in Figure 1. The axial, circumferential
and through-the-thickness coordinates are denoted by
x, and z, respectively.
A suitable micromechanical model should be used to
establish the effective materials properties of SMA
hybrid composite cylindrical shells. In this study, the
multi-cell micromechanics approach of Chamis37
is
used to determine the effective thermo-mechanical prop-
erties. The effectiveness of this approach in determin-
ation of the equivalent thermomechanical properties of
SMA hybrid composite beams has been claimed by
Birman et al.38
Based on this model, the resultant
expressions for the thermo-elastic properties of the
SMA hybrid composite are:
E11 ¼ Es
ð ÞVs þ E1mð1 VsÞ ð1AÞ
E22 ¼ E2m ð1
ffiffiffiffiffi
Vs
p
Þ þ
ffiffiffiffiffi
Vs
p
1
ffiffiffiffiffi
Vs
p
1 E2m
Es
ð Þ

2
4
3
5 ð1BÞ
12 ¼ 12sVs þ 12m 1 Vs
ð Þ ð1CÞ
G12 ¼ G13 ¼ G12m ð1
ffiffiffiffiffi
Vs
p
Þ þ
ffiffiffiffiffi
Vs
p
1
ffiffiffiffiffi
Vs
p
1 G12m
Gs
ð Þ

2
4
3
5
ð1DÞ
Gs
ð Þ ¼
Es
ð Þ
2 1 þ 12s
ð Þ
ð1EÞ
1 ¼
VssEs
ð Þ þ 1 Vs
ð Þ1mE1m
E11
ð1FÞ
2 ¼
E2m
E22
2m 1
ffiffiffiffiffi
Vs
p

þ
2m
ffiffiffiffiffi
Vs
p
Vs 2m s
ð Þ
1
ffiffiffiffiffi
Vs
p
1 E2m
Es
ð Þ

2
4
3
5
ð1GÞ
In equations (1), Vs is the SMA fibers volume fraction
and the subscripts ‘m’ and ‘s’ stand for the composite
matrix and SMA fiber, respectively. Furthermore,
Young’s modulus, shear modulus, Poisson’s ratio and
thermal expansion coefficient are denoted by E, G,
and , respectively.
Reported results by Shen39
reveal that for cylindrical
shells in moderately thick class of thickness, thermal
buckling phenomenon may not occur. Therefore, in
this study the attention is focused on thin shells that
obey the classical shell theory assumptions.
Donnell’s assumptions incorporated with the von-
Karman geometrical non-linearity, when are applied
Figure 1. Schematic, geometry and coordinate system of a SMA hybrid composite laminated cylindrical shell.
Asadi et al. 3
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to cylindrical shells, yield the strains in terms of the
mid-surface displacement components (u, v, w) as
xx

x
8

:
9

=

;
¼
u,x þ
1
2
w2
,x
1
R
v, þ w

þ
1
2R2
w2
,
1
R
u, þ v,x þ
1
R
w,xw,
8

:
9

=

;
z
w,xx
1
R2
w,
2
R
w,x
8

:
9

=

;
ð2Þ
where xx, and x are the components of axial,
circumferential and shear strain, respectively.
Additionally, a comma indicates partial derivative
with respect to the coordinate(s) appeared after
comma.
Considering T and T0 as the temperature distribu-
tion and reference temperature, respectively, the consti-
tutive law for the SMA fibers reinforced hybrid cross-
ply laminated composite cylindrical shell subjected to
thermal loading becomes
xx

x
8

:
9

=

;
¼

Q11

Q12 0

Q21

Q22 0
0 0
Q66
2
6
4
3
7
5
xx

x
8

:
9

=

;
0
B
@
T
xx

xx
ð Þ
8

:
9

=

;
1
C
A Vs
r
cos2
ð Þ
r
sin2
ð Þ
r
sinð Þcosð Þ
8

:
9

=

;
ð3Þ
where xx, and x are the components of axial,
circumferential and shear stress, respectively.
Furthermore
Qij are the transformed elastic constants
and defined in Appendix 1. In addition, r
is the recov-
ery stress generated by the temperature-induced reverse
phase transformation of the pre-strained SMA fibers
from detwinned martensite to austenite. Also, denotes
the ply angle in each lamina.
In the present work, the recovery stress of SMA
fibers is calculated using the simplified form of the
Brinson model.36
According to this model, the martens-
ite volume fraction ðÞ is separated into the
stress-induced ðSÞ and the temperature-induced
components ðTÞ
¼ S þ T ð4Þ
The constitutive equation of SMAs is written as36
¼ EðÞð LSÞ þ T ð5Þ
where L represents the maximum residual strain. The
Young’s modulus based on the Reuss model is
expressed as40
EðÞ ¼
EA
1 þ EA
EM
1

ð6Þ
in which EA and EM are Young’s modulus of the SMA
in the pure austenite and the pure martensite phases,
respectively.
The evolution equations to determine the
martensite fractions during heating stage for T 4 As
and CAðT AfÞ 5 5 CAðT AsÞ may be expressed
as36
¼
0
2
cos
Af As
T As

CA

#
þ 1
( )
ð7AÞ
S ¼ S0

0
ð7BÞ
T ¼ T0

0
ð7CÞ
where a subscript ‘0’ indicates the initial state of a par-
ameter and the constant CA is the slope of the curve of
the critical stress for reverse phase transformation.
Since the martensite fraction depends on the stress
and temperature, transformation kinetics must be
coupled with equation (5) to formulate a complete gov-
erning equation for SMAs.
Figure 2 demonstrates the stress–strain responses for
different temperatures with SMA properties provided
in Table 1. Experimental results given by Liang41
are
available for the loading curve at 50
C and the loading
and unloading curves at 10
C. There is a quite good
agreement between the experimental results41
and the
predictions of the current analysis. The stress–strain
Figure 2. SMA stress–strain curves calculated by the present
method.
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curve at 10
C indicates the shape memory effect
(SME) while the pseudoelastic effect is demonstrated
by the curve at 50
C. The SMA recovery stress versus
temperature are calculated with various pre-strains and
depicted in Figure 3. The SMA fiber is constrained to
maintain the deformation as the temperature is raised.
It is established that large internal stresses are produced
when the transformation to austenite occurs. In this
research, the residual recovery stress is used to improve
the adaptability of the structures by embedding SMA
fibers in the composite structure.
To derive the equilibrium equations of a shell, static
version of virtual displacements may be used.42
In the
absence of the external loads, both total virtual poten-
tial energy and the virtual strain energy of the shells are
equal. Therefore, in an equilibrium position, one may
consider
U¼
Z L
0
Z 2
0
Z h=2
h=2
xx xx þ þx x
ð Þdzddx¼0
ð8Þ
Substituting the strain-displacement relations from
equation (2) into the above equation, and applying
the Green-Gauss theorem to relieve the virtual displace-
ments, results in the three equilibrium equations as
Nxx,x þ
1
R
Nx, ¼ 0 ð9AÞ
Nx,x þ
1
R
N, ¼ 0 ð9BÞ
Mxx,xx þ
2
R
Mx,x þ
1
R2
M,
1
R
N þ Nxxw,xx
þ
1
R2
Nw, þ
2
R
Nxw,x ¼ 0 ð9CÞ
where in equation (9), the stress resultants are
defined as
Nxx, N, Nx, Mxx, M, Mx
ð Þ
¼
Z h=2
h=2
xx, , x, zxx, z, zx
ð Þdz ð10Þ
Definition of stress resultants in terms of mid-plane dis-
placement components are obtained by substitution of
equations (2) and (3) into the above equation
Nxx
N
Nx
Mxx
M
Mx
8

:
9

=

;
¼
A11 A12 0 B11 B12 0
A21 A22 0 B21 B22 0
0 0 A66 0 0 B66
B11 B11 0 D11 D12 0
B21 B22 0 D21 D22 0
0 0 B66 0 0 D66
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5

@u
@x
þ
1
2
@w
@x
2
1
R
@v
@
þ w

þ
1
2R2
@w
@
2
1
R
@u
@
þ
@w
@x
@w
@

þ
@v
@x

@2
w
@x2

1
R2
@2
w
@2

2
R
@2
w
@@x
8

:
9

=

;
Figure 3. SMA recovery stress versus temperature with
different pre-strains.
Table 1. Thermo-mechanical properties of the shape memory alloy SMA fibers (Nitinol).
Modulus, density
Transformation
temperature
Transformation
constants
Maximum residual strain,
material properties
EA ¼ 67GPa Mf ¼ 9
C CM ¼ 8 MPa=
C
ð Þ L ¼ 0:067
EM ¼ 26:3GPa Ms ¼ 18:4
C CA ¼ 13:8 MPa=
C
ð Þ s ¼ 10:26 106
1=
C
ð Þ
¼ 0:55 MPa=
C
ð Þ As ¼ 34:5
C 0 ¼ 0 s ¼ 0:33
Af ¼ 49
C
Asadi et al. 5
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NT
x
NT

0
MT
x
MT

0
8

:
9

=

;
þ
Nr
x
Nr

0
Mr
x
Mr

0
8

:
9

=

;
ð11Þ
where NT
and MT
are thermal force and thermal
moment resultants, respectively, which may be evalu-
ated as
NT
x MT
x
NT
MT

¼
X
N
k¼1
Z hk
hk1
Ax
A
1, z
ð ÞTdz ð12Þ
Here, T ¼ T T0 is temperature rise from the refer-
ence temperature T0 to an arbitrary temperature T.
Furthermore
Ax
A
¼

Q11

Q12 0

Q12

Q22 0
0 0
Q66
2
6
4
3
7
5

cos2 k

sin2 k

sin2 k

cos2 k

2sin k

cos k

2sin k

cos k

2
6
4
3
7
5
11
22
ð13Þ
Also in equation (11), Nr
and Mr
, respectively, refer to
the in-plane force and bending moment resultants
induced by the SMA fibers which can be expressed as
Nr
x Mr
x
Nr
Mr

¼
X
N
k¼1
Z hk
hk1
r
Vs
cos2 k

sin2 k

#
1, z
ð Þdz ð14Þ
Moreover, stretching, coupling stretching-bending
and bending stiffness parameters Aij, Bij and Dij, are
defined as
Aij, Bij, Dij

¼
X
N
k¼1
Z hk
hk1

Qij

k
ð1, z, z2
Þdz i, j ¼ 1, 2, 6
ð Þ
ð15Þ
Pre-buckling analysis
Unlike the plate structures which remain un-deflected in
pre-buckling state, cylindrical shells exhibit the lateral
deflection at the onset of thermal loading. Therefore,
bending deformations exist in both pre-buckling and
post-buckling equilibrium paths of a SMA reinforced
hybrid composite shell. Existence of lateral deformation
in the pre-buckling state makes the stability behavior of
cylindrical shell a complicated problem with a non-
linear pre-buckling path. However, pre-buckling path
is frequently linearized to obtain the simple solutions
for design purposes. Brush43
examined the effect of
pre-buckling rotations into the stability loss conditions
and concluded that, for a cylindrical shell under uni-
form pressure, when the ratio of the length to the
mean radius is greater than unity, ignoring the pre-
buckling rotations results in less than 5% error.43
Besides, as the L/R ratio increases, the error diminishes
permanently. Furthermore, in the analysis of linear
thermoelastic stability of functionally graded (FG)
cylindrical shells with simply-supported edges,
Bagherizadeh et al.44
reported a maximum of 6% devi-
ation from those reported by Shen45
based on the
non-linear equilibrium analysis. Consequently, for
moderately long class of shells, linear membrane pre-
buckling assumption results is an error which may be
ignored for design applications. In this study, the atten-
tion is focused on the linear membrane pre-buckling
analysis, which means that pre-buckling forces are
obtained neglecting the bending effects in pre-
buckling state.
Deformation of SMA reinforced cylindrical compos-
ite shell in pre-buckling state is symmetric. Therefore,
to obtain the pre-buckling forces, the symmetrical
linear case of equilibrium equations should be solved.
In this case, the second equilibrium equation vanishes.
From the third equilibrium equation, it can be con-
cluded that N ¼ 0. This condition in terms of mid-
plane displacements is rewritten as
A21u0
,x þ
A22
R
w0
NT
þ Nr
¼ 0 ð16Þ
where a superscript ‘‘0’’ indicates the pre-buckling
deformations. From the first equilibrium equation, it
is concluded that, N0
xx,x ¼ 0.
In this study, only the uniform temperature rise
loading is under investigation. In this case, the latter
condition in terms of mid-plane displacements takes the
form
A11u0
,xx þ
A12
R
w0
,x ¼ 0 ð17Þ
Integrating equation (17) with respect to x, solving for
w0
and inserting the result into equation (16), and inte-
gration again with respect to x results in
u0
¼ C1 þ xC2 x NT
Nr

ð18Þ
Considering immovable boundary conditions u0
0
ð Þ ¼
u0
L
ð Þ ¼ 0, yields C1 ¼ 0 and C2 ¼ NT
Nr

.
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Therefore, pre-buckling state of the SMA fibers rein-
forced cylindrical shell may be written as
u0
¼ 0, v0
¼ 0, w0
¼ R
A22
NT
Nr

N0

¼ 0, N0
x
¼ 0,
N0
xx
¼
A21
A22
NT
NT
x

þ Nr
x
A21
A22
Nr

ð19Þ
Since the pre-buckling study of this research is limited
to membrane analysis, the above-mentioned pre-
buckling deformation/forces may be used for arbitrary
classes of flexural edge supports.
Stability equations
Stability equations of SMA fibers reinforced hybrid
composite cylindrical shell may be obtained based on
the adjacent equilibrium criterion.46
To this aim, a
perturbed equilibrium position form a pre-buckling
state is considered. An equilibrium position in pre-
buckling state is prescribed with components
u0
, v0
, w0

given in equation (19). With arbitrary per-
turbation u1
, v1
, w1

shell experiences a new equilib-
rium configuration adjacent to the primary one
described with displacement components
u0
þ u1
, v0
þ v1
, w0
þ w1

. Linear stress resultants in
adjacent configuration are established as the sum of
stress resultants in pre-buckling state and perturbed
stress resultants generated due to the incremental dis-
placement field. The stability equations are then con-
cluded as
N1
xx,x þ
1
R
N1
x, ¼ 0 ð20AÞ
N1
x,x þ
1
R
N1
, ¼ 0 ð20BÞ
M1
xx,xx þ
2
R
M1
x,x þ
1
R2
M1
,
1
R
N1

þ N0
xxw1
,xx þ
1
R2
N0
w1
, þ
2
R
N0
xw1
,x ¼ 0 ð20CÞ
Considering the linearized form of the resultant-
displacement equations, three stability equations in
terms of perturbed components u1
, v1
, w1

are
A11u1
,xx þ
A66
R2
u1
, þ
A12 þ A66
ð Þ
R
v1
,x B11w1
,xxx

B12 þ 2B66
ð Þ
R2
w1
,x þ
A12
R
w1
,x ¼ 0 ð21Þ
A66 þ A12
ð Þ
R
u1
,x þ A66v1
,xx þ
A22
R2
v1
,

2B66 þ B12
ð Þ
R
w1
,xx þ
A22
R2
w1
,
B22
R3
w1
, ¼ 0 ð22Þ
B11u1
,xxx
A12
R
u1
,x þ
B12 þ 2B66
ð Þ
R2
u1
,x
A22
R2
v1
,
þ
B12 þ 2B66
ð Þ
R
v1
,xx þ
B22
R3
v1
, þ
2B12
R
w1
,xx

A22
R2
w1
þ
2B22
R3
w1
,
2 D12 þ 2D66
ð Þ
R2
w1
,xx
D11w1
,xxxx
D22
R4
w1
, þ N0
xxw1
,xx
þ
N0

R2
w1
, þ
2N0
x
R
w1
,x ¼ 0 ð23Þ
Solution methodology
In this section, thermoelastic stability of SMA fiber
reinforced cylindrical shell is investigated using the
HDQ method47
combined with the exact trigonometric
functions. A brief overview of the HDQ method is pre-
sented in Appendix 2. Before applying the HDQ
method, considering the periodity conditions of shell
in circumferential coordinate, and absence of pre-buck-
ling force N0
x, the following solution is assumed which
satisfies the circumferential constraints of shell
u1
x,
ð Þ ¼ u x
ð Þ sin m
ð Þ
v1
x,
ð Þ ¼ v x
ð Þ cos m
ð Þ
w1
x,
ð Þ ¼ w x
ð Þ sin m
ð Þ
ð24Þ
where m is the full-wave length in circumferential
coordinate. Substituting equation (24) into equations
(21)–(23) and utilizing the HDQ discretization for the
x-dependent functions, one may reach to
A11
X
K
j¼1
Cð2Þ
ij uj m2 A66
R2
X
K
j¼1
Cð0Þ
ij uj m
A12 þA66
ð Þ
R
X
K
j¼1
Cð1Þ
ij vj
B11
X
K
j¼1
Cð3Þ
ij wj þm2 B12 þ2B66
ð Þ
R2
X
K
j¼1
Cð1Þ
ij wj
þ
A12
R
X
K
j¼1
Cð1Þ
ij wj ¼ 0 ð25Þ
m
A66 þ A12
ð Þ
R
X
K
j¼1
Cð1Þ
ij uj þ A66
X
K
j¼1
Cð2Þ
ij vj
m2 A22
R2
X
K
j¼1
Cð0Þ
ij vj m
2B66 þ B12
ð Þ
R
X
K
j¼1
Cð2Þ
ij wj
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þ m
A22
R2
X
K
j¼1
Cð0Þ
ij wj þ m3 B22
R3
X
K
j¼1
Cð0Þ
ij wj ¼ 0 ð26Þ
B11
X
K
j¼1
Cð3Þ
ij uj
A12
R
X
K
j¼1
Cð1Þ
ij uj m2 B12 þ2B66
ð Þ
R2
X
K
j¼1
Cð1Þ
ij uj
þm
A22
R2
X
K
j¼1
Cð0Þ
ij vj m
B12 þ2B66
ð Þ
R
X
K
j¼1
Cð2Þ
ij vj
þm3 B22
R3
X
K
j¼1
Cð0Þ
ij vj þ
2B12
R
X
K
j¼1
Cð2Þ
ij wj
A22
R2
X
K
j¼1
Cð0Þ
ij wj
m2 2B22
R3
X
K
j¼1
Cð0Þ
ij wj þm2 2 D12 þ2D66
ð Þ
R2
X
K
j¼1
Cð2Þ
ij wj
D11
X
K
j¼1
Cð4Þ
ij wj m4 D22
R4
X
K
j¼1
Cð0Þ
ij wj þN0
xx
X
K
j¼1
Cð2Þ
ij wj
m2 N0

R2
X
K
j¼1
Cð0Þ
ij wj ¼ 0 ð27Þ
Here, Cð0Þ
ij is the Kronecker delta which is equal to one,
when i ¼ j, and otherwise is equal to zero. Also, Cð Þ
ij is
the weighting coefficient matrix of order differenti-
ation. The shell is divided into K grid points which
indicate the number of nodes in the x direction. Also,
HDQ discretization is applied to the boundary condi-
tions which gives us
For clamped end (C)
ui ¼ vi ¼ wi ¼ 0
X
K
j¼1
Cð1Þ
ij wj ¼ 0
ð28Þ
For simply supported edges (S)
ui ¼ vi ¼ wi ¼ 0
B11
X
K
j¼1
Cð1Þ
ij uj m
B12
R
X
K
j¼1
Cð0Þ
ij vj D11
X
K
j¼1
Cð2Þ
ij wj ¼ 0
ð29Þ
System of coupled differential equations (21)–(23) and
boundary conditions are converted to the set of coupled
algebraic equations by using the HDQ method as equa-
tions (25)–(29).
The new established system of equations contains the
unknown pre-buckling force resultants N0
xx, N0
, N0
x
which should be inserted from the equation (19). After
that, stability equations contain the unknown full-wave
number m and the temperature parameter T. To
obtain the critical buckling temperature difference of
the shell, starting from m ¼ 1, the full wave number of
the shell is assumed to be prescribed. The established
homogeneous system of equations is solved via the
standard eigenvalue techniques. The minimum positive
eigenvalue of T is associated to the critical tempera-
ture difference of the shell with the prescribed wave-
number m denoted by Tm
cr. It should be pointed-out
that, a convergence study to obtain the required number
of grid point is necessary at this step. Furthermore, since
the properties of the constituents are temperature-
dependent, an iterative algorithm is required to obtain
the Tm
cr. At first, properties are evaluated at the refer-
ence temperature T0 and the eigenvalue solution is per-
formed. Properties are then evaluated at the new
obtained temperature T ¼ Tm
cr þ T0. Such iteration is
repeated until Tm
cr converges. At this step, critical
buckling temperature difference of the shell under a pre-
scribed wave number for temperature-dependent (TD)
and temperature-independent (TID) assumptions of
material properties are detected. A simple search
through the values of Tm
cr, and collecting the minimum
one among them, yields the critical buckling tempera-
ture difference of the shell. The associated number m
indicates the number of full-waves in the circumferential
direction at the onset of buckling.
Results and discussion
The procedure outlined in the previous sections, is
implemented herein to study the effect of SMA fibers
on the thermal buckling behavior of composite lami-
nated cylindrical shells in the class-ply laminated
scheme. At first, to show the effectiveness and accuracy
of the present solution method, some comparison stu-
dies are performed. Afterward, a series of parametric
studies are presented to show the influences of various
involved parameters on the thermal stability and
buckled configuration of SMA hybrid cylindrical shells.
Comparison studies
Due to the lack of any data on the buckling tempera-
tures of SMA hybrid composite laminated cylindrical
shells, to verify the proposed approach, the case of the
S-S cross-ply composite laminated shell is investigated.
Properties of the shell are L=R ¼ 0:5, E11=E22 ¼ 10,
G12=E22 ¼ 0:5, 22=11 ¼ 2, 12 ¼ 0:25 and 11 ¼ 1:0
106
=o
C. Results of this study are compared with those
of Patel el al.48
based on finite elements method and
also the well-known numerical results of Thangartnam
et al.3
Comparison is accomplished in Table 2. Two
different cases of lamination scheme along with four
different cases of radius-to-thickness ratios are investi-
gated. Obviously, good agreement is observed at the
onset of comparison. It is worth noting that results of
Patel et al.48
are developed based on the linear
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membrane-bending pre-buckling analysis. Comparison
of the numerical results of this study and those of Patel
et al.48
demonstrates the influence of pre-buckling
rotations prior to stability loss. As one may conclude,
this influence is at most 3 percent. Furthermore, within
the studied range, the linear membrane assumption
does not violate the buckling pattern of the shell since
the full wave numbers of our results match well with
those of Patel et al.48
In another comparison study, to verify the iterative
process for temperature dependency assumption, a
simply-supported cross-ply laminated cylindrical shell
with lamination scheme 0=90
½ 2s is considered. Both
TD and TID cases of material properties are con-
sidered. The case TID indicates the conditions in
which properties are evaluated at T ¼ 0. TD proper-
ties of the constituents are assumed as follows
E11ðTÞ ¼ 150ð1 0:0005TÞGPa
E22ðTÞ ¼ 9:0ð1 0:0002TÞGPa
G12ðTÞ ¼ 7:1ð1 0:0002TÞGPa
11ðTÞ ¼ 1:1 106
ð1 þ 0:0005TÞ=o
C
22ðTÞ ¼ 25:2 106
ð1 þ 0:0005TÞ=o
C
12ðTÞ ¼ 0:3
ð30Þ
Numerical results are presented in Table 3 for three
different values of Z ¼ L2
=Rh. It is obvious that,
good agreement is observed at the onset of comparison
for both TD and TID cases. It is worth mentioning that
results of Shen7
are obtained based on the complete
non-linear membrane-bending pre-buckling approach.
However results of this study are confined to the case of
linear membrane pre-buckling approach. Nonetheless,
the established assumption in this study is valid and
yields the reasonable justification.
Parametric studies
In this section, numerical results on critical buckling
temperature of SMA hybrid composite laminated cylin-
drical shells are presented. Cylindrical shells made of
NiTi/graphite-epoxy are assumed. Material properties
of NiTi fibers and graphite-epoxy are considered to be
TD. Properties of NiTi fibers are presented in Table 1
whereas properties of graphite-epoxy fibers are given as
below
Graphite/epoxy properties.
E1m ¼ 155 13:53104
T

Gpa;
E2m ¼ 8:07 14:27104
T

Gpa
G12m ¼ 4:55 16:06104
T

Gpa;
1m ¼ 0:07106
11:25103
T

1=
C
ð Þ
2m ¼ 30:1106
1þ0:41104
T

1=
C
ð Þ; 12m ¼ 0:22
Table 2. Comparison of critical buckling temperature difference T ðo
CÞ of S-S composite laminated cylindrical shells with
L=R ¼ 0:5, E11=E22 ¼ 10, G12=E22 ¼ 0:5, 22=11 ¼ 2, 12 ¼ 0:25 and 11 ¼ 1:0 106
=o
C. Number in parenthesis indicate
the circumferential full wave number.
R
h
TCr

C
ð Þ
(0
/90
) (0
/90
/0
)
Present Patel et al.48
Thangartnam et al.3
Present Patel et al.48
Thangartnam et al.3
200 1018.63 (12) 1144.992 (12) 1188.950 (13) 1250.29 (11) 1199.88 (11) 1304.298 (11)
300 677.663 (14) 698.954 (15) 756.398 (15) 841.566 (14) 843.883 (13) 912.434 (13)
400 507.302 (17) 507.146 (17) 553.418 (18) 616.375 (16) 596.104 (16) 659.61 (18)
500 399.520 (18) 397.880 (19) 452.986 (19) 491.094 (17) 484.216 (17) 514.745 (19)
Table 3. Comparison of critical buckling temperature difference T ðo
CÞ of S-S composite laminated cylindrical shells with 0=90
½ 2s
layup, R=h ¼ 200 and various Z ¼ L2
=Rh ratios. Material properties of the constituents are given in equation (30).
TCr

C
ð Þ

Z ¼ 200
Z ¼ 500
Z ¼ 800
Present Shen7
Present Shen7
Present Shen7
TID 417.900 423.346 421.239 425.843 418.288 424.1917
TD 355.180 355.1019 356.698 356.6692 355.339 355.7793
Asadi et al. 9
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Total thickness of the shell is set equal to h ¼ 1 mm and
all of the layers are unified in thickness. Reference tem-
perature is set equal to T ¼ 20
C.
Table 4 presents the critical buckling temperature of
hybrid laminated cylindrical shell embedded with SMA
fibers. Only the TD case of material properties is exam-
ined. Three different cases of boundary conditions,
three different lamination schemes, two different
values of SMA fiber pre-strain and three different mag-
nitudes of SMA fiber volume fraction are examined.
Numerical results cover the case of shells with geomet-
ric parameter R=h ¼ 500 and L2
=Rh ¼ 500. As the
numerical results of this table accept, influence of
boundary conditions on the critical buckling tempera-
tures of the shell is almost negligible. The fact is, since
the shell is moderately long, the influence of boundary-
layer near the edge zones may be ignored. For shorter
cylindrical shells, the influence of boundary layers
becomes more important and the assumption of linear
membrane pre-buckling assumption cannot be justified.
Numerical results of Yamaki49
on the buckling of pres-
surized hydrostatically loaded isotropic homogeneous
cylindrical shells reveal the same. Generally, SMA
hybrid laminated composite cylindrical shells buckle
in higher temperatures in comparison to the associated
conventional composite laminated shells. Increasing the
SMA fiber volume fraction or SMA fiber pre-strain
results in the considerable enhancement of the buckling
temperature in the cylindrical shells. The reason is the
higher tensile stress of the SMA fibers associated to the
higher SMA pre-strain or SMA volume fraction.
Similar conclusions are reported by Asadi et al.30
for
SMA fiber composite beams and Li et al.25
for SMA
fiber composite circular plates. It is worth noting that,
in all of the three cases, SMA fibers are settled in lon-
gitudinal orientation. Since the longitudinal thermal
force is the dominant one in the pre-buckling state of
the shell, such choice is the most effective case. The
circumferential mode number seems to be independent
of the SMA fiber characteristics. A study on the
number of full wave through the circumferential direc-
tion reveals that shells with stiffer edge supports buckle
in higher number of waves. The reason is the higher
local flexural rigidity of the clamped edge in compari-
son to the simply supported edge.
Figure 4 demonstrates the position of SMA fibers on
the critical buckling temperatures of the composite
laminated cylindrical shells. An eight-layer cross-ply
composite laminated cylindrical shell with stacking
sequence 0=90
½ 4 is considered. Four different cases
for the position of SMA fibers are examined. In each
case, SMA fibers are settled in the lamina with the lon-
gitudinally orientated fibers. As seen, locating the SMA
fibers in the outer layers of the shell seems to be more
Table 4. Influence of SMA fiber characteristics on the critical buckling temperature difference of SMA hybrid cylindrical shells with
R=h ¼ 500 and L2
=Rh ¼ 500.
Boundary conditions Lay-up
TCr

C
ð Þ
Pre-strain Without SMA Vs¼5% Vs¼10% Vs¼15%
C-C 0SMA
=90

0 ¼ 2% 116.969 (21) 258.472 (21) 390.106 (21) 518.182 (21)
0 ¼ 3% 116.969 (21) 322.832 (21) 516.735 (21) 707.916 (21)
0SMA
=90=0

0 ¼ 2% 133.059 (19) 229.596 (19) 319.956 (19) 408.585 (19)
0 ¼ 3% 133.059 (19) 273.114 (19) 406.348 (19) 538.384 (19)
0SMA
=90=0=90

0 ¼ 2% 152.742 (18) 224.187 (18) 290.584 (19) 355.264 (18)
0 ¼ 3% 152.742 (18) 256.039 (18) 353.325 (18) 448.231 (18)
S-S 0SMA
=90

0 ¼ 2% 116.809 (18) 258.204 (18) 390.005 (18) 518.006 (18)
0 ¼ 3% 116.809 (18) 322.651 (18) 516.505 (18) 707.814 (18)
0SMA
=90=0

0 ¼ 2% 128.247 (16) 226.002 (16) 316.800 (16) 405.788 (16)
0 ¼ 3% 128.247 (16) 269.396 (16) 403.072 (16) 535.843 (16)
0SMA
=90=0=90

0 ¼ 2% 151.216 (14) 222.884 (14) 289.27 (14) 354.155 (14)
0 ¼ 3% 151.216 (14) 254.518 (14) 352.178 (14) 446.994 (14)
C-S 0SMA
=90

0 ¼ 2% 116.885 (20) 258.427 (20) 390.008 (20) 518.071 (20)
0 ¼ 3% 116.885 (20) 322.767 (20) 516.655 (20) 707.822 (20)
0SMA
=90=0

0 ¼ 2% 129.71 (18) 226.751 (18) 317.364 (18) 406.361 (18)
0 ¼ 3% 129.71 (18) 270.418 (18) 403.843 (18) 536.434 (18)
0SMA
=90=0=90

0 ¼ 2% 151.78 (17) 223.321 (17) 289.759 (17) 354.464 (17)
0 ¼ 3% 151.78 (17) 255.212 (17) 352.491 (17) 447.442 (17)
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effective on the thermal buckling delay. As expected,
shells with higher thickness-to-mean radius ratio
buckle in higher temperatures. The reason is the
higher flexural rigidity of the shell due to the decrease
in the shell mean radius.
Figures 5–8 are presented to investigate the SMA
fibers characteristics on the bifurcation temperatures
of the hybrid cylindrical shells. As one may conclude,
the higher the SMA fiber volume fraction, the higher
the critical buckling temperature. Such feature belongs
to the higher tensile force induced due to the higher
volume fraction of SMA fibers. Same conclusions are
reported previously by Asadi et al.30
in the analysis of
a SMA fiber reinforced composite beam. Dependency
of the shell critical temperature to the SMA volume
fraction seems to be approximately linear when other
characteristics are kept constant. Same as the SMA
fiber volume fraction, increasing the SMA fiber pre-
strain leads to the higher critical buckling temperature.
Variation of critical buckling temperature with respect
to the SMA fiber volume fraction also seems to be
linear.
Conclusion
A semi-analytical solution is presented to investigate
the thermal buckling behavior of SMA hybrid compos-
ite laminated shells with cross-ply lay-up. Properties of
Figure 4. Influence of SMA fibers position on the critical
buckling temperature of cylindrical shells with [0,90]4 layup
and various mean radius to thickness ratios.
Figure 5. Influence of SMA volume fraction on the critical
buckling temperature of cylindrical shells with [0SMA
, 90, 0, 90]S
layup and various SMA fiber pre-strains.
Figure 6. Influence of SMA fiber pre-strain on the critical
buckling temperature of cylindrical shells with [0SMA
,90,0,90]S
layup and various radius-to-thickness ratios.
Figure 7. Influence of mean radius to thickness ratio on the
critical buckling temperature of cylindrical shells with [0SMA
, 90,
0, 90]S layup and various SMA fiber pre-strains.
Asadi et al. 11
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the constituents are TD. Shell is assumed to be thin and
moderately long to justify the linear membrane pre-
buckling assumption. Brinson’s one-dimensional con-
stitutive law is implemented to model the behavior of
SMA fibers through the heating process. From the vari-
ous parametric studies, following general conclusions
may be presented:
. Increasing the SMA fiber volume fraction, or SMA
fiber pre-strain, results in the considerable enhance-
ment of the critical buckling temperatures.
Generally, cylindrical shells that are embedded
with SMA fibers buckle in higher temperatures
when are compared to the associated conventional
composite shells.
. While the characteristics of the SMA fibers affect
greatly on the critical buckling temperature, the
buckling pattern of the shell is almost independent
to the SMA features.
. The stiffer the edge supports of the shell, the higher
the full wave number of the shell at the onset of
buckling. Consequently, the circumferential wave
number of C-C shells is higher than C-S shells, and
the latter one is higher than S-S shells.
. Temperature dependency assumption plays an influ-
ential role in the prediction of critical buckling tem-
perature. Generally, when temperature dependency
assumption is established, critical buckling tempera-
tures are underestimated.
Funding
This research received no specific grant from any funding
agency in the public, commercial, or not-for-profit sectors.
Conflict of interest
None declared.
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Asadi et al. 13
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XML Template (2015) [17.2.2015–4:54pm] [1–14]
Appendix 1
Elastic constants are
Q11 ¼
E11
1 1221
, Q12 ¼
E2212
1 1221
,
Q22 ¼
E22
1 1221
, Q66 ¼ G12
In addition, transformed elastic constants are

Q11 ¼ Q11cos4
þ 2 Q12 þ 2Q66
ð Þsin2
cos2
þ Q22sin4

Q12 ¼ Q11 þ Q22 4Q66
ð Þsin2
cos2

þ Q12 cos4
þ sin4

Q22 ¼ Q11sin4
þ 2 Q12 þ 2Q66
ð Þsin2
cos2
þ Q22cos4

Q66 ¼ Q11 þ Q22 2 Q66 þ Q12
ð Þ
ð Þsin2
cos2

þ Q66 cos4
þ sin4

Appendix 2
In the HDQ method, the xth
-order derivative of an arbi-
trary function w at any discrete point can be approxi-
mated as
d
w xi
ð Þ
dx

X
K
j¼1
C
ð Þ
ij w xj

where (xi) is a discrete point in the solution domain,
w(xj) is the function value at point (xj) and C
ð Þ
ij are the
weighting coefficients for the xth
-order derivative of the
function in the x, respectively. The weighting coeffi-
cients can be determined by the functional approxima-
tions in the x directions. Usage of the harmonic
functions as the approximating test functions results
in the explicit terms for the weighting coefficients for
HDQ. The harmonic test function hk(x) used in this
approach can be defined as
According to the HDQ, the weighting coefficients
of the first-order derivatives Cð1Þ
ij for i6¼j can be
obtained by
Cð1Þ
ij ¼ 2
P xi
ð Þ
P xj

sin xi xj

2

where
P xi
ð Þ ¼
Y
K
j¼1,j6¼i
sin
xi xj

2

,
The weighting coefficients of second-order derivatives
Cð2Þ
ij for i6¼j may be determined by the following
formula
Cð2Þ
ij ¼ Cð1Þ
ij 2Cð1Þ
ij cot
xi xj

2

, i, j ¼ 1, . . . , K
And the weighting coefficients of the first and second
order derivatives C r
ð Þ
ij for i ¼ j are as follows
C r
ð Þ
ii ¼
X
K
j¼1,j6¼i
C r
ð Þ
ij , r ¼ 1 or 2 and for i ¼ 1, . . . , K
The weighting coefficient of third- and fourth-order
derivatives can be computed from Cð1Þ
ij and Cð2Þ
ij by
Cð3Þ
ij ¼
X
K
k¼1
Cð1Þ
ik Cð2Þ
kj
Cð4Þ
ij ¼
X
K
k¼1
Cð2Þ
ik Cð2Þ
kj
The accuracy of HDQ results is affected by the number
of grid points, K, and the relative spacing of the discrete
points. Choosing equally spaced discrete points would
be an easy option while non-uniformly spaced discrete
points may result in predictions with higher level of
accuracy. It is found that one of the optimal selections
of the discrete points is the zeros of the well-known
Chebyshev polynomials
xi ¼
L
2
1 cos
i 1
K 1

, i ¼ 1, . . . , K
ð Þ
hk x
ð Þ ¼
sin xx0
ð Þ
2

. . . sin xxk1
ð Þ
2

sin xxkþ1
ð Þ
2

. . . sin xxK
ð Þ
2

sin xkx0
ð Þ
2

. . . sin xkxk1
ð Þ
2

sin xkxkþ1
ð Þ
2

. . . sin xkxK
ð Þ
2

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